JAMRIS 2012 Vol 6 No 4 by JAMRIS

pISSN 1897-8649 (PRINT) / eISSN 2080-2145 (ONLINE)

VOLUME 6

N째 4

2012

www.jamris.org

JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS

Editor-in-Chief Janusz Kacprzyk

Executive Editor: Anna Ładan aladan@piap.pl

(Systems Research Institute, Polish Academy of Sciences; PIAP, Poland)

Associate Editors: Jacek Salach (Warsaw University of Technology, Poland) Maciej Trojnacki (Warsaw University of Technology and PIAP, Poland)

Co-Editors: Oscar Castillo (Tijuana Institute of Technology, Mexico)

Dimitar Filev

Statistical Editor: Małgorzata Kaliczyńska (PIAP, Poland)

(Research & Advanced Engineering, Ford Motor Company, USA)

Kaoru Hirota Editorial Office: Industrial Research Institute for Automation and Measurements PIAP Al. Jerozolimskie 202, 02-486 Warsaw, POLAND Tel. +48-22-8740109, office@jamris.org

(Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan)

Witold Pedrycz (ECERF, University of Alberta, Canada)

Roman Szewczyk (PIAP, Warsaw University of Technology, Poland)

Editorial Board: Chairman: Janusz Kacprzyk (Polish Academy of Sciences; PIAP, Poland) Mariusz Andrzejczak (BUMAR, Poland) Plamen Angelov (Lancaster University, UK) Zenn Bien (Korea Advanced Institute of Science and Technology, Korea) Adam Borkowski (Polish Academy of Sciences, Poland) Wolfgang Borutzky (Fachhochschule Bonn-Rhein-Sieg, Germany) Chin Chen Chang (Feng Chia University, Taiwan) Jorge Manuel Miranda Dias (University of Coimbra, Portugal) Bogdan Gabryś (Bournemouth University, UK) Jan Jabłkowski (PIAP, Poland) Stanisław Kaczanowski (PIAP, Poland) Tadeusz Kaczorek (Warsaw University of Technology, Poland) Marian P. Kaźmierkowski (Warsaw University of Technology, Poland) Józef Korbicz (University of Zielona Góra, Poland) Krzysztof Kozłowski (Poznań University of Technology, Poland) Eckart Kramer (Fachhochschule Eberswalde, Germany) Piotr Kulczycki (Cracow University of Technology, Poland) Andrew Kusiak (University of Iowa, USA) Mark Last (Ben–Gurion University of the Negev, Israel) Anthony Maciejewski (Colorado State University, USA) Krzysztof Malinowski (Warsaw University of Technology, Poland) Andrzej Masłowski (Warsaw University of Technology, Poland)

Patricia Melin (Tijuana Institute of Technology, Mexico) Tadeusz Missala (PIAP, Poland) Fazel Naghdy (University of Wollongong, Australia) Zbigniew Nahorski (Polish Academy of Science, Poland) Antoni Niederliński (Silesian University of Technology, Poland) Witold Pedrycz (University of Alberta, Canada) Duc Truong Pham (Cardiff University, UK) Lech Polkowski (Polish-Japanese Institute of Information Technology, Poland) Alain Pruski (University of Metz, France) Leszek Rutkowski (Częstochowa University of Technology, Poland) Klaus Schilling (Julius-Maximilians-University Würzburg, Germany) Ryszard Tadeusiewicz (AGH University of Science and Technology in Cracow, Poland)

Stanisław Tarasiewicz (University of Laval, Canada) Piotr Tatjewski (Warsaw University of Technology, Poland) Władysław Torbicz (Polish Academy of Sciences, Poland) Leszek Trybus (Rzeszów University of Technology, Poland) René Wamkeue (University of Québec, Canada) Janusz Zalewski (Florida Gulf Coast University, USA) Marek Zaremba (University of Québec, Canada) Teresa Zielińska (Warsaw University of Technology, Poland)

Publisher: Industrial Research Institute for Automation and Measurements PIAP

If in doubt about the proper edition of contributions, please contact the Executive Editor. Articles are reviewed, excluding advertisements and descriptions of products. The Editor does not take the responsibility for contents of advertisements, inserts etc. The Editor reserves the right to make relevant revisions, abbreviations and adjustments to the articles.

JOURNAL of AUTOMATION, MOBILE ROBOTICS & INTELLIGENT SYSTEMS VOLUME 6, N° 4, 2012

CONTENTS 33

Angle Measuring by MEMS Accelerometers Kamil Zidek, Miroslav Dovica, Ondrej Liska

Positioning and Control of Nozzles and Water Particles in Decorative Water Curtain and Water Screens Mahdi Hajiheydari, Sasan Mohammadi

Modeling of Circulatory System with Coronary Circulation and the POLVAD Ventricular Assist Device Alicja Siewnicka, Bartlomiej Fajdek, Krzysztof Janiszowski 13

PI Control of Laboratory Furnace for Annealing of Amorphous Alloys Cores Jerzy E. Kurek, Roman Szewczyk, Jacek Salach and Rafał Kloda

Stable Gait Synthesis and Analysis of a 12-degree of Freedom Biped Robot in Sagittal and Frontal Planes A.P. Sudheer, R. Vijayakumar, K.P. Mohandas 45

Intelligent Utilization of Waste of Electrical and Electronic Equipment (WEEE) with Robotized Tool Jakub Szalatkiewicz, Roman Szewczyk 50

Surface Topography Parameters Important in Contact Mechanics Paweł Pawlus, Wiesław Zelasko, Jacek Michalski

Influence of PWM to Trajectory Accuracy in Mobile Robot Motion Ryszard Beniak, Tomasz Pyka 57

Behavior Based Co-ordination of a Troop of Vehicles Targeted to Different Goals in an Unknown Environment Sourish Sanyal, Ranjit Kumar Barai, Pranab Kumar Chattopadhyay, and Rupendranath Chakrabarti 27

About Evaluation of Multivariate Measurements Results Zygmunt L. Warsza

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Robot for Monitoring Hazardous Environments as a Mechatronic Product Leszek Kasprzyczak, Stanislaw Trenczek, Maciej Cader 65

Analysis of Influence of Drive System Configurations of a Four Wheeled Robot on its Mobility Maciej Trojnacki

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Angle Measuring by Mems Accelerometers Submitted: 26th June 2011; accepted 23rd September 2011

Kamil Zidek, Miroslav Dovica, Ondrej Liska

Abstract:

This article contains the description of MEMS accelerometers implementation to device which is able to measure danger tilt. We can find out actual tilt in two basic axes X and Y, from -90° to +90°. Z Axis can only detect fall of device or in vehicle system very fast downhill grade during movement. For testing of the solution we select small mobile robotic carriage. Hardware and software part of solution are described. Because data from sensor are in raw format from analog MEMS Accelerometer, we use free C# library with Kalman Filter implementation to remove signal error. We can acquire next information from sensor data for example movement’s trajectory in X/Y axis (Cartesian system) and actual speed in all three axes. Fast alarm is provided by RGB led diode (red color is dangerous tilt.

• robots and automated devices with balancing function (segway), • controls with tilt measuring, • Auto pilots of aeroplanes, • car alarm systems, • car crash detection (used in airbag system), • monitoring of human movements (virtual reality gloves). Example of MEMS microstructure sensor magnified by microscope is displayed in Fig. 1 on the left side, right is displayed measuring principle.

Keywords: mems, Kalman filter, control.

1. Introduction to MEMS Sensors

Shortcut MEMS means micro electromechanical systems, marks mechanical and electromechanical construction of very small dimensions, and technologies used for their preparation too. MEMS technology is based on many tools and methods, which are used for creating very small structure with dimension of couple micrometers. An important part of technology was takeover from production of Integrated circuit (IC technology). Almost all of these devices are based on a silicon substrate. MEMS structures are realized from thin layer. There are produced by photo lithographic methods. Some other methods also exist, but they aren’t derivate straight from technology of IC. There are three basic steps of operation in MEMS technology for layer applying to silicon material to substrate. Process of MEMS is usually a structured sequence of this operation for creating real application. Real device, then, contains central unit for processing of data (microprocessor), and some other mechanical part which compose unit named micro sensor too [4].

Fig. 1. MEMS sensor and principle of microstructures Older accelerometers had big dimension and were very expensive. The construction was created from standard metal parts, springs and PCB. That was reason why at that time accelerometers were not used in electronics nor robotics. This situation was changed thanks to progress in MEMS technology. MEMS technologies reduce the price, energetic consumption and dimensions. Main usability is measuring of acceleration in three Axes: X – forward/ backward, Y – left/right, Z – up/down. For mobile robotics we can use this sensor for measuring acceleration or deceleration by movement front and back, second Axis for change direction of movement right or left, and third Axis for fall detection of device. Second method of usability is measuring of device tilt based on simple mathematics. Figure 2 shows MEMS Axis configuration and principle of tilt measuring with this sensor along y axis.

2. MEMS accelerometers

One of usual application for MEMS is sensor for measuring of acceleration. This MEMS sensor is usually named Accelerometer. They are divided to one-, two- or three Axes. Measuring of acceleration is possible to use in electronics and robotics for measuring: acceleration, deceleration, tilt, rotation, vibration, collision (crash) or gravitation. Accelerometers are used in many devices, special equipment and personal electrotechnics, for example:

Fig. 2. MEMS sensor: axis configuration, principle of tilt measuring Output information from accelerometer is voltage which depends on movement or tilt of sensor in space. A static characteristic of sensor is not exactly linear. For common application we can this nonlinearity omit. The acceleration is usually in MEMS application measured Articles

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in G unit. Expression “1 g = 9,80665 m/s2” means, that for every second, which passed the speed change will be 9,80665 meters for second. That is approximately speed 35.30394 km/h. The three Axis accelerometer can get null G on every Axis, if is in ballistic trajectory known as inertial or free fall. If we turn the accelerometer to 90 ° the output from one Axis will be exactly +1 g. In this situation, accelerometer measuring gravitation Force and can be in static position. Described characteristics for analogue MEMS sensor is depicted in Fig. 3. [1]

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Our values are counted according changed math, because we don’t know max and min values for actual accelerometer. This math get extreme values during accelerometer operation. incr = 180 * (H_max - H_min) (3) θ = incr * (H_nam - H_min) (4) where: H_max, H_min – initial value of accelerometer extreme H_nam – actual accelerometer value

3. Tested hardware platform

Fig. 3. Characteristics of MEMS accelerometer sensor with nonlinearity [2] Example of block scheme sensor connection to user application is displayed in the Fig. 4. Additional LCD display connected straight to microcontroller enabling testing of application without Computer necessity.

Introduced solution was tested on mobile computer with open source application in programming language C#. A prototype board contains Accelerometer MMA7341L (analog) and accelerometer MMA7455 (digital) from Freescale. Currently there is active only analog Accelerometer. Microcontroller computes values of voltage for all Sensor Axis with help of three 10 bits ADC converters. Data are coded to frames (9 bytes as string $XXYYZZ1310). Every axis has value coded to two bytes (Low and High 8 bites). First method of accelerometer communication is only for debug the application. Sensor is connected straight to PC. Data are sent thru serial line to serial port of PC. For implementation to mobile robot is used USART interface without UART/RS232 Transducer and communicate straight with High Level control system based on AT91 control board with Linux Embedded OS. These serial data are transferred to TCP packet thru ser2net command line application. Data are sent next thru wife interface to C# application. Block diagram of testing debug solution and mobile control system implementation is displayed in Fig. 5. Figure 6 shows is first prototype of sensor without RS232/USB transducer.

Fig. 4. Block scheme of complete application based on MEMS accelerometer sensor Sensitivity of measured values depends on sensor G range (most precise we acquire if sensor is set to ±1g). A disadvantage is that we cannot measure the higher values of acceleration. Common sensors are produced to ±5g and it is possible to switch between ranges during application activity. Computations of tilt angles θ are realized thru basic mathematics and goniometric function. V_out is actual value of voltage; V_offset is voltage by 0 g. Sensitivity of sensor is defined by technical documentation. In math is necessary find out positive or negative acceleration according to offset value. Datasheet math count according this: V_OUT=V_OFF+ ΔV/Δg*1g*sinθ θ=arcsin(V_OUT- V_OFF)/(ΔV/Δg) where: V_OUT – output of accelerometer (V) from ADC VOFF – acceleration 0 g offset ΔV/Δg – sensitivity 1 g – world gravitation θ – tilt angle 4

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Fig. 5. Block diagram of connection sensor to testing mobile control system

(1) (2) Fig. 6. Hardware of accelerometer. 1 – microcontroller; 2 – accelerometer MMA7341L (analog); 3 – accelerometer MMA7455 (digital); 4 – voltage regulator LF33CDT; 5 – I2C bus for LCD BO1602D; 6 – USB connector; 7 – RGB LED diode.

Journal of Automation, Mobile Robotics & Intelligent Systems

4. Software platform implementation

Software solution is based on an open source C# application, which is currently implemented to mobile solution Graphical Interface of solution is displayed in Figure 7. Left is displayed 2D graphics, tilt in x-Axis, left 3D graphics tilt in all three axis X,Y,Z. All values of real time tilt are displayed in graphical interface in text edit boxes. Basic value of danger tilt is set to value bigger than 40°. This value starts critical routine and block movement of mobile device to actual direction. Danger tilt value can be changed through graphical interface from 0–90°.

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K – Kalman gain, P – Covariance update, x – State update, F – State transition matrix, G – Noise coupling matrix, Q – Plant noise covariance matrix, H – Measurement model, R – Covariance of measurements, I – Matrix identify, z – Measurements of the system. Figure 8 shows graph of actual values when MEMS sensor is stand statically on the ground (blue plotline). Black plotline shown filtered value cleared from errors and noise from ADC transduction. There is used for testing application only 1D Kalman filter for filtering only actual acceleration value. Next extension will be implementation of 2D or 3D filter for all three Axes.

Fig. 7. C# application, 3D tilt X,Y Axis and configuration panel

5. Kalman filter implementation to smooth raw data

The Kalman filter is an efficient recursive filter that estimates the internal state of a linear dynamic system from a series of noisy measurements. The Kalman filter is used in sensor fusion and data fusion. Typically real time systems produce multiple sequential measurements rather than making a single measurement to obtain the state of the system. These multiple measurements are then combined mathematically to generate the system’s state at that time instant. Acquired data from MEMS sensor are in raw form with many disturbances, white noise etc. For testing solution we implement free C# Math. NET Neodym (Signal Processing) [8] with Kalman filter function to desktop application. Graphical interface provides settings of three basic values of Kalman filtering r, T, q which is necessary for customizing filter for real application. r – Measurement covariance T – Time interval between measurements q – Plant noise constant Discrete Kalman Filter consists of two parts: prediction and update. prediction: x(k|k-1) = F(k-1) * x(k-1|k-1) (5) P(k|k-1) = F(k-1)*P(k-1|k-1)*F(k-1) + G(k-1)*Q(k1)*G’(k-1) (6) update: S(k) = H(k)*P(k|k-1)*H’(k) + R(k) (7) K(k) = P(k|k-1)*H’(k)*S^(-1)(k) (8) P(k|k) = (I-K(k)*H(k))*P(k|k-1) (9) x(k|k)=x(k|k-1)+K(k)*(z(k)-H(k)*x(k|k-1)) (10) where: S – Measurement covariance,

Fig. 8. Kalman filtering for raw accelerometer data in static position In the Fig. 9 is displayed data from accelerometer during tilt to 90° to one side, next to static position and then tilt to opposite side. Reference signal is red plotline. Black line is Kalman filtered value.

Fig. 9. Dynamic data from MEMS Accelerometer sensor with Kalman filtering We experimentally find out constants for kalman filter with compromise of minimal displace during dynamic and static operation: r = 30.0, T = 2.0, q = 0.1. There is one problem in setup filter, when there is very fast acceleration and deceleration. This situation can occur when the real device fall or crash to the obstacle. We can avoid this situation by setting adequate value to danger alarm tilt and implementation of obstacles sensor detection (infra or ultrasonic) to mobile solution. Articles

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6. Conclusion

Introduced measuring solution is implemented to mobile device. Actual possibilities are measuring of tilt device – 90° to 90°. You can select bound angle for start indication of danger device tilt with next visual or sound alarm. We can improve precision data from MEMS sensor by using 12 bit ADC but then is necessary change the microcontroller. Next idea can be change of Accelerometer with digital I2C output, which removes error generated by ADC conversion. We are computing next values from acquired data for example: trajectory, deceleration, average and actual speed. Next work on this solution will be an implementation of Kalman filter to program of MCU firmware and display actual angle value and alarm on LCD display. This remove testing mobile computer from actual solution and application will be small and compact device. This researched accelerometer device will be used in rehabilitation system as safety circuit to monitor extreme acceleration and deceleration for fast action to stop device.

Acknowledgements

The research work is supported by the Project of the Structural Funds of the EU, Operational Program Research and Development, Measure 2.2 Transfer of knowledge and technology from research and development into practice: Title of the project: Research and development of the intelligent non-conventional actuators based on artificial muscles ITMS code: 26220220103

AUTHORS

Kamil Zidek*, Ondrej Liska, Miroslav Dovica – Technical University of Kosice, Faculty of mechanical engineering, Department of Biomedical engineering automation a measuring, Košice, 042 00, Slovakia, kamil. zidek@tuke.sk, ondrej.liska@tuke.sk, miroslav.dovica@ tuke.sk *Corressponding author

References

[1] Tuck K., “Tilt Sensing Using Linear Accelerometers”, Application Note, AN3461, Rev 2, 06/2007.http://www.freescale.com/files/sensors/ doc/app_note/AN3461.pdf [2] Clifford M., Gomez L., “Measuring Tilt with Low-g Accelerometers”, Application Note, AN3107, Rev 0, 05/2005,http://www.freescale.com/files/sensors/ doc/app_note/AN3107.pdf [3] “What is MEMS Technology?“ https://www. memsnet.org/mems/what-is.html, Online, cit. 8.2.2010 [4] Johnson C. D., “Accelerometer Principles”, Process Control Instrumentation Technology, Apr 14, 2009, 0-13-441305-9. [5] Zidek K: “Open robotics control system”, Technical University Kosice, Online, www.orcs.sebsoft.com 6

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[6] Saloky T., Piteľ J., Vojtko I., “Control systems design with reliability defined in advance”. In: Proceedings of the 1st IFAC Workshop on New Trends, Design of Control Systems, Smolenice, Slovakia, 7th–10th September 1994, pp. 404–407. [7] Zidek K., MEMS Accelerometer SVN, Google code, 2010. http://code.google.com/p/orcs/source/ browse/#svn/MEMS_Accelerometer_SVN2 [8] Christoph Rüegg, Math.NET Neodym 2008 February Release, v2008.2.2.364, “http://www.mathdotnet. com/downloads/Neodym-2008-2-2-364. ashx?From=NeodymCurrentRelease”

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Modeling of Circulatory System With Coronary Circulation and the POLVAD Ventricular Assist Device Submitted: 26th June 2011; accepted 23rd September 2011

Alicja Siewnicka, Bartlomiej Fajdek, Krzysztof Janiszowski

Abstract:

This paper presents an application of a numerical package for modeling and simulation of human circulatory system. The model includes a coronary circulation and the parallel heart assistance. The cases of the simulation of the proper and the pathology circulation conditions, such as left or right heart failure are shown. A description of the coronary circulation system is presented and obtained coronary sinus occlusion simulation results are included. An implementation of the whole package as a part of PExSim application is contained. The identification experiment for the ventricular assist device has been described and different methods of the artificial ventricle modeling are presented. An example of use of a fuzzy logic to presentation the dynamics of the POLVAD device is also included. Advantages of developed simulation platform are discussed. Keywords: PExSim, modeling of the circulatory system, modeling of the coronary system, modeling of the ventricular assist device, POLVAD.

1. Introduction The continuous development of technology enabled for the more common use of its achievements in medical applications. Therefore, in recent years many scientific projects were run to allow the use of technology to save human life and health. One of the biggest bio-engineering projects in Poland is the Polish Artificial Heart Program, whose aim is to develop of the construction and control algorithms for the heart assist device. Application of the developed solutions requires accurate testing. For this purpose the modeling methods are widely used. In recent years some different models of the human circulatory system were developed, both numerical and physical ones, for example electrical or hydraulic [1, 2]. All of them are widely used to reproduce hemodynamic conditions of circulation system. Besides this, they can be applied for the testing of medical devices such as the blood pumps or assist devices. For this purposes also models of the same new devices are created. This way the possibility of simulating of the influence of the heart support can be obtained without carrying out experiments on living organisms. This paper contains description of the developed circulatory model with the possibility of connection of the simply models of the extracorporeal ventricle assist device. In our case, the main aim of development the mathematical description of circulatory system was to create a research platform, for general purpose, which could be easily adopted to solve different problems. For

example it could be used for determination and testing of a Polish Ventricular Assist Device (POLVAD) control and diagnostic algorithms.

2. Model of the circulatory system The main part of the developed research platform is the mathematical model of the circulatory system. It is based on the description proposed by Ferrari [3] and implemented as a part of the PExSim application [4]. This software consists of predefined function blocks that represent basic mathematical and logic relationships, dynamic and static elements or support an input and output operations. The possibility of easy extension with user-written objects makes it a flexible tool that can be used for emulation of complex dynamic system. For the simple and clear presentation of circulation system and to ensure the ability of easy parameters changes, each of the blocks is responsible for reproducing behavior of different part of human circulatory system. They are grouped in Human Circulatory System library, which consists such elements and systems as: – left and right ventricle (LH, RH), – systemic arterial circulation (SAC), – systemic venous circulation (SVC), – pulmonary arterial circulation (PAC), – pulmonary venous circulation ( PVC). The model is based on a Starling`s law [3], which defines the conditions for a balance between the filling and ejection characteristics of the ventricle. The basic relation is a function that makes it possible to calculate the ventricular pressure [3, 5]: Pv (t ) = (Vv (t ) − Vv 0 (t )) ⋅ Emax ⋅ f (Vv (t ),V v (t ),Vv max (t )) + A ⋅ e k ⋅Vv ( t ) + B ⋅ e − j ⋅Vv ( t ) + C

(1)

where: Pv(t) – the ventricle pressure, Vv(t) – the ventricle volume, Vv0 – the ventricle rest volume, Ev(t) – the normalized elastanse function, Emax – the maximum value of the elastance (end-systolic), f (Vv (t ),V v (t ),V v max (t )) – the correction function dependent on the ventricle volume and ejection rate, and A, B, C, j, k – constant parameters. The values of the flows are calculated as the ratio of the proper pressure difference and the vascular resistance. The rest of the pressures values are calculated as the solution of the differential equation which defines the pressure derivative as the quotient of the sum of flows and the capacity of the system. In the PExSim modeling platform, the simple model of ventricular assist device (VAD), based on the ventricle activity description, was added. The parameters values were adopted to ensure Articles

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work consistent with the theoretical data. The model of the ventricular assist device is included into the system between the atrium and the arterial system, creating a bypass of the model of the natural ventricle. The mathematical equations in the circulation blocks were respectively adopted to join the parallel ventricle support. The proper connection of all elements creates a complete model of circulatory system with the external assist device (Fig. 1).

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3. The coronary circulation model The system described above did not contain the coronary circulation model. As the extension of functionality of the developed platform, the coronary circulation model was developed and included. This way a possibility of simulating the caval occlusion was achieved. The applied

Fig. 3. The electric analogue of the coronary circulation system

Fig. 1. The circulatory model in PExSim application with the parallel connection of ventricular assist device A full description of structure and function of the blocks one can find in [6]. As a result of this work we received a useful tool, that yields an investigation of the proper and pathological circulation conditions and the influence of the ventricle assistance for the hemodynamic conditions. For example, Fig. 2 presents the influence of the left heart support on the atrium (Pla) and arterial (Pas) pressures.

mathematical description is the combination of the models proposed in [7] and [8]. The schematic representation of this, as an analogy to electric circuit diagram, is shown in Fig. 3. It was added to the mentioned PExSim application as a new CC (Coronary Circulation) element. The driving pressure for the coronary circulation (Psqz) is taken as a proportional to left ventricle pressure. According to this, the input values for the model are: pressure values in aorta, left ventricle and right atrium (Pil, Plv, Pra). The others pressures are obtained by the equations:

Plca (t ) = Pil (t ) − Rlca ⋅ Qart (t )

Plcx (t ) =

Plad (t ) =

Fig. 2. The modeled waveforms of the atrium (Pla) and arterial (Pas) pressures for normal, pathological and pathological with left ventricle assistance (LVAD) conditions In the ventricular failure state the blood accumulates in atrium causing the rise of the pressure volume. At the same time the pressure in arteries is low due to insufficient stroke volume of the heart. In simulation we can observe the reduction of the atrium pressure value and increase of the arterial pressure as a result of left ventricle support which confirms medical observations. 8

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Pven (t ) =

Vven (t )

χ ven

Vlcx (t ) + Psqz (t ) Clcx

Vlad (t ) + Psqz (t ) Clad

⋅ exp[σ (Vven (t ) − Vven 0 )]

(2) (3) (4) (5)

where: Plca – the pressure in bifurcation of coronary arteries, Pil – the aortic pressure, Qart – the coronary arterial flow, Plcx and Plad – the coronary capillaries pressures (lad – left anterior descending artery, lcx – left circumflex artery), Rlca – the arterial resistance, Clcx and Clad – the coronary capillaries compliances, Vlcx and Vlad – the coronary capillaries volumes, Vven – the coronary veins volume, χven, σ, Vven0 – the parameters for venous compliance. The coronary arterial flow value (Qart) is obtained as a sum of the capillaries input flows (Qlcx1, Qlad1). Input and output flows are calculated based on pressures difference and blood vessels volumes as follows: flows in direction “to the coronary capilares system” (Plca≥Plcx, Plca≥Plad, Pven ≥Plcx, Pven ≥Plad, Pra ≥Pven):

Qlcx1 (t ) =

Plca (t ) − Plcx (t ) Rlcx1

(6)

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Qlad 1 (t ) =

Plca (t ) − Plad (t ) Rlad 1

(7)

Qlcx 2 (t ) =

Plcx (t ) − Pven (t ) Rlcx 2

(8)

Qlad 2 (t ) =

Plad (t ) − Pven (t ) Rlad 2

(9)

Qlcx1 (t ) =

Plca (t ) − Plcx (t ) Rlcx1 + β ⋅ (Vlcx ) −2

(11)

Qlad 1 (t ) =

Plca (t ) − Plad (t ) Rlad 1 + β ⋅ (Vlad ) −2

(12)

Qlcx 2 (t ) =

Plcx (t ) − Pven (t ) Rlcx 2 + β ⋅ (Vlcx ) −2

(13)

Qlad 2 (t ) =

Plad (t ) − Pven (t ) Rlad 2 + β ⋅ (Vlad ) −2

(14)

Qven (t ) =

Qex (t ) =

Pven (t ) − Pra (t ) Rven + β ⋅ (Vven ) −2

Fig. 4. The circulatory model in PExSim application with the coronary circulation system model As a result we gained the opportunity to simulate various dysfunctions of the coronary circulation system, e. g. coronary sinus occlusion. In Fig. 5 we can see the value of flow from the capillaries to veins in normal state and for the simulation of sinus occlusion.

(15)

Pven (t ) − Pra (t ) + Pvenoff

−2

Rex + ⋅ (Vven ) (16) where: β – the parameter for volume dependent resistances, Qex – the extraordinary venous outflow, Rex – the extraordinary venous resistance, Pvenoff – the venous pressure offset. The volumes of vessels are determined from the equations: (17) V lcx (t ) = Qlcx1 (t ) − Qlcx 2 (t )

2012

Pven (t ) − Pra (t ) Rven

(10) where: Qlcx1 and Qlad1 – the capillaries input flows, Rlcx1 and Rlad1 – the capillary resistances, Qlcx2 and Qlad2 – the capillary output flows, Rlcx2 and Rlad2 – the capillary resistances, Qven – the flow supplying the right atrium, Pra – the right atrium pressure, Rven – the coronary venous resistance. • flows in direction “from the coronary capilares system” (Plca<Plcx, Plca<Plad, Pven <Plcx, Pven <Plad): Qven (t ) =

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V lad (t ) = Qlad 1 (t ) − Qlad 2 (t )

V ven (t ) = Qlcx 2 (t ) + Qlad 2 (t ) − Qven (t ) − Qex (t )

(18) (19)

This complex model was added to the PExSim application as a new Coronary Circulation (CC) function block. In order to join the coronary circulation function block to the main model of circulation system, the modifications in equations for systemic arterial and venous circulation models had to be implemented. In SAC element the input for coronary arterial flow (Qart) was added and the description was modified. In SVC element two extra inputs were added, for the coronary venous and extraordinary flows (Fig. 4).

