Volume 14, Issue 2

ollision C

Volume 14 Issue 2

The International Compendium for Crash Research

Testing & Validation

of Toyota Vehicle Control History Speed and Acceleration Data

Driver Steering Demands

Following a Steer Tire Deflation on a Commercial Vehicle

Case Studies

Toyota Vehicle Control History

Ignition “Cycology:

Understanding Ignition Cycles collisionmagazine.com 14.2-FrontCover.indd 1

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Contents

Volume 14 Issue 2

inside 5

Collision Magazine Info and Advertiser Index

6

Remembering Tom Szabo

34

features 8

Nissan SUV EDR Recording Characteristics

by David M. Hallman and Billy S. Cox Jr.

18

A Review Of Engineering Mechanics Applied To Accident Reconstruction Part 3: Selected Applications

by Jai Singh

38

Driver Steering Demands Following a Steer Tire Deflation on a Commercial Vehicle

by Donald F. Tandy, Jr. and Jason W. Colborn

54

Crash·ol·o·gy: The Science of Crashes

by Wesley Vandiver and Robert Anderson

60

Ignition “Cycology” Understanding Ignition Cycles

by Kent Boots

72

Pedestrian Automatic Emergency Braking (P-AEB) And Its Use in Accident Reconstruction

by Alan Moore

78

Toyota Vehicle Control History Case Studies

by David M. Hallman, Robert D. Anderson and Billy S. Cox Jr.

92

High Speed Pedestrian Collision And Verification Of Car Collision Speed In Pedestrian Accidents Based On Biofidelic Dummy Injuries

by Annika Kortmann and Tim Hoger

102

Testing and Validation of Toyota Vehicle Control History Speed and Acceleration Data

by Robert Anderson and Wesley Vandiver

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92

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Accident Reconstruction ACTAR Accredited Biomechanics CDR / EDR Crush Analysis Drones Forensic Mapping Human Factors Motorcycles Vehicle Systems Visibility The Crash Hub is the leading online Accident

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Reconstruction Expert Directory. Crash Experts vary by experience, rate and specialties. When evaluating the right expert for your case, make sure you search, review and connect with the expert that best matches your case needs.

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ADVERTISER INDEX 4N6XPRT Systems

112, Inside Back

Crash Hub

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Crash Data Group

Back Cover

Engineering Dynamics

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Houston Auto Appraisers

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All rights reserved, © 2020 Crash Data Group Inc. The opinions and conclusions expressed in this publication and any associated data and throughout the articles attributed to specific authors are the opinions and conclusions of the authors noted and not necessarily the editorial staff or anyone else for that matter. While some articles have been reviewed for content, the accuracy of reprinted models or equations cannot be fully guaranteed. It is the responsibility of the reader to apply critical thinking to an individual review of the content and make their own personal judgements as to its value to them, individually.

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Remembering Tom Szabo We hear it all the time now: “these uncertain times…”

But nothing reminds us of what uncertainty truly means like the unexpected passing of a longtime friend and colleague. In August (2020) Tom Szabo passed unexpected while hiking in Phoenix.

Tom was born in and grew up near Montreal. An avid hockey player and standout baseball player scouted by a major league baseball team while in college, Tom chose a different route but never gave up his love of hockey. He started his college carrier by earning a Bachelor’s of Science Degree in Kinanthropology (Kinesiology) from Ottawa University. He then went on to McGill University where he earned his Bachelor of Engineering Degree in Mechanical Engineering.. Among his other professional accomplishments, presentations and testing conducted all over the world, not to mention authoring enough papers to fill a small library, Tom designed and tested a biomechanically correct breakable knee for the Hybrid III dummy and established design criteria for biomechanically correct breakable hip, femur and tibia elements for the Hybrid III dummy for use during motorcycle crash testing. He participated in the design and validation testing of a biomechanically correct thorax and the design of an anthropomorphically correct neck for use with the Hybrid III crash test dummy.

Tom – together with Jud Welcher, as well as the whole Biomechanical Research & Testing crew – were always an important and integral part of the ARC-CSI Crash Con-

ferences from the start not only as part of the crash crew but as valuable speakers on a variety of topics, not the least of which was explaining the intricacies of the voluminous crash data they collected in the end of conference review.

With the BRT team, Tom’s testing and research insight and his invaluable contributions to the conference testing series and presentations ultimately contributed to success of the 16-year running Crash Conference and inspired many in the field.

“Tom was my friend for over 30 years and business partner for 25 years. He made us laugh, he was equally capable of listening as he was of offering thoughtful and insightful advice. He unselfishly shared of his wealth of knowledges and understanding. The loss of his insight, straight forward and objective view of things is a loss to us all. Myself personally, and to all of those at BRT, we owe a large measure of our success to Tom, specifically his love of knowledge, writing skills, deep understanding of the sciences, passion for what he believed in, and his willingness to always bring up others with him. There are few like him. He will be greatly missed. Until we crash again - ” – Judson Welcher “Tom left the accident reconstruction, biomechanics and testing community better than you found it. I will miss the example by which he led. His bluntness was balanced by both his playfulness and his alternative vernacular. I am a better person for having known him” – Bob Anderson “Tom's untimely passing is truly a loss not only to the industry but to many of us personally who knew him both as a friend and mentor in many ways. He taught me more about testing than I could have learned in a lifetime of classes. I will miss his infectious smile, insightful and thoughtful approach to crashes and unfailing willingness to help and contribute. I will mention and remember him often and keep his memory alive.” – Rusty Haight “Tom Szabo was the kind of man that you wanted to be around. Or you wanted to be him. Or you wanted his opinion. He was just that kind of a man – charismatic, charming, wicked smart, generous, confident yet humble, tough but fair.

There will never be another like him. He was an extraordinary engineer and an even better friend. I will miss his presence for the rest of my life.” – Adelino Yung “Tom was my friend, a mentor and a very respected colleague.

He helped me improve on a Low Velocity Impact Simulator

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I nicknamed, “LVIS.” Tom was one of the first occupants we tested. I was honored that he took an interest in the project and to have him and Jud Welcher contribute suggestions for improvements. We were friends and worked together for many years running crash tests for the participants of a variety of crash conferences. He was a prankster with a quick wit and quirky sense of humor. He was legendary in many respects and will be greatly missed by all who were lucky enough to work with him and know him.” – Billy S. Cox, Jr.

It goes without saying that Tom will, of course, be remembered as a true leader in the community but also one of those genuinely good guys. His untimely and early passing will leave a hole in our ranks which will be near impossible to fill.

“Tom was a giant, in every sense of the word. He filled every room with his presence, laughter and incredible insight. It is hard to comprehend the amount of knowledge and information that he has added to the reconstruction community. I feel very fortunate to have known him and to have been able to learn from him.” – Jeff Suway “I am very grateful that I had the privilege of having Tom as a mentor in the early years of my career. I have always had the utmost respect for him professionally and truly appreciated his sense of humor. His passing is a great loss to the accident reconstruction and biomechanics community. He left far too early and will be greatly missed.” – Michelle R. Hoffman “Tom has been a part of my life longer than I can remember.

From a very young age, he was a tremendous influence on my life personally and professionally. He was a great mentor for many engineers including myself - always willing to provide valuable advice to anyone who would ask. Tom's legacy will not be forgotten and those who knew him will surely miss him.” – Sean Haight

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Nissan SUV EDR Recording Characteristics David M. Hallman M.S., P.E. Billy S. Cox Jr. bstract To date, there have been multiple published studies of the EDR data recording characteristics for vehicle manufacturers such as GM, Kia, and Toyota. These authors are not aware of any published studies of the EDR characteristics of Nissan / Infiniti vehicles. The purpose of this investigation was to develop and execute a repeatable test method for determining Nissan’s non-deployment trigger threshold and evaluate / confirm the published non-deployment event recording order.

A

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Data imaged from multiple Nissan passenger cars and sport utility vehicle airbag control modules (ACMs) by these authors and others has produced stored non-deployment events with reported longitudinal and lateral delta-v values below the Code of Federal Regulations Title 49 Part 563 (1) (49CFR563) trigger threshold.

Data imaged from several Nissan ACMs by others has produced some non-deployment events with longitudinal and lateral delta-v values of zero. This would appear to indicate that the Nissan ACM will record a non-deployment event below the 5 mile per hour or 8 kilometer per hour trigger threshold as defined in 49CFR563.

Reviewed information posted on user groups devoted to the general use and interpretation of vehicle crash data seemed uncertain as to the non-deployment event recording threshold and event recording order for crash data imaged from Nissan ACMs. The consensus of the information posted appears to be that Nissan ACMs can record an event any time the crash sensing algorithm wakes up. If this is correct, it implies that Nissan ACMs will record an event below the 5 mile per hour or 8 kilometer per hour trigger threshold defined in 49CFR563.

Some posts on the EDR users group also indicated that Nissan has no special rules specific to recording order.

The Nissan data limitations as stated at the beginning of the Nissan specific Bosch CDR Report specifies the event recording order and rules. This published recording order will also be evaluated.

Introduction Crash data imaged from vehicle ACMs is useful information which is generally available for crash investigation and reconstruction. The Bosch Crash Data Retrieval Tool is the most widely utilized system for crash data imaging as most vehicle manufacturers work with Bosch for data imaging and translation. Each manufacturer provides data as required by 49CFR563, but the source of that data within the vehicle, how it is sent to the ACM and, more importantly, the correct translation and interpretation of that data can vary widely between manufacturers. Extensive research has been performed by others (2,3,4) on the crash data provided by various manufacturers including, but not limited to, GM, Toyota and Kia. Published information on the evaluation of Nissan crash data has been limited to date.

Crash data imaged from Nissan passenger and sport utility vehicles in the course of normal crash reconstruction casework by these authors and others was

found to include stored non-deployment events with both the longitudinal and lateral delta-v values below the 49CFR563 defined trigger threshold of 5 mph (8 kph). In some cases, stored non-deployment events were found with both longitudinal and lateral delta-v values of 0 mph and 0 G acceleration values as well.

Based on this non-deployment event data, assumptions have been made by crash investigators that the minimum Nissan non-deployment event recording threshold is less than 5 mph (8 kph). Prior to this investigation, there has been no published systematic testing to investigate and confirm the non-deployment event recording trigger threshold for Nissan vehicles.

The primary purpose of this investigation was to begin to evaluate the trigger threshold for non-deployment event generation. Secondarily, the event recording order was also evaluated and compared to the Bosch provided data limitations. This investigation utilized crash event data from a single Nissan sport utility type vehicle and future testing and investigation is planned with a Nissan sedan.

Materials and Test Equipment These tests were conducted in advance of the 2020 SATAI conference in Glendale Arizona. A total of fifteen low-speed crash tests were conducted for the purposes of this investigation. The Nissan was driven by an instrumented human volunteer for all fifteen tests. Six tests were vehicle-to-vehicle in the forward direction.

Six tests were vehicle to barrier in the forward direction. Three tests were vehicle to barrier in reverse.

The test vehicle was a 2013 Nissan Pathfinder, VIN: 5N1AR2MM6DC642452 which was purchased as a used vehicle. After being purchased, repairs were made to the Supplemental Restraint System (SRS) including, but not limited to, replacement of the ACM. The tests in this investigation were all conducted with the same measurement equipment installed. This equipment is as follows: DTS Tiny Data Acquisition System (TDAS) Pro, Human Subject Instrumentation, Racelogic Video VBOX, Triaxial Accelerometers, IST EDR 3C – 10 G, Slam Stick S, Speed traps on the ground, Garmin Virb, Garmin Virb Elite and Garmin Virb XE Action cameras, High Speed Video Cameras, DSLR Still Cameras, and various other video and still cameras. The ACM was imaged through the data link connecter (DLC) with the Bosch CDR kit after every test to determine if an event was stored. A stationary 2005 Lincoln Navigator utilized for the first six tests was also occupied by an instrumented human volunteer. The test results

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from the human volunteer instrumentation will not be discussed here.

Tests Conducted The first six tests were vehicle to vehicle tests conducted with the Nissan Pathfinder moving forward. The Nissan was idled into the back end of a stationary 2005 Lincoln Navigator equipped with a receiver hitch. The photo in Figure 1 shows the test set-up.

The Nissan was allowed to roll in at idle from increasing distance from the Navigator and therefore increasing speeds for the first five tests. The accelerator was briefly applied for the sixth and final test of this series. Some of the results from this testing are found in Figure 4. Test 6 was the only test in this series which resulted in a stored non-deployment event.

The next six tests (Tests 7 through 12) were vehicle to concrete barrier tests in the forward direction. The Nissan was idled into a stationary concrete barrier.

The Nissan rolled in at idle from increasing distance from the barrier for these tests. The accelerator was not applied.

Some of the results from this testing are found in Figure 5.

Tests 11 and 12 both resulted in stored non-deployment events.

The final three tests (Tests 13 through 15) were also vehicle to barrier tests, but in the rearward direction. The Nissan was idled into the stationary concrete barrier in reverse.

The Nissan rolled in at idle from increasing distance for two of these tests. The accelerator was applied for the final test (Test 15) of this series. Some of the results from this testing are found in Figure 6. Test 15 resulted in a stored non-deployment event.

Methods The information in this paragraph describes the collection / presentation method for all test data presented in the results section. Vehicle speed reported is the approach speed of the Nissan measured with a speed trap placed on the ground immediately prior to the impact with the rear of the Navigator or the barrier. The maximum longitudinal acceleration reported is the Class 180 filtered peak (5) from the accelerometer connected to the TDAS, the longitudinal delta-v was calculated from the filtered TDAS acceleration data and the EDR delta-v was reported directly from the imaged data as listed in the System Status at Event Chart in each EDR report.

Results The first six tests resulted in a single non-deployment event which occurred during Test 6. This test is highlighted in yellow in the Figure 4.

In test six, the EDR reported delta-v was -4 mph but the calculated delta-v from the acceleration data was -5.3 mph.

Tests 7 through 12 resulted in a two non-deployment events which occurred during Tests 11 and 12. These tests are highlighted in yellow in Figure 5.

Figure 1: Test Set-up, Tests 1 through 6.

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Figure 2: Test Set-up, Tests 7 through 12.

Figure 3: Test Set-up, Tests 13 through 15.

In Test 11, the EDR reported delta-v was -5 mph and the calculated delta-v from the acceleration data was -5.5 mph.

In test 12, the EDR reported delta-v was also -5 mph and the calculated delta-v was also -5.5 mph.

This offset was determined by finding the location in the IST and TDAS acceleration data which matched the first ACM recorded acceleration data. This offset accounts for the difference between the EDR reported delta-v and the calculated delta-v. The offset resulted in an EDR reported delta-v below the 5 mph 49CFR563 trigger threshold, when the actual vehicle delta-v was greater than the part 563 trigger threshold at -5.3 mph.

The final three tests resulted in a single non-deployment event which occurred during Test 15. This test is highlighted in yellow in the Figure 6.

In Test 13, the EDR reported delta-v was 5 mph and the calculated delta-v from the acceleration data was 6.5 mph.

The chart shown in Figure 7 is a summary of the nondeployment events captured during this investigation including the write locations as indicated by the EDR data imaged after each test.

Analysis Test 6, Non-deployment Event 1 Test 6 was the first non-deployment event which was recorded during this investigation. This test was part of the crash series of the Nissan into the rear of the Navigator. As expected, the event data was written in Location 1 in the ACM as both data locations were empty prior to this event being stored. The maximum EDR reported delta-v from this event was -4 mph. The delta-v calculated from the TDAS measured and filtered acceleration was -5.3 mph.

In reviewing the individual acceleration data points in the EDR data (Figure 8) which are recorded once every 10 ms, the first recorded data point was found to be at -3 G as opposed to zero.

When the EDR acceleration data from Test 6, Event 1 is compared to the measured acceleration data (Figure 9), there is an approximately 42 ms offset between the start of the crash as recorded by the IST and the TDAS accelerometers and the start of the recorded EDR acceleration.

Tests 11 and 12, Non-deployment Events 2 and 3 Tests 11 and 12 were the second and third non-deployment events which were recorded during this investigation.

These tests were part of the Nissan front end to barrier series. The event data for Test 11 was written into Location 2 in the ACM resulting in both locations being filled as the event generated in Test 6 was still present in Location 1. The non-deployment event data for Test 12 was written in Location 1 and replaced the non-deployment event data from Test 6. The maximum EDR reported delta-v from the Test 11 event was -5 mph. The delta-v calculated from the TDAS measured and filtered acceleration for Test 11 was -5.5 mph. The maximum EDR reported delta-v from Test 12 was -5 mph. The delta-v calculated from the TDAS measured and filtered acceleration for Test 12 was -5.5 mph. In reviewing the individual acceleration data points in the EDR data for both non-deployment events, the first EDR recorded acceleration data point was found to be at -3 G as opposed to zero. The EDR report graphs for both the Test 11 and Test 12 events were nearly identical to the graph shown in Figure 8.

When the EDR acceleration data from Tests 11 and 12, Events 2 and 3, is compared to the measured acceleration data (Figure 10), there is an approximately 45 ms offset between the start of the crash as recorded by the IST and the TDAS and the start of the recorded EDR data meaning that the ACM did not record the first 45 ms. Again,

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this accounts for the difference between the EDR delta-v and the delta-v calculated from the filtered and processed accelerometer data. The EDR reported delta-v is -5 mph for both events, and the delta-v calculated from the filtered TDAS recorded acceleration is -5.5 mph for both events.

A comparison of the data for Test 10 and Test 11 (Figure 5) shows that the difference between setting and not setting a non-deployment event. The TDAS calculated delta-v for Test 10 was 4.85 mph and the ACM did not store an event. The maximum measured longitudinal acceleration for Test 10 was -5.2 Gs at approximately 85 ms and the maximum measured longitudinal acceleration for Test 11 was -6.5 Gs at approximately 78 ms. Further evaluation of this threshold will be conducted in future testing.

Test 15, Non-deployment Event 4 Test 15 was the fourth and final non-deployment event which was recorded during this investigation. This test was part of the crash series of the Nissan backing into the barrier. The non-deployment event data was written into Location 2 in the ACM and replaced the non-deployment event data from Test 12. The maximum EDR reported delta-v from this event was 5 mph. The delta-v calculated from the TDAS acceleration was 6.5 mph. In reviewing the individual acceleration data points in the EDR data (Figure 8) which are recorded once every 10 ms, the first recorded data point was found to be at 9.5 G as opposed to zero.

Figure 4: Chart 1, Summary of Test Results, Tests 1 through 6.

Figure 5: Chart 2, Summary of Test Results, Tests 7 through 12.

Figure 6: Chart 3, Summary of Test Results, Tests 13 through 15.

Figure 7: Chart 4, Summary of Test Results, Tests 13 through 15.

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Figure 8: Test 6, Event 1 EDR Longitudinal Acceleration.

Figure 9: Test 6 EDR Acceleration Data Offset.

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When the EDR acceleration data from Test 15, Event 4 is compared to the measured acceleration data, there is an approximately 12 to 15 ms offset between the start of the crash as recorded by the IST and the TDAS accelerometer and the start of the recorded EDR data. The ACM did not record the first roughly 12 to 15 ms of acceleration data.

This explains the difference between the EDR reported delta-v of 5 mph, and the 6.5 mph delta-v calculated from the accelerometer recorded value.

The time duration of the missed data for Crash 15, Event 4 is less than the forward crashes. This is reasonable as the EDR measured acceleration at algorithm enable was

9.5 Gs as opposed to the 3 Gs that were seen in the nondeployment events set in the forward direction. The higher accelerations were due to several factors regarding this crash. This crash was into a concrete barrier as opposed to another vehicle, the forward impacts were into a trailer hitch which, prior testing has shown, tends to produce a long duration event (6). The vehicle approach speed was higher than prior barrier tests at 5.1 mph. Lastly, the rear of this vehicle was found to have a higher stiffness than the front of the vehicle. The acceleration from the rearward impact increased more rapidly due to the combination of the higher speed and increased vehicle stiffness.

Figure 10: Tests 11, 12 EDR Acceleration Data Offset.

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A comparison of the data for Test 14 and Test 15 (Figure 6) again shows that the difference between setting and not setting a non-deployment event. The TDAS calculated delta-v for Test 10 was 4.93 mph and the ACM did not store an event. The maximum measured longitudinal acceleration for Test 10 was 6 Gs at approximately 28 ms and the maximum measured longitudinal acceleration for Test 11 was 10 Gs at approximately 20 ms. Further evaluation of this threshold will be conducted in future testing.

Conclusions The initial purpose of this testing was to determine the minimum non-deployment event recording threshold for Nissan vehicles. Based on an evaluation of previously collected case related EDR data, this threshold appeared to be less than the 5 mph or 8 kph described in 49CFR563. In this investigation, one non-deployment event was stored with an EDR reported delta-v of below the threshold at 4 mph. However, that event was found to have an onboard accelerometer measured and calculated delta-v of 5.3 mph. Based on that data and the testing described here, the minimum event recording threshold appears to be 5 mph or 8 kph on the longitudinal axis in both the forward and reverse direction for this vehicle. During this testing, no non-deployment events with zero acceleration or zero delta-v were stored. The testing in this analysis, however, was limited to longitudinal crashes only.

aligned with the vehicle x-axis resulting in purely longitudinal accelerations. It is also currently unknown if the same behavior would be observed in passenger cars. As such, there are plans to augment this research with additional testing and vehicles. Future testing is planned for August 2020 using a Nissan passenger car and adding a lateral acceleration. Additional testing is also planned for January 2021 using the Nissan Pathfinder utilized in this investigation. This future testing will be the subject of future publications.

Acknowledgements The authors wish to thank SATAI for providing a location for this testing and Robert D. Anderson, Wesley Vandiver, Michael Rosenfield, Michelle Hoffman, and Kurt D. Weiss for their assistance during testing. Thanks also to any others whom the authors may have missed in these acknowledgements.

References 1. “49CFR563” 49CFR563 - EVENT DATA RECORDERS.NHTSA, 1 Oct. 2011. Web. 13 Dec. 2014.

2. Anderson RD, “Estimated Likelihood of Obtaining the Non-Deployment Event of Interest and Pre-Crash Data Guide for The Toyota Gen1 Airbag Control Modules”, Collision, The International Compendium for Crash Research, Volume 12, Issue 2, 2018

During the testing conducted for this investigation, it was confirmed that the event recording order follows what is published in the data limitations in the EDR report. The 3. Haight WR & Anderson RD, “Toyota Gen1 EDR oldest event is overwritten by a new event. If the oldest Event Recording Logic,” “Case Problem,” and “Case event is Event 1 and Event 2 is more recent, Event 1 is Problem Solution,”, Collision, The International overwritten by a new non-deployment event leaving the Compendium for Crash Research, Volume 13, Issue more recent event in place. Future work will involve con2, 2019. firming that this recording order does not change when a 4. Vandiver, W., Anderson, R., Ikram, I., Randles, B. deployment occurs. et al., "Analysis of Crash Data from a 2012 Kia Soul Also, during this testing, it was determined that the ACM Event Data Recorder," SAE Technical Paper 2015-01does not record the start of the crash pulse during a low 1445, 2015, doi:10.4271/2015-01-1445. speed non-deployment event. In the low speed frontal non-deployment event setting crashes into a second vehi- 5. SAE J211-1 (1995): Instrumentation for Impact Test, Part 1, Electronic Instrumentation, 49 CFR 571.202a cle and into a barrier, the ACM did not record up to 45 ms of the crash pulse and in low speed rear non-deployment 6. Cox BS, “Comparison of Accelerations Data and Huevent setting crash the ACM did not record 12 to 15 ms man Subject Kinematics Data in Bumper-to-Bumper of the crash pulse. vs. Bumper-to-Hitch Low Speed Collisions”, Proceedings of the American Academy of Forensic Sciences Future Work 60th Annual Meeting, Volume 14, 2008.

Presented here are the results from a single vehicle test series. It is currently unknown if the observed recording threshold behavior is influenced by off-longitudinal axis or roll axis acceleration as all the impacts were essentially www.collisionmagazine.com

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A REVIEW OF ENGINEERING MECHANICS APPLIED TO ACCIDENT RECONSTRUCTION Jai Singh, BS, MS, MA, ACTAR

Biomechanical Engineering Analysis & Research, Inc.

A

PART 3: SELECTED APPLICATIONS

bstract This paper is the final paper, in a series of three papers, that collectively provide a review of engineering mechanics as applied to the field of accident reconstruction. The first and second paper, in this series, reviewed the fundamental and foundational aspects of rigid body kinematics and rigid body dynamics, respectively. The most salient findings from the first two papers are presented within the introduction of the subject work. This is followed by the presentation of the translational and angular impulse-momentum forms that derive

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directly from the translational and rotational rigid body equations of motion. The planar reduction, which greatly simplifies the problems of interest, is one that is found in multiple accident reconstruction applications. These simplifications, which are consistent across the applications of interest, are also presented as being introductory. The body of the work focuses on four selected applications: rigid vehicle dynamics, rigid body accelerometry, the Calspan Reconstruction of Accident Speeds on the Highway (CRASH3) modeling approach for oblique impacts and planar collision analysis.

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Introduction Brief review of kinematics and dynamics The first two papers in this series of papers introduced a number of important findings in regards to rigid body kinematics 1 and dynamics. 2 These findings are of direct relevance to the subject work. However, it is first important to review the subscript and superscript system employed. For translational vector kinematic quantities, the previous work employed a single right subscript to denote the body being referenced, the point being referenced or both. Portions of the subject work involve multiple bodies, multiple points of interest and multiple points in time. Consequently, the right subscript on translational vector variables includes up to three terms of the form ‘iq(t)’ where i refers to the ith body, q refers to the point q (the symbol, o, in this position, is a reference to the mass center of the body) and (t) refers to the point in time that is of relevance (the symbol, (o), in this position, is a reference to an initial value and the symbol (f ), in this position, is a reference to a final value). Translational kinematic vector variables also have a left superscript and potentially a left subscript. The left superscript refers to the expression frame of the vector. For example, a translational kinematic vector variable with a left superscript of Bi is a vector that is expressed in terms of components along the unit vectors of the ith body frame of reference. The left subscript, when present, refers to the frame-referenced time derivative (i.e. the frame in which the frame referenced derivative is taken). Second frame-referenced time derivatives were presented with a double left superscript and subscript (the inner superscript and subscript pair referencing the expression and derivative frames for the first derivative and the second pair referencing the second derivative). For the subject work, having previously presented the full pair forms, the following shorthand notations are employed, for the sole purpose of presentation simplification, for any relevant translational kinematic vector ζ and frames A and B: (1) Angular kinematics vector variables were presented with a right subscript, left subscript and left superscript. If these terms, respectively, are A, B and X, the interpretation is one of the kinematic variable of the A frame, about the B frame, expressed in the X frame. Finally, because all of the relevant frames of reference are fully specified for each vector kinematic variable, the use of dot notation (single and double dot notation for the first and second time derivatives) is equivalent to the use of alphabetic notation (v and a for velocity and acceleration) irrespective of expression frame.

The first paper presented the primary equation for relating the inertial frame of reference (G frame) position of a point, q, to the position of the same point in a body frame (B frame) and the G frame position of the origin of coordinates of the B frame.

(2) In equation (2), R was defined as the direction cosine matrix (DCM). The DCM transforms the coordinates of a vector from the right subscript frame into those of the left superscript frame. The DCM was shown to be an orthogonal matrix (R-1 = RT where the first is the inverse and the second is the transpose). The DCM can be parameterized in any number of ways (e.g.

Euler angles, quaternions, etc.). Equation (2) can readily be expressed in two alternative forms through simple algebraic rearrangement.

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i  rio  R Bi of riqthe operand on the left Equation (5) was generalized for any vector expressed in theriqframe of reference G G dt dt d d shown Bi the of each equality, as the derivative transport theorem. The G derivative of equation to be G (2) was G r  r  GR riq G G G B B i i iq Bi ion (2), R was defined as the direction cosine matrix (DCM). The DCM transforms the coordinates      R   Bi riqio  r r r dt following: iq io Bi dt iq G B i or from the right subscript frame into those of the left superscript frame. The DCM was shown to G riq  G rio  G R Bi Bi riq  BGi Bi  Bi riq thogonal matrix (R-1 = RT where the first is the inverse and theGsecond isGthe transpose). The DCM d Gin the B di frame, When q is fixed the position of q in the Bi frame is time-invariant and the first term in t Bi G arameterized in any number of ways (e.g. Euler angles, quaternions, Equation riq etc.).  rio  G R(2) riq readily Bi can parentheticaldtin equation such a case: dt (7)q is (7) When is zero-valued. fixed in the BFor i frame, the position of q in the Bi frame is time-invariant a ssed in two alternative forms through simple algebraic rearrangement.

G G Bi Bi riq parenthetical rio  G R Bi Bi in riq equation  G Bi  (7) riq is zero-valued. For such a case: Bi Bithe four translational For a two Greference riq  G rio frame  G R Bi problem, riq Bi Bi Bi G G G G G G  Bi  (3) eight riq  R G riq  rio riq  rio  R Bi riq (3) velocity solutions and translational acceleration solu-Bi Bi G G G When q is fixed in the Bi frame, the position of q intions the were Bi frame is time-invariant Bithe riqvelocity  the rio first transformation Rterm  riq Bi G in presented.

The angularand For (7) a two reference frame problem, parenthetical in equation is zero-valued.

For such a case: the four translational velocity solutions and eight translation relationships were presented in Appendix B of the first paThe first paper also introducedacceleration the concept solutions of the framewere presented.

Theframe angular velocitythe transformation relationships presenta 1 For a two problem, four translational velocitywere solutions per. reference referenced time derivative. Simple derivatives de- paper Bi Bi G G in Appendix B ofGwere the first [1].   rwere presented. The angular velocity (8) r  r  R solutions acceleration transformation relatio

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o introduced the concept of the frame-referenced time derivative. Simple derivatives in Appendix B of the first paper [1]. as the expression frame of the vector quantity serving as G  Bi  G R Bi BGi   Bi Bi R G BGi   Bi  Bi R G GG   Bi G R Bi (9) hose for which the derivative frame was the same as the expression frame of the vector G For a two reference frame problem, the four translational velocity solutions and eight translational the operand for the derivative operator.

G G Bi Bi Bi Bi s the operand for the derivative operator.  i  R Bi G   Bi were   Bi G R Bi R G presented R G GG  were presented. angular Simple velocityderivatives transformation G Brelationships G Bi  st paper also introduced theacceleration concept of solutions the frame-referenced time The derivative.

The paper primary findings of the second paper findings were the presentation of the Newton-Euler primary pre- equations of motio in derivative Appendix B of thewas first efined as those for which the frame the same[1]. as the expressionThe frame of the vector of the second paper were the G nd law of motion, was show representing the G frame expression of Newton’s 2 d G b G  Bi d Bi b Bi  The translational dynamics, sentation of the Newton-Euler equations of motion.

The of the Newton-Eul The of the second paper were the presentation (4) primary findings y serving as the operand for  the derivativeoperator.

(4) Gb Bi b G Bi Bi translational Bi Bi dynamics, G G representing the G frame expresto be: G  dt dt    Bi dynamics,  Bi representing R Bi The RG G  RG G  R Bi (9) of Newton’s 2nd law the G frame expression G Bi  G Bi translational nd o introduced the concept of the frame-referenced time derivative.

Simple derivatives G G Bi i to be: sion of Newton’s 2 law of motion, was shown to be: dwas b the d Bas btheBiexpression G G vector ose fordefined which as thethose derivative frame frame  GGdefined  Bi b in b same (4)G  were in which the derivative frame differed the the   Gd  Gd G  d d 2frame Mixed derivatives were as those which the de-of G from G expression G G G G The operator. primary findingsdt of the second paper were the presentation the Newton-Euler equations rio (1 Fi derivative. pi  m rio   mof rio   m i  vio   m i of vmotion. per also introduced the concept ofdt the frame-referenced time Simple derivatives   i i io  m i a io  m i stity theserving operand for the derivative 2 as the operand for the differed derivative operator.

G G 2 G G rivative frame from the expression frame of the dt dt dt dt nd             d d d d law of motion, was shown The translational dynamics, representing the G frame expression of Newton’s 2 G G G d as those for which the derivative frame was the same as the expression frameG Fof the vector pi  m i  2 G rio   m i  rio   m i  vio   m i G v io  m i i vector quantity serving as the operand for thediffered derivative lso introduced the concept of the in frame-referenced time frame derivative.

Simple derivatives to be: dt dt dt dt ving as theGwere operand for derivative operator. derivatives defined as those the derivative from the expression frame G the G Biwhich BiG       Bi B i dB b Gframe d BG d bthe Boperator. d bbthe i  G as the i same hose which was Bb G expression frame of the vector i  i   b (4) ector for quantity serving as the operand for the derivative operator.    Bderivative      b b b b (5) G B i G G Bi i dtG Bi G G operator. dt as the operand derivative (10) dt for the dt G G 2  Gd G   Gd G  d b GG  Bi ddBiGb Bi   d G  G G G    (10)    Fi  m r m r m v  m v  m a  m r  Gb  b (4)       G Bi Bp G i i io i io i io i io i io i io 2 i dG which b d Bibdiffered dt dtframe Bi B Bidt G G expression  Gdt product  dtbetween  frame  dt (5)   derivative   cross i the G  B B were defined as those in from the i i b b  b b   denotes the angular velocity vector.

The ability to rewrite the any d b Bi G  G d Bi b G G Bi Bi  dt forthe dt operator.

(4) the operand G bderivative Bi b (5) dityc serving as the as vector-matrix product of the antisymmetric (i.e. skew-symmetric) matrix dt dt vatives were defined as those in which the derivative frame differed from the expression frame e first vector (b) and the second vector (c) was introduced. The relationship between the rtion quantity as operand forBithe operator.

G derivative (5),  Gserving denotes thethe angular velocity The ability to rewrite the cross product between any equation d Bi bframe d vector. b Gthe equation ωderivative denotes angular velocity vector.

The BIn B i which B(5), G two vative the derivative (for  and ) wasIn shown to (10), p is the translational (i.e. linear) momen i referenced i time sctors wereband defined as those in the frame differed from the expression frame         Gframes b b b b (5) and c as the vector-matrix productG of the (i.e. skew-symmetric) matrix Bi G Bi G Biantisymmetric ability tofor rewrite the product between any two vec- tum. The corresponding Bi frame expression was shown dt dt cross ntity serving as the the second derivative operator.

G operand Bi Bi G ted with the first vector and the vector (c) was introduced.

The relationship between the dtors b (b) d b G i  i by applying (i.e. premultiplying) the Bthe b BBand  Bvector-matrix  GGproduct b cBGias b b  GG Biof the b antisym- as being obtainable (5) i i d time derivative and the frame referenced time derivative (for two frames  and ) was shown to dt dt G B B G DCM from the G frame to the Bi frame to both sides of  skew-symmetric) with thebetween any denotes the angular velocity(i.e. vector.

Thei dability rewriteassociated the cross product d i b metric bd  rGqto matrix Bi d B Bi G G i           b b b b (5) (6)(10). This application was referenced as applyBi G G G Bi (c) R Bi (b) rq and of R d c as the vector-matrix product the (i.e. matrix first vector the second vector wasskew-symmetric) introduced. equation  antisymmetric dt angular dt dtsecond dt introduced. denotes velocity vector.

The to rewrite the crossand product e(5), first vector (b)the andThe the vector (c) was Thederivative relationship between the any ing abetween kinematic transform. The rotational dynamics (Euler relationship between theability standard time (10), p is the translational (i.e. linear) momentum. The corresponding Bi frame expre   In equation b andand c as the vector-matrix product of derivative the (for antisymmetric (i.e. skew-symmetric) matrix d r equation) were derived the body frame.the This was shown d the frame referenced time (for two frames A and vative the frame referenced time derivative two frames  and ) was shown to q    shown as being obtainable by applying (i.e.inpremultiplying) DCM from the G frame to the B denotes the angular velocity vector.

The ability to rewrite the cross product between any Rof reference rframe  was generalized any (b) vector the of the on the left with the firstfor vector andexpressed the secondinRvector introduced.

Theoperand relationship between theto(6) q (c)  to be preferable the derivation in the G frame due to a kinematic transfo B) was shown to be: dt dt both sides of equation (10).

This application was referenced as applying nd c as the vector-matrix product of the antisymmetric (i.e. skew-symmetric) matrix s the derivative transport theorem.

The time G derivative of (for equation (2) wasshown towas be the me derivative and the frame referenced derivative frames and ) shown to matrix associated equation (10),rotational ptwo is the translational (i.e. linear) The corresponding Bi frame expression the fact that momentum. the with mass moment dynamicsbetween (Euler equation) were derived in the body frame.

Thisof was shownwas to be pre he first vector (b) and the second vector (c)Inwas introduced.

The relationship the frame to of iner shown as being obtainable by applying (i.e. premultiplying) the DCM from the G frame to the Bi moment   inertia tensor isthe not time-varying in the B frame with but ismass on (5) was generalized for any vector expressed in(for thetwo frame of reference of the operand ontothe left the derivation in the G frame due fact that the matrix associated d r rivative and the framedreferenced time derivative frames  and ) was shown to q    sidesGof equation This application referenced kinematic transform. The (6) was R both The r  theorem.

(6) (2) equality, as the GderivativeG R transport derivative of(10). equation was to be time-varying in the G frame.as applying is not time-varying in the Bshown frame but isthe time-varying in the Ga frame. d G dt d  G q G Birotational dt  dynamics (Euler equation) were derived in the body frame.

This was shown to be preferable to ng: d r riq  r  R r d io B iq q dt dt (6) R   rqi the  R derivation in the G frame due to the fact (7) that  G the matrix associated with mass moment of inertia tensor dt G  Bi the  Bdt eneralized for any Gvector of theBoperand onisBthe left d Bi L  id i reference of i time-varying i  Equation wasGin generalized expressed in but  is Bframe not in the frame inBGthe Birtime-varying riq d Gexpressed rGiod(5) L GBiframe.

I BGi  Bi  BGi Bi  Bi I BGi Bi  Bi riq  BGi Fi qforriqany vector R Bi riq  G G Mi  i d B G G G (6) R  r rq The  R i dt the derivative transport theorem.

G derivative of equation (2) was shown to be the  r  R r the dt frame of iqreference ofio dt the operand on the left of each Bi iq (11) dt dt ) was generalized for any vector expressed in the frame of reference of the operand on the left (7) equality, as the derivative transport theorem.

The G Gfirst ded G in the G Bi frame G Bitime-invariant Bi Bi B B B B B B B B B B the B frame, the position of q is and the term in the i i i i i i i i i i i  ality, as the derivative transport rtheorem. equation was shown the    vector R Bi G derivative of rio  The riq  G In riq G Mi (11), Li the L to be I momentum  I G and riqthe Fi  (2) L  angular  Gmatrix (11) equation is associated with the mas iq  Bi  I is of equation (2)the was shown beBi the following: generalized anyrivative vector expressed in frame of to reference of the operand on Gthe left G Bi G Bi dt uation (7) is for zero-valued.

G G For such a case: of inertia tensor.

The body frame expression of I was shown to be a diagonal matrix (conta dG d G BiG derivative of equation (2) was shown to be the as the derivative transport The riq  theorem. rio  G R riq Bi principle momentsand of inertia the diagonal and with all product of inertia terms being zer q is fixed in the dt Bi frame, the position of qIninequation the Bi frame is the first term in dt (7)along Bi G G (11),when L time-invariant is the angular momentum and is the matrix associated with the mass moment Inwas equation (11), Lthe isI the angular momentum vector andcoordinates iq dGGG rio G G d (8)vector R Bi GBiFor  Bi rBaiqi case: the body frame chosen as a central frame (i.e. one with origin of at the m G G r Bi G  Bsuch i Bi hetical in equation (7) is zero-valued. riq  rioriq R Bi rioriqofGRinertia BiBi riq rtensor. body frame of Iassociated was shown a diagonal (containing the iq I is the matrix withtothebemass momentmatrix of inertia of The the body).

The expression corresponding dt G dt (7)G frame expression could be obtained by applying the (7) G dG d GG principle moments of inertia along the the allexpression product ofofinertia terms being tensor.

The and bodywith frame I was shown to be zero-valued) Bi B G G i i Bdiagonal i frame and the G frame. The last term in equation (11) is the applied torq riq four Grio nce frame problem, the velocity andbetween eight translational r translational rio GGRR riq  BBGii  Bi solutions  BBitransform riq BiB  rGiqR (8) rth i when the was chosen as a central frame (i.e. one with origin of coordinates at the Bis G body Bi  frame iq dtiq q inriqthe a diagonal matrix (containing the principle moments of mass center the Bi frame, thedt position of Bri ioframe time-invariant and the first term in the (7) i i relationships body due towere the moment ons were presented.G The angular velocity transformation presentedof applied force on the body. When more than one force is ap G G Bi Bthe Bi of body).

The corresponding G frame expression could be obtained by applying the the kinematic i uation (7) is zero-valued.

For such a case: inertia along the diagonal and with all product of inertia riq  rio  R Bi riq  G Bi  riq individual force in equation (10) is replaced by the sum of all applied forces and individual m he first paper [1]. the ixed in the B frame, position of q in the B frame is time-invariant and the first term in the transform between the B frame and the G frame.

The last term in equation (11) is the applied torque on the i i i wo reference frame problem, theinfour velocity eight translational When q is fixed the Btranslational frame, the position of solutions qofinequation the Biandterms zero-valued) when the bodymoments. frame was chosen the right (11) isbeing replaced by the sum of applied i th al in equation (7) is zero-valued.

For such a case: i body due to the moment of applied force on the body.

When more than one force is applied, the B B G G G i ation solutions presented.

The transformation relationships were presented frame is rtime-invariant andivelocity theG first in the parenthetias a central frame (i.e. one with origin of coordinates at G were G Bi Biangular Gterm R Bthe n the Bi frame, of qR in frame and the first term in(8) thethe io  G  the position BBii Bindividual time-invariant RriqBi BGi  i Bi RrGiqis R Bi in equation (9) i G G B i  Bi G G Biforce (10) is replaced by sum of all applied forces and the individual moment on endix B of the first paper cal in [1]. equation (7)aiscase: zero-valued. For such a case: the mass center of the body). The corresponding G frame quation (7) is zero-valued.

1.2 Translational and angular impulse-momentum forms Bi equation GFor such G G the Bright i of (11) is replaced by the sum of applied moments.

(8) be obtained by applying the kinematic r  rio  R Bi G Bi  riq expression impulse-momentum could ce of frame problem, theGwere fouriqthe translational velocity solutions and eight and translational relationships can be derived directly from G Bi Bi Bi The G translational G ngs the second paper presentation ofBi the Newton-Euler equations ofangular motion.       R  R   R  R (9) and the G frame. The last transform between the Bi frame B iG BBi i G B i G Bi G G Bi Bi G G angular GBvelocity i G ons were presented.

The transformation relationships were presented (10) and (11), respectively.

For the translational case: nd   (8) r  r  R   r (8) ynamics, representing the G frame expression 2 law and of motion, was shown iq io Bi G Bi 1.2 ofiqNewton’s Translational angular impulse-momentum formstorque on the ith body term in equation (11) is the applied reference frame translational he first paper [1]. problem, the four translational velocity solutions and eight Thepresentation translational angular impulse-momentum relationships can beon derived directly from equations solutions wereofpresented. angular velocity transformation relationships were due to presented the of moment force the body.

When G of applied mary findings the secondThe paper were the ofand the Newton-Euler equations motion.

Fi dt   G d G pi  m i  G d G vio ence problem, the four translational velocity solutions andFor eight translational  (10) and (11), respectively. the translational case: nd G G B B B B G G xnslational B offrame the first paper [1]. more than one force is applied, the individual force in i dynamics, G 2 law of motion,   Rrepresenting  Gi i R G theGi   Bframe   G itransformation Rexpression  R Bi of Newton’s (9) was shown GG  i G B i  ions were presented. angular relationships were presented d 2 BGi  BThe d G  velocity d GG G  Bi G G G G (10) p  m  2 rio   m i  rio   m i  vio   m i v io  m i a io  m i rio G G G G G thei first ipaper Bi G term on the left of the first (12), by definition, is the G frame translation dt Bi  (12) Fi dt  equality d(9)pi inmequation  Bi  G R   Bi R G  dt  The  dt [1]. GG  Bi R G GG  R Bi i  d vio B i G Bi G Bi  Bi  ngs of theGsecond paper were the presentation of the Newton-Euler equations of motion. impulse.

Evaluation of the integral in the term on the right of the second equality yields the dif GCollision 2 G G 18 Magazine Volume 14 Issue 2 www.collisionmagazine.com  d BG  Bi d G Bi of Newton’s  d 2nd law ofG motion, was dGG G G G i G frame ynamics, shown velocity Bi m  Bi R Gmexpression   R GG   G RGBthe  i G Rthe  (9) rio between (10) the terminus and start of the event.

Fi  representing pG i v Gframe m i v translational B i 2 G rBioi   i G Bi rio  Gm i Bi io  m i a io  m i i io  The term on the left of the first equality in equation (12), by definition, is the G frame translational (linear) y findings paper were the presentation of the Newton-Euler equations of motion. dtof the second dt dt dt       of the integral in the was termshown on the right ofG the second equality yields the difference in ional dynamics, representing the G frame impulse. expressionEvaluation of Newton’s 2nd law of motion, G G Pi  and m i start vio of m vio  f   G vio  o  ings of theVolume second paper were the presentation of the Newton-Euler equations of motion. i the G frame translational velocity between the terminus the event.

18 G 10/9/2020 3:42:59 PM  G d 2 14G Issue 2.indd the  d G  expression  G d G of Newton’s nd G G of motion, dynamics, (10) p  m representing r  m G frame r  m v  m G v 2 mlaw a  m G r was shown

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equation (10). application wasBThe referenced as applying a kinematic transform.

B i i  i i eof translational (i.e.BiThis linear) momentum. corresponding was (11) The ying inBGthe is time-varying in the GBiframe. frame  Bmatrix I BGi  Bi  I BGi Bmass  BiB  BGi Fi expression Mi B Lbut riqi frame i  G Lthe G B i  i frame due to the fact that associated with moment ofshown inertiatotensor ynamics (Euler equation) were derived in the body frame.

This was be dt ble by applying (i.e. premultiplying) the DCM from the G frame to the Bi frame topreferable to he in B frame but is time-varying inthat thethe G frame. on G frame the fact matrix associated with mass moment of inertia tensor G due to was (10). the This application referenced as applying a kinematic transform.

The d Bi Bi Bi  Bi Bi Bi B i Bi Bi Bi varying in the B frame but is time-varying in G frame.       I   I   M L L r F (11) momentum may be written in terms of the body frame expre L is the angular momentum vector and I is the matrix associated with the mass moment frame the angular G i were derived i G in the G Bi G Bi Bi body iq i expression uler equation) body frame.

ThisGThe was shown toG be preferableofto G dt dto .frame body frame expression of I was shown to be a diagonal matrix (containing the B i Thedue Bi the B B B B B B B B with i  that i the imass i matrix associated of inertia tensor I Gi  Bmatrix  Gi Bassociated  i I Gi Biwith  of  Gmoment Li fact riq the Fi (11)the mass moment of inertia tensor and the body frame expression o G Mi  GGL  i i nts offrame inertia along the and with all product of inertia terms being zero-valued) dt but d Bidiagonal velocity he B is time-varying Bi Bi in the Bi BG Bi B angular Bi Bi Bi of the body frame about the G frame. The former is time-invariant. The body f i frame. i body expression momentum 1), L is the angular momentum isthe matrix the angular mass moment  Bione  I(i.e. I GThe I frame  riq  with Li  G frame Lvector Fi of (11) may be written in terms of the body frame expression G M i  as G and B i  origin G of Bassociated i coordinates rame was chosen central with at the mass center expression of theG linear impulse can be written as the transformed G frame expression. dtaexpression of the to matrix with the mass moment or. The body frame of I was shown be a associated diagonal matrix (containing the of inertia tensor and the body frame expression of the The corresponding G frame expression be obtained by applying the kinematic G he angular vector(10) andisI replaced is the could matrix associated the mass moment by product the of allwith applied forces d Bmomentum Bents Bi equation Bi diagonal Bi and with Bsum Bi inertia of the body frame about the G frame. The former is time-invariant. The body frame of inertia all terms being zero-valued) i i i i angular i velocity  BIi  vector along the I BGof  Bi term I Bto equation  of Mthe Liangular riqmatrix (11) en Bframe and G frame.

The in (11) is the applied torque on the body expression was shown be a diagonal matrix (containing the Bmass Bi moment i frame G i  G Lthe G B i last G B GF i associated i i (11), L is the momentum and I is the with the individual on the right ofcoordinates equation (11)  Bi  f   BGi Bi  oG  Bi riq  I  written Bi  asBithe I BGitransformed  Bi R G GG Pi  expression of the linear impulse canmass be expression. y frame dt was chosen and as a the central frame moment (i.e. one with origin of at the center G  frame the moment of applied force on the body.

When more than one force is applied, the inertia along the diagonal and with all product of inertia terms being zero-valued) ensor.

The body frame expression I was shown to be a diagonal matrixthe (containing is replaced byexpression the of sum of could applied The corresponding G frame bemoments. obtained by applying kinematicthe (17) in equation (10) isalong replaced by the sum of all applied and the moment on was chosen as a central frame (i.e. oneand with origin of forces coordinates atindividual the mass center oments of inertia the diagonal with all product of inertia terms being zero-valued) B B Bi Bi Bi Bi G e angular vector and IThe is the associated moment i i torque B i frame and the G frame. lastmatrix term in equation with (11) isthe themass applied on the ween the Bimomentum (17)      R I  I   r P  G the  G G i tionframe (11) iswas replaced byas the sum ofand applied moments.

Bi  G Bi  f  G Bi  o  iq orresponding Gchosen frame expression could be obtained by applying the kinematic 1.3more Planar simplifications Translational angular impulse-momentum forms ody a force central frame (i.e. one origin of coordinates frame expression of I wasonshown to be a with diagonal matrix (containing themass center obody the moment of applied the body.