Fig. 5. The modeled coronary flow from the capillaries to veins under variation of perfusion pressure for the normal (A) and coronary sinus occlusion (B) conditions For the coronary vein occlusion conditions we can observe the increase of the invert flow value as a result of vascular obstruction.

4. The POLVAD modeling The next step for development of complete simulation platform was to create a model of the real assist device based on measurement results. For this purpose, an identification experiment of an extracorporeal ventricular assist device (POLVAD) was carried out at the Foundation of Cardiac Surgery Development in Zabrze (Poland). The Polish Cardiac Assist System POLCAST undergoing testing consists of the pneumatic driving unit POLPDU-402 and the pneumatic drive pulse pump POLVAD-MEV with disc valves and two chambers separated by a flexible membrane (Fig. 6). The blood flow is a result of membrane movement, which is caused by air pressure changes in the pneumatic chamber. For the Articles

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in vivo experiments, instead of blood, a fluid with similar properties was used. The assist device was connected to a hybrid circulatory system model, which represents the circulatory system.

Fig. 6. The extracorporeal heart support ventricle POLVAD: a) pneumatic ventricle, b) blood ventricle, c) pneumatic pipe, d) input cannula, e) output cannula, g), f) disc valves The following variables were measured: a supply pressure (Ppn), and pressures and flows in input and output cannula (Pin, Pout, Qin, Qout). As a result of the experiment, data series for different supply and load conditions were obtained. They were used as a basis for modeling the dynamics of the ventricular assist device which is not a trivial task. The change in parameters strongly affects the nature of the process. What is more, a model of mechanical valves has to be developed. The next problem is a strong dependence of the process state: full filling and emptying of blood chamber. For cardiological reasons, the most important value is the volume of the cardiac output. This could be estimated from the output flow value. For this purpose, proper modeling of the output flow is the basic factor in a control system design. The value of the input flow is also very important. That is why we have been trying to develop a model of the structure presented in Fig. 7, in which Pmax, Pmin, %SYS, BPM are the maximal and minimal values of the control pressure, the percentage ratio of the systolic phase time to the cycle time, and the beats per minute, respectively.

Fig. 7. A schematic diagram of the dynamics of a ventricular assist device The original idea for more accurate mathematical description of the chamber was to modify the shape of the elastance function. The valves, usually modeled as ideal diodes [3], were replaced by inertia elements. The method of determining the pressure in the blood chamber was also modified [9]. The unknown parameters of the description were determined using dedicated program called PExSim Optimizer [10]. The waveforms obtained by modeling approximately reflected the nature of the 10

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measured flows values. However this model was independent from the control pressure value. It means that description would never be universal – we would have to adjust parameters to every particular situation. Another approach to the proper description of the dynamics of the assist device was an attempt to modeling using the fundamental principles of hydraulics and pneumatics. First, the pressure in the blood chamber was determined as a function of the control pressure. An algorithm was obtained to detect changes in the control pressure phases, which facilitated the modeling the opening and closing of the valves. The output flow was calculated on the basis of the pressure balance and changes in the valve resistance. The parameters of the process were selected manually. The exemplary result of the simulation is presented in Fig. 8.

Fig. 8 The modeled and measured assist device output flow (Qout) The obtained waveforms are not fully identical with real values, but they reflect the nature of the appropriate flows. The advantage of this description is that for the same parameter values it gives a similar quality of rendering for different supply parameters. In this model, the control pressure is one of the model inputs and it provides information about the extreme values and the filling ratio of the control signal and a BPM value. Different approach to the problem was the fuzzy modeling. It is based on an approximate description of the dynamics using a set of transfer functions, being the sum of the impact of individual inputs to the model output. For example, in our case, the output flow value, for zero initial values, can be represented according to the relation: Q ( s) = G ( s) P ( s) + G (s)Q (s) + in in QoutPin QoutQin out (20) +G (s)Pout (s) + G (s)Ppn (s) QoutPout QoutPpn

Partial transfer functions have the form (21) of general transfer function with unknown coefficients α and β and the delay values Tj: − sT j [β + β s + ... + β sn ]e nj j j 0 1 G ( s) = QoutVj 1+ α s + ... + α nsn 1

(21)

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Determination of the coefficients is performed mostly in an indirect way of parametric identification method. On the base of sampled data the discrete transfer functions GQoutVj ( z −1 ) are determined in such a way that the ~ modeled signal, Q out ( kD ) , where Δ is the sampling step, should be as closely as possible to the measured one. We get a description consisting of a vector of inputs to the model v(kD), containing the signals measured in the respective moments of time, and the vector q of unknown coefficients of the investigated transfer functions.

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As a result of the presented fuzzy modeling method the nine partial models had to be determined. The measurements were carried out for five values of pressure difference. For the estimation of the models the measurement data for maximum, medium and low pressure differential were used. To verify the obtained model, the remaining series were used. The sample waveforms of the modeled and measured output flow are shown in Fig. 10.

Q (k D) = v(k D) ⋅q out

(22) Vector of transfer function coefficients can be estimated using various methods, for example, by the smallest sum of squared errors (LS). The dynamic model determined in this way usually accurately reproduces the dynamics of the process but is very local. This means that the change of supply parameters makes it necessary to re-selection transfer function coefficients. That why, the fuzzy modeling is very useful method. For the same structure of the system (22) vector of coefficients θ is dependent on some fuzzy variables, defined by membership functions. For example, separate models can be determined for the value of low, medium and high value of the fuzzy variable. The model is the weighted sum of the partial models and the corresponding membership function values. General fuzzy model can be determined as a linear combination of several local models set for the different intervals of membership function. Estimation algorithms of partial models coefficient vectors are more complex and require the simultaneous calculation of the vectors for all the partial models. The basic difficulty is also the designation of the proper shape and number of membership functions. In our case, in order to obtain a general description of the dynamic properties of the device, as a fuzzyfying variables were used two signals: output flow value Qout and a set pressure different DP. The first signal determines different valve states (Fig. 9a): positive flow (ejection, valve open), backward flow (closing of the valve) and the closed valve phase (no flow). For the second variable the membership function was divided for three areas (Fig. 9b): low, medium and high values of pressure.

Fig. 9 The membership functions for the fuzzy variables: a) output flow value, b) pressure difference

Fig. 10 Sample measured and modeled flow waveform for the verification data set The estimated parametric fuzzy model reproduces output flow value relatively well both, for the data on which he was appointed and the verification series. It yields a good representation of the dynamics of the POLVAD artificial ventricle. The extended description of the fuzzy modeling applied to determine the assist device model one can find in [11].

5. Summary The paper presents a new numerical library for modeling and simulation of human circulatory system with the extension of coronary circulation model and possibility of the parallel assist device connection. The main result of the coronary model addition was to allow simulation and verification of the influence of the left ventricle assistance on the coronary flow conditions. The whole library was implemented as a part of PExSim application. Modular construction of the plugin ensures flexibility because it can be easily modified to solve different problems within modeling and support of the human circulatory system. The modeling methods of the ventricular assist device were presented. We received an approximate representation of the output flow value dependent on the supply pressure. Method based on fuzzy modeling made it possible to achieve better results. However, in the case of even small modification of the device construction, the measurements and the whole modeling procedure will have to be carried again. Also, we do not receive the direct dependence of the output signal from the power supply parameters. Work on the determination of a better model of the mechanical construction of the ventricular assist device is still carried out. However, as a result of presented work we have the useful tool, which gives the opportunity to study functions of individual components of the human circulatory system as well as the system as whole. It enables a simulation of the proper and the pathology circulation conditions, such as left or right heart failure. Implemented model of the ventricle assist device Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

gives the opportunity for modeling the influence of the heart assistance on the hemodynamic co

Acknowledgements This work was partially supported by the National Centre for Research and Development (NCBiR) in Poland under Project “Development of metrology, information and telecommunications technology for the prosthetic heart” as part of the „Polish Artificial Heart” Program.

AUTHORS

Alicja Siewnicka*, Bartłomiej Fajdek, Krzysztof Janiszowski – Warsaw University of Technology, Faculty of Mechatronics, Institute of Automatic Control and Robotics, ul. Św. Andrzeja Boboli 8, 02-525, Poland, E-mail: alicja.siewnicka@gmail.com, b.fajdek@mchtr. pw.edu.pl, kjanisz@mchtr.pw.edu.pl *Corresponding author

References [1] M. Korda, S. Leonardis, J. Trontelj, “An electrical model of blood circulation”, Medical and Biological Engineering and Computing, vol. 6, 1968, pp. 449–451. [2] M. Sharp, R. Dharmalingham, “Development of a hydraulic model of the human systemic circulation”, ASAIO J., vol. 45, 1999, pp. 535–540. [3] G. Ferrari, “Study of Artero-ventricular Interaction as an Approach to the Analysis of Circulatory Physiopathology: Methods, Tools and Applications”, Ph. D. dissertation, Consiglio Nazionale delle Ricerche, Rome, Italy.

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[4] K. Janiszowski, P. Wnuk, “A novel approach to the problem of the investigation of complex dynamic systems in an industrial environment”, Maintenance Problems, vol. 4, 2006, pp. 17–36. [5] De Lazzari C., Darowski M., Ferrari G., Clemente F., Guaragno M., “Computer simulation of haemodynamic parameters changes with left ventricle assist device and mechanical ventilation”, Computers in Biology and Medicine, vol. 30, 2000, pp. 55–69. [6] B. Fajdek, A. Golnik, “Modelling and simulation of human circulatory system.”, Methods and Models in Automation and Robotics (MMAR), vol. 15, 2010, pp. 399–404. [7] W. Shreiner, F. Neumann, W. Mohl, “The Role Of Intramyocardial Pressure During Coronary Sinus Interventions: A Computer Model Study”, IEEE Transactions on Biomedical Engineering, vol. 37, 1990, pp. 956–967. [8] K. M. Lim, I. S. Kim, S. W. Choi, B. G. Min, Y. S Won, H. Y. Kim, E. B. Shim, ”Computational analysis of the effect of the type of LVAD flow on coronary perfusion and ventricular afterload”, The Journal of Physiological Sciences, vol. 59, 2009, pp. 307–316. [9] A. Siewnicka, B. Fajdek, K. Janiszowski, “Application of a PExSim for modeling a POLVAD artificial heart and the human circulatory system with left ventricle assistance”, Polish Journal of Medical Physics and Engineering, vol. 16, no. 2, 2010, pp. 107–124. [10] M. Stachura, “Application of the PExSim package in identification of multi-dimensional model of a waste water treatment plant”, Pomiary Automatyka Kontrola, vol. 55, no. 3, 2009, pp. 156–159. [11] K. Janiszowski, “Fuzzy identification of dynamic systems used for modeling of hearth assisting device POLVAD”, Pomiary Automatyka Robotyka, 11/2010, pp. 90–95.

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PI Control of Laboratory Furnace for Annealing of Amorphous Alloys Cores Submitted: 29th May 2012; accepted: 21st June 2012

Jerzy E. Kurek, Roman Szewczyk, Jacek Salach, Rafal Kloda

Abstract:

There are presented theoretical and practical aspects of automatic control of resistive furnace for thermal annealing of magnetic cores made of amorphous alloys. Process of annealing requires specific conditions both from the point of view of temperature and its changes. Solutions presented in the paper create possibility for low value of error as well as fast achievement of set value. Keywords: PI controller, resistive furnace, temperature control.

1. Introduction New soft magnetic materials – amorphous alloys based on iron, nickel and cobalt gives new possibilities for design of inductive components [1], magnetic field sensors [2], magneto-mechatronic sensors [3], and heat transportation devices [4]. However, production of amorphous alloys cores requires precise thermal relaxation (core’s annealing) [5]. This process is usually realized in 1 hour in argon protective atmosphere in order to avoid quick corrosion of cores surface. The relaxation improves cores magnetic permeability and reduces its coercive force. Thermal relaxation in amorphous alloys, if performed correctly, enables fabrication cores with relative permeability magnitude greater than 2×106. This makes the amorphous alloys one of the best magnetic materials, with highest magnetic permeability. This paper describes a control system for resistive furnace for annealing of amorphous alloys cores in the laboratory of Institute of Metrology and Biomedical Engineering, Warsaw University of Technology.

2. The furnace, measurement system and control system equipment The thermal relaxation process of cores is realized in a small laboratory resistive furnace which mass is approximately 3 kg. The furnace has canal winding, installed in chamotte corpus covered by thermal isolation with mineral wool. Inside the furnace there is a long quartz pipe with 40 mm diameter, which is filled by argon with pressure slightly higher than atmosphere pressure during the relaxation stage. Argon atmosphere protects the core during relaxation process. The relaxation process begins with heating the furnace to the relaxation temperature. Then, there is inserted capsule with room temperature having inside the annealed core and it is heated in the furnace during required time in the relaxation temperature. The temperature of amorphous alloys cores relaxation is equal to 345ºC and the

relaxation time is equal to 60 minutes. When the relaxation is finished the capsule is taken out of the furnace and it is cooled inside the cold part of the quartz pipe. Therefore, also cooling in argon protective atmosphere is performed. The controlled output signals are the furnace temperature measured by thermocouple type K and then the capsule temperature measured by thermocouple type J. Thermocouples are connected with temperature transducers AR-580 of Apar firm. Temperature measured range is from 0 to 500ºC and transducers output range is voltage from 0 to 10 V. Both sensors have linear characteristics. The furnace is powered by pulse wide modulation power controller EJ1P50E of Carlo Gavazzi firm. Control output of the controller is voltage from 0 to 10 V and full pulse control period is 3 sec. The furnace, temperature transducers and power controller are connected with PC computer by data acquisition card NI-USB-6361 of National Instruments firm. The control of the furnace temperature is realized by computer controller implemented on the connected PC computer. The controller program was prepared using the LabView software and implementing the PI controller. Block diagram of the laboratory furnace for annealing of amorphous alloys cores with measurement and control system is presented in Fig. 1, and the furnace laboratory stand is shown in Fig. 2, where 1 – capsule with core, 2 – furnace, 3 – quartz pipe, 4 – temperature transducer, 5 – argon inlet, 6 – data acquisition card and 7 – PWM power controller. Transducer AR-580

Thermocouple type J

NI Card USB-6361

PWM Power Controller

Resistive Furnace

PC Computer

Transducer AR-580

Thermocouple type K

Annealed Core

Fig. 1. Block diagram of the installation for annealing of amorphous magnetic cores

Fig. 2. Laboratory stand of furnace for cores annealing Articles

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3. Requirements of the control process

5. Control algorithm

As mentioned before, the controlled process consists of (i) heating stage of the empty furnace and (ii) core relaxation stage in the relaxation temperature. The control input is a voltage of power controller and the output signal is in the first stage temperature inside the quartz pipe and in the second stage the annealed core temperature. For the furnace heating stage it is only required relatively short heating time and limitation of control input magnitude and change speed because of furnace properties, step response time for the furnace is approximately 3 hours. Then, for the annealing stage it is required 1. annealing temperature equal to 345ºC, 2. annealing temperature errors should be less than ±5ºC, 3. time of core heating from temperature 320ºC to 345ºC no longer than 15 min, 4. no overshoots in the core heating stage to 345º C. The third requirement is rather important since usually annealing process starts in temperature 320ºC and if core stay too long in the annealing temperature but less than required annealing temperature its properties are different to the required ones.

Accordingly to the requirements indicated in Section 3, two stages of the process were identified: heating stage and annealing stage. Considering step response time for the furnace (which is approximately 3 hours) two control algorithms were proposed: 1. linear control of the furnace heating stage, and 2. linear control for annealing of the core with nonlinear phase after insertion of capsule with core into the furnace. In both cases we have used PI linear controller

 1   R( s ) = k P 1 + (2) T s I   Settings of the controller were chosen based on the calculated model (1) of the furnace. Calculating the settings in such a way that overshooting of the process is equal to zero, ϰ = 0, one obtains [6]

k P = 0.6

Next, we have modeled control system [Fig. 4] with PI controller and calculated settings.

4. Identification of control plant

Model of the furnace has been calculated based on step response of the furnace for change of voltage of power controller from 0 to 0.7 V in the form

(1)

where k is model gain, T time constant and T0 time delay. In the identification we have found the following values of the parameters k = 500 [ºC/V], T = 3450 [sec], T0 = 840 [sec] In Fig. 3 there are presented response of the furnace and response of the model. It is easy to see that the calculated model is quite good. It should be however noted that because relatively big mass of the capsule with core (0.25 kg) with respect to mass of the furnace (3 kg) insertion of the core in room temperature approximately 22ºC into warm furnace with temperature 345ºC really influent temperature of the furnace.

Furnace + u

–

PI Controller

Tr +

Fig. 4. Block diagram of furnace temperature control system, T – furnace temperature, u – control input voltage, Tr – reference temperature, z – disturbance Unfortunately, in the contradiction to setting base we have obtained small overshooting for step change reference temperature Tr, Fig. 5. However, overshooting for the disturbance which modeled insertion of capsule with core into the furnace was quite small. Therefore, we have decided to apply for control of the furnace the PI controller with calculated settings. 500 400 T (oC)

k G(s) = e −T0 s Ts + 1

T = 0.0049, TI = 0.8T0 + 0.5T = 2397 (s) kT0

300 200 Tr T

100 0 0

400

5000

10000

15000

20000

25000

30000

t (s)

T ( oC) 350 2.5 2.0

250

1.5

U (V)

300

measurements

200 150

1.0 0.5 0 -0.5 0

100

5000

10000

15000

20000

25000

30000

t (s)

50 model

Fig. 5. Furnace model control response with PI controller

0 0

5000

10000

15000 t (s)

Fig. 3. Furnace step response: measured temperature and model response 14

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PI controller has been used for control of the furnace and core temperature for (i) shortening of the furnace heating and (ii) control of the furnace heating after insertion of

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the capsule with core and (iii) control of the core temperature in the annealing process. The nonlinear phase of control algorithm after insertion of the capsule with core into furnace was as follows: 1. before insertion of the capsule with core automatic control was changed into manual control with constant control input voltage, 2. after insertion of the capsule into furnace there was added one triangle control input impulse with magnitude 0.6 V and time 600 sec (10 min) to constant control input; the triangle input was designed based on practical experiments and in the control process it was automatically generated by the controller software LabView, 3. after the triangle impulse the control was changed from manual mode into automatic mode with PI controller with calculated settings but also with annealing core temperature as the controlled output signal. In the control we do not use PID controller because in the control system we have quick measurement disturbances which generate quite big control input changes calculated by PID controller since derivative action D of PID controller implemented in the software has big dynamic derivative gain.

6. Experimental results Designed control system has been applied for control of the furnace for annealing of cores in the Institute of Metrology and Biomedical Engineering of Warsaw University Technology. In Fig. 6 there are presented temperature of the furnace, temperature of the core and control input voltage obtained by PI controller and triangle impulse in the insertion of capsule with core into the furnace. Controller settings were as we calculated before.

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It is interesting to note that the core temperature is lower than the furnace temperature in the annealing process. In the annealing process we have obtained maximal core temperature error 2ºC. The core heating time from 320ºC to 345ºC was 12 min, less than it was maximal allowed value 15 min and annealing time in temperature equal to 345ºC was 60 min.

7. Concluding remarks Proposed PI control system allows conducting annealing process according to requirements – quickly and in the required temperature without overshooting and without presence of operator, operator action was only required for short time in the moment of insertion of capsule with core into furnace. In laboratory conditions the proposed control system has shorten the annealing time about 70% comparing with annealing process in the manual mode and also improved quality of the annealing because less annealing temperature errors. Moreover, the annealing was automatic and no operator assistance was required. Presently we work on improving the automatic annealing process and shortening assistance of the operator. The research on magnetic cores was founded in 20102012 as a research project.

AUTHORS Jerzy E. Kurek* – Institute of Automatic Control and Robotics, Warsaw University of Technology, Warsaw, Poland, jkurek@mchtr.pw.edu.pl. Roman Szewczyk – Industrial Research Institute for Automation and Measurements, Warsaw, PL 02-486, Poland. Jacek Salach and Rafal Kloda- Institute of Metrology and Biomedical Engineering, Warsaw University of Technology, Warsaw, Poland. *Corresponding author

References

Fig. 6. Heating and annealing process with PI controller and triangle impulse control input: a) furnace and core temperature b) control input voltage

[1] O’Handley R., Modern magnetic materials – principles and applications. John Wiley & Sons, 2000. [2] Ripka P., Magnetic Sensors and Magnetometers. Artech, Boston, 2001. [3] Bienkowski A., Szewczyk R., “The possibility of utilizing the high permeability magnetic materials in construction of magnetoelastic stress and force sensors”, Sensors and Actuators A113, 2004, p. 270. [4] Kolano-Burian A., Kowalczyk M., Kolano R., Szymczak R., Szymczak H., Polak M., “Magnetocaloric effect in Fe-Cr-Cu-Nb-Si-B amorphous materials”. J. Alloys Comp. vol. 479, 2009, p. 71. [5] Bieńkowski A., Szewczyk R., Salach J., Kolano R., Kolano-Burian A., “Influence of thermo-magnetic treatment on magnetoelastic properties of Fe81Si4B14 amorphous alloy”, Journal of Physics – Conference Series 144, 2009, 012070. (http://iopscience. iop.org/1742-6596/144/1/012070) [6] Pułaczewski J., Układy regulacji z regulatorami typu PID, Poradnik Inżyniera Automatyka, WNT, Warsaw 1973, pp. 571–635. (in Polish) Articles

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Surface Topography Parameters Important in Contact Mechanics Submitted: 2012; accepted: 2012

Pawel Pawlus, Wieslaw Zelasko, Jacek Michalski

Abstract:

The random surface models are important to many statistical peak-based contact models of rough surfaces. Statistics of 3D surface topographies and 2D profiles are compared and their interrelationship examined for generated and measured common random engineering surfaces. The applicability of the spectral moments approach to random surface specification is checked. Parameters important in contact mechanics, like summit density, summit curvature and summit height obtained by their definitions and predicted by the spectral moment approach, as well as calculated directly from profiles are compared. Also, the values of plasticity index are computed using various methods. Good agreement is found between theory and measurement.

sented by McCool [5]. Surface and profile measurement and their resultant statistics were compared and their interrelationship examined for several common engineering surfaces [6]. Good agreement was found between theory and measurements over a large range of sampling intervals. Yu and Polycarpou [7] compared the summit density and summit radius obtained from numerically generated isotropic Gaussian surfaces.

2. Connections between summit parameters and spectral moments

Spectral moments m0, m2 and m4 can be obtained from profiles. They are equivalent to the mean square height, rms. slope square and second derivative of profile. The areal (3D) surface summit density is given by [5]:

Keywords: surface topography, contact mechanics, spectral moments.

1. Introduction

All engineering surfaces are rough and their description is important to the study of many interfacial phenomena, such as friction, wear, electric al and thermal contact resistance, etc. Surface topography is recognized as being an important factor in determining the nature and extent of contact. Because surfaces are rough, the true area of contact, which is much smaller than the nominal area of contact must support very large pressure. Two types of parameters were advocated for contact and wear prediction: parameters based on peak (summits) and parameters based on plots of material ratio. The pioneering contribution to this field was made by Greenwood and Williamson [1], who developed a basic contact model (GW model) of isotropic surface. Chang et al. [2] put forward an elastic-plastic contact model for rough surfaces on the basis of volume conservation of plastically deformed asperities. These models have been extended by many researchers. Parameters connected with peak as peak radius, peak height and peak curvature were used. These parameters are based on a 2D profile. However the statistic of the areal (3D) surface and the statistics of a 2D profile of the surface are not the same. It is necessary to distinguish a peak on a profile from a summit on the surface. A detailed comparison was made by statistical approach. Rough surfaces were modeled as two dimensional, isotropic, Gaussian random surface by Nayak [3]. Dependencies between profile spectral moments and parameters important in contact mechanics were also developed by Bush et al. [4]. They were pre16

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Spd =

1 m ( 4). 6 p 3 m2

(1)

The mean summit curvature averaged over all summit heights is [5]: Spc =

8 m4

3 π

(2)

The variance of the summit height is [5]: s 2s = (1 −

0.8968 ) m0 . a

(3)

The distance between the mean of the summit height distribution and the surface mean plane is [5]: ys = where: a=

4 m0

πα m0 m4 . m2 2

3. Calculation procedure

(4) (5)

Isotropic surfaces of Gaussian ordinate distribution were generated, using the procedure developed by Wu [8]. Each surface of this type is characterized by correlation distance (in which the autocorrelation function decays to 0.1 value) and standard deviation of height. In addition, some measured isotropic Gaussian surface topographies were analyzed. The values of their texture parameter Str were higher than 0.8. These surfaces were measured by stylus 3D Talyscan 150 equipment with nominal radius of tip 2 µm. The initial numbers of the measured points

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Fig. 1. Various summit identifications

Fig. 2. Scheme of contact of two rough surfaces a)

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were between 401 x 401 to 601 x 601. The sampling intervals were 5 and 10 micrometers. However in order to decrease correlation length sampling interval sometimes increased and the number of points was reduced. For each of measured surfaces, the form was eliminated by a polynomial of the 2nd degree. Digital filters were not used. For each surface, parameter connected with summits were calculated. For areal measurements, the mean radius of each summit R was computed as reciprocal of mean arithmetic average curvature in orthogonal directions. Summit curvature was calculated on the basis of three-point formula [9]. The summit identification is a real problem. Usually surface point is a summit if its ordinate was higher than ordinates of four or eight nearest neighbors (see Figure 1). The second possibility was accepted by the present authors. This criterion was based on works of Greenwood [10] and Sayles and Thomas [6] as well as our previous research. Areal density of asperities Spd, standard deviation of summits heights σs and distance between the mean of asperity heights and that of surface ordinates ys (see Figure 2) were obtained from their definitions directly from areal surfaces. The parameters characterized summits were also determined on the basis of 2D profiles. Sets of parallel profiles were obtained from measured surfaces and average profile spectral moments m0, m2 and m4 were calculated according to procedure presented in paper [11]. Parameters characterized summits were obtained using equations (1) – (5). It is also possible to estimate parameters characterizing summits from profile peaks analysis (summits are local maxima on the surface, as distinct from peaks, which are local maxima on a profile). Therefore peak density, average peak curvature, standard deviation of peak heights and distance between the mean of line of peak heights and mean profile line were calculated for set of parallel profiles and mean values were taken into consideration. As recommended by Nayak [3] summit density was computed as square of peak density multiplied by 1.2. The well-known plasticity index postulated by Greenwood and Williamson (GW) Ψ [1] in 1966 is widely applied in studying the contact of rough surfaces. The basic assumptions were adopted in GW model: – asperities are spherical near their peaks (summits), – there is no interaction between asperities,

Fig. 3. Modeled isotropic surface topography (a), profile from this surface (b) Articles

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– only the asperities deform during contact, – all peaks (summits) have the same radius R. The contact between two rough surfaces is modeled by contact of single rough surface with a smooth plane. Figure 2 shows the geometry model of contacting rough surfaces, z denotes the height of asperity, d separation of the surfaces measured from the summits mean plane, but h is the separation of the surfaces based on surface heights (ordinates). The plasticity index postulated by Greenwood and Williamson was defined as: y=

E ' s s 0.5 ( ) , H R

2012

dinates the errors of summit density was also high based on profile peaks analysis. For the other cases application of this method led to overestimation of density; however it was found that summit density should be equal to square of peak density on profile, in this case the error of summit density was smaller than 6% for the ρ coefficient not higher than 0.77. Mean radius of summit curvature was accurately predicted by spectral moments approach, independently on the ρ value. The errors were smaller than 10%, only for highly correlated points case (ρ = 0.99) deviation was 24%. Estimated value was usually smaller than that obtained by definition. However the analysis of profile peaks led to overestimation of mean summit radius; the errors were in the range: 16–35%. Relative difference between standard deviation of summit height was usually smaller than 5% but not higher than 10% when spectral moments approach was used. Higher errors occurred for measured surface topog-

(6)

where H is the hardness of the softer contacting materials, and 1 − n12 1 − n 2 2 −1 E' = ( + ) E1 E2

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(7)

Ei and νi (i = 1, 2) are Young’s moduli and Poisson’s ratios for the two contacting elements. In this work the plasticity index was Table 1. Surface topography parameters and plasticity indices calculated using calculated for various methods of comdifferent methods puting contact parameters. The following material properties were selected (conSurface ρ σ, µm σs, µm Spd, 1/µm2 R, µm ys, µm Ψ tact of steel-on-steel elements) E1 = E2 = M1s 0.85 0.176 0.158 0.00104 157.6 0.17 1.85 2.07 x 105 MPa, Brinell hardness H = 200 0.164 0.00211 161.5 0.15 M1m (1960 MPa), ν1 = ν2 = 0.29. These proper0.174 1.84 ties were also used in paper [2]. M1p 0.174 0.169 0.00231 208.3 0.07 1.64 Figure 3 shows example modeled surM2s 0.4 0.176 0.132 0.000191 1024.5 0.25 0.65 face M1 of correlation ρ = 0.85 between M2 0.174 0.137 0.000187 976.6 0.25 0.68 neighboring points and profile from this m M2 0.174 0.152 0.000235 1351.4 0.12 0.61 surface. p

4. Results and discussion

The results of calculations of selected surface topographies are listed in Table 1. Index s means calculation of contact parameters from the areal (3D) surface, m – using profile spectral moment and p – basing on the profile peaks analysis. Surfaces 1-5 were modeled, 6-10 measured. ρ means average value of correlation between neighboring points (ordinates) obtained from 6 profiles. It is evident from the analysis of the simulated and measured surfaces that high values of the ρ parameter (not smaller than 0.85) correspond to large errors of summit density Spd prediction using spectral moment approach. The errors were bigger than 100%; summit density was overestimated. So the error in obtaining summit density on the basis of profile measurement can be large. For ρ values between 0.25 and 0.77 the deviations of summit density was smaller than 10%; for non-correlated neighboring points (ρ between 0.1 and 0.12) application of spectral moments method caused underestimation of summit density – errors were between 15 and 18%. For high correlation between neighboring or18

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M3s M3m

0.12

M3p M4s M4m

0.91

M4p M5s M5m

0.65

M5p M6s M6m

0.5

M6p M7s M7m

0.25

M7p M8s M8m

0.99

M8p M9s M9m

0.77

M9p M10s M10m M10p

0.1

0.176

0.123

0.174 0.174

0.13 0.142

0.93

0.000061 0.000052

3076.5

0.254

0.36

0.000077

2816.9 4000.1

0.269 0.14

0.39 0.34

0.82

0.000232

177.3

0.70

3.92

0.91 0.91

0.87 0.91

0.00047 0.000531

176.8 232.5

0.56 0.28

4.04 3.61

0.93 0.91

0.81

0.000063

714.2

0.95

1.94

0.91

0.83 0.87

0.000061 0.000074

713.9 952.4

1.06 0.48

1.92 1.74

0.66

0.46

0.000046

1383.1

0.79

0.99

0.64 0.64

0.5 0.51

0.000042 0.000052

1220.2 1724.0

0.89 0.45

1.18 0.99

0.82

0.49

0.000026

1666.7

1.1

0.99

0.8 0.8

0.56 0.61

0.000024 0.000029

1538.4 2173.9

1.24 0.64

1.13 0.97

0.57

0.56

0.00038

436.7

0.134

2.07

0.56 0.56

0.00081 0.0007

333.5 512.8

0.15 0.061

2.44 1.91

2.28

1.78

0.000019

714.2

3.17

2.23 2.23

1.81 1.99

0.0000199 0.000022

746.3 833.3

3.07 1.75

2.88 2.84 2.82

1.07

0.64

0.000017

1897.5

1.45

1.05 1.05

0.69 0.79

0.000014 0.000019

1724.13 2380.9

1.72 0.83

1.03 1.15 1.05

Journal of Automation, Mobile Robotics & Intelligent Systems

raphies. When the σs parameter was calculated directly from profile based on its peaks, the deviations were usually higher (up to 24%). Estimation of summit standard deviation height on the basis of profile analysis using 2 applied methods led usually to overestimation of σs. For highly correlated points (ρ = 0.99) no difference was found after application of three analysed methods. In this case standards deviation of summit height was equal or very close to standard deviation of ordinates. When spectral moments method was used, the predicted ys distance was higher than value obtained from the analysis of simulated and measured surfaces for correlation ρ smaller than 0.77, however for larger ρ values it was usually smaller (except for M8 surface) but differences were not higher than 20%. Calculation of the ys parameter directly from the profile peaks caused its underestimation (1.7-2.5 times). Generally when spectral moments were used, good agreement was found between the theory and the results of the areal (3D) surface topography analysis except for summit density which could be overestimated by theory for comparatively high ρ values. However these parameters cannot be calculated on the basis of profile peaks analysis; the errors were higher, particularly for the ys parameter. Only summit density can be calculated without large errors as the square of peaks density on the profile when correlation between neighboring points was not too high. Therefore when summit contact parameters are estimated from profile spectral moments, ρ values higher than 0.85 should be avoided. Application of spectral moments method led to correct estimation of plasticity index for modeled surfaces; the errors were not higher than 8%. Differences were larger for measured surfaces (up to 20%). However plasticity index can be determined on the basis of profile peak analysis – the errors were not larger than 10% and for measured surfaces they were smaller than those obtained after using spectral moments approach. The reason of such low deviations is that as a result of application of profile peak analysis both σs and R values were overestimated. Decrease of correlation length ρ causes increase in the distance between the mean of asperity heights and that of surface ordinates ys and decrease in standard deviation of summit height σs. Mean value of standard deviation of ordinates is a little smaller than standard deviation of height of areal surface; differences were a few percents.

5. Conclusion

Applicability of the profile spectral moment approach to areal random surface specification was checked. Good agreement between the analysis of modeled and measured surfaces and theory was generally found. The errors of calculation of parameter important for contact mechanics after the analysis of profile peaks, particularly for the distance between the mean of asperity heights and that of surface ordinates ys, were higher than those after using profile spectral moments. However the errors of computing the plasticity index on the basis of profile peaks analysis was small, especially for small correlation between ordinates. Summit density can be overestimated by the profile analysis (using both applied methods) for comparatively high correlation between neighboring points ρ. Therefore when summit contact parameters are

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estimated from 2D profiles, ρ values higher than 0.85 should be avoided. Summit density can be calculated as the square of peaks density on the profile when summit was identified based on its eight neighbors for not too high correlation between ordinates. Decrease in the ρ parameter by increase in the sampling interval caused increase in the distance between the mean of asperity heights and that of surface ordinates ys and decrease in standard deviation of summit height σs.

AUTHORS

Paweł Pawlus*, Jacek Michalski – Rzeszow University of Technology, Faculty of Mechanical Engineering and Aeronautics, Al. Powstancow Warszawy 8, 35959 Rzeszów, Poland. E-mails: ppawlus@prz.edu.pl, jmichals@prz.edu.pl Wiesław Żelasko - Upper-Secondary Technical School Complex in Lezajsk, ul. Mickiewicza 67, 37-300 Lezajsk, Poland. E-mail: wzelasko@2be.pl Corresponding author

References [1] J. A. Greenwood and J. B. P. Williamson, “Contact of nominally flat surfaces”, Proc. Roy. Soc. (London), A295, 1966, pp. 300-319. [2] W. R. Chang, I. Etsion and D. B. Bogy, “An elasticplastic model for the contact of rough surfaces”, ASME Journal of Tribology, 109, 1987, pp. 257263. [3] P. R. Nayak, “Random process model of rough surfaces”, ASME Journal of Lubrication Technology, 93, 1971, pp. 398-407. [4] A. W. Bush, R. D. Gibson and G. P. Keogh, “The limit of elastic deformation in the contact of rough surfaces”, Mech. Res. Commun. 3, 1976, pp. 169174. [5] J. I. McCool, “Comparison of models for the contact of rough surfaces”, Wear, 107, 1986, pp. 3760. [6] R. S. Sayles, T. R. Thomas, “Measurements of the statistical properties of engineering surfaces”, ASME Journal of Lubrication Technology, 1979, 101, pp. 409-417. [7] N. Yu and A. A. Polycarpou, “Extracting summit roughness parameters from random surfaces accounting for asymmetry of the summit heights”, ASME Journal of Tribology, 126, 2004, pp. 761766. [8] J. J. Wu, “Simulation of rough surfaces with FFT”, Tribology International, 33, 2000, pp. 47-58. [9] D. J. Whitehouse, “The digital measurement of peak parameters on surface profiles”, Journal Mechanical Engineering Science IMechE, 20/4, 1978, pp. 221-226. [10] J. A. Greenwood, “A unified theory of surface roughness”, Proc. Roy. Soc. (London), A393, 1984, pp. 133-157. [11] J. I. McCool, “Finite difference spectral moments estimation for profiles: the effect of sample spacing and quantization error”, Precision Engineering, vol. 4, no. 4,1982, pp. 181-184. Articles

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Behavior Based Co-ordination of a Troop of Vehicles Targeted to Different Goals in an Unknown Environment Submitted 26th January 2012; accepted 16th August 2012

Sourish Sanyal, Ranjit Kumar Barai, Rupendranath Chakrabarti, Pranab Kumar Chattopadhyay

Abstract:

The issue of coordinated operation of multi-vehicle for a variety of tasks is getting increasing attention day by day and standing as a major research field due to their increased capacity and flexibility they can offer as a team. This paper presents a novel algorithm for multi-vehicle navigation, based on exhaustive search to avoid a set of randomly generated obstacles, predict the approximate position of other vehicles and thus keeping a safe distance to avoid collision and to maintain a formation amongst them while targeted towards the assigned goals. The proposed algorithm uses two optimizing functions in deriving drive commands, direction and turning, for a troop of vehicles. This particular algorithm is similar to the artificial potential field (APF) method which is widely used for autonomous mobile robot path planning due to its simplicity and mathematical elegance. In this work we have taken a behavior based reactive scheme together with artificially generated perturbation as the vehicles are running in a real time environment. Simulations have been carried out for a group of four vehicles, paired in two groups, approaching two different targets avoiding eight randomly generated obstacles, and keeping proper coordination between the members of intra and inter groups. The effectiveness of the proposed approach has been shown by some simulation results. Keywords: behavior-based collision avoidance, randomized obstacles, multi-vehicle coordination, particle swarm optimization.

1. Introduction The challenge that a troop of multiple uninhabited autonomous vehicles (UAVs) would be able to adaptively react to their environment, whether known, unknown or uncertain, and learn about their surroundings while following either an individual or a communal agenda is an intriguing field of research. Achieving such a degree of control and producing such sophisticated behavior remains an elusive goal that demands considerable attention and this is inherently a complex task. The problem of multi-vehicle coordination and control has been receiving an exquisite amount of attention during the past few years due to critical importance of the field in wideranging applications [8]. In many practical applications of autonomous vehicles multiple teams are to be used. Such teams have many potential benefits, including faster completion through parallelism and increased robustness through redundancy. 20

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Further, teams of vehicles can increase the application domain of autonomous vehicles by providing solutions to tasks that are inherently distributed, either in time, or in space, or in functionality. Since the 1980s, researchers have addressed many issues in multi-vehicle, or multirobot teams or automated guided vehicles (AGVs) [12], such as control architectures, communication, task allocation, swarm robots, learning [25]. A critical issue in these mobile robot teams is coordinating the motions of multiple vehicles interacting in the same workspace. Regardless of the mission of the vehicles, they must be able to effectively share the workspace to prevent interference between the team members. Solutions to the motion coordination problem are approached in a variety of ways, depending upon the underlying objectives of the vehicle team. In some cases, the paths of the robots are explicitly planned and coordinated in advance, as might be needed in a busy warehouse management application. In other cases, planning is relaxed and emphasis is placed on mechanisms to avoid collision, applicable for tasks such as automated hospital meal deliveries. In yet other situations, the robots could have mechanisms with little pre-planning that focus on coordinating vehicle motions in real-time using reactive, behavior-based, or controltheoretic approaches, such as would be used in a convoying or formation-keeping application. Existing work on multi-vehicle control focuses receding-horizon planning (an optimization method) and hierarchical structures. The receding-horizon trajectory planner based on Mixed Integer-Linear Programming (MILP) is capable of planning planner-based trajectories directed to a goal [14,15,16]. The goal is constrained by no-fly areas, or obstacles, and is free from leader-follower architecture which is adopted by model predictive control (MPC) [17]. Game-theoretic approach is also adopted by different co-ordination schemes for decision making of the multi-vehicle problem [18,19,20]. A disjoint path algorithm for a reconfiguration of multi-vehicle was also proposed [21]. A class of triangulated graphs for algebraic representation of formations have been introduced to specify a mission cost for a group of vehicles [22]. The present work focuses on simultaneous movement of a troop of vehicles from their initial locations towards different targets in such an environment where obstacles are generating stochastically based on the Artificial Potential Field (APF) approach. The basic idea of the APF approach is to fill the robot’s workspace with an artificial potential field in which the robot is attracted to its target position and is repulsed away from the obstacles [4]. This method is particularly attractive because of its elegant mathematical analysis and simplicity. The application of

Journal of Automation, Mobile Robotics & Intelligent Systems

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APF for obstacle avoidance was first developed by Khatib [3]. In the past decade this method has been studied extensively for autonomous mobile robot path planning by many researchers [5-7]. This is a new approach where the troops are divided into two groups and set out for their own targets, maintaining a formation amongst them. This work is an extension of the work done by Kevin Passino [2] on obstacle avoidance of a single vehicle in presence of a number of fixed obstacles.

2. Problem description A. Cooperation of multi-vehicles

The word cooperation means interaction or integration of multiple vehicles [11]. In a cooperative team the vehicles have to communicate, exchange information or interact in some way to achieve an overall mission. The term cooperation has been widely discussed in different scientific community and different definitions have been proposed.

B. Multi-vehicle path planning problem

It is defined as follows: given a set of m vehicles in k-dimensional workspace, each specified with an initial starting configuration (e.g., position and orientation) and a desired goal configuration, determine the path each vehicle should take to reach its goal, while avoiding collisions with obstacles and other vehicles in the workspace. More formally, let A be a rigid vehicle in a static workspace W = k [18,19], where k = 2 or k = 3. The workspace is populated with obstacles. A configuration q is a complete specification of the location of every point on the robot geometry. The configuration space C represents the set of all the possible configurations of A with respect to W. Let O ⊂ W represent the region within the workspace populated by obstacles. Let the close set A(q) ⊂ W denote the set of points occupied by the vehicle when it is in the configuration qÎC. Then, the C-space obstacle region, Cobs, is defined as [1]:

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of X corresponding to robot A with the obstacle region O, is i (5) X obs = {x ∈ X | Ai (q i ) ≠ Φ} The subset jof X corresponding to robot Ai in collision with robot A is ij X obs = {x ∈ X | Ai (q i ) ∩ A j (q i ) ≠ Φ}

(6)

The obstacle region in X is then defined as the combination of Equations (5) and (6), resulting in m

i i X obs = (∪ X obs ) ∪ ( ∪ X obs ) i =1

(7)

ij ,i ≠ j

With these definitions, the planning process for multivehicle system treats X the same as C, and X obs the same as Cobs ,where Cinit represents the starting configuration of all the robots, and Cgoal represents the desired goal configurations of all the vehicles. The APF uses two types of potential field, namely a repulsive potential field to force a robot away from obstacles or forbidden regions and an attractive potential field to drive the robot to its goal. The robot moves under the action of a force that is equal to the negative gradient of that potential, and it is driven towards the positions with the lower potential. In this paper, we consider the robot as one particle that moves under the action of the composition of forces Ar , which is the summation of goal’s attractive force Frg and the obstacle’s repulsive force For as shown in Fig. 1.

Cobs = {q Î C | A(q ) ∩ ο ≠ Φ }

(1)

The set of configurations that avoid collision (called the free space) is: (2) C free = C \ Cobs . A free path between two obstacle-free configurations Cinit and Cgoal is a continuous map:

τ [0,1] → C free

(3)

such that τ (0) = cinit and τ (1) = cgoal .. For a team of m vehicles, define a state space that considers the configurations of all the robots simultaneously: X = C1 × C 2 × ... × C m .

(4)

Note that the dimension of X is N, where N = m dim(C i )The C-space obstacle region must now be i =1 redefined as a combination of the configurations leading to a robot-obstacle collision, together with the configurations leading to vehicle to vehicle collision. The subset

∑

Fig. 1. Virtual attractive force of robot in APF Typically, optimization criteria guide the choice of a particular solution from an infinite number of possible solutions. Examples are: minimal path lengths (local or global), minimal time to reach targets, and minimal energy consumption to reach the goal. Presence of constraints brings forth more complexity. Such constraints arise from navigational restrictions e.g. limitation on the maximum angle of rotation, restrictions on maximum slope, inability to traverse rocky terrain, etc. or the need for a vehicles to move in tandem. Since general optimal solution for multiple moving objects is computationally difficult, sometimes intractable [24], local optima is sought for instead of global optima in the path planning problem. Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

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{

3. Problem formulation and the proposed scheme

T M t ( x1 , x2 ) = 1, [ x1 , x2 ] ∈  xti , xti 

For convenience and for reducing complexity, the scope of the present work has been limited to the motions in only 2-D space. It is assumed that the obstacles are not dynamic but they are randomly generated in the workspace. No vehicle is stationary with respect to another. So it is obligatory to keep a safe distance between mobile vehicles to avoid collision and maintain a formation. It is assumed that there are m no of vehicles and the ith one follows a discrete time kinematic model given as:

xvi 1 (k + 1) = xvi 1 (k ) + d cos(θ vi (k ))

(8)

xvi 2 (k + 1) = xvi 2 (k ) + d sin(θ vi (k ))

(9)

θ vi (k + 1) = θ vi (k ) + f θ v (u i (k ))

(10)

where k is the discrete time index taking values of nonnegative integers {0,1,2,3,….}(in the present problem k denotes the number of search steps); θ vi is the orientation of the ith vehicle; f θ v can be a nonlinear function encoding kinematic restrictions on the vehicles; u i is the local controller corresponding to ith vehicle. For convenience, let (11) xvi = [ xvi , xvi ]T , and xvi = [( xvi )T , θ vi ]T . p

It has been assumed that the controller has prior access to the information on randomly generated obstacles but not to the vehicles. The vehicles are to communicate with the controller (distributed controllers, dedicated one for each vehicle like an embedded system) to update the information on their positions at every iteration before taking the next move. The environment is modeled as a 2D, plane, having four quadrants (upper right and left and lower right and left) of a Cartesian coordinate system with axes (x1,x2). A Gaussian profile map has been set up which is accessible to all the vehicles through their controllers. It encodes the possible obstacle locations xsi = [ xsi 1 , xsi 2 ]T , i =1,2,…n obtained from sensory data which act as centers of the Gaussian peaks. It is assumed that the number of the obstacles is n (n = 8 for this case). Considering initial position to be [x1,x2], the mathematical description is as given below: n  ( x1 − xsi1 ) 2 + ( x2 − xsi2 )  M p ( x1 , x2 , k = 0) = ∑ ci exp −   vi2 i =1  

(12)

There may be some uncertainty in the data for the distances measured by the sensors. The uncertainty can be encoded with variation in vi . Then uncertainty of prior information having a peak width of vi and the distances of the real obstacles from the centre of the peak in terms of vi may be clubbed together. Furthermore, a specific priority can be assigned intentionally to a particular task by assigning different values as weights to ci . In this approach, all the vehicles share the common map Mp (x1,x2) at every iteration. The vehicle (controller) sensor samples the Cartesian plane to get information on updated positions of obstacles and other vehicles and derive the drive command. The output is in the form of binary i.e. an output of 0 means no obstacle or and an output of 1 means an obstacle in near proximity. 22

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}

(13)

= 0 otherwise

Emphasis has been given on moving the vehicles in discrete steps as if moving from cell to cell rather than moving along a smooth curve. Random velocities have been assigned to the vehicles. No restriction has been imposed on maximum angle of rotation in one step but in reality a sharp turn may adversely affect the stability of the vehicles. This problem can be redressed by slowing down the vehicles. For most of the practical situations, a vehicle located somewhere in the terrain is unable to locate all the obstacles and other vehicles at a time. In order to account for this inability, an artificial perturbation has been added to the output vector [9, 10]. The problem can now be formulated by slightly modifying the above-mentioned kinematic problem:  xvi 1 (k + 1)   xvi 1 (k ) + λ cos(θ vi (k ) + T β ki )   i   i i i   xv 2 (k + 1)  =  xv 2 (k ) + λ sin(θ v (k ) + T β k )  ≠ i i  θ vi (k + 1)    θ v (k ) + T β k    

(14)

The nonlinear function f of the previous model has been reduced to a linear incremental function having a step increment in sample time T and λ is the minimum incremental distance that any vehicle traverses before scanning its world map for the next time slot, β ki is the steering angle and θ vi (k ) is the orientation with respect to X-axis for the last update. So the concise form of the model for positional and angular updating is given as:

{

 xvi 1 (k + 1), xvi 2 (k + 1)  = Γ  xvi 1 (k + 1), xvi 2 (k + 1)  , uki

}

(15)

where xv1 (k+1) and xv 2 (k+1) are the updated states of X-Y coordinates in the Cartesian coordinate system at i time t for the of the ith vehicle and uk is the drive commands generated by the controller – the minimal positional and steering rate update at Tth discrete sample time, where Γ is the mapping function. The goal of vehicle coordination is to derive a sequence of controls for each vehicle i.e. (16a) u i = u0i , u1i ,...., uki

{

}

such that the trajectories are:

{

}

x i = x0i , x1i ,....xni

(16b)

4. The real-time problem taken for path optimization The specific scenario and the considerations behind experimentations and finding out the simulation results are worth mentioning at this point. Four vehicles paired in two different groups have been considered – they have set out for two different targets of same preference. • There are eight obstacles randomly generated in the workspace whose locations can be traced out by the sensors of the distributed controlling mechanism of each individual vehicle.

Journal of Automation, Mobile Robotics & Intelligent Systems

• Each vehicle can sense their present location in Cartesian X-Y coordinate system while- they also have prior information about their starting locations. • The controller of each vehicle can communicate with that of other vehicles and can distinguish between a moving object and a static obstacle. Against these considerations in the backdrop, the real time algorithm used in this experimentation has been framed. Three functions have been taken: one for obstacle generation, a Gaussian profile function to get estimate of the obstacle positions and a goal function to find out the best possible position and orientation for gradually getting nearer to the goal.

4.1. The proposed algorithm

Three functions have been used in this multi-vehicle path planning viz. ‘obstaclegeneration’, ’obstaclefunc’, and ‘goalfunc’ along with the main program. Main Algorithm Step 1: Segregate the 360o contour of the robots world map into N no. of segments. Step 2: Do while count<preset value Step 3: Loop for K=1 to N (incrementing step angle), calculate the Gaussian profile obstacle distance from each point of the circular trajectory by calling the function ‘obstaclefunction’ and find the furthest obstacle distances from each point and follow the same for finding the Euclidian distance to goal from all of those points by calling the function ‘goalfunction’. Step 4: Add the return array functions so as to treat this function as a composite one. Step 5: Minimize the composite function so as to get the best angle to move. Step 6: Orient the robot towards that best found direction and move minimum incremental distance i.e. proceeding cell by cell (as if repulsed by the random obstacles and attracted towards the goal). Step 7: Repeat step 2 to step 5 for all subsequent robots. Reactive Behavior Scheme: Step 8: Loop for i = 1 to 3 Step 9: While Robotix , y or Roboti +1x , y ≠ Goalx , y do Step 10: If Robot1x , y − Robot2x , y ≤ predefined thresold give either an X-axis shift or Y-axis shift accordingly. Step 11: Endwhile Step 12: Endfor Add Artificial Perturbation Step 13: generate delta-increment and delta-angle by random function generation Step 14: Add them with goal function and obstacle function array Step 15: Go back to step 2 Step 16: End-while Analysis: As it can now be seen that the above algorithm shares similarities with the approach of Artificial Potential Field algorithm, first proposed by O. Khatib classically for stationary obstacles and goals. In the present problem the obstacles are generated randomly but after that they are stationary for the entire run. There is also a reactive behavior amongst the motions of the robots as each robot considers the others like obstacles and keep safe distance as well as a specific formation. The reactive behavior is

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also exhibited while the robots are repulsed from the obstacles. The troop is also attracted towards the goal more aggressively than they are being repulsed from the obstacles. This weighted approach is taken to find nearer space to global optimal solution while optimizing the composition of the goal function and maximum distance Gaussian profile obstacle function. The higher aggression to reach the goal reduces the probability of being confined to local minima and forces it to follow a much straighter path as can be seen from the traced out paths of the robots through the resulting diagrams (viz. Fig 6 and Fig 9). Three functions have been used in this multi-vehicle path planning viz. ‘obstaclegeneration’, ’obstaclefunc’, and ‘goalfunc’ along with the main program. The main program executes the simulation loop of the constrained optimization problem and derives drive commands for the troop. The pseudo-code of the main program is given below: loop for i=1 to size(sampled contour) % Starting of the Main Program theta(i,1) ¬ theta(i-1,1) + angular increment end of loop Set xgoal 1,2¬ assign goal coordinates 1,2 % Assign two goals and four initial locations Set initial1,2,3,4¬ assign initial locations coordinate 1,2,3,4 % of the vehicles Call obstaclegeneration function % to generate eight random obstacles and to display Loop for k=1 to size(iterations) Set x1,2,3,4(min,max) ¬ Workspace(min,max) % to keep vehicles within the workspace Loop for m=1 to size(sampled contour) Xs1,2,3,4(:,m) ¬ x1,2,3,4 (rows,k)+increment(rad,thetam) Go(m,1) 1,2,3,4¬ call obstaclefunc(Xs1,2,3,4(:,m),w1) Gg(m,1) 1,2,3,4¬ call goalfunc(Xs1,2,3,4 (:,m),xgoal1,2,3,4,w2) Ggo(m,1) 1,2,3,4¬ Go(m,1) 1,2,3,4+ Gg(m,1) 1,2,3,4 End of inner loop minvalue1,2,3,4¬ min(Ggo(m,1)1,2,3,4 % minimum value and its sequence minvalueseq1,2,3,4¬ sequence(min(Ggo(m,1) 1,2,3,4) x1,2,3,4(rows,k+1)¬ x1,2,3,4(rows,k)+ increment(incr, theta(minvalueseq1,2,3,4) deltaincr ¬ 0.1*incr*random generaration % To generate artificial pertubation deltaangle ¬ 2*pi* random generaration x1,2,3,4 (rows,k+1)¬ x1,2,3,4(rows,k+1)+increment(delta incr,theta(deltaangle)) Loop i=1 to (number of vehicles-1) % Formation & Coordination amongst the vehicles delta1,2,3¬ xi+1(x,y)-xi(x,y) if (delta<mags(safedistance) then xi(x,y) ¬ xi(x,y)+shift Plot online path tracing End of last inner loop End of the main simulation loop Plot some results of the troop movement End of the Main Program Three functions were called from the main program of which the first one is obstacle generation. This function has not taken any input from the main program and not also returned any value but generates eight random obstacles in the world map of the vehicles. Articles

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5. Simulation results The results obtained from the simulations are given in this section. Fig. 2 shows four vehicles at their starting points. Fig. 3 shows the input obstacle functions of the workspace. Fig. 4 shows the upper left and Fig. 5 the upper right goal functions estimated from the sampled workspace Fig. 6 shows the online tracing of the traversed path Fig. 7 shows the output vector from goal optimizing function Fig. 8 shows the output vector from weighted combination of Gaussian profile (obstacles) optimization and Fig. 9 shows the final paths traced out by the vehicles Fig. 5. Upper right goal function estimated from the sampled workspace

Fig. 2. Four vehicles just about to start in two different groups in the given workspace

Fig. 3. Input obstacle functions estimated from the sampled workspace

Fig. 4. Upper left goal function estimated from the sampled workspace 24

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Fig. 6. Online tracing of the traversed path of four vehicles to two goals in a group of two

Fig. 7. Output vector from goal optimizing function for every iteration

Fig. 8. Output vector from weighted combination of Gaussian profile (obstacles) optimizing and goal optimizing function

Journal of Automation, Mobile Robotics & Intelligent Systems

Fig. 9. Final paths traced out by the vehicles for a given random obstacle map and goals

6. Conclusions There are many open issues in multi-vehicle path planning and coordination which are yet to be addressed. Currently used techniques are not suitable for very large number of vehicles and for 3-D trajectories (aerial vehicles). Another difficulty is faced in practical implementation of the real time mobile vehicles. It requires incorporating practical motion and sensing constraints of physical vehicles in 2-D space. As already mentioned this technique is an application of Artificial Potential Field Approach in static environment. This approach could be extended to control and coordination of mobile vehicles in highly stochastic and dynamic environment but that would be a slight deviation from classical APF and its complexity is higher. It may require online path planning and coordination strategies. Motion coordination of multiple vehicles in a shared workspace has large scale practical values. Example applications include container management in ports, extra-planetary explorations, search and rescue, mineral mining, transportation, industrial and household maintenance, construction, hazardous waste cleanup, security, agriculture, and warehouse management. Due to complexity and cost, relatively few real-world implementations of these systems have been accomplished till date. It is expected that such systems will have wide-spread use in near future as the technology continues to mature. Because of the need for motion coordination of multivehicle systems, the work described in this paper is of critical importance. As multi-robot systems can operate in stochastic and unpredictable settings, the study of the interaction dynamics of these settings may have broader impact in a wide range of applications. One possible solution of multi-vehicle problem has been presented in this paper. The task is to find out the optimal path towards goals avoiding obstacles by learning through random search in an unknown environment. A Gaussian Profile Map function optimally directs the vehicles away from the obstacles as if the Robots are repulsed from the obstacles. The vehicles are more aggressive towards the goals rather than to the obstacle avoidance phenomenon in this project. The time taken by the troop to reach two different goals in two pairs is less than a minute as being

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observed. This algorithm, if compared with others, the time efficiency could be found to be better to some extent. The obstacles are not dynamic in the present work but they may be stochastically generated at any location in the workspace. Moreover the troop is to maintain formation and coordination amongst themselves. A comprehensive result has been achieved by simulating the algorithm in MATLAB® environment. In highly stochastic environment, a more robust and adaptive algorithm may be required. Application of Neuro-Fuzzy or NeuroGA system could be very useful in this context. A simple algorithm based on random search has been used which is very easy to implement. It is based on APF approach and also shares some similarities with evolutionary computation techniques. The system is initialized with a population of random solutions and searches for optima by updating generations. The potential solutions, in this algorithm fly through the problem space following current optimal output. They are taken as the optimally best possible directives for movement of each individual vehicle. The controller updates the parameters in accordance with the optimal directives generated by the algorithm. Still then, for very complicated, dynamic and stochastic environments an expert system with leader-follower architecture may be even a better alternative.