When than one force at is applied, the 2. body The body frame expression of torque the angular momentum may be written in terms frame expression Bi frame corresponding and the G frame.GThe last expression term in equation (11) is the applied on the the Abenumber of by accident reconstruction evaluations are based in ofℝthe In other words, they are p y). frame could obtained applying kinematic nertia along the diagonal and with product of inertia terms zero-valued) e in The equation (10) isThe replaced by the all sum ofangular all applied forces andbeing the individual moment onof inertia tensor and the body frame expression of the translational and impulse-momentum relaof the matrix associated with the mass moment oment of applied force on the body.

When more than one force is applied, the evaluations.

The typical planar evaluation involves the case in which G frame vector components a frame and the G frame.

The last term in equation (11) is the applied torque on the etween the B ational and angular impulse-momentum forms i Planar simplifications 1.3 simplifications was chosen asreplaced a centralby frame (i.e. one with origin of Planar coordinates at the mass center uation (11) is the can sum ofderived applied moments. tionships be directly from equations (10) and angular velocity of the body frame about the G frame.

Thezero-valued former is2 time-invariant.

The body frame abou ation (10) is replaced by the sum of all applied forces and individual moment on unit vector, corresponding to the Z axis, are while G frame vector components the E 3the e to the moment of applied force on the body.

When more than one force is applied, the lrresponding and angular impulse-momentum relationships can be derived directly from equations G frame expression could be Aobtained the kinematic evaluations are based in ℝ . In other words, they are planar number by of applying accident reconstruction expression of linear impulse can be written as the transformed G frame expression.

(11), respectively.

For the translational case: 11) is in replaced by the sum of applied moments.

A the number of to accident reconstruction evaluations are based and E2applied unit vectors, corresponding X-axis and Y-axis, respectively, zero-valued.

This is mir E1 the The body frame expression of individual the angular momentum may be written in terms of theare body frame expression equation (10) is replaced byterm the sum of all applied forces and the moment on pectively.

For the translational case: the G frame.

The last in equation (11) is the torque on Borce evaluations.

The typical planar evaluation involves the case in which G frame vector components along i frame and 2 lational and angular impulse-momentum forms for all for each body frame and for all other frames of relevance.

The result is that the Z-axis, z of the matrix associated with the mass moment of inertia tensor and the body frame expression of the .

In other words, they are planar evaluations.

The in R i-axis equation (11) is replaced by the sum of applied moments. oment appliedimpulse-momentum force on the body. relationships Whenthe more than force is applied, vector, corresponding to theequations Z Baxis, are Bzero-valued while G frame vector components about the E3 unit Bi Bi from B Bi Bi G nal andofangular can beone derived directly iaxes i remain i coincident frames and all other ‘vertical’ throughout the evaluation.

For any position angular velocity of the body frame about the G frame.

The former is time-invariant.

The body frame G G G G G (17) I    I     r  R P typical planar evaluation involves the case in which G moment  G Garei  zero-valued. This is mirrored v al and angular impulse-momentum G Bi  onto X-axis G G Y-axis, iq Bi  f  and Bi  o  (12) Fi dtsum   ofd allpiapplied  m iforms dand vioEand ation (10) is replaced by the thevectors, individual corresponding respectively, E1forces (12) 2 unit   espectively.

For the translational case: and its frame-referenced time derivatives: expression of the linear impulse can be written as the transformed G frame expression. unit frame vectorof components along the angular impulse-momentum relationships can be derived directly equations 1) is replaced by the sum of applied moments. for each body frame andfrom for all other frames relevance. The result isE3that thevector, Z-axis,corzi-axis for all body anslational and angular impulse-momentum forms responding to the Z axis, are zero-valued while G frame ely.

For the translational case: frames andcan other ‘vertical’ axes from remain coincident throughout the evaluation. For any position vector tional angular impulse-momentum relationships be derived directly equations G equation G (12), G G Planar Gequality left of and the first equality in by definition, isall the Gequation frame translational (linear) 1.3 simplifications  The term the first in B(12), Bi G Fi dton  the dleft pi of  m i i i   Bi E  i  d vio rEG2eunit vectorBi components (17)  I  derivatives: I BG(12)  Bi  f   BGi about rBi e o 1 the riq1 2and R  BIn 1Pi  vectors, corand its frame-referenced time ), respectively.

For the translational case: G  Bi   G ation the integral in the term on the right of the second equality yields the difference in l andofangular impulse-momentum forms A number of accident reconstruction evaluations are based in ℝ . other words, they are planar by definition, is the G frame translational (linear) impulse.    are     G G G G G responding to X-axis and Y-axis, respectively, zero-valr   r  e2   r    r  e2  (12) Fi dt   the d relationships pterminus m  d viostart slational velocity and of theThe event.  between ngular impulse-momentum can derived directly from equations evaluations. typical planar evaluation involves the case inwhich G frame vector components along Evaluation ofi the iGintegral in be the term onthe theG right oftranslational the  mirrored G by definition, G G e left of the first equality in Gequation (12), is frame (linear) ued.

This is frame and   r body  for all other   for while 0 each 0 vector r  e eG1 frame (12) F dt  d p  m d v ely. For the translational case:  unit vector, corresponding to the Z axis, are zero-valued components about the the E i i i io   3 equality  the right thePlanar 1  second yields the1.3 difference in thesimplifications G frame translauation of the integral in the term on of second equality yields the difference in     frames of relevance.

The result is that the Z-axis, zi-axis for     G G Gis the G frame translational (linear) r  the first equality inGtional equation (12), by definition,   (18) r e r r e    2 and E unit vectors, corresponding to X-axis and Y-axis, respectively, are zero-valued.

This mirrored E 1A number (13) Pi  mvelocity vioterminus  m i vand start v2 ioof 2  based ℝ of accident reconstruction evaluations areall In other words, theyis are planar between the andevent. start of the event. all body    in 2 . i the io  f  terminus  o  the anslational velocity between frames and other ‘vertical’ axes remain coinciG GonGthe right of G the G f the integral in the term second equality yields the difference in -axis for all body for each body frame and for all other frames of relevance.

The result is that the Z-axis, z     (12) Fi dt   d in piequation  m i  d(12), vevaluations. n the left of thefirst equality istypical the G frame translational (linear) The planar evaluation involves theunit casevector in 0which G frame vector components along subs In the two equations under (18), e refers of  frame and iwith the numerical io by definition,  0 to the the  the dent throughout evaluation.

For any position vector nal velocity between the terminus and start of the event. frames and all other ‘vertical’ axes remain coincident throughout the evaluation.

For any position vector valuation of the integral in the term on the right of the second equality yields the difference in unit vector, corresponding to Z axis, are zero-valued while G frame vector components about the the E 3 G Gfollowing G l case, the simplest approach equality from (11): to theequation appropriate under (18),  refers to the derivative fr anddirection. its frame-referenced timeequation derivatives: (13)In the second Pi  m i isGto viouse  mthe vioits vioreferring i and  f  frame-referenced  oof  the time(13) derivatives: translational velocity between the terminus and start event.

E and E unit vectors, corresponding to X-axis and Y-axis, respectively, are zero-valued.

This is mirrored 1 2 he first equality in equation (12), by definition, is the G frame translational (linear) equations underin(18), e refersequation to the unit vectorto ofany the order  frame and with the numerical subscript The result shown the second extends derivative involving any number of rel G G G G In the two m iright vioof vsecond body frame andthe fordifference all other (13) frames of relevance. The result is that the Z-axis, zi-axis for all body G i  m i v io  io  oeach the integral inPthe term on the thefor equality yields in f   d frames of reference.

For the angular velocity and its frame referenced derivatives:   B B B appropriate direction.

In the equation (18),  refersFor to any the derivative frame. i i ireferring G G use G G to nal case, thebetween simplest approach to following equality from  coincident    r  eunder  the evaluation. frames and allthe other ‘vertical’ axes (11): remain throughout position vector r  esecond (14) For the simplest approach is to equation use the Pi rotational  mis Land riqthe mthe (13) al velocity the terminus start the i v iocase, GF iv 1 1 iof io  fevent.   v io  o  dt i         The result shown in the second equation extends to any order derivative involving any number of relevant its frame-referenced time derivatives: (18) r   r  e2   r    r  e2  following equality from and equation (11): (18) the simplest approach is to use the following equality from equation (11):   derivatives: 0 referenced 0  G frames of reference. For the angular velocity and itsframe d G G G G B B B     iv i i 0     0     Pi simplest differential m i  vapproach tionalby case, isiointegrating following equality from equation(13) (11): (14) f  use L riqvthe ough the the time dti and the resultant: io  m io  oG F i to i     0   r  e1   0   r  e G dt 1 d Bi     �      �-    L i  Bi Griq  BGi Fi (14) (14)  - 3 r e20    (18) r   r  e02    r 3  Bi Bi Bi Bi G Bdt i d     andtowith InBi the equations under (18), to unit vectorunder  frame the numerical subscript the simplest equality equation (11):e refers BiFi dt B (15) d Lisi to  usedt rthe  following riqtwo Ffrom  the of the (18), In the two equations e refers the unit vector i iq and G Gresultant: i dt  differential  rough by theapproach time integrating the    L r F (14) (19)     0 0     i iq G i    0     0   second dt of(19), thethe Athe frame and with the numerical subscript referring   referring toIn thethe appropriate direction.

In equation under (18),  refers to the derivative frame.     two equations under term following the dash in the right subscript refers to the approp  - 3  G dt and integrating - 3   � y the time differential thethe resultant: Multiplying through by time differential dt and inte- to the appropriate   � In direction. the second equation under Bi Bi the in G dBthat BiBB BiB i expression of moment arm is time invariant. y follows from the fact the i Bi i i frame The result shown the second equation extends to any order derivative involving any number of relevant of the expression frame. result shown in the second equation extends to any order deriv d LL F dt  riq  axis riqrdt (14) The (15) i  iqand G Fresultant: i dt i  GGFi i integrating  grating  resultant: the gethrough the differential the In theof two equations under (18), eofrefers to unit vector of the frame. frame andresult with of theNewton’s numerical (18), B the refers to frame the derivative The shown in integralby on thetime leftdt ofB the first equality leads to the solution ofthe theangular change in angular frames reference.

For velocity and its referenced derivatives: law of mech involving any number relevant frames of reference.

The expressions 2ndsubscript B B B G Bi i i i i the two equations under (19), the term following the dash in the right subscript refers to the appropriate In (15) d L  r  F dt  r  F dt the second equation extends to any (18), orderderivative iq G ithe last equality iq referring G i is simply e integral term oni the right of the Bi frame expression the   to the appropriate direction.

In of the second equation toinvolvthe derivative follow the form of the equations under (18) while the under expressions ofrefers Euler’s equation followframe. the form o Bi B i expression Bi the Bthe axis expression frame. result shown inofthe second equation extends to any derivative moment arm isThe time invariant. follows from the fact that the i (15) L iintegrating   BBii rframe FThe dt result  of riqequations of ing any number relevant frames ofNewton-Euler reference.

Forequations theorder yty the time differential dtG dand the iq  G i resultant: G Fi dt shown under (19).

This reduces the six coupled of motion to       (15) shown in the second equation extends to any order derivative involving any number of relevant 0 0 nd anysolution numberofofthe relevant frames of reference. The expressions of Newton’s 2 law of mechanics the integral on the left of the first equality involving leads to the change in angular angular its frame referenced derivatives:  velocity itsand   derivatives: of the momentequations arm isthe time invariant. ws from the fact that the Bi frame expression  velocity  coupled (two translational and one rotational). frames of reference.

For angular and frame referenced expressions   0the 0 of Euler’s follow the form of the equations under (18) while the equation follow the form(19) of the The integralG term on the right of the last equality is simply the B frame expression of i       Bi Bi B Bi B   B B B i i i i i The equality follows the factofof that the Bi frame gral on the left first equality leads the the change in angular G Fi Lthe Pfrom d from Lofi the rlast dt riqGto Fi dtsolution frame the moment is (15) time uality follows the fact i Biriq iGexpression (16)  iq  that    invariant.    equations shown under (19).arm This reduces the six coupled Newton-Euler equations of motion to three  �- 3  � - 3    angular  gral term on theonright of the last equality is simply the Binvariant. expression of the expression of the moment arm is The time Evalui frame planar case reduces the problem of orientation to rotation about a single axis. The of the integral the left of the first equality leads to the solution of the change in   kinematics  0rotational).

0  coupled equations (two translational and one ation of the integral on the left ofthat thesimply first equality leads expression transforms coordinates to any other  frame, therefore, reduces to a function . The on the right of the is of one the frame  from any  Bi lastBiequality i frame of the moment armthe is B time invariant. ws fromintegral the factterm that the Bi frame   the0right (19) 0 the  Bi Lexpression riqIn the (19)  following    (16)   in i  G Pi two equations under (19), the term   subscript dash the appropriate to the solution of the change in angular momentum.

The single angle.

For example, the DCM that transforms coordinates from refers the G to frame to the Bi frame red lse.   ral on the left of the first leads to The the solution of the change angularof orientation planar case reduces thein problem kinematics to rotation about a single axis.

The DCM     Bi equality Bi Bi  � axisof ofthe thelast expression The result shown in the second extends to any are: order derivative - 3This - 3  inverse � equation  L i term  r  the (16) integral isframe. simply the to any and (t).

DCM the transform to equality a Bfunction of the heading angle, i G Pi right ral term on the right of the lastiqon equality is that simply the expression the transforms coordinates fromofany oneframe other frame, therefore, reduces a function of a i frame nd lawto B B B involving any number of relevant frames of reference.

The expressions of Newton’s 2 mechanics i i i linear impulse.

Bi frame expression  L i of  the riqsingle  G Pi angle.

(16) coordinates from the G frame to the Bofi frame For example, the DCM that transforms reduces follow the form of the equations under (18) while the expressions of Euler’s equation follow the form of the In the two equations under (19), the term following dash the right to theappropriate In two under (19), the subscript term following the inverse   cos cos the equations sin the and 0in refers  sin i  t   0      DCM transform are: to a function of the heading angle,  i  tThis i  t  the i  t  i(t). equations shown under frame.

(19). This the Newton-Euler of axis motion to three  reduces  refers  equations axis of the expression The resultinshown incoupled the second equation to any order derivative dash the six right subscript to theextends appropriate  Bi L i  Bi riq  BGi Pi coupled equations (twoBitranslational (16) R G  t(16)    sin i one t   rotational).  cos  i  t   0  G R Bi  t     sin  i  t  nd  cos  i  t   0     and involving any number of relevant frames reference.

The expressions of Newton’s 2 law of mechanics of the of expression frame.

The result shown in the second    cos  iunder  Euler’s t   0(18) sin while i  t   the cos  iequation t   0 sinfollow    t   00  of1the 00 expressions 1 follow the form of the equations of  equation extends to Gany order derivative  involving iany the form    B i The planar case reduces problem oforientation single axis.

DCM The body frame expression of the angular momentum may equations shown reduces three R t   the (19). sin cos  ofithe t  sixkinematics 0coupled R Bi Newton-Euler tto cos  of (20)  t  number  relevant  rotation equations   0The  iThis  aexpressions  frames   sin  about  to G under i  tThe i  tmotion of reference. transforms coordinates from any one frame to any other frame, therefore, reduces to a function of a     be written in terms ofthat the bodyequations frame expression of the coupled (two translational and one rotational). nd 0angle of 0 as remaining 0thethe 0 duration When the heading is taken constant temporal follow form of the 1 of the planar Newton’s 2 1law  the  of mechanics  over angle.

For example, DCM that transforms coordinates from the G frame to the Bi frame reduces matrix associated withsingle the mass moment of inertia tensor under consideration, the DCMs shown in equation (20) reduce to constant valued equations under (18) while the expressions of Euler’s equa- matrices. The fin This DCM andkinematics the inversetotransform are: a single axis. The DCM toThe a function of the heading angle, planar case reduces the problem of orientation rotation about and the body frame expression of the angular velocity of i(t). tion follow the constant form of the equations shownduration under (19).the planar event When the heading angle is taken as remaining over the temporal thatGtransforms from any one frame to any other frame, therefore, reduces of to a function of a the body frame about the frame. Thecoordinates former is timeThis reduces the six coupled Newton-Euler equations of The findings underangle. consideration, the DCMs shown in equation (20) reduce to constant valued matrices. reduces single For DCM that transforms coordinates from the G frame to the B0i frame   cos    t  sin  t 0  cos  t  sin  t invariant. The body frame expression of theexample, linear impulse          the        i i to three coupled equations (two i i motion translational and   G the inverse   transform are: to a function of the heading angle,  Bi frame i(t). This DCM and can be written as the transformed G expression.

R G  t     sin  i  t   one cos rotational). i  t   0  R Bi  t     sin  i  t    cos  i  t   0  (20)     0 0 1 0 0 1    cos  i  t   The 0  reduces the problem  sinplanar  cos  i of t   orientation  sin  i  t ki  0   i  t  case    Bi a single axis. The DCM R G  t     sin  i  t   nematics  cos  i to t  rotation 0  G Rabout  cos i that t  0 (20)  Bi  t     sin  i  t   When the heading angle as remaining constant over the temporal duration of the planar event  is taken   0 0 1 0 0 1 findings under consideration, the  DCMs shown in equation (20)  reduce to constant valued matrices. The 

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Collision Magazine - Volume 14 Issue 2 19

When the heading angle is taken as remaining constant over the temporal duration of the planar event under consideration, the DCMs shown in equation (20) reduce to constant valued matrices. The findings Volume 14 Issue 2.indd 19

10/9/2020 3:43:00 PM

ular velocity and its frame referenced derivatives:

  �  - 3  

      �- 3 

In  the  equations  under 0 two 0  (19), the term following the dash in the right subscript refers to the appropriate  of the  expression   axis The result shown in the second equation extends to any order derivative      0 frame.    0    relevant frames of reference. The(19) involving any number of expressions of Newton’s 2nd law of mechanics   �        �   3  3  follow the form ofthe equations under (18) while the expressions of Euler’s equation follow the form of the equations shown under (19). This reduces the six coupled Newton-Euler equations of motion to three transforms coordinates from any one frame to any other the expected result of the angular velocity about the axis the term following dash in(two the right subscriptand refers the appropriate coupled the equations translational onetorotational). frame, therefore, reduces to a function of a single angle. orthogonal to the plane remaining unchanged irrespective he result shown in the second equation extends to any order derivative For The example, thethe DCM that transforms fromto rotation of the frame is achieved. nt frames ofThe reference. expressions of Newton’s 2nd lawcoordinates of kinematics mechanics planar case reduces problem of orientation about a single axis. The DCM frame reduces to a function of the the G frame to the B i equation under (18) that while the expressions of Euler’s of theframe, therefore, reduces to a function of a transforms coordinates from any onefollow framethe to form any other Selected applications (t).

This DCM and the inverse transform heading angle, θ his reducessingle the six coupled Newton-Euler equations of motion to three from the G frame to the Bi frame reduces i angle. For example, the DCM that transforms coordinates are: of the heading angle, i(t). This DCM and the inverse transform onal and one are:moving in an inertial frame of reference to arotational). function Rigid vehicle  

One of the simplest modeling approaches when it comes blem of orientation kinematics to a single  rotation cos  i  t about i  t  axis.

0  The DCM   cos  i  t    sin  i  t   0     sin  m any one frame to any other frame, therefore, reduces to a function of a  to modeling the dynamics of    a vehicle traveling within an Bi 0  Greduces R Bi  t     sin i  t frame  cos i  tthat R G  t   from  sin the i  tG frame  costo the (20)   0treats      i  t B CM that transforms coordinates frame inertial is one the vehicle as being rigid, i  2. 0 Selected0 applications    1 0 0 suspension 1 and tire forces and in which transform are: e, i(t). This DCM and the inverse excludes driveline,   2.  Selected applications

 

2.1

the assumption is made that the vehicle traverses a flat sur-

Rigid vehicle moving in(20) an inertial frame of reference

face. For duration this problem, there are two frames of relevance:   cosis taken  sin  When 0  heading angle  sin 0  i  t   constant the remaining over thewhen temporal of the planar event i  t  i  t   as approaches it comesintoan modeling the dynamics of a vehicle traveling 2.1 vehicle moving inertial frame of reference  One of the simplest modeling  Rigid  G the inertial frame G (OXYZ) and the body frame B (oxyz), consideration, DCMs reduce constant valued matrices.

The findings within Selected  cos  under 0 R Bi  t   the sin cosinframe 0  (20) (20) to  t shown  isapplications  2. inertial  equation one that treats the vehicle as being rigid, excludes driveline, suspension and tire of a vehicle t i  t  i an i  tOne of the simplest modeling approaches when it comesframe. to modeling the dynamics which is taken as being a central The dynamics  forces 2.assumption applications is made that the vehicle traverses a flat surface.

For this problem, there suspension 0 1 0 and in which 0 thewithin 1  anSelected inertial frameproblem is one that treats the vehicle as being rigid, driveline,    is typically posed as starting with the excludes body frame are two frames of relevance: the inertial frame G (OXYZ) and the body frame B (oxyz), which is taken as forces and in whichinthe assumption is made vehicle 3,4traverses a flat surface. For this proble 2.1 Rigid vehicle moving an inertial frame ofofthat reference Ifwith one the considers aframe translational velocity thethe vehicle. being One a central frame.

The dynamics problem is typically posed as starting body 2.1 Rigid vehicle moving in an inertial frame of reference frames of relevance: the frame G (OXYZ) and body B traveling (oxyz), which is of simplest modeling approaches when itinertial comes to modeling the dynamics of compoaframe vehicle n as remaining constant the temporal duration ofare thetwo planar event point, q,afixed to the vehicle, the Gthe frame Whenover the heading angle is the taken as remaining constant translational velocity of the vehicle [3-4].

Ifgeneral one considers general point, q, fixed toposed the vehicle, the and Gwith One of the simplest modeling approaches when it comes to modeling the dynamics oftire a the vehicle 2.

Selected applications being a central frame.

The dynamics problem is typically as starting bod within an inertial frame is one that treats the vehicle as being rigid, excludes driveline, suspension shown in equation (20) to constant valued The under findings nentsofofthe thepoint B frame position vector of the point are: over reduce the temporal duration of thematrices. planar event conframe components of the B frame position vector are: within an inertial frame that vehicle being excludes driveline, translational velocity of is theone vehicle [3-4].the If one considers a rigid, general point, q, fixed there tosuspension the vehicl forces and in which the assumption is made that the treats vehicle traverses aasflat surface.

For this problem, sideration, the DCMs shown in equation (20) reduce to 2.

Selected applications forces and inthe which is made that vehicle traverses a flat surface.

For this frame components ofthe the B frame position vector ofbody the point are: are frames of relevance: inertial frame G (OXYZ) and thethe frame B (oxyz), which is taken as proble 2.1two Rigid vehicle moving in anassumption frame of reference Ginertial G B constant valued matrices.

The findings shown under equar  R r (22) are two frames of relevance: the inertial frame G (OXYZ) and the body frame B (oxyz), (22) q B q being central frame.

The approaches dynamics problem is typically posed the as starting body framewhich is One ofathe simplest modeling when it comes to modeling dynamicswith of a the vehicle traveling tion (20) apply to any angle, β(t), that represent the angle G G typically B being a central frame.

The dynamics problem is posed as starting withG the bod translational velocity ofRigid the vehicle [3-4].

If vehicle one in considers a rigid, general point, q, fixed tosuspension the vehicle,and the rqframe excludes R rq reference 2.1frame vehicle moving anbeing inertial B of within an inertial is one that treats the as driveline, tire between the appropriate corresponding axes of any two translational velocity of the vehicle [3-4].

If one considers a general point, q, fixed to the vehic frame components of the Bsimplest frame position ofderivative the point applications The2.first G Selected derivative of equation (22), employing equation (6), is: Thevector first Gthe ofare: equation (22), employing equaOne of the modeling approaches when it comes tosurface. modeling the dynamics a vehicle tr forces and in which the assumption is made that vehicle traverses a flat For this problem,ofthere reference frames in question. There areframe a number of otherof the components B frame position vector of the point are: tion (6), is: within an inertial frame is one that treats the vehicle as being rigid, excludes driveline, suspension a are two frames of relevance: the inertial frame G (22), (OXYZ) and the equation body frame The first G derivative equation employing (6), B is:(oxyz), which is taken as G G B simplifications that arise from the planar These aretheof G case.

G G B G r  R r (22) forces and in which assumption is made that the vehicle traverses a flat surface.

For this problem dmoving rq The d Grqposed q B a central frame. dynamics is typically asB starting with the body frame dproblem 2.1being Rigid vehicle inGan inertial frame of qreference G B reserved for the context specific applications in which (23) vq  G Gthe rq  Gframe rrqq the body  G rqof they R Binertial RB  GGRRBBGBand r(OXYZ) q  G B toframe aremodeling two frames relevance: G B (oxyz), is t G velocity of the vehicle [3-4].

If one considers a general point, q, fixed the vehicle, thewhich G Onetranslational of the simplest approaches when it comes to modeling the dynamics ofd arvehicle traveling dt dt d rq G dtd G B q G G G B arise. One final point, however, that is worthy of note,   being a central frame.

The dynamics problem is typically posed as starting with the body  r  v  R r  R  R r frame components of the B frame position vector of the point are: q q B q B B G q own under equation (20) apply to any angle, (t), that represent the angle between the appropriate The an firstinertial G derivative equation (22),the employing equation (6),dtis: within frame isofone that treats vehicle as being rigid, excludes driveline, dt suspension and tire dt herein, is the simplification of the matrix associated with ofofthe translational velocity vehicle [3-4].

If one considers a(6), general point, q, fixed to the vehicle The first Gare derivative equation (22), employing is: responding axes of any two reference frames a number other forces andininquestion. which theThere assumption is madeof that the simplifications vehicle traverses aequation flat surface.

For this problem, there Taking another G derivative: (23) the body frame expression of the mass moment of inertia G position G Bvector of frame components of the B frame the point are: G G G B t arise from the planar case. These are for the context specific applications arise. d rframe B (oxyz), which is taken as(22) rq  they Rand arereserved two frames of relevance: therqinertial frame in GGwhich (OXYZ) B rq the body Taking another G Gderivative: q G G G d G B G denoted as the axis that isd the nortensor. zi axisofisnote, G B  rq the  problem vmatrix  rq associated rq with  G R B GB rqwith R RGB Bd as starting G simplification G The dynamics G posed e final point, however, thatIfisthe worthy herein, is the of d d rthe being a central frame. is typically bodyB frame(23) q B G q   d r d r G G G BB G G d dt dt dt G G G q G q G in the G G B G v  r  R G B   mal to the plane of planar simplification, the steps r r rqthe G   R  R B G (24) R r q BB qq B to the vehicle, zrqi axis ais r qaxisRathat rq fixed   [3-4].

Ras R body expression of the moment ofangle inertia tensor.

Ifequation the denoted the isis: translational of the vehicle IfG one considers q point, q  B general B G q, G G dt B dt applyframe to any angle, (t), thatmass represent between the appropriate Thethe first Gvelocity derivative of (22), employing (6), Gdt q B G Gqequation  d r d r dt dt dt simplification are as follows: d G G q G normal to theframes plane of simplification, steps in are vector as follows: frame components ofthe thesimplification B simplifications frame position of the rq point  G a qare:  R B GB rq  G R B  G R B GB rq wo reference in planar question.

There are athe number other Taking another of G derivative: Taking another G derivative: dt dt dt Gderivative G B The first G of equation (22), employing equation (6), is: se. These are reserved for the context specificthe applications in right which another Garise. derivative: dofGthey rthe r Expanding term toTaking the last equality Gusing the derivativedtransport theorem, given by the d q G 0q  G vq GrGq  GGRRB BB Brqrq  G R B G B q  G R B GB rq  I xis     (22)(23) r  I I I 0 0 0 0 G G hat is worthy of note, herein, the simplification of the matrix associated with x x y x z x x i i i i i i i i   first equation under (5): d r d r dt dt d of     dt Gthe Expanding to Gthe transport theorem, give q term Gusing q G G theB last equality GG Gthe Bderivative G G Gright GGB B G    G (24) of the mass moment ofBiinertia If the z axis is denoted as the axis that is r a r r    R  R  R (21) d r d r I I 0 I 0 0 0 0 I  I yi xtensor.   dB G B B rq G G B GG Gq d Gq rq G G B B yi yi yi zi  i  yi yi  dt  underq (5):  q d Gqdt G BGG q i first equation qr  vqa   d G R qr dt r r r   R  R   R r  R  R B q B B G lanar simplification, the steps in the simplification are as follows: q is: B G q B B G q  I z x I z y The  G derivative  0 (22),  B B employing first of equation (6), ITaking 0 0G derivative: I zi zequation dtdt dtdt  q zi zi  another  0 d GI zriqzi  dtdt i i  ii G i  BB rq  GB B  GB rq (25) a q  G R B (21)  GB B  GB rq   GBR B B BG  of  using (24)Bgiven by the Expanding the term toG the right theorem,  transport dtthe lastG Gequality d G rqthe Bderivative G G B B G G BB B   I x i x i I xi y i I x i z i   I xi xi  0 0 0  Taking 0 0 G rq G d Gterm derivative: d Grq Busing r   the equality R BG rq  G B transport  G rq d G G Rofis Expanding the theorem, give e first in theunder origin ofanother coordinates ofto the Baright frame not last located B B B derivative equation (5): d Gthe the  simplification  isfor the planar case  first  which (23)(24) rq Bdt rq B r rq Grq Gr GvqG a iq d RGBBR  GR BG GR B dGGrqqG R BG GR  r  (21)  I I I  0 I 0  0 0 0 first equation under (5): q q B G q B B G q the is fordtadtcentral frame. dt A dt number of other dt dt y icenter yi y i ziof  ymass  the  body.yi yiThe second   simplification  i xi The G frame expression 2nd law, as the B frame expression, is: G B  G Gtransformed G  I z x I z y that  for are  onNewton’s q GHowever, B Bdbasis. r d G rq G mplifications arise the planar case reserved an application-specific one I 0 0 I 0 0 I d G G  z z z z z z  G B G d r   i i  i i i i  i i   ii G q   G G B B G BB B BR nd r a r   R   R B GB rq is: (25) B B The G frame expression Newton’s 2 law, as the transformed B frame expression, q q B G q B    a  r r  r  R    R    ditional point is worthy of note at thisTaking stage. another The firstGequation under (9) relates the G frame expression  derivative:   d r q G BG G B G q B BG q G B G q dt dt dt B G equality G B Bderivative R dt  GGR qthe B G BB theorem, Bgiven Expanding the term to the rightF of the last frame F using m aGq right q  the rq  GB using  (26)by the a qm Gathe Rq Bof the transport R Gexpression B G  BExpanding  same. to last equality B  G rof B BG B  G rq he of thethe body frame the inertial frame to thenot body the term  theangular planar velocity case in which origin of about coordinates the Bi case frame located dtG first under (5):in iswhich The first simplification is equation for theofplanar the G B G  G B Fabout  R agiven  mby Rthe G result GF  the right derivative transport equathe subject planar simplification case, based onisequation (20), the expected of the angular G velocity Bthe Gtheorem, q the B G afirst q d GG rthe d GB rqm Expanding term to the last equality using derivative transport theorem, given ody.

The second for aofcentral frame.

A number of other d Gof the q is not located at origin of coordinates the Bi frame G the G B G G B nd rq is   The G frame Newton’s 2 law, as the transformed B frame expression, is: (24) a r r    R  R  R Substitution ofirrespective the expression result from equation (25) into equation (26): tion under (5): s orthogonal to the plane remaining unchanged of the frame achieved. q B G q B B G q B B first equation under (5): r the planar case aremass reserved on an application-specific basis.

However, one center of the body. The second simplification for a  d GNewton’s dt G is dt  rqdt B The G frame expression 2nd G B law, Gas the BBtransformed B B B frame expression, is: rq  G B (26): Substitution from into equation (25) B  G rGq (25) the R result  equation  R  G rq f note at this stage. central The first equation underof (9)other relates the G frameathat expression qof B G B BG frame.

A number simplifications arise F dtGG R BGF  G G B G mB BB aBqr mB GR BGBBarq  (26) B G  (27) F  R F  m a  m R  e body frame about the inertial frame to the body frame expression of the same.   G q BGdBG G BB G q G rqqG Rderivative G theorem, theonterm to the right ofB the last Gequality transport given by the for the planar caseExpanding are reserved an application-specific G using BB B FBGthe B m RB GB a Bq   a q GRm GGF   BGB  BBB  BG rq B G  B B G rq B G rq  G based on equation basis.

(20), the expected result of theunder angular velocity aboutatthe a q G F R  BGR B dt first equation However, one additional point is(5): worthy of note G F  m a q  m R B  BG rq  G  B  G rq    The expression Newton’s 2nd itlaw, the transformed B frame Substitution of therelates result from equation (25) into equation (26):any remaining unchanged of the frame isarbitrary achieved. qG is frame an body-fixed point, canasbe replaced with otherexpression, body-fixedis: point,(25) such as o. this irrespective stage. TheBecause first equation under (9) the G frame Substitution of the result from equation (25) into equation (26): B vectors B The component forms for each of the three in the parenthetical, for o, are:  dG G rNewton’s  point, expression of the angular velocity of The the body frameanabout Because arbitrary body-fixed can be with any other body-fixed point, su q G B G G transformed B replaced G q isexpression B B2nd law, BB G frame asit the B  R G BB Fa m aq BBGBrrq B frame expression, is:(25)(27) (26) a q  GG R rqm rR BGBGB GaGR  G R BFGfor m r B G qR BG G B of G F Bforms the m  qthe BBvectors BG q q G B Gparenthetical, q Theexpression component each three in the for o, are: the inertial frame to the body frame of  dt F   BB  B r  v e GFv e G R  BvGBeF  m G a q  m G R B  BG rq  GB B  GB rq  (28) same. For the subject planar case, based on equation (20),G o x 1 y 2G F z G 3R B GB F  m G a q  m G R B GB a q B(26): Substitution of the result from equation (25) into equation o with Because q is an arbitrary body-fixed point, it can be replaced point, such as o. v x e 1 any  v y other e 2  v zbody-fixed e3 Gr nd law, The G frame expression Newton’s 2arbitrary as body-fixed the transformed Bitframe expression, is: any other body-fixed point, s BB Because q is an point, can be replaced with The component forms for each of the three vectors in the parenthetical, for o, are:     r  v e  v e  v e (29) BG o from x 1equation y 2 z 3 into equation (26): Substitution ofGwww.collisionmagazine.com theforms result (25) B Gof the three BB The component each for o, are:  Bin theBparenthetical,  20 Collision Magazine - Volume 14 Issue 2 (27) F G G R B GGfor F  m G RBBB Gvectors B m aq G BG rqB  G  B  G rq  v x eq 1  v y e 2  vz e 3 F  R BB G F  m a q  m (26) BG roR B Ga  r  v e  v e  v e (28) B G oG x G1 y 2 z 3 G BB B (30) e 3B m G  B  x e 1Fy e R2 GB Fz   GB rq   G avqx e1m  vR eB2 BGvrzbody-fixed qe G ro with yother 3 G B Because q is an arbitrary body-fixed point, it can beB replaced any point, such as o.

B Substitution of the result from equation (25) BB into equation (26): G  B  x e 1  y e 2  z e 3 The component forms for each of the three vectors for o, are: Volume 14 Issue 2.indd 20 10/9/2020 3:43:00 PM  in the  parenthetical,  (29) BG ro  v x e 1  v y e 2  v z e 3 Because q is an arbitrary body-fixed point, BBitrcan  e any other body-fixed point, suc  vbee replaced  v e  vwith

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d G rq Gdt G d G rq G G z x x z  y d G dt B rq (24)  rq  a q  R B GB dt  Gthe R B result can  Rbe rq and  vequation B Gsubstituted erivative: tensor into the to the   v   v x left of the last equality in equation (11). z x y y dt dt Euler’s equation, in dtthe body frame, is given by equation  (11). The collision partner specific subscript can rivative: term to the right of the last equality usingCarrying the derivative theorem, given by theto the following result. out thetransport vector-matrix operations leads erivative: readily be dropped on the body frame expression of the matrix associated with the mass moment of inertia G(5): G G B G under d term rq to d and rqequation, d ing the the right of the last equality using the derivative transport theorem, given by the G G Euler’s in the body frame, is given by equation (11).

The collision partner specific subscript can G G G B G G B tensor the result can be substituted into the equation to the left of the last equality in equation (11).

G G (24)2 2 d GGrq  G rq  G a q  GGd G R B BG rq  G R B GGd GBBrq  G R B BG rq       I   I   I   I   I    I    I    I    I uationGunder readily be dropped on the body frame expression of the matrix associated with the mass moment of inertia x xx y xy z xz y zy z yz x y zx x z yx y z yy y z zz dtG rq  (5): dt dt  d d r    operations leads to the following result.

(24) RB  Gthe R B Gvector-matrix rq  a q B B d GR B GBrq  GCarrying G qout Brq   B 2 2  rq B  (24) R B G rq   Rtensor  BR dt G G rq G G aq d dt dt BBand G rq  x can  y Isubstituted  B B result G into the (11).

M   equation  left  xlast z I zzequality  y z I xyin equation (32) I yx be I xx the G yy  z I yz   x I(25) zxthe z I xz  x to y I zy x zof  rq the  r  dt a q  R B  dt B GB B B  GB rq   G R BdtBG G B G q  equality  2leads to 2the following result.

2 2 Carrying out the vector-matrix operations dt   to the right of theG last using the derivative transport theorem, given by the   I zx y Iyxy z Izxz I  I zzy xI zy I yxyI yz I xyx  I  x  I  x  I  y  I      G  d G rq B  B   G x BB Bx Ixxx y Iyzxxx x  z Iyyxyy y  z Izyyyz y  z Izzzxz  ndBG  (25) aThe B  rq   Rtransport rq  Gas Bthe rq zy R frame  Gthe  GBtransto the right of the last using derivative theorem, given bythe the2case ofzthe q  equality B G B2 G expression Newton’s law, For B frame being a central frame, the prod (5): B 2   dtusing the derivative m to the right of the last equality transport given  theorem,   by the 2  z I xz (32)  x y I zy  x z I xx  x z I zz  y z I xy  frame  G M   x I yx  y I yy  z I yz  x I zx 2 (5): formed B expression, is: uct of inertia terms are zero-valued.

Equation (32) simpli     nd  I   I   I   I   I    I    I    I    I xpression Newton’s 2 law, as the transformed B frame expression, 2 yframe, 2 the x xxis: being y xy a central z xz zy z yzproduct x y zx x z yxterms y are z yy zero-valued. y z  zz (5): For the case of the B frame of inertia Equation  x I zx    y I zy    z I zz  x I yx2  y I xy2  x y I xx  x y I yy  x z I yz  y z I xz   B  fies  B d GB rq B nd B  G (32)  x I yx    y I yy    z I yzto the (32) Mthe   Ifollowing:  z I xz  x y I zy  x z I xx  x z I zz  y z I xy  G BB simplifies B Bto following: G  B B law, as the transformed B frame expression, is: rame expression 2    (25)x2 zx a q  G R B Newton’s  r r  r    R     d r GG B B G q  B BG qG GB B G q G G G q 2 B B G G B R BF  G a BB B a B     F  m m R (26) dt  (26)      x I zx  y I zy  z I zz (25)  x I yx  y I xy  x y I xx  x y I yy  x z I yz  y z I xz  rq  GBB  Brq B frame G q  d G rq   BG r   R Bq BGBBthe Ga q  GR B  being a central frame, the product of inertia terms are zero-valued. Equation  caseB ofGthe  (25) a q  R B dt GGB GBB GGB Gqrq B G RFor B BGGrq  G  B G G rqB    x I xx  (26)  dt B G F  F  R m a  m R a y z I yy  y z I zz q B G q (32) simplifies to the following:     B nd Substitution of the result from equation (25) into equation the result from equation (25) into equation (26): sion Newton’s 2 law, as the transformed BFor frame is:frame being a central  y I yy  the (33) the expression, case of the B inertia terms are zero-valued.

Equation  x product (33) G M   frame, z I xx  x of z I zz  nd as the transformed B(32) frame expression, is:following: ion Newton’s 2 ndlaw, (26): simplifies to the    I  x z Iyyy I xxy  I yy  ution of the result (25) into equation (26):  sion Newton’s 2 from law, equation as the transformed B frame expression, is: x yzz x Izxxzz y  zI  B G G G G G G F G RG BRGBB FGB F R BB ar  GB B  GB rq (26)  (27)   F mm aG qa mm R B q  G G B G B BBG BG qq  (33)   y I yy  x z I xx  x z I zz  G M(26) F  GGR B GBFG  m a  m R a q B G q G G G B   x I xx   B G BB B B  y z Izz  dynamics can readily be obtained by F  FR B G FR m a Gm Rm (26) y z I yy rotational  the translational (27) rexpressions frame dynamics and frame TheB G expressions of the translational dynamics and B G G aR q B BG    I    I    I B G F  qm a qThe q  G  B  G rqof (27) zz x y xx x y yy    z GR ,such from equation (25) into equation (26):applying (33) I yy frame x can readily x z I zzbe obtained by applying nesult arbitrary body-fixed point, it can be replaced withthe anykinematic other body-fixed point, to the body expressions. transform, yo. z I xx BG M rotational asdynamics esult from equation (25) into equation (26):  tesult for each ofbody-fixed the three in can the parenthetical, for any o, are: as x   GRx  I the body frame expresthe kinematic , to from equation (25) intovectors equation (26): z I zz transform, qforms is an arbitrary point, it be replaced with body-fixed point, o.y I xxrotational Gsuch B y yy dynamics The GBframeB expressions of the translational dynamics and can readily be obtained by body-fixed point, itother can be reG G BBecauseGq is an Garbitrary BB G B G G B   (27) F  R F  m a  m R r    r F  R F M  R M (34) B G q B BG in q the G parenthetical, B G q mponent forms of the for o, are: GRsions.

B G frame expressions.

B G G Gfor each B GB three Gvectors BB B B kinematic , to the body applying the transform, B placed with any other body-fixed point, such as o.

The (27) (28) F  GR B GBF  m GGaroq  v mx eG1RB v yBGeBB2rqvGzBe3B  GBrq G The GGframe  expressions of the translational (27) dynamics and rotational dynamics can readily be obtained by F  R B Gcomponent F  m a q forms  m Rfor B each BG rq  B three G rq vectors in the parof the B G G further B G Bthe following: G For the planar case, the B frame expressions simplify  R , frame applying the kinematic transform, trary body-fixed point, it can be replaced  v any e  other v z e 3 body-fixed point, suchBas (28) G ro  v x ewith M  Gexpressions.

Fto o.the R Bbody Rto (34) GF B GM enthetical, for o, are: 1 y 2 (34) BB be replaced with any other body-fixed point, such as o. trary body-fixed point, it can  ms for each of the three vectors in the parenthetical, for o, are:    r  v e  v e  v e (29) BG o x 1 y 2 z 3 trary body-fixed point, it can be replaced with any other body-fixed point, such as o. G G B G ms for each of the three vectors in the parenthetical, for o, are: v y G M  to 0following:  v xRB  B GB M  BB F further Rthe (34) zsimplify GF ro parenthetical, ms for each of the threeB vectors in BG the for o, are: case, the B frame expressions   v xFor e 1 the v y eplanar  v e (29) 2 z 3    0 expressions  B B For the planar case, the B frame further sim r  v e  v e  v e (28) B  (35) F  m v   v M  (28) G o x 1 y 2 z 3 G y z x G B  (30)    G  B  x e 1  y e 2  z e 3 plify(28) to the following: GBro  v x e 1  v y e 2  v z e 3For the planar case, the B frame expressions further simplify to the      0z I zzfollowing: 0  r  v e  v e  v e (28)  B y 2 v   v      G o x 1 z 3 x z y (30) G  B  x e 1  y e 2  z e 3 BB     B B     (29)  (35) M 0  BG ro  v x e 1  v y e 2  v z e 3 G F  m  v y  z v x  BB r  v x e 1  v y e 2  v z e 3 (29)  v x  z v y  G  0  (29) BG BB o        z I zz    e3 (29) 0 Rigid body accelerometry BG ro  v x e 1  v y e 2  v z2.2 B (35) F (30)  m  v y  z v x  GB M   0  B (35)upon data from Gmass    e   e   e The determination of the center of kinematics and dynamics of a rigid based G B x 1 y 2 z 3 B  z I zz common and important application (30)of the rigid e3 0 GB B  x e 1  y e 2  zaccelerometers    positioned in the periphery body is another (30) 2.2 Rigid body accelerometry G  B  x e 1  y e 2   ze3 (30) of engineering mechanics. Contextually, a problem involving the motion of a single rigid body in ℝn will The determination of the center of mass kinematics and dynamics of a rigid based upon data from involve translational of freedom and 0.5n(n-1) rotational degrees of freedom. As a result, a single Rigid body 2.2 nRigid body degrees accelerometry accelerometers positioned in the periphery of theaccelerometry rigid body is another common and important application 3 requires a minimum of six independent measurements to determine the state of rigid body problem in ℝ Substitution of these component definitions into the cordetermination of theContextually, center of mass kinematics and dynamics uponindata from ofThe engineering mechanics. a problem involving the motionofofaa rigid singlebased rigid body ℝn will Theindetermination the iscenter mass kinematics and the body while a rigid body problem ℝ2ofrequires aof minimum ofofthree independent measurements to responding vectors in equation (27) followed by expansion accelerometers positioned in the periphery the rigid body another common and important application involve n translational degrees of freedom and 0.5n(n-1) rotational degrees of freedom. As a result, a single dynamics of a rigid based upon data from accelerometers determine the state of the body.

For the presentation of the subject application, the following assumptions leads tointo the following result for the bodymechanics. frame 3 requires of engineering Contextually, abyproblem involving the motion of a single rigid body ℝn will a minimum of six independent measurements to determine the in state of rigid body problem in ℝexpression se component definitions the corresponding vectors in equation (27) followed are made.

The first is that all peripheral accelerometers are rigidly mounted tofreedom. theisbody.

This assumption positioned in0.5n(n-1) the periphery of the rigid of body another of the force.

2 involve n translational degrees of freedom and rotational degrees As a result, a single the body while rigid body problem in ℝ requires a minimum of three independent measurements to the following result for the body frame expression of the aforce. makes the body frameℝposition time-invariant and reduces the body to frame referenced time andofimportant application of engineering me3 requiresvectors acommon minimum sixofindependent measurements determine the state of rigid body determine theproblem state of in the body.

For the presentation the subject application, the following assumptions derivatives of the same to zero valued vectors.

The second assumption is that the measurements from chanics. arigidly problem involving the body. motionThis of 2Contextually, the body while a rigid body problem in ℝ requires a minimum of three independent measurements to are made.

The first is that all peripheral accelerometers are mounted to the assumption  v x  y vz  peripheral  n z vy single axisofaccelerometers of the acceleration along theassumptions individual will involve nthe translational degrees a represent single rigidpoint bodymeasurements inofRthe determine the state the body. vectors For the presentation subject application, following   the body makes frame position time-invariant and reduces bodythe frame referenced time B  (31)      F m v v v axis of eachThe uniaxial accelerometer.

The third assumption is that for a peripheral point at which the (31) G z x xare z made.  y of freedom and 0.5n·(n-1) rotational degrees of freedom. is that all peripheral accelerometers are rigidly mounted to the This assumption derivatives of thefirst same to zero valued vectors.

The second assumption is that thebody. measurements from  v z  x v y  y v x  3 As a result, a single rigidand body problem R requires makes the body frame position vectors time-invariant reduces theinbody frame time   peripheral single axis accelerometers represent point measurements of the acceleration alongareferenced the individual minimum of assumption sixThe independent measurements to determine derivatives of the same to zero valued second is that thepoint measurements from axis of each uniaxial accelerometer.

Thevectors. third is assumption that for a peripheral at which the Substitution component definitions into the corresponding vectors (27)afollowed by therepresent state theequation body while rigid body problem inalong R2 the individual peripheral single axis point measurements of the acceleration n the body frame, isEuler’s givenof bythese equation The collision partner specific canof in equation, in (11). the body frame, is given byaccelerometers equationsubscript leads result forsubscript the body frame expression of the force. axis of each uniaxial accelerometer.

The assumption that for a peripheral point at which the requires athird minimum of threeis independent measurements d on the bodyexpansion frame (11). expression ofthe thefollowing matrix associated with the mass moment of inertia Thetocollision partner specific can readily to determine the state of the body. For the presentation ult can be substituted the equation to the left of the last equality in equation (11). be into dropped on the body frame expression of the matrix v x  yand vz the z v y of the subject application, the following assumptions are ector-matrix operations leads with to thethe following result.of inertia tensor associated mass moment   B accelerometers are x v z made. The first is that all peripheral (31) result can be substituted into the equation left G F  mto y  z vof x the  vthe 2 2 rigidly mounted to the body.

This assumption makes the    x I xx    y I xy    z I xzlast    equality I yzequation  x y I zx(11).  x Carrying I the  I vectory I zy  zin z I yx  y zout  vyyz  xy v yz zz y v x   2 2 body (32) frame position vectors time-invariant and reduces leads to the following result.  x I yx    y I yy    z I yzmatrix  x Ioperations zx  z I xz  x y I zy  x z I xx  x z I zz  y z I xy   the body frame referenced time derivatives of the same to 2 2    y I zyEuler’s  z I zz equation, I xy body I xx  x  I yy  by I yz  y z(11).

I xz  The collision   x I yx  iny the x y frame, isygiven partner specific subscript can x zequation   x I zx   zero valued vectors.

The secondofassumption is that the readily be dropped on the body frame expression of the matrix associated with the mass moment inertia measurements from peripheral single axis accelerometers tensor and the result be substituted the equation to the Equation left of the last equality in equation (11).

B frame being a central frame, thecan product of inertia into terms are zero-valued. he following: Carrying out the vector-matrix operations leads to the following result.