Acknowledgement

This work was inspired by the prior research work of Kevin Passino [2] and O. Khatib [3]. The authors wish to acknowledge their contributions. The authors are also indebted to the Head of the Dept. of Electrical Engg, Jadavpur University, for allowing this research work to be conducted in its Mechatronics Laboratory.

AUTHORS

Sourish Sanyal* – Electronics & Communication Department, College of Engineering & Management, West Bengal University of Technology, West Bengal, India. E-mail: sourish2007_may@yahoo.co.in Ranjit Kumar Barai, Rupendranath Chakrabarti, Pranab Kumar Chattopadhyay – Electrical Engineering Department, Jadavpur University, West Bengal, India. E-mails: ranjit.k.barai@gmail.com; rupen_chakrabarti@yahoo.co.in; pkchattopadhyay47@hotmail.com *Corresponding author

References [1] L. E. Parker, “Path Planning and Motion Coordination in Multiple Mobile Robot Teams”, Encyclopedia of Complexity and System Science, Robert A. Meyers, Editor-in-Chief, Springer, 2009. [2] K. Passino, Biomimicry for Optimization, Control, and Automation, Springer, 2005. [3] 0. Khatib, “Real-Time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics Research, vol. 5, no. 1, 1986, pp. 90–98. Articles

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[4] M. Gerke, “Genetic path planning for mobile robots”. In: Proceedings of the American Control Conference 1999, vol. 4, 1999, pp. 2424–2429. [5] J. Latombe, Robot Motion Planning, Kluwer Academic Publishers, Boston, 1991. [6] 0. Khatib, “Real-Time Obstacle Avoidance for Manipulators and Mobile Robots”, International Journal of Robotics Research, vol. 5, no. 1, 1986, pp. 90–98. [7] Zhang Pei-Yan, Lu Tian-Sheng, Song, Li-Bo Source, “Soccer robot path planning based on the artificial potential field approacch with simulated annealing”, Robotica, vol. 22, no. 5, September/ October, 2004, pp. 563–566. [8] C. Zhang, R. Ordonez, C. Schumacher, “Multi-Vehicle Cooperative Searchwith Uncertain Prior Information”. In:Proceedings of the 2004 American Control Conference. [9] N.M. Kwok, Q.P. Ha, G. Fang, “Motion Coordination for Construction Vehicles using Swarm Intelligence”, International Journal of Advanced Robotic Systems, 2012. DOI: 10.5772/5672 [10] N.M. Kwok, Q.P. Ha, V.T.Ngo, S.M. Hong, “Particle Swarm Optimization of a Group of Construction Vehicles”. In: ISARC 2006. [11] F. Arrichiello, Coordination Control of Multiple Mobile Robots. Dissertation at Universit`a Degli Studi Di Cassino, Dipartimento Di Automazione, Elettromagnetismo, Ingegneria Dell’informazione E Matematica Industriale. [12] Ping Ping Khaw, W.S. Wijesoma, Eam Khwang Teoh, “Intelligent Control And Navigation of an Outdoor AGV”. School of Electrical and Electronics Engineering, Intelligent Machines Research Lab. Nanyang Technological University, Singapore. Available at: http://www.araa.asn.au/ acra/acra1999/papers/paper43.pdf [13] D. Rathbum, S.n Kragelund, A. Pongpunwattana, B. Capozzi, Metron Aviation, Inc., Herndon,VA. [14] J.S. Bellingham, A. Richards, J.P.How, “Receding horizon control of autonomous aerial vehicles”. In: Proc. American Control Conference, Anchorage, Alaska, May 2002. [15] J.S. Bellingham, M. Tillerson, M. Alighanbari, J.P. How, “Cooperative path planning for multiple UAVs in dynamic and uncertain environments”. In: Proc. of 41th conf. Decision Contr., Las Vegas,Nevada, USA, 2002. [16] Y.K.M. Alighanbari, J.P.How, “Coordination and Control of multiple UAVs with timing constraints and loitering”. In: Proc. American Control. Conf., Denver, CO, 2003. [17] W.B. Dunbar, R.M. Murray, “Model predictive control of coordinated multi-vehicle formations”. In: Proc. of 41th Conf. Decision Contr., Las Vegas, NV, 2002. [18] S. Ganapathy, K.M. Passino, “Agreement strategies for cooperative control of uninhabited autonomous vehicles”. In: Proc. American Control conf., Denver, Colorado, June 2003. [19] Y. Liu, M.A. Simaan, J.J.B. Cruz, “Game theoretic approach to cooperative teaming tasking in 26

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[21]

[22]

[23] [24] [25]

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the presence of an adversary”. In: Proc. American Control Conf., Denver, Colorado, June 2003. Q. Li, S. Payandeh, “Planning for dynamic multiagent planar manipulation with uncertainty: A game theoretic approach”. In: Proc. American Control Conf., Denver, Colorado, June 2003. M.E. Broucke, “Disjoint path algorithms for planar reconfiguration identical vehicles”. In: Proc. American Control Conf., Denver, Colorado, June 2003. R.O. Saber, W.B. Dunbar, R.M. Murray, “Cooperative control of multi-vehicle systems using cost graphs and optimization”. In: Proc. American Control Conf., Denver, Colorado, June 2003. J-C. Latombe, “Robot Motion Planning”, Kluwer Academic Publishers, 1991. S. M. LaValle, “Planning Algorithms”, Cambridge University Press, 2006. J. E. Hopcroft, J. T. Schwartz, M Sharir, “On the complexity of motion planning for multiple independent objects; PSPACE-Hardness of the ”Warehouseman’s Problem”, International Journal of Robotics Research, vol. 3, 1984, no. 4, pp. 76–88.

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About Evaluation of Multivariate Measurements Results Submitted: 14th Nov. 201; accepted 22nd March 2012

Zygmunt L. Warsza

Abstract:

A brief review of few problems arising in the correct numerical expression and evaluation of results of indirect multi-parameter measurements is given. There is included a theoretical basis for determining the estimates of values, uncertainties and correlation coefficients of the indirectly obtained multi-measurand, which are processed from data of the simultaneously measured set of variables. The alge-bra of random vectors is used. A numerical example illus-trates the linear transformation of two variables and the types of improperly evaluated results – that may occur with over-rounding. There are given thresholds of the safe uniform rounding of mean vector and its scatter ellipsoid. There is proposed an upgrading of the GUM Example H.2 and of the uncertainty equation for nonlinear functions. It is also evidenced that correlation matrix of current 2010 data of fundamental physical constants recommended by CODATA has non-negligible negative eigenvalues. In the end of this work it is argued for the urgent needs of stand-ardization of e-publication of the experimental data in two parts: e-printed traditional narrative part, and an attached computer readable file with all numerical input data and results, to allow “fast” numerical peer review of the proposed publication reporting new measurement results. This work is a result of an inter-disciplinary cooperation of a metrologist and a nuclear physicist. Keywords: uncertainty, indirect measurements, multimeasurand, correlated data

1. Introduction

Simultaneous measurements of several statistically related quantities, i.e. correlated, are performed in science, education, technology, economy and many other disciplines. From the digitally processed on- or off-line data of m variables, directly measured on input, the n other variables (in physics called as observables) are determined indirectly on output, if their mutual relation is known. In addition to estimators of values and uncertainty the knowledge about correlation coefficients of output quantities also is of special importance for some or all of these variables to be jointly processed further. Accuracy of evaluated output multi-measurand data depends on the statistical uncertainties of given parameters of input multi-measurand, as well as on the accuracy of their processing. Final rounding of indirectly obtained data of output multi-measurand must depend on a uncertainty of the input data [6]. The “safe rounding” of the digitally processed multi-measurand data should be done in such a way that they are not be damaged. If the accuracy of final uncertainties or number of repetitions of raw meas-

urements are not given in publication of input data then it should not be assumed that the values of estimators of standard deviations and correlation coefficients of the initial variables are correctly found from measurement data and are as their values for whole populations. In indirect multi-dimensional measurements there are two border types of relations of the uncertainty both components uA and uB [1]. First case: uncertainty uA<<uB. In such situations it is enough to provide the necessary instrumental resolution and accuracy for measurement of input values and to determine cross-links to the output. Second case: uA>>uB when all environmental effects interacting on input measurements are carefully eliminated and the uncertainty of type B is small compared to the range of random scatter of observed variables. Here one should achieve maximum accuracy in measurements and then the information obtained in the experiment is not partially lost in the processing of the random input data and in the rounding of the obtained results. The number of observations should be as large as possible to minimize the statistical type uncertainty uA.

2. Theoretical backgrounds in short

In multivariate indirect measurements the input multimeasurand can be expressed by random vector X=[X1, X2,... Xm]T and output one – by vector Y =[Y1, Y2, ... Yn]T. These random vectors X and Y of dimensions m and n, respectively, can be described by the multi-dimensional distributions. In general case the relation between them can be expressed by Y = F(X) If F is a linear operator, then Y = S·X Where: S is matrix of dimensions n x m and n ≤ m. Two examples of multivariate indirect measurements are given in Fig 1.

Fig. 1. Examples of indirect evaluation of measurement data of 2 jointed output variables Y=[Y1, Y2]T from measurements of 3 input variables X=[X1, X2, X3]T: a) no correlated, b) correlated X1, X2 [10] Articles

Fig. 1. Examples of indirect evaluationVOLUME of measurement 2 6, N° data 4 of 2012 jointed output variables Y=[Y1, Y2]T from measurements of 3 input variables X=[X1, X2, X3]T: a) no correlated, b) correlated X1, X2 [10]

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The basic structure of the numerical estimation of the multi-measurand should contain averaged components of the random vector and a description of the multidimensional scatter region of it. The accuracy of both these data should be also known. Even if the relation Y = F(X) is nonlinear, in the most cases for small deviations of the random vectors X and Y their scatter regions can be defined by a model of joint n-dimensional normal probability distribution. Then, for a given probability density p0 the distribution region for p ≥ p0 takes the form of a n-dimensional hyper-ellipsoid with its center at the end of the average vector. Relations between covariance matrices of hyperellipsoids of the output and input measurands (in a linear approximation of the observables in the vicinity of the end of mean vector Y ) are described analytically by

y √2∙0.345

1.155

1.845

√2∙1500 ζ η

(√2)·(1.500

0.100)

(√2)·(0.345

0.001)

r (ζ, η) =

1.0

0.0

1.0

x = (ζ +η)  (√2) y = (ζ −η)  (√2) x y

1.845

0.100

1.155

0.100

r (x, y) =

1.00000

0.9998

1.0000

Fig 2. Linear transformation vector Fig 2. Linear transformationofof2D 2D random random vector [5] [5]

c Y = S c X ST Where: is the matrix of linear sensitivity coefficients. The matrix r of the correlation coefficients is defined by the relation

y A. x = 1.845(100) Basic equations for the processing of 2D random vecA y = 1.155(100) tors are 1.155given in Table 1 and typical distortions of outB. x = 1.84(10) put data by not proper – too high rounding are shown in y = 1.16(10) Fig. 3 [5].

r = σ c σT

1.845 x

x = 1.8(1) y = 1.2(1)

2. Correlator 3. Correlator 4. Mean vector over-rounded scatter region moved ignored over-rounded to 1

It is called also shortly as the correlator. A multidimensional distribution is normal if matrices c and r are positive definite, i.e. their eigenvalues λi, which are the roots of the characteristic equations

1.155

det[c-λ1]=0, and det[r-λ1]=0,

1.155 1.845

should be positive [3]. This requirement was not included in Supplement 2 of GUM [1] in above form. So, to express correctly the result of measuring or evaluating random vector quantity the minimal data structure should contain: Mean vector, Vector of standard deviations and their uncertainties (or number of measurements in each sample), Positive definite correlation matrix and uncertainties of their elements, Machine precision used to compute vector parameters and eigenvalues of correlation matrix. With these data the user will have complete information to plan and control the safe usage of data in next computations. In Fig 2 there is shown an example of a linear transformation of two dimension (2D) “Greek” vector X=[h, z]T to “Latin” vector Y=[x, y] T [5].

B A

y 1.155 1.845

1.845

Fig 3. Cases of improper presentation of correlated 2D data

Fig 3. Cases of improper presentation of correlated 2D data To express difference ΔY = Yi – Y between the center of the ellipse of transformed original raw data Y and the end of rounded vector Yi Mahalanobis distance χ is used, which is given by [12]

χ 2 = ∆Y

1 r ( x, y ) −1 ∆Y T σ xσ y

Let us consider rounding of data Y given in Fig 2 [5]. Raw data Y rounded to 3 digits after decimal point:

Table 1. Basic formulas of the 2D random vector transformation of Fig 2 Covariance matrix

Transformation x  y = S  

ς  1 η  = 1   

1 − 1 

ς  η  =  

ς + η  ς − η   

Standard deviations s x = s + s + 2 s V s h rVh 2 V

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2 h

1  σ ς2 1 cxy =   1 − 1 σ ς σ η ρς η Correlator

s y = s + s − 2 s V s h rVh 2 V

2 h

1 rxy =   ρ xy

ρ xy  1 

1  σ x2 σ ς σ η ρς η  1 σ ς2 − σ η2  =    σ η2  1 − 1 σ ς2 − σ η2 σ y2 

Where: correlation coefficient

c & r positive define det(A- λI)=0 λi ≥ 0

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(ρxy=0,9998),

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rounding of the multivariate random vector data. All such safety thresholds of rounding are expressed in terms of decimal numbers [6], i.e. for: standard deviations Ui of vector components Vi

Y=[1,845(100); 1,155(100)]

B. Rounding of Y to 2 digits after decimal point:

1 AiU ≥ AiUth = Upper Integr [ log10 ( 2

Y1= [1,84(10); 1,16(10)];

ΔY1= Y1–Y= [–0,005; 0,005]; χ12 = 25 > 1

n  U  4λ T  i   unit i 

)]

2 min C L

values of the components Vi of the average vector elements of the correlation matrix AiV ≥ AiVth = AiUth –

C. Rounding of Y to 1 digit after decimal point:

 AC ≥ AiCth ≠ j = Upper Integr log10 

Y2= [1,8(1); 1,16(1)]; Y2=Y2–Y= [–0,045; 0,045]; χ 22 = 2500 > 1 For the ellipse border the Mahalanobis distance χ=1. Then the ends of both rounded vectors Y1, Y2 situated are outside of the ellipse of transformed original random input data Y. If the assumption proposed by V. Ezhela in [2– 4] for safety processing of random vectors is to be satisfied, these ends has to be situated inside of this ellipse. Then in both cases the results are over-rounded. Such assumption can be valid only for the absolutely accurate input statistical data from the whole random populations of Xi or for very large going to infinity, number N of sample elements, and when instrumental errors are negligible. That condition is not fulfilled in many existing in measurement situations, where data samples of a small number of elements are only possible to obtain in a limited time of observation. Then all estimators of mean values, standard deviations and correlation coefficients obtained from the samples of limited number N of multivariate observations have their own uncertainties, which are quite high for small N. From above it follows that two different type requirements for precision of processing and rounding procedures of multivariate data can be used: – very high for safety numerical processing of input random vector of statistical parameters treated as absolutely accurate (if they are given for whole population or for being safe if accuracy of them is unknown), and – lower, dependent on given or possible to be estimated accuracy of statistical parameters of the input multimeasurand. In the second case thresholds for limiting the rounding of output data parameters have to be established as dependent on the accuracy of statistical parameters of input samples.

3. Thresholds of the safe rounding of transformed multivariate data

Applying spectral theorems of matrix theory V. Ezhela in [4] proposed thresholds of the number of digits after decimal point for fully safe independent uniform Table 2 Parameters

σx

σy

Raw results

0,3242

0,0664

0,1555

0,0256

A. rounding to 3 digits

0,324

0.067

0.156

0,026

B. rounding to 2 digits

0.32

0.07

0,16

0.03

C. rounding to 1digit

0.3

< 0.1

0,2

< 0.1

 n −1   2 λmin

  

Where: λmin – minimal eigenvalue of the correlation matrix, – “tolerance” factor at defined confidence level. Above formulas look sophisticated, but are quite easy to be used – see examples in [4], [6] and [7]. For data of the Example H.2 of GUM [1] (the same as in chapter 9.4 of Supplement 2 [11]) after high precision processing of the input data correctness of the rounding of the multivariate output vector Y=[R, X, Z], when TCL=1 (one standard deviation) are the following numbers of digits [4], [7]: Upper Integer{Ath[R]} = 5 Upper Integer{Ath[X]} = 4 Upper Integer{Ath[Z]} = 5 Correlator of the output vector [R, X, Z]T has elements uniformly rounded to 9 digits and the smallest eigenvalue λmin=2,22711×10-8 and for this λmin the minimum number of digits of the correlation coefficients is: AC(CorH3) = 8 As large number of significant figures are obtained from proposed by V. E. “thresholds of uniform rounding”, such as for scalars by GUM, valid with assumptions: • input data are treated as absolutely accurate, • scatter region for p≥p0 is kept as n-ellipsoid, • numerical processing is “safety rounded”, and • the output vector must be maintained inside of the scattered area of the transformed raw input data. Then such of many digits thresholds are not needed in measurements. They are valid only for the save digital processing of random vector itself because of the assumption that mean values of components the input vector, their standard deviations and correlation coefficients are absolutely accurate known, which is not happen for any real experiments. Obtained experimentally measurement data are not absolutely accurate as number N of observations in samples are limited (uncertainty type Ais rising with decreasing of N) and unknown instrumental errors are not negligible (represented by uncertainty type B). Then in requirements for processing and rounding the uncertainties of estimators of mean value (or other the most probable values of measured vector components e.g. mid-range for uniform distribution and for trapeze distributions of the ratio of their bases from 1 to 0,65. The accuracy of stand2σ y 2σ x ard deviations and of correlation coefficients have to be also taken in con0,1328 0,0512 siderations. Then the rounding of real 0,133 0,051 multivariate measurement data has to 0,14 0,05 be done below thresholds given by V. <0,2 <0,1 Ezhela and dedicated for the save data Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

processing itself only. Approximately it should be enough if precision of processing is the one digit more than the accuracy of measured input data.

Fig. 4. Rounding with constant correlation coefficient ρxy An graphical illustration of such rounding of data given in table 2 for constant correlation coefficient ρxy is shown in Fig. 4. The largest ellipse C obtained after rounding standard deviations σx, σy to 1 digit do not fully cover the primary ellipse A. But it was checked also that for larger ellipses tangential to the rectangular of twice larger sides ±2σx, ±2σy after their rounding to 2 or to 1 digit it is ok. For example for such ellipses A2, B2 (not given in Fig. 3) it is 2σ (x B ) − (x − x B ) > 2σ (x A ) 2σ ( y B ) − ( y − y B ) > 2σ ( y A .)

Rounding of the correlation coefficients depend on their value and on accuracy. Then special care, not considered here, is needed, The different rounding of multivariate data with changes of correlation coefficients values ρij, also should be applied. Two following methods of such rounding below thresholds are preliminary tested: • Method 1 (of Z. L. W) – author proposed to maintain a constant values of non diagonal elements of the positive covariance matrix, i.e.:

Where: signs in the upper index indicates the direction of change. • Method 2 (of V. E.) – V. Ezhela proposed to use truncation, i.e. to omit further digits after no changed the last accepted digit [6]. Both methods are used for the output data of the Example H.2 GUM for rounding them to 3 and 2 digits after decimal point [6]. Results are given below in Table 3. 30

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It was checked: – positive definite of the rounded correlator; – the relative distance by Mahalanobis χ2 [12] between the end of rounded vector and the center of raw original data after transformation. Correlators of both methods are positive definite, but the full theoretical justification of the method 2 is not given jet. Method 2 gives a smaller values of Mahalanobis distances of ends of the rounded vectors from the ellipse center of transformed raw data, but its smallest eigenvalue is closer to zero than obtained in method 1. Conclusion: since in multivariable measurements the rounding level of output vector Y depends not only on the precision of digital processing but mainly on the uncertainties of all statistical parameters of input vector X, then additional formulas of rounding thresholds then given by V. Ezhela [4], [6] are also urgently needed.

4. Upgrading the GUM proposals for multivariable measurements

GUM [1] and other official metrological documents about uncertainty are applied up to now only in measurements of the single quantity. But statements in the main text of GUM was formulated in such a manner that the reader gets the impression that a generalization to the multivariate case is straightforward. That was considered in details in Example H.2 of GUM which illustrates clauses 7.2.5 and 7.2.6. Supplement 2 to GUM [11] – about extension of evaluation uncertainty of measurements to any number of quantities, has been published just now and time to implement its clauses in practice is not jet long enough.

About Example H.2 of GUM

In Table H.2 of GUM [1] there are given five (and in Supplement 2 – six) raw simultaneous measurements of input vector X=[U, J, Φ]T and vector Y=[R=UcosΦ/J, X=UsinΦ/J, Z=U/J]T is evaluated. The results are presented there in Tables H.3 and H.4. Rounding of correlation coefficients is not properly done there, since the smallest eigenvalue of correlator matrix is negative and so the scatter region is not of the 3D-ellipsoide form. Also final output data of Example H.2 does not satisfy “physical law” of impedance of the two-terminal passive element which is: X2+Y2=Z2 as σ2= -71,5 [3], [6], [7].

Table 3 Method 1 (of Z.L.W) Rounding to 3 digits Mean Standard Correlator value Deviation 127,732 0,160 1 –0,586 –0,483 219,847 0,661 –0,586 1 0,991 254,60 0,529 –0,483 0,991 1 Eigenvalue: [2,39983; 0,598631; 0,00154204] χ2 8,39

Rounding to 2 digits Mean Standard Correlator value Deviation 127,73 0,16 1 –0,59 –0,48 219,85 0,66 –0,59 1 0,99 254,26 0,53 –0,48 0,99 1 Eigenvalue: [2,39972; 0,598797; 0,00148649] χ2 284,0

Method 2 (of V. E.) Rounding to 3 digits Mean Standard Correlator value Deviation 127,732 0,160 1 –0,588 –0,485 219,847 0,661 –0,588 1 0,992 254,260 0,529 –0,485 0,992 1 Eigenvalue: [2,40297; 0,596499; 0,000533094] χ2 1, 108

Rounding to 2 digits Mean Standard Correlator value Deviation 127,73 0,16 1 –0,58 –0,48 219,85 0,66 –0,58 1 0,99 254,26 0,53 –0,48 0,99 1 Eigenvalue: [2,26846; 0,657234; 0,0743036] χ2 6,445

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For σ2>0.1 (i. e. σ>0.316) is obtained u2(F)<0. So, the additional component of the uncertainty formula of nonlinear function given in Notice to clause 5.1.2 of GUM should be corrected by removing from the sum in brackets the second component with the third derivative.

For establishing requirements of safety digital processing purposes according, the input multivariate measured data of H.2 Example are firstly treated as absolutely accurate data. Then, for such theoretical case, according thresholds given by V. Ezhela [4] (see chapter 3) the required digit numbers after decimal point are as follows: for mean values and standard uncertainty of R and Z – 5 digits, of X - 4 digits and for correlation coefficients – 8 digits ! [4, 7]. Therefore, high numbers of digits cannot be accepted for describing the measurement results as they are obtained under the assumptions that: input data are treated as fully accurate, numerical results of processing are safety and uniformly rounded to maintained vector end in the scattered area of the transformed original input data. They can be used only as a reference of processing of the absolutely accurate data of the random vector of similar component values as parameter estimators of samples of the input vector X in H.2 Example. All these samples have only N=5 ! (or 6 in supl. 2) measurements each and the accuracy of SD of each variable and of correlator elements is very poor. Relative uncertainty of SD of such small samples is about 36% – see Table E.1 in GUM [1].

5.2. Nonlinear processing of input vector

In case of the nonlinear processing of input vector the widely used as approximation differential “linear uncertainty propagation law” does not work properly in more accurate calculations for highly nonlinear functions. The nonlinear uncertainty propagation should be used with the obligatory positivity constraints

Ci, [δCa, δCb ]

Fk(Ci), [δFm, δFn ]

Cоmpоnent of X (input)

Cоmpоnent of Y (output)

[δFi , δF j ] = ∂l Fj ∂ k Fi {δcα1 …δcα k , δcβ1 …δcβ l } ∂cβ1 … ∂cβ l α 1 … ∂cα k

∑ k!l! ∂c

k , l =1

5. About Notice GUM on the nonlinear uncertainty propagation

Covariance matrix [δFm, δFn] is non-degenerate and positive definite if dimensions dim(Ci)=m, dim(Fk)=n and the order T of component of the Taylor polynomials approximated the measuring vector function Fk obey the inequality

In the multivariable case the linear propagation of errors from m-dimensional vector X to n-dimensional vector Y in some cases is misleading for n>m, i.e. for nonlinear functions.

5.1. Single measurand case

n ≤ nth =

In the Notice to clause 5.1.2 of GUM [1] is determinate the uncertainty of highly non-linear single function y=f(X) only. There is recommended that the linear propagation of variance is supplemented by higher-order components, i.e.

V. Ezhela noticed in [4] that calculations of the variance u2(y) according Table 4 to this formula may give a false negaElementary charge tive value because of the component in parentheses with the third derivative. Planck constant This is illustrated below by the examElectron mass ple of the single nonlinear polynomial function 1/fine structure const

N° 4

(m + T ) ! −1 m !T !

Where: m

nth

Commentary on fundamental physical constancies The adjustments of the fundamental physical constants (FPC) are regularly performed by the Fundamental Constants Data Center at NIST and recommended by CODATA as the unique source of the current FPC values. There are 325 adjusted quantities, from which 79 are e

1.602 176 565(35)⋅ 10-19

6.626 069 57(29)⋅ 10-34

1.0000

-31

0.9998

1/ α(0)

9.109 382 91(40)⋅ 10

137.035 999 074(44)

0.9999

− 0.0145 − 0.0072

0.0075

F(x) = 1- x + 2x2 + 3x3 + 4x4, If measurand x is normally distributed around x= 0 with variance σ2, then the variance u2(F) calculated according to the recommendation 5.1.2 is u 2(F) = σ2F’(0)2 + 1/2σ4F’’(0)2 + σ4F’(0) F’’’(0) = = σ2 [1+ σ2(16/2 -18)] = σ2 [1-10 σ2]

called basic algebraically independent constants CαB. As an example there are listed in Table 4 last values of four FPC given by CODATA 2010 [9] in SI units and below – their correlation matrix. Eigenvalues of above correlation matrix are: [2.99942, 1.00006, 0.000719993, - 0.000202165]. The last eigenvalue is non-negligible negative. Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

Only two of constancies FPC can be used together in joint precision calculations. More about negative eigenvalues in CODATA publications is given in [3], [7], [8].

6. Application of e-publishing in multi-variable measurements

The traditional form of the scientific communication based on the paper oriented e-publications is now not the proper way to present and to exchange the multidimensional experimental data. After V. Ezhela in [3] the standardization of two-component forms of the scientific publication is unavoidable. First component will be the traditional descriptive scientific text already well formalized by publishers. The second part should be computer readable file with all numerical input data and results to allow “fast” numerical peer review of the publication reporting new results. It is discussed in detail in [3] together with given four dozen examples of “bad practice” of physical publications in journals of high “impact factor” and in other sources. Some particular problems connected with that proposal are in [8] and [10].

7. General conclusion

Some of the presented problems of the evaluation of results and uncertainty of the indirect multivariable measurements still need farer investigations to obtain clear enough backgrounds for the common international acceptance of the rounding and digital presentation methods of multivariate data results and of the calculation uncertainty of highly nonlinear related multivariate data. These problems are not yet fully included in just finished the first version of Supplement 2 to GUM [1] about the extension of expressing uncertainty to any number of quantities [11]. Then its recommendations should be corrected and included in the next upgraded version of Supplement 2 and taken into account also in other post-GUM documents.

AUTHOR

Zygmunt Lech Warsza – Industrial Research Institute of Automation and Measurement (PIAP), Al. Jerozolimskie 202, 02-486 Warszawa. Mob. phone (+48)692 033 661, e-mail: zlw@op.pl

References [1] Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM 1995 with minor corrections), JCGM-100-2008. [2] Ezhela V.,“A multi-measurand ISO GUM supplement is urgent”. CODATA DSJ 6, pp. 676-789 [Errata: DSJ 7, (2007) E.2-21, [3] Ezhela V., “Physics and Metrology”. In: Proceedings of V Congress of Metrology KM 2010, CD. Technical University, Lodz, Poland. [4] Ezhela V.: “Comments on some clauses of GUM which provoking the incorrect presentation of mul32

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tidimensional measured data”. In: Proc. of V Congress of Metrology KM 2010, CD. Technical University, Lodz, Poland. [5] Warsza Z., Ezhela V.: “Zarys podstaw teoretycznych wyznaczania i numerycznej prezentacji wyników pomiarów pośrednich wieloparametrowych” (Outline of the theoretical backgrounds and numerical presentation of multivariate indirect measurement results), PAK, 2011, no. 2, pp. 175–179. (in Polish) [6] Ezhela V., Warsza Z., “Przetwarzanie wyników w pomiarach wieloparametrowych” (Evaluation of results of the indirect multidimensional measurements), Electrical Review, 2011, no. 2, pp. 236–241 [7] Warsza Z., Ezhela V., “Sugestie kilku uściśleń w Przewodniku GUM-2008 i zaokrąglanie wyników pomiarów wieloparametrowych” (Some upgrading suggestions of the GUM-2008 and rounding rules for multivariate measurement results), PAK, 2011, no. 3, pp. 291–296 (in Polish). [8] Ezhela V., Warsza Z., ”Nieścisłości stałych podstawowych i propozycja standaryzacji dualnego publikowania wyników pomiaru multi-mezurandu” (Inaccuracy of fundamental constancies and proposal of dual publication form of multi-measurand data). PAK, 2011, no 5, pp. 486–490 (in Polish) [9] http://physics.nist.gov/cuu/Constants/index.html [10] Warsza Z. , Ezhela V., „O wyrażaniu i publikowaniu danych pomiarów wieloparametrowych – stan aktualny a potrzeby” (About evaluation and publication of multivariate measurement data - current status and needs), PAR, no. 10, pp. 68–76. (in Polish) [11] Evaluation of measurement data, Supplement 2 to the Guide to the expression of uncertainty in measurement, Extension to any number of output quantities, JCGM 102 2011. [12] “Mahalanobis distance”. Wikipedia, the free encyclopedia.

Journal of Automation, Mobile Robotics & Intelligent Systems

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Positioning and Control of Nozzles and Water Particles in Decorative water curtain and water screens Submitted 21st February 2012; accepted 7th July 2012

Mahdi HajiHeydari, Sasan Mohammadi

Abstract:

LED and LCD displays, laser show devices and animators, musical fountains, water curtains and artificial waterfalls, etc., are decorative instruments. Some of them were developed for intelligent pouring of water droplets as particles in a space (particle screen). They have nozzles in the line that generates pattern with programmed on or off of its valves. This article introduces an improvement of water curtain (waving pouring water) and water screen (stenciled water screen by use of on-off valves), and combines them together.

each of them can move in cross direction (say y axis) which each of them have on/off valve to control pouring the water or not. In the design process it should impose restriction of nozzle movements in real world to imaginary world. A snapshot of the developed software using openGL in C++ is viewed in the Figure 1.

Keywords: water nozzle, curtain, decorative, simulation, openGL.

1. Introduction Decorative and special effect equipments are used everywhere – in hotels, airports and public places, etc. Many years ago water curtain and water screens have born as a decorative equipment for information or funny multimedia screens. Smoke screens [9] and then recently fog screens [10] emerge after that. Smoke screens reveal laser or light beam projected to a curtain generated by use of smoke particles as a screen. Fog screens do something like that, but they use small water droplets instead to accomplish it. Water curtain is simply a curtain but its brother, water screen, is an intelligent machinery to produce patterns as a media. Single in line (2D) water screens can generate patterns with placing particle (water droplets) in a plane. At the 2006 virtual reality conference a paper was been proposed about making 3D patterns with water drops [1], after that scientists pursued to study of this type of screens. Then modifications and improvements in usage [4] or in development [5] (used as computer interface and similar usages [7]) were being proposed in the literature. Finally, Peter C. Barnum, Srinivasa G. Narasimhan, Takeo Kanade at 2010 proposed a“multi layer display with water drops” [8]. This article introduces an improvement of water curtain and water screen, and combines them together.

2. Machine at the simulation and study phase Simulation software can be the first step in development to investing reality requirements, problems and abilities of a machine or process which will be designed. Here it will be used to design surface generated with the machine physically. In this article we show how it has been used to design and simulate a set of nozzles in a line (say x axis) and

Figure 1. Simulator Snapshot Some other views of the simulated curtain and cut logo can be seen in the Figure 2.

Figure 2. Other views of the simulated curtain Control panel can be seen in the left most of the window. At the middle of it design pane are placed. And simulation animated preview at the right most. 2.1. Design toolbox Design toolbox is in middle vertical section of the snapshot above. It can be seen at the top of the design portion we have a time line. Below of its nozzle position a design pane and then On/Off design pane placed. A user defines position of nozzles in each step of motion, and intermediate steps will be interpolated. We can say in each frame because in simulation these positions are used to make animation. Application imposes restrictions to position definition of nozzles. For example, if a nozzle is going to the positive direction with +20 velocity at Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

the origin, next position of the nozzle can be extremely at, for example, +40 and only -10 because of negative acceleration needed to stop and go back in negative direction. These restrictions make a legal zone to definition of nozzle positions. It shows to the user with a green zone, and user has to define and adjust control points in that area. Movement restriction defines based on acceleration restriction of nozzles which depends on manufacturing of the machine and will be defined in next development steps. User can define intermediate frames. An application generates jumped frames accordingly with interpolation. It maybe seems a user-friendly approach to design. It can improve in next versions with a 3D design medium approach. At the bottom of the design section On/Off pane can be seen. In this section a pattern draws or loads from a bitmap file. This pattern will be ended with the watery curtain. Corresponding nozzles at appropriate steps of time Offs if corresponding point is black or make it On if it’s white. Then the pattern designed as a black region will produce as a cavity in the generated surface. 2.2. Physical simulation Here a particle based simulation is used. A number of water drops injected to the air with nozzles in each step of time and simulate its motion till receives to the ground. Height of water and many other parameters can be controlled via control panel in the left most of the window (depicted above). We use dynamic relations to simulate movement of water drops including gravitation force, air drag, and cohesion between water drops. Look at the Figure 3.

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And Fy like that, which G* is modified gravitational force constant. By use of these relationships and FBD and knowing kinetics relationships, motion of drops in the air can be simulated. But each water drop should have a delivery point. Where is that? Drops initially have position of nozzle jet tip coordinate and its horizontal velocity and delivering vertical velocity based on pressure of water in the jet. Then we need simulate motion of nozzle, too. We have positions of nozzle in known time steps as motion descriptor of the nozzle. In real world nozzle has a complicated motion (positive or negative acceleration and…) but we simulate the motion as depicted in the Figure 4.

Figure 4. Ideal and simulated velocity-time relations Which as depicted below traversed distance of two approach (theoretic and simulated reality) is same as depicted in the Figure 5.

Figure 5. Equal deltaX in ideal and simulated relations

Figure 3. Forces on water drop used in simulation Components of air drag force with a simple calculation (based on F=-kV2 relation) obtained as:

FDrag x = kvx vx 2 + vy 2

Then “actual velocity” in constant velocity portion is approximated and time of constant acceleration obtained. Now simulation of delivering of water drops done based on this nozzle positioning simulation. 2.3. Water pressure In vertical movement of drops of water they don’t move with constant velocity, they have constant acceleration. Then a stream of water doesn’t have a continuous flow line from top to end shape of pattern (when a continuous water flow is emerging). Its shape is as Figure 6.

FDrag y = kvy vx 2 + vy 2 For simulation of cohesion force between water drops we use gravitational force relationship between them. As gravitational force two drops exert cohesion force to each other if they are very near to each other. Then it is proportional to square of distance too:

Fx =

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(

G * m2 dx 2 + dy 2

)

( dx )

Figure 6. Drops distance increases in vertical movement

Figure 7. Vertical pattern stretching

Journal of Automation, Mobile Robotics & Intelligent Systems

According to above descriptions on falling of drops of water, it can be seen that generated patterns stretches vertically like Figure 7 This stretching is not linear due to gravity acceleration. We don’t have any control on the water after ejection from nozzle. This stretching is inherent of this media but if the initial vertical velocity increases, it can eliminate this effect of acceleration. To increase of the initial vertical velocity we should prepare pressure by use of a pump or height of water in reservoir as in Figure 8.

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4. Conclusion Next step is mechanical development of the machine. It has some challenging aspects. Because of number of nozzles and then number of linear position controllers (belt and pulley assembly and DC motor drivers) cost, electrical power consumption and other aspects of design is important. It may be influence on using which motor, stepper or dc, and kinds of them.

AUTHORS

Mahdi Haji Heydari* – Department of Mechatronics, Islamic Azad University – South Tehran Branch, Tehran, 1767969473 Iran. Emial: mhheydari.m.e@gmail.com.

Figure 8. Pressure of water reservoir

Sasan Mohammadi – Department of Mechanics, Islamic Azad University – South Tehran Branch, AHANG AFSARIE Tehran, 17776113651 Iran. E-mail: s_mohammadi@azad.ac.ir *Corresponding author

References 3. Mechanical implementation There are many possible approaches to implement of movement of the nozzle. At the first it seems that maybe use of belt and pulley is a good approach as shown in Figure 9.

Figure 9. Belt and pulley assembly

Figure 10. Data transferring to the nozzles Rack and pinion or simple caret on the rails considered, but they have some drawbacks (e.g. large mass, cost or…) then we didn’t mention them here. The block diagram of the process is depicted in Figure 10. Shift registers can cascade data with high speed (e.g. 300 Mbps theoretically). But any serial connection has a draw back. If an intermediate member lost its connection all others to the end of the chain lost their connections. But it can be developed appropriately.

[1] S. Eitoku, T. Tanikawa, Y. Suzuki, “Display Composed of Water Drops for Filling Space with Materialized Virtual Three-dimensional Objects”. In: IEEE Virtual Reality Conference 2006. [2] S. Eitoku, K. Hashimoto, T. Tanikawa, “Controllable water particle display”. In: Proceedings of the 2006 ACM SIGCHI International Conference on Advances in Computer Entertainment Technology. [3] S. Eitoku, K. Nishimura, T. Tanikawa, M. Hirose, “Study on design of controllable particle display using water drops suitable for light environment”. In: Proceedings of the 16th ACM Symposium on Virtual Reality Software and Technology, 2009. [4] K. Obana, T. Okumura, T. Kanaoka, K. Takano, K. Sato, “Electro-holography system using flow controlled device attached water particle 3D screen”. In: International Workshop on Advanced Image Technology (IWAIT 2004). [5] J. Tao, Z. Geng, Q. Fan, “A Digitized Water Display System Based on RS-422 Bu”, International Conference on Electrical and Control Engineering, 2010. [6] P.C. Barnum, S.G. Narasimhan, T. Kanade, “A multi-layered display with water drops”, ACM SIGGRAPH 2010 papers. [7] L. Shuai, Kuang Ying-hu, Z. Zhou, “Research on robot tactile display based on water jet technology”, IEEE International Conference on Robotics and Biomimetics 2005 (ROBIO). [8] L. Shuai, Ch. Jiang, Z. Zhou, Y. Hou, “Ramp Tactile and the Study on the Technology of Water-Jet Tactile Display”, Measurement & Control Technology, no. 7, 2008. [9] W. Fu, “Smoke screen technology and its development”, Electronics Optics & Control, 2002, no. 03. [10] L.D. Paulson, “Displaying data in thin air”, Computer, IEEE Journal , vol. 47, issue 3, March 2004.

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Journal of Automation, Mobile Robotics & Intelligent Systems

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2012

Stable Gait Synthesis and Analysis of a 12-degree of Freedom Biped Robot in Sagittal and Frontal Planes Submitted: 1th March 2012, accepted: 22nd August 2012

A.P. Sudheer, R. Vijayakumar, K.P. Mohanda

Abstract:

Legged machines have not been offered biologically realistic movement patterns and behaviours due to the limitations in kinematic, dynamics and control technique. When the degrees of freedom (DOF) increases, the robot becomes complex and it affects the postural stability. A loss of postural stability of biped may have potentially serious consequences and this demands thorough analysis for the better prediction and elimination of the possibility of fall. This work presents the modelling and simulation of twelve degrees of freedom (DOF) biped robot, walking along a pre-defined trajectory after considering the stability in sagittal and frontal planes based upon zero moment point (ZMP) criterion. Kinematic modelling and dynamic modelling of the robot are done using Denavit-Hartenberg (DH) parameters and Newton-Euler algorithm respectively. This paper also proposes Levenberg-Marquardt method for finding inverse kinematic solutions and determines the size of the foot based on ZMP for the stable motion of biped. Biped robot locomotion is simulated, kinematic and dynamic parameters are plotted using MATLAB. Cycloidal gait trajectory is experimentally validated for a particular step length of the biped. Keywords: Denavit-Hartenberg parameters, sagittal and frontal plane, zero moment point (ZMP), LevenbergMarquardt algorithm, Cycloidal gait trajectory.

Nomenclature θi ith Joint angle ai Legth of ith link di offset distance of ith link Twist angle about xi axis miR ; miL Masses of ith right and left leg qi ith joint parameter B, L Breadth and length of the foot Foot pose function Hessian matrix Jacobian λ Damping factor Inertial/mass Matrix Coriolis and centrifugal matrix Gravity matrix D Coefficient matrix of torque τ R Orientation matrix Moments at the ith joint Xzmp; Yzmp; Zzmp X, Y and Z coordinates of ZMP Force at the ith joint 36

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Fg; Fr ; Fi Total inertial and gravity force; Reaction force; inertial force Mg; Mr ; Mi Total inertial moment; Reaction moment; Inertial moment. g Acceleration due to gravity Rolling angle of circle SL Step length vi Instantaneous velocity of ith link Ii Inertial matrix of ith link, ωi Angular velocity of ith link x , y , z x,y and z accelerations

1. Introduction

Robots of current generation have been used in various fields isolated from the human society. They suffer major shortcomings because of their limited abilities for the manipulation and interaction with humans. Humanoid/biped robots are better suited for working in human environment and have a better degree of mobility, especially in environment with obstacles. The main motive behind the development of bipeds is its adaptability to human environment, so that there is no need to make special working environment for bipeds. Early studies on bipeds were mostly on its locomotion and not on its real industrial applications. Now it has reached the level of designing customized bipeds for specific applications. Still, there are issues yet to be addressed, among them the most basic being stable dynamic locomotion and gait synthesis. Bipeds can perform both Static and dynamic walking. In static walking, the complete system stays balanced by always keeping the centre of mass (COM) of the system vertically over the support polygon formed by the feet during locomotion [1]. In dynamic balance or walking the vertical projection of COM does not stay within the support polygon during the motion, i.e. during motion the COM may leave the support polygon for certain periods of time. Therefore some complicated and coordinated movement of other body parts only can balance the biped. This makes the dynamic walking more difficult from a design point of view. Location of COM and ZMP are the two important issues in biped locomotion. The concept of ZMP was put forward by M. Vukobratovic et al. [2] which revolutionized and accelerated the studies in dynamic walking of bipeds. ZMP is termed as the point on the ground about which the robot’s resultant moments at the ground is zero. This is used as a stability criterion for dynamic walking in this work. If the ZMP is inside the support area, the walking is considered dynamically stable, be-

Journal of Automation, Mobile Robotics & Intelligent Systems

cause the foot can control the robots posture. The ZMP criterion cannot be applied to biped robots that do not continuously keep at least one foot on the ground or to those which do not have active ankle joints. The motion of a humanoid comprises of time-functions of angular positions and velocities of the joint angles of the robot. The straight forward approach is to generate the joint time trajectories by solving inverse kinematics, to maintain the physical stability of the humanoid. It becomes increasingly difficult to compute the inverse kinematics as the DOF of the biped increases. However, such an approach is suitable for off-line generation of joint trajectories. Generation of low-energy gait is an open and nontrivial issue over a considerable period [3] during the motion of robot. This paper mainly concentrates on inverse kinematics and dynamics by using Levenberg-Marquardt algorithm (LMA) and Newton-Euler algorithm (NEA) respectively to analyze the stability of biped locomotion during dynamic walking. It also proposes a methodology to find the foot size for the smooth motion of joints. In almost all previous works related to humanoid walking analysis and synthesis, stability in sagittal and frontal planes are analyzed separately with the kinematic modelling based on the geometrical approach. DH parameters are used in the present work for the kinematic modelling and ZMP concept is used for the stability analysis in sagittal and frontal planes.To the best knowledge of the authors, no work based on LMA and DH modelling for analysis and synthesis has been reported in the area of humanoid or biped robots with the minimization of foot size. This paper is structured as follows. In section 2, kinematic, LM algorithm and dynamic modelling are described. The fundamental theory of the centre of mass (COM) and ZMP, in single and double support phases are given in section 3. Section 4 deals with the simulation of the gait trajectory for stepping motion with stability in both sagittal and frontal planes. Results and discussion are shown in section 5. Section 6 presents the concluding remarks with outlook.

2. Modelling of biped robot

Fig. 1 shows a 12 DOF biped robot modelled in SOLID WORKS which is having six DOF per leg, two at the ankle, one knee and three at the hip. The ankle joint of both legs have yaw and pitch motions, the knee is having only pitch motion and hip joints of both the legs have roll, pitch and yaw motions. The proposed model consists of seven links in order to approximate the locomotion characteristics similar to those of the lower extremities of the human body. The complete walking cycle consists of three single support phases (SSP) in which only one leg is on the ground while the other swings forward and four double support phases (DSP) in which both legs are on the ground. The stance leg in contact with the ground carries the whole weight of the robot. During the transition from single support phase to double support phase, swing leg decelerates to zero velocity. As a result of this, huge impact forces are developed at the contact phase for a short period of time. At the end of the DSP the swing leg accelerates which creates jerk in the joints and links of the robot. In DSP,

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Table 1. Biped mass and dimensions Mass Kg Link length

m0R

m1R

m2R

m2L

m1L

m0L

0.025

0.2

0.25

0.30

0.25

0.2

0.025

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0.042

0.098

0.090

0.060

0.090

0.098

0.042

Fig.1. Twelve DOF biped robot model the robot will be stable when the projection of COM stays within the support polygon. As a result of transition between the single support and double support phases, the problem of instability of the humanoid arises. The contact phase in walking is almost 20% of the total gait period [4]. As it is difficult to find out the reaction forces accurately, it is assumed that the impact of swing leg is perfectly inelastic while ensuring that no slippage occurs. Another important assumption made is that during the SSP, stance foot remains in flat contact with the ground. In SSP, the robot will be stable when the ZMP stays within the support foot polygon.

2.1. Kinematic modelling

Kinematic diagram of the 12 DOF biped is given in Fig. 2. Biped robots can be described kinematically by using joint-link DH parameters namely joint angle (qi), link length (ai), offset distance (di) and link twist (ai). Table 1 shows link dimensions of the biped robot. Four DH parameters corresponds to each link of biped are given in table II based on the frame assignment as shown in Fig. 3. Forward kinematics determines the pose of robot end effector as a function of its joint and link parameters where as the inverse kinematics gives the values of the joint variable corresponding to end effector or foot pose. 2.1.1. Inverse kinematics A suitable step length is assumed for the biped walking analysis from the idea that the step length for minimum energy consumption is about 60% of hip height [5]. Cartesian space trajectory is planned to get the trajectory of the swing foot which follows a cycloidal trajectory profile during the motion of robot [6]. As this profile is made by superposition of linear and sinusoidal function, Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

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tion and the steepest descent direction methods because it uses a search direction that is a cross between the GaussNewton direction and the steepest descent direction. Table 2. D-h parameter table Link(i) đ?&#x153;˝đ?&#x153;˝ đ?&#x2019;&#x2026;đ?&#x2019;&#x2026; đ?&#x2019;&#x201A;đ?&#x2019;&#x201A;

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it has a property of slow start, fast moving, and slow stop. This reduces the jerk during the start and end of walking. This characteristics can reduce the over burden at instantaneous high speed motion of the actuator. Points on the swing gait trajectory are taken as poses of the foot for getting the joint variables of each leg of the robot. Final pose matrix of the biped robot model 0T12 is equal to the pose of the foot on the swing gait trajectory. 2.2. Levenberg-Marquardt algorithm (LMA) The Levenberg-Marquardt method is a search method which gives the advantages of both Gauss-Newton direc-

Fig. 3. D-H modelling-Frame assignment for 12-dof biped 38

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Fig. 2. Kinematic diagram of biped robot

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LMA is used for finding inverse kinematic solution in this work. Solving inverse kinematics involves solution of twelve nonlinear equations with trigonometric functions. Six independent equations, three for orientation and three for position are to be solved. Since it is a bit laborious to do the inverse kinematics of the 12 DOF robot manually for the whole interpolated points, inverse kinematics is carried out in MATLAB and optimized results satisfying the boundary conditions are obtained. Levenberg-Marquardt iterative method is used for this purpose. It is a modification of Newton-Euler algorithm and gradient descent method. It is also called damped Gauss-Newton method, as it uses a damping factor to decide the accuracy level of solutions when the search approaches the minima. For starting, an initial guess is to be provided. An advantage of this method is that, the search direction is independent of the initial solution set given and it gives the actual minima even if the initial assumptions are far from the global minima. LM algorithm for the present context is explained below. Min F(q)=[F1 (q)â&#x20AC;ŚF12 (q)]; Sub to Ď&#x20AC;/4â&#x20AC;&#x2030;â&#x2030;¤â&#x20AC;&#x2030;qâ&#x20AC;&#x2030;â&#x2030;¤â&#x20AC;&#x2030;Ď&#x20AC;/4, Where, qâ&#x20AC;&#x2030;=â&#x20AC;&#x2030;[q1 q2 ...q12]â&#x20AC;&#x2030;T. The coefficient of the quadratic term of local Taylor series expansion of a function is, Yâ&#x20AC;&#x2030;=â&#x20AC;&#x2030;fâ&#x20AC;&#x2030;(qâ&#x20AC;&#x2030;+â&#x20AC;&#x2030;dq)â&#x20AC;&#x2030;â&#x2030;&#x2C6;â&#x20AC;&#x2030;fâ&#x20AC;&#x2030;(q)+ J(q)dq+dqâ&#x20AC;&#x2030;T H(q)dq. The convergence criteria is fâ&#x20AC;&#x2030;(qâ&#x20AC;&#x2030;+â&#x20AC;&#x2030;dqâ&#x20AC;&#x2030;)â&#x20AC;&#x2030;â&#x2030;&#x2C6;â&#x20AC;&#x2030;fâ&#x20AC;&#x2030;(qâ&#x20AC;&#x2030;); Therefore, J(q)dqâ&#x20AC;&#x2030;+â&#x20AC;&#x2030;dqâ&#x20AC;&#x2030;T H(q)dqâ&#x20AC;&#x2030;=â&#x20AC;&#x2030;0 dqâ&#x20AC;&#x2030;=â&#x20AC;&#x2030;â&#x20AC;&#x201C;H(q)â&#x20AC;&#x201C;1â&#x20AC;&#x2030;J(q). Modified Hessian is H(q,â&#x20AC;&#x2030;l)â&#x20AC;&#x2030;=â&#x20AC;&#x2030;2Jâ&#x20AC;&#x2030;T Jâ&#x20AC;&#x2030;+â&#x20AC;&#x2030;lI 1. Set damping factor l=â&#x20AC;&#x2030;0.001 2. Solve dqâ&#x20AC;&#x2030;â&#x20AC;&#x2030;=â&#x20AC;&#x2030;â&#x20AC;&#x201C;H(q,â&#x20AC;&#x2030;l)â&#x20AC;&#x201C;1g 3. If (qn+â&#x20AC;&#x2030;dqâ&#x20AC;&#x2030;)â&#x20AC;&#x2030;>â&#x20AC;&#x2030;fâ&#x20AC;&#x2030;(qn), increase lâ&#x20AC;&#x2030;(xÂ 10 say) and go toÂ 2. 4. Otherwise, decrease Îť(x 0.1 say), let qn+1â&#x20AC;&#x2030;= qn+â&#x20AC;&#x2030;dq, and go to 2. When qn the algorithm has converged set lâ&#x20AC;&#x2030;=â&#x20AC;&#x2030;0 and compute the final solution Where qn is the initial vector assumed and Hessian and Jacobian matrices, H(q)&J(q) are given in equation

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1. a & 1.b respectively.

( )

⋮

( )

⋮

( ) ⋯ ( )

⋱

⋯

⋮

( ) ( )

( )

(1.a)

(1.b)

( )

2.3. Dynamic modelling

The main challenges of gait planning are learning which includes the selection of specific initial conditions, constraints and their associated gait parameters [4]. In this section, different methodologies are adopted for dynamically constrained locomotion of biped robot in sagittal and frontal planes. From a control point of view, the inverse dynamics problem is of solving the joint torques from the joint angles along with their first and second order derivatives. In this work, we have used the Newton-Euler recursive algorithm for dynamic analysis. Since stance foot is assumed to be in flat contact, resultant ground reaction force/moment and torques can be computed using Newton-Euler algorithm [10]. The process consists of 2 iterations, (i) forward iteration to compute link velocities and accelerations and (ii) backward iteration to get the torque variation at joints. Initially velocity and acceleration of base frame is taken to be zero. While the stance leg is in motion no external forces are acting on it, except gravity loading. It is also assumed that the centroid of the link and the centre of mass of link coincide. A general dynamic model for biped walking related to the joint coordinates vector and joint torque vector without considering the friction and other disturbances is given below. ••

(

)

•

M (θ )θ + C θ , θ θ + G (θ ) = Dτ

(2)

Where, is the 12 x 12 inertia matrix, is the 12 x 12 coriolis and centrifugal matrix and is the 12 x 1gravity vector, is the 12 x 12 coefficient matrix of joint torques. Joint torques at different joints are determined through backward iteration by using the set of equation 3 and 4. t = i hiT i Ri -1 zˆ 0 ηi = Ri+1 i+1ηi+1 + iR0i-1Di x iRi+1 i+1fi+1+

(3)

(iR0i-1Di+iR0i –ri) x iFi + iNi

(4)

Where, i = 12,11…,1; ihi is the moment exerted on link i by link i–1 and iRi-1 is the orientation matrix, iFi is the total external force acting at the centre of mass of the link, iNi is the total external moment acting on link at its centre of mass, is ifi the force exerted on link i by link i R0i–1Di is the coordinates of the ith joint when referred to frame i and i r– is the centre of mass of link referred to frame i.