 



 







 



 







 



 

 xI xxI    y I xy    zI xz  y 2 I zy  z 2 I yz  x y I zx  x z I yx  y z I yy  y z I zz    x I xx  y z yy  y z I zz   BM   B  x I yx    y I yy    zI yz  x 2 I zx  z 2 I xz  x y I zy  x z I(33) G xx  x z I zz  y z I xy  G M   y I yy x z I xx  x z I zz    I   I   I  x 2 I yx  y 2 I xy  x y I xx  x y I yy  x z I yz  y z I xz     z I zz  x xyzxI xx  yxzyy I yyz zz

(32)

(32)

For the case of the B frame being a central frame, the product of inertiaby terms are zero-valued. Equation ssions of the translational dynamics and rotational dynamics can readily be obtained (32) simplifies to the following: G matic transform, RB, to the body frame expressions. www.collisionmagazine.com

  x I xx  y z I yy  y z I zz  (34)    y I yy  x z I xx  x z I zz  M   Volumeexpressions 14 Issue 2.indd 21further simplify to the following:   , the B frame   z I zz  x y I xx  x y I yy  G

G

F R

B B G

F

G

G

M R

B B G

M

B G

Collision Magazine - Volume 14 Issue 2 21

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 dt ‘k.’ theorem.  frame, is obtained by ‘q’ applying the derivative transport dt replaced dt by the subscript dt noting subscript being that the B derivative of Br The first G derivative,   is zero, is given by (8). Taking the G derivative of equation (8): G equation B B

d B body dframe, d B frame, expressed B B B B B in the body The G derivative of the angular velocity G of the B B B  about  the  0   frame,  inertial B inertial  B (3 d Bvelocity d B Gthe G  B d of G  Bbody B the B G B B G B about B G frame, B G expressed The G derivative of the angular in the body G G G G G  G      0          (37) dt dt dt frame, is obtained by applying the derivative G B transport G d G rk B theorem. d Bro GG B   d GB B B G B B Gd BBrk  dt the derivative dt dt G B   rk  G B  frame, is obtained by applying (36  transport  Rtheorem.  B  dt

 dt 

dt

 

dt



G B B The G derivative frame expression vector rk, is already known as per equatio d B of the d body d B of the Bposition B BB B B Gbody  vector known(37) as per equation The G derivative of the frame expression of     0 dGBto     the position  B Brk, is already d d G B G B G B G B G(36) Bresults into B equation B B B leads B BtheGfollowing B (5). Substitution of these result:frame, the The of the angular velocity the body Gframe, inertial expressed in the body B  B following  of  0about  G (37) dt Gofderivative dt results dt G  B  into Gequation G B(36) G B  B  G B (5). Substitution these leads to the result: dt dt dt frame, is obtained by applying the derivative transport theorem.

represent point measurements of the acceleration along the G G B B rkofGthe ro position  B B B rkrkB, is (3  G R BB GBvector   B rk   as per G derivative of the body frame expression already equation (38) G B B G  Bknown G individual axis The of each uniaxial accelerometer.

Thebody thirdframe G  B   Br rk   B position (38)   roB  G R BB d r   The G derivative of the expression of vector r , is already known as per equation   d d k GBthe k G B G B k  B B B result: B B (5).

Substitution of these results into equation (36) leads to the following   B  G B  G  B  (37 assumption is that for a peripheral point at of which ac- into equation G B  G  B  0  G B  G  B (5). Substitution thesethe results (36)G leads to the following result: dt dt dt BR to this equation: B celeration is measured, the measurement from three Applyingcomes the kinematic transform Gthe kinematic transform R to this equation: Applying G G BBRG to B this Bequation: B the kinematic ro  Gof  B frame (38)  rkmutually  Gtransform Rthe r  BG  B B GB of the rkposition uniaxial accelerometers thatApplying are arranged in Ga derivative  B  Gvector B  body The G G k B rk  roB  RBB  Gexpression BB  rkB B GB  (38)   rkB   rk, is already known as per equation  G BB BB (5).assumption Substitutionisof these results to the following result: orthogonal triaxial arrangement. The final r  into r equation   r(36) (3   leads k B G  B B G  B B rk  B G k B G o B G BB    (39) (39) G rk  G ro  G  B  rk  G  B   G  B  rk  that each block Applying of uniaxialthe accelerometers, containing uniB kinematic transform RG to this equation: G G BR to this Applying the kinematic transform r equation: r  G R B  GB   B  B rk  GB  B   GB  B  B rk   G (38 axial accelerometers arranged in configuration per Atthe times, it may beasdesirable to writek theo G frame expression of the acceleration as the sum of t Atacceleration times, it that may be desirable to write the G frame expression of the acceleration as the sum of the Bsensing B B B B B B the third assumption, are mounted such the beGfdesirable writeB theto Gas frame expres- force rk  G ro and atimes, due toGgravity term, isB referred the specific (39)at k. At  rk it may   , which  rk to G B second BGB B G B G k B B ro term, RBGtofkthis due to gravity aB rsecond the specific force at k.

(39) ofGBthe r, kwhich  equation:  referred rkto as Applying the kinematic axes of the block are alignedacceleration with the coordinate axes ofandsion G k transform G G B is G B  acceleration asthe sum of the acceleration due GG G G the body frame. In cases when the last two assumptions to gravity and a second fgkacceleration , BwhichBis referred to as the   term, (4 fk Gthe  B B rk G B At times, it may be desirable to write the G frame expression rk G  ro GBf of (39  r   G B  B as rk  the sum of the (40) r  g G G B k  G  Bof don’t hold, additional DCMs are required to transform the k k At times, it may be desirable to write the G frame expression the acceleration as the sum of the specific force at k.

G acceleration due to gravity and a second term, fk, which is referred to as the specific force at k. relevant instrumentation frames of the reference into the body acceleration to gravity and a second term, Gfk, which is referred to as the specific force at k.

For Bdue frame: At times, it may write theinG frame expression of the acceleration as the sum of the For three the B uniaxial frame: ofaccelerometers of reference. comes For thefrom sake of simplicity presenta- be desirable ration is measured,frame the measurement that aretoarranged G Gf , which is referred to as the specific force rk and  G fak second  GG g term, acceleration due to gravity (40) (40)at k. k G tually orthogonal tion, triaxial The final is there that each block ofB runiaxial the arrangement. problem is presented for theassumption case in which (40) (4  Gg B B G G k B Rfk G   r  f f  R r  g  k B GG kG B G k B G k B GG rometers, containing arranged for in the thecase configuration as per the third  are nuniaxial blocks ofaccelerometers n uniaxial accelerometers of Rn G G   G (41) r  f  R f R r  g  G k G k G k  (40 rk  Gfk  kg For the Baxes frame: For the B frame: mption, are mounted such that the sensing of the block are aligned with the coordinate axes of (e.g. three triaxial blocks for a problem in three dimensions thethree B frame: leration is measured, the measurement comesFor from uniaxial accelerometers that are arranged in Substitution from equation (38) intoare equation (41):to ody frame.

In casesand when lastblocks two assumptions hold, additional DCMs required two the biaxial forThe a Substitution problem in two dimensions). utually orthogonal triaxial arrangement. final don’t assumption is of For theequation B frame: from into equation (41):G r  G g B that(38) B each B block G B uniaxial rk  of (41) f  R f  R Gsake  orm the relevant instrumentation frames of reference into the body frame reference.

For the G G k G k G k B B Furthermore, body frame isarranged taken as a in central lerometers, containing uniaxial the accelerometers the frame. configurationB r as per (41) f the R third f  B R  G r  G g  (41)





G kG G B B G k G G kG B B B G G mplicity presentation, the that problem is presented forthe theblock case in there rk Bare ro G ofgn Bruniaxial  k BB r  GBr RnGGblocks RB fkof  GB B Br Brk  mption,ofare mounted such the sensing axes of arewhich aligned the coordinate axes RBBR B Bg B Gwith B GG fkB B RkGB G G k  GGG k    th  r  R r  g  R R  r     n    G k G o G B G B k G B G B k {k: 1 ≤ k ≤ n} accelThe G frame position vector of the k rometers for the case ofwhen ℝ (e.g. triaxial blocks a problem three dimensions and twoBbiaxial Substitution fromfor equation (38)ininto equation body frame.

In cases thethree last two assumptions don’t hold, additional DCMs to B (41): G areB required B B B   R      f  r   r Substitution from equation (38) into equation (41):   G o G B k G B G B k B aSubstitution G B frame.

B B Binto equation (41): equation s for a the problem in two dimensions).

Furthermore, the body frame taken as central erometer block isframes given of byreference equation (2) with the is subscript sform relevant instrumentation into the body frame the  For Substitution from equation equation of Rreference. from   GB (38) rk sake G fo (38) G  Binto G  B (41): B  rk  B B BG BB BB ‘q’ being replaced is bypresented the subscript ‘k.’ case first G BThein Bwhich GderivaG are B B   BB r  mplicity of presentation, the problem for the there n blocks of n uniaxial      f  r    k BB GG rkB BBGGB BBB G k B B B rk  B g G GoRB GGGRBBB GB G rk  R G  B ro  B th {k: 1 ≤ k ≤ n} accelerometer RGBby g   fo equation rk(2) GBRthe  BGBGR BrkrkGB BG BBrk  GGBBB BGBrk B  B rk     block is given Glerometers frame position kthat BG  B GRG B r  r  is zero, isingiven by tive, noting the B derivative of Barkproblem (e.g. three triaxial blocks for three dimensions and two for thevector case ofof ℝnthe G k G o G r  R G  Bro with biaxial gBBGBGR B G B B G k B that riptfor‘q’a being replaced by the(8). subscript ‘k.’

The first derivative, the  R Gisnoting B G BBBof rBkBrk B B B B ftaken rk BBGderivative  ks problem in two dimensions).

Furthermore, theGbody frame as frame. equation Taking the G derivative of equation (8): o  GB B a G central B B G   R   f  B B rkGGBB rkGB  B rk   R G o G B G frok GG o, is given by equation (8). Taking the G derivative of equationB(8): B B B B B  B B rk  BG B BB G B BB rk B B B B B  G fo  G  with  r the  G  is given G frame position vector of the kth {k: 1 ≤ k ≤ n} accelerometer block  B  Grfko (2)  G foby  Gequation B rk G B  B rk   G G BB   kG B (42) G G G G G B G B cript ‘q’ being replaceddbyrk the subscript noting G derivative, d ro G ‘k.’ The d rk that the B derivative of rk d first B B B (36)  R B   rk  G (8):   equation G Bof B  ro, is given by equationdt (8). Taking dt the G derivative dt  Equation (39) may be expanded. For the case of R3, this    dt









(41

(4 (42) (42)

results in three coupled equations. Solely for the purpose

(36)

G B d G rk G d G ro G   G d B  B d rk  B of simplifying thebody presentation, the frame references are G derivative of the angular velocity of theRbody frame, about the inertial frame, expressed in the (36)  r        B  G B  k G B dt of the The dt G the derivative angular of the body dt frame,    dt velocity  , is obtained by applying derivative transport theorem. temporarily dropped.

G

about the inertial frame, expressed in the body frame, is 5 Alem and Holstein G B transport obtained byB dapplying the derivative G derivative of the angular of the body frame, thetheorem. inertial frame, expressed in the bodytake the approach of writing equad B velocity dabout B B B B B B   derivative B  G B theorem. tion (43) for three (37) triaxial accelerometer blocks and deG B G  B  Gtransport G  B  0  G B  G  B me, is obtained by applying the dt dt dt fine the difference between the two sides of the resultant G B B equation as a non-zero error vector. The least squares erd frame d B d B vector rkB, is already B B B of the B position G derivative of the body expression known as per equation          (37) G B G B G B G B G  B  0  G B  G  B ror minimization approach that is taken by those authors dt ubstitution of these results intodt equation (36) leads todtthe following result: is to define the translational center of mass and angular (37) accelerations as the independent variables followed by as per equation G derivative of the body frame vector r G Gexpression G B of the B position B B kB, is already known The G rderivative body frame expression of the posi (38)  ro  ofRthe B  G  B  rk  G  B   G  B  rk   equating the partial derivatives of the error vector, with Substitution of these resultsk into equation (36) leads to the following result: tion vector rk, is already known as per equation (5). Subrespect to the independent variables, to zero. The center stitution of results intoB equation (36) leads to the BRGthese ying the kinematic transform to this equation: G G B B B B G of mass translational acceleration is taken as the average of rk  ro  R B  G   B  rk  G  B   G  B  rk   (38) following result: the translational accelerations from the peripheral blocks B k  GB ro  GB   B  B rk  GB B   GB B  B rk  (39) while the angular acceleration is determined from the least G rB RG to this equation: lying the kinematic transform squares solution. Park and Hong 6, on the other hand,

3, this results in three coupled equations. Solely for the be expanded.

For the case of mes, it may beEquation desirable(39) to may expression ofBℝ the acceleration as the sum of(39) the BwriteB the BG frame B B B    r  r    r      r   the G k G o the G Gpresentation, B k G B the G frame B k references purpose of simplifying are force temporarily ration due to gravity and a second term, fk, which is referred to as specific at k. dropped.

2 2 as the sum of the imes, it may be desirable to writerthe G frame expression the G   2  rk 2   of  racceleration 3   k1 3 k 1 2  rk 1 3  rk 2 1 2  rk 3 1(40) rkro1G fk rkG3g G  the specific  fk, which isreferred to as force at k. leration due to gravity and a secondterm,     2 2                 r  r r r r r r r k2 3 k1 1 2 k 3 2 3   k 2   o2   k 3 1 k1 3   k 2 1 rk 3  Gro3  G   rkG2   r     r  2  rk 3 2 2  rk 1 1 3  rk 2 2 3  e B frame: rk  fk  g 1 k 1 2   k 3 1 (40)

(43) (43)

 G fk take rk writing (41)accelerometer blocks  Rthe g  equation (43) for three triaxial Alem and Holstein G rk  [5] G fapproach k  R G  of and define the difference between the two sides of the resultant equation as a non-zero error vector. The 22 Collision Magazine - Volume 14 Issue 2 www.collisionmagazine.com least squares error the translational center Bequation B B G approach B Gthat is itution from equation (38) into (41):  minimization  G taken by those authors is to define (41) G rk  G fk  R G fk  R G  rk  g  of mass and angular accelerations as the independent variables followed by equating the partial derivatives of Bthe error vector, with respect toB the independent to zero. The center of mass translational B G G B B B variables, B   (38) ro equation  R22G into  Gg  BR titution from G rk 2.indd G R B G B  rk  G  B   G B  rk  Volumeequation 14 Issue acceleration is taken as the(41): average of the translational accelerations from the peripheral blocks while the B

he B frame:

B

B

G

B

G

B

B



B

G

B

B

B

G



10/9/2020 3:43:01 PM

(42)(42

algorithm can readily be characterized as linear and uniaxial. In the first, the force associated with a collision is modeled as being a linear function of the depth of residual damage. In the second, the uniaxial structural stiffness for each region of a vehicular collision partner is arrayed along the relevant undeformed principle axes. One reason, albeit circular, for this stiffness array arrangement is that the primary data source for quantifying vehicular collision partner platform and regional stiffness data employs only axial component reporting of nodal residual damage depth data. Thirdly, the central collision model is based on the uniaxial collision between two collision partners with each collision partner modeled as a single define the vector oflumped system variables consisting of thelinear ity,stiffness. at the terminus of theis closure phase, foronly the model center of mass andasa single massless The model a closure phase and assumes center of mass translational acceleration, the angular ac- mass both collision The closure phase velocity the achievement of a common velocity, at theofterminus of thepartners. closure phase, for the center of mass of both collision partner, for the caseforofthe a central celeration and the squared velocity components. change for the ithfor partner, case of a central collisionangular partners.

The closure phase velocity change the ith collision impact, is mathematically represented as: This approach increases the minimum number of required impact, is mathematically represented as:

independent measurements from six to twelve. Interested readers should refer to the cited references for further details.

vi 

CRASH3 model for oblique collisions



2 Ei  E j





(44)



1 mi 1  mi m j

(44)

subscripts in this equation referisto the corresponding partners denotes the internal work The damage analysisThe algorithm of the CRASH3 model The subscripts in collision this equation referand to Ethe correspondabsorbed by the referenced collision partner.

For oblique collisions, it is assumed that a common a long-standing and well-established mainstay in the field ing collision partners and E denotes the internal work velocity is reached at amodeling peripheral point rather at the mass center of each collision partner. Figure 2 of McHenry of accident reconstruction.

The approach em- than absorbed by the referenced collision partner. For oblique [7] depicts an offset frontal collision as an example type of collision. This figure is partially replicated ployed within the damage analysis algorithm can readily collisions, itofisthis assumed that a common velocity is reached as Figure 1 of the subject work. be characterized as linear and uniaxial. In the first, the at a peripheral point rather than at the mass center of each force associated with a collision is modeled as being a collision partner. Figure 2 of McHenry 7 depicts an off<<Note to editors: please insert Figure 1 here>> linear function of the depth of residual damage. In the set frontal collision as an example of this type of collision. second, the uniaxial structural stiffness for each region of This figure is partially replicated as Figure 1 of the subject The original reference makes no explicit reference to the inertial frame of reference or global coordinate a vehicular collisionsystem. partner isRather, arrayedsuch along the relevant work. in form of the relevant derivations. The original reference appears to be implied undeformed principle axes.

One reason, albeit for utilizes a subscript of p forcircular, the peripheral point. Equations 18 and 19 of the original reference represent the this stiffness array arrangement is that the primary data The original reference makes no explicit reference to the source for quantifying vehicular collision partner platform inertial frame of reference or global coordinate system. and regional stiffness data employs only axial component Rather, such appears to be implied in form of the relevant reporting of nodal residual damage depth data. Thirdly, derivations. The original reference utilizes a subscript of the central collision model is based on the uniaxial col- p for the peripheral point. Equations 18 and 19 of the lision between two collision partners with each collision original reference represent the propounded translational partner modeled as a single lumped mass and a single and rotational dynamics relationships, respectively (the massless linear stiffness. The model is a closure phase only variable ψ was used to denote the heading angle). model and assumes the achievement of a common veloc-

Figure 1: A partial replication of Figure 2 of McHenry 7 showing an offset frontal impact. The center of mass of each collision partner is represented by the larger colored circle. The center of mass velocity change is denoted as ∆v1 and ∆v2, respectively. The peripheral point at which the common velocity is reached at the terminus of closure is shown as the smaller circle. The velocity changes at the peripheral point are denoted as ∆v1’ and ∆v2’. The scalar magnitude of the moment arm of the collision force is denoted as h1 and h2. www.collisionmagazine.com

Volume 14 Issue 2.indd 23

Collision Magazine - Volume 14 Issue 2 23

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the dynamic center of mass of the two collision partner system is defined as:

m 1 G r1o  m 2 G r2 o (49) m1  m 2 ounded translational and rotational dynamics relationships, respectively (the variable  was used to teand therotational heading angle). dynamics relationships,The respectively (the variable was usedyields to the velocity of the dynamic center of mass. first G derivative of thisequation propounded translational and rotational dynamics relationships, respectively (the variable  was used to .

G denote the heading     1  m 1  1 2   1 Fx  mangle).

Fx h 1  I 1   m G r  m(45) 1 x 1  m 1 x p  h 1  1 G 2 r2 o (50) rCM  1 1o 2 (50)      (45) Fx  m x   m x  h  F h   I    m   tional and rotational dynamics relationships, respectively (the variable  was used to m1  m 2 1 1 1 p 1 1 x 1 1 1 1 1 1 (45) 2  1 Fx h 1  I 1    m  1   1 (45) Fx  acceleration: m 1  x1  m 1  xp  h 1 angle). ng the second fordynamics the angular nslational andequation rotational relationships, respectively (the variable1  was1 used to Solving the second equation for the angular acceleration:externally applied force to the two collision partner system during the When there is no net, unbalanced, on for the angular acceleration: ding angle).

When there is no net, unbalanced, externally applied force 2 second equation for the angular acceleration:     F h (45) Fx Solving m 1  x1 the  m x  h  F h   I    m   closure phase and the sole force is the collision force, which is internal to the system, the dynamic center of x 1 1 1 1 p 1 1 x 1 1 1 1 to the two collision partner  1   (46) system during the closure phase  2 2 during is zero-valued.

This is consistent with the CRASH3 modeling   1 mass  1the closure phase 111  m 1  1  (45)force is the Fx  m 1  x1   m1 F xxph1 h 1  Fx h 1acceleration  m I1  and the sole collision force, which is internal to  (46) (46) 1 2 F h approach.

A result of this approach is that the dynamic center of mass velocity is constant during the x 1 m  quation for the angular acceleration: the system, the dynamic center of mass  1   1 1 (46) acceleration during  2 closure phase and can equally be written in terms of the individual collision partner velocities at any point m  itution of the result from equation (46) into the second equality of the first equation under (45) 1 1 the closure phase is zero-valued.

This is consistent with the ond equation for the angular acceleration: during the closure phase. wed by equation rearrangement: F h from (46) into the second equality of the first equation under (45) CRASH3 Substitution x 1 result from equation (46) into the sec  of the (46) modeling approach. A result of this approach is  2 Substitutionond of equality the 1result equation (46) into(45) the followed second equality of first equation (45) is constant during nt: mfrom that thethe dynamic center of under mass velocity of the first equation under by h 1 1 F 2 2 2 (46)  1   F x 1 2  followed byrearrangement: rearrangement: F  F   1 closure h   h 1  phase and can equally be written in terms of the  1    12   (47) Fx  m 1  xp  h 1  x p   mx 1 1h 1  xp   x 2 1  2 12    x  the  2     Fx dynamics Fm F m m h h     individual result and fromrotational equation (46) into relationships, the second equality of the first equation under (45) tional respectively (the variable  was used x 1 1 x 1  1 1  1  1 to  collision partner velocities at any point during  1    1   xp   xp    h1  h1  1  2   F   (47) 2    12  h 12  Fx  (47) Fxphase. h 1 2 closure gement: angle). the m p  hthe 1     1 the 1 F  m 1   (47) the result from m equation (46) xp1 equality xfirst  x m1h1of     (45)  1  under   x 1 xinto 1  1 second p  equation 2  m1 m 1of equation m 1  12  1  (47): the first equality in the first equation under (45) coupled with the result rrangement: 2 2 2  FxI 1  result F  (47): h 2of1 equation (45) m G r Fx equation m 1  x1 under m 1 Fx x(45) Fxwith h 1  the the first coupled 1  h1 p h 11 1  1m 1 1x  m 2 G r2o  o  G m 1 G r1o  f   m 2 G r2o  f  G     1 x  h   x    h   x   1    1o  o  G (47)     1 p 1 1 p 1 1 p 2 2 2   2 2 2 rCM  o of equation (47):  rCM  f    rCM From the first m equality first (45) coupled with the result m1  F1  Fx equation  under F inx the 1 1 h 1 m 1  Fx 1  1  h 1  m  m m  m  (48) x p   1 xp   1   1  1 2 1 2 (47) xp  h 1  x p   x 2   h11  xp  2 x  12   m 1   2 2 acceleration:  F quation for the angular m   h (51)   (48) x 1   x   2m11 2   x p   11 x p  1m 1  1F 1    2m 1G  1 G G G first coupled equality with in xthe first (45) ty in the first equation under (45) the result of equation (47): x equation 1m under 1o 2oframe 1othe 2o(48) mFrom r  m r m r  m r   the 1 1  h1     x   x 1 2 1 2 The form of the G expression of position of point q (the point of common velocity o o f f         G 1 1 p 2 2  p (51) rCM o 1 rCM duration  Gthe   G rCM coupledtounder with the of masses. equation m ofhof F(45) hresult f 1(47): 1  xeffective 1 coupled erms 1 in and 2first are referred as the Integration equation (48) over of quality the equation with the result equation (47): equation (2).

The form of the G frame expression of the velocity at point q, noting that the p  1   2 2 m1  m2 m1  m2 (46)   v  1 = of Fx  Integration m of v analysis for the of second collision partner osure to phase yields the relationship 1’. A similar erred as the effective (48) over the of from theduration origin each body frame coordinate axis to the point is taken as constant, is given  (48) x 1   masses.   2 1 1 12   x p  11  xequation p 2 The terms  and  are referred to as the effective masses.

Integration of equation (48) over the duration of the start to the result of v =  v ’.

The velocity change during the closure phase, at the peripheral point, is point 1 2 h 1 form (8). collision Writing this partner phase: 2 1 =2m Gequation for G each collision G G 1v ’.xA analysis the second he relationship of v The of thefor 1 1F 1 similar G frame expression of partner the point q (the common given by  at m 1 position r1o  o form  mof rthe mof  m 2 velocity), r of theisclosure 1 expression GThe G 2of 1 Gr1oof 2o  o G frame f  the position   (48) x      x x (48) G G     m r  m r m r  m 2 G r2o2off ofGGpoint 1 p 1 p =  v ’.

A similar analysis for the second collision partner the closure phase yields the relationship of v result (46)minto the second equality of the under (45) replacing mi and in equation with  and , respectively.

The center of mass 2 2(44), 1m 1 first 1jmequation j, during i i j    1 2 1 =ned 2vby The equation velocity change the closure phase, at the peripheral point, is (51) r r r    1o o 2o o 1o f       2’.from G G The form of the G frame ofpoint the velocity at  point that the vector m 1 equation 1  h 1 (2). f  velocity),  o  expression qby (the of common is given by position equation  (51) rCM G rCM  q, noting B rCM m m m CM CM  othe CM  f  G point,  ofclosure 1phase, 2 at the peripheral 1  mB21 ity changes are obtained by multiplying the peripheral point velocity changes the corresponding G is gement: leads to the result of v =  v ’.

The velocity change during 2 2 2 1 and m , in equation (44), with  m and  m , respectively.

The center mass j i i j j m  m m  m   re referred to as the effective masses.

(48) overcoordinate the duration of fromIntegration the origin of of equation each body frame axis to the point is taken as constant, is given by equation r  r  R  r  1 2 1 2 B G 1q 1q  oG 1o  o  expression o (2).

Thenoted form that ofThe the ofB1 the velocity  frame 1 ive The effective mass concept was explored byfor Rose et al.corresponding Those authors the obtained by of replacing m1i (8). mjvelocity , in equation (44), with i[8]. mcollision and jmpartner center of mass icollision j, respectively. nedmass. by relationship multiplying the peripheral point byfor the Writing thischanges equation each at the start of the closure phase: ’.and A similar analysis the second partner elds the v 1 = 1γv and γ are referred to as the effective masses.

The terms d  are referred to as the effective masses.

Integration of equation (48) over the duration of at point q, noting that the position vector B2from the G G G 2 2 2 2 G position G G of The of authors the G expression ofm the point point given by 1 due nal formulation was incomplete to2 by theet of form inclusion centripetal acceleration term.  q o(the changes are obtained multiplying peripheral velocity changes R GB2origin rof r2qvelocity), is  common ive massvelocity concept was explored byduring Rose al.

[8].

Those noted the The the the that  position  Gof F Flack Fof h frame  hpoint by m 2the r2ocorresponding B 2q opoint o 2 r o  Gr2o  m vyields The velocity change the the point, isthe 2 m form of the frame of of q axis (the point is given by x x closure 1 phase, xatG 1 peripheral 1 expression 1 r1o 1 2 r1o 2 = 2v 2’.relationship 2o  f  velocity),  o frame  f  Bcommon G G      Integration of equation (48) over the duration of the clo=  v ’.

A similar analysis for the second collision partner se the of v of each body coordinate to the point is taken asthe (47) x h x h x 1              1 1 1 CM         r r r equation The form the G frame expression of the velocity at point q, noting that position vector ( 1 p 1 1 due to p 1inclusion 1 p mass 2  (2).

2 of effective mass.

The effective concept was explored by Rose et al.

[8].

Those authors noted that the CM incomplete the lack of of the centripetal acceleration term.

CM  f   o  mass B1 B1 velocity Gofcenter G m The ng of miv and (44), with and jmjclosure ,of (2).=phase, The the GGof frame of the at Writing pointmq,1 this that the position vector j, in equation i m mgiven noting m 2equation 1 1 equation 1respectively.

1∆v ∆v  form  o  frame r1qperipheral r1o  expression Ris r1qterm.

1Gm ult =m framework The velocity change during the γof ’.at A1the similar surem phase yields theim relationship constant, equation (8).

2original 2v2’.

B1is B1 2by o  to  ocoordinate  point, from the origin of body axis the point is taken as constant, is given by equation irst subject specific evaluation is the evaluation the terminus of closure phase common formulation was incomplete due to the lack inclusion of the centripetal acceleration 1 1of 1each obtained multiplying the peripheral point velocity changes by thebody corresponding from the origin ofto each frame coordinate axis to the point is taken constant, is given (52)by equation At the terminus offor closure: placing mi by and equation (44), with imfrom jmkinematics The center of analysis for the second collision partner leads the each collision partner the of start theas closure phase: j, in i and j, respectively.

(8).

Writing this equation for each partner at the start theofclosure phase: B2 to B2 at G result G Itcollision Gmass ityspecific vector.

Thism can beis accomplished purely the perspective. is important note rk evaluation the evaluation of the terminus of closure phase common    R  r r r effective massequation concept was explored

by Rose etthe al.

[8]. The Those authors noted that (8).

Writing this equation for each of the phase: B2 partner G the form of the G of of point q closure (the point of common velocity), is given 2q 2oexpression o  the B2  oposition  o frame collision at the2qstart in the first under with result of equation (47): sty are obtained by multiplying the’.coupled peripheral point velocity by the corresponding = γ(45) ∆v The velocity the ofvector ∆v he common velocity is clearly an change expression the Gclosure frame.

The G frame of The first subject framework specific evaluation is during the evaluation of the terminus ofposition closure phase common 2 condition 2the2kinematics be accomplished purely from perspective.

It changes isin important to note was incomplete due to the lack of inclusion of the centripetal acceleration term.

B1 point q, G G the velocity G equation (2).

The form of the G frame expression of at that the position vec The effective mass concept explored by et al. The [8].

Those authors B1  f   B1 noting r1q  f  B1 r1o to note r1q phase, at the point, obtained by replacing mi noted B1 R B G that the ynamic center ofvector. mass ofThis the two collision system is defined as: velocity canperipheral be accomplished purely from the kinematics perspective.

It  is Gimportant  vector condition is clearly anwas expression inpartner theRose G is frame.

G frame position of 1q  oframe  GGr1o Bto1  ofthe r 1 isG taken B B Gr GR B1 axis from the origin of each body coordinate point as constant, is given by equat G 2 1 1 1q o     1o o  G frame r1qframe.

R B1 Gposition B1  o   ofr1q ation was incomplete due to lack of ofof centripetal acceleration the terminus closure:  system Fmxvelocity At , inthe equation (44), with γisthe m and γan mjexpression , respectiveand that the common vector condition in common theterm.

G  o   rThe 1 (52) ss of the two collision isinclusion defined as: Bthe B2 phase: G partner G at G j  i clearly i (8). of j closure 2    (48) Writing this equation for each collision the start of closure x 1  partner x   x (52) mework specific evaluation isthe evaluation of the terminus phase      R  r r r B B p 1 p (52) G G G 2 2 2 G B2  f  2q 2q  f  2 o f   2 r B2 G G   r  r  R  ly.center The center mass velocity changes are obtained by m hof mass of the dynamic of the two collision partner system is defined as: B B G G G m r  m r B G 2q 2 2 2q o 2o o B o   1 1 1       2 G kinematics 2 1 1o perspective.

2 2o   s framework can be accomplished purely from the It is important to note   R  r r r  B G B o 2q (49) revaluation 2q  G o 2o o G G specificmultiplying evaluation is the of the terminus of closure phase CM  Gperipheral point velocity changes by r1qthe r1o  f common ofR B1 G BG1 B1  f G B12 r1q G 2   B m 1 G r1o  the m r2 o mthe f   ocity an2from expression in G 2frame.

The G frame position G 1 m r1q  o   r1o o   R B G B  o   B r1q This vector can be condition accomplished purely the kinematics perspective. is important to note GThe twoIt Gequations (49)  is clearly rCMcorresponding (53) condition at( under equal to each other when the common velocity mass.

The effective mass concept m r(48) closure: m remass referred to as the effective masses.

Integration ofr terminus equation over duration ofG (53) are G 1o 2 G r2the o At the of of of the two collision partner system is defined as: B2 B2 m  m 2effective 1clearly  G1 frame. the terminus n velocity vector condition is an expression in the The G frame  GAt R Bof rholds. r2position r2q G R B(49) B G G of closure: G 8 the At terminus ofclosure closure: CM of 2qthe f o f  B    Those authors noted that was explored by Rose et al.

2 r  r  2 f r B G B o 2q analysis for dynamic themsecond partner elds the relationship of v 2q  o  2o  o  1 = 1vyields 1’. A similar irst G oftwo this equation the velocity of the of mass.

1  mcenter 2 collision nter ofderivative mass of the collision partner system is defined as: original formulation was incomplete due to the lack of inG G the B B G G G v =  v ’.

The velocity change during closure phase, at the peripheral point, is 2 2 2 m 1 ofr1othe m his equation yields theG velocity dynamic center of mass. 1q  f   Gr1o  f   GR B1 BG11GB  f   BG11r1q G 2 r2 o Gr  (49) rCM of r1q  f  when r1o  fthe  common R B1 G r1qB1velocity  r2q clusion the centripetal acceleration term.

The two equations under (53)terminus are equal to each other G m , respectively.

G f   f   rq  f  at the terminus 1qcondition equation (44), with  The center of mass ng mi and At the of closure:  j, in G imi and j j m Themfirst derivative ofm this the velocity of the dynamic center of mass.

Gequation G yields   1 f (53) m r r  m G r 2  m 1 r 1o 2 2o m 1 B B (53) G G G G 2o r1o  2holds.

2 of1G closure (53) B1 B1  B2 r G G B B2 CM point 2q  fG r(49) 2 oGf R G(50) G obtained by multiplying the peripheral velocity changes by the corresponding  R r r r  2 B G G B G 2 q CM  2 2 B f      r  r  R  r  2 mfirst rm m 1  mevaluation 2    r  r  R  r B G 1q B G 2q G 1o f B f 2o f B f The specific is the evalu1 rsubject 1o  m 2 framework 2o  m 2         1 2 2 B2 1 G B  f  2q 2q  f  G 2 o f  1 Rose 2 et al. [8]. Those G G noted (50) r  was explored by effective mass concept that the 1o authors r1q  f   G r1o  f   G R B2 BG B  f   B r1q mcommon  m 2velocity r2 o G phase 1of rmass. e of this equation CM yields the velocity of theofdynamic center mthe G G G ation of terminus closure  1 m 2 (50)  r    r  f   r2q  f   rq  f  CM ( was incomplete due to the lack of inclusion of the centripetal acceleration term. m B B Gduring Gthe The two purely equations under (53) are 1q equal each common condition (54) at the terminus n thereof is this no net, unbalanced, externally applied force to center the m two collision partner system 1  2 vector. can be accomplished from the kineThe velocity changes at qto are: under  Gthe Rare velocity r2qother rwhen r2 q other vative equation yieldsThis the velocity of the dynamic of mass.

The two equations under (53) are equal to each other when the velocity condition at the terminus B common G  B  f to f 2 o f     The two equations (53) equal each when G G B1 B1 G B2 G G of which closurepartner re phase and the sole forceperspective. ism the isholds. internal system, the dynamic of B2 collision r1othat to Rthe r1q  G r2o fcenter mIt r2 oimportant balanced, externally applied force tothe collision matics isforce, to note the comG B1during G B1  common f closure fthe 1 revaluation 1o 2 two   R B2 G B2  f   r2q   (50) ofthe closure holds. mework specific evaluation of terminus ofsystem phase CM is the r the common velocity condition at the terminus of closure acceleration during the phase iscondition zero-valued.

Thisthe is consistent with the CRASH3 modeling theremon is noclosure net, unbalanced, externally applied force to the collision partner system during the G theis G G isWhen the collision force, which is to system, dynamic center of velocity vector clearly expression intwo m Ginternal m 2 o perspective. m r1okinematics 11the 2  m2 r s force can be accomplished purely from Itequations is important to(53) note  Gtov1q rq  f G when r1o  o the  Gcommon R B1 BG1 velocity  B1 rcondition Thean two under are equal each at the termin G G G  other 1q holds.

B1  o      (50) r  oach.

Aclosure result of this approach is that the dynamic center of mass velocity is constant during the r  r  r phase and the sole force is the collision force, which is internal to the system, the dynamic center of G G G the closure phase is zero-valued.

This is consistent with the CRASH3 modeling CM the isGclearly frame. an The G frame position of the dynamic center 1q  f   2q  f   q f     The velocity changes at q are: ocity vector condition expression in the G frame.

The G frame position of of closure holds. r r r mterms f 2q Gf  qG f 1  m 2 of B2 B2 G (54) re approach phasemass and can equally be written the individual partner velocities at1qthe any point acceleration during theinclosure phase is system zero-valued.

This is during consistent with  modeling  is that themass dynamic mass velocity constant during the r2q B 1 G vCRASH3 of of the center two collision partner iscollision defined as: G G B1 B2 R B2 B2 B  o   2q G  rq  f   G r2o  o   G (54) t, unbalanced, externally applied force toofthe two collision system the of mass of theistwo collision partner system is defined as: ispartner 2  r  R  r  r  R  r   (54) G G B B G G B B B1 velocity G1 B1  f  1q G G2 B2  f  1 constant 2 2q 1o  f of mass 2o during fG   RB2the ng thebe closure phase. approach.

A result of this approach is that the dynamic center is   G r  R  r  r  r   ually written in terms of the individual collision partner velocities at any point BG1 G B1 Gf  B G 2q 1o G f 2o f B f    B1q B G dynamic eo sole force is the collision force, which is internal to the system, the center of     r  r  r 2 1 1 2q f 1q  f   v1q system rcollision  r1o partner  R r1q  pointq f  2 net, unbalanced, externally force to theintwo collision partner G  B1  o   q  f  during o  the B1velocities closure phase and can applied equally be terms of thethe individual at any G Gwritten ( uring the closure phase is zero-valued.

This is consistent with CRASH3 modeling m r  m r (55) The center of mass velocity changes are: G G B B G nd the sole forcethe is the collision which 1 force, 1o 2 2 ois internal to the system, the dynamic rG1ocenter  Rof r1qB2  G r2o f   G R B BG B  f   B r2q during closure phase.

The velocity changes at q are: G G GB (49) r  G BB2  f   (49) f   CM   of during this approach is that theisdynamic of mass velocity is constant  during  R B2 G B2  o   r2q v2q rq  f   modeling rthe The changes at q CRASH3 are: 2o  o  m 1  mcenter on the closure phase zero-valued.

This isvelocity consistent with the 2 n equally be written in terms of the individual collision partner velocities at any point  G v  G r1oB f1  G r1o o esult of this approach is that the dynamic center of mass velocity is constantG during the G The velocity changes at Gqv1q are: q  f   GGr1o  o   GGR B1 BBG11o B1  o  B1r1q  hase.

Gr 1 nd be written in G terms of of the individual velocities atvany  point rq  f  changes  r1o  o   The first derivative of this yields the velocity The center ofequation masscollision velocity changes are:of The e ofcan thisequally equation yields the velocity the dynamic center ofpartner mass.

1qvelocity B1 G vG  B1 G o r 2o f  r1q G r2o o  at qR are: (55) 2o B2 B2 (55) G G G G ure phase. the dynamic center of mass.

B B G r   Gv2q  GrqGvf   GrG2or o   GGR  B B 2 2q  r1q G v2q G rq  f 1q G r2oqof  rR1oBB2o2  GG2 RBB2B2oo  G  B r2q o G G  v1o  r1o f   r1o o m r  m 2 r2 o ( G (50) rCM  1 1o This problem may also be approached by considering at point q. Starting w B the acceleration B G G G G (56)   v r r r      R  G G G 2q B G 2q 2o  o  B o m1  m 2  The center of mass velocity changes  cross product v2oare: r2o f   with r2oqofthe (8), replacing the equivalent vector-matrix product using the antisymm  24 Collision Magazine - Volume 14 IssueThe 2 center of mass www.collisionmagazine.com velocity changes are: the angular velocity vector and using the first equation under (3) to substitute for the B fram Thepartner center of mass velocity changes G are: G G t, unbalanced, externally applied force to the two collision system during of theq leads 1ofollowing 1o opoint the position result:  the the Gv1oto Gr This problem may also be of approached byvector considering acceleration q. Starting with equation  f   Grat  e sole force is the collision force, which is internal to the system, the dynamic center of G v1o  G r1o f   G r1o  o  (56) (8), replacing the cross product with the equivalent vector-matrix product using the antisymmetric form of G G G (56) 2ov1of   Grr2o o   r Gr Volume 14 Issue 2.inddis24zero-valued. This is consistent with the CRASH3 modeling  Gv2o G  uring the closure phase B i 1o  o  B i G G f  G G 10/9/2020 3:43:02 PM 1o   v r r   substitute for the 2o r(3)2o  to  2o R  R B frame r  expression r f r the angular velocity vector and using the first equation under o G

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differentiation:   G iq G io G Bi  iq io G Bi iq io  r2q  f   G r2 o f   G R B2 BG2 B2  f   B2 r2 differentiation:  dt   dt  q G G G G derivative of the second equation G G G G G (58) G G Gthe product G Grule of Taking the shown under and using    r  f  G r2q  f   rq  f   Bi  Griqd  rio riq  rio  GGBi  riq  rio G  Gd G  G riq G rio GG G  r1q  f   G r2q  f1q rq  f  G G G G   differentiation:  expression  d G acceleration riq velocity rio first dG (59) of the bod The parenthetical term the frame of the angular    r    (54) G the G riois G G riq G rio     G G G B iq G B ations under (53) are equal to each other when the common condition at terminus i i G B2   G rio    riq  rio  GG Bi   riq  G rio  dt G riq(54) G B r1oB1f   G R B1B1 BG1 B1  fG  B1 r1q G G r2oB2f   G R B2 B2 BG2 the  r i dt   G G 2q term, using theG second under (3), extract B2  f  G frame. For the last parenthetical lds. r1o f   R B1 G B  f   r1q  r2o f   R B2 G B  f   r2q G G G G Gdt  dtequation  (58) Taking G GG derivative (58) and using the prod GBi G riq  G of riq  the rio   rio the riqsecond  G rioequation rshown rio under 1 2 G B G iq  taking the B frame of q and changing the exp G G derivative Gd G  ofG the remaining G G G i d Gposition G vector G  G G r G Gr  Gr Gr G  G r       (58)  r    r  r differentiation: io  G Bi io iq io G rB i  rioiq  expression rthe toangular rthe rioiq body  io frame about (59) parenthetical term of isG  the ofriqiqthe iq G G frame Gbefore Bi  io following: G acceleration Bi   iq of the G The first G G resultant as leads ty changes at q are:    dt dt r  r  r The first parenthetical term is the G frame expression of the angular acceleration of the body fra     1q  f  2q  f  q f  s at q are: the GTaking frame.

For last parenthetical term, using the (54) second under extracting the DCM, the Gthe derivative of the second equation shown equation under (58) and(3), using the product rule of the G frame.

For the last parenthetical term, using the second equation under (3), extracting G G G B1 G B2 G derivative B2 the of GBthe second equation and using product  dG the  rule of t dGGGshown G G position G G G G under G G (58) G the G GGTaking derivative of remaining vector of qand G changing GG expression G frame G r1o f   G R BG1 BG1 B1G f  taking r1qG differentiation: the r2o  GR the r2qG derivative BB2the B 1 B2 iqframe Giorthe       r  r   r  r r  r   r  r Gtaking  d d d 1f  fR  r   r  r   r  r     of remaining B frame position vector of q and changing B B G G G G G G G B iq io iq io G B iq io       v  r  r    r iq G angular Bi iq io Gi of Bi the ioabout i acceleration i The term is the G frame expression B o1  first B1 G B1  o  B1parenthetical 1q f G the 1o riqGiio rioofdt the  G iqframe riqthe  G rexpress differentiation: as leads to the following: body Bi  io  G v1q  G rq  f  1qG r1o  o  qof R B1 resultant rbefore R Bi riq  R Bi  dt riq dt G  B1  o   1q (55) of the resultant as before leads to the following: dt dt (55) G (3), Gextracting  the GGframe. term, using the second equation the DCM, G G G G G G G under B2 For the B2 last parenthetical G G G G G G (55)      G riq G rio G riq  rdio G G Bi G riq  rio (60) r   d rGof  v  r  r B o2   R B2  o  the Gr2qG derivative G io B GBi second B2 B2G changes atqGare: Taking using the frame product(58) rule the q and the expression G frame G changing  of the rioGremaining (59) iq riq position  Grioequation  Gvector runder rio (58)  the  shown v2q  G rq  f  2qG r2o  o  q f G R B2 2otaking rG2qGderivative d G is  of  Gand G riq  G G iq  G  B2  o   dGGthe  d G The d d G G G B B i term G G Biexpression Gdt G angular the G frame of acceleration of the body f G B Gfirst parenthetical G G G G dt i i  riothis (59)   r r   r  r   the    differentiation:   of the resultant as before following: d d d (60) of result into equation (59): riq Substitution rioleads r R  G riqto  r   r  r     GR G B iq io G B iq io Giq G iBi  Giq  BiG Bi G G G iG Bi iq ioBi r r r r  r r   R  R    dt dt the G frame.

For the last parenthetical term, using the second equation under (3), extractin dt dt dt       iq io B iq B iq G B iq io B B G G G G  dt i 1 Taking the derivative of dt the second equation shown underi (58) and usingi the product rule of r of mass velocity changes The v center  are: rq  fof r1o  ovelocity  1 rG  dt vector  of q and changing the expre G  B1  o are: 1q  mass   R B1changes elocity changes are: 1q taking the G derivative of Gthe remaining B frame position G Gof the G term is the G frame G Gangular G G G G G G body G Gabout (55) The firstdifferentiation: parenthetical expression of the acceleration frame     d d   d d d   r  r   r  r   rio    G G G G G G B B G resultant G G Gthe G B2 B2 of the G G iG iq toG i  io  G rBi  iq  io  GG B i GG r G  B i Gr riq  as before leads following: riqthe rRioframe   r   (60) about r r r r  r r    R    2o the The first parenthetical term is G expression of the angular acceleration of the body frame Gr GGRframe.

G r   G v2q  G rq  f   Substitution       G G G B iq io G B iq io   of this result into equation (59): iq io B iq B iq G B iq io Substitution of this result into equation (59): For the last parenthetical term, using the second equation under (3), extracting the DCM, i i 2q i i i v1oGo r1o fB2  G r1oB2oo  Substitution dt dt G G dt dt of this result Ginto equation term,  (59):  under (3), extracting   dt   v1o  r1o f   taking r1o o the G the G G derivative frame. Forofthe last parenthetical using the second equation the the remaining B frame vector of Gq and changing the expression frame DCM,  (56)  d position d GG G G GG G G G G G G (56)G  (56)    (59)e r  r  r  r   r  r  r2oresultant r  frame expression of this equation can be obtained by using the second The corresponding B i taking the G derivative of the remaining B frame position vector of q and changing the expression frame   d d d     G G  v2oof G the G G G G G G G G G G iq G io G B iq io G B iq io f 2o o  as before Bi Gr Gi   G G G G G i iq  leads rio  to  following:  the v2o  r2o f   r2o o     are: dt mass velocity changes  toriqisthe GGframe  Brof dtBBi  GGform G G G G (61) i rioangular velocity rioiqproducts riqrthe riqofG the riqbody rioframe Gr Bi term Gr G  B iGR iq GGthe R GiqBangular ioB i expression Bi antisymmetric first parenthetical G acceleration ab   Substitution of this result into Gequation (59): iofr the  r  r  r  r (3),Gbefore replacing the cross using the vect     of theThe resultant as leads following: iq io G B iq io G B io dt dti i  dt i  the G frame.

For the last parenthetical term, Gusing the second equation under (3), extracting the DC first equation and applying a kinematic transformation to the resultant.

(61) G Gunder (9) lem may also be approached  byG vconsidering the at point q. Starting with equation G G  d Bof  using dq.expression d G Starting G term G G be G vector G Gand also r1obe r1oacceleration i G G G with Gthe GBi r Gexpression G G  Gsecond The parenthetical is G frame the angular acceleration of the body frame the G derivative of the remaining B frame position of q changing the expression fra lso be approached by considering acceleration atBpoint equation of this equation can obtained by the equation under The corresponding 1othe G G G i frame f    ofirst taking (60) r r r  r r   R  R   This problem may approached by considering the ac  (61) about riqd GioGthe rof Bi drexpression  equation rioB  Bof  equation ofthis  Bdi BGi  Brcan iq iq io under  ing the cross product with the equivalent vector-matrix product using antisymmetric beriqioobtained by The corresponding B i G i G io  B iqiq B(59): G iq  G i frame GG G form G G G using the second equat i i this result into (56) i dtSubstitution dt dt (60) r r r r  r r   R  R   the G frame.

For the last parenthetical term, using the second equation (3), extracting the DCM, G (3), replacing G G   of the resultant as before leads to the following: ss product with the celeration equivalentat vector-matrix product using the antisymmetric form of the cross products using the antisymmetric form of the angular velocity vectors, using the B B B G B G B B B B   i i i i i i i i iq io B iq B iq G B iq io q.r2oStarting equation (8), i R i r rio  the  point v2o equation r2owith  Bi   Rform Bi   G velocity  Giusing R G antisymmetric  Gangular riq vectors, ar velocity vector and using the first to substitute for B frame replacing thethe cross products of the iq Bi  dt replacing dtG riqexpression  f   under  o  (3) (3), i G i  Bdt of q Band taking the G for derivative of theexpression remaining B frame position vector changing the expression frame vector and using the equation under (3)equivalent to substitute the B frame first equation under (9)vector-matrix and applying a kinematic transformation to the resultant. thefirst cross with the product ition vector of q leads to the product following result: first equation under (9) and applying a kinematic transformation to the resultant.