3. Zero moment point and gait trajectory

Static walking stability condition is sufficient to en-

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sure locomotion for very slow motion of biped robot. Some of the drawbacks of this technique of motion planning are the discrete nature of the motion of the robot and the time required for taking a single step being unusually long. It is not always necessary for the centre of mass (COM) of the robot to lie vertically above the base polygon. Another method of analyzing stability is based on the Zero moment Point criterion [6]. Zero Moment Point is defined as the point about which the moment of all the active forces acting on the robot turn out to be zero. In static gait planning problems, biped robot is stable if the projection of the centre of mass (COM) falls within the convex hull of the foot support polygon. In dynamic locomotion, link acceleration, inertial forces, and ground reaction force are also to be considered and the ZMP should be within the convex hull of the foot polygon for satisfying the stability criterion. The dynamic locomotion is highly nonlinear and difficult to analyze in real environment. The condition in the static and dynamic stability of the biped during the single support phase is the location of the ZMP must be inside the convex hull of the supporting foot. In double support phase ZMP or projection of COM should lie within the convex hull of support polygon formed by left and right foot. According to D’Alembert’s principle, if all forces are balanced, the motion of the biped is physically realizable. By D’Alembert’s principle the total forces and moments acting on the biped must be zero. This is given by:

Fr + Fe = 0 : Mr + Me = 0

(5)

Where is the ground reaction force, Fi is the total inertial and gravity force acting on biped, Mr is the reaction moment and Me is the inertial moment acting on biped. Let Fi be the inertial force, Mi be the inertial moment, and mi be the ith mass of the th segment (i = 1...n) . We have:

Fe =

i =1

∑ (−Fi − mig) =∑ −mi(v i + g) Me =

i =1

∑ Mi =∑ −

d (Ii wi ) dt

(6) (7)

Where, vi is the instantaneous velocity, Ii is the inertial matrix, wi is the instantaneous absolute angular velocity of ith link at its COM, terms are relative to the fixed reference coordinate, say O as in Fig. 4 [10]. The balancing problem of the biped system can be reduced at an assigned ground point (x’= 0, y’, z’) called the ZMP, where the resultant moment (M) at the ground plane is zero (Mx’= My’= 0).From the relation of the equivalent force moment, one obtains:

Fig. 4. Reference co-ordinate system for foot base Articles

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Fe =

∑ m (v + g) i

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(8)

i =1

∑

(9) ∑

∑

(10) ∑

(11)

Where F ′ and M ′ are the resultant force and moment at the ZMP (0, y′, z′) respectively, and (xi, yi, zi) is the vector from the origin O of the fixed reference coordinate O to the COM of link considered. x, y, z, are the corresponding components of accelerations in respective directions. From the above equations, one obtains: Xzmp = 0

(12) (13)

Fig. 5. Cycloid curve

4. Simulation

Numerical simulation of 12 DOF biped walking is done using MATLAB™.3-D kinematic pattern of the biped for a single step is shown in Fig. 6. Variations of kinematic and dynamic parameters and ZMP are plotted. Initially both legs of biped are in stance position then the left leg is stepping a length of 10 cm in 1 s. Simulations are carried on a biped robot having hip height of 25 cm and mass of 1.7 kg. Kinematic and dynamic modellings help to synthesis and analyze the biped robot at different scaled dimensions based on stability.

(14) The constraint on the dynamic motion of the biped during the single-support phase is the location of the ZMP (0, Yzmp, Zzmp ) must be inside the convex hull of the supporting foot. In the single-support phase the stable convex hull is same as the area occupied by the supporting leg on the ground. Therefore, Zmin< Zzmp< Zmax and Ymin< Yzmp< Ymax where, we assume that the supporting foot is rectangular, parallel to the fixed reference coordinate O, and between points (0, Ymin ,Ymax) and (0, Zmin ,Zmax). Mathematical interpolation is one of the simplest methods for providing suitable gait trajectory in accordance to the given boundary conditions. Cartesian space trajectory planning is carried out to get the trajectory of the swing foot. Generally human’s ankle joint motion trajectory is a cycloidal profile in normal walking (Kurematsu, Kitamura & Kondo, 1988) Cycloidal profile reduces effects of sudden acceleration at the beginning and deceleration at the end during the gait generation. Hence the cycloidal profile is used for the trajectories of the swing foot. As this profile is made by superposition of linear and sinusoidal function, it has a property of slow start, fast moving, and slow stop. This avoids the jerk that can happen during the start and end of walking. This characteristics can reduce the over burden at instantaneous high speed motion of the actuator. Equation of a cycloid in parametric form for selecting break points on the trajectory is given in equations 15 and 16. Gait trajectory pattern is shown in Fig. 5. xi= SL (ji– sin ji)/2p

(15)

zi= SL (1– cos ji)/2p

(16)

Where i = 0,1...N, the number of poses of foot on the trajectory and SL is the step length. 40

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Fig. 6. 3-D Biped walking pattern

5. Results and discussion

Stable gait generation of a 12 DOF biped robot is demonstrated in this paper. Variations in parameters like joint angles, link velocities and link acceleration are plotted during the stable motion of biped. Torque and ZMP variations are also analyzed here. The variation of joint angles at ankle, knee and hip for right and left leg are varying smoothly and continuously for a single step as shown in Fig. 7 and this assures a smooth transition of the robots motion. Rolling angular variations at the ankle joint of the left leg and hip joint pitch angular variations (4th and 9th joints)of both legs are high compared to other joint angle variations. Because these two joints plays vital role in the stability of biped motion in this analysis. Initially, when the left leg is about to lift, both hip and knee joints should have some angular variations for bringing the COM within the support foot polygon. Link velocities and accelerations at the COM are given in Fig. 8 and Fig. 9, respectively. First link is fixed at the ground during the walking so the velocity and accelerations are zero. Velocity is maximum for the swing foot (link 12) and minimum for the lower part (link 2) of the stance leg. All other links the velocities are varying approximately in between 0 and 25 cm/s.

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Fig.7. Joint angle variations of left and right leg

Fig. 8. Link linear velocity Variations in accelerations are smooth but there are up and downs because the biped is moving in high speeds with step length of 10 cm. There are some values of accelerations at the beginning and end of the gait trajectory so that the jerk will be the minimum at these two locations. Up and downs of velocities result in irregular variations in the accelerations as shown in Fig. 9. This creates jerk at the joints and links of the robot at intermediate positions and biped can mostly be suited at slow speeds for small step lengths and moderate speed at higher step lengths. Jerk will be reduced for higher step lengths in moderate speeds but stability will be achieved with larger foot size. This will be clear from the ZMP variations plotted in Fig. 11. It is clear from the velocity and acceleration diagrams that the velocity and acceleration variations are same for link 3, 4 and 5. Similarly velocity and acceleration variations are same for links 6, 7 and 8. This is because of the

assumption that the joints 4th, 5th and 6th are at the same origin and also the joints 7th, 8th and 9th are at the same origin in modelling. Fig.10 shows the continuous variations of torque for all joints. Starting torque for the first joint is high because this joint is only making the robot walk by swinging the whole system in the frontal plane. Geared motor can be used for getting high torque at joint 1. Torque is smallest for the ankle joint of the swing foot. By changing various kinematic and dynamic parameters it is possible to bring the ZMP within the limited size of Y-Z plane for attaining stable walking, and variations are plotted against the stepping time as shown in Fig. 11. Variation is more in Z direction compared to because the Z component of acceleration has an more effect on shifting of ZMP. The inertia components are small here due to the small size of biped. However those inertial Articles

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Fig. 9. Link linear acceleration

Fig. 10. Joint torque 42

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Fig. 11. ZMP variation in first walking phase

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terms will not be negligible in case of fast bipedal activities like running and jumping, or when the link masses and dimensions are comparable to those of the actuators. There will not be any difference if we neglect the inertial effects in slow motion. ZMP moves in Y-Z plane approximately in a parabolic path within the foot base. The maximum approximate range of Yzmp and Zzmp are -2.4 cm to 0.7 cm and -2 cm to 3 cm respectively. Fig.1 1 shows the movement of ZMP on the foot of the stance leg. This plot gives the feasible size of foot of stance leg for a particular step length. The resultant values of ZMP variations are represented graphically for step length of 0 to 20 cm in Fig. 12. This helps to decide the foot size for a range of step length based on kinematic and dynamic constraints. As per the Fig.â&#x20AC;&#x2030;12, foot size of 10 cm x 10 cm is required for biped walking through a cycloidal gait for a step length of 20 cm.

Fig. 14. Snapshots of biped walking

Fig. 12. ZMP Vs Step length

6. Experimentation

Walking gait generation is simulated and the results involving the relevant variables are analysed in the previous section. In this section, gait generated for a step length of 10 cm is experimentally validated in a 12 DOF biped. The experimental validation is done by matching simulated cycloidal trajectory with real time gait trajectory. Validation can also be done experimentally by evaluating and comparing ZMP variations along with the gait trajectory. Computer / Processor is interfaced with the biped robot through a mini maestro 12 channel servocontroller for controllling actuators for the required cycloidal trajectory. ProcessorJoint Angles

Servo Contro ller

Biped Robo t

ated real time gait trajectory is compared with the cycloidal gait trajectory determined from the simulation result. Robot stable motion and Real time gait trajectory are shown in Fig. 15 and Fig. 16 ,respectively. Joint angles are fed to the biped for getting the cycloidal trajectory with a fixed step length. One of the experimental real time cycloidal trajectory is given in Fig. 16. Experiment is conducted five times for the same joint angles and steplength. Average step length obtained in the real time gait genaration is approximately 10.3 cm instead of 10 cm. A cycloid is constructed corresponding to the step length of 10 cm and its calibrated image is superimposed on the plot of 19 instataneous poses of swing foot. The dots in Fig. 15 are the instantaneous poses obtained during the experimentation.

Cycloidal Gait

Fig. 13. Data Flow diagram The block diagram shown in Fig. 13 depicts the details of data flow for testing and validation of bipedal gait. Joint angles corresponding to a single step swing foot trajectory is determined using MATLAB and the signals are sent to the biped for the required motion. Fig 14 shows the snapshots of 12 DOF biped robot walking for a step length of 10cm. Instant motions are captured for the gait analysis during the foot step movement. Evalu-

Fig. 15. Points on the stable gait trajectory Articles

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form. This research was supported by National Institute of Technology Calicut, Kerala under the provision of Faculty Research Grant (FRGph02/07/03-04). This support is gratefully acknowledged.

AUTHORS

Sudheer.A.P* – Assistant Professor in the Department of Mechanical Engineering, National Institute of Technology Calicut, Kerala, India, 670601. E-Mails: apsudheer@nitc. ac.in,apsudheer@rediffmail.com

Fig. 16 Real time gait trajectory Fig. 16 depicts the variations of points on the real time stable gait trajectory with the theoretical cycloidal gait. In this particular real time gait trajectory the step length obtained is approximately 10.1 cm instead of 10 cm. This Analysis shows the correctness of modelling and gait trajectory of the 12 DOF biped robot.

7. Conclusion and outlook

Stability analysis of a twelve dof biped robot in the sagittal and frontal plane for a cycloidal gait is presented in this paper. Generation of a gait for the stable walking based on zero moment point within a particular foot size is attempted. The inverse kinematic solutions are found by using Levenberg-Marquardt iterative method. Motion of the robot is constrained because of the limited number of DOF. The present synthesis and analysis gives idea of foot size for stable biped walking. Experimentation for determining the real time gait trajectory is attempted and trajectory is compared with theoretical trajectory. This experimental result authenticates the suitability of the model for the synthesis and analysis of biped robot. At present researchers are trying to minimize the foot size to avoid self collision and flexibility with higher level of stability during walking. This work gives a clear direction for generating an optimum gait trajectory based on ZMP with minimum foot size. It is expected that this will lead to optimization of the foot size by considering all the kinematic and dynamic constraints for achieving better stability. It may be considered that the synthesis and analysis procedure can be refined in many ways: some of them being, optimum smooth gait planning with minimum energy consumption using traditional and soft computing techniques by changing the walking parameters like step lengths, stepping time and height of the trajectory. Simulation can be extended to various structured and unstructured environments. Authors are attempting to verify experimental results by evaluating and comparing ZMP variations along with the gait trajectory in a complete walking cycle.

Acknowledgements

The authors are thankful to the referees and the editor for their constructive suggestions and comments which have immensely helped to bring this paper to the present 44

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R.Vijayakumar – Professor in the Department of Mechanical Engineering, National Institute of Technology Calicut, Kerala, India. K.P. Mohandas – retired Professor, Electrical Engineering, National Institute of Technology Calicut, Kerala, India. *Corresponding author

References

[1] M. Raibert et al., “Legged robots that balance”. MIT press Cambridge, MA, 1986. [2] M. Vukobratovic and B. Borovac, “Zero-moment point-thirty five years of its life, ”International Journal of Humanoid Robotics, vol. 1, no. 1, 2004, pp. 157–173. [3] P. Vadakkepat and D. Goswami, “Biped locomotion: stability, analysis and control,”Robotica, vol. 27, no. 1, 2009, pp. 355–365. [4] T. Zielinska, C. Chew, P. Kryczka, and T. Jargilo, “Robot gait synthesis using the scheme of human motions skills development”, Mechanism and Machine Theory, vol. 44, no. 3, 2009, pp. 541–558. [5] F. Silva, T. Machado et al., “Energy analysis during biped walking”. In: Robotics and Automation, Proceedings of IEEE International Conference, vol. 1. IEEE, 1999, pp. 59–64. [6] Z. Tang, C. Zhou, and Z. Sun, “Trajectory planning for smooth transition of a biped robot”. In: Proceedings of IEEE International Conference on Robotics and Automation, ICRA 2003, vol. 2, IEEE, 2003, pp. 2455–2460. [7] M. Vukobratovic, D. Andric; B. Borovac, ”How to achieve various gait patterns from single nominal”, International Journal of Advanced Robotic Systems, vol. 1, no. 3, 2004, pp. 99–108. [8] A. Takanishi, M. Ishida, Y. Yamazaki, and I. Kato, “The realization of dynamic walking by the biped walking robot WL-10RD”. In ICAR’85, 1985, vol.1, pp. 459–466. [9] H. Miura and I. Shimoyama, “Dynamic walk of a biped,” International Journal of Robotics Research, vol. 3, no. 2, 1984, pp. 60–74. [10] W. Spong, M. Vidyasagar, Robot dynamics and control, John Wiley& Sons, New York, 1989.

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Intelligent Utilization of Waste Of Electrical and Electronic Equipment (Weee) with Robotized Tool Submitted: 18th May 2012; accepted: 30th August 2012

Jakub SzaĹ&#x201A;atkiewicz, Roman Szewczyk

Abstract:

The article examines the solution of intelligent robotic utilization of electrical waste and electronic equipment (WEEE), as means of materials recovery. The paper provides criteria for selection, identification and waste analysis, enabling application of robotic dismantling solutions. Based on proposed criteria, the waste of computer hard disk drives (HDD) was identified, analyzed, and their characteristics described. Furthermore applying described approach to robotized dismantling, the complete process of waste HDD processing, covering all the procedural steps was presented for the selected disk. Described intelligent utilization process can be performed with use of 2 robots equipped with, 2 tools and one conveyor handle. Authors also briefly presents issues related to proposed process approached during HDD waste disassembly studies. In addition, paper presents the quantities of materials to be recovered from the HDD waste. Keywords: WEEE, robotic, dismantling, disassembly, waste, recycling, recovery of resources, HDD.

1. Introduction Waste of electrical and electronic equipment (WEEE) is a global concern. In the 27 EU countries it is estimated that the weight of produced waste WEEE in 2005 was 8.3-9.1 million Mg (tones), 25% of which is collected and processed, while remaining 75% is not registered and does not occur in collection points [1, 2]. Such state of waste management system can be caused by lack of processing capacities and suitable technologies which can utilize WEEE effectively. The amount of WEEE rises continuously [3, 4]: in 2008 Sweden collects 16.7 kg per person of WEEE, Britain 8.2 kg per person, Austria 6.5 kg per person [5]. Moreover, European Commission proposes rising collection targets from 4 kg per person to 65% [6] of average mass of electrical and electronic equipment placed on market. WEEE has to be utilized, but it also can become a source of valuable resources. The factors described above reveal the need for development of new ways to process WEEE effectively allowing recovery of valuable raw materials. It is believed that intelligent robotized disassembly can be the technology that will take a part in solving WEEE processing problem.

2. Currently used technologies for processing WEEE waste There are two main WEEE waste treatment methods: manual dismantling, and mechanical methods based on shredding, and multistage separation of materials. Man-

ual dismantling and mechanical processing approaches differs between each other, on the degree of recovery of raw materials from WEEE waste. Recovery ratios are presented in Table 1. Table 1. Recycling methods recovery ratio [7] Washing Machine

Oven

Shredding

44.1%

74.9%

Manual system

95.6%

90.6%

Manual dismantling is the most flexible way to process wide range of different electrical and electronic equipment waste as well as have the highest recovery rate of raw materials. However, manual dismantling is very extensive and requires direct human contact with waste. Manual dismantling is based on removing the components from the devices and theirs segregation accordingly to the materials they are made of. Often manual removal is the first stage in the process of mechanized waste treatment technology (shredding), to extract the hazardous substances and components which cannot be processed together. Mechanical treatment of waste is based on shredding process, after which, shredded residue is separated in multistage process to obtain rich fractions of resources. This method is useful to process large quantities of mixed WEEE waste, but its disadvantages are: high energy demand associated with the shredding of waste, lower level of resources recovery and impurity of recovered raw materials. Nowadays, works are being carried out to robotize WEEE waste processing methods, as an alternative to existing processes, and as supporting solutions for others existing ones. The example of such works is modeling of the dismantling line to utilize LCD monitors [8]. Another type of work being carried out is to robotize dismantling process of desktop computers [9]. In literature are also presented works covering prototype automation solutions replacing certain activities in existing WEEE processing plants i.e. Automatic unscrewing of extracted washing machines engines, transporting released parts using robotic arms equipped with grippers, etc. [10]. Intelligent robotized dismantling is the alternative to traditional technologies, (shredding and manual processing). It combines the advantages of traditional technologies, theirs capability to adapt to various wastes, with large processing capacity. Intelligent robotized utilization allows for obtaining high purity of raw materials recovered from waste, with minimal energy input and no human labor (no human contact with waste). Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

3. Overview of intelligent robotized utilization technology Robotics applications in waste dismantling are based on well known and applied in modern industry robotics and automation solutions. Intelligent robotized dismantling can be described, as reversed process of robotized manufacturing, or as replacing human labor by robot in repeatable operations during dismantling process. Knowing the dismantling steps, actions and procedures performed during manual WEEE processing, it is possible to replace them by robotized tools. Complete process can be organized in form of demanufacturing line, equipped with robotized tools. On this line, step by step performed are successive actions of dismantling process and resources are being recovered. Due to the speed of robots, repeatability and positioning precision, it is possible that robotized disassembly unit will be small, and it will process large quantities of waste. The main advantage of robotized disassembly is its energy efficient method used to separate screwed together components. Energy efficiency is obtained by: applying automated screwdrivers controlled by robot arm to free screwed together parts, and robot arm grippers to remove freed parts, and carry them to right container. Unscrewing and carrying operation, are the least energy consuming actions, so they offer great energy savings in WEEE processing, combined with obtaining the highest possible purity of recovered resources. Such approach differs from shredding method that consumes huge amounts of energy, required for cutting metals and plastics into pieces. Unfortunately due to extreme variety of WEEE not every waste can be processed in robotic disassembly unit. That is why it is fundamental to arrange ways allowing categorizing which of WEEE can be processed in this particular way. Moreover there are two approaches to robotic disassembly of WEEE that differ due to the level of theirs complexity.

4. Complexity issue in robotized disassembly – opposite approaches The robotized disassembly can be performed in two ways: fully programmed, determined algorithm of operations, and on the other hand, cognitive algorithms, organizing the operations accordingly to the data collected from sensors. Applications of cognitive algorithms, self adapting to the data collected from sensors, is very complex and multidimensional. Self adapting approach requires advanced sensors, operations on collected data and analysis of possible operations to be performed this makes it difficult and expensive. Due to such high application requirements, dynamic approach is not analyzed in this paper. However in future it is possible that this complex approach will may occur effective in mixed waste processing, or as a part of other systems. On the other hand intelligent robotized disassembly can be based on determined algorithms, allowing effective and simple application of this technology in WEEE processing plants. Each step, in the disassembly process, is strictly programmed and is performed in its time, embedded in the chain of disassembly program. However this approach can be only applied to uniform wastes, where once programmed operations can be performed on large volume of devices, leading to the same constant and predictable effect. Determined algorithms simplify the complexity of robotic solutions, decreasing the necessary data collection to minimum. Disassembly algorithm is in form of basic pre46

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programmed, repeatable operations, in a result of which appliance disassembly is performed. Once programmed disassembly unit, will carry out the program on any number of identical devices. However due to wide variety of WEEE, even in categories of waste i.e. IT equipment – laptops, routers, differ between models and versions, so it is necessary to equip disassembly unit in optical recognition system (bar code reader, or camera) to identify its brand, model, version to choose the right program for identified device. The drawback of simplicity of robotic disassembly based on determined algorithms, is that the disassembly unit is designed for specific waste (certain tools are used), so only similar wastes can be processed. That is why it is not possible to process i.e. LCD, laptops and ovens on one robotic disassembly unit. However, it is possible to process i.e. all models of all manufactures, of hard disk drives. Identified specificity, of the intelligent robotic disassembly approach, based on determined algorithms, requires methods of selection and analysis, which of the range of WEEE can qualify, and is reasonable to process in this intelligent technology.

5. Criteria for identification and selection of waste qualifying for intelligent robotized utilization, based on determined algorithms Not every waste can be processed in robotic disassembly unit, due to its characteristics and other factors that can limit applications of this method. To identify WEEE qualifying for being processed in intelligent robotic disassembly technology, certain criteria were developed and verified. The criteria are: – Physical characteristic: similar construction, standardized external dimensions, standardized placement of mounting holes, construction easy to dismantle, small amount of homogenous parts, good physical shape, no deformations, no dirt, way of identification i.e. by barcode. – Market factors: life cycle of the product on the market, quantity and mass on the market, upward trend in sells. – Resources to be recovered: amount of valuable/dangerous resources, which recovery is economically reasonable/ necessary (law). – Environmental impact: decrease of amount of waste, recovery of the resources decreases use of natural resources. – Additional criteria: the ease of extraction of identified item from collected WEEE, currently ineffective recycling

Figure 1. Example of computer Hard Disk Drive (HDD) 3.5” view from electronics side

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methods i.e. high energy consumption, low recovery rate, contamination of recover resources. Applying above criteria to WEEE, several types of wastes have been identified and qualified for intelligent robotic utilization method. Detailed analysis was carried out on computer hard disk drive, which was chosen, as a waste appliance that meets all above criteria (Figure 1).

6. Characteristics of hard drives as a waste eligible for processing in the intelligent robotized utilization technology Hard disk drives can be found in computers and other devices, like IT or broadcasting equipment, often HDD’s are found alone in waste, due to rapid obsolescence of IT solutions. Given the number of hard drives sold worldwide in 2009 (550 million units), estimated mass of waste of HDD every year is 280 000 Mg [11]. The mass of HDD waste is arising annually, and is significant enough to justify materials recovery from it and search for the new technology to process this appliance waste alone. Computer hard drive has standardized external dimensions (few models such as 2.5”, 3.5”). The design is simple and based on uniform chassis in which components are installed. Extraction of the hard drives from computers is a standard procedure in many companies that process WEEE. Their physical condition is good - without deformation. Also important parameter is what kind of materials can be recovered from HDD waste. In the case of hard drives mainly they are: aluminum, ferromagnetic metals (and magnets), stainless metals and other mixed scrap (Table 2).

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and only partial dismantling will take place i.e. to remove dangerous substances, covers, and other easy to remove parts, as the pre-treatment of waste method. Some of the raw materials will be recovered, overall mass decreased, labor saved, and energy demand for further processing decreased. Both variants, full, and partial disassembly, are described on example of computer hard drives: Partial dismantling – is the removal of the external screws and external electronics, as well as the closure of the disk, followed by several internal bolts. It is possible to carry it out using two tools – a screwdriver, and a gripper. As a result of the partial dismantling, 8% weight reduction was achieved (removed were: bolts, electronics, and closure of the disk). In addition, the source of heavy metals and organic matter – (electronics) is removed. This situation is presented in Figure 4. Full dismantling – is the removal of every component from the HDD chassis. This procedure allows reducing the mass of waste HDD by 87% and recovering pure resources like aluminum, stainless, and ferromagnetic metals. The rest, 13% or the input mass is not processed, this “other” components group contains: plastic (2%), disk heads (3%), motor (8%). All components extracted from disk are shown in the Figure 2.

Table 2. Average mass and groups of resources in hard disk drives 3.5” [9] Type of material

Average mass of mateContent % in rial in 3,5” HDD HDD mass Mass (g)

Overall HDD

515

100,0

Aluminum

264

51,3

Ferromagnetic metals

17,5

Stainless metals

10,3

External electronics

7,9

Other components

12,8

Literature shows that, research is performed to robotize dismantling of computer [12] components. This will allow in future for integration of robotized computers disassembly, with intelligent robotized utilization of every component, directly recovering raw materials in one process.

7. Complete and partial disassembly of WEEE Performed research showed [9] that the dismantling of waste can be performed in two ways, full and partial disassembly of appliance. The level of dismantling depends on quantity and purity of recovered raw materials, and on the other hand is limited by its difficulty. In some cases the complexity of disassembly of appliance will be too high for its complete robotic processing,

Figure 2. Fully disassembled hard disk drive

8. Intelligent robotized hard disk drive utilization – process simulation based on determined algorithm Robotized process organization example, is presented on Western Digital Caviar WD400JB-00ENA0, hard disk drive manufactured in 2003 with a capacity of 40 GB. The block diagram with dismantling steps and percentage of resources recovered from waste is presented in Table 3 with described steps of disassembly process. Hard drives, extracted from computers, are introduced into the disassembly line directly on the conveyor belt. HDD’s are being queued and positioned on it, to allow each disk to be grabbed by the handle, enabling the spatial manipulation (1st step). Construction of hard drives is based on the aluminum chassis fitted with standardized threaded holes for mounting the drives. In the case of the disassembly those holes can be used to hold the disks in the handle and manipulate them. Placement of the holes in the chassis is shown in Figure 3. Next step (2nd step) is identification of the hard drive. Model of the HDD is identified by reading barcode or other OCR method. On this basis program is selected and Articles

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uploaded. In the case of absence of the program for a given drive, or the inability to read the model, the disk is forwarded to the interim storage.

Figure 4. Exposed components after disassembly of cover plate

Figure 3. Placement of mounting holes in chassis of disk – pointing by arrows Immobilized in the handle hard disk is transported to the robots operation area, where the first stage of dismantling is being performed (3rd step). Threaded joints are disconnected by first robot equipped with automatic screwdriver. This operation frees electronics. Second robot equipped with gripper removes electronic board from the drive and moves it into the container collecting those parts. Next, the drive is turned 180 degrees around in the holder and in the same manner, the removal of the top plate is executed. Threaded connections of the cover to the chassis are disconnected and the cover is transferred to the container collecting aluminum. Exposed components after disassembly of cover plate are shown in Figure 4. Following, joints are unscrewed inside the disk (4th step), holding magnetic shields with magnets on them (Figure 5), than plates on the rotor, data 1’st step: connector and finally heads arm. Introduction of waste to In the case of plates mounted on robotized disassembly unit 100% - of waste mass the impeller it is necessary to immobilize the motor so the screws can be localized and unscrewed. This operation can be impleMaterials recoverd: mented using the gripper from „Printed circuit boards” the second robot arm. However, Electronics 8% - of waste mass pinpointing the exact location of the bolts is only possible via optical system or proximity sensors Materials recoverd: „Ferromagnetic metals” in which the disassembly unit magnetic shields, has to be equipped. Another elemagnets, closure 17% - of waste mass ment to disassemble is the disk head, this part is mounted using other type of screw connection, therefore it is necessary to replace the tip of the screwdriver Materials recoverd: „Stainless metals” on robot arm to perform this opbolts, data plates hold eration. The last operation is to 10% - of waste mass dismantle the motor. During each unscrewing operation, second robot equipped with gripper, simultaneously picks and moves released components to right container, collecting each 48

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Figure 5. HDD during components disassembly – arrows pointing glued magnets

2’nd step - Identification: Of waste model and upload of disassembly program 100% - of waste mass

3’rd step - Dissasembly: Removal of external parts: External bolts holding electronics, closure plate, and theirs transport to right containers

Table 3. Block diagram of disassembly process, with percentage of raw materials recovered from HDD’s. Blue fields represent each step of disassembly process. Green fields represents type of materials recovered. Materials recoverd: „Other parts” Plastics, heads arm, motor, data connector, other 13% - of waste mass

4’th step - Dissasembly Removal of internal parts: bolts, magnetic shields, heads arm, data plates, motor, and theirs transport to right containers Materials recoverd: „Aluminum” Chassis, closure, data plates 51% - of waste mass 5’th step Releasing of empty chassis from conveyor handle to container collecting aluminum

Journal of Automation, Mobile Robotics & Intelligent Systems

raw material. After disassembly of all components from the disk, its empty chassis made of aluminum, is released from the handle above the container collecting aluminum components (5th step). This operation finishes the process of intelligent utilization of the hard disk. All dismantled components from hard disk drive are presented in Figure 2. Table 3 presents block diagram of the process described above, together with the participation of recovered materials [11]. Described investigation of disassembly process of HDD, covers all identified standard procedures and proposes solutions for process design. However, not all computer hard drives are made in the same way, some components are assembled by negative allowance or glued together, also some drives are sealed by aluminum seal tape etc., those non-standard construction solutions used in hard disk drives, complicate and require introduction of additional tools to disassembly process. In the case of uncommon, complex and uneconomical to process lots of drives it is possible to carry out a partial disassembly of them, and forward them to another process that can handle the rest of the process i.e. manual dismantling. Currently manufactures of many appliances are foreseeing the utilization issue, at the end of life of theirs products, and the products are design much simpler than it was in the past. This approach brings savings during production, and utilization. The general trend is, the newer the appliance the less complicated its construction and more uniform each type of waste collected.