G equation G G be obtained G G by using G GsecondGequation G expression of this can the under The corresponding Bi frame The corresponding Bi frame expression of this equation can   G leads to G Gio   G    r  r   r  r   r  r of the resultant as before the following: r of q leads to the following result:    Substitution of this result into equation (59): G iq G io G B iq B G B iq io  d Bi  i G i d G d G using the form of thethe angular velocity vector Bi i G G G G (3),the replacing products using the antisymmetric ofRBBthe angular velocity vectors, m may also be approached byantisymmetric considering acceleration Starting with equation riq Bibe robtained rBform rreplacriousing the (59): Rusing  [4]: This equation use following Substitution ofatthis result into equation the second equation under (3), B i cross Bpoint Bq. G makes Gof the Bby B i riq   io Bi iq G  Bi  iq  i i i i ii    (62)    r  r  R R  R r    r Bi B iunder dt dt dt G G G equation G (3) G and using the first to substitute for the     B B B G B G B B B Bi G iq (9) G and io G BiaGkinematic Bi Btransformation G Bii G the Bi resultant. i i i iq iiq  i i  i      first equation under applying to g the cross product with the equivalent vector-matrix product using the antisymmetric form of (57) r  r  R  R r  r    r  r  R R  R r    riq  G   the products the form B i io B i G G Giq B i G Bi G G iq io G Ging iqGG G cross io Bi using Bi G iq  G B Gof Bthe G G G G GG  BG G antisymmetric i d i i d d    (57) riq  B rframe  R  R r  r frame expression of this equation can be obtained by using the second(60) equ The corresponding B (61)    r  r   r  r    r  r B B G G G G G G G i     of theiqunder position of Gq iqleads to the i G i iq io B i G Bi G io G the io G frame Bi iq io B G G Bi io G B B B G i i i i   R  R    r r r r  r r velocity vector and using theexpression first equation (3) vector to substitute for B expression G iq G io angular GB G using Gthe G iqBi first iqGvectors, G BiG r RiqGunder velocity equation (9) velocity vector i i  R riqthe rio  GGdt (61) G B i GB Bi G G Gr B G angular Bio     r  r    i i     (3), replacing cross products using the antisymmetric form of the G G B iq io B B iq io dt dt i following result:  Bi i B i of Bi result Bi [4]: G equation Bii G Bi Substitution This equation makes use of the on vector of q leads to the result: i iqfollowing  in this under Rinto Rof  (59): rkinematic   BGi  Bi  Bto riqthe and applying aRkinematic x multiplication shown, offollowing the three terms that precede theThis parenthetical the   toresultant. equation use G requation G riomakes Gprevious Band G equation, Bfollowing Bi a [4]: iq   transformation G  Bi transformation i the i first (9) applying the resultant. (62) ation shown, of the three terms that precede the parenthetical in the previous equation, The corresponding B i frame expression of this equation can be obtained by using the second equation under on (9), results in the G frame expression of the angular velocity ofsolution the body frame about the G shows the centripetal acceleration termusing should be present when under one is c G B iGthat B GG expression of this equation can obtained the equation corresponding Bresult GGequation GBi G G by G G second i frame ults in the G frame expression of the angular velocity of the body about the Substitution of This this into Bi B i The G G Gcross the products using the antisymmetric form ofBbe the velocity using the (63) rioBG G B B angular  rR (59): GBGirR  rGiqFor vectors, rio  Bplanar  Breplacing (57) riq  G rio  G R Bi (3),  R r  r G B i  G  B GG G B GG G rio B i B i i i  iq G B iq G B G B i B i the i (57) GThis G iq io G B B B B i i i  R   R  approach that involves acceleration of a peripheral point. the case, using i i B i i i equation makes use ofapplying the following [4]: G iR Bi Gi B of G the  thei rantisymmetric (3), replacing the cross products using form vectors, R BBii to RBBii angular riq  GG Gvelocity i Bi   Gi  riq using equations:  the a first equation under (9) and a kinematic the G riq  Gtransformation io  G G B ii resultant.

Bi  angle, equation (62) reduces to the following two G applying G G first equation under (9) and to the resultant.  G GG rio  GGGakinematic (61) BiG GG   G r should Bi transformation rio  be BiGGpresent riq isG rconsidering present (62) G riq acceleration Bi term B i   one This solution shows that the centripetal when an equation B i  ofiq frame expression this equation can be obtained bybeiousing the second un The corresponding B multiplication shown, of the three terms that precede the parenthetical in the previous equation, i   R    R  (63)is consid shows the should when one Bthree B Bthat Bi B icentripetal BiG G B i term Bi Bi The matrix multiplication shown, of This theThis terms G that BGi G BGi G Bi Bacceleration i solution i i i riqofequation cross of following [4]: (62) using  Bthe the  makes Rproducts Ruse point.

R riq B the  Gplanar BBi  form using riq angular 2  approach that involves the Gacceleration of For case, as the heading G rio Ga peripheral Babout Biantisymmetric G B BB B Bivelocity i Gusing i the i of (3), replacing the the vectors, (9), results in the Gprecede frame expression of the angular velocity the body frame G i i i i i B B B G B G B B B B i i i  approach involves the acceleration arBioperipheral point. using  (62) as the riqB xi of  R riq Giy  ziiFor riq i xriqiplanar the parenthetical in the previous equation,that by riq equa Bi the  Gi rio two R equations: R Bi GGi  G G Bi case, zi  xi riq  obtained i G G Bi Bi  G i i angle, equation reduces to the following expression of this equation can be bythe using thesecond equation under The(62) corresponding Bi Gframe first equation under (9) and applying a kinematic transformation to resultant. angle, equation (62) reduces to the following two equations: G G B i present Bi Bi G is considering tion (9), results in the G solution frame expression of the angular 2 This shows that the centripetal acceleration term should be when one an Bi Bi  Bi  R  B Bi (3), replacing theofcross products using the antisymmetric the vectors, using the G Bangular G  Gyi Bi BGi rio Ryi Bform i GBiriqBiof riqGvelocity equation makes use the following [4]: equation  xi G  Bii  z4i :  yi i G Bi  zi G riq This makes use of the velocity of the bodyThis frame about the G frame. approach that involves the acceleration of a peripheral point.

For the planar case, using  Bas the heading 2following Bi Bi Bthe BBii Bi Btransformation Bi G This use of following [4]: Bkinematic G Bi Bi resultant.

Bi firstequation equation makes under (9) and applying a to the i i i i 2  iofollowing  the   BBi i rGiq xBi i GB riqxi   G rriqioyi two R r  G riq  Bi B   GB G riq  xi to G rG GBR  zi R i B Bi  angle, equation (62) reduces riqGiequations: rio  xiBacceleration  i rGBiqi Bzyiii BGi iqBiterm riqi  xi Gibe  G G BG Bi G  Bipresent  xi   zi  should  zi i i (64) This solution shows that the centripetal when   R    R  (63)one isofcon 2 G B B G B G B G G B Finally, the equations The G frame expression th i B i B Bi Newton-Euler B i Bi i of Bmotion Biiare considered.

G G G G Gi Bi G iG G i Bi B 2 Bi  Bri  B G B B B   Bi G riq  G rio r   r   R    R  (63) ithat i i i i i (58) riq  G rio  GG  riq  G approach rio GiGGriqByi i B r r Bacceleration Bi B Bi For Bithe (63) a point. planar case, using  as  y the  yof G ioinvolves iqR xBi i G Bi  zBi i  iq GBBiperipheral Bri  zi B  i R i G G B G G G iq io (62) r r   R   r i i i i i 2      expression is obtained dynamics

by Gequation (10). corresponding GB iq is given G ioB BGi rio  y The Bi riq iq G Bi  GiqB iq    r  iyBiframe iy G BiG riqB B B B      x G B z G B z This equation makes use of [4]: i i the following i i i i i  i i i i i (58)  G rreduces riqto ythe  Bi  zi i riq two Gr iq  xi (62) io  xi  xi Gequations: Bi  z i angle, equation i G following transform: This solution shows that the centripetal acceleration term should be present when one is considering an (64) G G G G G G G G G G 2  (58)the riq therioG derivative riqG  ofrioGthe riqG  rGio equation  G Newton-Euler GThis riq  rGiounder G G (58) GBi are Biacceleration Bi G The B that should be when one is heading considering an aking shown and using rule G Bi B i  solution Finally, motion considered.

G frame expression ofusing the translational i of iplanar G product Bterm Bi B Bi present G acceleration equations ithat  This solution shows the centripetal term (58) involves the ay peripheral point.

For the case,  as the riq  rio second rthe riothatequation riq Finally, rshows  acceleration BGiof ry the centripetal rof riqGthe  of following [4]:  r  r   r    R    R  G Bi approach iq  This iomakes Gthe B iuse io Newton-Euler equations of motion are considered.

The G frame expression of the tran  iq      G io iq x G B z iq y G B z 2 G B B ii Bi BBi Bi G G G B i i i B ii G iG i BBii i case, Bi using BBi i G Bkinematic i peripheral i the i i expression is obtained by dynamics given by equation (10). corresponding fferentiation: approach that involves theThe acceleration of aGpresent point.

For the heading i frame riq B rioGcorresponding Bplanar  zi  as riqconsidering frame Fi Bi is m m angle,isequation (62) reduces to following two beequations: when an approach xi The GR xi one zii iq xi expression Gi  i yi RGG Birio  GrF G Briiothat is obtained by k dynamics is the given by should equation (10). transform: angle, equation (62) to the following Gof G two B i equations: Bi Bi G tive of the secondTaking equation under (58)second and using thereduces product rule 2 involves ofBi BaiG peripheral point.

ForBithe the Gshown derivative equation shown unthe Racceleration  Brule Rshould (63) transform: Bi considered.

Bi G  Bexpression aking derivative the equation shown undershows (58) and using the product of G G G G Finally, Gof the Gsolution G B i are B G G i theG This Newton-Euler equations of motion The frame theplatranslational i G i  that the centripetal acceleration term when one is considering  G G   Bi G riq  G rof r  r  r 2Bi izi be  Bpresent riq  yi of (58) riq  the rio GGG  riqsecond r    r  r    G iq y G io y iq x G G Bi  zi io io G B iq io    d d Bi planar B Bi B(65) Bi heading Bi angle, i reduces i to the i i i i nar case, using θ as the equation (62) reduces For the case, equation following equations: G G G G G G G G der (58) and using the product rule of differentiation:  x(10). ioacceleration fferentiation: the rGiq  yi Gof  zperipheral  Bi(59) rBiqframe expression   obtained byusing kinematic dynamics given equation The corresponding riq isapproach rio  Gby that GB iriq r G rio Bi  z i B  xiBiB For theis2planar case,  as the head i  xi i B G riq  G rio   G B i   Bi   involves rio BBiiBrGBi aiFtwo  BGpoint.

(65) (64) R Giqi G  FBii  R iGrfollowing  i i B Gm Bi i the i G rio Bi i r G  riq m GBi F equations: rio one is considering  dt transform:  This angle,  dt R  R Gcentripetal xto G io xi iq  ytwo Gi Bi m iq rxio G acceleration  zi i should i  the i F i  i  z i m i when 2 present solution shows that the term be an G G G equation (62) reduces to following equations: Bi Bi Bi Bi Bi G ve of Gthe second under rule  Newton-Euler  of Bi equations the of motion considered. 2The G frame expression of(64) the t  G d G equation  G shown (58)  the Gproduct riq  yi are d Gand using G G G G G Finally, G riq  yi B G rio  yi B riq  xi GB Bi  zi  G Bi  zi B B B     approach that involves the acceleration of a peripheral point.

For the planar case, using  as the heading d G   (59)i r rio dGG Bi   G rGiq  rioG  i G i i riqrthe G riq  G rio   G B iG riq  xi GThe  (59)  Bi riq  yi Gi Bi B  GriqrioG dynamics is rgiven byfollowing equation (10). corresponding the planar equation  dtrcase, rio the  Gangular Bi (65)   reduces Gito   zii frame expression is obtained b G io Byii  equations: G Bi B i  zi For Griq isG the G Bexpression iq  of iq B ioyGiof i 2 he first parenthetical term frame acceleration the body frame i   angle,  dt   For the planar case, equation (65) reduces to the following equations: equation (62) reduces to the equations: (65) R  m R r  F  BF B Bi about Bi m ifollowing i Gtwo dt dt G i i io G      riq  yi G  Bi i Gzri io Bi riq  xi BGi Bi  zi transform: G riq  xi  G rio  xi  e G frame. For the last parentheticalFinally, term, using the second equation under (3), extracting the DCM, the Newton-Euler equations of motion are considered.

The G frame expression of the translational (59) (64)of the translational 2  GG dframe expression  GFinally, the equations motion are considered.

TheB G frame expression dacceleration 2 Bi corresponding Bi BB Bi king G derivative of remaining B frame position of body q and changing the expression frame iB G term Gis the G G G of the G Newton-Euler frame expression isBi obtained by kinematic isG rgiven byequation (10). ofequation the frame about Bi The Bi  Bi  i r B i rof rioBframe i B i riabout  the parenthetical For planar (65) reduces following equations: (59)  the ris  GGdynamics angular  the case, riovector G G iq  Bi G Bi  zi Bi  G to iq yi G yr iq y the  xi G BB i zr i r r  dynamics  he term therG the angular acceleration of the body G riqfirst Gr io   G B i  iq  io frame Bi expression iq of i i i i obtained G iq  xi G io  x iq R GBi  zF  yi G FBi  zm RiqG xirioexpression i Bii frame by kinematic is given by equation G i G i  m iisG r io the as before leads to the the second following: dt dt transform: lastresultant parenthetical term, using equation under (3), extracting the (10).

DCM, The icorresponding     (64) e G frame. For the last parenthetical term, using the second equation under (3), extracting the BDCM, B 2 transform: Bi Bi Bi B i i i ve of the remaining B frame position vector of q and changing the expression frame   r  r  r   r   yi motion G equations iq  yi G ioexpression iq  xi are G B iq  yi The G BG king the G derivative of the remaining B frame position vector of planar q and Bchanging the frame i  zi i  zframe i Finally, the Newton-Euler considered. the translatio G equation Bi of G reduces Bi Bi following G G i For the case, (65) equations: Finally, the Newton-Euler equations of motion areexpression consid- of (65)  rio ore leads toGthe following: firstGparenthetical term is theG G frame expression of the RGG about F F tomthe  G dof dThe d angular term is the frame expression of the frame Bi acceleration Bithebody iBG mG i R G rioB  G i G G leads G G the resultant as before to the following: G (60) Bi Bi frame expression is obtained by kinem isriq given by equation The B i i r (10). i corresponding riq  rio   r R Bi riq  dynamics R Bi        G G frame.

Bi iq ered.

(65) RioG The Fi Gmframe rio  G F of mthe translational dynami R G expression i G rio angular acceleration the body frame about the ast parenthetical dt term, using the second under (3), extracting the DCM, dt of equation  dt  Finally, the Newton-Euler equations of motion are considered.

The G frame expression of the translational transform: G G frame ics is given by equation (10).

The corresponding B the position last parenthetical using the second equation of the remaining frame ofterm, q and changing the by expression frame vector For planar case, equation (65) reduces to theThe following equations: d GBGFor d the G G G is G given equation (10). corresponding Bi frame expression is iobtained by kinematic  rGiqd dR BGi Bi riq G  G R BiGd G Bi riqBi For (60) riq  G rio  BGi planar dynamics Bi rio equation G G G G the case, (65) reduces to the following equations: expression is obtained by kinematic transform: re leads to the following: under (3), extracting the DCM, taking the G derivative of (60) r r r r  r r   R  R    B G B G dt result into i iq io Bi  iq  G Bi iq io ubstitution of this equation (59):Bi  iqtransform:  dt R G Fi  m i i R G rio  BGi F  m i BGi rio dtthe remaining  dt  changing Bdtframe position vector of q and

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G G Bi G Bi G Bi theBGiexpression frame of the leadsG to the  resultantGG as before d (59): r(61) G G R Greduces Fi  mto m i BGi rio sult riq GGequation rioR GG  Bdi(59):  Bi riqG riq GGGrioB For G rGGioBi case,  riqequation  G rio (60) (65) the equations: i R G following io  G F  riq into riqBi planar Rfollowing:  G rioequation  this Gresult  Gthe ubstitution of Bi G into iq Bi i dt  dt 

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(65)

G G G G G G G ForG the planar case, equation (65) reduces to the following equations:   (61) riq expression rGio  GGGBof this G G rio G G using G riq  G rio  G  B GG B i  G riq  For the planar case, under equation equation can be obtained by second equation he corresponding Bi iframe i  (61) (65) reduces to the following  r  r  r  r  riq  G riothe G iq G io G Bi iq io G Bi  G Bi  ult into equation (59): ), replacing the cross products using the antisymmetric form of the angular velocity vectors, using the equations: st equation underof(9) and applying kinematic to theequation resultant.under frame expression this equation canabe obtainedtransformation by using the second G G G G G G of equation can obtained by using the second equation under heGcorresponding Bi frame riq  GG rio using riq  expression rio  GG Bi form  this rio be G  Bi the G of Bi  iq  s Gproducts antisymmetric the rangular velocity vectors, using(61) the ), replacing the cross products using the antisymmetric form of the angular velocity vectors, using the Bi Bi transformation Bi G Bto G Bi Bi Bi Bi i 9) and applying a kinematic the resultant.   (62) R Ga kinematic R Bi G  Bi transformation R Bi riq  GtoBthe  resultant. riq G rand iq  applying G rio  G  Bi  rst equation under (9) i ame expression of this equation can be obtained by using the secondwww.collisionmagazine.com equation under Collision Magazine - Volume 14 Issue 2 25 Bi Bi G Bi G Bi i i products ofBBGithe   BGi rio using (62) Bi angular  makes R G the RBantisymmetric  following RBiBi Bform riqG [4]:   GBGi  Bi Bivelocity riq Bi vectors, Bi using B i the G riqequation G the BBi i  iBi of his use  (62) R G R Bi G  Bi  R Bi riq  G Bi  G  Bi  riq G riq  G rio  and applying a kinematic transformation to the resultant.

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G G R Bi BGi  Bi BGi  Bi  Bi R G GG  Bi use of the Volume following [4]: GB B i  14 Issue 2.indd 25the Gfollowing Bi Bi makes G use Bi of Bi Bi Bi his equation [4]: i riq    r  R R  R r  riq G io G Bi G B i Bi iq G  Bi  G  Bi 

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10/9/2020 3:43:04 PM

(65)

uniaxial form that is shown by equation (48). The previous (66)two equations reduce to the CRASH3 uniaxial form (66) B B  F  m r when the centripetal acceleration component is discountG iy G io  y ed and the body frame acceleration along one axis and the B B  G Fi  x  m G rio  x corresponding moment arm component about the other (66) dynamics is given by For the planar case and with a central body B Bdynamics rotational is given by equation Theequation Bi frameB(11).

B G  F  m r   G i y io y   x Fi  x  m uces to a single non-zero G rioand (11). valued For theGrelationship: planar case with a central body frame, planar body frame principle axis are zero valued.

Bi G

Fi  xi  m BGi rio  xi

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(66)

B F y  m this equation reduces to ar single non-zero relation m BG  rio  x G Fi  x valued Planar collision B B B given B by iequation B me rotational dynamics is (11).

For the planar case and with a central bodyanalysis (66) I z ship: riq  x  G Fi  x riq  y (67) G B z z  G Fi  y B B  equation reduces to a single non-zero valued relationship: G Fi  y  m G rio  y B accident reconstruction literature contains numerous   x with The Fi  x  m rotational dynamics is given by equation (11). For theBGplanar case a central body G rioand he radius of gyration and the equation explicit and implicit references to the(66) term planar collision B B relationships B B from B quation reduces to a single non-zero valued relationship: BB (66): B B rio  y(67) rotational I z z dynamics  G Fi by riqis xgiven (67) Friiqyy  m G(11).

The BB For the planar case and with a central body iFframe Gm B  z G Fi  y x Gequation  r G i x G io  x analysis.

These references refer to a particular methodolog(66) frame, equation reduces toBBa singleB non-zero valued relationship: B this B B B BB B B ical approach, r rIiqyz z x  GGFriioyx rriqiqxy  G Fi  x riq  y m (67)all of which share certain commonalities.

G Fi  y G GrB io yzG io  rotational dynamics isradius givenofby equation (11).

For the planar case and with aascentral efinition ofBThe the Bradius of gyration and the relationships from equation i frame (68) of the gyration and the (66): relaThese can be enumerated follows:body G  B  z  Using the definition 2 B B B B B B B  I  F r  F r (67) frame, this equation reduces to a single non-zero valued relationship:  G B  z z z G i  y iq  x G i  x iq  y   tionships from (66): zequation z mics is given equation (11).

For the planar case and with a central body finition of theby radius of gyration and the relationships from equation (66): 1. The collision problem under consideration is modeled  BGrio y BGB FriiqxBx mBGBrGiorBioxxB riq  yB  B o a single non-zero valued relationship: B B B 2 (66)    I  F r  F riq as ). This eliminates (67) the terms along, but planar (68) (R the definition of(64) theand radius the from (66): G under B z Gof Bgyration  zfor z z and i  y relationships iq  x G i  xof yequation B solving B2 expression the lt into the Using two equations Bi Gframe  the B m B B G Fi  yB  G rioB y     r r r r  z z  G io y iq  x G io x iq  y  not about, the E3 unit vector of the G frame and the B theBfollowing: I z z  BG Fi ByG B Briq z x  BG Fi  x B riq  y (67)B (68) (68) G B z 2 B B B unit vector e B   Using the definition of the radius gyration and equation (66):of each B frame. The corresponding rio  y theriqrelationships riqfrom  zof  x  G rio  x y  3  B z  (11). G frame rotational dynamics is given by equation For the planar case and with a central body   (68)each z axis. These B B under B(64) frame expression of the nof this result Binto the and solving for the B B two B 2 equations coordinate axes are the Z axis and i G riqnon-zero riq  y valued rio  x riqto ya single  2  B riq  x 2 rioB y zrelationship: 1 x G G B this equation reduces of gyration and the relationships from equation (66):        nius at q yields the following: (69) z z B B B B axes are treated as being mutually coincident during  B   G rB  z  r B  r 2B 2 B B 2 riq  y  F xB two Gthe into  riq m G io  equations yfor  iq  xriq G io  x expression G of the of under (64) solving the B  result the two under G rio x equations  yi Substitution iyframe rio  x ofB this Br G ioand x  z z  into    this    B result  zz the collision event.

(68) G BB  z  B B B B 2 (66) BB  B accelera- B B B F (64) B Bfor 2the at q yields the following: Fiirframe rBequations riqthe (67)expression of the B and solving expression of B B m r G B  zB r I zinto z  GB y iqr  x  GBFi  xunder  y (64) rGio  yi 1yB riqof rresult   r  G io y frame Substitution this the two and solving for the B             2 r r r r i z z G x G io x iq y    iq  x iq  y G iq  x G io  x iq  y G io  y B  BG   (68) B 2.iq  x An (69) tionsolution the following: analytical approach is used  Bin zterms  qB yields   of B   expressed  impulse-momentum G riq  G B  z   in B the 2center 2for 2 acceleration at2at written ofB the thefollowing: B q yields B Bmass acceleration    r r r r riqthe B B B B B 2 B y   rG iq  y   z z  rGz zio y r iq  x  r  zrz    rG io  x       and coupled with certain constraint 2 riq  x for the B frame expression of the conditions.

1 radius the of the of from (66): iq  y and  x relationships G io iq  y into  x Substitution  x gyration G io  yunder micsdefinition is iqgiven by equation (11).

For theand planar with a central body B equation ofGthis result theiqthe twocase equations (64) and solving i

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i i i i i i i riq   Bi  (69)  B BBi   BGi i Bi  zBi i  Bi B i  2 2  Bi Bi Bi r Bi 2 Bi 2 Bi   to a single non-zero valued relationship: r r i riq  xi riq yi riq3.Gi yriioThe  Gyiriq  xiqithe xi following:  G iq  yiacceleration  2  riq  x  zi zi Bi at Gq ioyields   1 zi ziGrio  xi  iGrriqioyxi i  yi Bi impulse due to the collision, internal to the coli  r      B B B B frame expression of the thecan two equations under (64) and solving for the B (69) 9) be rewritten in terms of the solution for the center of mass acceleration expressed in the 2 i         i i i i G G Bi  z i Bi Bi Bi 2 Bi 2 Bi iq Bi B iB Bi 2r Bi   Bi r  G rBioi r xi 2riq  yi Bi   Gr iq xi rio G r r r  i io  yi r    r r r B lision partner system, is the sole impulse considered.   G iq  y  y iq  x iq  y       G io  x x G iq  yi Bi iq iq  y  i zBi zi (67) B i  i  i  B owing: BGi iq  xi Bi I iq  xi Bi F zBiizri GiB (68) Bi B 2 Bi BGziii F riqzyi zii i BBii i i i 2i G B i  zi zi zi G i  yi iq xii  i  xi r r       2 r r r r r 1 i iq x iq y   ) can in terms of Gthe for the Gcenter acceleration expressed inBi the iq  solution xi yi iq  xi G io  yi Bi  be rewritten i 2 ioz zxi ofiq mass Bi  (69) time step event.

(69)2 i 4.  collision  as a single G Bi  zi is modeled Bi 2   Bi r Bi riq 2xi 2 GriqBBi  riq yBii2Br Bi zi  2  Bi  i i Bi The   z B Bi x    r r r r Bi Bi Bix  2 iq B i iiq  y  i Bi i2 i i   G G io y iq iq   yi     G io x    iB i i   for the i      Bi (69) Bcan 2 rewritten Bi  B i the Bi r r        Equation be in terms of solution center of mass acceleration expressed in the 2 r r r (70)  i riq  i z z z z 1  of gyration    dius i i i G io  xi iqand  y io  y G Bri  ziqi  xi Bi ri the riq 2xxii Biq  yi zi Gzfrom riq(66): Momentum transfer occurs instantaneously.

A conseii equation relationships x iq  y i  i riqBGi y i i riq  yi  (69) Bi B  G BG 2 iq xBi i the    B i  z i i  Bi  tution Bi this 2into BBi i r 2i   BG2i rof   i B frame. r   r r i frame expression of the result two equations under (64) and solving for the B i r r r  y iq iq G iq r yi  x  zi ziBi G iq  xi G2 iqxxi iq  yi i quence of this is that the pre-impact and post-impact io  y iq  y    io 2 zi zi B i  B i  B i i   Bi2 Bi  B2 i i i zi zi Bi rBiqi  yzi zBi of i q yields  rcan  Bi i 2BGiterms riq the riq  yi   for the center ration following: r2iqBi rewritten solution mass acceleration in the   at  Equation   BGBi i rthe iq  x(69) iq  xi be xi 2  rof xBi ziBzii r2iq  yin i 2 Bi B differ butexpressed the position and orientation of  i i   iq  xvelocities G rio  xi B  i i i r r  r r rio  BiB B frame.

Griq ioxyi  iq rxiq xiz z iq  yi 2   Gi B  z (70)  i  y G io     Bi 2 Bi Bi  i i  Bi 2 i i i Bii(68) Bi Bi   i  G  2 i    Bxii Bi riq  xBi i  Bi zi zi   G riq  yi riqBixi riqriq yeach rB  zyii   Bi collision partner (during the collision event) do 2 22 BiBi i Bi i yi G r2iq   yi Bfor in terms solution the center mass in the  rziqiBzof 2 Gzriacceleration riqBixri riqexpressed Briiqr  BGi rGio ioxi of  Bthe 2  riq  x Bi   i  ii y i ziB ixin  Griq iq i x iqzy(69) ri iq terms iBGi rcan yBiGiriosolution  (70) Bi  Equation be rewritten of the for 2  r r r i not. i  G  1 z x iq y   iq x io x iq y iq x     y  B B i i i i    i i i i   i i2    BGii riq      Bi Bi 2 2 Bi 2 2 GBi B   (69)Bi i  zi 2   B2i2  BB G B z  Bwhich B B 2 B  2 r r eo last term, derives from the centripetal acceleration, the biaxial forms of i i B B B B B   i i i i i r  r  of the 2 x Bi i iniq Bi frame. i i B iq irBri iqr x2 i rBi Bi r expressed BGi r  iqBi yri  riq y Bi Bcenter rio Gmass 2  riq  x   y zi zi B i r r2i iozyzBi acceleration ri ioB  G io  yi  Griq Giyri iqthe i r i  i ii  r r  xi   r  G x        yiB i  iq y G iq x iq x iq y i i            G iq x iq x z z G iq y iq x iq y   z z iq x iq y ii i i i i ii i i   i zii zi (70) zi zi iBi thei corresponding i i   that is i shown  Bi isi resolved  are more complicated uniaxial form by 5.

The collision at a single fixed point G Bi  zimpulse G rio than   i 2 expression of o the Btwo equations under (64) and solving for B  riq  yi  r Bi Bi the Bi Bi B2i the Bi 2 i frame 2 2     G io y   2 Bi BBi  Bi i  B Bi       r r    r r r i B 2 2 B 2 B 2 B i i i two i i  yi iq  y Githe xicentripetal iq  xi iqalong llowing: common Bi collision interface at the point of rvious riq  xi equations   zi zi reduce  GrriqBiqito riq  xCRASH3 ri iq Gyriiqiqyiyizuniaxial riq  xi form  zii zriiqwhen iqzi(70) iq  xi yi the 2  riq  x   the yi zi i zi of Grxioi  xi Bi Bi i dropping term, which derives from the centripetal acceleration, the biaxial forms of canthe belast rewritten in terms of the solution for the center mass acceleration expressed in the  the tonis(69) discounted and body frame acceleration along one axis the 2 and (70)contemporaneous    maximum engagement  (i.e. at the    G B z G rio   Bi   B B B B 2 2 2 i i i i i i 2 Bi 2 Bi 2 Bi  r r r r Bi Bcorresponding B i  z z form Bi Bi 2  x B B i i i     iq iq y iq y   G io y 69) and (70) are more complicated than the uniaxial that is shown by r  r    me. i i i 2  i  i  i     zi zi Biplanar   arm component frame principle axis are zero valued.   BGbody  r r r r r    r iq2 yithe terminus of closure and start of separation phases).

Bi other Bi iq  xi Bi about B zi zi iiq  xi iq  xi iq  xi iq  yi Biqi  yi i iq  y i riq  xderives riqreduce   xi rterm,   ythe  which  centripetal 2 acceleration, riq  xi Gform ropping the biaxial forms of (70)   y i from i the G rio last iq  yi G rioto B 8).1 Thethe previous two equations CRASH3 when the centripetal i G Buniaxial i Bi  z i i 2    2(69) Bi  Bi 2 are  G i BiBri rthe corresponding  2 iq  yi 2 Bi Bi  zi2  i  BiBrriuniaxial  Bi along andBi2r(70) complicated form that shown by rBiqi component  Bi rB i Bthe riq22than r  Bmore riqdiscounted n9) the body frame one the last  yi x 2 which iq z xi  BG 6. isaxis There is no explicit force-deflection i io  yi iswith i zxi i  dropping  iqriqfrom  the io acceleration, the biaxial forms of or force-crush reBi riq derives B By  zi zicentripetal  xiacceleration yzi analysis  iiqterm,  and i   GyEven Bi r G iqiBxi and i  iiyi  zi zi  on  r    r r r i zi      G iq x iq x z z G iq y iq x iq y ).ingThe previous two equations reduce to the CRASH3 uniaxial form when the centripetal i i i i i i i (69) corresponding moment arm component about the other planar body frame principle axis are sponse zero valued.

2 model employed in the formulation. and (70) are more complicated than the uniaxial form that is shown by  Bi Bi equations Bi 2 2 tion literature explicit and implicit references to along the planar zand riq  xisicontains discounted riq ynumerous  term 2 component the 2body frame acceleration one axis and the i zi i Bi Bi  equations 2 Bi Bi B equation (48).

The previous two reduce to the CRASH3 uniaxial form when the centripetal i riqlast of rplanar ziderives with the term, from the centripetal the biaxial forms of Even io component  a particular  dropping  share  acceleration,  en refer to methodological approach, all of which certain riqvalued.

2zero  xother iq  ywhich zi frame in of the center mass expressed in i i acceleration G rthe xi solution for  xi greferences moment arm about the body principle B i theare Bterms i The presentation is forand a system comprised of rio  acceleration (70)  axis  (69) component is more discounted and the body frame one axis the corresponding  Btypical  along  as anar analysis G Bi acceleration  zi Bi equations and (70) are complicated than the i uniaxial form that is shown by an beGcollision enumerated follows: 2 rio  yi  from riq  yi  partners and a single collision event. This Bi Bi Bi Bi Bi Bi 2  corresponding two collision Gderives  term, which the centripetal acceleration, the biaxial forms of     r r    r r r component the planar body frame principle axis are zero valued.

G iqmoment contains yi iq  yarm zi zi equations Gabout iq and xi reduce iq  xother iq references ythe nt reconstruction literature explicit implicit to the term planar equation (48). previous two to uniaxial form when the centripetal i numerous i i  theThe  CRASH3 problem involves four relevant frames of reference. These more complicated than corresponding uniaxial form that is shown bywhich nar collision analysis 2 2 alysis.

These references refer to a particular methodological approach, all of share certain acceleration component is discounted and the body frame acceleration along one axis and the 2 roblem the terms   BBi i Bi planar 2 Bi as 2 Bi(ℝ ). This eliminates Biunder 2 consideration Bi Bi is modeled  i  r  r   frames are the G frame (OXYZ), the two two equations reduce to the CRASH3 uniaxial form when the centripetal r r    r r r reconstruction literature contains numerous explicit and to the term principle planar axis are zero valued. body frames B1 y iq zi zi implicit references i of corresponding G iq  xi These xcan zi z G iq as iqxxi i component iq  yiq B frame. ties. bevector enumerated follows: i  i e unit moment arm the other planar body frame about, theiqE the Gyi frame and the vector of each The 2.4 Planar collision analysis 3i unit 3 about discounted and the body frame acceleration along one axis and the (o x y z ) and B2 (o2x2y2z2) and the normal-tangent frame, ysis.

TheseB references refer to a particular methodological approach, all of which  contains 2 1share 1 1 1certain coordinate axis and each z axis.

These axes asand being 2 are Bi the 2Z B The accident reconstruction literature numerous explicit implicit references to the term planar i axes i Bi are treated r  r     omponent about other body frame principle axis valued. riqzero 2 are iq  xi the iq  yi planar zi zifollows: es.

These can be enumerated as Nthe (qntk).

Both bodyofframes are central frames. The point  xi 2acceleration, B ithe with dropping the last term, which derives from centripetal biaxial forms e collision problem under consideration is modeled as planar (ℝ ).

This eliminates the terms ident during the collision event. collision analysis.

These references refer to a particular methodological approach, all of which share certain (70)   2.4 Planar collision analysis   Bi  G Bi  z i 2 q is the point at which the collision impulse is resolved and r B B B B B 2 ong, about, the Eapproach G frame and the of each frame.

The by ons (69)i not (70)i are more corresponding uniaxial form Bthat is shown iomentum i complicated i i 3 unit  vector yi  certain analytical is used and coupled with constraint  the commonalities. can bethe enumerated as follows:  e3 iqunit  but vector riqand  These riqof riqthan contains numerous explicit and implicit references to the term planar The  xiliterature  yi G riq  yi yi accident zi zi reconstruction G riq  xi 2). This collision problem under consideration is modeled as planar (ℝ eliminates the terms represents the origin of the N frame.

The Z, z1, z2 and k  rresponding coordinate axesequations are the Z reduce axis and z axis. These axesform are treated as being alysis on (48). The previous two toeach the uniaxial when the centripetal collision analysis.

These2 references refer to aCRASH3 particular methodological approach, all of which share certain  B Bi E3 unit g, but not about, and the e unit vector of each B frame.

The 2 the 2 Bivector of the G frame 3 i axes remain coincident during the collision event.

2). This ration component isriq discounted and the frame acceleration along one (ℝ axis and eliminates the utually the event. erature coincident contains numerous explicit and implicit references toderives theisterm planar collision riq  x1. during The under consideration modeled as planar the terms ycollision zi zi problem Even dropping the lastbody term, which from the commonalities.

These can be enumerated as follows: i i with esponding coordinate axes are the Z axis and each z axis.

These axes are treated as being nponding impulse-momentum analytical approach used with constraint arm component about the E other planar body principle axis zero valued. nces refermoment to a particular methodological approach, all ofand which share certain along, but not about, the unit vector of coupled theframe G frame andcertain the e3 are unit vector of each B frame. The 3is centripetal acceleration, ually coincident during the collision event.the biaxial forms of equations (69) In terms of the derivation, it is useful to first consider the nditions. enumerated as follows: coordinate axes are thethe Z biaxial axis and eachas axis. These are treatedthe as being 1. corresponding The collision problem under consideration is modeled (ℝ2). axes This eliminates andanalytical (70)the are centripetal more complicated than corresponding term, which derives from acceleration, forms ofz planar polar constraint representation of the positionterms vector of q. The body impulse-momentum approach is used andthe coupled with certain mutually coincident during the collision event.

Planar collision analysis along, but not about, the E unit vector of the G frame and the e unit vector of each B frame.

The 3 3 more complicated than the corresponding uniaxial form that is shown by ditions.

2). This eliminates the terms mcident underreconstruction consideration is modeled as planar (ℝ 2.

An impulse-momentum analytical approach is used and coupled with certain constraint literature contains numerous explicit and implicit references to the term planar corresponding coordinate axesform are the Z axis each z axis. These axes are treated as being s two equations reduce to the CRASH3 uniaxial when the and centripetal theanalysis.

E3 unit vector of G frame and e143 unit vector of each B frame.

Theall of which share certain 26theCollision Issue 2the www.collisionmagazine.com conditions. on refer to- Volume athe particular methodological approach, mutually coincident during collision event. discounted These and references the body Magazine frame acceleration along one axis and the nate axes These are the ZAn axis and each zasaxis.

These axes are treated as being onalities. be enumerated 2.can impulse-momentum approach used and coupled with certain constraint component about the other planar bodyfollows: frameanalytical principle axis are zeroisvalued. during the collision conditions. event. umThe analytical approach is used and coupled with certain constraint under consideration is modeled as planar (ℝ2). This eliminates the terms Volume 14 problem Issue 2.indd 26 10/9/2020 3:43:04 PM nalysiscollision along, but not about, the E3 unit vector of the G frame and the e3 unit vector of each B frame. The

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as a single time stepcontemporaneous event. Momentumterminus transfer occurs instantaneously. agement (i.e. at the of closure and start of 3.pre-impact The impulse impulse due to theatcollision, internal theposition collision partner system, is theatsole The collision resolved a single fixed point along the common collision interface the impulse that the andispost-impact velocities differ buttothe and he typical presentation is for a system comprised of two collision partners and a single collision considered. point of maximum (i.e. do atemployed the contemporaneous terminus of closure and start ofevent. ion partner (during theengagement collision event) not.

-deflection or force-crush response model in the formulation. his problem involves four relevant frames of reference.

These frames are the G frame (OXYZ), the two 4.at aThe collision is modeled a common single time step event. transfer occurs instantaneously. separation phases). eresolved collision, internal to the collision partner system, is the soleMomentum impulse single fixed point alongas the collision interface at the (o x y z ) and B (o x y z ) and the normal-tangent frame, N (qntk).

Both body frames areand ody frames B 1 1 force-deflection 1 2 2is 2that 2or 2 the A 1consequence of this pre-impact and post-impact differ but the position Therecomprised is(i.e. no explicit force-crush response model employed in the formulation. agement at 1the terminus closure and start ofvelocities system ofcontemporaneous two collision partners and internal aofsingle collision event.

3. orientation The impulse due to the collision, to the collision partner system, is the sole impulse entral frames.

The point q is the point at which the collision impulse is resolved and represents the origin of eachMomentum collision partner (during collision event) do not. as aframes single of time step event. transfer occursthe instantaneously. ant reference.

These frames are the G frame (OXYZ), the two considered. fpical the N frame.

The z1,impulse and kisaxes remain coincident during the collision 5.

The collision resolved atof adiffer single point along the common collision event. interface at the presentation isZ,and for az2response system comprised twoBoth collision partners and a event. single collision that the pre-impact post-impact velocities but the position and -deflection or the force-crush model employed infixed the formulation. o normal-tangent frame, Nsingle (qntk). body frames are 2x2y2z2) and 4. point Thefour collision is modeled as areference. time step event.

Momentum transfer occurs instantaneously. of maximum engagement (i.e. at the contemporaneous terminus of closure and start of roblem involves relevant frames of These frames are the G frame (OXYZ), the two ion partner (during the collision event) do not. point at which the collision impulse isthat resolved and represents thereadily origin bevelocities differ but the frame expressions of this position vector can A consequence of this is the pre-impact and post-impact position nresolved terms of the derivation, it is useful to first consider the polar representation of the position vector of q.  and separation phases). (o x y z ) and B (o x y z ) and the normal-tangent frame, N (qntk).

Both body frames rames B  cos   sin 0 1 1 1 1 1 2 2 2 2 2    cos     sin    0    are at a single fixed along the common interfaceevent. at the system comprised of twopoint collision and acollision single collision k axes remain coincident during thepartners collision event. written in coordinate form as: orientation of each collision partner (during the collision event) do not. he body frame expressions of this position vector can readily be written in coordinate form as:     N G 6.

There is no explicit force-deflection or force-crush response model employed in the formulation. frames.

The point q is the point at which the collision impulse is resolved and represents the origin agement (i.e. the contemporaneous terminus of closure and the starttwo of ant frames of at reference.

These frames are the G frame (OXYZ), R G    sin     cos    0  R N    sin     cos    0  5.TheThe collision impulse is resolved at a single fixed point along the common collision interface at the N frame.

Z, z , z and k axes remain coincident during the collision event.

2 (o2x2yto the 1normal-tangent frame, N (qntk).

Both bodyvector framesofare 2zfirst 2) and seful consider polar representation of the q.is the impulse 0 0 0 1  T T sole e impulse due totothe internal the collision system, sole impulse point of maximum engagement (i.e. atposition contemporaneous of closure and0 event. start 1of he impulse due thethe collision, internal the collision partner system, is terminus the   B 2 the B2collision B 2partners Bcollision, Bfor B1 to to The typical presentation is a system comprised ofpartner two 1 1    e-deflection or force-crush response model employed in the formulation. r 0 r  r r2q  y 2 0 and a single collision(71) r  r point at which the collision impulse is resolved and represents the is position vector can readily be written in form as: 1q 1q  x1 1q coordinate y1 2q 2q  x2 origin (76)    nsidered. separation nsidered.

This problem involves four relevant framesevent. ofthe reference. frames G frame the ms of theremain derivation, it is phases). useful tothe first consider polar representation position    sin  of are 0the cos(OXYZ),  sin   cos  the  vector These (71)   of q.   two0  axes coincident during collision ek collision is modeled as a single time step event.

Momentum transfer occurs instantaneously.

6.

There is no explicit force-deflection or force-crush response model employed in the formulation. he collision is modeled as a single time step event.

Momentum transfer occurs instantaneously.

The N frame expression of the G frame expression of q can be obtained b    y1this z1)collision and B2 (o xB 2y2z2)can and normal-tangent frame, N (qntk).

Both body frames are of the position vector body frames B1T(oof Nreadily Gform as: 1x1of 2vector dy frame expressions position written in coordinate Tthe system two partners and aGsingle collision event.

(76)   cos     sin    B B2 vector, B1 comprised R   be sin   expression differ cos but 0frame, Ris:  sinand    cos    0  sin    0       2position 2 and post-impact he magnitude of each being independent of the  cos consequence of this is that the pre-impact velocities the position  N  and     (71) r  r 0 r r 0 consequence of this is that the pre-impact and post-impact velocities differ but the position the appropriate kinematic transform.

1q central x1 1qframes.  y1 2q 2qthe  x2 point 2qat  y 2 which point the collision impulse and represents the  Thethe q isrepresentation  the vant frames of reference.

These frames are the frame (OXYZ), theof seful to first consider polar ofG position vector q.is resolved 0 origin1  event) entation of partner (during thethe collision event) not.

0two 1T  0 N Rcollision The typical presentation is for a system of 0do two collision partners and a single event. ientation ofeach eachcollision collision partner (during collision do not.the T comprised     cos    0  G R N    sin     cos    G    sin of the N frame.

The Z, k axes remain during collision event.

B1 normal-tangent B1 z1, z2 and B1frame, B 2 coincident B2 body B2 (o x y z ) and the N (qntk).

Both frames are 2position 2 2 2 impulse iscollision vector can readily be written in coordinate form as:     (71) r  r r 0 r  r r 0 e is resolved at a single fixed point along the common collision interface at the B 2 B 2 B 2 B 2  x1 atrelevant  y1  x22 common 1q 1qfour 2q frames 1 N  d frame,  2qthe  are interface he collision impulse is resolved a1 single along at the This problem involves frames of2qis:reference. the G frame (OXYZ), (72)  expression r1q fixed r1q point d r2qThese  y 22 rcollision 1   i  i   0  cos  i 0 i   0 two  sin   0 0   cos   cos ector, being independent of1q the 1qis  x1resolved  yand 2q   x2 origin 2q y 2riq  n  the 1 point maximum at which the collision impulse represents the int engagement (i.e. at the contemporaneous terminus of frame, closure and startstart of ofbody The magnitude of each position vector, being independent   (o x y z ) and B (o x y z ) and the normal-tangent N (qntk).

Both frames are body frames B intInof of maximum engagement (i.e. at the contemporaneous terminus of closure and      1 1 1 The 1 1 it 2 2 to 2 2first 2 consider N frame expression of the G frame expression of the N N ofThe N Gposition N frame expression of the G frame expression the position vector of q can be obtained by applying terms of the derivation, is useful the polar representation of the vector of q.

T T k axesBremain coincident during theframe, collision event. riq(71)   riq  t   R G riq  d iq   sin     cos    0   sin  i  i    diq  sin  i  i   B2 B2 B2 1 phases). paration of the expression is:    r 0 r  r r 0 agnitude of each position vector, being independent of the expression frame, is: paration phases). central frames.

The point q is the point at which the collision impulse is resolved and represents the origin coordinate cos   vector  form sin of as: 0  be obtained  sin  the   cosby   y 22 vector 1qBThe x1 2q the 2q this x22 2qkinematic of   applying  0apposition q can 2 expressions  appropriate  transform. body of position can readily written informulation.

2 1q the By11frame B 2 polar angle position from the origin ofbe coordinates of the B point i 1denotes The N frame expression the vector of0q can i frame to the  ere is no explicit force-deflection force-crush response model employed in the (72) explicit d 2q Z,ofzB2the rz or force-crush r2qvector 0 the 1   of the 0 position  0 GG frame 0expression  xthe 1q  y 1 force-deflection xor  y 2 remain N  collision  transform.

1 2 sin of no N rframe.

The k axes coincident the event. herer1q is response modelduring employed in formulation.

1, 2q 2 and propriate kinematic (76 R     cos  0 R   sin   cos  0         seful to firstfrom consider polar of position the position vector ofpoint q. G appropriate N , measured the xthe thenrepresentation the body frame vector of the q can be written as: transform. the kinematic i axis,    2 B2 2 B1 2 B1 2 T B2 T N vector, being independent of the expression frame, is: B B B B B B (72)  1 r comprised rbe r1qin r2q   2 rrcos d 2q 0 form 2   00  0event. sin   cos0 i  i1  cos  i     2 r2qcollision 0 i     0 1  his position vector readily 1q 1q  written x11 r two y1r 2qa  ysingle presentation is forcan ad system of (71) riq1q ncollision r2qxas: 1coordinate 2  and  x22  y2 1q  of 1q  x1 y 1 the 2q Bi partners the ofT last position vector from origin coordinates of frame to point The form on the right of the equality vector makes use function is an eve In terms ofisthe it is first consider the polar representation position of presentation forderivation, a the system comprised a single collision of  collision  Gand the  event.   q.of the fact that  the cosine Nuseful Ntotwo G partners (77) TN frames Nsin    B1 B2 m involves four relevant frames of reference.

These are the frame (OXYZ), the two (77) r r r R d sin cos 0 d sin              (72)  sin coordinate 0the function  i as: cos 0   cos Equation i  i   (77), function, t 0 iqsin  iq G r iq d iq cos i   (73)iq  i i    r  d 1qvector  Bcos  of of vector G in   sine These =  riq two  sin   = -sin().     n n the body frame position point be written as: The frame this position can readily be form 1 2the 1 2q can 2q 2q 2written 2that em relevant frames of reference. frames are the frame (OXYZ), the cos(-) cos(), and is an odd sin(-)  B1 involves 2body B1four 2T 1qexpressions B T 2 N cos sin  body  cos  sin    0  of the   0(72)  00BGNiframes    of B B22 r from  Nare 0point (othe x111ry B2 (o y the normal-tangent frame, (qntk).

Both s B1r1q  obtained    d r  1xB 1z1)rand 2 2zposition 2)Band 22x 2 N G the position     notes polar angle of the vector the origin coordinates the frame to 1q  y 2q 2q  x 2q  y   0 1 0 The N frame expression of expression vector of q can be (71) res 0 r r r 0 1 2 2 The ofB2each independent ofof expression frame, riqpartners,  is:  R G riq  d iq   sin     cos    0   sin  i  by i  applyin  diq   y2 1q B x11 (omagnitude 2q  xvector,  (qntk).   riq tare y1zy11) and (o2q y2z2) and Nthe Both body the two collision is: 2qbeing impulse each  the 1x11q 2x2position 2 the normal-tangent  the  N R frame, G frames  mes.

The point q is point at which the collision is resolved and represents origin (76)   sin   cos  0 R   sin   cos  0  kinematic  and represents  be written     suredThe from theqxiisTaxis, then Bthe body frame position vector ofresolved the point q can N T as: the appropriate transform.

TGT Bis B of these   B1 the B1 B2 mes. point the collision impulse 0  (i i and  (71)  0 0 1   0 2 he frame position obtained by applying a kinematic  coincident  B2 r2q  origin (73) 1 G sinZ, 1zexpression r2q  remain d1at cos r1q2 vectors r2q1qwhich r1q2is 0  collision r2q0  event. r2q1y 2 transform 0 the and k2 point axes the The eme.     1, z0 2 y 1 0   x20 x1 sin during    0    0 0 1is  cos   sin   cos   sin  0            B1the right 2 Bof 2 last Bthe 2 Buse 2 position vector from the origin of coordinates the B frame to point N N function, i   1of 2 2 ame.

The Z, z , z and k axes remain coincident during the collision event.

The form on the equality makes of the fact that the cosine function an even 1 2 vector, being independent of the expression frame, is: re constant valued for thedenotes evaluation).

If i the position d 1q the  polar r1q  xangle  rof d 2q  vector r2q  x2 from  r2q  y 2T r1q  n      cos  2  2     1 T  cos  1  1 (72)  G r2q  n B1 B 2 1q  y 1 N n the body frame position vector of the q that can be as: 2Nof  position cos 0cos i    0  riqpoint  sin i  R  of Rsin(-)  cos (73) r1qit  duseful 0and  of r2qrepresentation  written dsine cos the Nq, 0function, vector   sin  cos  icos   first  1 point       N N 0 (77), Nwritten cos(-) = cos(), the function is = -sin().

Equation for n an sin iN   sin 1q  1 of sin 2q  2odd G the origin coordinates the B frame to  cos  the derivation, is to consider the polar the q.  The right ofsin the that cosine G  on r1qthe  equality r2qof  dfact r1qform r2 q  use  1last  the i 1q  1      makes 2 q  sin 2  2    func  t   t the The magnitude of each position vector, being independent of expression frame, is:dof iNthe  the position vectors is obtained by applying a kinematic transform (  i and N G frame N position G the derivation, it is useful to first consider the polar representation of vector q.

The N frame expression of the expression of the position vector of q can be obtained by applying    0 cos    00 sin1i  i   diq 0sin  i  0i     1  (77 B1 expressions 2 B1 2the Bx 2 then ofvector the two collision partners, is: Rcos riqcos(), coordinate  form  d iqand  sinthat 2 ame of2 this can be written axis, the frame measured B2from rangle position cos readily  sin iT body 0 rin cos ri iqposition  iq  t(72) G as:            r1qthe d2qposition each rof r cos(-) = the sine function is an odd function, sin(-) = -sin().          E    i iq i i i X the vector from the origin of coordinates of the B frame to the point Ifr1q idenotes 0 0 0 0 x1   y 1 Tpolar 2q  x 2q  y i ation).

2 2 B2    as:   rame expressions ofvector this vector can readily written form the appropriate kinematic transform.  Gr2qposition vectors   in coordinate  of (73) the sin 0Grof axis, dthe then cos sin sin  0be position        1q, measured  1  from    of point q written as: frame expression these position isbe obtained by applying a0 of kinematic transform (  (74) 0 is: 1   0 0 2q 2  can 2 cos i0i and each the two collision partners,  r  d     0 sin   d sin    the x the body frame vector point q can be written as:        B 2 B 2 B 2 B 2 i   1 1 iq iq  Y  iq  i i i2  iq  2 i i N    T T N d 1q  r x1  B rr1q  y1  cos r2qx2 0 r2q  y 2 of  sin 02   of B for the B Bd 2q  y The  G frame  expression (71) rcos  cos  the position vector of q can be obtai nstant valued 0sin 2(72)  B revaluation). 1 point r1q from 0  0T B1qr2q   the r2qnB 1q 1 2q the n the r2q Bcos N frame ofexpression  x 00 1q 1q  yof x iframe 1The 0two equations T e position origin to0the N of icoordinates   cos  vector  Brcos  Bcos         the under (78) are the moment components that NNi 0 G B B2 i        i  i  arm i B1  sin i 1  1 2 2 components N N N  cos  r  cos   sin  0 cos     distance              Br11q  x1 by  r r2q iqThe G transform (76) rtT2qxB2sin 2cos 1 the R i2 qcos (71) r1q is obtained r2q  sin  N duse nr i   0 R (78)  N ris d2 on 0of 0 lastkinematic 2that 1 the r2 qN   rTsin the transform. i 1q  y 1  0  a kinematic y position vectors ( and  sin  even  body  rapplying i1appropriate cos herein,      function  cos right equality makes function 1q q 1q 1q the 2q 2fact  form  t  r1q  2 can written n the frame position vector of the point be as: 1 cosine 2q an n  n of the (73) r  d  cos   0  r d  cos  sin  0   by and Brach [9] and with the preceding steps, showing how the terms are deriv             N d sin N N G2q Brach (74)          d sin cos 0 sin sin 1q 1q 1 1 2q 2 2       G     iq  If i denotes i i  ithe i polar angle of coordinates of frame to0the  an N  (77) being    cos i position rvector dand sin sin riqiqsin of 1i sine  Nthe00B0iis point i  =i -sin().   N Equation d10iq  sin      N  odd theriqvector, iqthe 0ifrom cos riqi  cos i cos R 0the origin  function ation).     Gcos(), iq 0 i  ifunction, t 0   = that the sin(-) (77), written ude of each position independent of expression frame, is:  icos(-) X    r  d sin       r  d sin r r as well2 qas its  fo preliminary note ispoint that the from DCM between Bi andN1 frames are:  vector 1  frame  1q trigonometry,  1q  tas: 2q  t  2q  2 he form of the makes and difference identities, (73)  the 1q  inverse  q, measured  the   use from  1last 0onGthe right 0 then 0of the equality Texpression sum N q can be the xiaxis, body position the G0  of  written   (74)   T r vector, 0 0 0 1 0 0 tude of each position being independent of the frame, is:   r  cos   sin  0 cos    cos       each of the two collision partners, is:   d  sin   cos  0 sin   d sin    r        B                i i  i  iq 0 iq  Y2 rtwo iq i i 0   i  iq (73) i i iq  n      0 0 0 which states that for any angles  and :  sin    d  cos  sin   The  1  expression  2  vectors  2  isobtained G frame these position by applying a kinematic (i iand    1 2q  G    B2q 0of  N  transform   N B    2 2  1the   cos  i  dsin  2 cos  cos N i 2TGB 0frame (78)N are frame expression the vector ofiqqcan applying riq0 of the riqposition R components sinbe are 0 shown sin diqsin0i  0  r1qThe obtained   cos by   i sin (72) irof r1q0i  y expression d 2qof 0 r2q  y under components theBtwo moment arm that G rdistance iq d i     t   cos   sin  0 cos      T 1q  i for  x  2q  xequations        are constant valued the evaluation).

B B i i i i 1 2  makes use of the Bsum  appropriate identities, N   N quality from trigonometry, 2 Br2 (73) d2 1qdifference Bform sin   2last  d 22q (74) cos 0 the 1and Bof  the 1 2 transform. use 0an The on the last equality makes use the 1q 1d kinematic 2q  1right the   of rB2qi  nderived.      sin  even 2cos 1cos The the right equality of0form fact that cosine function is  showing  1rthe cos diq   sin 01qand sin sin  of  function,  0 0 Nrsteps, (72)  i  isdcos 1qyon r dsin r02qiwith rpreceding by Brach Brach the herein, how are A 00 under 1q 2n  1q  i  r x 2q[9]  x2   2q sin   cos  iand these i cos the  ismakes The 1 1   2 R  sin i of1the two cos ithe   terms R N(78)  are  sin 2 0 2 cos  distance components equations arm    final   position obtained by applying aiqkinematic transform (Niysin i and  B imoment i      function, les  andvectors :   the   com i The G frame expression of position vectors obNis an odd N N even (75) fact that the cosine function is an cos(-β) cos(-) = cos(), and that the sine function function, sin(-) = -sin().

Equation (77), written for preliminary note is that the DCM between the B and N frames as well as its inverse are:      iframe (78 G  r  d sin       r  d sin      r r 0 0 1 0 0     srm the polar angle of the position vector from the origin of coordinates of the B to the point on the right of the last equality makes use of the sum and difference identities, from trigonometry, i 1q 1q t 1q 1 1 2 q 2q t 2 q 2 2   by Brach and Brach [9] and with the preceding steps, herein, showing how the    0  riqapplying  kinematic by  0 i    sin    0φcos ation). collision   i γare  =cos   i0 and cos sin cos 0 1  1  ter cos by i  sin i   Xcos   a N tained transform (θ and odd each of the two partners, cos(β), that the sine function is an function,  r  cos   sin  0 cos    cos     i is: i can       The G frame expression of these position vectors is obtained applying a kinematic transform (  and          i i       dsstates from the x axis, then the body frame position vector of the point q be written as:           iq n i i i i  i that for any two angles  and : G G 0 0 0 0 polar the vector from the of the B thethe note ispoint that DCM between the frames well as its inver right lastthe equality makes use of the as cosine is a i and  i of   Nthat  function frame  Bthe (74)   cos  cosof i preliminary on rvalued diq the 0  The diiqthe riq position  iorigin coordinates  to sinthe  constant sin  of  are  angle  cos sin constant evaluation). sinform i cos    written  fact iq Y i Nfor N N G sin  sin     Equation sin(-β) =-sin(β).

(77), eachsin(-) of the = -sin(). Equation valued forbody the evaluation). iqsin  =be cos (77) riq position riqthe ridentities, cos R  trigonometry, dpoint sin 0written   as: 0that sin dsin sin iodd  0ifunction,  for        d from the x axis, then the frame vector q (75) iof i can i  i an i iq t G i i iq  cos(-) cos(), and the sine function is quality makes use of the sum and difference from         cos  sin   0to sin 0cos 0cosits    the  T N B and  inverse  follow  the 0formT of dynamics  equations Npartners, he DCM from G frame the the 1translational 0 for planar case cos generates four for two collision partn i frame i Ni 0  the    equations cos  B is: B   i cos    cos i     and Nsin The under (20). rcos   10  two r2q  i Rcollision R  cos   1q n n is: 2  2    cos sin (79)   cos  that The sin cos 01the under partners, sin 0 distance (73) r1q: d 1q  cos  1  sin sin d   0cos 0iof 0collision 0 sin 2 the components the two 0cos  cos sin 0  (78) Br i  N 1 i  are i  0 1 2q  2   sineach    show  of two equations the moment arm components are    i2q  i i i  law (74) (74) frame expression is an obvious choice for translational dynamics.

Newton’s third ofsin mo          dles sin cos 0 sin d sin G                Ni  N N (75) iq  i i iq cos i T   cos   riq  X  i  T  B2N r1q iby  sin  0 cos                   B1 (78)  r  d sin     r  d sin      r B     N i i i i     i 1q t 1q 1 1 2 q 2q t 2 q 2 2   0 0 1 0 0 1 Brach and Brach [9] and with the preceding steps, herein, showing how the terms are derived.

A fina         (73) cos cos cos sin sin          r  d  cos  sin  0  r  d  cos  sin  0                    R   sin     cos    0 R   sin     cos       i that the G frame collision force, and therefore the impulse, is equal in magnitude and opposite     1q  0 G 1 1  2q 0 2q   2 2  form    of  the B i i i   difference   0 1q on its 01equality  trigonometry,  N N  the rightr of the makes sum e NThe follow form equations N ifrom  Gvectors the  of equality  and (74) is under transform DCM  i the rlast d sin cos 0and sin dofiqidentities, sin (20). the  frames as its  by use      0 0 0 0 sin     and sin cos cos sin     note is that the between B N as well inverse are: Y  iq   i i i r cos      r cos         frame   inverse iq The iqthe  obtained e expression of these position applying a kinematic (  and i       form on the right ofipreliminary the last makes use fact that the cosine function is an even function, i i    with respect(75) 1qpartners. n 1 1 2q  n 2 2 to the two collision 0 1 0 0 which states that for any two angles and :    0 0N N  (77),  N N odd function, 0 =sin 0 that the 0thethe 1   case t valued evaluation). cos(-) cos(), and sine function is0 the angenerates Equation written for ( rsin(-) d1q-sin(). sin      r  r  d sin       its r1q four   =    cos cos   for  the   cosposition the  sin  obtained CM from the G frame to Ntranslational frame and inverse follow form of equations under The dynamics for planar equations for two collision partners.

The G       1qand  t(20).

1 1 2 q 2q  t 2 q 2 2 e expression of these vectors is by applying a kinematic transform  i i        quality makes use of the sum and difference identities, from trigonometry, G G each ofexpression the two collision partners, is: for under cos iare   thedynamics. moment  Newton’s 0distance cos that   are states sin   0   translational   law translational  imotion  i equations The components of obvious the two equations (78) shown sin0 i  arm frame is  an choice the third for Pcomponents P2of 0 0 t valued for the evaluation).

1   case The dynamics the planar generates four sin    sin  cos   cos  sin    for0 two c              G gles  and : r  how  opposite Bi N that costhe ilast equality sinand i collision 0   cos i of cos i difference the            by Brach Brach [9] and with the preceding steps, herein, showing the terms are derived.

A final the G frame force, and therefore impulse, is equal in magnitude and in direction (75)  i iq X The form on the right of makes use the sum and identities, from trigonometry, (78) R  sin     cos    0 R   sin     cos    0  Newton’s(79         e N frame and form of the equations Bi i N i i frame is an obvious for dynamics. th  choice  the translational  G its inverse  follow thecos  under sin(20).   iexpression G cos  the note  is dcos sin N    cos N     preliminary that the DCM between the B and N frames as well as its inverse are: G (74) riq  states r  d  sin   cos 0 sin   sin   with respect to two collision partners. i            r     r cos      which that for any two angles  and :            i   0 iq equation, i  i Y   iq iq    cos i   This using equation (13), can be expressed as the following:  1q n i  1 1 2q  n 2 2  0 0 1 0 0 1    r  sin cos cos        that the G frame collision force, and therefore the impulse, is equal in magnitude    The components of the two equations i i  i  under (78)  arethe moment arm distance  componena sin      sin iq X cos     cos   i sin    N i  N  N N   0   r 0   d   sin   0(75)   respect    0 1 G G0 (78)    r  r  d sin      r   with to the two collision partners. the 1qsin  t    1q  d bysin 1Brach 2 q [9] 2q (74) twith 2 q  preceding 2 2 steps, Gand i Brach riq   riq  Y   diq   sin  i   cos 1q 0  its  and  under  1iform  herein, showing how the terms are i  follow iq  G equations cosThe cosG frame    cos from inverse the of0the   DCM   the   sin to   the  sinN  frame   i  and P1note  P2that (80) G   G  (20). i   sin cos    G the sin im  GN 0   G vas 0cos  is sin  sin  cos  preliminary cos sin            i   i i and the DCM between B well as its inverse are: 0 0 0 P  P m v   0        The translational dynamics for the planar case generates four equations two1o collision partners.

The 1 1 1o 1 0 0 N  1   0   0 f o   B i    vG1oframes (75)   for    G (79) R   sin     cos    0 R   sin     cos    0 n the right of the last equality makes use of the sum and difference identities, from trigonometry,        cos cos cos sin sin          P   P           B i i N i i frame expression is an obvious choice for the translational dynamics.

Newton’s third law of motion state  equality makes use of the   1 i last 2 The form on the right ofofthe N frame and itsangles inverse follow the form the equations under (20). equation, using equation (13), can be expressed as the following: se that for any two This and :     0 0 1 0 0 1 that the G frame collision force, and therefore the impulse, is equal in magnitude and opposite in directio 0  under cos  i     sin  i      cos Gare  idistance  equations and from trigonometry, which G Gi    G The components ofGP the sin two (78) Thedifference components of the two equations under (78) are the moment arm components that and  trigonometry, n the right of the lastsum equality makes useidentities, of the sum difference identities, from P m v are shown v 2 o  othe 2 2  v 2 o  m 2 equations  the Bi 2 o  f  as N ofpartners. with respect topreceding the two collision This equation, using equation (13), can be expressed The DCM from the G frame to the N frame and its inverse follow the form the under (20). by Brach and Brach [9] and with the steps, herein, showing how the terms are derived.

A final states that for any two angles α and β: moment arm distance components that are shown R   sin     cos    0 R   sinfollowing: i     cos  i        by Bi i N s that for any two angles :    cos     cos    sin  G   (81) G G G Gi sin   and    sin P  P  m  v  m v  v 9as 1the planar 1o 1 1o f 1o o preliminary note is that the DCM the1 case Bi and N frames as well its inverse are:     The translational dynamics forbetween generates four equations for two collision partners.

The and with0 the preceding steps, herein, Brach and Brach  (75) 0 1  0G 0 G G G frame expression of the collision impulse asthe transformed G Gcan also Gbe expressed G G cos    frame sinan sin  The  cos expression   cos     is obvious      P   P (80 choice for the translational dynamics. law of motion states P P  m  v  m v v showing how the terms are derived.

A final preliminary 1Newton’s 2  third 1 1 1o 1 1o f 1o o     sin       sin    cos     cos    sinexpression  of the same.

(75) G therefore G  G that the G frame collision impulse, equal and opposite in direction as well for two collision is that the Bi and GP and sin (75)  0  note vi the in magnitude  sin  the 0  N frames   cos  iforce,  G the DCM between (82) P itranslational m v v2iso cos for i  The dynamics four equations 2 o  m 2 2 o  o planar case generates and N cos follow sinthe  form sin2 of   with   cos the equation,  using  two 2under  bef are:  rom the G frame to thecos N frame its inverse the equations (20).

Bits respect to collision partners.

This equation (13), can expressed as the following: i as inverse R Bi    sin  i     cos   0 R  obvious  sin  i choice  Gcos PG m 0 G N  m(79)G v Newton’s G  i expression  i translational G  N frame isNan law   G v 2 o  othird P1  Gfor RPN2 the P2 2 G RvN2dynamics. oG P2 2 2 of   G P1     that the G frame collision force, and therefore the impulse, is equal in magnitude and opp 0 0 1 0 0 1 The G and frame expression the collision be expressed as the transformed N frame(80) G G Gcanunder  rom the G frame to The the N frame itsGinverse follow form of its theimpulse equations DCM from the frame to theofNthe frame and inverse G also P(20).

(81 P  m 1  Gpartners. v1o  m 1 G v1o  f   G v1o  o  1 P 1  2 collision with respect toPthe two expression of the same. follow the form of the equations under (20).

The G frame expression of the collision impulse can also be expressed as th The translational dynamics for the planar caseexpression generates of four equations for two collision partners. The G the same.

G G This equation, using equation (13), can be expressed N G as the G following: N G G G P1of G vmotion P2 (83) P1 the  GR P22 dynamics. frame expression is an obvious choice Gfor translational states (82 P  GRPN G P m2 2 Newton’s v 2 o  m 2third v2law N G P1 of  2 oo G opposite G N G G N that the G frame collision force, andGtherefore the impulse, is equal in magnitude and in direction G G P1  R N G P1 P2  (81) R N G P2 P1This  G Pequation,  m 1  G vusing v1o  f  (13), v1ocan 1o  mequation 1  o  be expressed as the following: with respect to the two collision partners.

The G frame expression of the collision impulse can also be expressed as the transformed N fram expression of the same.

G G G G G www.collisionmagazine.com Collision Magazine - VolumeG 14 Issue2 G v 27 G G  P G 1 v 2 (82) P2   G P  mP2  v  GPv1  P  m 1  v1o  m 1 v1o  f(80) 1o  o   2o  m 2 1

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P1  R P1 P2  R P2 This equation, using equation (13), can be expressed as the following: G G The G frame expression of the collision impulse can also be expressed transformed frame P   P as  mthe v2N  G v2 oo 2  v2 o  m 2 o  f  10/9/2020 Volume 14 Issue 2.indd 27 3:43:06 PM expression of the same.

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

 rsin  d cos    sin P r   N riq  t   N R G G riq r d  sin RG v1o(77)     1 001sin  1 ii d2m R G v1o f   R G v1o o  m  1 R G v1o f(78) diqiq   diqiq rR  sin   1qcos ii   iir2q   d  G t equation.

1q  t  q ii1   2 2(77)   v1o  o   m 1  0sin i sin 2qsin ,rris:  riq  t   R G riq 1qcos   sin  cos   cos                   i i i    0   1  N0  0 0 0  0  10  N 0 0   NN v G0  N v G  0  N 0 GNP 0 N riq  d iquse  R G G makes  cos sin  function i  R iG G P is1dan sin  Gvi 1o 1  m G    fact   the00cosine  Gfunction, 1o  f  1o  o v e rlast equality of the that even R m 11 GR (77) R G v N  f   NGR G G v1oG o iq m i   sin    N 1 G N v1o G 1o  oN  R m N  R G f  P1  m  G1 v1o G m 1 1o R G v1o f   v1o o  m(84) RG G v   1 G 1 cos      r cos                     he sine function is an odd function, sin(-) = -sin(). written for 1 1 2q  n 2 2 Equation (77), 0 0 0 1 0 0 One may also wish to transform the entirety of each translational dynamics equation under  fact(78)  N arm Ndistance  even on of the makes useof thatare the cosine function,  two equations The components of the components that are shown last equality NG Pthe moment m 1 function v1o  f G Gis onthe the right right the makes of the the the function isvan an even function, NG N 1cosine GN(78) 1o  o  Nis achieved artners, is:1  of    lastN equality  duse 2fact  2that   r2 q   N r2q   G  NP N N G expression.

This by applying kinematic both sides of th 1q  sin  1  the t  an 2 q  sin is  m v  cos(), and that sine function odd function, sin(-) = -sin().

Equation (77), written for  R P  m R  v  m R v GGvv1o2o oo  transformation  m 2 N R G G v2oto G 1 1 G 1o  f

by Brach andsine Brach [9] andiswith the function, precedingsin(-) steps,=herein, showing derived.

A2oaf final G 2 how 2 thewritten Gterms 2o are 2 G   f   R G v2 o  o  cos(), and that the function an odd -sin().

Equation (77), for       equation.

0 0 0 the lastpartners, equality makes useDCM of thebetween theBcosine function even function, eeright two collision  of  note is: fact thatthe  N  the preliminary is that frames is asan well as its inverse are: i and two collision partners, is: N N N N  G    N G P m N v G  N G  that  N rfunction,   v G cos sine 1  function 1     is an odd the  cos  G 2 n 2q  n 2P  m o  f  v G 2o  o v and sin(-) =2R-sin().

Equation (77), written R G  v2o  m22 GR G2for mG2 N R G G v2o N N  f   G R G v2 oG o  N G 2 2 2o 2o  oN Gf P  m  R         N N  R v  m R v  v  o  Gm(85) RNG G v the four N frame expressed start of closure velocity compoN G G 2 N 2o G 2 GN 2 G GG 2 2o  f   m 2oN (78)  is:   moment arm dsin   are rdistance sinNNr2  02that r2qcos ollision  R P  m R  v  m R v  v R v  RG under the components shown  NNrr11qare  t  d 1q partners, 1 2q t 2 q ions  cos    cos                sin(78)    G 1 1 G 1o 1 G 1 G 1o f 1o o 1o f      cos     sin    0        2  n  n N  i  r2q 22 Nivnents  i dynamics  N frame the 11i  wish    to transform translational  cosmay  cosNvof  11 also   One equation into the N frame terminus of separa1q  n  2q  nG 2 each P2 Bentirety m  N steps, N N expressed N 2 AG G 2 o o  and the four  preceding  2 o f  N N terms Nare         the herein, showing how the derived. final 0 0 0 N These four equations, excluding , contain ten variables consisting of the two components of the    i      P  m v  v dR1qexpression. rr2q applying d 2R sin       rr2 q  N N cos   (78) N 2 0 N G 2 o  f  (79)  sin 22sin  NNrr1q  G    isachieved cos to 2 oo 1  t0  q  t   akinematic both 2sides of theG equality for each (78)   NBrr1q sin   1iThis i Nby   2 i transformation sin  Nsin icomponents.

Bi    velocity tion applications, it may t   d 2 q  t   G P1  m 1 G v1o  f  G vFor 1o  o  certain  inverse M between Ndframes  1q  1q  N2rq2q  nare:2q impulse, i and r1q the 1  1as 10well  2four  0 N2 frame  cos  1 as its  cos  2 the expressed start of closure velocity components and the four N frame e 1qequation.  n    0 0    0 0 1 0 0 1  0            be preferable to write at least one of the velocity vectors inbe   N   0 0 terminus 0,tocontain  wish  ofFor N   of  ten(78) velocity components. certain it may preferable   d1q(78)  sinare four equations, variables consisting oftranslational theapplications, two components ofequation the N frame transform the entirety each dynamics into to thewri N  1moment     Narm One also  separation (79) shown d 2 q may sin excluding requations r2These  1the   These 1q   r1q  tunder q distance 2that 2are   r2q  t components  four equations, excluding , contain ten variables consisting ofth the Ntwo terms of the transformed body frame components.

For the N G N G N G G N G one of the velocity vectors in terms of the transformed body frame components.

For the i collisio the four N frame expressed start of closure velocity components and the four N frame expressed expression.

This is achieved by applying a kinematic transformation to both sides of the equality fo  sin   0  steps,0herein, iN  impulse,  sin    0    cos showing 0ipreceding     the     d with how the terms are derived.

A final i  R P  m R  v  m R v  v  m R v  R 0 N arm 0N components G G G  impulse, NGframe G 2oexpressed N G start th Gthat G 2arefour 2 N 2 G 2 G G 2o  o  velocity 2o  f  the of components and  f  closure (78)  The  two onents of the equations under are the distance shown collision at the  R P1 moment m 1case Rseparation arm vthe jthm v1o f  i v1o partner m Rpartners. v1oj th time: R GitG v1o o2o translational for the planar four equations for two collision The  Bi dynamics  Gterminus Ggenerates 1o 1 R G components.

1 G time: at o are f of velocity For certain applications, may be preferable to write at least of the two equations under (78) are the moment distance components that shown equation.   eonents DCM the B and N frames as well as its inverse are: (79)  cosbetween    0 R sin cos 0          One may also wish to transform the entirety of each translational dynamics equation into the N frame       i N i i i terminus of separation velocity components.

For certain applications, it may b    (84) N N and [9] and preceding herein, showing how dynamics. the terms are derived.

A frame is an obvioussteps, choice for the translational Newton’s law motion states  P2 third m v2 oof  NG v2 o components. one of the velocity in by terms of the transformed body Forofthe ithequality collisionfor partner and Brach Brach [9] expression and with with the the preceding steps, showing terms are derived.

A2 final final N herein, NThis is vectors Nhow the G G f  frame o  to both sides expression. achieved applying a kinematic transformation the each    0 1 0 0 1  P  m v  v one of velocity in terms Nof the transformed ry is DCM between the B and N as inverse are: 1 frames 1 G the ofnote the two equations (78) moment distance components that are shown 1o well fimpulse, 1oits  under  are G the G Bi vectors in Bi N body Bi frame component   Gas  o is that thethe G frame collision force, therefore equal magnitude opposite N G in N G and N N GR N G time: atG the jth arm ry is that that DCM between thethe Bii and and frames as its inverse are:   GNGvvdirection  G vR G G RBi m equation.

G vth Bi m G vio  R G vio  j R B Gv j j G io j   v1oR   G v1o  o  [9]  with  sin i the  to 0  two collision showing    sin how 0  R are   cos  inote the preceding  i N  well as terms atmthe j ioA G Pderived.

1  1 R Gtime: 1 1 G v1oi  f  io  jR 1o f  1o  o  i  ach and with steps, herein, the final respect the partners.  Bi   (86) of the two compone equations, excluding , contain tenGNvariables consisting (79) iplanar    the case cos i between The R    sinframes   as  cos Gipartners.

0G These for the generates e is that the BNiand N well as its are: generates   0 translational Gi two planar case N N Bi N G Bi Bi he equations for collision The  cos   sin 0 cos m  sin  0 N N N GN N G N DCM R vNv N  dynamics viiinverse GGGfour four  PiiR Bim v 21o i  i  (86) NfourB v  v v  R R v  R NGcos G N G G GG N GB G vio N N G N  cos     sin   0    sin  0 R P  m   R v   m v  R v  G 1 1 G G      1o f o G G B     G G G G B G io j io j io j io j j         i i i G 2 2 2o 2 G G i i the N oGframe start closure velocity InofPcomponent form:  f  four2o  f  partners.

R0GGP GThe m R Gframe  v2o R v1oexpressed  v1oGvo2o m 1of R 2 ov o   R j  components  translational  collision R(80) v1oR oG G R Bi BGiand vio jthe  R Bi G Bi third two G equations for 1impulse, hoice the dynamics.

Newton’s law states 1motion 1o  m 1 f f  G vio G 0 NN for 0   four 1 0 1  Bi GGvio  j1o io j  P     (79) B R   sin    cos    0 R   sin     cos    0 (85)  ii    0N i R NN  N sin 1 iiN   2 terminus  ii of separation Bi    sin  i     (79) (84) R  cos  cos 0  velocity components.

For certain applications, it may be prefera B i expression the for i nd therefore magnitude and ino direction   impulse,  sin  isi equal  is 0in cos opposite  0Ptranslational  sin 0N v 1    cos  obvious m 1 G P2 choice ithe  0 an  i Nv  0dyi  2  Gv G v2m 2 o  f N 0 N  o1one Bi transformed Bi body N G of G0 N G N G the G N frame G N Bi G G0 1  G the velocity vectors in terms of components.

For the     1o f 1o o     In component form:     0 0 1 1     B    In component form: viothat i    GGviojv2oyi osin partners.

Newton’s third ofexpressed  i v2osin  m mioj2   xi Rcos i i collision The 0cos i2  RG 0 v2oG fv ioj  vxi2o  o  G v Gthe 2 i RG (79) n 2G G v 2o iR InmG component form:  the  This sin cos  four 0 equations R N can law besin    states  cos Gthe   Pth  f   R  using (13), as following: equation,  namics.  i equation motion jat sBfor planar generates two partners. i  case  for  i      time: the j   Bi N Bi Bi frame collision force, and therefore the impulse, is equal ious choice0for Gthe translational dynamics.

Newton’s third law ofio0jmotion states  cos v cos  i     G vioj   xi sin  i  Nexcluding N  i     the  Gfor  N 0 1 equations, 0equations 1sin i Nv  G v  B i frame y t  BThe four , contain variables consisting of2 o the two0Bcomponents ten   yi   2   G vioj Ncos m2partners. i v G These ational dynamics case generates four two collision N G N G G N i G GP GN 2 o  f   GThe v Gin oG Gof    iojN sin  i Bi     G   Gi v  inthe magnitude and opposite direction respect to the P1 for the P2 planar (80) ational dynamics for the planar is case generates four for two collision partners.

G N Bi ioj  v          cos sin 0 Gv G v ioj  iv Gv G in magnitude G with G G           nequations  xi v 2xi ioj  ioj R P  m R  v m R  m R R     orce, and therefore impulse, equal and opposite in direction i i G G B N N G   G 2 2 G 2o 2 G G G   Nv  Bixvcos  i  2o f 2o o 2o f o  o    0R GR vioj yBii2xv (81) m m 1 0 of v1oclosure impulse, thetranslational fourP1Nframe expressed start velocity N frame  NP dynamics.  v1o Newton’s and io0 jcos RiBi1Bthe Gi viofour j 0sin v  j  expressed Gv 1  iojcomponents ression is an obvious choice for the motion v   1o  n BGstates  fthird   vlaw  o0 of  i io j 0G Rio i    G G B G G i io  j  B io          Bji (85) i i i i two collision partners. ression is an obvious choice for the translational dynamics.

Newton’s third law of motion states     ision partners. v sin cos 0 v v cos v sin                       (87)  G ioj it  beG preferable  equations N N For N Gvelocity i i applications, i write at G Bileast i Bi terminus of separation certain to  t  components.  ymay io j equal ioj   yi ioj   xi sin NinP magnitude direction i collision force, and therefore the impulse, is and opposite in lframe dynamics for the planar case generates four for two collision partners.

The G           v cos 0 v v cos   m v  v      iN      i G two G ioj 2magnitude 2equations, G 2 oand G frame collision force, therefore the impulse, equal in direction t ioj   yicomponents 2 o  o G ioj four excluding ten of the the  ,  f  opposite  isin These  i variables  For theconsisting    yi of  3), can be expressed asand the following: Ginof 0third 0body 1  contain one oftranslational the velocity vectors terms the transformed frame components. ith collision partner     0 0 0 ect to the two collision partners. n is an obvious choice for the dynamics.

Newton’s law of states G G G G G motion          frame (17) 0 components dynamics is given equation, in the body1frame, by equation ect to the two collisionG Ppartners. form: (82) the four of0closure velocity and the four N exp P2   P  m 2impulse,  The v 2 o rotational m vcomponent v 2 o  oexpressed 0 bystart 0as given Euler’s   2 In 2 o(80)  the  G the Pjth2 time: (80)  fN  frame   1at collision force,Gand therefore impulse, is equal in magnitude and opposite inthe direction this equation for each of two collision partners (as well as replacing the body frame impulse exa G G G terminus of separation velocity components.

For certain applications, it may be preferable to write These four equations, excluding consisting of the two components of the N frame (81), contain ten variables P1 two  Pcollision  m1 v v1o  f   v1oGG oP 1o  m 1   G he partners.

(80) G P2 th collision Nvectors Bgiven Bi Bi 1  rotational with the transformed G frame expressions): The dynamics is given by Euler’s equation, in the body frame, as by equation (17).

Writing i one of the velocity in terms of the transformed body frame components.

For the i p P P (80) Bi N N Bi Bcomponents  (86)  Gexpressed impulse, four frame expressed of the four NG in expressed viojThe vframe cos 1collision 2G the i  frame, expression of theequation also be the cos 0Euler’s Gasclosure  jvelocity  N R  sin ioiN  frame  and ion (13),The canGbeframe expressed as the following: G v iojequation, i    as Gv  (as given n R ioj io  rotational the giv vimpulse  G RN  start   xi    xi body This equation, using can beBcan expressed iiotransformed G(13), G vio  j   as G vthe G R Bi G vdynamics Bi is Gv io  j  at io  j   well  j bythe ijth      this equation for each of the two collision partners as replacing body frame impulse expressions time: the  For for  Bi partners terminus of separation velocity components. it may be preferable to write at least theBi bod N Bi well G of the same. this equation eachapplications, ofcos the two collision replacing G expression G sin (as v iojG as cos (87) 1certain  i     G vio P1 Gv G Pwith (80) Bth expressions): B B1  i    B1 0   G v ioj B G v ioj   t   B i    B using equation expressed yi 2 as the transformed  1 yri the (82) 1 1 1G G frame Ption, Pm  G v 2 ofollowing: (13), m 2 can 2  using I    I      R P o  f G 2 oo one ofthe thefollowing: velocity vectors in terms of the transformed body frame components.

For i collision partner tion, equation (13), canv2be be expressed following:  as     G G2 G G G B1 G B 1  fexpressions): G B 1 o  1q G 1  with the transformed G frame     (81) P1  P  m 1  v1oIncomponent m 1 v1o  f   v form: 1o  o  th 0 G v   G R 0 Bi v  NG vio0 j   N R G1 G R Bi BGi0vio j  NR Bi BGi vio j 0 Gj time: at the (83) P1 G G R N NG PG1 G P2  BG RN NBG P2 G  ioB j B Bi G ioB j G G the following: B1 B1 G sing equation (13), canGGbe expressed as 1 1 1 1 1 (81) P  P  m  v  m v  v G G G G (81) (88) I    I     r  R P 1  be  GGBi B1B B2 I GBNi B21 f  (81) 1o  f   collision impulse can Palso as 1the vtransformed  o  N frame B1 B1 G BB21  o  B 2 B 1 P expressed m 11  v1o v1o B2 G 1 1o  m 1 1o  f  1o  o  I 1qB1  r1q   B1 R G G P N IG G  r2Bq1 f  B2 R yRGoBB1 iIv1inBiG12the G N G G  o2   B2 cos GB G B1 P i 1NB 2 sin R   G v v vGioioj j is   BGBi2v fioBGi Gv  (86) by equ v  v  R R  (82) P2   G P  m 2  G v 2 o  mG2vioGj v 2no f  Gvcos         sin 0 G i       io j  x x io j The rotational dynamics given by Euler’s equation, body frame, as given      In component form: G B G G G B B G i i i i i io j io j j io j 2 o o        i i i G G   Gv   Gv   B Bi P1  GG P  mG1  N v 1o  m B(81) i of the two collisionBpartners i 1 1o f 1o o     G G G this equation for each (as well as replacing the body frame im B B B B B B G  v cos    v 2 o f cos  i    2 r2GBqvioj 2 xRi sin  i    (87)B 2  2Gvioj   2o sin PvNiojm G P2  G G G v i    2 I 0  yi 2 I  2 G ioj   y(82) (89)the  may i G (83) B 2 with G G Balso G B 2  f  by G B 2 P2 Bimpulse (82) G G N GP2   G P  B 2  o  B 2 the m 22t   G v v2 o  m m 22 i G v v 2 o  oo  with 2 2 2 be written replacing frame expressions 2 o  f  Each N 2 oequation BiI   G2body B B   I     r  P the transformed G frame expressions):      P  R P P  R P (82) i i  B2 R Gtran 2q B o  component 1 N Gimpulse 1 2 can also N G 2be expressed  G v0ioj   x B2   G vioj G  x Bcos  G 0vioj   n  1N In form: of the collision transformed frame 2  f    G  2  0 as the          cos sin 0 i G v ioj   y i sin  i   0 0     i i i i       Nframe expressions:  G      G G G G       P2 the  P  m 2  vimpulse v2 oalso v 2 oexpressed B Bi 2 o  m 2 can  f   be  o   Nalso  the  sin    (82) iBbody    Bframe   cos 0  Bi vimpulse v j G xi sin  i   ame as transformed N  i replacing frame iB1the  Gi v iojB1  yBi cos Each equation may written by the with transformed B1   yi B1 expressions G v ioj  be ioj t  1 1  B G  ame expression expression of of The the collision collision impulse can also be expressed as the transformed N frame N B I    I    r  i i i G frame expression of collision impulse can also  GB1 Rioimpulse Each equation may also the expre G be G 1qframe G P1  B1G v BB1G1 written  Gthe   cos  o io  jBbody viois vioby   GB1 BWriting sin equation, byBreplacing 1 if (17).

B 1body B1 given Bcos N  i           sin 0 n of the same.

G1 v     jexpressions: n io j x j x given     1 1 y i        N frame i i The rotational dynamics by Euler’s in the frame, as equation 0 0 1 i i The rotational dynamics is given by Euler’s equation, in G G N G G N I    I     r  R P n of the same.

0 0 0             G B G G 1q N G 1      B1  f  B1  o  N frame expressions: 1 be expressed as the transformed N frame expression of the (83) P  R P P  R P       N B B B 1 N G 1 impulse 2 N G also 2 i i xpression of the collision can be expressed as the Nbody this equation for each ofvthe two (as well replacing body frame impulse  collision  sin  transformed   Gi vby  expressions 0  frame, sin  i     (87)   cos  frame  i(17).   Writing the i partners i  as G v ioj  the G v ioj   xi this yi given ioj equation as   yi cos same. GG P  GG R NN P GGGP ioj GG tR NNP B2 B B2 B2 B 2B B2 B2 B B B B B N e same.    (83) 1 1 1 1 1 1 1 I    I     r2asq   B2 RB G G(90) P2 B with the G2 frame expressions):  I r1qcollision for GP2 each I   G B10equation  BI2 1  Gof Gthe B two 0 B 2N fpartners B 2 (as o  Bwell  P11 transformed R NN GG P11  P  RNNThe B1B2G R BG1 P1 G B0 B1 G f  0(83) B21 Euler’s BB B 22B 1 N by equation 12 1 1 2 0 G 2rotational dynamics 21  o   2 is given by equation, in the body frame, as given    I     r   (17).

R N NGW P    I   G B2    I G  B2 f  G GB  r  R P    G G 1q B f B o  2 q 1  with N G1  2 1 B2  o  replacing the body frame(as impulse expressions the G N G G N G this equation for each of the two collision partners well as replacing the body frame impulse expre B B B B B B B G (83) P1  R N G P1 P2  R N G P12I  1   1 I 1 (83)  1transformed 1 (88)impulse expressions wi  may r1q R G expressions): P Gframe  dynamics B2 1the Each equation also be written by replacing theN body frame G the B1  transformed G 1 B B 2 f G  expressions): frame 2 G 2 2 The rotationalwith is Bgiven body frame, equation (17). Writing (91) I  BB1G2by BEuler’s  B12 Io equation,  in r2Bqvariables,   Bas Rgiven P2 B2by  B2 N G  Ban G B2  o  B2four NBG 2 2 for 2 2  f  additional These twoNequations introduce the planar consisting I   G B2   I G  B2  f   BG2  Bcase, r2 q   B2 RofN th frame expressions: GP   2 o this equation for closure each of the two collision partners (as well as replacing the body frame impulse expressions and terminus of separation angular velocities for each collision partner.

The final equa One may also wish towith transform the entirety of each transB B B B B B B G B2 B2 B2 B2 G 1 1 1 the transformed frame expressions): I o G1four B2Bconstraint   1BI2 R G1   coefficient  case, consisting I   BG2constraint  B2  G  I equations. r1 2qvariables, Pthe  1 R(89) G planar G Pin 1 of B1  o  G  B 2an G 2 f the  f  additional AB2typical ofrB1q restitution the normal These into two equations introduce the start ofdirectio B1 B1 isB 1for B1 B N 1 1 1 lational dynamics equation the N frame expression.

I  introduce   I     B1 r1q   Bfor R Nthe   velocca These two equations an additional four G B1 G B1  f  G B1  o  variables, G P1planar (88) fixes the ratio of the normal direction separation velocity to the normal direction closing closure and terminus of separation angular velocities for each collision partner.

The final equations are This is achieved by applying a kinematic transformation B1 B1 to closure B1 B B B B G terminus separation 1 angular velocities for each collision part 1 (88) such I   Gthe   B2 G I and  1B  1 Bof ofa 1restitution r1qB2with R in PBtransformed B2 The ris2 q the peripheral point q.B1body velocity peripheral fixed the body which frame, constraint equations.

A typical constraint is coefficient 2the GB 12the Bimpulse o2for may also be written by replacing 1  f   G 2 IBexpressions Iframe frame   B2  o  Gthe point, normal R GinGdirection, P bothEach sides equation of the equality for each equation.   Gequations.

B2  G  B 2  f   G constraint 2 constraint coefficient of restitution in B2 B2A typical B2 B2 B2 B2 B N 2 fixes the ratio of the normal direction separation velocity normal I   G toB2 the I G  B2direction  G  B2 closing  r2velocity R atG Pthe N frame expressions:  direction q  2 f o separation fixes the ratio of the normal velocity to theN normal dire q. The frame velocity for a peripheral point, body frame, such as q, was These four equations,peripheral excluding point γ, contain tenG B 2 variables B2 B2 B 2 fixed B2 in the (89) I may  BG2   I BG2 point the rbody R G G Pimpulse q. frame point, fixed in the    a peripheral Each equation also be written frame expressions with the transf B2peripheral G  The 2 q  velocity 2for B 2 by B2  o  G  f  replacing B B B B B B B N 1 of the 1 1 1 1 1 1 two components impulse, I N  frame N expressions: IThese  B1  o   introduce requation R Nan frame additional four variables,(90) for the planar case, consis One may consisting also wishoftothe transform the entirety translational into Gof B 1each G  B 1two G dynamics 1q   GP 1 the N frame  f   equations closure and terminus of separation angular for each collision partner. The expression. This is achieved by applying a kinematic transformation to both sides of the equality forvelocities each Each equation may also be written by replacing the body frame impulse expressions with the transformed equations. of Nrestitution in the norm equation.

B1 B 2A typical B1B2 B1constraint B 1 is the coefficient B1 B B2 B2 B 2 constraint B2 B2 B1 N N frame expressions: I   G B2   I G  B2  f   G  B2I o G Br1 2q   I R NG GBP1 2 f  G  B1  o   r1q   1 R(91) G P1  fixes the ratio of the normal direction separation velocity to Nthe normal direction clos N G N G N G G N q.G The G N G velocity for a peripheral point, fixed in the body fram peripheral point frame  R G P1  m 1 R G  v1o  m 1 R G v1o f  B1 Iv1o oB1  m1 B1 IR GB1 v1o f   B1  R G v1o oB 1 r  B1 R N P (90) B2  B 12 the start  B2 I  GBfor 2 B 1 the G Bvariables, GB 2 B1 Bo2 case, N Gof f These two equations introduce an additional four planar consisting 1 I G  B2  f  1q r2 q   B2 R NofNG P2  G B2   G  B2  o(84)   (84) N N N closure of separation angular velocities for each collision partner. The final equations are  G P1 and m 1 terminus G v1o  f   G v1o  o  constraint equations. A typical constraintB2 is the coefficient restitution in the normal direction, which N (91) of the s I   BG2 Bintroduce  BG2  B2  o  four  B2 variables, r2 q   B2 R N for  planar   B2 I toBG2ofan These two equations case, consisting G P2the B 2 additional f 2 fixes the ratio of the normal direction separation velocity the normal direction closing velocity at the N G Nclosure G G N G N Gvelocities for each collision partner. The final equatio terminus angular  N peripheral R G G P2  mpoint m 2frame R G velocity v2oand v2oa o peripheral  of m 2separation Rpoint, v2 o body q. vThe frame, such as q, was 2 RG 2o  G G v 2o  ffixed G the o  f   for   Rin (85) constraint equations.

A typical constraint is the coefficient ofcase, restitution in the normal (85) These two equations introduce an additional four variables, for the planar consisting of the startdirection, of N N N  G P2  m 2 G vclosure  v fixes the ratio of the normal direction separation velocity to the normal direction closing velocity G and 2 o f  2 o  o  terminus of separation angular velocities for each collision partner. The final equations are peripheral The G frame velocity for aofperipheral in the body frame, constraint equations.

A point typicalq.constraint is the coefficient restitutionpoint, in thefixed normal direction, whichsuch as

iq N iq iq  n N iq  t

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fixes the ratioten of variables the normal directionofseparation velocity to of thethe normal direction closing velocity at the These four equations, excluding , contain consisting the two components N frame peripheral point q.

The G frame velocity for a peripheral point, fixed in the body frame, such as q, was impulse, the frame- Volume expressed components and the four N frame expressed 28 four CollisionNMagazine 14 Issue 2start of closure velocity www.collisionmagazine.com terminus of separation velocity components. For certain applications, it may be preferable to write at least one of the velocity vectors in terms of the transformed body frame components. For the ith collision partner at the jth time:

Volume 14 Issue 2.indd 28

10/9/2020 3:43:07 PM

                     



1 i i i 1 i  N  B  i  N   Bi Bi 0    sin   0   cos 1   i v 1q  o  NNG(87) v1o o   NN R B1 BBG11 B1  o   BB11 r1q 0 0  N G v 0 v v cos v sin                  otational      G i i G G i G i frame, as given by equation (17). ioj  t dynamics is given by Euler’s equation, iojin ythe body io j   y j G  xi v21qN oWriting G v  oB1  G R B1 B1G GB1  o   B2 r1q i G B2  NG vioj  n    cos       sin      0   BGi vioj   xi  GGBGv v2qiojo  xiiNcos  sin RvG2 oGGovi 1q RB2BNGi RvBGioio2Gimpulse (93)  oGframe  G Bexpressions ri2 qR G  GGRvB21o BG1 vo 2 o f  r1q R r2q j y 1o  o  G Bv G B2 G  B2  f   1 q  f G i   2 i o  0 i partners 1   (as0welli as replacing for each of0 the two collision the body 0 0 quation    (94)    Bi     N Bi Bi N Writing  given  sinGby frame   expressions):  cos   as cos  i Nv  (17). sin NiRN   B1G(87)  i   in 0the Nioj the transformed dynamics Euler’s equation, frame, G v io i  G v ioj   G vgiven Gv j   t  is  yi  ioj   yi by equation xGi v    v1o   body  BNN1 rR1qG GG R B BBG11 B  f   BB11 r1q  G 1q  o   GG B1R GGGvB1o  f NR G  o1q N 1 o  f  1 r each ofthe two collision     R  R  Ra kinematic R 1  r1q  v v  transform. beframe transformed by partners as replacing the can body impulse expressions Gthe G N G G applying 0 0 (as well 1 These 1q  fframe  0equations 0byinto    body  N 1o  f  N G B1 B1 G BB11  f  rotational isBgivenB by Euler’s equation, in the frame, as given equation  0  dynamics N(17). Writing B1 1 1 1 v1o  f   N R B1 BG1 B1  f   B1 r1q ormed G frame expressions): N G v1q  f   N G(88)  B1   B1partners I BG1 B1  f (as  Bwell  B1 r1q  Nthe R GG G P1 frame   Gcollision G  B 1 as o  replacing R B1 B1G B1  f   r1q vexpressions equation for each of the Itwo impulse N G N  f  G G v1o  fB1  G 1q v v R GN body  R R R G N G N G 1q  f  Rgiven  G GR G1oG fv (17). G R GB1G RGB1BB1G1 f B1  o  r1q B1  r1q  G G v1q the transformed G frame expressions): 1o  o  Writing  o  equation rotational dynamics is given by Euler’s equation, in the body frame, as by (95)(94) B1 B1 B1 B1 B1 B1 B1 G (88) I    I     r  R P B B N N N     B2 B2 N G N G N G 1 1 B1B 2 twoB 2collision G G B 1  o  (as Bwell 1q as replacing 1 BG2 B2 G 1  f B 2partners  vBB211o fo NrR RNBN1RRGGG(89) 2 equation for each Gof the the body frame impulse expressions N f R N v r     1 GR Gv GGv 1q B 1q 1o f     R I   G  B2 B I BG2  r P     B B N G G B G 2q   2q o B o v  v r     2 2 G B2  o  2q G 2 B2  f  G 1q R G 1o o  o R GBG1 v2Go oB R B22 G B22  o  r2q  1  o  R G 1q B1 B1 B1 G G G v 2q  o  the transformed G frame (88) I  expressions): I BG1 B1  f   BG1 B1  o   B1 r1q   B1 R G(89) P1  G  B1   B2 B2 N N N B2 B2 B2 B2 G N G v 2q  o   N G v 2o  o   N R B2 BG2  B2  o   B2 r2 q (89)  B2 I BG2by body   B2 R impulse I  B2replacing r2 q frame v R B2 B2G B2  o   r2 q  G N with G the N o  GG v 2o  oB 2  G be B2 written G P2 N o (96)  f   G  B 2 the G 2q equation may also expressions transformed BB21 RB1G NGRvG2qG ov  r2qB1 r  B1 B1 B1 B1 B1 B1 BG o   R GNGRvG2 oG  o   R GN RRG(88) 2 RG I    I     r  R GPG1  1q  f   G v1o  f   B1 G B1  f   1q    B B B B B B B G G B G G 1q G B f B o     2 2 2 2 2 2 2 1 (96)(95) 1 1 me expressions: (89) I   G  B2   I G  B2  f   G  B2  o   r2 q   R G NP2  B2G BN N N B2 B2 N N G G 2 B B v  v    r R N N N R  R  R R   v v  r 1 1 G the B2 RGGG B2o 2 qG r may also be written Each by replacing body impulse expressions 2q  ov o  transformed   f  N B2 2q N G G2o G  ov G B2 BG equationthe may alsoframe be written by replacing thewith body B2  f  2   fB f NR 2 G 1q R G 2 q1o G 1q f f R GB1G v2o r2q  G G v2 q  f  G R B2 G  B2  f   f 1 B 12 B12 BB12 BB12 BB12 BB1 2 BB1 2 NG sions: frame impulse expressions with the transformed N frame (90) I    I     r  R P B2 B2 N N (89) N I     r  R P       G B G G 1q N G 1 G B G G 2 q G 2 B f B o B f B o         equation may also be written by replacing 1the body12frame impulse expressions with theNG transformed 12 2 v2q  f   NG v2 o f   N R B2 BG2 B2  f   B2 r2 q v N f  G G v2 o fB 2  R B2 B2G B2  f   r2 q N G N G expressions: ame expressions: R  R  r2qB2 r  NG v G N G v G G 2q R (97) G G 2 q f 2o f f v  v B1 B1 B1 B1 B1 B1 B1 N R  R GN RRGBG2 R BG2BB2G2  G G G G 2q 2q  o  2 o o B2  o  (90) I  also  BI2 Gby B1replacing r   RBNimpulse (97)(96) G be B 1B2 written G P1  B2 expressions B 2 1q frame N  f B 2 GB  body 2 B1  othe 2 equation may with the transformed (91) I   G B2   I G  B2  f   G  B2 The  previous r2 q   four RN GP B2equations.