9. Conclusion Presented in the article approach of identification of waste eligible for processing in technology of intelligent robotized utilization by determinate algorithms, allows recognition of fields of application of this technology, to provide significant reduction of cost in waste processing. Based on presented case of HDD intelligent utilization steps, whole process can be designed and calculated for research trails and testing purposes. HDD case shows that it is possible to implement whole disassembly process using: conveyor handle, and two tools mounted on two robotic arms. Intelligent robotized WEEE utilization technology is new and effective way to process this group of waste. Its high efficiency and speed, allows to process large volumes of selected groups of wastes, recovery of valuable and high purity resources, and to avoid human contact with waste. Estimated on example of robotic disassembly of hard disks, based on the volume of the world production of HDD’s (550 million units = mass of 280 000 Mg), it is possible to recover: 143 600 Mg of aluminum, 49 000 Mg ferromagnetic metals, 28 800 Mg stainless metals, 35 800 Mg of mixed metals, and 22 100 Mg of printed circuit boards, each year. With additional treatment of extracted components, it is possible to recover raw materials like rare-earth metals and certain quantities of precious and heavy metals embedded in released parts. This allows for almost 100% level of recovery in the case of presented hard disk drives. The authors believe that the use of intelligent robotized utilization technology, on industrial scale is reasonable. It is believed that robotized disassembly is also the future of the waste processing industry, and a unique opportunity for companies offering robotic solutions to arise on the markets of WEEE processing.

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Acknowledgement

Works were financed of research project, Ministry of Science and Higher Education no. N R03 0083 10.

AUTHORS

Jakub Szalatkiewicz* – Industrial Research Institute for Automation and Measurements, Warsaw, PL 02-486, Poland, jszalatkiewicz@piap.pl, jakub.szalatkiewicz@gmail.com Roman Szewczyk – Industrial Research Institute for Automation and Measurements, Warsaw, Warsaw, PL 02-486, Poland, rszewczyk@piap.pl *Corresponding author

References [1] J. Huisman, “2008 Review of Directive 2002/96 on WEEE, Final Report”, United Nations University. [2] M. Cobbing, “Toxic Tech: Not In Our Backyard”, Greenpeace.org, 2008. [3] J. Lee, H. Song, J. Yoo, “Present status of the recycling of waste electrical and electronic equipment in Korea”, Resources Conservation & Recycling, no. 50, 2007, pp. 380 –397. [4] H. Kang, J. Schoenung, “Electronic waste recycling: A review of U.S. infrastructure and technology options”, Resources Conservation & Recycling, no. 45, 2005, pp. 368– 400. [5] R. Wawrzonek, „Praktyczne aspekty funkcjonowania systemu gospodarowania zużytym sprzętem elektrycznym i elektronicznym”, Elektro Eco, 2009. (in Polish) [6] “Environment Commission proposes revised laws on recycling and use of hazardous substances in electrical and electronic equipment”, http://europa.eu/rapid/ pressReleasesAction.do?reference=IP/08/1878&form at=HTML&aged=1&language=EN&guiLanguage=en [7] T. Yuksel, I. Baylakoglu, “Recycling of Electrical and Electronic Equipment, Benchmarking of Disassembly Methods and Cost Analysis”. In: Proceedings of the 2007 IEEE International Symposium on Electronics & the Environment, Orlando, pp. 222 – 226. [8] H. Kim, S. Kernbaum, G. Seliger, “Emulation-based control of a disassembly system for LCD monitors”, Int. J. Adv. Manufacturing Technologies, no. 40, 2009, pp. 383 – 392. [9] F. Torres, P. Gil, S. Puente, J. Pomares, R. Aracil, “Automatic PC disassembly for component recovery”, Int. J. Adv. Manufacturing Technologies, no. 23, 2004, pp. 39 – 46. [10] B. Basdere, G. Seliger, “Disassembly Factories for Electrical and Electronic Products To Recover Resources in Product and Material Cycles”, Environmental Science & Technologies, vol. 37, 2003, pp. 5354 – 5362. [11] J. Szałatkiewicz, “Resources recovery from computers hard disk drivers”, Inżynieria Ekologiczna, no. 23, 2010, p. 77–87 (in Polish). [12] L. Ching-Hwa, C. Chang-Tang, F. Kuo-Shuh, C. TienChin, “An overview of recycling and treatment of scrap computers”, Journal of Hazardous Material B114, 2004, pp. 93 –100. Articles

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Influence of PWM on Trajectory Accuracy in Mobile Robot MOTION Submitted: 30th March 2012; accepted: 12th June 2012

Ryszard Beniak, Tomasz Pyka

Abstract:

The paper compares simulation results for direct and PWM control of DC motors in a tri-wheel mobile robot with a castor sliding wheel. Our aim was to determine to what extent PWM control changes the trajectory accuracy. For this purpose, we compare kinematic and dynamic control. To make the model more realistic, we considered the impact of viscous and rolling friction of driving wheels on the motion along the trajectory. We conclude that dynamic control is of higher quality as compared to kinematic control, and that there is a significant impact of PWM control on the trajectory accuracy.

2. Mobile robot model

Simulations were carried out for the model presented in Fig. 1:

Keywords: mobile robot control, kinematic and dynamic control, castor sliding wheel, pulse width modulation.

1. Introduction

The issue of trajectory accuracy has not been yet resolved for mobile robots with to a satisfactory extent. The regulator is incapable to determine the trajectory accurately enough given surface irregularities impacting the behaviour of wheels. For this reason, the theoretical and empirical trajectories might deviate from each other significantly. The impact of slips on the trajectory has been considered in [1, 2, 3, 4] for tri-wheel robots, in [5] for four-wheel robots and in [6] for unicycles. Not considering wheel slips might lead to significant trajectory errors [7, 8, 9, 10]. Yet, for simulation of trajectory of tri-wheel robots, the castor free wheel is assumed to be passive [11] and not to have any impact on the trajectory. In this paper, we consider a tri-wheel robot with a castor sliding wheel [12]. The innovation is that the castor wheel does exert an impact on the entire mobile robot. Viscous and rolling frictions do influence the motion of the castor wheel. Friction coefficients are assumed to have normal distribution N(0.001, 0.00058). The simulation was carried out for an example trajectory as in [12] rotation around a fixed axis (Fig. 5), i.e. around a characteristic point A (Fig.1). For simulation purposes, we assumed the parameters of a Dunkermotoren GR63x25 motor with a PLG52 1:36 transmission. Next, we compared the kinematic and dynamic control for both direct (without any modulation) and PWM control types [13]. [6] contains a comparison between kinematic and dynamic control, though without any restriction imposed on torques and forces. Kinematic [14] and dynamic control are still applied in robotics [10], though they are increasingly often combined with adaptive [11] and robust control. 50

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Fig. 1. Tree-wheeled robot with a castor rear wheel where S – center of frame mass, A – characteristic point, l – distance between the characteristic point and arm of the castor wheel, l1 – distance from characteristic point to driving wheel, l2 – distance between center of frame mass and the characteristic point, l3 – length of the free wheel arm, β – rotary angle of the frame, φ – angle of the castor wheel along the axis y3, α1, α2, α3 – rotary angles of individual wheels along axis z1, z2 and z3, respectively. Mobile robot dynamic equation is described as follows:

+ C(q, q )q = B(q )τ + JT (q )λ , M (q )q

(1)

where: T =  β ψ ϕ α 3  , q q T =  β ψ ϕ α 3  ,

 a11 a M =  21  a31  0

a12 a22 a32 0

a13 a23 a33 0

0  c11 c  0 21 , C=   c 0 31   0 0 

c12 c22 c32 0

c13 c23 c33 0

0 0  , 0  c44 

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The elements of matrix M are described as:

a11 = m3 (l 2 + l32 + 2l3l cos(ϕ )) + I y 3 + 2h12 (m1r12 + I z1 ) + 2 I y1 + m5l22 + I z 5 , a12 = a21 = −m3 r1l3 sin(ϕ ), a23 = a32 = m3 r1l3 sin(ϕ ), a33 = m3l32 + I y 3 , a22 = m3 r12 + 2m1r12 + 2 I z1 + m5 r12 , a13 = a31 = −m3 (l32 + l3l cos(ϕ )) − I y 3 . The elements of C matrix are calculated as follows: c11 = m3 (−3ϕ l3l sin(ϕ ) + β l3l sin(ϕ )) + ( N1 f1 + N 2 f 2 )h12 ,

c12 = m3 (−r1l3 cos(ϕ )(ϕ − β ) − ϕ r1l3 cos(ϕ )) + ( N1 f1 − N 2 f 2 )h1 , c = m ϕ l l sin(ϕ ) + D , c = m r l cos(ϕ )(ϕ − β ) + D , 13

3 1 3

c21 = m3 ( β l3l sin(ϕ ) − ϕ l3l sin(ϕ )) + ( N1 f1 − N 2 f 2 )h1 ,

c22 = −m3 r1l3 cos(ϕ )(ϕ − β ) + ( N1 f1 + N 2 f 2 ), c31 = m3 β l3l sin(ϕ ), c32 = m3 β r1l3 cos(ϕ ).

c33 = Dϕ , c44 = D3 r32 cos ϕ + Dα 3 ,

( M 1 − M 2 )h1 − N 3 f3l3 sin(ϕc − ϕ )υ  M + M − N f l sin(ϕ − ϕ )υ  1 2 3 3 3 c T B(q)τ + J (q)λ =  − N 3 f3l3 sin(ϕc − ϕ )υ   υ Dϕ r3 cos ϕ where ϕc is the natural (non-sliding) angle of the castor wheel with sliding being exempted, R is the current radius of the trajectory and υ is the desired velocity of the castor sliding wheel. N1, N2, N3 are the pressure wheels of the ground, respectively, M1, M2 are DC torques, f1, f2, f3 are wheels coefficients friction of the ground, respectively and Dφ, D3, Dα3 are damping factors. Variables υ , R and ϕc are calculated from the formulas below: Fig. 2. Algorithm of DC controller

Fig. 3. Dynamic controller of the robot Articles

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Fig. 4. Kinematic controller of the robot Inducted voltages are equal to:

 −l l + (l 2 R 2 + R 4 − R 2 l 2 )  3 3 ,   l 2 + R2  

ϕc = a cos  R=

υ=

e1 = kΨ1α 1 , e 2 = kΨ2α 2 ,

( x 2 + y 2 )3/ 2 , x y x y

( r ψ 2 1

The motor armature resistances and voltages are calculated as:

Vd 1 = Ra1M d 1 / (kΨ ) + kΨα 1 , Vd 2 = Ra 2 M d 2 / (kΨ ) + kΨα 2 ,

(2)

To the robot dynamics equations (1) described above we added equations describing DC motors [15]:

= (V − R Q − e ) / L , Q 1 1 a1 1 a1 a Q = (V − R Q − e ) / L , a2

where Q 1 , Q 2 – currents, ea1 , ea2 – inducted voltages, La – armature inductance. 52

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i =1

j =1

i =1

j =1

Rc1 = ∑ δ Ti RTi + ∑ δ Dj RDj ,

(3)

Rc 2 = ∑ δ Ti RTi + ∑ δ Dj RDj . 4

i =1

j =1

∆Vc1 = ∑ δ TiVC (TO ) + p ∑ δ DjVF (TO ) , 4

i =1

j =1

∆Vc 2 = ∑ δ TiVC (TO ) + p ∑ δ DjVF (TO ) , V1 = V − ∆Vc1 , V 2 = V − ∆Vc 2 ,

where Ψ1 , Ψ2 – exciting flux, k – gear, α 1 = ψ + β h1 and α 2 = ψ − β h1 – driving wheels velocities.

(5)

)

+ l 2 β 2 + l32 (ϕ − β ) 2 + 2l3 (ϕ − β )(r1ψ sin(ϕ ) − l β cos(ϕ )) .

To make the model more realistic, we also modeled two transistor bridges H, each consisting of four transistors and four diodes. The algorithm which sets the instantaneous value of PWM voltages was presented in Fig. 2., where K is the duty cycle, DTK – 1/20 of PWM impulse duration , Vd1, Vd2 – voltages set by the regulator based on the set-torque Md1 and Md2, Vmax – maximum voltage (Vmax = 24V), Vmin – minimum voltage (Vmin = 0V). Moreover, we assumed that the duration of a single PWM impulse Tc is 2 ms and that we do not take commutation process into account, dt – timestep. The voltages Vd1 and Vd2 are calculated as follows:

(4)

(6)

(7)

(8)

(9) (10)

where Ra1 – resultant resistance of motor 1, Ra2 – resultant resistance of motor 2, Ra – armature resistance, Rc1, Rc2 – 1st and 2nd H-structure inverter resistance, Vc1, Vc2 – first and second H-structure inverter voltage reduction, V – voltage value at source, VC(TO) – forward voltage of transistor, VF(TO) – forward voltage of diode, δ – Kronecker delta, i, j – indexes, T – transistor, D – diode, where V = 24V while two transistors or two diodes conducting (factor p = –1) and V = 0V in other cases (factor p = 1). If any δT = 1 or δD = 1, that means transistor or diode conducted.

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ωref , ω ref are reference values vectors of general coordi-

The torques used in (1) are calculated as: M 1 = kΨQ 1 , M 2 = kΨQ 2 .

N° 4

(11)

For dynamic and kinematic control (Fig. 4 and Fig. 5) x d , y d are desired velocities of the characteristic point A as in Fig. 1., x , y are output velocities of the characteristic point A, xd, yd are desired positions of the characteristic point A, ω is the vector of general coordinates,

nates, I is the vector of currents of DC motors, uref is the vector of reference voltages of the motor and Θ is the vector of disturbances (vector of varying coefficients of the friction between wheels and ground). Figures 3 and 4 present the controllers structure for dynamic and kinematic control, respectively. Kinematic controller is assumed to be as in [14].

Fig. 5. Trajectory of angular velocity β for robot moving around the characteristic point

Fig. 8. Movement of the characteristic robot point A – dynamic control

Fig. 6. Trajectory of first and second wheel for robot platform moving around the characteristic point A

Fig. 9. Torque in kinematic trajectory control

Fig. 7. Movement of the characteristic robot point A – kinematic control

Fig. 10. Torque in dynamic trajectory control Articles

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Fig. 11. Desired torque in dynamic trajectory control

3. Simulation

The model calibration was inspired with Pioneer 2DX [16] technical parameters:

For variable f1 and f2, they are assumed to have normal distribution N(0.001, 0.00058). The simulation is carried out for an example trajectory as in [12] and for the robot platform (Fig. 1.) moving around the characteristic point A (Fig. 5). In the first second, the robot accelerated and then next 20 seconds it was moving with the constant velocities. We also analyze for both control types in direct torque or voltage control (as exemplary testing base) and the widely known PWM control. This control (with H Transistor Bridge) is the most popular for changing voltages in DC motors. Figures 6, 7 and 8 show cases when PD controller with gains Kd = 141.0 and Kp = 473.0 is applied. These gains were calculated by means of Hooke-Jevees optimization of the trajectory accuracy [12]. Fig. 7 and Fig. 8 show the movement of the characteristic point A.

4. Simulation results

The errors of the trajectory as in [12] and, for the trajectory with the robot platform moving around the characteristic point A (Fig. 5), were presented in Tables 1 to 4. The errors for the whole trajectory are calculated as follows:

∆β =

1 1 2 2 ( β di − βi ) , ∆ψ = ∑ (ψ di −ψ i ) , ∑ n i =1 n i =1 n

2 1 1 2 ∆β = β di − β i , ∆ψ = (ψ di −ψ i ) , ∑ ∑ n i =1 n i =1

(

)

(13)

(14)

where: ∆β – root mean square (RMS) errors of angle β , ∆ψ – RMS error of angle ψ , ∆β – RMS error of velocity β , ∆ψ – RMS error of velocityψ , i – an individual 54

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Fig. 12. Torque in kinematic control – moving around The control types are denoted as follows. 1 stands for direct control, 2 denotes direct control with varying friction coefficients, 3 is PWM control and 4 – PWM control with varying friction coefficients.

Fig. 13. Torque in dynamic control-moving around Table 1. Root mean square errors for given in [12] trajectory Kind of control

Root mean square errors

∆ψ ∆β [rad] ∆ψ [rad] ∆β [rad/s] [rad/s]

Kinematic direct_f

9.19·10-2

3.62·100

9.58·10-3

2.61·10-1

Dynamic direct_f

5.70·10-4

1.47·10-4

5.52·10-3

2.84·10-3

Kinematic direct

9.19·10-2

3.62·100

5.98·10-3

2.61·10-1

Dynamic direct

5.71·10-4

1.48·10-4

5.52·10-3

2.86·10-3

Kinematic PWM

9.97·10-2

3.63·100

1.02·10-3

2.62·10-1

Dynamic PWM

8.08·10-4

2.49·10-3

5.70·10-3

4.13·10-3

Kinematic PWM_f

9.97·10-2

3.63·100

1.02·10-2

2.62·10-1

Dynamic PWM_f

6.55·10-4

1.76·10-3

5.70·10-3

4.27·10-3

step, n – number of steps, index d stands for desired values. The maximum errors were calculated as H – errors, i.e. as a maximum absolute error for the whole trajectory. Angles β and ψ and their angle velocities are equal to:

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∆β max = ∆β ∞ , ∆β max = ∆β ,

(15)

∆ψ max = ∆ψ

(16)

∞

∆ψ max = ∆ψ ∞ ,

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where index max denotes the maximum value. Index f in tables 1 to 4 means that in this simulation we analyzed the varying wheel friction of the ground. Table 2. Maximum errors for given in [12] trajectory Kind of control

Maximum errors

Fig. 14. Desired torque in dynamic control – moving around

∆ψ ∆β [rad] ∆ψ [rad] ∆β [rad/s] [rad/s]

Kinematic direct_f

1.45·10-1

5.94·100

2.50·10-1

3.70·10-1

Dynamic direct_f

2.16·10-3

1.39·10-3

2.50·10-1

1.08·10-1

Kinematic direct

1.45·10-2

5.94·100

2.50·10-1

3.69·10-1

Dynamic direct

2.16·10-3

1.39·10-3

2.50·10-1

1.08·10-1

Kinematic PWM

1.57·10-3

5.95·100

2.50·10-1

3.76·10-1

Dynamic PWM

3.06·10-3

6.21·10-3

2.50·10-1

1.08·10-1

Kinematic PWM_f

1.57·10-1

5.95·100

2.50·10-1

3.75·10-1

Dynamic PWM_f

2.83·10-3

5.19·10-3

2.51·10-1

1.10·10-1

Table 3. Root mean square errors for trajectory of robot moving around the characteristic point Kind of control

Root mean square errors

∆ψ ∆β [rad] ∆ψ [rad] ∆β [rad/s] [rad/s]

f indicates that in the given simulation we analyzed the varying wheel friction of the ground. Tables 3 and 4 show errors of general coordinates for robot platform moving around the characteristic point A. For this type of motion, the characteristic point A should not move. But with the impact of the sliding castor wheel being considered, the point A moved in the first phase of the motion. Table 4. Maximum errors for trajectory of robot moving around the characteristic A point Maximum errors Kind of control

∆β [rad]

∆ψ [rad] ∆β

Kinematic direct_f

2.88·100

Dynamic direct_f

∆ψ

[rad/s]

8.03·10-2

1.72·10-1

4.95·10-3

8.69·10-3

3.83·10-3

1.46·10-2

6.34·10-3

Kinematic direct

2.88·100

8.03·10-2

1.72·10-2

4.22·10-3

Dynamic direct

8.57·10-3

3.77·10-3

1.47·10-2

6.01·10-3

Kinematic direct_f

1.64·100

4.57·10-2

1.39·10-1

3.87·10-3

2.89·100

5.32·10-3

1.74·10-1

1.17·10-2

Dynamic direct_f

Kinematic PWM

8.36·10-5

3.67·10-3

1.91·10-3

8.51·10-4

8.60·10-3

6.91·10-3

1.70·10-2

1.52·10-2

Kinematic direct

Dynamic PWM

1.64·100

4.57·10-2

1.39·10-1

3.87·10-3

2.89·100

5.46·10-3

1.74·10-1

1.18·10-2

Dynamic direct

Kinematic PWM_f

8.36·10-5

3.67·10-3

1.73·10-3

7.67·10-4

8.66·10-3

6.79·10-3

1.73·10-2

1.51·10-2

Kinematic PWM

Dynamic PWM_f

1.65·100

3.64·10-3

1.39·10-1

8.26·10-4

Dynamic PWM

8.36·10

6.68·10

1.81·10

1.60·10

Kinematic PWM_f

1.65·100

3.71·10-3

1.39·10-1

8.26·10-4

Dynamic PWM_f

8.34·10-3

6.47·10-3

1.98·10-3

1.84·10-3

-3

Large errors achieved for simulations with the kinematic control indicate that this control type is insufficient for the mobile robot. As shown in Tables 1 and 3, there is also a significant difference between PWM and a direct voltage control. However, in real models we can use only PWM, as direct control is strictly theoretical. The index

Results shown in Table 3 and 4 are encouraging, because, for the robot moving around the characteristic point A, the varying friction does not influence the trajectory. In figures below, M1 direct, M2 direct stand for torques for theoretical direct voltage control. M1 PWM and M2 PWM denote torques for PWM voltage control. Even though the simulation results for mobile robot moving along the trajectory as in [12] seem to be almost identical as presented Fig. 9-11, results presented in tables 1 and 2 leave no doubt that the kinematic control causes greater deviations compared to dynamic control. For comparison, we present the set torques (Fig. 10). Fig. 12, 13 and 14 show the torque of mobile robot when rotating around its a fixed axis (characteristic point A). Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

Fig. 12 and 13 depict theoretical torques for kinematic and dynamic control, from which, when compared to Fig. 14 showing the set torques, we can conclude that for kinematic control the torques are less oscillatory as compared to dynamic control. The reason for this is the impact of angle velocity regulation in dynamic control, which causes a greater sensitivity of the automatic regulation system.

5. Conclusion

In Fig. 9 to 14, we show torques for kinematic and dynamic control types, with and without PWM. We also show the impact of the varying friction coefficient on the trajectory accuracy. The kinematic controller is not suitable for high accuracy control of the mobile robot, because in our model with a castor sliding wheel the kinematic controller cannot stabilize the motion, as the errors in kinematic control are too large. Such magnitude of errors might have been triggered with too large time constant of the regulator. Given this, the kinematic control should not be applied for mobile robots, as it brings about significant trajectory errors. Our results are conductive for improving the robot simulation procedure so as to achieve results closer to robot behavior in reality.

Acknowledgements

The research was co-financed by the European Social Fund

AUTHORS

Ryszard Beniak* – Opole University of Technology, Faculty of Electrical Engineering, Automatic Control and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland, e-mail: r.beniak@po.opole.pl Tomasz Pyka – Opole University of Technology, Faculty of Electrical Engineering, Automatic Control and Computer Science, Prószkowska Street No. 76, 45-758 Opole, Poland, e-mail: t.pyka@doktorant.po.edu.pl *Corresponding author

References

[1] R. Balakrishna and A. Ghosal, “Modelling of Slip for Wheeled Mobile Robots”, IEEE Transactions on Robotics and Automation, vol. 11, no. 1, February 1995. [2] Xiaocai Zhu et al., “Robust Tracking Control of Wheeled Mobile Robots Not Satisfying Nonholonomic Contraints”. In: Proceedings of the 6th International Conference on Intelligent Systems Design and Applications (ISDA’06), 2006. [3] Y. P. Li, T. Zielinska, M. H. Jr Ang and W. Lin, “Vehicle dynamics of redundant mobile robots with powered caster wheels”, Proceedings of the Sixteenth CISM‑IFToMM Symposium, Romansy 16, Robot Design, Dynamics and Control, Warsaw: Springer, 2006, pp. 221–228. 56

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[4] K. Zdanowska, A. Oleksy, “Motion planning of Wheeler mobile robots subject to slipping”, Journal of Automation, Mobile Robotics & Intelligent Systems, vol. 5, no. 3, 2011. [5] Ch. C. Ward and K. Iagnemma, “Model-Based Wheel Slip Detection for Outdoor Mobile Robots”, IEEE International Conference on Robotics and Automation Roma, Italy, April 2007. [6] D. DeVon and T. Bretl, “Kinematic and Dynamic Control of a Wheel Mobile Robot”, Proceedings of the 2007 IEEE/RSJ International Conference on Intelligent Robots and Systems San Diego, CA, USA, 29th Oct. – 2nd Nov. 2007. [7] N. Sidek and N. Sarkar, “Dynamic Modeling and Control of Nonholonomic Mobile Robot with Lateral Slip”, Proceedings of the 7th WSEAS International Conference on Signal Processing, Robotics and Automation (ISPRA’08), 2008. [8] Yin-Tien Wang,Yu-Cheng Chen and Ming-Chun Lin, “Dynamic Object Tracking Control for a NonHolonomic Wheeled Autonomous Robot”, Tamkang Journal of Science and Egineering, vol. 12, no. 3, 2009, pp. 339–350. [9] N. Sidek, N. Sarkar, “Inclusion of Wheel Slips in Mobile Robot Modeling to Enhance Robot Simulator Performance”, The 3rd Int. Conf. on Mechatronics (ICOM 08), 18th–20th Dec. 2008. [10] R. Dayal Parhi and B. B. V. L. Deepak: “Kinematic model of three wheeled mobile robot”, Journal of Mechanical Engineering Research, vol. 3(9), 2011, pp. 307–318. [11] F.N. Martins, W.C. Celeste, R. Carelli, M. Sarcinelli-Filho, T.F. Bastos-Filho, “An adaptive dynamic controller for autonomous mobile robot trajectory tracking”, Control Engineering Practice, vol. 16, 2008, pp. 1354–1363. [12] R. Beniak, T. Pyka, “Selection of controller setpoints for a tri-wheel robot taking into account the impact of the dragged wheel and varying coefficients of wheel friction”, The Measurements, Automation and Monitoring, no. 11, 2010. pp. 1390–1395. (in Polish) [13] M. Kaźmierkowski, R. Krishnan, F. Blaabjerg, “Control in Power Electronics”, Academic Press an impring of Elsevier Science, 2002. [14] L. Gracia, J. Tornero, “Kinematic control of wheeled mobile robots”, Latin American Applied Research, vol. 38, no. 1, pp. 7–16. [15] R. Beniak, T. Pyka, “Control stepsize optimization for tri-wheel mobile robot”, 3rd International Students Conference Electrodynamics and Mechatronics, Opole, 2011. [16] M. J. Giergiel, Z. Hendzel, W. Żylski, “Modeling and control of wheeled mobile robots”, Warsaw: Polish Scientific Publishers PWN, 2002. (in Polish).

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Robot for Monitoring Hazardous Environments as a Mechatronic Product Submited 10th July 2012; accepted 7th September 2012

Leszek Kasprzyczak, Stanisław Trenczek, Maciej Cader

Abstract:

The mining mobile inspective robot is designed for monitoring explosive hazardous environments, where unknown gas mixtures consisting of explosive, toxic and suffocating gases can be present. This situation is often observed in mining hard coal excavations where inertia processes are carried out during or after a fire. Then the gas mixtures include, among others, significant concentrations of carbon monoxide, carbon dioxide and methane. In the paper the principle of operations and the robot assemblies which consist of mechanical, electronic, pneumatic and software elements were described. Furthermore in the paper initial traction tests of the robot were discussed. Keywords: mobile robot, hazardous environments, mine.

1. Introduction In the hard coal excavations, toxic and explosive gas mixtures can exist. In order to isolate the hazard zone from the remaining parts of a mine it is necessary to build isolated dams in the inlet and outlet sides. Fig. 1 shows the longwall as an example of isolated excavation, where valuable mining machines were trapped, like a shearer, conveyor belts, powered roof supports. In order to enable accurate measurements of dangerous gases in an isolated excavation, the Institute of

Innovative Technologies EMAG and the Industrial Research Institute for Automation and Measurement PIAP elaborated a mobile inspective mining robot GMRI for working in explosive hazardous zones. The robot, as a vehicle, is equipped with appropriate sensors for measurements of some gases concentration, cameras, transmission system, control and visualization systems as well as electrical accumulators. The device is built according to explosion-proof techniques [13–15]. The first and most important issue which had to be solved was determining the functional assumptions and the robot technical concepts solutions. In the paper [1] minimal dimensions of the robot were specified, as well as the dimensions of solid and water obstacles. Moreover the range of operation, kinds of analyzed gases, extreme climatic conditions and selection of suitable materials were discussed. Selection of sensors for measuring parameters of a mining atmosphere and the analysis of regulations and standards concerning the measurements accuracy and explosion-proof aspects were described in the paper [2].