B2 N2  N are all N equations vector The first component term, in each o B B R v  v    r N N N 2 2 B1 B1 four are allr2 qvector equations. term, The in each equation, G 2q v Gprevious Gequations. f The 2 o f all vector B2   f  equations are first component equation, B2 Requations 2 qThe ame expressions: I   BG1 B1   B1 I BG1 B1  f   BG1  B1 The  previous r1q  subscript R N NG P G 2q  o   G v B2(90) G B2 velocity.

2o  o  o  B1 four 1 referenced o respective normal direction It is the first component term, from first component term, in each equation, is the respective B2 B2 B2 B2 B2 B2 B2 N respective subscript referenced normal direction velocity.

It is the first component term, from (91) I   introduce IanGadditional  B2  f   G  Bfour  r2 q equation, R N the  for  B1 isis used the following definition the coefficient of restitution.

G B2   G P2that 2 o e two equations variables, planar case,in consisting of the startnormal of of (90) B1 B1 B1 The B1 B B N subscript referenced direction velocity.

It is the equation, that used in the following definition of the coefficient of restitution.

1 1 fourequations all vector Nequations.

The first componentB term, in each equation, is the (90)    B2 I BGvelocities Bprevious  BG2  r   B2 Rare NN G G1  N G B 2 I   BG NP 2 B1  oeach   Bcollision 2 2 1q 1 f re and terminus of separation for The final are R direction R G GG vequations R G(91) RItB2is Beach  2component B2 first r2q  I   G2 BB21angular  r2 q   Rpartner.

P   I respective G 2v2 q  f first G component from equation, that is used 2ovelocity. f G  B 2  f   Gsubscript N G G  f   term, B2  o  referenced normal the term,in from each N N v  v (97) raint equations.

A typical constraint is the coefficient of restitution in the normal direction, which ations introduce an additional four variables, for the planar case, consisting of the start of N N G 1q  f   n G 2 q f  n the following definition of1qthe coefficient  Gr v 2 q  f   n of restitution. equation, that is used in the followingNdefinition coefficient restitution.

N of the N R  BNG2 v f  n B2 N of v  v   B B B B B B B N the ratio the normal velocity the closing G velocity G B2  f  2q 2 2 2 2 2 2 normal 2 The 2q  f  2are o f  minus of of separation angular each partner. final equations at B2the I direction  Gan velocities I Gfor  Gcollision r2 q planar Rdirection P2  Gconsisting four  B2  oto  in  separation N G v 1q  o   n  N G v2 qo  n Badditional N G B 2  f  variables,   e two equations introduce for the case, of the start(91) of 2 G v 1q  o   n  G v 2 q  o   n heral q. The G frameisvelocity for a peripheral point,infixed the body frame,which tions. point A typical constraint the coefficient of restitution the normal direction, N such as q, N was ure and terminus of separation angular velocities for each collision partner. The final equations G v 1q q f  n  f   n  G v 2are of the normal direction separation velocity to previous the normal direction closing velocity at the (91) (98) The four equations are all vector equations.

The first component term, in each equation, is This the con    (98)coefficient.

N N A second constraint condition is the commonly traint equations.

A typicalan constraint is the coefficient of restitution incase, the normalthat direction, which e two equations introduce additional four variables, for the planar consisting of start of employed is the impulse ratio  v v nt q. The G frame velocity for a peripheral point, fixed in the body frame, such as q, was G G A second constraint condition that is commonly employed is the impulse ratio coefficient.

This con 1q   o n 2 q o n     respective subscript referenced normal direction velocity.

It is the first component term, from each posits that the ratio of theThe tangential component theand ratio of the normal direction separation velocity to thecollision normal direction closing at the ure terminus ofThese separation angular velocities for each partner. finalvelocity equations are of the impulse to the normal component of the imp two equations introduce an additional four variposits that in the ratio of the tangential component of the of impulse to the normal component of the imp equation, that is used the following definition ofcomponent the coefficient restitution. a constant value.

The first and second terms in the N frame expression of the collision im pheral point q. The G frame velocity for a peripheral point, fixed in the body frame, such as q, was traint equations.

Aables, typical constraint is the coefficient of restitution in the normal direction, which for the planarAcase, consisting of the start of closure a constant value.

The first and second component terms in the N frameemployed expression of the collision im second constraint condition that is employed is the directions, impulse coefficient.

This are constraint Aclosing second constraint that isratio commonly are incommonly the normalvelocity and tangential respectively.

They used in the foll the ratio of the normal directionofseparation velocityvelocities to the components normal direction atcondition the and terminus separation angular for each colliN N are the components in the normal and tangential directions, respectively.

They are usedisin the foll posits that the ratio of theof tangential component ofGthe impulse to the normal component of that the impulse v1qratio was isratio the impulse This constraint posits G v2 q f  n definition the impulse coefficient. n coefficient.  f  q, pheral point q. Thesion G frame velocity for a peripheral point, fixed in the body frame, such as partner. The final equations are constraint equations. coefficient.  ofNterms definition thesecond impulse ratio a constant value.

The firstofand component in the expression the collision N N frame the ratio the tangential component of theof impulse to impulse(98) v  v G G 1q o  n 2 q o  n     A typical constraintare is the of in restitution in the the coefficient components the normal and tangential respectively.

They used value. in the following 1the impulse N N is a are the normaldirections, component constant   NNG Pt NNG Pof n 1  N G Pt   N G Pn  0 normal direction, which fixes of thethe ratio of theratio normal di- The first and second   G Ptcomponent P  P   P  0 definition impulse coefficient.

G n G terms t G in n the N frame exA second constraint condition that is commonly employed is the impulse ratio coefficient. This constraint rection separation velocity to the normal direction closing pression of the collision impulse are the components in These constraint conditions the of normal equations to eight. of For fourteen kine N N raise 1 of Nthetotal N number posits that ratio of the tangential impulse to the component thethe impulse velocity at the peripheral point q. the The Gtwo frame velocity (99) is kine  the component P  P   P  0 These two constraint conditions raise the total number of equations to eight.

For the fourteen GP t G n G t G n normal and tangential directions, respectively.

They unknowns, prescribing any combination of six of the same allows for the remaining eight to deter a constant The first and component terms in the N frame of the collisioneight impulse for a peripheral point, fixed in thevalue. body frame, such as second q, any unknowns, prescribing combination of six of definition the same expression allows for the remaining to deter are tangential used in thedirections, following of the impulse ratio analytically. are the components in the normal and respectively.

They are used in the following was given by equation (8).two Writing this equation for each analytically.

These constraint conditions raise the total number of equations to eight.

For the fourteen kinematic coefficient. definition oftime, the impulse ratio coefficient. collision forunknowns, each pointprescribing infor the folanytocombination ofeach six ofpoint the same allows given by equation (8).partner, Writing this equation eachleads collision partner, for in time, leadsfor to the the remaining eight to determined lowing four equations: analytically. following four equations: (99)   NG Pt NG Pn 1  NG Pt   NG Pn  0 (99) is equation for each collision partner, for each point in time, leads to the given by equation (8).

Writing this equation for each collision partner, for each point in time, leads to the G G G G (92) v1q  o   GG v1o  o   G R B1 BG1 B1  o   B1 r1q R B1 BG1  B1  f   B1 r1q G v 1q  f   G v 1o  f   following four Gequations: These two constraint conditions raise the two totalconstraint number of equationsraise to eight.

Fornumber the fourteen kinematic These conditions the total of (92) his equation for each collision partner, for each point in time, leads to the unknowns, prescribing any(92) combination of six the same allows for the remaining eight to determined G G equations toof eight.

For the fourteen kinematic unknowns, R B1 BG1 B1  o   B1 r1q v1o  f   GGR B1 BG1B B1  f   B1Br1q GGGv 1q  f   GGG G G analytically.

B21 B21 G G B12 BB12 by equation (8). Writing this equation for each collision partner, for each point in time, leads to the (93) (92) v  v  R   r v  v r v  v  R   R   r prescribing any combination of six of the same allows for GG 2q BB21 G G 21q G BB12 ff 2q G BB21oo G 21oo f  B12 G 1q 21o o oo qf  1qoo by equation (8). Writing this GGequation for each collision21qqpartner, for each point in time, leads to the wing four equations: the remaining eight to determined analytically.

GG BB2 wing 2B1 1 (93) RRB2B BGfour B2B oequations: B2B1r2r1q vv2 1q  GGGvv21o  GGR  BB21 r 1q (92) q GG q  ff   GG o  f f   GRBB2 1 GBG BB2 ff  B r2q  G These G o   G G G 1 1 1 equationsvcan be transformed the N frame byvapplying a kinematic 2 2 (93) R BB21into RBB12 BG2transform.  B2  f   B2 r2q G G G 2q  o G G v 2Bo1 o   G B2 G 2q G G 2 q  fG  G vB21o  f   Discussion  ov   r (92) v  v  R   r v  R   r B1 BG1 B1 BG1 B1  o  B1  f  G 1q  o  G 1o  o  G B1 1q G 1q  f  G 1o  f  G B1 1q (92) R B1 G B1  o   r1q R B1(93) r1q G v  o   G v 1o  o   G v 1q  f   G v 1o  f   G  B1  f   ed byG vapplying B2 the N1qframe B2 G a Btransform.

B1 B1 N kinematic GG G B2 N G 2 N (93) The presentation of R B2into  r  v  R   r R  R  R R   v v  r G 2 q G G B G 2q 2 q f 2 o f B f B o G G G G G B G 1q         1q  o o 2 into the o 2 2 1 These equations can be transformed N1oframe by applying aB1 kinematic transform. the four applications, in the subject G G G B2 B2 G G G B2 B2 (94) work, should not be as the presentation of (93)interpreted v  v  R   r v  v  R   r B G G 2q  o  GN 2 o  oG B2 GN 2 q  f  GN 2 o  f  B1 G B2 B 2  oB NB 2 2 q  G B BG BG B 2 2q N 2 2 1 rB 2  f  1 1  vr R  v    (93) v  v  R  v  v  R   r  R  R R   v  r G 1q G 21o B G B1 B o2  G B1q the o o    G G B G 2 q G G 2q 1 2q o 2 o o B o q f 2 o f f full scope and range of the applications of engineer          2 G G G B G 1q q o 1o o B o 2 2   into the N frame    kinematic can 1 B1 by B1 N G be transformed N G frame med by applying a1 R transform.

These equations R G GG v1o into  Nthe RG N R B1 (94)  G G v1q  o   G B1  o  ingr1q o mechanics to the field of accident reconstruction.

The B1 B1 applying a kinematic transform.

(94) eNGequations be into frame by applying a kinematic transform.

R v1q  o   NG vcan  Ntransformed r1qthe N B1 G B 1  o   1o  o  B B N N N B B N G N G N G 1 1 e equations can be transformed into theBN frameR by applying a kinematic 1  transform.

1 r specific 1 Gv  R r1q applications chosen for presentation in the subject G 1o1o  o   GGGvv  of  RRBG B1B1  oG  B1  f 1q  N R G GG v1o  o   N R G G R B1 BG1 B1R oG G v r1q f  1q 1q 1 1q  o  work were based upon their exemplary nature, their rela(95) N G B1 v1qB1o   BNN1 R GB1GG vN1ov o   NNRNGvGG R B1  BBNG11 R B1  o B1 BB11 r(94) N G N N N N RGGBG 1q  1 r tive commonality, in the experience of the author, as seen R Gvv1o RGBv G Grv1qG  B1 G R BGR BN1  f rR1q 1q   f  RGG 1oR fB  B1r1qfB1  1q B1 Go qG 1oof   1q  o  G G  f v 1 B 1 GG (94) G 1 o 1q R G GGNv1q1o f  o NN R G GG vB11o  f1  N RBBG11 oG R B1 (95)    r N in the field and relative simplicity.

In regards to the latter, G B1  f  1q (94) v B1o   NG v1o o   N R B1 BG1 B1  o   B1 r1q N N N (95) R B1 BG1 B1 NGGf  v1q r R  v    r (94) G v1q  f   G v1o  f   1q the BN 1q 1qN o  B1  oNN B1 N o  1G G B 2 B1 B2 simplifying assumption of planarity, as applicable to GGB 1o N G 1 G v r2q  G G v 2q  N R G GG v1o  f   N R G G R B1 BG1 BR  f  RGGGvv1o2 o fo RRB1G GRB2B1  f GB2 ro1q three r1q  o  1q 1q  f  of the four applications presented, can readily be dis1 f (96) B1 B (95) N G N G N G B2 1 r B2 N N N  R R  v carded for more complex modeling endeavors in R3.

N G N NR GGB1G v1qB2f  B1N R G BG N N N G B G 1q f B1 N v2q  vG R o1  B1   r RBB12  f  G  R v1oo fo  R RBG1R GRGGB2vB1q1  Gff  G G2 rvG2q1o  B2  orR o  RG 2o Br2  o   2 q  G v 1q qo 1q  f  G G 2 (95) G B G 1q B2 B2   N r2q  R G GGNv2q 1oo f NNR G GG vB21 o o1   N RBB11G f G R B2 (96) N G B2  o   (95) applications presented, in the view B2 N N N The simplest of the four G v1qB2f   N G v1o  f   N R B1 BG1 B1  f   B1 r1q N (96) R B2 G B2 Go v1q f r2q G v1o f  R B G B f  r1q G v 2q  o   G v 2o  o   B2 N  N1 G   N B2 B2 the author, is that of the equations of motion for planar GB2 1 N B2 N R G v B2 v  N R N G N G of  v    r R  R R   v  r (95) BG G B2B2  oG  o  GGG 2o2o  of  2q  R G G v2 o o  R G R B2 G B2  oG G 2Grq2q 2 B2  f 2 q  f  2q 2q  o  (97)nature of the applicavehicle dynamics. The simplified B2 B2 (96) N G N G N G B B N N N v2qB o2   BN2 R GBG2 v2 ov o  NR GvG R B2  BG2 R B2  o 2 B2 r2q 2 r N N N RGGBG N NG N 2  R R   v  r G G B G 2 q v  v    r R 2q f 2 o f B f        R v2 o o N R GG R B2 GNB22  oG  2r2qB2  BGR qG 2oof B2 G  f  2q  o  G GG 2o 2 2q 2 G G (96) GBv oo BN2  f  2 qGG G 2q B2 R G GNv2 q  f   N R G G vB22o  f   RBG2 R B2 (97) r2q  N G  B2  f   (96) v  v    r R B2 B N N N G 2o  o  B2  o  B2 2 q  2o r N N B2 BG2 (97) R B2 G B2 NGGf v2q www.collisionmagazine.com Collision Magazine - Volume 14 Issue 2 29  v    r R G v 2q  f   G v 2 o f   2 q G all G B2  oN 2Bq2 first component 2q  o  are 2o B2 N o vectorBN 2equations.

The term, in each equation, is the B2 B2 NThe Gprevious N four G equations R v  v    r G 2q G 2 o f  B2 G B2  f  2q f   R  R R   v  r G G G B G 2q 2 q f  2o  f  subscript 2 referenced B2  f  respective normalNdirection velocity.B It is the first component term, from each B2 N G G 2 (97) Rused BNN2 following R G GG v2oterm,  NR R B2ofequation,   v2in  r N N N allN vvector equations.

The in is the GB2Gfirst G each G 2q q  fcomponent f  definition B f Bthe B     G equation, that is the coefficient of restitution.

2  G v2 o f   R B2R G GGvB22q ff  rR  R Gequations.

R B2 G2 B22The  r2q component G 2q  f The 2are qG G v 2o vector f  is the  f   first previous four equations all term, in each (97) equation, is the normal direction velocity.

It first component term, from each Volume 14 Issue 2.indd 29 10/9/2020 3:43:08 PM B2 B2 N N N (97) R v  v    r G G B G 2 q 2 o f  B2  fvelocity. respective subscript referenced It is the first component term, from each BN  B2 N N 2q N normal N direction 2 2 wing definition of the coefficient of f restitution.

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tion, however, was still useful in regards to showing the process of developing the relevant kinematic relationships, applying the equations of motion, working between the reference frames of a simple two frame system and showing that the body frame expression of Newton’s 2nd law of mechanics is not simply one of force being equal to mass of the object multiplied by the body frame expression of the translational acceleration vector. This simple model can readily be extended by considering forces such as those generated by the driveline, suspension and tires. Increasing model complexity does not change the underlying form of the applicable vector-matrix relationships but does require the following threefold change in the modeling approach. For the planar case, the use of a single angle, the heading angle, is sufficient for parameterizing the DCM between the inertial frame of reference and the body frame (as well as its transpose). In R3, however, there exist three rotational degrees of freedom that must be accounted for within each DCM for each frame to frame coordinate transformation. While the details were not presented in the subject series of papers, such a parameterization can be

The presented forms of the CRASH3 model for oblique collisions and for planar collision analysis share certain similarities.

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readily achieved through the use of Euler angles, quaternions or other approaches. The choice of parameterization does not change the underlying validity of the kinematic and dynamic relationships in regards to the general DCM. The second change is the required inclusion of additional frames of reference. As an example, increasing the model complexity by accounting for tire forces (beyond the simplest inclusion as externally applied loads) would require the establishment of a tire fixed frame of reference for each tire coupled with the development of the DCMs for transforming the tire fixed frame kinematic quantities into the body and inertial frames of reference. Finally, increasing levels of complexity typically generate additional model parameters that require quantification (e.g. roll stiffness for a model incorporating rotational motion of a vehicle body about its longitudinal axis).

The application example falling on the opposing end of the simplicity scale, in the view of the author, was that of rigid body accelerometry. This opinion is based upon the fact that the equations developed were for the case of R3.

The equations developed for this example application were bounded in that the presentation was purposefully terminated at showing the relationship between the measured translational acceleration of any peripheral body-fixed triaxial accelerometer block and the translational acceleration at the center of mass of the body. The actual application of this relationship clearly requires the presentation of additional theory, which extended beyond the scope of the subject work. Such theory includes, but is not limited to data acquisition, signal filtering, methods for using redundant signal data and determining the correct sign for the angular velocity vector components. This information is readily covered elsewhere. 5,6,10 One point of note is that certain methods employ an accelerometer array that use biaxial accelerometer blocks for at least one peripheral location on a rigid body. Such an approach does not invalidate the equations developed in the subject work in that the equation and terms corresponding to the excluded axis can simply be set to zero.

The presented forms of the CRASH3 model for oblique collisions and for planar collision analysis share certain similarities. Both approaches are used for modeling collisions, utilize equations of mechanics and use a single time step in modeling the collision response. They are not, however, equivalent and alternative represents of the collision modeling problem. The CRASH3 model inherently uses a linear empirically derived relationship between the collision force magnitude and the resultant planar residual damage profile to determine an estimate for the internal work absorbed, for each collision partner, while the planar collision analysis method is silent on such a relationship.

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Secondly, the presented form of the CRASH3 model only considers the closure phase while the planar collision analysis method, specifically when the constraint condition for the coefficient of restitution is used, models both the closure and separation phases. In practice, the geometric centroid of the area encompassed between the direct damage portion of a residual damage profile and its normative position is used as the location of the point q, in the body frame of each collision partner. This differs from the planar collision analysis approach, which places the point q on the common collision interface at the terminus of closure. Finally, the presented formulation for the CRASH3 model, in terms of its derivational approach, does not account for the centripetal acceleration component nor the second component of the cross product term between the collision force and the moment arm of the same. The original paper 7 also postulated an approach for evaluating oblique collisions that involved using a collision impulse ratio constraint. A problematic aspect in such regards does not rest with the theory of such an approach but rather in regards to the standard reporting of residual damage profile data from controlled collision testing (measurements reported along the relevant undeformed test vehicle principle axis only rather than a reporting of the biaxial nodal residual deformation). Additional evaluation beyond the scope of the subject work is most likely required.

The collective form of the planar collision analysis equations presented herein is not singularly unique. The translational and angular impulse-momentum relationships given under equations (84), (85), (88) and (89) involve, as system unknowns, the translational impulse components expressed in the G frame. Ishikawa, 11,12 in his derivations, starts with the N frame expression of the translational impulse components. The N frame, for the single time step approach, undergoes no acceleration or rotation during the collision. This meets the requirements for an inertial frame of reference. A consequence of this finding is that the direct use of the N frame as the inertial frame of reference and not using a separate G frame is an entirely valid approach. Ishikawa, rather than using the impulse ratio coefficient for the second constraint condition, uses a tangential coefficient of restitution followed by the determination of the former. Carpenter and Welcher 13 present a method for distinguishing non-sliding collisions from sliding collisions. In their method, the second constraint condition, initially, consists of a tangential velocity constraint. The normal and tangent components of the collision impulse that are determined from the initial evaluation are checked against the stick-slip condition of |NPt| ≤ µs |NPn| (where µs is the static friction coefficient). Failure to meet this condition is indicative of a sliding collision and results in replacement of the second constraint condition with NPt =

α µ NPn, where µ is the kinetic friction coefficient and α = ± 1 is used to evaluate the correct direction. Zhou et al. 14 present an extension of the planar case with the introduction of a roll degree of freedom. Their modeling approach requires separation of the total mass of a vehicular collision partner into sprung and unsprung components along with the quantification of the lumped parameter equivalent torsional spring and torsional damper that are used to model the suspension coupling between the sprung and unsprung masses. This approach can readily be extended, fully for R3, as per a number of commercially available and commonly utilized, impulse momentum based, accident reconstruction and simulation software packages.

This represents the conclusion of the subject series of papers and one would be remiss to not provide a few conclusory comments. The subject series of papers were not intended to and did not present novel applications or novel findings. Instead, the objective was the presentation of the long-established and well known principles of rigid body kinematics and dynamics, starting from basic principles, and a select set of applications of these principles to the context of accident reconstruction. Presentations of the kinematics and dynamics can readily be found in a virtually innumerable number of journal articles and textbooks.

A number of presentations, however, employ symbolic nomenclature systems that do not include, within the variable names, the explicit references to the derivative frames and the reference frames in which the variables are being expressed. Such presentations, in the view of the subject author, make it particularly difficult and unnecessarily so, when it comes to understanding the relevant derivations and to implementing the results for context specific applications. It is hoped that the nomenclature system employed in the subject work, while increasing the complexity of the appearance of the variables and equations, aids in the preclusion of confusion as to the derivative frames and expression frames that are being employed.

The importance of proper frame referencing can especially be seen when working with multiple frames of reference and when it comes to defining the various forms of the derivative kinematics expressions. These expressions are different for the general case of a two reference frame system in which a point in question is fixed in neither of the two frames. A number of these definitions become reductive when the position of a point is fixed within one of the frames (e.g. within a body frame). However, in the view of the subject author, and contrary to traditional pedagogical approaches employed within most engineering curricula, the derivation of simplified relationships from more complex forms rather than simply providing the simplified relationships produces a more thorough understanding of the

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topic, generally, and of the limitations incumbent within the simplified formulations. As a contextual example, one often finds planar collision mechanics presentations in the form of a listing of component equations without an explanation as to how the basic equations were obtained from the more general case of R3. Such an approach, while meeting an objective of presenting component equations, does not aid in understanding the derivational basis of the same nor the mechanics by which one would remove any or all of the simplifying assumptions when it comes to the evaluation of situations in which those assumptions are not necessarily valid. The presentation of the relevant equations in vector-matrix form, as appropriate, mitigates a good portion of this issue secondary to the fact that such equations are the same in vector-matrix form, irrespective of the spatial order under consideration. The generation of the resultant component equations simply requires the ability of being able to carry out the incumbent vector matrix operations and by knowing which terms should be zero valued for simplified cases.

2. Singh J (2020) A review of engineering mechanics applied to accident reconstruction – Part 2: dynamics.

Collision 3. Jazar RN (2008) Vehicle Dynamics, Theory and Practice. New York: Springer, pp. 219-279.

4. Jazar RN (2011) Advanced Dynamics, Rigid Body, Multibody and Aerospace Application. Hoboken, New Jersey: John Wiley & Sons, Inc., pp. 695-864.

5. Alem NH and GL Holstein (1977) Measurement of 3-D motion. UMTRI Paper No. UM-HSRI-77-46, International Workshop on Human Subjects for Biomechanical Research, New Orleans, Louisiana, October 18, 1977.

6. Park S and SK Hong (2011) Angular rate estimation using a distributed set of accelerometers. Sensors 11: 10444-10457.

7. McHenry RR (1975) A comparison of results obtained with different analytical techniques for reconstruction of highway accidents. SAE Technical Paper The key points of the dynamics work is threefold. The No. 750893. first is the fact that the statements of Newton’s 2nd law of 8. Rose NA, SJ Fenton and RM Ziernicki (2004) An motion as either F = ma or in its translational impulseexamination of the CRASH3 effective mass concept. momentum form is that such a statement is valid only SAE Technical Paper No. 2004-01-1181. for an inertial frame of reference. The second is that the equation for rotational dynamics, Euler’s equation, is most 9. Brach RM and RM Brach (2007) Vehicle Accident Analysis and Reconstruction Methods (1st ed.). easily implemented in the body frame secondary to the Warrendale, Pennsylvania: Society of Automotive Entime-invariance, for a rigid body, of the matrix associated gineers, pp. 103-152. with the mass moment of inertia tensor. Finally, the expression frames of dynamics equations are readily change- 10. Kok M, JD Hol and TB Schon (2017) Using inerable by means of the application of the appropriate kinetial sensors for position and orientation estimation. matic transform (i.e. premultiplication by the appropriate Foundations and trends in signal processing 11(1-2): DCM).

1-153.

It is hoped that the applications discussed, while limited in number, showed not only the broad applicability of engineering dynamics to the context of accident reconstruction but also showed the details, starting from basic principles, of the manner in which one obtains the final forms of the relevant relationships that are found elsewhere in the literature.

References 1. Singh J (2020) A review of engineering mechanics applied to accident reconstruction – Part 1: kinematics.

Collision

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11. Ishikawa H (1993) Impact model for accident reconstruction – normal and tangential restitution coefficients. SAE Technical Paper No. 93065.

12. Ishikawa H (1994) Impact center and restitution coefficients for accident reconstruction. SAE Technical Paper No. 940564.

13. Carpenter N and J Welcher (2019) Inter-vehicular sliding friction and crush energy losses in impulse momentum planar collision. SAE Technical Paper No. 2019-01-0422.

14. Zhou J, J Lu and H Peng (2007) Collision model for vehicle motion prediction after light impacts. 20th IAVSD Symposium, Dynamics of Vehicles on Roads and Tracks, Berkley, California, August 13-17, 2007.

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Driver Steering Demands Following a Steer Tire DEFLATION on a Commercial F. Tandy, Jr. & Jason W. Colborn Vehicle Donald Tandy Engineering & Associates, Inc.

A

bstract This paper discusses the results of known scientific studies which have been conducted where a steer tire on a commercial vehicle was purposely aired out in order to evaluate vehicle controllability. Additional tests on a semi-tractor without a trailer were also performed specifically for this paper.

The first test was conducted by deflating a tire of a semi-truck while at rest before accelerating the vehicle up to speed and documenting the disintegration of the tire, using on board video equipment and a computer with transducers to measure driver inputs as well as vehicle responses during the testing. In the second test, an air out condition was created using a special test device that fired five twelve-gauge shotgun shells at a steer tire sidewall of the semi-truck while it was traveling at freeway speeds. Results from these tests, as well as other studies, show that a rapid steer tire (i.e. a tire on the front axle) air out event creates a slight pull to the side of the deflated tire which requires a corrective steer to maintain a straight-ahead course. Absent an odd occurrence such as a tread wrapping an axle or a pre-existing mechanical problem with the vehicle, a steer tire air out is a controllable event.

Introduction Commercial vehicles like medium duty and semitrucks require a large number of tires due to their size and weight. Like any other vehicle component, a tire can be disabled and separate or, most commonly, completely air out. There have been several recent articles published concerning the effect of a tire tread 34 Collision Magazine - Volume 14 Issue 2

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separation on vehicle handling of various utility vehicles, pickup trucks, vans, and passenger cars 1-12 and a paper written specifically discussing tire air outs on light vehicles. 13 One paper used computer modeling to study the effects of wheelbase and steering compliance on the ability to control a heavy commercial vehicle. 14 However, there have been fewer studies on what occurs when a steer tire is disabled in the real world on heavy vehicles such as a commercial vehicle or a semi-truck. What is known is that when a steer tire airs out, it leaves evidence on the roadway due to the weight of the vehicle pushing the rim down on the deflated carcass which moves against the ground making what has been referred to as a “waffle mark” (see Figures #1 and #2 for real world examples). In support that a steer tire disablement on a commercial vehicle is controllable, the Commercial Motor Vehicle Driver’s handbook for various states discuss what to do in the event of a steer tire disablement. An example of this comes from the Texas 2014 Commercial Driver’s License Manual and instructs drivers to “hold the steering wheel firmly” and to “stay off the brake” which is typical language for the other states.

This is good advice, namely to obtain directional control first before braking. So, this begs the question: “Why tell a driver what to do if a tire disablement is uncontrollable?” This paper will discuss what scientific tests have been conducted over the years in order to determine driver demands during a steer tire disablement on a commercial vehicle. Unscientific information such as internet videos found on sites like YouTube that some may claim show that a steer tire deflation on a commercial vehicle is uncontrollable will also be discussed.

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Figure 1: Example of a “waffle mark” created by a deflated steer tire on a commercial dump truck.

Figure 2: Example of a “waffle mark” created by a deflated steer tire on a commercial concrete mixer truck.

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Scientific Studies Conducted to Evaluate the Controllability of a Steer tire Air Out on a Commercial Vehicle There have been studies over the years on instrumented and non-instrumented commercial vehicles to evaluate the driver’s inputs and vehicle responses during a steer tire disablement. Michelin Tire Corporation ran subjective tests on several heavy vehicles. Government agencies such as the Federal Highway Administration (FHWA) and the National Transportation Safety Board (NTSB) have conducted test programs involving steer tire disablements on heavy vehicles. In addition, companies such as SEA, Inc. and Bridgestone Americas have conducted their own tests on semi-tractor trailers where steer tires were deflated at speed and drivers were instructed to steer to maintain their lane of travel. Tests were also conducted by these authors on a ‘bobtail’ semi-tractor (i.e. tractor without a trailer attached). All of these studies are discussed below and readers should refer to those studies which are published in the references for more information.

Figure 4: Image taken from Michelin training video showing the medium box truck after the right steer tire was aired out.

Michelin Tire Series Called “The Critical Factor” Michelin Tire Corporation has also put forth training videos 15 which show how to handle a steer tire air out on a motorhome (see Figure #3), a medium truck (see Figure #4), and a semi-tractor trailer (see Figures #5 and #6).

These videos, which Michelin purports to be applicable for all motor vehicles, show that when a steer tire airs out, it creates an increased rolling resistance or drag which applies a yaw moment on the vehicle. Maintaining the lane of travel requires the driver to input a counter steer away from the disabled tire. Although these videos did not emFigure 5: Image taken from Michelin training ploy instrumented vehicles, analysis of the videotape of the video showing the semi-truck and trailer that was driver’s steering inputs show that in these tests the drivers tested. were able to maintain their lane of travel. Michelin also recommended not to apply the brakes but rather apply some throttle during the event.

Figure 3: Image taken from Michelin training video showing the moment the motorhome left steer tire was aired out.

Figure 6: Image taken from Michelin training video showing the aired out right steer tire on the semi-truck and trailer.

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Federal Highway Administration Testing on Loaded and Unloaded Semi-Trucks and Trailers In 1974 and 1975, The FHWA contracted Ultrasystems, Inc. under U.S. D.O.T. contract DOT-FH-11-8562 to study and published the results 16 of steer tire aired out tests on a loaded and unloaded semi-truck (see Figure #7 for photograph of test vehicle and Figure #8 for photograph of tire air out test device).

capability study by Taborek 17 which concluded that drivers could input up to 80 foot-pounds or 960 inch-pounds of steering torque. Figure #9 shows the test data for a baseline run with a bias ply tire and Figure #10 shows the test data for a baseline run with a radial ply tire. Note the low levels of yaw rate which confirms the vehicle was under control during the event. The bar chart in Figure #11 shows the average steering torque required in all runs without aftermarket equipment or alignment changes.

After running 53 different test events, some of which were run to study the effect of aftermarket devices on the controllability of the combination, the authors concluded that: “It was discovered that for a flat, dry surface with the vehicle driving straight, the driver could control the combination vehicle which was loaded up to the gross combination weight of 73,820 pounds.”

Figure 7: Figure 5-1 from 16 reproduced showing semi-truck and trailer combination that was tested.

The Ultrasystems, Inc. study also published crash data for the years 1968 to 1972 where there were 160,000 truck crashes, a mechanical defect was reported in 11,119 or 7% of all crashes and 1,633 of those or 1% of all truck crashes involved a steer tire failure. Since police reported crashes exceeded 30,000,000 for that time frame, it is clear that commercial truck steer tire disablement crashes are rare events.

National Transportation Safety Board Testing on a Loaded 55 passenger Motorcoach In April 2007, employees of the National Transportation Safety Board (NTSB), along with personnel from tire and vehicle manufacturers, conducted fifteen tests at Continental’s Uvalde proving Grounds in Uvalde, Texas. 18 In these tests, the steer tire on a fully loaded 55 passenger Motor Coach Industries, Inc. model 102DL3 motorcoach (see Figure #12 for a photograph of an exemplar 102DL3 motorcoach) was induced to experience a tread belt separation or was purposely aired out rapidly with an explosive charge on the sidewall (see Figure #13).

Figure 8: Figure 5-2 from 16 reproduced showing tire deflation device.

The metrics used to evaluate the controllability of the combination were steering torque and path deviation. In the three baseline tests (i.e. vehicles without aftermarket devices installed), the lateral path deviations were low and the average steering torque was 23 foot pounds or 276 inch-pounds which was well below values cited by a driver 38 Collision Magazine - Volume 14 Issue 2

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Figure #14 shows the steering torque data recorded when the tire rapidly aired out in a right hand curve in Blowout Test 5 and Figure #15 shows another test where the tire aired out in a left hand curve in Blowout Test 8 (refer to 18 for data plots for the other test runs). The time channel was adjusted so that time equals zero is when the tire was aired out. The measured results showed that, although there was some variation in the magnitudes, the amount of steering correction needed to keep the motorcoach in the travel lane after the steer tire was disabled was similar to the steering needed to negotiate gentle curves or to counteract sustained crosswinds in most trials. This study also

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Figure 9 (above): Figure 6-9 reproduced from 16 showing test data for a baseline test with a bias ply tire.

Figure 10 (right): Figure 6-13 reproduced from 16 showing test data for a baseline test with a radial ply tire.

Figure 11: Bar chart comparing driver capabilities to steering torque demands for what is required to keep a commercial vehicle under control for the baseline runs. www.collisionmagazine.com

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concluded that on this specific vehicle, that application of brakes did not cause the driver to have to input more steering wheel angle.

SEA, Inc. Testing on a Loaded Semi-Truck and Trailer

Figure 12: Photograph of an exemplar 2009 Motor Coach Industries, Inc. model 102DL3 similar to the one that was tested.

In 2013, Anthony Cornetto, et al. of SEA, Inc. published the results of a study 19 where the left steer tire of a 2006 International 9400i tractor towing a loaded 48 foot flatbed trailer (see Figure #16). While driving at 55 miles per hour, the left steer tire was blown out with an explosive charge. The driver did not brake but steered to maintain control. The steering wheel angle data in Figure #17 shows that the driver made an initial steer of approximately 40 degrees to the right to compensate for the drag of the air out and then settled in at around 25 degrees in steady-state driving (the vertical black line indicates when the tire was aired out at time equal to zero). It should be noted that part of the SEA study involved comparing actual vehicle test data to computer simulation results and that is why there are multiple curves.

Figure 13: Figure 2 from 18 reproduced to show the shape charge and blasting cap used to initiate a rapid air loss.

Figure 15: A portion of Figure 18 from 18 reproduced to show the steering torque time history data for Blowout Test 8.

Figure 14: A portion of Figure 15 from 18 reproduced to show the steering torque time history data for Blowout Test 5.

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Figure 16: Figure 1 from 19 reproduced to show the test vehicle with deflated left front steer tire.

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Figure #18 shows that the time history of yaw rate, shown in red as “experimental,” has a very low magnitude. This confirms that the driver was able to maintain control and proceed basically straight during the event. In fact, the authors of 19 concluded that: “The tests results showed that the sudden failure of the left steer axle tire of the test vehicle was a controllable event.”

Bridgestone Americas Tire Corporation Testing on a Loaded Semi-Truck and Trailer In November 2015 Bridgestone Americas conducted tests which have been disseminated but not yet been published where Figure 17: Figure 2 from 19 reproduced to show steering wheel the left steer tire was disabled on a 2005 angle time history (note that “Experimental” shown in dark Freightliner while towing a 53’ Great Dane trailer loaded to a combined weight of apblue is the data from the actual vehicle test). proximately 56,000 pounds (see Figures #19 and #20). The truck was instrumented with video and data acquisition equipment to record channels such as speed, translational accelerations, angular rates, steering wheel angle and steering torque (see Figure #21 for a photograph of the steering wheel transducer). While driving between 55 and 59 miles per hour, the left steer tire was disabled with a sliding spike that would retract away from the truck after impact with the tire (Figure #22 and #23).

Figure 18: Figure 6 from 19 reproduced to show yaw rate time history (note that “Experimental” shown in red is the data from the actual vehicle test).

Figure 19: Test vehicle and trailer. www.collisionmagazine.com

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The testing began with a series of lane changes lane through a 1000 foot slalom (Trial 1). Then multiple runs were made to hit the spike and initiate a rapid air loss. Figure #24 shows an example of how fast the air loss occurred and Figure #25 shows the combination at its point of rest with the left steer tire deflated and debeaded. Figure #26 shows the left steer tire post-test. Attempt 5 (Trial 2) was successful in producing a rapid air loss at a time of 131 seconds (see Figure #27 in for driver inputs and vehicle response data and Figure #28 for steering torque time history).

In summary, during the three rapid air loss tests the peak steering magnitude required to maintain the lane was between 20 and 60 degrees. The peak steering torque ranged between 50 and 60 inch-pounds. These are deCollision Magazine - Volume 14 Issue 2 41

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Figure 20: Test trailer loading.

Figure 21: Steering angle and torque transducer.

Figure 22: Tire disablement spike and gate with direction of travel.

Figure 23: Tire disablement spike with direction of travel.

Figure 24: Distance from spike to 70% air loss is approximately 12 feet.

Figure 25: Test vehicle and trailer at point of rest after rapid air loss in right steer tire.

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mands that are well within that of human capabilities.

The low levels of yaw rate and lateral acceleration confirm that the driver was able to keep the lane of travel.

Figure 26: Left steer tire after removal.

A tire change was performed and the following run (Trial 3) produced a slow air out. A tire change was performed again and a rapid air loss was then achieved on the next attempt at a time of 566 seconds (Trial 4). See Figure #29 for driver inputs and vehicle response data and Figure #30 for steering torque time history for this run. Another tire change was performed and a rapid air loss was achieved at a time of 528.5 seconds on the final trial (Trial 5). The time history data plots are for this run are shown in Figure #31 and Figure #32.

It should be noted that the act of running over the spike apparatus at 55 miles per hour creates a disturbance in itself. This was investigated via an overlay of three runs aligned in time to the largest rapid air loss event. The

Figure 28: Steering torque data from Test 2 (Attempt 5).

Figure 27: Driver input and vehicle response data from Test 2 (Attempt 5). Note the ~40° right steer followed by a ~60° left steer and 3 degrees/sec yaw rate over 4 seconds after the rapid tire air out.

Figure 30: Steering torque data from Test 4 (Attempt 1).

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Figure 29: Driver input and vehicle response data from Test 4 (Attempt 1). Note the ~20° right steer followed by a ~45° left steer and 3 degrees/sec yaw rate over 4 seconds after the rapid tire air out.

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Figure 31: Driver input and vehicle response data from Test 5 (Attempt 2). Note the ~20° right steer followed by a ~30° left steer and 2 degrees/sec yaw rate over 4 seconds after the rapid tire air out. www.collisionmagazine.com

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results showed that running over the spike without a tire disablement imparted a small 10 degrees of steer magnitude but a torque of approximately 20 inch-pounds or about one third of that experienced during a rapid air loss.

The results can be seen in Figure #33.

Testing on a Freightliner ‘Bobtail’ Semi-Truck Testing was conducted by these authors specifically for this paper on a 1995 Freightliner FLD120 tractor (see Figure #34). The tractor was equipped with data collection and video equipment in addition to the driver. No trailer was connected to the tractor resulting in what is commonly called a ‘bobtail’ configuration. This was a high mileage vehicle and was tested in this condition (i.e. the steering gear and other parts were not replaced before the test).

Figure 32: Steering torque data from Test 5 (Attempt 2).

Tests were performed on the back straightaway of Texas World Speedway in College Station, Texas. The back straightaway is essentially flat and approximately two thousand feet long. The non-banked or apron portion of the turn before the straight was used to accelerate to the desired test speed (see Figure #35).

Two tire disablement tests were conducted on this vehicle.

The first test involved driving the vehicle with a deflated

Figure 33: (Overlay plot of Test 2, Attempt 3,4,5). Negligible steering and ~60 inch-pounds peak steering torque during rapid air loss and ~20 inch-pound traversing spike without air loss.

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tire from a stop. The second test involved rapid air loss while driving in a straight line.

Accelerating from a Stop with the Left Steer Tire Fully Deflated For the first test, the valve core of the left steer tire was removed while the tractor was at rest. The tractor was then accelerated first through a left-hand turn and onto the straightaway. At the beginning of the straight section the truck reached 50 miles per hour and the tire began to disintegrate. Approximately 35 degrees of steering to the right was required to drive the vehicle straight during this time. The driver lifted off the throttle and the truck was brought to a stop in a straight line. Figure #36 shows the vehicle and tire post-test and Figure #37 shows the test data time histories. Peak steering angle was approximately 80 degrees during straight line deceleration with light (~0.15g) braking. Higher magnitude of steering was required during braking since the radius of the deflated

Figure 36: Post-test condition of the left steer tire and vehicle body damage after the run-flat test.

Figure 34: Freightliner in ‘bobtail’ configuration as tested.

Figure 37: Driver input and vehicle response time histories for the run-flat tire test on the ‘bobtail’ semi-truck.

Figure 35: Back Straightaway of Texas World Speedway. www.collisionmagazine.com

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tire is less than that of the inflated tire on the other side of the axle. With equal braking torque, a higher force at the ground is generated on the left side, thereby creating a moment on the vehicle as well as an unbalanced force which turns the front wheels to the left through compliance in the steering system. This data supports the theory that a truck driver should not apply brakes in the event of a tire disablement.

Demands During a Tire Disablement Compared to Driver Capabilities

A study by the National Highway Traffic Safety Administration 20 found that peak steering torques of 33.9 footpounds (over 400 inch-pounds) were recorded in tests of drivers in limit handling maneuvers. Figure #40 is a bar chart that summarizes the comparison of test results to driver studies. The data clearly show that, absent an odd occurrence, responding to an air out of a commercial truck Rapid Steer Tire Air Out Test at Speed steer tire is well within the capability of a normal driver.

In order to initiate rapid air loss for the second test, a mul- This finding is not surprising since, according to the U.S. tiple barreled canon was manufactured which used several Department of Transportation, 21 there were approximateexplosive charges to propel lead shot into the sidewall of ly 415,000 police-reported crashes involving large trucks the tire. This device was electrically controlled via a foot and buses in 2015. Over 75 percent of these crashes were pedal switch operated by the driver. In this particular test, coded as a collision with a vehicle in transport as the first the device produced complete loss of air within a tenth of harmful event. Vehicle-related factors were found in apa second. proximately 6 percent of all of the 4,050 large truck faThe truck was first accelerated through the turn and onto tal crashes of which only 49 or 1.2 percent were related that the straight for the second test with all tires inflated. The to tires. This is a very small number considering the 22 that the U.S. Department of Transportation reported in straightaway was entered at approximately sixty miles per hour and the blowout device subsequently activated. there were nearly 6,300,000 police reported crashes in to¬¬The left steer tire deflated quickly causing the truck tal in this time frame of which 32,166 involved a fatality. to pull slightly to the left at a peak yaw rate of approxi- The Use of YouTube and Other Internet Type Videos to mately three degrees per second. In response, the driver Assess Vehicle Controllability slowly steered to the right to a peak of approximately 75 degrees. The tractor was then allowed to coast for twenty- Today’s internet provides a host of media platforms from two seconds which required an average steering input of which videos of vehicle crashes can be viewed and downapproximately 40 degrees. Light braking was then applied loaded. Figures #41, #42, and #43 show a time sequence to bring the truck to a stop. Figure #38 shows the vehicle of what is described as a right steer tire blowout on a truck and tire post-test and Figure #39 shows the test data time on the YouTube site. It is clear that the truck does veer to histories for this test. Note that the yaw rate trace shows the right, strikes a guardrail, and overturns. Such videos the driver exiting the turn and going straight at just over cannot be used to determine if the event was in fact con147 seconds at which time the air out device was triggered. trollable. There are three factors that must be studied in any crash are; (1) driver, (2) vehicle, and (3) environment: The Driver: Just because a driver loses control in a given situation does not mean that the situation is uncontrollable. Drivers may be tired, not paying attention to the driving task, or they may not do what is proper in a given situation. These videos provide no information to important factors such as what the driver is doing with the steering wheel, how fast they are traveling, and how hard they hit the brakes if they hit the brakes at all to name a few.

In order to remove this unknown, scientific testing such as that described in this paper records using video and/or instrumentation what the driver is doing during the test.

Figure 38: Post-test condition of the left steer tire after the rapid air out test.

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The Vehicle: The vehicle is an important part of the system. In the tests described above, the vehicle condition is a known quantity. Test drivers evaluate the vehicle prior to the test to make sure there are no issues with the condition of the unit. A quick clip from a YouTube video provides no

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Figure 39: Driver input and vehicle response time histories for the rapid air out test on a ‘bobtail’ semi-truck. www.collisionmagazine.com

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Figure 40: Bar chart comparing driver capabilities to steering torque demands for what is required to keep a commercial vehicle under control in the event of a steer tire air out. insight into the condition of the vehicle or if any components broke or were contacted by a tire tread during the event.

The Environment: Scientific tests are usually run on a test track where the slope and condition of the roadway is known. Real roads have slopes and surface irregularities that sometimes cannot be controlled for in evaluating vehicle or driver performance. Objects in the road and weather can also be factors as are other vehicles on the roadway.

Based on all the unknowns discussed above, it is clear that using a crash video from the internet to claim a vehicle or situation is uncontrollable is unscientific. It would be the same as claiming that the the two screen captures found in Figures #44 and #45 MUST indicate that the brake system on the blue sedan has a defect since the vehicle crashed into the side of the white car as opposed to other causes such as, possibly, an innattentive driver.

Figure 41: Screen capture from a YouTube video showing a truck in the right lane.

Summary/Conclusions 1. Crashes involving truck steer tire failures are rare events. For the years 1968 to 1972 there were 160,000 truck crashes. A mechanical defect was reported in 11,119 or 7% of all truck crashes and 1,633 of those or 1% of all truck crashes involved a steer tire failure. More recently in 2015, there were 425,000 crashes involving heavy trucks in all of the nearly 6,300,000 police reported crashes. Only 49 or 1.2% of the 4,050 fatal truck crashes involved a tire event.

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Figure 42: Screen capture from a YouTube video showing a truck moving to the right.

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Clearly commercial vehicle tire disablement crashes are rare events.

2. Drivers with commercial driver’s licenses have special training. The CDL handbook for most states inform these drivers to hold the steering wheel firmly and to not apply the brakes in the event of a tire disablement.

Figure 43. Screen capture from a YouTube video showing a truck rolling over a guardrail.

Figure 44. Screen capture from a YouTube video showing a blue sedan just before crashing into the side of a white car after running a red light.

3. Scientific studies conducted over the years show that, under normal situations, an air out of a steer tire on a commercial vehicle can be controlled with an input of steering. The steering magnitudes and torques required are well within the capabilities of normal drivers.

However, special situations (i.e. suspension or steering parts breaking, or if the vehicle is in poor condition, or if the tread from the disabled tire wraps any of these components, etc.) can make it more difficult or in some cases impossible for a driver to maintain control.

4. If a driver applies the brakes in the event of a commercial truck steer tire air out, even more steering is typically required to maintain the lane of travel. The higher magnitude of steering is required since the radius of the deflated tire is less than that of the inflated tire on the other side of the axle. With equal braking torque, a higher force at the ground is generated, thereby creating a moment on the vehicle as well as turning front wheels through compliance in the steering. These data support the theory that a truck driver should not apply brakes in the event of a tire disablement.

5. The use of internet crash videos and the like is not a surrogate for real vehicle testing.

These YouTube type videos cannot be used to opine about controllability in a given situation. There are just too many unknowns in the events captured on these videos.

References

Figure 45. Screen capture from a YouTube video showing a blue sedan just after crashing into the side of a white car after running a red light. www.collisionmagazine.com

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1. Dickerson, C.P., Arndt, M.W., and Arndt, S.M., “Vehicle Handling with Tire Tread Separation,” SAE Technical Paper No. 199901-0450, 1999, doi:10.4271/199-01-0450.

2. Arndt, S.M. and Arndt, M.W., “The Influence of a Rear Tire Tread Separation on a Collision Magazine - Volume 14 Issue 2 51

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Vehicle Stability and Control,” ESV 2001 Paper No. 258, June 2001.

3. Gardner, J., “The Role of Tread/Belt Detachment in Accident Causation,” ITEC Paper 27A, 1998.

4. Arndt, M.W., Rosenfield, M.J., Arndt, S.M., and Stevens, D.C., “Force Response During Tire Tread Detachment Event,” SAE Technical Paper No. 2004-011075, 2004, doi:10.4271/2004-01-1075.

5. Arndt, M.W., Rosenfield, M., Arndt, S.M., “Measurement of Changes to Vehicle Handling Due to Tread Separation Induced Axle Tramp,” SAE Technical Paper No. 2006-01-1680, 2006, doi:10.4271/2006-011680.

6. Fay, R.J., Robinette, R.D., Smith, J., Flood, T., and Bolden, G., “Drag and Steering Effects from Tire Tread Belt Separation and Loss,” SAE Technical Paper No. 1999-01-0447, 1999, doi:10.4271/1999-010447.

7. Baker, J.S., McIlraith, G. D., “Tire Disablements and Accidents on High-Speed Roads,” Committee on Highway Safety, 48th Annual Meeting.

8. Klein, E.Z., Black, T.L., “Anatomy of Accidents Following Tire Disablements,” SAE Technical Paper 1999-01-0446, doi:10.4271/1999-01-0446.

9. Tandy, D.F., Tandy, K.T., Durisek, N.J., Pascarella, R.J., Carr, L., Liebbe, R., “An Analysis of Yaw Inducing Drag Forces Imparted During Tire Tread Belt Detachments,” SAE Technical Paper 2007-01-0836, 2007, doi:10.4271/2007-01-0836.

10. Tandy, D.F., Granat, K.J., Tandy, K.T., Durisek, N.J., Baldwin, J.W., Pascarella, R.J., “Vehicle Response Comparison to Tire Tread Separations Induced by Circumferentially Cut Tires and Distressed Tires,” SAE Technical Paper No. 2007-01-0733, 2007, doi:10.4271/2007-01-0733.

11. Tandy, D.F., Pascarella, R., Ault, B.N., Coleman, C., Tandy, K., “Steering and Handling Performance During a Full Tire Tread Belt Separation,” SAE Technical Paper No. 2011-01-0973, 2011, doi:10.4271/201101-0973.

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12. Beauchamp, G., Koch, D., and Thornton, D., “A Comparison of 25 High Speed Tire Disablements Involving Full and Partial Tread Separations,” SAE Int. J. Tran. Safety, 1(2):364-385, 2013, doi:10.4271/201301-0776.

13. Tandy, D.F., Ault, B.N., Colborn, J., Pascarella, R., “Objective Measurement of Vehicle Steering and Handling Performance When a Tire Loses its Air,” SAE Technical Paper No. 2013-01-0748, 2013, doi:10.4271/2013-01-0748.

14. Bernard J.E., Shapley, C.G., “The Effect of Vehicle Design on Post Blowout Controllability,” SAE Technical Paper No. 791042, 1979, doi:10.4271/791042.

15. Michelin Tire co. (1987). The Critical Factor. Available online at: http://www.michelintruck.com/assets/ movies/The_Critical_Factor.mov.

16. R.L. Anderson, R.A. Nidley, G. McCormick, F. Russoniello, “Control of Large Commercial Vehicle Accidents Caused by Steer tire Failures – Final Report,” DOT Contract No. DOT-FH-11-8562, August 1975.

17. 17. Taborek, J.J., “Steering Forces and Stability,” Machine Design, June 27, 1957.

18. Sind-Prunier, P., Yohe, L.L., “Human Performance & Vehicle Group Chairmen’s Report of Operation Testing,” National Transportation Safety Board, 2007.

19. Cornetto, A., Bayan, F., Dunn, A., Tanner, C., Wahba, R., Suway, J., et al., “Tractor-Semitrailer Stability Following a Steer Axle Tire Blowout at Speed and Comparison to Computer Simulation Models,” SAE Technical Paper No. 2013-01-0795, 2013, doi:10.4271/2013-01-0795.

20. Forkenbrock, G.J., Elsasser D., “An Assessment of Human Driver Capability,” NHTSA Report No. DOT HS 809 875, June 2005.

21. The National Highway Traffic Safety Administration, “Large Truck and Bus Crash Facts 2015,” FMCSARRA-16-021, November 2016.

22. The National Highway Traffic Safety Administration, “Traffic Safety Facts 2015,” DOT HS 812 384, 2017.

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crash·ol·o·gy THE SCIENCE OF CRASHES Wesley Vandiver

Robert Anderson

Collision Forensics, Inc.

Biomechanics Analysis

Toyota Vehicle Control History - The Basics of What You Need to Know

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ince their introduction in the 1990s, Event Data Recorders (EDRs) have become a common source of important data for those investigating or analyzing crash events. Unlike EDR recordings that are triggered as a result of an impact, Toyota Vehicle Control History (VCH) recordings are triggered by events other than impacts, such as driver behavior and activation of safety systems. In response to these triggers, the VCH acts much like a traditional EDR in that it reports pre-trigger data parameters (vehicle speed, etc.), but also reports posttrigger data parameters, that often provide fifteen seconds of data surrounding the triggering event.

The VCH data becomes important in situations where EDR data may not be available, including noncontact situations, as well as low speed and low deltaV impacts, such as pedestrian and bicyclist collisions, which often do not qualify for an EDR recording.

The VCH data can also add to an EDR report’s precrash data set. Example real-world cases when VCH data has proven useful are covered in the Toyota VCH Case Studies paper in this issue of Collision.

As outlined in the Testing and Validation of Toyota VCH Speed and Acceleration Data paper in this issue of Collision, the Toyota VCH was first introduced to the vehicle crash forensics community at the 2014 CDR Summit by Dave Hallman, of Hallman Engineering. Published details regarding the Toyota VCH were provided by representatives from Toyota Motor Corporation and Toyota Motor North America in 2016 and 2019 SAE papers. 1,2 Figure 1 lists the model start year and production start date of each Toyota, Lexus, and Scion vehicle

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equipped with VCH. This coverage information was determined by the model year start and production date start for VCH listed in the Repair Manuals on Toyota Technical Information System (TIS).

The VCH is a function that records control data (record data) when triggered by specific vehicle and driver behavior. The VCH is recorded in different storage areas designated by trigger group and it is possible to save records for up to a total of 140 triggers. If storage space is filled, the data is overwritten on a FirstIn, First-Out (FIFO) basis. Each trigger receives time information from the main body Electronic Control Unit (ECU), which is the time elapsed since the engine switch was turned on for the given ignition cycle.

Absolute time may be recorded for vehicles with factory-installed navigation systems.1 Like EDR data, the VCH data are recorded in nonvolatile memory within the Airbag Control Module (ACM). VCH data can be retrieved using Toyota Techstream, but it cannot be cleared using this application and disconnecting power from the vehicle will also not result in VCH data loss. 1,2 In addition to VCH data, additional Pre-Collision System operational data (PCS-O) and image data (PCS-I) became generally available in 2017 with Toyota Safety Sense (TSS). Any images stored as the result of a specific trigger, such as Automatic Emergency Braking (AEB), are stored in the camera sensor nonvolatile electrically erasable programmable read-only memory (EEPROM). Figure 2 shows an example of the multiple low-resolution images (15kb) that are stored in the forward-facing camera. 2

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Figure 1: Start of VCH coverage by production date and model year for Toyota, Lexus, and Scion vehicles Both the VCH and PCS data stored in the ACM, and the PCS image data stored in the camera sensor, can be retrieved using Techstream and a Vehicle Interface Module (VIM) such as the MongoosePro Toyota 2 cable [3], Nexiq USB-Link [4], or the Bosch CDR 900 [5]. The equipment set-up for obtaining VCH and PCS data through the DLC is shown in Figure 3.

As shown in Figure 4, and in the Toyota VCH Case Studies paper in this issue of Collision, VCH data can also be obtained directly from an ACM (without connecting to the DLC) using Techstream, a VIM, and a newly-developed cable designed by Crash Data Specialists. Stored images cannot be obtained when using the direct-to-module

method because, as previously discussed, the images are not stored in the ACM. The Bosch Crash Data Retrieval System and many other EDR tools are read-only systems that can collect recorded data but do not erase or write new data. Techstream is primarily a read-only tool; however, it is possible to erase images stored in the camera memory. This process is outlined in Figures 5, 6, and 7, and is an important piece of information for an investigator examining a Toyota that has been subjected to a prior VCH download.

For vehicles equipped with TSS 1.0, VCH data and PCS image recording is only triggered with AEB. For newer vehicles equipped with TSS 2.0, image recording has 6 dif-

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Figure 2: Images from AEB activation approaching a stationary target at 40 kph 2

Figure 3: Obtaining VCH data via the DLC

Figure 4: The cable for acquiring VCH data directly from a Toyota ACM.

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Figure 6: Techstream Event Viewer showing stored images on Key Cycle 11225

Figure 7: Techstream Event Viewer showing no images after using clearing utility

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Figure 8: Toyota Safety Sense 2.0 vehicles and features 7 ferent triggers, including PCS without AEB. The vehicles which may be equipped with TSS 2.0 are shown in Figure 8.

As addressed in the Testing and Validation of Toyota VCH Speed and Acceleration Data paper in this issue of Collision, a discrepancy has been observed between the ignition cycles reported by the EDR and VCH. It was thought that this discrepancy was the result of a longer time to boot up and increment the EDR counter. However, this behavior has not been demonstrated in testing. Rapid sequences of alternating ignition ON and OFF cycles have produced identical increases in the ignition cycle counts in both the EDR and VCH. Any ignition activation that increased the ignition cycle count in one record also increased it in the other. The testing did not reveal any method of increasing the ignition cycle count in the VCH that did not also increase the ignition cycle count in the EDR. This leads to reasonable speculation that the consistently observed disparity between the total ignition cycles reported in the VCH and EDR may stem from the vehicle manufacturing process.

As also outlined in the Testing and Validation of Toyota VCH Speed and Acceleration Data paper in this issue of Collision, the VCH time stamp depends on the time zone, daylight savings and auto adjust settings of the vehicle’s clock.

With the number of electronic control units (ECUs) in modern vehicles rapidly growing, the potential for stored data extends beyond EDR, VCH and PCS. For example, 58 Collision Magazine - Volume 14 Issue 2

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Figure 9 shows the operational history of the door locks in a 2017 Land Cruiser. This type of data reflects the continually expanding information stored in vehicles and accessible to investigators who are equipped with the knowledge and tools necessary for acquisition.

The concept of collecting digital data from vehicles involved in collisions largely started with the introduction of the Vetronix Crash Data Retrieval System in the year 2000. The availability of this system launched the need for investigators to acquire state-of-the-art equipment and attend new types of training. Since that time, crash inves-

With the number of electronic control units (ECUs) in modern vehicles rapidly growing, the potential for stored data extends beyond EDR, VCH and PCS.

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Figure 9: Techstream Event Viewer showing other types of stored data tigators have been continuously exposed to new technology for acquiring electronic data from vehicles, including Toyota Techstream, Berla iVe, diagnostic scan tools, etc.

Each of these technologies evolves rapidly and requires an ongoing commitment that includes equipment purchases and training for one to stay current. The adoption of these technologies into an investigator’s workflow, if ignored, places him/her in the position of not collecting information that could be relied upon in forming his/her opinions, and more importantly, could be key evidence in a case.

2. Lance Lewis, Barry Hare, Harold Clyde and Robert Landis, “Vehicle Control History: Data from Driver Input and Pre-Collision System Activation Events on Toyota Vehicles,” SAE International, Technical Paper 2019-01-5094 3. https://techinfo.toyota.com 4. http://www.drewtech.com/mongoose/toyota2.html 5. https://www.nexiq.com/Product/Detail/125032

References

6. https://www.crashdatagroup.com/cdr900/

1. Motomi Iyoda, Tom Trisdale, Rini Sherony, Daniel Mikat, and William Rose, Event Data Recorder (EDR) Developed by Toyota Motor Corporation, SAE International, Technical Paper 2016-01-1495

7. https://www.cmtoy.com/which-toyota-vehicles-havesafety-sense/

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Ignition “Cycology” Understanding Ignition Cycles Kent E. Boots

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bstract The majority of the vehicles supported with the Bosch Crash Data Retrieval (CDR) tool do not provide a date or time for recorded events in CDR reports. Most vehicles do however keep track of ignition cycles. The ignition cycles at the event and at investigation (data collection) are both part of the CDR report from many manufacturers even prior to 2013. Since 2013 each vehicle equipped with an EDR must record the ignition cycles at the crash and at the download (data collection). 1 The minimum range is 0 to 60,000 ignition 60 Collision Magazine - Volume 14 Issue 2

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cycles with an accuracy of +/- one cycle, and a resolution of one cycle. 2 Knowing how many ignition cycles there are at the event can be like a date stamp when you know the number of ignition cycles for a given vehicle per day. In most cases, first responders typically do not have any issues relating a CDR event to their crash.

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However, many times in civil cases investigators may not get vehicle access for months or even years after a crash.

When there is a non-deployment (ND) event present, the question has to be answered as to whether or not the event data is related to the crash. It might not be as simple as saying the CDR data matches the circumstances of my crash so it must be related data.

Definitions An ignition cycle is an incremental counter in the airbag control module (ACM) for the lifetime of the vehicle. The Bosch CDR Help file defines an ignition cycle as “Ignition power applied to and removed from the ACM.” It also defines “powered-up” as “the act of applying approximately 10V - 16VDC power source to the appropriate pins on a specific control module.” 3 The following definitions appear in the Bosch CDR Help file as well as NHTSA Part 563. 4,5 •

“Ignition Cycle, Crash means the number (count) of power cycles applied to the recording device up to and including the time when the crash event occurred since the first installation of the EDR. Also seen as “Ignition cycle at (event name).”

•

“Ignition Cycle, Download - The number of power cycles applied to the recording device at the time when the data was downloaded since the first installation or use of the EDR. Also seen as “ignition cycle at investigation.”

General Motors defines a power cycle as Off – Run – Crank – Run. 6 The ignition key is normally in the “Off” or lock position (See Figure 1). The driver turns the key to the “Run” or on position (See Figure 2). Then the driver turns the key to the “Crank” or start position until the engine is running (See Figure 3). The driver lets go of the key and it returns to the “Run” or on position (See Figure 2). When the driver reaches their destination, they turn the key to the “Off” or lock position (See Figure 1) which starts the next power cycle. Push-button start vehicles are similar in that the driver presses the button to turn the ignition from off to on, then to start the engine, and finally to turn the vehicle off.

Example The author investigated a case where the driver alleged he was traveling on the freeway when his tire went flat, he lost control, and the vehicle went into a ditch. The driver is the original owner and there were no other reported losses with the vehicle. The vehicle was repaired and being driven prior to crash data being collected. There was one nondeployment event present. The pre-crash data was not

consistent with the driver’s statements. There were 3552 ignition cycles at the event, and 7251 ignition cycles at the investigation, a difference of 3699. With an average of 6 to 7 ignition cycles per day that gives a date range of January 25th to April 25th, 2015. This is over a year prior to the alleged loss on July 7th, 2016. It would take an average of 35 to 49 ignition cycles per day just to be in the month of July of 2016 and take an average of 37.4 ignition cycles per day to “match” the date.

Background The CDR Software Help File has vehicle coverage notes that are specific to each Automobile Manufacturer. Starting in model year 2004 with the Chevrolet Malibu, the General Motors vehicle coverage notes included Note 6 which states, “A key is not required to image these vehicles through the DLC, so you can leave ignition off or on during imaging.” 7 Almost all currently CDR Tool supported General Motors vehicles, as of the 2020 model year, are Note 6 vehicles. Note 6 vehicles typically have two fuses that protect the ACM. One of the two fuses provides power to the ACM regardless of whether the ignition is on or not. Those fuses are sometimes labeled with the word “battery” in the description. The second fuse only provides power to the ACM when the ignition is turned on. Those fuses are sometimes labeled with the word “ignition” in the description. The ignition feed is a backup to a message from the CAN bus that the vehicle engine is running. Some newer Note 6 General Motors vehicles (e.g.

2016 Chevrolet Malibu) only have one fuse that protects the ACM. This single fuse provides power to the ACM regardless of whether the ignition is on or not. These vehicles (just like two-fuse Note 6 vehicles) require a power cycle and to receive a message from the CAN bus that the vehicle engine is running to increment the ignition cycle counter.

Note 6 vehicles do not require the ignition switch to be turned on for data collection, but it can be. Whether the ignition is on or off, the process of Crash Data Retrieval will not add an ignition cycle. If data collection is conducted by going direct-to-module no ignition cycles are added either. The only way to increment the ignition cycles is by going through a power cycle as previously described. The author has an ACM (SDM40) from a 2016 Chevrolet Malibu with a deployment event. The data limitations from the CDR File states that applying and removing battery power will not increment the ignition cycle counter (See Figure 4). This supports the fact the author has imaged that particular ACM direct-to-module numerous times without incrementing the ignition cycle counter.

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Figure 1: Ignition Switch in “Off” or Lock Position

Figure 2: Ignition Switch in “On” or Run Position point that non-deployment events could be cleared after 250 ignition cycles. Clearing non-deployment events after 250 ignition cycles only applies to certain General Motors vehicles. Older General Motors vehicles indicated in the data limitations that non-deployment events “will be” cleared after approximately 250 ignition cycles (See Figure 5). Starting in Model Year 2010 General Motors began using airbag control modules (SDM10) that do not clear non-deployment events after 250 ignition cycles. This is module family specific and not model year specific. Some 2010 model year vehicles may still clear non-deployment events. Earlier versions of the data limitations discussed overwriting events with another non-deployment event after approximately 255 ignition cycles (See Figure 6). The same CDR File viewed in the current version of the software discusses overwriting a non-deployment event if all records are full and the event is older than approximately 250 ignition cycles (See Figure 7).

Figure 3: Ignition Switch in “Crank” or Start Position Can a recorded event be “cleared” after a certain number of ignition cycles or are they there forever? 2013 and newer vehicles cannot overwrite deployment events. They must store up to two non-deployment events subject to certain criteria, and they can only be overwritten by other subsequent events. 8 You might have heard or read at some 62 Collision Magazine - Volume 14 Issue 2

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Discussion The ACM is “on” or powered-up when it receives approximately 12-volts (ignition on) from the vehicles electrical system. The circuitry becomes active (capacitors, transistors, ICs) and an ACM diagnostic runs. The ACM may receive a signal from the body control module or similar component that the ignition system is active. Typically, we would know an ACM diagnostic has run when the airbag light illuminates and then goes off (See Figure 8 & 9). The time it takes for the diagnostic to run varies by manufac-

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Figure 4: 2016 Chevrolet Malibu – SDM40 - CDR Report Data Limitations Excerpt Figure 5: 2009 Chevrolet Silverado – SDMCG - CDR Report Data Limitations Excerpt

Figure 6: 2010 Chevrolet Camaro – SDM10-Continental – CDR Report Data Limitations Excerpt, Revision 002 (CDR Tool 3.6)

Figure 7: 2010 Chevrolet Camaro – SDM10-Continental – CDR Report Data Limitations Excerpt, Revision 021 (CDR Tool 19.4.2) turer. One of the author’s personal vehicles takes approximately three seconds for this to occur. The ACM is “off” or powered down when there is no longer 12-volts present (ignition off). The circuitry becomes inactive and capacitors drop below a certain level. The ACM may receive a signal from the body control module or similar component that the ignition system is inactive.

The Bosch CDR Tool help file talks about EDR data volatility and states that when collecting data directly from an ACM it should be “…powered down and left to discharge any stored electrical power for at least 2 minutes after power is removed from the CDR Tool.” 9 This time period allows the capacitors to discharge.

For most vehicles, the process of collecting data adds one ignition cycle to the ACM. The ACM requires an active source of power for the purposes of crash data retrieval. Normally the preferred method of data collection is through the diagnostic link connector (DLC). During this process, the ACM receives power when the ignition is powered up either by turning on the ignition key, pushing the start button, or energizing an electric vehicle by some method such as stepping on the brake pedal. Turning the ignition key on to collect data would add one ignition cycle. As previously discussed, General Motors, Note 6 vehicles are an exception to this.

Push-button Start Vehicles Remote keyless systems (RKS) or remote keyless entry (RKE) vehicles use an electronic remote control as a means

to unlock a vehicle’s doors. The remote is sometimes called a key FOB. These vehicles have an electronic keyless ignition and a Push-button start. The transmitter is required to be in (or near) the vehicle for the system to operate.

Starting the vehicle with the Push-button or pushing it sufficient times for the ignition to be on will increment the ignition cycle counter. When and only when the Pushbutton is pushed is the ignition turned on (diagnostic) such that we could collect data or start the vehicle.

Start/Stop Vehicles A vehicle with a start-stop system or stop-start system will automatically shut down an internal combustion engine when a vehicle comes to a stop. The system then restarts the engine when the driver is ready to move again. The important thing to remember with this type of system is the ignition is still on even though the engine may shut down.

Other systems in the vehicle, including the ACM, remain powered because the ignition is still on. Consequently, a vehicle with this type of system does not increment the ignition cycle counter as long as the ignition remains on, even when the engine pauses and then restarts during a stop.

Remote Start Vehicles Some vehicles have a remote vehicle start function which allows the engine to be started from outside the vehicle.

This allows the engine to “warm up” and also allows for climate control systems to warm or cool the vehicle. Once inside the vehicle the driver must turn the ignition on

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Figure 8: SRS Light Illuminated During Remote Starting

Figure 9: SRS Light Off After Diagnostic During Remote Starting

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either by pushing the start button or inserting and rotating the ignition switch to the on position. The questions the investigator needs to be able to answer are two-fold. The first is whether or not remote starting a vehicle adds an ignition cycle. The second is whether the subsequent turning on of the ignition adds an additional ignition cycle.

Figure 10: Digital Meters Showing ACM Circuit Voltage with Ignition On

Figure 11: Digital Meters Showing ACM Circuit Voltage with Engine Cranking

Figure 12: Digital Meters Showing ACM Circuit Voltage with Ignition on, Voltage with Engine Cranking

Two different FCA (Fiat Chrysler Automobiles) were examined and tested. The first was a 2008 Jeep Wrangler Rubicon which has remote start capabilities and an ignition key. The Rubicon did not have any EDR events present so the ignition cycles at investigation were unknown. Initially the author remote started the vehicle while observing the dash lights. The SRS light illuminated and then went out. The ignition key was then inserted and turned to the on position.

The SRS light illuminated again and then went out.

It appeared that two diagnostics had run, but how many times had power been applied to the ACM? The Wrangler has an ACM that is protected by two separate fuse circuits; both located in the power distribution center (PDC) under the hood. Fuse M16 is the Run/Start fuse. It provides power to the ACM when the key is on, as well as when the engine is being started (cranking). Fuse M32 is the Run fuse and provides power to the ACM only when the key is on.

When the engine is being started (cranking) no power is provided to the ACM on that circuit. Both fuses were removed, and two digital volt meters were connected on the line (from power) side of the empty fuse sockets. The vehicle was remote started while observing the meter displays.

Both meters initially displayed approximately 11.5 volts DC (ignition on) when the ignition turned on (see Figure 10). When the engine turned over the M32-Run meter (shown on the right) dropped to effectively zero volts (see Figure 11). Once the engine was running both meters initially displayed approximately 11.5 volts (battery voltage) and then the voltage increased to approximately 14.5 volts as the alternator began producing power (See Figure 12).

Ultimately the key was placed in the ignition and turned on while the meters were observed; there was no visible change in the voltage. During remote starting, how many times was 12-volts potentially applied to the ACM; only once. So, in theory the process of remote starting the vehicle, and subsequently turning the ignition switch on would only add one ignition cycle. This theory was unable to be verified since there were no EDR events present.

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The second vehicle that was tested was a 2016 Dodge Charger SXT. This vehicle has remote start capabilities, remote keyless entry, and a Push-button start. The Charger was known to have a Non-Deployment (ND) event present so the number of ignition cycles at investigation was available. The ignition was turned on, CDR data was collected from the vehicle, and the ignition cycles at investigation were recorded. The ignition was turned off. The Charger was then remote started and after the engine was running the remote was used to turn the vehicle off. The ignition was turned on and then CDR data was collected.

The ignition cycles at investigation had increased by two; one for the remote start, and one for data collection.

The Charger ignition was turned off, and then the vehicle was remote started again. This time the push button was pushed sufficient that the vehicle could be driven. The ignition was then turned off. Following that, the ignition was turned on, CDR data was collected from the vehicle, and the ignition cycles at investigation were recorded. The ignition was turned off. The ignition was turned on and then CDR data was collected. The ignition cycles at investigation had increased by two; one for the remote start, and one for data collection. This showed that pushing the Push-button start after the engine was running did NOT add an additional ignition cycle. This is consistent with the voltage testing done on the Wrangler because once the 12-volts was applied to the ACM it was never removed until the engine was turned off.

One final test was conducted with the Charger. The vehicle was remote started and then CDR data was collected from the vehicle (while the engine was running) without turning the ignition on. This time the ignition cycles at investigation had only increased by one. It only increased by one because the vehicle had been started since the last data collection, but there was no additional ignition cycle added for data collection.

Electric Vehicles Some vehicles such as the Tesla Model 3 support three types of “keys”; an authenticated phone (app), a key card (proximity card), or key FOB. If there is no authenticated phone app or FOB in proximity, the driver can power up the vehicle by pressing the brake pedal within two minutes of scanning the key card on the center console. 10 Once the driver powers up the vehicle the ACM begins a boot sequence and when complete adds an ignition cycle to the internal count. For Tesla vehicles this process occurs in less than one second after power up. If the driver waits longer than two minutes to power up the Model 3 after applying

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the key card, authentication has to be done again in order to restart the two-minute period. 11 The Model 3 will power off automatically after being in Park for 15 minutes, even if the driver is sitting in the driver’s seat. The vehicle can also be manually powered off while sitting in the driver’s seat using the user interface (screen). The vehicle powers back on again if the driver presses the brake pedal or touches the touchscreen. 12 If the vehicle powers down, then so does the ACM. To increment the ignition cycle counter in the ACM the driver needs to authenticate and press the brake pedal to turn on the drive rails and power up the ACM. If the vehicle shuts down the driver would have to re-authenticate and step on the brake pedal to turn on the drive rails and power the ACM. Direct to Module When data is collected directly from the ACM (in or out of the vehicle) the power for the ACM comes from the CDR Tool. When the CAN+ interface is connected to an ACM via a direct-to-module cable the ACM receives power as soon as the power cord is plugged into the CAN+ interface. When collecting data from ACMs that require the “387 adapter” (F00K108387) power is applied to the 387 adapter and not the CAN+ interface. Power is only being sent to the ACM when the green LED illuminates on the 387 adapter. The author has seen instances where the green LED illuminated, went out, and illuminated again during the data collection. If there was any sequence of power up, power down during your imaging process, each has the potential to add an ignition cycle. When data is collected directly from the ACM using the CDR 900 interface, power is only supplied to the ACM when the LED on the tool transitions from yellow to green.

Typically, the data collection process adds an ignition cycle, however General Motors Note 6 vehicles are the exception to this. The author also used the Tesla EDR Tool connected to an ACM. The power was left connected and data was collected multiple times over a period of time. The ignition cycles at investigation did not increment because power was never removed from the ACM. When power was removed from and then reapplied to the ACM, the ignition cycle counter increased accordingly. The author did a similar test with the Bosch CAN+ CDR tool. General Motors Note 6 airbag control modules (SDM10 and newer) did not increment under any circumstances during data collection. Airbag control modules from older General Motors vehicles and other manufacturers incremented the ignition cycle counter with each data collection as long as power was removed and then reapplied.

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Power Testing

To investigate this, the author conducted a series of two separate sets of tests using airbag control modules from six In many crashes one or both of the battery cables are cut different vehicles. Each ACM was imaged using the Bosch by the fire department to reduce the risk of an electrical CDR tool and the ignition cycles at investigation were refire. Some CDR Reports have ignition cycles at investiga- corded (baseline). A 12-volt power supply was connected tion that are significantly higher than at the event. This to a timer circuit. The timer circuit was connected to each has sometimes been attributed to cut battery cables com- ACM using the Bosch CDR Tool direct to module cable. ing into contact with each other. When battery cables A total of five power on cycles was completed. Each ACM are cut, they typically have a relatively smooth end with was imaged a second time, and the ignition cycles at invesno exposed wire and are no longer in close proximity to tigation were recorded. It was expected that each ACM each other (See Figure 13). The cut battery cables would would increment by one due to data collection. have to touch each other end to end and remain in contact long enough for the ACM to power up. The cables would For the first series of tests the time was set to approximately then have to separate long enough for the capacitors to 2.5 seconds power on, followed by 1 second power off. discharge. Then the cables would have to touch end to end The first set of tests showed only two of the six airbag conagain. This process would have to repeat over and over to trol modules added ignition cycles: Nissan and Subaru (see increment the ignition cycle counter. Is this likely to hap- Table 1). pen; you decide. It is the author’s opinion adding ignition cycles under these circumstances is pretty unlikely.

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*One ignition cycle was added Chevy Silverado (2005) Chrysler 300 (2011) Ford Fusion (2010) Nissan Sentra (2015) Subaru Legacy (2017) Volvo XC90 (2019)

Ignition Cycles Baseline Ignition Cycles After Test 829 830 10,154 10,155 17,551 17,552 13 15* 136 138* 310 311

Table 1: The first set of tests showed only two of the six airbag control modules added ignition cycles: Nissan and Subaru For the second series of tests the time was set to approximately 1.5 seconds power on, followed by 0.25 seconds power off. The second set of tests showed three of the six airbag control modules added ignition cycles: Chevy, Nissan, and Subaru (see Table 2).

The Nissan Sentra requires the use of the “387 adapter” (F00K108387) for data collection. The author noticed during data collection for the Nissan ACM the green LED on the adapter turned on, off, and then back on again.

Each time the LED came on the ACM got power. This could account for two of the ignition cycles for the Nissan ACM in both series of tests. Further testing is necessary to confirm this suspicion.

It seemed that the shorter “off” time created a situation in which the capacitors did not fully discharge. This could account for the additional ignition cycle(s).

Daily Ignition Cycle Research The only other published data on ignition cycles was conducted by David M. Little, P.Eng. He presented his research at the 2007 CDR User’s Conference in Houston, Texas. He subsequently published his research in Collision Magazine. 13 His research was based upon 135 Gen*One ignition cycle was added **Two ignition cycles were added Chevy Silverado (2005) Chrysler 300 (2011) Ford Fusion (2010) Nissan Sentra (2015) Subaru Legacy (2017) Volvo XC90 (2019)

eral Motors vehicles. He used several different start dates including the door date and warranty start date (when the vehicle was sold). He used the first of each month for the door date. 14 The end date was the crash date.

Some of the vehicles in his sample base were commercial vehicles. He separated out the non-commercial vehicles older than one year. Those results showed both a median and an average of approximately 6 (based upon 39 samples) to 7 ignition cycles per day (based upon 84 samples) (see Table 3).

The author began doing research on daily ignition cycles in January of 2020 and presented his initial data two months later at the 2020 EDR Summit in Houston, Texas. A total of 172 non-commercial vehicle samples had been analyzed at that time based solely on the door date regardless of age.

A minimum value for each sample was established by using the first day of the manufacture month. A maximum value for each sample was established by using the last day (28, 29, 30, or 31) for the manufacture month.

At the time this article was written a total of 275 samples have been analyzed, however the research is ongoing.

Eliminating the thirty vehicles less than one year old made little difference in the daily average.

Ignition Cycles Baseline Ignition Cycles After Test 830 10,155 17,552 15 138 311

832* 10,156 17,553 18** 140* 312

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Average Standard Deviation Median Max Min n

Average Daily Ign. Cycles Based on Door Date Warranty 6.1 7.2 2.2 2.3 5.9 6.9 13.9 15.7 0.5 1.4 39 84

Average Standard Deviation Median Max Min n

Average Daily Ignition Cycles Based on Door Date Min. Max. Average 5.20 5.38 5.29 2.38 2.43 2.41 4.98 5.08 5.03 24.01 24.40 24.21 0.82 0.83 0.82 275 275

Table 3: Average daily ignition cycles based on door date and warranty

Table 4: Daily ignition cycles based on door date

The lowest value of less than one ignition cycle per day was a vehicle that is not normally driven and only used on occasion. The second lowest value of one to two per day was for a 74-year-old driver who had only owned the vehicle for two months when the crash occurred on about the 80th ignition cycle. The highest value of twenty-four per day was a student who was also a part time realtor.

The crash occurred approximately five years after the vehicle was purchased on about the 45,000th ignition cycle.

The highest value was close to 64,000 ignition cycles on a 22-year-old vehicle.

in July of 2014 and involved in a crash in March of 2017 (950 days later). The ignition cycles at event was 7733.

This is an average of 8.14 ignition cycles per day. The vehicle was repaired and sold shortly after the crash. The ignition cycles at investigation in January of 2019 (676 days later) were 8858; a difference of 1125. That is an average of 1.66 ignition cycles per day with the new owner. The overall average from the date of manufacture to the date of investigation is 5.35 ignition cycles per day. An analysis of ignition cycles from a subsequent crash with the new owner would result in a combined average of two households.

One thing to consider is the possibility of a vehicle being repaired after a crash, sold, and put back into service with a new owner. For example, a vehicle was manufactured

A summary of the analyzed samples is shown in Figure 14.

The distribution of the analyzed samples is shown in the histogram (in Figure 15).

Figure 14: Summary of analzed samples of average daily ignition cycles www.collisionmagazine.com

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Average Daily Ignition Cycles

Figure 15: Histogram of average ignition cycles per day

In the current study, based upon the door date alone, the average daily ignition cycles used for non-commercial purposes was 5 to 5.4

•

If the vehicle is not a General Motors Note 6 vehicle the ignition cycle counter will most likely be incremented by any of the following: turning the key on, pushing the start button, using the remote start, energizing an electric vehicle, collecting data direct-to-module.

•

If the vehicle is a General Motors Note 6 vehicle the ignition cycle counter will most likely NOT be incremented by collecting CDR data whether it is through the DLC or direct to module whether or not the ignition is turned on.

•

In the current study, based upon the door date alone, the average daily ignition cycles used for non-commercial purposes was 5 to 5.4.

The investigator should test an exemplar ACM in his/her crash to see what does and does not increment the ignition cycle counter.

Conclusions •

Clearing non-deployment events after 250 ignition cycles only applies to certain General Motors vehicles: typically, 2009 model year and older.

•

•

If the vehicle is from a manufacturer other than General Motors the ignition cycle counter will most likely be incremented by any of the following: turning the key on, pushing the start button, using the remote start, energizing an electric vehicle, collecting data direct-to-module.

Acknowledgements

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The author would like to thank the following individuals for contributing data for the ignition cycle research.

•

Anthony Andreoni – A2 Consulting Engineers

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•

James “Doug” English – James D. English & Associates

•

Sgt. Colin Foster – Calgary Police Service, Canada

•

Patrick Lynch – Moreno Valley PD (CA) – Riverside Sheriff’s Department

•

Mike Merolli – Mapfre Insurance

•

Sgt. Thomas J. Potter – Salt Lake City PD ¦ Kent E. Boots & Associates

•

D. Patrick Ryan, P.Eng. – Graham Ryan Consulting Ltd., Canada

•

Detective Ronald W. Sanders – Seattle PD ¦ Collision Analysis & Reconstruction Services

•

Sgt. Timothy S. Seeley – Lubbock PD ¦ Tim Seeley & Associates

References 1. NHTSA, Title 49, CFR Part 563.6, and 563.7(a) Table I 2. IBID, 563.8(a) Table III

4. IBID 5. NHTSA, Title 49, CFR Part 563.5(b) 6. Don Floyd, General Motors LLC (personal communication, December 19, 2019) 7. Bosch CDR Tool Software Version 19.4.2 Help File (Bosch Diagnostics, 2020) 8. NHTSA, Title 49, CFR Part 563.9(a)(b) 9. Bosch CDR Tool Software Version 19.4.2 Help File (Bosch Diagnostics, 2020) 10. Tesla Inc., Tesla Model 3 Owner’s Manual, Software version: 2020.20, p. 9 11. IBID, p. 10 12. IBID, p. 55 13. Little, David M. “Average Daily Ignition Cycles in SDM-Equipped General Motors Vehicles.” Collision Magazine, Volume 2 - Issue 2, Fall 2007, p. 42-45 14. David M. Little, Baker Engineering (personal communication, February 3, 2020)

3. Bosch CDR Tool Software Version 19.4.2 Help File (Bosch Diagnostics, 2020)

Loss of Use

Diminished Value & Total Loss Claims

Manufacturers Defects

ACV Valuation Disputes

Luxury Motorhomes & RV’s

Classic Cars / Sports / Exotics

Vehicles Damaged in Transit

Divorce / Estate / Probate DTPA - Dealer Fraud Blackbox / CDR Downloads

HOUSTON AUTO APPRAISERS

Repair Shop Negligence

Bonded Titles & Mechanics Liens Expert Reports, Photos & Documentation

Automotive Litigation Support & Expert Witness Services Roy Theophilus Bent, Jr. 25+ years Plaintiff / Defense Expert Witness Mediator, Arbitrator, Umpire Trial Court Testimony & Depositions

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www.HoustonAutoAppraisers.com

1300 Rollingbrook Suite 406 Baytown, Texas 77521 Info@houstonautoappraisers.com 1-877-845-2368

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Pedestrian Automatic Emergency Braking (P-AEB) And Its Use in Accident Reconstruction Alan Moore, P.E., ACTAR

A.B. Moore Forensic Engineering, inc.

A

dvanced Driver Assistance Systems (ADAS) have advanced rapidly in production cars in recent years. In model year 2020, every major auto manufacturer has some level of ADAS functionality available in their vehicles. IIHS & HLDI predicted in 2018 that, by approximately 2022, 25% of vehicles on the road will be equipped with ADAS, specifically frontal crash prevention technology. With some risk of extrapolation, this implies that half of all two-vehicle crashes will include an ADAS component by 2022. Some evidence suggests that the adoption curve is moving faster than HLDI predicted.

The future of accident reconstruction will include the need to understand and evaluate ADAS technology.

Pedestrian Automatic Emergency Braking (P-AEB) is a combination of several technologies: •

Forward Collision Warning (FCW)

•

Brake Assist, or Dynamic Brake Support (DBS)

•

Automatic Emergency Braking (AEB)

•

Pedestrian Detection

Forward Collision Warning uses a windshield camera and/or radar to monitor objects ahead of the vehicle.

When it detects a potential for impact with a Time To Collision (TTC) of less than approximately 3 seconds*, it may provide the driver with a collision warning consisting of visual (lights and displays), audible, and haptic (vibration or force) alerts. An interesting example of a haptic alert is the seat cushion vibration 72 Collision Magazine - Volume 14 Issue 2

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provided by some General Motors vehicles. Most pedestrian detection systems use a windshield-mounted camera, which allows an investigator a preliminary assessment of whether the vehicle is equipped with ADAS. Note the triangular opening(s) in the windshield frit (Figure 1), which is indicative of a camera. The square opening is a rain sensor.

In many AEB-equipped vehicles, the Brake Assist function is enabled shortly after a collision warning is provided.

Brake Assist, also known as Dynamic Brake Support, monitors the brake pedal travel and how quickly it is depressed. After it receives a collision warning and senses the driver depressing the brake pedal, Brake Assist quickly ramps up brake force to near the maximum braking capability1 of the vehicle.

Figure 2: Automatic Emergency Braking (in blue) compared to Brake Assist (in green) in a 2018 Toyota Camry.

If the driver does not respond to the collision warning, vehicles so equipped may engage Automatic Emergency Braking. This function autonomously applies the brakes without any driver involvement. An AEB engagement produces less deceleration than a Brake Assist engagement. In comparison to a Brake Assist engagement, autonomous AEB braking occurs later, ramps up slower, and uses a lower maximum acceleration (Figure 2 through 4).

Sensing the pedestrian Before issuing a collision warning or initiating automatic braking, an ADAS-equipped vehicle must first identify and analyze potential hazards. To detect a pedestrian, a

Figure 1: Triangular opening(s) in the windshield frit is indicative of a windshield camera.

Figure 3: Brake Assist – driver applied brake after warning in Japan-spec Subaru Levorg, with Eyesight 32.

Figure 4. AEB braking – no driver involvement in Japan-spec Subaru Levorg, with Eyesight 32.

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vehicle must be able to identify a pedestrian’s location, path and speed. Image classification methods are used to identify pedestrians, which is a challenging task that is affected by factors such as the following: •

Lighting

•

Sun glare

•

Background objects and visual noise

•

Biofidelic motion

•

Contrast

•

Structure from motion

Radar can also be used for pedestrian detection. Pedestrians reflect radar, although not as well as a motor vehicle.

An interesting characteristic occurs in pedestrians walking toward or away from the vehicle, in which the Doppler effect of swinging arms and legs is visible (Figure 5). 3 Since Brake Assist requires the driver to depress the brake pedal, possibly as a result of a collision warning, the driver’s reaction time (in responding to the collision warning) is important to consider. Research has shown that a driver’s reaction time to a known sound, with a known response, is approximately 0.8 seconds. 4,5 Note that this assumes that the driver knows the sound and its significance, which in this case would be the audible collision warning. A driver in an unfamiliar vehicle may have a longer reaction time.

Vehicle testing Standardized testing on P-AEB has frequently used a minimum of 3 scenarios, 6 depicted in Figure 6.

•

An adult walking across the lane, perpendicular to the vehicle’s path

•

An adult walking in the lane, parallel to the vehicle’s path

•

A child appearing from behind a stopped vehicle, walking across the lane, perpendicular to the vehicle’s path

Testing reported by NHTSA 7 has shown inconsistent performance in P-AEB systems, which has also been observed in other frontal crash prevention systems. In a series of tests, one vehicle avoided impact in two out of seven identical tests, but struck the pedestrian target in the remaining five. Another vehicle avoided impact in five out of 7 tests, striking the target in two tests. These were standardized tests under controlled conditions. Real-world performance of P-AEB is likely to be less consistent.

Testing conducted by AAA 8 also found that vehicles varied in their response to a pedestrian. In a 20 mph approach to a pedestrian walking across the lane, perpendicular to the lane, a 2019 Toyota Camry issued a collision warning approximately 1.2 seconds before potential impact, and was able to brake to a stop in 5 out of 5 tests. In contrast, a 2019 Chevrolet Malibu issued a collision warning much earlier, 2.1 seconds before potential impact, but did not autonomously brake. However, the Malibu did autonomously brake in a subsequent test at a higher speed of 30 mph. A 2019 Honda Accord braked to a stop in 3 tests, and struck the pedestrian target in two tests. AAA also tested vehicles at night, and none of the vehicles were able to warn or brake for a pedestrian at night. AAA’s report concluded that “drivers must not rely on assistance from

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Figure 6: Three test scenarios used by IIHS. 6 current pedestrian detection systems during nighttime driving or other environments with reduced visibility”.

al, and option codes listed in the glove compartment or on the VIN door sticker can be helpful references.

IIHS conducted pedestrian detection tests on a variety of vehicles in 2018, showing greater consistency in performance in each vehicle but continued variability between vehicles. A 2019 Subaru Forester, in the adult crossing perpendicular to the lane scenario, was able to avoid impacting the pedestrian in every test. 9 Conversely, the same tests on a 2018 BMW X1 resulted in impact in every test with less than 0.5 seconds of automatic braking. 10

The second step is to determine if the vehicle performed as designed. To answer this question, EDR data is usually necessary. Traditional EDR data, such as that from a Bosch CDR Tool download, is often insufficient if it doesn’t provide information on the target and doesn’t differentiate between driver actions and system driving inputs. Notable exceptions exist, as described below.

Reconstruction of a P-AEB crash A reconstructionist needs a method to evaluate a pedestrian accident involving a P-AEB equipped vehicle. The first step is to determine if the vehicle was equipped with P-AEB, which is not a simple task. Manufacturers equip vehicles with different levels of ADAS technology, and the presence of one ADAS feature does not guarantee that it is equipped with pedestrian detection. The manufacturer’s build sheet for the vehicle, sales brochures, owner’s manu-

Figure 7: A flowchart for evaluation of a P-AEB accident.

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Figure 8: General Motors EOCM data from a Bosch CDR Tool download, where driver override parameters are shown.

The last step is to determine if the P-AEB system could have mitigated or prevented the accident. Currently available data suggests that while P-AEB can do so, there are many scenarios where it may not respond at all, even under controlled test conditions. Real-world performance is likely to be less effective.

* The TTC at which FCW can trigger may vary greatly based on speed, conditions and manufacturer algorithms. The indicated value of TTC is a generalization that may not apply to a specific accident scenario.

References

Newer Toyotas may be equipped with Vehicle Control History, which records a wide variety of vehicle operations.

In a crash or near-crash scenario, it may record a multitude of events including hard braking, ABS operation, and PCS (Pre-Collision Safety) system operation. In doing so it provides data on the target type, target speed and direction, response of the P-AEB system, and the driver’s inputs.

1. 2019 Ford Edge Owner’s Manual

2013+ Cadillacs and some other General Motors vehicles record ADAS operation within the Active Safety Control Module (ASCM) or External Object Calculating Module (EOCM). Some, but not all, are downloadable by the Bosch CDR Tool version 19.3 or newer. The General Motors data includes parameters that describe why an ADAS system may have not responded or ceased responding due to an override by the driver (Figure 8).

4. IDRR Software and Related Research, Jeff Muttart and Swaroop Dinakar, 2019

Subaru’s Eyesight system also has the capability of recording ADAS-related data. 1 Efforts by others in the accident reconstruction community at obtaining the data through Subaru have generally been unsuccessful, and no commercially available download tool currently exists. Nissan’s Consult III software also has the ability to download ADAS-related data.

Information in the ADAS field is changing rapidly, and that which has been described above may not be applicable to future vehicles and accidents. Within the accident reconstruction community, the road to success on keeping up with the changes is by sharing test and research data, and regularly presenting and/or publishing the results. Reconstructionists are encouraged to contact the author with questions and new information on the subject.

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2. Accident Reconstruction with Data Recorded by Electronic Control Units in Vehicles with a Pre-crash Safety System, Oga, et al, SAE 2018-01-1440 3. Surface Vehicle Recommend Practice, SAE J3116, June 2017

5. Large-Scale Field Test of Forward Collision Alert And Lane Departure Warning Systems, DOT HS 812 247, 2016 6. Pedestrian Autonomous Emergency Braking Test Protocol (Version II), February 2019, IIHS 7. Objective Test Procedures for Pedestrian Automatic Emergency Braking Systems, Heath Albrecht, NHTSA, SAE Government Industry Meeting, January 2527, 2017 8. Automatic Emergency Braking With Pedestrian Detection, AAA, October 2019 9. IIHS P-AEB Test VRU18004, 2019 Subaru Forester, December 2018 10. IIHS P-AEB Test VRU18006, 2018 BMW X1, December 2018

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Toyota Vehicle Control History

Case Studies David M. Hallman M.S., P.E., Robert D. Anderson M.S.,

Billy S. Cox Jr.

I

ntroduction Starting in vehicle Model Year 2013, Toyota quietly added a data recording feature to the Toyota Rav4 which collected a large amount of data based on a wide variety of possible trigger events defined by Toyota. Similar to EDR data, when a trigger event occurs, data preceding, and post trigger is captured and stored. This data cannot be manually erased but will eventually be overwritten by newer events as the write locations fill up. The recording time, frequency, number of data points captured, and number of events stored prior to overwrite depends

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on the type of event and is defined in the Toyota Information System (TIS) available by subscription through the Toyota Technical Information website. The data is stored in the airbag control module (ACM) and can be imaged through the DLC using the Toyota Tech Stream software, also available through www.TechStreamSupport.com for an annual fee.

Since the introduction in the 2013 Rav4, this recording capability has been added to virtually all Toyota and Lexus models as well as one Scion model. Tables listing the production date and model years that VCH coverage began

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for each North American vehicle in the Toyota family of vehicles is included in crash·ol·o·gy in this issue of Collision (Vol 14, Issue 2). As this data has become widely available across all Toyota models, there are increasing opportunities to put this data to use as a reconstruction tool.

This article will give an overview of seven case studies from the authors where the VCH data was utilized with and without EDR data to assist in determining the facts of, and reconstructing, a crash. The purpose of this article is to demonstrate some of the potential uses for this data as another tool for the reconstructionist’s toolbox.

Vehicle Control History Case Study #1, Mercedes v.

Camry A low speed rear-end impact is an example of a type of vehicle collision which may not be sufficient to qualify for

event recording. In this case study, a Toyota Camry was stopped and waiting to exit a parking lot when it was rearended by a Mercedes-Benz C-Class.

According to the police report, it was daylight, the weather was clear, the roadway was dry, level and straight, and no unusual roadway conditions were noted. The vehicles were not disabled and both sustained only minor damage.

The responding officer’s estimate for the impact speed was 2 mph. A crash scene diagram was not prepared. An aerial photograph of the accident site is shown in Figure 1.

The Mercedes driver stated that her foot slipped off the brake pedal, and that she then idled forward and bumped the Toyota.

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Figure 1: An aerial photograph of the accident site The Toyota driver stated that she was stopped when she was rear-ended multiple times and pushed into the middle of the road.

The Mercedes C-Class sustained scuffing over its central bumper cover and its lower grille was separated and displaced rearward as shown in Figure 2. The Toyota Camry had various nicks and gouges on its rear bumper cover as seen in Figure 3. Unfortunately, there was insufficient resolution or detail to identify a pattern that would indicate the area that was contacted in the collision under investigation here.

A repair estimate for the Toyota Camry listed rear bumper cover replacement. A vehicle inspection revealed evidence of repairs including tool marks on the bumper bracket mounting bolts and writing on the inside of the bumper cover. The thin foam cushion absorber still exhibited some crinkling and the black trim below the bumper cover was cracked.

The Toyota Camry’s airbag control module (ACM) was interrogated for stored crash related events using the Bosch Crash Data Retrieval (CDR) Tool. The Event(s) recovered was “None” indicating that the parking lot accident did not qualify for recording.

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can be obtained using the Toyota Techstream tool using the CDR 900, Mongoose cable or other vehicle communication interfaces. Within Techsteam, the Vehicle Control History Report, is an index of the recorded Triggers. The first page of the VCH Report obtained from the Toyota Camry is shown in Figure 4.

The Toyota Camry’s repair estimate, 10 days post-crash, reported an odometer reading of 8157 miles. As seen in the VCH Report shown in Figure 4, there was a sudden braking history trigger at 8053 miles. Prior to that trigger the brake pedal was released with a simultaneous speed increase occurred with no accelerator pedal application (Figure 5), which is characteristic of rear-end occupant kinematics.1,2 The Mercedes’ closing speeds was calculated to be above 7 mph, which is outside the typical range of idle out speeds.3 As such, it appears that the Mercedes driver’s foot did not simply slip off the brake so that the vehicle idled forward a short distance into the rear of the Toyota.

The pre-trigger speed data shows that the Toyota was rolling forward slowly and coming to a stop before being struck. This is consistent with the Toyota driver pulling forward, which the Mercedes driver mis-interpreted as the Toyota initiating its turn. However, because the Toyota stopped again, the Mercedes struck it while the driver was pulling forward and looking left to make sure it was clear to turn.

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Figure 2: Mercedes C-Class sustained scuffing over its central bumper cover and its lower grille was separated and displaced rearward

Figure 3: Toyota Camry had various nicks and gouges on its rear bumper cover

Figure 4: The first page of the VCH Report obtained from the Toyota Camry www.collisionmagazine.com

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Figure 5: Prior to that trigger the brake pedal was released with a simultaneous speed increase occurred with no accelerator pedal application The Lexus ES 350’s extent of damage was generally consistent with the -16 mph longitudinal Delta V and +7 mph lateral Delta V reported in the EDR report, the ignition cycles at crash and download were 105 and 118, respecThe three distinct impacts with diminishing peak speeds tively. The odometer reading in both the EDR report and is consistent with the Toyota’s post-impact braking while the VCH match the odometer reading observed on the inthe Mercedes proceeded at its post-impact speeds. This strument panel, which was 275 miles. The lack of deployallowed the Mercedes to catch up strike the Toyota with ments matched the lack of commanded deployments listed progressively lower impact speeds. in the EDR report, and the reported seat belt use matched the subtle evidence seat belt use for both driver and front Vehicle Control History Case Study #2, Lexus ES 350 passenger. Therefore it is reasonable to conclude that the Intersection collisions are an example of a type of vehi- EDR recorded events were related to the crash of interest. cle crash where the VCH data can expand and augment the EDR’s pre-crash data by providing pre-crash data at The EDR report provided pre-crash data in half second ina higher sample rate and over a longer time. In this case crements beginning with T = -4.95 seconds. In addition to study, a new Lexus ES 350 (Figure 6) was turning left at providing the ignition or key cycles at crash, the VCH data an intersection when it was struck in a head-on side-swipe shows that according to the vehicle’s clock the crash ocorientation, which did not deploy supplemental restraints. curred at 6:46 p.m. on 3/1/2019 (VCH data provides GPS date and time information when the vehicle is equipped The EDR report had three recorded events, TRG1 (side), with a factory navigation system). The VCH reported key TRG2 (front/rear/side) and TRG3 (rollover). Because cycles at crash was 119, which is 14 ignition cycles higher Toyota’s number their events sequentially starting with than the EDR reported ignition cycles at crash. However, TRG1, every event to which the Lexus was subjected to as was presented and discussed at the 2019 EDR Summit in its service life that qualified for recording was still re- and the 2020 EDR Summit, it is not unusual for the EDR corded. In addition, because all three TRG’s were within a reported and VCH reported ignition cycles to be different.

1/10th of a second, they were all from the same crash.

Figure 7 shows a table of combined pre-crash data selected from the EDR and VCH data. It should be noted that the The post-trigger data shows the Toyota’s speed rising and falling to zero three times, which coincides with the Toyota driver’s description of three rear-end impacts.

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Figure 6: Lexus ES 350 time interval between rows is not necessarily uniform due to the different sample rates.

The braking, accelerating and steering data shown in Figure 7 is consistent with the Lexus driver slowing to enter the intersection, initiating a left turn, but then steering out of the turn and braking in response to an oncoming vehicle. It is noteworthy that looking at the EDR data alone would have missed the Lexus’ initial braking and slowing from 34 mph to 28 mph.

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As discussed in crash·ol·o·gy in Collision Vol 14, Issue 1, it is important to note that airbags must be predictive, meaning that the decision whether or not to deploy cannot be made on Delta V, because the decision must be made early in the collision before the Delta V’s occur. Given the experience of injuries and fatalities related to the deployment of airbags when they were first introduced, combined with smart airbag technology which allows airbag deployments to be suppressed until higher severities for seat belted occupants for a net overall gain in safety, it is not a surprise that the Lexus frontal, side and curtain airbags were not commanded to deploy.

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Vehicle Control History Case Study #3, Kawasaki v.

Corolla In collisions with only one surviving driver and no witnesses, the narrative can be skewed. The example demonstrates that the objective electronic evidence can be one of the only ways to obtaining a correct and full understanding of the circumstances of a fatal crash. This case study involves an intersection collision between a 2016 Toyota Corolla (Figure 8) and a 2010 Kawasaki motorcycle which resulted in a fatality. EDR and VCH data were not collected at the time of the initial investigation and crash reconstruction.

As no available data was collected, the motorcycle rider (a teenaged young man) was deceased, and there were no other witnesses, the reconstruction which was performed was based solely on the Corolla driver’s testimony and im-

Figure 8: 2016 Toyota Corolla

properly interpreted scene evidence. The original investigators determined that the deceased motorcyclist must have been speeding and was therefore at fault for the crash.

The Corolla, which was sold for salvage by the insurance company after the crash, was located in a salvage yard several months later (Figure 9) and the airbag control module was found to still be in the vehicle.

The module was collected from the remains of the vehicle and the crash data and the VCH data were both imaged from the ACM direct to module as discussed in crash·ol·o·gy in this issue of Collision. The EDR data showed one side event which deployed the side curtain bag and the pretensioner. The imaged VCH data showed two related events (Figure 10) as determined by comparing the

Figure 9: 2016 Toyota Corolla located in a salvage yard several months later

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imaged crash data to the raw VCH data. The speeds between the two data sets matched and approximately three 0.15 second increments after the Sudden Brake and ABS event (which occurred simultaneously) in the VCH data, a lateral acceleration was reported in the VCH data (Figure 11).

At issue in this case was the fact that the Corolla driver reported that she stopped fully at the stop sign and then proceeded to move cautiously forward into the intersection where she was impacted by the “speeding” motorcycle who did not have a stop sign facing his direction. The crash data indicates that the Corolla coasted down to 10 to 12 mph and then accelerated to approximately 30% of full accelerator pedal travel prior to the impact. Per the EDR data, the brakes were not applied until 0.3 seconds prior to the impact which would have placed the vehicle past the stop sign at brake application. Based on the fact that brake oil pressure becomes active approximately 0.45 seconds prior to lateral acceleration being detected and reaching a maximum value (Figure 11), the VCH data support this as well. In this matter, the combination of the EDR data and the VCH data allowed for a correct crash reconstruction.

It is important to note (Figure 11) that the EDR time zero does not occur at the same time as the VCH trigger. The VCH trigger is set by the application of the brakes (Sudden Brake event) and the activation of the ABS Braking. These VCH recorded events both occurred prior to the actual impact as indicated in the EDR pre-crash data. The investigator in a case such as this will need to account for and explain this time difference. VCH is simply an additional tool that must be interpreted by the investigator in order to be used properly.

Vehicle Control History Case Study # 4, Chrysler v. Rav4 As this example demonstrates, the EDR can only provide information when there is a crash, but as the VCH can show a pattern of aggressive or risky driver behavior. This case study involves an intersection collision between a 2013

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Chrysler 200 (Figure 12) and a 2016 Toyota Rav4 (Figure 13). Tthis crash occurred at an intersection with traffic lights. The main issue in this matter was statements made by the parties in the case. Both parties indicated that they had a green light when they entered the intersection.

The speed limit in the area of this crash was 30 mph. The Chrysler 200 operator indicated that he had stopped at the red light and then proceeded to drive across the intersection after the light facing his direction turned green. The Rav4 operator stated that she had a green light and the Chrysler 200 pulled out right in front of her vehicle.

Both vehicles were CDR supported and both were imaged. The Chrysler was imaged direct-to-module and the Rav4 was imaged via the DLC. VCH data from the Rav4 was also imaged through the DLC. The Chrysler deployed supplemental restraints and the Rav4 did not.

The Chrysler had a single deployment event stored which indicated a -12.4 mph lateral Delta V and a -6.8 mph longitudinal Delta V. The pre-crash data indicated that the vehicle was stopped and proceeded to accelerate slowly up to 18 mph over approximately 4.5 seconds until time zero. A time-distance study was conducted from the EDR data and the vehicle would have travelled approximately 75 feet based on that analysis. Coincidentally, the calculated 75 foot distance was the exact distance from the intersection stopline to the point-of-impact.

The Rav4 data showed three triggers, TRG1 (front-rear), TRG2 (side) and TRG3 (rollover). The TRG1 pre-crash data indicated that the Rav4 was travelling at a steady 44.7 mph until 1.35 seconds prior to the first trigger. At that time, the operator applied the brakes and brake Collision Magazine - Volume 14 Issue 2 85

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Figure 12: 2013 Chrysler 200

Figure 13: 2016 Toyota Rav4

pressure peaked approximately 1 second later. At the same time, per the EDR data, the operator steered hard right.

The front / rear trigger (TRG1) occurred 0.35 seconds after peak brake pressure was achieved. At a minimum, the Rav4 operator was travelling well over the posted 30 mph speed limit for the area.

times per the data. This appears to indicate that the operator routinely drives the vehicle in a way that causes multiple VCH triggers to be set. This would be considered aggressive driving.

The vehicle control history for the Rav4 was also evaluated (Figure 14). Interestingly, there was not a Sudden Brake or ABS activation event stored in the VCH that could be connected to the crash. However, the data still told an interesting story about the vehicle owner’s general driving habits and the driver behavior immediately prior to the crash. Nearly four pages of events were recovered from the vehicle with only 33,898 miles on the odometer. The VSC system operation event location had been over written 488

As shown, there were multiple VSC events in the approximately 3 minutes prior on the same key cycle as the crash.

This appears to indicate that the operator was driving the vehicle aggressively in a way that repeatedly activated the VSC system multiple times in the three minutes prior to the crash. The Rav4 operator was also speeding at the time the crash occurred. The VCH data and EDR data from both vehicles certainly cannot indicate whom had the green light, but in this case, it could be interpreted to support the statements made by the operator of the Chrysler 200.

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Vehicle Control History Case Study #5, Sienna Van v. ???

In this example, the VCH data demonstrates an unmistakable pattern of vehicle abuse in the context of significant damage without any EDR data. This case study was a damage evaluation of a single vehicle which is a 2019 Toyota Sienna with less than 6,000 miles on the odometer.

The story told to the insurance provider by the owner was that the vehicle was parked on the street in Chicago and left there for a period of time. The owner stated that when the vehicle was parked in Chicago it was in perfect condition. The vehicle was then picked up by unknown parties and delivered to Minneapolis and the damage (Figures 15 and 16) was “discovered” at that point. The vehicle had reportedly not been driven between the time it was parked and subsequently picked up.

No EDR events were retrieved from the vehicle when the ACM was imaged. The VCH (Figure 17), however, told an interesting story for the vehicle. In the last ten miles of the recorded history over just a few key cycles, the vehicle was accelerated hard immediately after shifting into reverse

Figures 15 and 16: 2019 Toyota Sienna

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(note damage to the rear) on multiple occasions and many other trigger events were stored. Key cycle 392 is particularly interesting as it appears to show that the vehicle was subjected to multiple “neutral drops” which is defined as revving the engine in neutral and then shifting the transmission into drive or reverse with it revved to attempt to spin the tires or to accelerate rapidly.

Based on the VCH data as shown, it is clear that the vehicle was being driven aggressively in a way that would damage the drivetrain and, most likely, the vehicle itself.

All of this aggressive operation happened in roughly the last ten recorded miles of the vehicle’s life. Even though no EDR data was recovered, the recovered VCH data gives some indication about how the vehicle was being operated prior to the imaging.

Vehicle Control History Case Study #6, Tundra v. Dog Pedestrian and animal impacts are examples of low Delta V collisions that may not be sufficient to qualify for re-

cording. In this case study, a 2019 Tundra struck a dog, estimated to weigh 60 lbs, on the highway on March 20, 2020. The contact damage that resulted was some slight deformation of the front license plate and splitting of the vertical portions of the right and left side of the front license plate bracket (Figure 18).

The following day the Tundra’s airbag control module (ACM) was interrogated for stored crash related events using the Bosch Crash Data Retrieval (CDR) Tool. The Event(s) recovered was “None” indicating that the collision with the dog did not qualify for recording. The ignition cycles at download were 1762 and the odometer reading was 34,665 miles.

As shown in Figure 19, the VCH Report shows four triggers: sudden braking, ABS operation, PCS operation, ABS operation. All four triggers occurred at 5:01 p.m. on March 20, 2020, as the driver had reported. The VCH showed the key cycles at 1782 at the time of the event and 1783 at the time of download, which is 19 ignition cycles

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Figure 19: VCH Report showing four triggers: sudden braking, ABS operation, PCS operation, ABS operation higher than that listed in the EDR report. The VCH reported mileage matched the odometer reading of 34,665 miles.

Figure 20: 2019 Toyota 4Runner

In the VCH the overlapping recordings resulted in 12 seconds of speed which began at 110 kph (68 mph). Then the brakes were applied, which deactivated the cruise control.

ABS was active for a couple of seconds while the brake oil pressure reported the maximum reading. The brake oil pressure decreased, and the vehicle was brought to a stop. Many other vehicle parameters familiar from EDR reports were also reported, including steering wheel angle, accelerator percent, engine RPM, shift position, yaw rate, longitudinal and lateral acceleration, gear position, and individual wheel speeds.

Vehicle Control History Case Study #7, 4Runner

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Another example of a situation in which the VCH data added details to the EDR data is a 2019 4Runner that was rear-ended on May 1, 2020. The vehicle inspected 26 days later on May 27, 2020. Upon inspection the vehicle was found to have significant right rear damage, as shown in in Figure 20. The driver’s seat belt was found in an extended position with loading marks found on the plastic of the latch plate, the seat back was twisted inboard, the Collision Magazine - Volume 14 Issue 2 89

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Figure 22: VCH indicated a sudden braking trigger at 9 p.m. on May 1, 2020, which matched the subject collision

Figure 23: Rear-end collision is evidenced by the sudden increase in speed without accelerator pedal application

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head restraint was broken off, and there was evidence consistent with the driver being ejected into the rear occupant compartment. The odometer reading was 13,937 miles.

The 4Runner’s airbag control module (ACM) was interrogated for stored crash related events using the Bosch Crash Data Retrieval (CDR) Tool. The ignition cycles at crash were 667 and 671 at download. TRG1, TRG2 and TRG3 were side, front/rear and rollover recordings, respectively, all within a tenth of a second of each other confirming that all the EDR recordings were related to the subject crash.

The pre-crash data shows that the vehicle was stopped over the span of the pre-crash data, and the Delta V was 35 mph, which is consistent with the extent of vehicle damage.

The pre-crash accelerator pedal and brake oil pressure are plotted in Figure 21. Interestingly, the accelerator pedal was applied twice in the 5 seconds leading up to the crash while the vehicle remained stationary. The driver maxed out the brake oil pressure reading just before impact, indicating awareness of the impending collision. The driver reported that when it became evident that his vehicle was not moving forward when he applied the throttle, he applied the brakes firmly as he anticipated the impending crash.

As shown in Figure 22, the VCH indicated a sudden braking trigger at 9 p.m. on May 1, 2020, which matched the subject collision. The key cycles at crash were 689, which is 22 greater than the EDR reported 667 ignition cycles at crash. The vehicle mileage of 13,937 miles matched the odometer reading.

As shown in Figure 23, the rear-end collision is evidenced by the sudden increase in speed without accelerator pedal application. With three times the sample rate of the EDR pre-crash data, the VCH data provides better details or higher resolution of the pre-crash accelerator pedal and brake application.

Conclusions The purpose of this article was to provide examples of field applications that demonstrate the general usefulness of the VCH data as provided by Toyota and Lexus vehicles. As demonstrated in this case studies, this data can provide valuable information in the absence of, or in addition to, EDR data.

evant VCH data to the event(s) of interest. As such, it is important to keep in mind that: •

Date and time, when reported by the VCH, have been found to depend on the vehicle clock settings.

•

Odometer readings reported by the VCH, EDR and instrument panel display have been found to coincide.

•

VCH key cycles have been observed to be about 20 greater than the ignition cycles reported by the EDR.

For crashes where there is both EDR and VCH data, the VCH data trigger and the EDR time zero will seldom (if ever) coincide. Therefore, the reconstructionist must be prepared to explain the time difference between the two in order for the data to be utilized appropriately and be relevant to the analysis.

As shown in these case study examples, the VCH data can provide valuable information for crashes and other incidents involving Toyota vehicles as a useful addition to EDR data, or as a source of evidence of vehicle and driver behavior that might not otherwise be available.

References 1. Anderson RD, Welcher JB, Szabo TJ, Eubanks JJ and Haight WR, “Effect of Braking on Human Occupant and Vehicle Kinematics in Low Speed Rear-End Collisions,” SAE paper 980298, Society of Automotive Engineers, Inc., Warrendale, PA, 1998.

2. Ivory MA, Furbish CJ, Hoffman MR, Miller ER, Anderson RL and Anderson RD, “Brake Pedal Response and Occupant Kinematics During Low Speed RearEnd Collisions,” SAE Paper 2010-01-0067, in Accident Reconstruction - 2010 (SP-2267), Society of Automotive Engineers, Inc., Warrendale, PA, 2010.

3. Randles B, Voss DP, Ikram I Furbish CJ, Welcher JB and Szabo TS, “Acceleration Testing and Modeling of Vehicle Kinematics Under Idle Conditions,” SAE Paper number 2014-01-0484, in the Society of Automotive Engineers International Journal of Transportation Safety, Volume 2, Issue 2, July 2014.

Date/time stamps, odometer readings and ignition/key cycles are useful in determining the relationship of any rel-

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High speed pedestrian collision and verification of car collision speed in pedestrian accidents based on biofidelic dummy injuries Annika Kortmann and Tim Hoger ecent studies have shown 1 that the construction and use of the biofidelic dummy leads to much more realistic vehicle damage in passenger car/pedestrian collisions than in collisions with conventional dummies. When comparing the longitudinal throwing distances of biofidelic and conventional dummies, no significant difference occurred in contrast to the damage caused to the passenger car. Often, when investigating pedestrian collisions with regard to the collision velocity, the pedestrian injuries remain disregarded.

The relationship between collision speed and pedestrian injury characteristics was investigated on the basis of real accidents by Appel et. al. 2 Whether the biofidelic dummies present corresponding “injuries” in a collision as a function of the passenger car collision velocity can be checked by undergoing an “autopsy” on the dummy after the crash tests.

R

For this purpose, a series of crash tests were carried out with the biofidelic dummy from crashtest-service.com

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GmbH using the same vehicle model in the speed range from 28 to 109 kph. A collision speed of 109 kph represents the highest speed in currently available crash tests between passenger cars and pedestrians in relevant databases.

Introduction Generally, the more indications available for resolving the course of events in car-pedestrian accidents, the more the results can be restricted. In accident reconstruction, the extent of the damage to the vehicle and, when present, the throwing distance of the pedestrian are used for determining the vehicle’s collision speed. The injuries of the pedestrian are often only taken into account in regard to the impact direction, and where appropriate, in cooperation with forensic scientists. The relationship between collision speed and injury characteristics according to Appel et. al.2 offers information on the injuries of the pedestrian which likewise provides the possibility to estimate the collision speed.

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This is particularly interesting at lower vehicle velocities, since there is no or very little damage occurring to the vehicle, whereas the pedestrian can bear severe injuries as a result.

available crash tests. Up to a collision speed of about 80 kph, corresponding crash test attempts and the evaluation of throwing distance of pedestrians exist in relevant databases.

The crash test series with several VW Polo 6R and the biofidelic dummy 1 have shown that the conventional dummies presented weaknesses due to its hard construction and the restricted rigid mobility, also the damage caused to the vehicle and the movement sequence of the pedestrian during the collision was not realistically represented.

Through the dynamic separation of the biofidelic dummy during the collision extensive damage occurred which can also be observed in real accidents.

To also be able to provide information in the high speed range (v > 100 kph), as part of an expert seminar by crashtest-service.com GmbH, the crash test series with the VW Polo 6R and the biofidelic dummy was supplemented with a collision speed of 109 kph. Thus, the relationship between passenger car damages and pedestrian injuries be can extended. Additionally, it is possible to check whether at high collision speeds the postulate tears according to Appel et. al. 2 between the leg and torso also occurs to the biofidelic dummy. Similarly to the previous crash test series, the dummy was also impacted laterally in the high speed crash test. The point of impact on the passenger car was situated in the centre of the bonnet, figure 1.

Originally, the biofidelic dummy was developed as a pedestrian surrogate in order to realistically reconstruct the damage to the vehicle in a collision. Furthermore, the construction of the biofidelic dummies 3 (due to the achievable production accuracy) offers the possibility, for the first time, to reproduce the pedestrian injuries from a collision.

Taking into account the damage to both vehicles, for example in a crash attempt recreating a passenger car versus passenger car collision, the solution of the road traffic accident appears to be obvious. This is now also possible for passenger car/pedestrian collisions by means of a subsequent examination of the biofidelic dummy.

Supplement to the crash series VW Polo impacting a biofidelic dummy in the high speed range The passenger car collision velocity could be further and more precisely restricted with the increasing amount of

Changes in the rolling behaviour of the pedestrian in high speed attempts The rolling movement of a pedestrian in a collision with a passenger car is known, as shown in the side view in figure 2 in individual images. The crash attempt shows a biofidelic dummy, which was laterally impacted at a velocity of 80 kph. The rolling behaviour of the dummy begins when the dummy is positioned with the hips almost half way between the vehicle front and the lower edge of the windscreen. A typical lateral shearing movement of the head occurs in the direction of the windscreen.

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Figure 1: Crash vehicle VW Polo 6R (left) and impact situation with Biofidel dummy (right) for the high-speed test with v = 109 kph

Figure 2: Motion sequence and rolling behavior of the Biofidel dummy at v = 80 kph.

For this purpose, a comparison is shown in figure 3 of the collision sequence under the same test conditions at a passenger car collision speed of 109 kph. In the top row of images, it is clearly recognizable that the typical rolling behaviour of the pedestrian does not take place due to the mass inertia of the dummy and the high collision speed.

the head impact occurs mainly on the roof of the vehicle.

The roof is massively deformed and the dummy penetrates the vehicle interior with head and torso. During the collision phase, there is a notable elongation/stretching of the dummy’s torso, which is particularly significant, evident in the lower row of movement sequence images.

After the undergoing of the feet, the torso remains almost upright during the first collision phase, until the hips of the dummy almost touch the front windshield. Only thereafter does the torso tilt towards the windshield, whereby

Velocity dependant damage to the VW Polo 6R

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The damage caused to the passenger car at a collision speed of 109 kph is exemplarily portrayed in the two images in

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Figure 3: Motion sequence of the Biofidel dummy at v = 109 kph.

Figure 4: Hood and roof damage on the VW Polo at v = 109 kph. figure 4. In the left image the hip impact of the dummy is visible, which ends just before the lower edge of the front windscreen. Up until this point, the upper body of the dummy tilted less than 5°, so almost upright.

The front bumper covering appears to be undamaged except for the missing grill and lighting units. The head impact on the roof can be retraced through the deep deformation as shown in the right image of figure 4.

Upon disassembly of the front bumper cover, the profound intrusion of the vehicle’s crossmember is apparent, see figure 5. The lower crossmember, which should prevent the undergoing of the feet, was (in the direction of travel) bent inwards as far as the radiator at the right side. The left side of the lower crossmember remained fully intact, figure 6.

The exact stand position of the dummy at the first point of contact can also be seen through the deformation of the upper main front crossmember. The disassembly of the front bumper cover is therefore essential in order to identify the exact impact configuration. Figure 7 shows the damage incurred to the VW Polo of the previous crash test series with increasing collision speed in the range between 28 and 80 kph, which is supplemented now by the high speed crash attempt at 108.7 kph.

As a result of the largely elastic deformation of the front bumper cover, the grill and the lighting units came off the vehicle’s front. The higher speed results in a significant intensive deformation of the bonnet, a penetration of the front windshield and compression of the roof.

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Figure 5: Damage of the VW Polo after dismantling the bumper cover

Figure 6: Side-dependent deformation of the bow to prevent the feet from being subjected to it As previously discussed, at lower velocities of around 30 to 50 kph there is no visible damage to the front bumper.

In a high speed crash test attempt, a throwing distance of the biofidelic dummy of around 62 m longitudinally and 3.1 m laterally was reached. Accordingly in figure 8, the throwing distance diagram from 1 was supplemented with the high speed crash attempt.

Based on the motion sequences shown in figure 3, it becomes clear that at higher speeds different structures of the vehicle are affected, which may also have a different stiffness. Also, the angle upon impact of the dummy changes, since the dummy no longer collides exclusively with its upper body against the inclined surface of the windshield, but collides also almost horizontally on the vehicle roof.

Whether the air resistance during the flight-phase of the dummy at a speed of over 100 kph is a relevant factor also needs to be investigated.

The determined throwing distance range derived by Focken 4 as a function of the collision speed is also still valid even for high collision speeds. The crash test carried out at a collision speed of approximately 109 kph lies in the lower range of throwing distances according to Focken [4].

Whether the lower limit hereby is only to be clarified by the (usual) tolerances, or whether other effects due to the higher velocity occur, which lead to an actual smaller increase in the throwing distance with increasing velocities over 100 kph, needs to be further investigated.

When using a biofidelic dummy in crash tests it is possible to compare the “injuries” of the dummy with the collision speed, even though the dummy was only initially developed as a pedestrian surrogate for realistic damage generation on vehicles. After a collision the dummy undergoes an examination in the dummy laboratory at crashtestservice. com GmbH. The dummy is hereby disassembled, so that every component including soft tissue, tendon and liga-

Supplement to the longitudinal throwing distance diagram

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Injury characteristics of the biofidelic dummy in comparison to real accidents

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kph (figure 10, left), damage identified on the dummy consisted primarily of multiple fractures to the ribs, a fractured pelvis, fracture of the femur and arm on the impact side and several torn ligaments. By increasing the collision speed of the passenger car to 109 kph, multiple fractures to the same bones occurred (figure 10, right). Additionally, fragmentation of the hip bone and a ripping out of the femur from the hip socket were recorded. In the hand area was such a deep incision that in a real accident a complete separation of the right hand could have occurred.

Analysing the movement sequences of the biofidelic dummy at a collision speed of approximately 109 kph in regard to the massive damage to the hips and femur, it becomes apparent in figure 11 that the hip area of the dummy is extremely elongated between torso and leg. Due to the high elasticity of the dummy’s skin, the upper body had not been separated from the legs, although the hip bones were shattered and the femur bones placed out of the hip socket. In the case of real accidents, it can be assumed that with the same injuries it would have resulted in an amputation of the legs during the collision. There is still development potential in this area, so that a tearing of the dummy can also occur.

Figure 7: Change in the damage pattern (impact against Biofidel) on the VW Polo 6R with increasing speed from 27.5 to 108.7 kph in the front and side view ment injuries and also fractures of the bone can be documented. Photographic material of a typical dummy examination is shown in figure 9.

The detected damage to the dummy was detailed on to a drawing of a human skeleton, figure 10. Noted were cuts, fractures and torn ligaments. The illustration of the skeleton was compared with the construction of the biofidelic dummy (figure 10, centre). At a collision speed of approximately 70 www.collisionmagazine.com

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A likely amputation of limbs from the collision can currently only be concluded through a dummy examination and the identified bone injuries. Noticeable is that the biofidelic dummy did not sustain a fracture of the skull in any of the collisions, although the head collided partly at higher speeds with the roof or against the front windscreen. The cause of this is the (still) relatively rigid and massive head of the dummy.

Whether the “injuries” to the biofidelic dummies are equivalent to the injury characteristics of a human at similar collision speeds can be compared in the diagram according to Appel et. al. 2 (figure 12), which characterises the relationship between impact velocity and the degree of Collision Magazine - Volume 14 Issue 2 97

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Figure 8: Diagram with longitudinal throw distance of the different dummy types as a function of collision speed-trend line after Focken [4] injury of a pedestrian. For clarity, the identified injuries of dummy at a collision speed of 68 km/h (green) and at 109 kph (red) were supplemented in the diagram according to Appel et al.

This analysis has shown that the injuries sustained by the biofidelic dummies approximated those reported by Appel et al [2] with respect to pedestrians in real accidents. Rib and femur fractures and a fractured pelvis were identified on the dummy at a velocity of around 70 kph. A likely separation of the upper and lower body in the impact range of around 110 kph could be concluded by the analysis of the massively damaged bones in the pelvis area of the dummy.

Injuries to the skull of the biofidelic dummy have not yet been present and are (so far) not expected due to the structure.

Conclusion To expand the crash test database, a passenger car collision with a biofidelic dummy at a collision speed of 109 kph was carried out, which is currently the highest speed documented in relevant databases. The movement sequences and the damages can be well comprehended by using a biofidelic dummy.

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The movement analysis of the biofidelic dummies at this speed has shown that the mass inertia of the dummy and the high velocity of the vehicle lead to a substantially delayed rotation of the dummy relative to the vehicle. Subsequently, a head impact occurs on the vehicles roof. During the collision, the dummy is significantly elongated in the torso area, without any separation.

The throwing distance of the dummy at a collision velocity of around 109 kph lies in the lower throwing distance range derived by Focken 4. Whether it is a tolerance, or whether the higher speed causes other effects such as impact level, or the air resistance, or the structure rigidity of the passenger cars roof area take effect, which results in an actual smaller increase in the throwing distance with increasing velocities over 100 kph, needs further investigation.

Through an examination of the dummy (similar to an autopsy of humans) after the crash tests, the established relationship by Appel et. al. between impact velocity and the degree of injury in pedestrians can essentially be comprehended. The expected limb separation in the high speed range could not be reproduced using the biofidelic dum-

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Figure 9: Investigation of a Biofidel dummy mies due to the skin being too elastic, so there is also still a need here for further development.

An analysis of the bone injuries of the dummy allows us based on the shattered pelvis and the detachment of the femur from the hip joint to conclude an expected separation.

A head fragmentation could not be observed in any of the crash attempts, which due to the massive head structure, and is also not expected.

Future work In this article, the application of the biofidelic dummies in pedestrian collisions was discussed. Currently, the company crashtest-service.com GmbH is developing in cooperation with the HTW Dresden and the TU Berlin, among other parts, a variable structure for the buttocks area of the dummy as well as an improved cervical spine and a new rib cage.

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Figure 10: Injury image of the Biofidel dummy at a collision speed of 68 kph (left) and 109 kph (right) – design of the used Biofidel dummy

Figure 11: Massive hip damage (left) and severe dummy elongation during collision (right) 100 Collision Magazine - Volume 14 Issue 2

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Figure 12: Relationship between impact velocity and degree of injury in pedestrians according to Appel et. al. 2 (translated into English) About the Authors Dipl.-Phys. Annika Kortmann, road traffic accident expert at engineering office S+B in Münster.

Dipl.-Phys. Dr. rer. nat. Tim Hoger, publicly appointed and sworn expert for road traffic accidents working at engineering office S+B in Münster.

References 1. A. Kortmann, T. Hoger, Pkw-Beschädigungen und Längswurfweiten bei Verwendung von Biofidel-Dummys und konventionellen Dummys im Vergleich, VKU 10/2018.

2. H. Appel, U. Wanderer, S. Meißner, G. Schmidt, J. Barz, D. Kallieris, R. Mattern, F. Schüler, Mechanik und Biomechanik des Unfalls, H. J. Wagner (Hrsg.): Verkehrsmedizin, Springer-Verlag, Berlin 1984.

3. A. Kortmann, Crashverhalten im Crashvergleich: der neue Biofidel-Dummy bei unterschiedlichen Szenarien von Pkw-Fußgängerunfällen, VKU 03/2018.

4. U. Focken, Experimenteller Vergleich des gebremsten und ungebremsten Anstoßes bei der Kollision zwischen PKW und Fußgängern; Diplomarbeit, Fachhochschule Osnabrück, 1998.

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Testing and Validation of Toyota Vehicle Control History Speed and Acceleration DatA

I

Robert D Anderson

Wesley Vandiver

Biomechanics Analysis

Collision Forensics, Inc.

ntroduction

Unlike traditional Event Data Recorders (EDRs), which have become a vital source of information for those analyzing collisions, the Toyota Vehicle Control History (VCH) can include data recorded from a number of events/triggers that are not collision related. Acquiring VCH data involves different procedures than acquiring traditional EDR data, but yields some familiar data elements, including vehicle speed, engine RPM, accelerator pedal percentage, etc. A series of instrumented tests were conducted to examine the potential data available from the VCH, and the accuracy of speed and acceleration data 102 Collision Magazine - Volume 14 Issue 2

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obtained. VCH data from a dozen different trigger types was obtained and evaluated, and some best practices became apparent to assist an investigator in evaluating and explaining recovered data.

Toyota has been one of the most forthcoming manufacturers with its EDR data in terms of historical coverage and the volume of data available to investigators. In addition, Toyota recognized the importance of recording vehicle behavior and driver operations leading up to crashes and non-crash events, such as evasive maneuvers, aggressive driving, and crashes that do not qualify for an EDR recording, such as pedestrian and low speed impacts.

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The recording of VCH data began with the 2013 RAV4 and 2014 Highlander. This feature has subsequently been added to virtually all Toyota and Lexus models, and one Scion model. Tables listing the production date and model years that VCH coverage began for each North American vehicle in the Toyota family of vehicles is included in crash·ol·o·gy in this issue of Collision (Vol 14, Issue 2).

VCH was first introduced to the forensic automobile crash community by Dave Hallman at the 2014 CDR User’s Summit.[1] Two years later, Iyoda et al. outlined Toyota Motor Corporation’s EDR program which began to be applied to its vehicles in August 2000. In addition to crash triggered recordings of the now familiar EDR, non-crash triggered event recording in the VCH was outlined.[2]

Xing et al. investigated the accuracy of vehicle speed and longitudinal acceleration data in the VCH of a 2017 Corolla during Autonomous Emergency Braking (AEB).

It was found that vehicle speed was consistently underreported, but was generally within 2 kph of the actual speed, and that speed errors increased with the level of braking. The average longitudinal accelerometer error was -0.006 g’s and the sensor saturated so that its minimum reading was -0.91 g’s.[3] Lewis et al. from Toyota Motor North America and Toyota Motor North America Inc. presented VCH data and images obtained from AEB testing with a 2017 Corolla.

The testing was performed to demonstrate the usefulness of the data. Although external instrumentation for comparison was not used, it was observed that the VCH recorded data was consistent with the testing conditions.[4]

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Figure 2: The test vehicle with instrumentation installed

Figure 1: The 2019 Camry test vehicle

Figure 3: Racelogic VBOX Sport and Video VBOX Pro In this project a series of instrumented tests was conducted in July 2019 to examine the potential non-AEB data available from the Toyota VCH and the accuracy of that data.

As promised in crash·ol·o·gy in Collision, Vol 13, Issue 2 and the 2020 EDR Summit, the results and analysis of that testing are presented here.

(VBS) shown in Figure 3. Speed is calculated from the Doppler shift of GPS radio signals. Speed is reported in 0.01 kph (0.006 mph) resolution and the specified accuracy is 0.1 kph (0.06 mph) when averaged over 4 samples.

The longitudinal and lateral acceleration is derived from the speed data. It is reported with a 0.01 g resolution and the specified accuracy is 0.5%. The VBS has an internal Instrumentation, Equipment and Methods battery and GPS antenna, records data at 20Hz, and it Testing was conducted on a large asphalt track in Glen- records no video. The VVBP uses an external battery and dale, Arizona. The test vehicle shown in Figure 1, was a GPS antenna, records data at 10Hz, and records up to 4 2019 Toyota Camry, VIN: 4T1B11HK8KU219703. The synchronized videos. interior of the test vehicle and instrumentation used are The VBOX data was visualized and processed using the shown in Figure 2. latest version of Racelogic’s VBOX Test Suite. UTC Time, Reference speed and acceleration data was obtained from Racelogic Video VBOX Pro (VVBP) and VBOX Sport 104 Collision Magazine - Volume 14 Issue 2

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speed and acceleration data were exported in CSV format so that they could be imported into Microsoft Excel and HydeSoft Computing’s DPlot for processing and graph-

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Figure 4: Example Video VBOX Pro view captured during a hard right turn during testing ing. Since the VCH does have not a universal time stamp, such as UTC, the VCH data was aligned with the VBOX data by visually aligning peaks in the data. The resulting time offset was applied to the entire VCH data set within each phase of testing and was not adjusted further during any part of the analysis. The correlation between the VCH speed and longitudinal acceleration and the reference speed and acceleration was evaluated by calculating the correlation coefficient (R) and the Root Mean Square Error (RMSE).

The driver’s actions and vehicle responses were documented in the VVBP captured video, which included a forward view through the windshield, the driver’s actions, the infotainment screen, and the instrument panel as shown in Figure 4.

Additional video was obtained with a Garmin VIRB camera, which can display GPS data, as well as CAN bus data elements, including engine RPM and vehicle speed via a Bluetooth connection to the DLC. An example of the view from this video is shown in Figure 5.

The Bosch CDR 900 was used with current and activated Toyota Techstream software to obtain VCH and PCS data, and the Bosch CANplus Vehicle Interface Module was used with current and activated Bosch Crash Data Retrieval Tool software to obtain CDR reports, which included ignition cycles.

The test vehicle was put through a number of maneuvers organized in three phases with the goal to set as many different kinds of non-AEB VCH triggers as possible: •

Phase 1: for triggers associated with hard acceleration from low and high speed, rapid braking, and simultaneous accelerator and brake pedal application, the vehicle was driven predominantly forward in straight lines back and forth across the track.

•

Phase 2: for triggers associated with hard acceleration after shifting into drive or reverse, hard acceleration while traveling at medium to high speed or the cruise control is set, the vehicle was driven predominantly in straight lines across the track.

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Figure 5: Example view from the Garmin VIRB video during testing

Figure 6: Speed and Longitudinal Acceleration from VBS in blue plotted over VVBP in red •

Phase 3: for triggers associated with sudden turns, traction control, and vehicle stability control, the vehicle was driven through a series of sudden turns and hard swerving

Evaluation of Speed and Acceleration Data As demonstrated in Figure 6, the speed and acceleration output from the VBOX Sport and Video VBox Pro were essentially indistinguishable. The only difference observed was that when zoomed in as shown in Figure 7, the VBS data has slightly better resolution due its higher sample rate of 20Hz versus the 10 Hz of the VVBP. For the remainder of the analysis, the VBS data was used as the reference for 106 Collision Magazine - Volume 14 Issue 2

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the VCH data, while the VVBP was used to confirm or correlate the driver’s actions and vehicle response that correspond with a set of position, speed and acceleration data.

The testing produced a dozen different kinds of Toyota VCH triggers, which are listed in Table 1. It was common to have overlapping triggers. During Phase 3, overlapping and sequential triggers created more than two minutes of continuous VCH data.

The VCH speed data points plotted over the VBOX speed data for all three phases is shown in Figure 8. Over all three testing phases, the combined data set had a correlation coefficient, R, and RMSE of 0.993 and 0.69 mph,

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Figure 7: Zoomed in over plot of VBS in blue on VVBP in red Trig ID Trigger Description (abbreviated) 2-0 3-0 4-0 5-1 5-2 6-2 8-0 10-0 12-0 13-0 16-0 17-0

accelerator high at low speed accelerator high at mid to high speed accelerator high after brake release accelerator med or higher after shift R accelerator med or higher after shift FWD FWD position with med or high accelerator simultaneous accelerator & brake pedals medium or high accelerator while in N VSC TRC sudden turn accelerator high with cruise control

Number of Triggers Recorded 4 5 13 3 5 3 3 1 5 3 4 3

Time Step (sec) 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.15 0.15 0.15 0.5

Pre and Post Trig (sec) 7.5 7.5 7.5 5 5 5 5 5 5 5 5 5

Table 1: Toyota VCH Triggers Created respectively. For Phases 1 and 2, the R was 0.999 and 0.998, respectively, and it dropped to 0.973 for Phase 3.

Similarly, the RMSEs were 0.60 and 0.68 for Phase 1 and 2, and 0.91 mph for Phase 3.

Accelerating and decelerating in the forward direction produced VCH and VBOX longitudinal accelerations with the same polarity, but accelerating in reverse produced longitudinal acceleration with opposite polarity as shown in Figure 9. Accounting for the polarity difference, Phase 1 and 2 testing produced R and RMSE values of about 0.99 and 0.028 g’s, respectively. For Phase 3 testing, the R dropped to 0.967 and the RMSE increased to 0.038 g’s.

The lateral acceleration data obtained from the VCH appeared to have an offset of approximately -0.025 g’s as demonstrated during portions of the testing where the lateral acceleration was essentially zero, as shown in Figure 10. During the more dynamic portions of the testing, such as hard steering, the VCH lateral accelerations generally tracked the VBOX data with some over-shoot on the peaks, as shown in Figure 11. The correlation coefficient, R, was low for Phase 1 and 2 testing, 0.35 and 0.46, respectively; but it climbed to 0.93 for Phase 3 testing. The overall RMSE for all three test phases combined was 0.07 g’s.

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Phase 1

Phase 3

Phase 2

Figure 8: VCH speed data plotted on the VBOX data

Forward

Reverse

Figure 9: VCH longitudinal acceleration plotted on VBOX data from Phase 2 testing Ignition Cycles Baseline downloads of the test vehicle revealed that the VCH key cycle count at 1274 was 25 ignition cycles higher than the CDR ignition cycle count of 1249. Throughout the test day, these counters remained 25 ignition cycles 108 Collision Magazine - Volume 14 Issue 2

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apart. Throughout the day, ignition cycles were intentionally created in rapid succession. These tests were conducted such that the ignition key was started in the OFF position, cycled to the ON position and back to OFF position five times in rapid succession. These key cycle ON events did not allow sufficient time for the instrument cluster lights

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to illuminate and turn off as one would expect during a normal diagnostic check of the vehicles systems. The EDR was downloaded after every series of ignition cycles. Interestingly, each series of five ignition cycles did raise the number of recorded ignition cycles in the EDR download, but not always by a count of five.

Increased counts of three, four, and five ignition cycles were observed. Equally interesting was the fact that no matter the increase in the EDR download, the VCH data reported the exact same increase.

Time stamps Prior to testing and the baseline download of the VCH, the in-vehicle clock was observed to display the correct time for Arizona, which is UTC -7. The vehicle was driven for approximately 30 minutes while VCH triggers were created by hard acceleration, rapid deceleration, etc.

When the baseline VCH download was conducted approximately one hour after acquiring the vehicle, it included data for triggers with date and time stamps that were approximately one hour in the future and approximately two hours after the triggers actually occurred. Checking the clock settings, it was determined that while the correct time was being displayed, it had been set manually. In other words, the setting for “Auto Adjust

by GPS” was turned off. The time zone was set to Eastern Time without compensation for Daylight Savings Time. The Eastern Time Zone, without adjustment for Daylight Savings Time is UTC -5, two hours different than the current time at our test location. As seen in Figure 12, the download of the VCH data involves the collection of time information from the vehicle navigation system; however, location data did not appear to be considered – only the time information relative to the indicated time zone was reported.

Later testing conducted with all clock settings properly adjusted yielded timestamps that were appropriate for the time and time zone during those tests.

Figure 10: VCH lateral acceleration plotted over VBOX data during straight line driving

Figure 11: VCH lateral acceleration plotted over VBOX data during hard steering Discussion As a means of comparison of the VCH speed and longitudinal and lateral acceleration with reference instrumentation, the correlation coefficient (R) and root mean square error (RMSE) were determined. The correlation coefficient is a measure of how closely two signals track one another and R values close to 1 are desirable. The root mean square error is a measure of the variance between the data sets and can be thought of as www.collisionmagazine.com

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Figure 12: Screen showing time information being collected from the navigation system the standard deviation of the errors. RMSE values close to 0 are desirable.

The speed data from the VCH matched reference speed data very well with correlation coefficients above 0.99 and RMSEs of less than 1 mph. A reduction in speed accuracy was observed during hard, acceleration, braking and yawing, which is to be expected during any event involving tire slip.

The reference VBOX acceleration was calculated from the motion of the antenna over the ground or x-y plane. Accelerometers are subject to shifts in readings due to vehicle pitch and roll due to the component of gravitational acceleration that is introduced. The -0.025 g offset observed in the lateral accelerometer VCH data during straight line driving, shown in Figure 10, could be accounted for by a 1.5 degree roll angle of the vehicle. Similarly, the overshoot observed in the VCH lateral acceleration during hard steering, shown in Figure 11, could also be accounted for by vehicle roll during those maneuvers.

Despite the inaccuracy introduced in the VCH accelerometer data from changes in vehicle orientation favorable results were obtained. Compared to the reference instrumentation, the longitudinal and lateral accelerations’ correlation coefficients were above 0.97 and 0.93, respectively, and the RMSEs were less than 0.04 g’s and 0.07 g’s, re110 Collision Magazine - Volume 14 Issue 2

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spectively. As such, the VCH reported acceleration can be considered a fair representation of the vehicle’s actual acceleration.

Although the testing in this study was limited to a single vehicle, the results are within the expected accuracy of EDR data that has been observed by researchers in other studies. Perhaps this is not surprising, since the source of the data, the vehicle CAN bus, is the same for both VCH and CDR. As such, accuracy of CDR and VCH data can be thought of as different approaches to assessing the accuracy of vehicle CAN data.

Prior to this study, it was thought that the discrepancy observed between the CDR reported ignition cycles and the VCH key cycles was due to how cycles were counted.

It was thought that the discrepancy could be accounted for if the ACM counter was incremented only when the vehicle was started and ran long enough for the ACM to boot up, while the VCH counter increments even in shorter start-off cycles. However, this supposed behavior was not observed during this test series as a constant difference between VCH and CDR reported ignition cycles was observed regardless of how long the vehicle was turned on or run. The behavior of this vehicle implies a constant difference between the two ignition cycle counters during the vehicle’s service life presumably dating back to the factory.

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While only one vehicle was tested, because the vehicle CAN bus is the data source for both VCH and EDR data, the subject study can be thought of as adding to the ever growing body of research assessing the accuracy of vehicle CAN bus data.

Overlapping VCH trigger data can extend the duration of recorded data. In fact, more than two minutes of continuous VCH data was obtained due to over-lapping and sequential triggers.

Figure 13: The Toyota Camry’s clock settings screen Although early examples of VCH data provided only ignition cycle counts for an investigator to rely upon in determining when a trigger occurred, later versions include date and time stamps. These data are of tremendous use to an investigator, but could be problematic if the reported data is not what is expected for the incident location. Under the circumstances of this testing, a possible source of unexpected data in the date and time stamps was found.

The system reported data as occurring in the time zone for which the system was set. Based on this study, a few basic “best practices” were identified that will assist investigators in proper and accurate evaluation of VCH data: •

Document the vehicle clock settings to explain discrepancies in the time stamp for VCH triggers related to time, time zone, DST and auto adjust settings.

•

Obtain the CDR report and VCH data on the same key cycle in order to document the difference in key cycle counters.

Conclusions The accuracy of vehicle speed and, longitudinal/lateral acceleration reported by the Toyota VCH for a dozen types of non-AEB triggered recordings was quantified.

The test vehicle’s VCH reported speeds and accelerations were found to have a good correlation with reference instrumentation.

The time stamp of VCH triggers was found to depend upon clock settings. Therefore, a best practice of documenting vehicle clock settings is recommended to enable the investigator to explain possible time stamp discrepancies.

A constant difference between the VCH key cycle and CDR ignition cycle counters was observed. A best practice based upon this observation is to obtain the CDR report and VCH data on the same key cycle in order to determine the difference in key cycle counters.

References 1. David Hallman, “Unique Features of Toyota GEN3 Data,” presented at the 2014 CDR User’s Summit, Houston, Texas.

2. Motomi Iyoda, Tom Trisdale, Rini Sherony, Daniel Mikat, and William Rose, Event Data Recorder (EDR) Developed by Toyota Motor Corporation, SAE International, Technical Paper 2016-01-1495 3. Peter Xing, Mike Yang, Brandon Tsuge, Thomas Flynn, Jonathan Lawrence and Gunter Siegmund, “The Accuracy of Toyota Vehicle Control History during Autonomous Emergency Braking,” SAE International, Technical Paper 2018-01-1441 4. Lance Lewis, Barry Hare, Harold Clyde and Robert Landis, “Vehicle Control History: Data from Driver Input and Pre-Collision System Activation Events on Toyota Vehicles,” SAE International, Technical Paper 2019-01-5094

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