2. The review of existing mining robots The leading world organizations attempt to construct mining inspection and rescue robots. The most recognized solutions are the following: 1. NUMBAT a mine reconnaissance robot designed in the 1990s by the Australian Commonwealth Scientific and Industrial Research Organization (Fig. 2) [3–5],

E C D

Fig. 1. Scheme of the isolated excavation monitored by the mobile robot A/B – explosion-proof inlet/outlet dams with the 80 cm diameter sluice, C – shearer, D – conveyor belts, E – powered roof supports, F – operator’s console, G – mobile robot

Fig. 2. NUMBAT [3] 2. REMOTEC WOLVERINE V-2 robot developed by the Mine Safety and Health Administration of America (Fig. 3) [5, 6], 3. GROUNDHOG robot developed by Carnegie Mellon University Robot Research Center USA (Fig. 4) [5, 7] Articles

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4. GEMINI-SCOUT Mine Rescue Robot – Sandia National Laboratories USA (Fig. 5) [8] 5. Chinese constructions: CMU-1 developed by China University of Mining and Technology, LURKER-1 and LURKER-3 developed by the Robotics Research Centre of Shandong University and others [9–12]. The above listed robots are driven by electric DC motors. It is often difficult to obtain their full technical data. Sometimes only media reports exist and there are no scientific descriptions in the form of available articles, like in the case of Gemini-Scout [8].

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it is constructed according to the M1 category in compliance with EN 50303 [14]. The device has the M1 category if it has double security or the “ia” intrinsic safety. Taking into account the above information, the robots with DC engines should have double security in order to be continuously utilized in the explosive atmospheres of coal mines. The use of a flameproof casing only qualifies the device to the M2 category which means that a machine has to be turned off once dangerous concentration of methane is detected. In the quoted articles [3–12] sometimes there is no information about categories of robots and in some cases the authors admit that their solutions do not have proper anti-explosion measures [9].

Fig. 3. REMOTEC WOLVERINE V-2 [5] Fig. 6. Prototype of GMRI

Fig. 4. GROUNDHOG [7]

This was achieved through the use of the compressednitrogen pneumatic drive and through limiting the speed of movable elements. Decompressed nitrogen cools the inside of the actuators which becomes an extra security measure against overheating of friction elements. Electronic assemblies of GMRI were designed for the “ia” intrinsic safety category in compliance with EN 6007911 [15]. The prototype of the robot, along with technical documentation, was tested and analyzed by an independent notified body for compliance with the ATEX directive and received the M1 category.

3. Robot measuring functions

Fig. 5. GEMINI-SCOUT [8] In the light of Polish and European laws as well as standards harmonized with the 94/9/EC (ATEX) [13] directive, the device can work continuously in an atmosphere with methane and/or coal dust explosion hazards if 58

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The main task of the robot is transmitting data regarding the composition of mining excavation atmosphere to an operator. Values of concentrations of the following gases are transmitted: – explosive – methane CH4, – toxic – carbon monoxide CO, – and others – carbon dioxide CO2 and oxygen O2. For the sake of limited oxygen content in the tested gas mixture which is due to the inertia process (with nitrogen), it was obligatory to apply sensors which do not need the presence of oxygen or they need very small amounts of it. Moreover the robot is equipped with temperature and humidity sensors for measuring the ambient gas mixture. The robot also has to transmit video from cameras to enable the operator to make an evaluation of excavation conditions and to control the robot remotely. The user interface includes software for the visualization of films from cameras and values of measurements from all sensors.

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Fig. 7 presents a view of the robot chassis model with the jib as a pipe, through which the tested atmosphere is sucked into the gases meter. In the isolated region there is high humidity usually close to 100%. If we do not fulfil the requirement of 95% RH without condensation, then the life expectancy and accuracy of sensors will decrease. Therefore a drier system was elaborated in EMAG which dries the tested gas mixture before introducing it to the sensors. The block scheme of the system is shown in Fig. 8. In the elaborated equipment (E) the mining atmosphere (MA) is sucked in from the jib thanks to a gas pump (GP). The gas mixture is at first dried in the dryer (D). Then the tested gas is introduced to the sensors (S) and leaves the equipment.

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3.1. Infrared sensors

The range of infrared radiation which is typically used in the sensors, it is between 3-5 μm. In this band there are absorption lines of many essential gases, and for 4 μm there is no absorption line. The radiation of 4 μm wavelength is used as a reference signal in NDIR sensors. Besides, simple bulbs with filaments emit radiation between 2 to 5 μm (depending on the filament temperature and bulb enclosure material). Moreover within the range 3-5 μm there is a “water window”, which means that water steam does not absorb radiation in this band and thereby does not influence the measurements values. Table 2 shows the lengths of waves of infrared radiation which are absorbed by gases typically occurred in mining atmospheres. The sensors manufacturers give different values because an absorption line of a gas is usually not a single value but a narrow band. Table 2. Values of absorption lines/bands of selected gases [16-19]

Fig. 7. Model of the GMRI robot chassis with jib (designed by PIAP) E MA

MA D

MA S

Fig. 8. Block scheme of a drier system for measuring gases concentrations of mining atmospheres with high humidity (designed by EMAG) In table 1 the measuring ranges of the robot sensors were given. The non-dispersive infrared sensors (NDIR) for measuring methane and carbon dioxide were applied. Electrochemical sensors were used for detecting oxygen and carbon monoxide. In the next sections of the paper the operating principles of the sensors were described. Table 1. Measuring ranges of the GMRI sensors Tested gas

Measuring range

Methane CH4

0 … 100%

Carbon dioxide CO2

0 …5%

Carbon monoxide CO

0 … 1000 ppm

Oxygen O2

0 … 25%

Temperature

-40 … ~+120°C

Humidity

… 100%

Gas

Wave length [μm]

Methane CH4

3,32 or 3,4

Hydrocarbons CxHy

3,0 – 3,5

Carbon dioxide CO2

4,2 – 4,3

Gases, with particles consisting of two or more different atoms, absorb appropriate infrared wave lengths which are compatible with their natural frequency vibrations. The amplitude of the interatomic bonds vibrations increases along with the temperature of the particles (Fig. 9). However other gases, like nitrogen, oxygen and noble gases cannot be detected using this method. a)

Fig. 9. Increase of the amplitude of interatomic bonds vibrations after absorption of the IR radiation [16] Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

An operating principle of gas sensors which uses a bulb as the radiation source and a pyroelectric detector is shown in Fig. 10. The bulb is supplied with a rectangular signal with frequency between 1 to 10 Hz. At the pyroelectric detector output, near sinusoidal voltage appears with the same frequency as the rectangular signal. When the concentration of the target gas increases, the amplitude of the output signal decreases, according to exponential expression described by the Beer-Lambert law [17–21].

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gas concentration is calculated from the ratio of the two output peak to peak signals. In practice more complicated equations are used for measuring concentrations, which compensate the influence of temperature and sometimes pressure.

3.2. Electrochemical sensors

Electrochemical sensors use chemical reactions for measuring target gas concentrations. A typical electrochemical sensor consists of a sensing electrode (working) and a counter electrode which are separated by an electrolyte (Fig. 12). At first a gas passes through the diffusion barrier, charcoal filter and hydrophobic membrane and then the reaction on the sensing electrode begins.

Fig. 12. Structure of a typical electrochemical sensor [16]

Fig. 10. Principle of operations of gas sensors with the IR radiation [19] Pyroelectricity is the ability of certain materials to generate a temporary electrical potential when they are heated or cooled. The electric potential disappears after a relaxation time, that is why the fallen radiation has to be changeable. Thus the bulb command signal is a period rectangular one. All pyroelectric materials are also piezoelectric ones, so it is necessary to protect sensors against forces. In Fig.11 the structure of an IR gas sensor with two measuring channels is shown. In the active channel there is an optical filter which passes radiation with wave length as the wave length for the target gas. In the reference channel the optical filter passes radiation with wave length 4 μm. The signal amplitude of the active sensor decreases together with the increasing concentration because more energy is absorbed by the gas particles. The reference channel is used to compensate for changes in source intensity, optical degradation and temperature to some degree. The amplitude of this detector will not show any changes due to the effects of the target gas. The target

Fig. 11. Structure of an IR gas sensor [17] 60

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The target gas reacts at the surface of the working electrode involving either oxidation or reduction reactions. These mechanisms are catalyzed by the electrode materials specifically selected for the tested gas. Through an external resistor which connects working and counter electrodes proportional to the concentration electrical current flows [16, 17, 22]. It is required to have a stable and constant potential at the working electrode. In reality the sensing electrode potential does not remain constant due to the continuous electrochemical processes. That is why the metrological parameters deteriorate. In order to improve the performance of the sensor, a reference electrode is introduced between the sensing and counter electrodes, and to the working electrode, a fixed stable constant potential is applied. No current flows between the sensing and reference electrodes. The hydrophobic membrane enables a protection against humidity and prevents the sensor from drying out as well as leaking the liquid electrolyte. Moreover it allows enough gas molecules to reach the working electrode. The membrane is made of thin, porosity Teflon. The filter is installed in order to eliminate cross-sensitivity for unwanted gases. It is usually made of activated charcoal, which filters out most chemical compounds except carbon monoxide and hydrogen gases. By correctly selecting the filter, the sensor is more selective. The electrodes are usually made of noble metals such as gold or platinum. Every electrode can be made of different material, which enables suitable chemical reactions with a target gas. The electrolyte allows to carry the ionic charge across the electrodes. Typical chemical reactions for the measurement of oxygen and carbon monoxide are as follows: – for an oxygen sensor [17]: – cathode: 4H+ + O2 + 4e- → 2H2O,

Journal of Automation, Mobile Robotics & Intelligent Systems

– – – – –

anode: 2H2O → 4H+ + O2 + 4e-, for a carbon monoxide sensor [22]: anode: CO + 2OH-→ CO2 + H2O+2e-, cathode: ½ O2 + H2O + 2e- → 2OH-, total reaction: CO + ½ O2 → CO2. As we can see from the above reactions, the carbon monoxide sensor needs small amounts of oxygen for correct functioning. Too low an amount of oxygen causes significant shortening of the sensor life (even to several hours). The oxygen sensor fills up with water thanks to absorption of steam water from an ambient environment. Temperature is a factor which has a big influence on the measurement accuracy. So it is necessary to compensate it [16, 17, 22].

4. Robot transmission system

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two intrinsically safe circuits which are galvanically separated by optocouplers.

5. Robot supply system The robot is equipped with two accumulators which supply two separated intrinsically safe circuits. The CAD prototypes of accumulators designed by EMAG are shown in Figs. 14 and 15. The main issue was to design accumulators which are intrinsically safe and can work properly up to +60°C. The cells should not cause a fire even in failure situations. Thus copper plates and encapsulation were used for dispersion of potential heat especially in damage situations. Accumulators have an electronic output current limiter as well as overcharge and overdischarge protection.

On the robot platform there are 4 programmable controllers which are placed in a metal casing (Fig. 13). For communication with the operator’s console the main microcontroller is applied. It receives commands from the operator’s console and distributes them among other controllers. On the robot board there are also: – Valves controller which is designed for robot motion control and data conditioning from state sensors, – Sensors microcontroller which is used for triggering and conditioning measurements of the atmosphere parameters, – Auxiliary controller for remote switching of front and rear cameras with lighting system and for controlling the state of a lit-ion accumulator. Fig. 14. CAD model of the lit-ion accumulator

Fig. 13. Transmission system of the robot GMRI The transmission line has two-wire pairs. One pair is used for half-duplex V23 modem and the second pair is used for data streaming from the cameras in the RS485 standard. Commands concerning the robot motion are sent through V23 modem from the operator’s console to the valves controller (through main µC). Also queries about measuring data are sent from the console to the sensors µC as well as commands which camera with lighting should be switched to the auxiliary µC. All circuits of the robot are intrinsically safe because the robot is designed for working in explosive zones, so the circuits of the console are galvanically isolated from the robot circuits. Moreover the robot is equipped with

Fig. 15. CAD model of the Ni-Cd accumulator The lit-ion accumulator contains 4 cells. This accumulator is not charged in explosive atmospheres. It supplies cameras with lighting, electrovalves, state sensors, valves controller and auxiliary microcontroller. The nickel-cadmium accumulator contains 6 cells. It supplies transmission circuits and atmosphere parameters sensors with their microcontroller. The accumulator is charged through the modem line which has certain DC level voltage. The charge and discharge characteristics of singular lit-ion and Ni-Cd cells are shown in Figs. 16 and 17. It is visible that the electrochemical processes are different. Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

6. Traction tests The drive structure of the robot is based on system of movable supports (legs) and system consists of wheels – two widely spaced wheels in the back and two narrowly spaced ones in the front (Fig. 16). This solution was elaborated by PIAP and submitted to the Polish Patent Office. Drive system of the robot GMRI are supplied by compressed nitrogen from a tank placed in the robot platform (Fig. 17). Pneumatic actuators of the drive system are controlled by 2-state valves. Nitrogen is a neutral gas which does not have any influence on fire progress and measurements. The reserve of the bottle volume is enough for the distance of 500 m.

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The robot is able to widen/narrow the back wheels independently (Fig. 18). When the back wheels are widened, the stability of the robot is better on an irregular floor of an excavation (max. width is 1.1 m). When the back wheels are narrowed, the robot can be transported along an 80-cm diameter pipe. The support system enables to lift or lower the platform evenly or non-evenly. The even lift is performed when the whole mobile body is lifted parallel to the floor up to the given height (max 0.5 m). The platform can be positioned in 4 heights – minimal, maximal and two medium ones. The non-even lift enables to lift only one side of the mobile body – back or front, left or right (Fig. 19). This functionality meets the requirements of the floor conditions (longitudinal and lateral slopes) which the robot has to cope with. Thanks to this functionality the mobile platform of the robot can remain in a horizontal position irrespective of the floor conditions in a mine. Additionally, the robot has the jib for taking samples of a gas mixture. The probe is driven by a rotational pneumatic drive. The length of the probe is 1 meter. The robot can lift its whole body so the range of the probe is extended. Manipulating the robot body is aimed at directing two immovable wide-angle cameras to a desired place. The robot can lift its front wheels with the use of its front legs and a special actuator connected with the

Fig. 16. Motion systems: 1 – wheel system, 2 – support system

Fig. 19. GMRI with a platform in a first medium position

Fig. 17. Pneumatic scheme of the robot drive system

Fig. 18. Widening back wheels of GMRI 62

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Fig. 20. GMRI turns left

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structure of the front bridge. Thanks to that the robot can pass over the obstacle or mount it with the front wheels. The turning of the robot is performed by the front bridge which is connected with linear actuators by means of joints. When the robot turns left (Fig. 20), the piston rod of the right actuator is moved forward and the left one retracted simultaneously. The robot moves forward or backward thanks to linear pneumatic actuators and interlocking brakes which are turned on in proper sequences and time intervals. A typical movement sequence on a flat ground (Fig. 21) lies in blocking of the brakes in the back wheels and releasing the brakes in the front wheels. Then the piston rods of the actuators of back wheels are moved forward. This way the robot can move 30 cm forward.

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(Fig. 22), water obstacles of at least 10 cm, lengthways slopes of at least 30° (Fig. 24) and go up side slopes of at least 30° (Fig. 23). The rear wheels are independent and can adapt to the ground shape. Moreover the robot dimensions should enable it to move through an 80 cm diameter sluice of dams (Fig. 25).

Fig. 24. Going uphill of a 30° slope Fig. 21. Simulation of the GMRI movement (30 cm forward) The functional assumptions described in the paper [1] defined the traction requirements. Generally, the robot should overcome solid obstacles at least of 20 cm

Fig. 25. Moving through an 80 cm diameter sluice

7. Summary Fig. 22. Overcoming a solid obstacle

Fig. 23. Driving through a 30º side slope (rear view)

Designing and building a mobile robot for working in explosive zones is very difficult because one has to fulfil the ATEX directive and harmonized standards. Additionally, the communication between the operator’s stand and the robot is limited only to wired connections because radio waves are attenuated by rock mass and isolated dams. The robot prototype was subject to traction tests at EMAG’s and PIAP’s headquarters with a view to prepare it for testing in a real mining excavation. The tests were carried out in a working excavation on the level of 726 m in the Bobrek coal mine of Kompania Węglowa S.A. with positive results [23]. The ergonomics of the remote manual control of the robot has to be improved. It is necessary to improve the robot lighting system for better recognition of encountered obstacles in completely dark excavations. In a future project we intend to build a mining robot with electrical drives for better passing over more demanding obstacles and on expanding the operations range of the robot. The project to construct a mobile inspection robot is important for the mining industry due to the possibility of Articles

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using the robot to assess the state of the excavation and atmospheric parameters. Thanks to this, it will be possible to make faster and more accurate decisions about further operations related to the dangerous area of the mine.

AUTHORS

Leszek Kasprzyczak* – Institute of Innovative Technologies EMAG, Leopolda St. 31, 40-189 Katowice, Poland, E-mail: kasprzyczak@emag.pl Expert in mechatronics, functional safety, measurement systems, construction and operation of machinery Stanisław Trenczek – Institute of Innovative Technologies EMAG, Leopolda St. 31, 40-189 Katowice, Poland. E-mail: s.trenczek@emag.pl Deputy Director for Research, a specialist in the field of mining aerology. Maciej Cader – Industrial Research Institute for Automation and Measurements, Al. Jerozolimskie 202, 02486 Warsaw, Poland. E-mail: mcader@piap.pl Expert in mobile systems construction, simulation and Rapid Prototyping technology. *Corresponding author

References [1] L. Kasprzyczak, S. Trenczek, Z. Borkowicz, M. Cader, “Functional assumptions and concepts of technical solutions of mobile inspective robot for working in explosive hazardous environments”. In: Proc. of EMTECH 2009 Conf. ”Supply, computer science and automatics in excavation industry”, Poland, 2009, pp.99-105 (in Polish). [2] L. Kasprzyczak, P. Dzierżak, “Sensors for measurement mining atmosphere parameters of mobile inspective robot, Mechanizacja i Automatyzacja Górnictwa, vol. 466, no. 12, 2009, pp.19–30. [3] D. W. Hainsworth, “Teleoperation user interfaces for mining robotics”, Autonomous Robots, no. 11, 2011, pp. 19-28. [4] J.C. Ralston, D. Hainsworth, “Recent advances in remote coal mining machine sensing, guidance, and teleoperation”, Robotica, no. 19, 2001, pp. 513-526. [5] R. Murphy, J. Kravitz, S. Stover, R. Shoureshi, “Mobile robots in mine rescue and recovery”, IEEE Robotics & Automation Magazine, 2009, pp. 91–103. [6] R. Murphy, J. Kravitz, “Preliminary report: Rescue robot at Crandall Canyon”, Utah, mine disaster. IEEE Int. Conf. on Robotics & Automation, Pasadena, 2008, pp. 19-23. [7] S. Thruny, D. Hahnel, “A System for Volumetric Robotic Mapping of Abandoned Mines”. In: Proc. of ICRA 2003. [8] https://share.sandia.gov/news/resources/news_releases/miner-scou/ 64

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[9] X. Rongb, R. Songb, “Mechanism and explosionproof design for a coal mine detection robot”, Advanced in Control Engineering and Information Science, no. 5, 2011, pp.100-104. [10] YunWang Li, Shirong Ge, Hua Zhu, “Explosionproof design for coal mine rescue robots”, Advanced Materials Research, 2011, pp. 1194-1198. [11] W. Wang, Z. Du, L. Sun, “Kinematics Analysis for Obstacle-climbing Performance of a Rescue Robot”, IEEE Int. Conf. on Robotics and Biomimetics, 2007, pp.1612-1617. [12] J. Gao, X. Gao, J. Zhu, “Coal mine detect and rescue robot technique research”, IEEE Int. Conf. on Information and Automation, 2009, pp. 1068–1073. [13] Directive 94/9/EC of the European Parliament and the Council of 23 March 1994 on the approximation of the laws of the Member States concerning equipment and protective systems intended for use in potentially explosive atmospheres (ATEX) [14] EN 50303 Group I, Category M1 equipment intended to remain functional in atmospheres endangered by firedamp and/or coal dust [15] EN 60079-11 Explosive atmospheres. Equipment protection by intrinsic safety „i” [16] www.intlsensor.com [17] www.e2v.com [18] www.citytech.com [19] www.dynament.com [20] S. Lang, “Pyroelectricity: from ancient curiosity to modern imaging tool”, Physics Today, vol. 58, no. 8, 2005, pp. 31-36. [21] J. Piotrowski, A. Rogalski, “Semiconductor infrared detectors”, WNT, Warszawa 1985 (in Polish) [22] www.figaro.co.jp [23] Report from traction tests of the GMRI mobile inspection robot, Mining Rescue Center, 2010, p. 9 (in Polish).

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Analysis of Influence of Drive System Configurations of a Four Wheeled Robot on its Mobility Submitted: 24th July 2012; accepted 7th September 2012

Maciej Trojnacki

Abstract:

The work covers analysis of mobility of a four-wheeled robot based on its dynamics model. Several configurations of robot’s drive system are considered: one driven axle, two independently driven axles and drive transmission from one axle to another by means of a toothed belt. The analysis of robot’s mobility is limited to cases of its motion with constant velocity on the ground with various inclinations and mechanical characteristics. It is assumed that robot’s wheels roll without sliding. In the conducted investigations of robot’s mobility, limitations resulting from wheels’ interaction with the ground are taken into account. Based on results of the investigations, advantages and drawbacks of each of drive system configurations of the robot are discussed. Keywords: mobile robot, mobility analysis, dynamics model, computer simulation.

1. Introduction

Motion of a robot on diverse terrain depends on its movement abilities, in the literature often termed as ‘mobility’. It can be defined as robot’s ability to move with desired parameters of motion in defined conditions of environment, with limitations of the robot itself taken into account [3]. For determination of the mobility, analysed are robot’s motion on the ground with various mechanical properties and inclinations and its ability of negotiation of environment obstacles of various shape and height (e.g., kerbs, stairs). Robot’s movement abilities on particular terrain are affected by a number of factors, including: • geometry and type of a locomotion system (e.g., wheeled, tracked, hybrid, legged, jumping), • properties of effectors (e.g., tyre type for wheeled robots), • mass properties of a robot, • constraints resulting from characteristics of drives (e.g. power, maximum rotational speed, maximum driving torque), • battery parameters (e.g., maximum continuous discharge current), etc. So far, a very popular locomotion system intended for moving in diverse terrain was the tracked system. However, observation of mobile robots’ market reveals that even more often the tracked locomotion system gives way to the wheeled system [4] and the hybrid system, that is, the one which incorporates features of both continuous and discrete locomotion [5]. The analysis of mobility of a particular robot can be

carried out for two principal purposes, that is, for the purpose of: designing and testing of robot’s mechanical structure, synthesis of robot’s control system. At the stage of mechanical design, the mobility analysis based on robot’s dynamics model allows testing of fulfilment of the imposed requirements and design optimization. The analysis can be repeated on the robot’s physical prototype in order to verify results of the analysis carried out using the robot’s virtual model. An important assumption associated with the analysis of robot mobility is whether sliding of robot’s wheels is taken into account. The problem of modelling of motion of mobile robots including wheel slip was discussed, for instance, in [6]. From previous research it follows that the longitudinal slip of wheels should be taken into account mainly in case of significant accelerations and sometimes also in case of transition from one type of ground’s material into another (e.g., from concrete onto ice). On the other hand, side slip of wheels should be considered during negotiation of a curved path always if the robot is of the skid-steered type. Side slips tend to increase with increasing robot’s velocity of motion and with decreasing radius of curvature of the path. In the present work, mobility of a four-wheeled robot is analysed for different configurations of the drive system and including constraints which follow from robot’s dynamics and type of terrain on which the motion takes place. Aim of the analysis is discussion of advantages and disadvantages of different configurations of the robot’s drive system. The analysis is limited to the case of translational motion of the robot’s body with constant velocity. For this reason, the occurrence of wheel slip is neglected.

2. Model of the robot

The subject of this paper is a small four-wheeled mobile robot whose parameters are based on the PIAP SCOUT mobile robot (Fig. 1). This robot was designed for quick reconnaissance of field and places difficult to access, such as, the bottom of vehicle’s chassis, spaces under seats in means of transportation, narrow rooms and ventilation ducts. The robot is equipped with the hybrid locomotion system which combines tracks and wheels. Back wheels of the robot are independently driven with two servomotors. The drive is transmitted from the back wheels to the front wheels via two toothed belts, which also play the role of caterpillar tracks. In the present work, the following configurations of robot drive are considered: Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

• single-axle drive, • drive transmission from one axle to another by means of a toothed belt, • independent two-axle drive.

Fig. 1. PIAP SCOUT mobile robot The robot’s model (Fig. 2) consists of the frame (0) and driven wheels (1, 2, 3, 4). Additionally, the front wheels can be connected with the back wheels by means of toothed belts. Origin of the coordinate system of the

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robot is chosen at the point R, which is located in the middle of the distance between the front and back wheels and in the middle of the distance between wheels of the left-hand side and right-hand side. For particular pairs of wheels the following subscripts are introduced: l – wheels of the left-hand side (l = { 1,3 } ), r – wheels of the righthand side (r = { 2,4 } ), f – front wheels (f = { 1,2 } ), b – back wheels (b = { 3,4 } ). The following symbols for the ith wheel have been also introduced in the robot’s model: Ai – geometrical center, rgi = rg – geometrical (unloaded) radius, θi – rotation (spin) angle. The distance of the front axle from the back axle (wheelbase) is denoted with L, whereas the distance of the left-hand side wheels from the right-hand side wheels (track width) with W. For the mobility analysis assumed are the following values of robot geometric parameters: L = 0.35 [m], W = 0.42 [m], rg = 0.085 [m], and mass parameters: • robot’s total mass: mR = 13.73 [kg], • coordinates of the mass centre: xCM = 0 [m], yCM = 0 [m], • mass moment of inertia of the wheel about its spin axis IWy = 0.006 [kg m2].

2.1. Dynamics of the robot

Within the present work robot motion on inclined terrain will be analysed, which results in the robot being tilted about x- or y-axis (see Fig. 2a-b). The analysis will be constrained to the case of translational motion of robot’s body. It is assumed that robot’s wheels roll without sliding, so the following relationships are satisfied:

vR = vRx = θ i rg => θ i = vR / rg = θ i = v R / rg , (2.1)

where: vR – value of velocity of the characteristic point R of the robot. Dynamic equations of motion, for the case when robot is tilted about y-axis with pitch angle β = const, have the form (see Fig. 2b):

mR xCM = ∑ i =1 Fix + mR g sin( b) = 4

= 2 Ffx + 2 Fbx + mR g sin( b) mR zCM = ∑ i =1 Fiz − mR g cos( b) = 4

= 2 Ffz + 2 Fbz − mR g cos( b) = 0 I Ry b = − ∑ i =1 Fix h − ∑ i =1 Fiz xi = 4

= − (2 Ffx + 2 Fbx ) h − 2 Ffz x f − 2 Fbz xb = 0

(2.3) (2.4)

where: h = rg + zCM, xf = L/2 – xCM, xb = –L/2 – xCM, xCM and zCM – coordinates of the robot’s mass centre in the robot’s coordinate system, GR = mR g, mR and GR – respectively mass and value of gravity force for the robot, g = 9.81 [m/s2] – the acceleration of gravity, IRy – mass moment of inertia about the axis parallel to y and passing through the mass centre of the robot. Dynamic equations of motion for the robot’s wheels can be written as:

Fig. 2. A simplified model of the robot moving on the inclined terrain resulting in robot’s tilt about its x-axis (a) and about its y-axis (b), driving torque as well as reaction forces and moments of forces acting on the wheel (c) (l ={1, 3}, r = {2, 4}, f = {1, 2}, b ={3, 4}, L = Af Ab = 2AfR = 2AbR, W = Al Ar = 2AlR = 2ArR) 66

(2.2)

Articles

IWy q i = t i − Fix rg + Tiy ± Tt

(2.5)

where: Iiy – mass moment of inertia of a wheel about its spin axis, τi – driving torque, Tiy = – Fiz fr rg – rolling resistance moment, fr – coefficient of rolling resistance, Tt – moment transmitted to the wheel by the toothed belt.

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Moment of force Tt occurs only in case of transmitting drive from one wheel to another by means of the toothed belt. The “+” sign in the above equation pertains to the wheel being driven, and “–“ to the driving wheel. In equation (2.4), the rolling resistance moments of wheels were neglected, because they have only minor influence on dynamics of the robot treated as a whole as compared to other moments of forces. In turn, dynamic equations of motion of the robot, for the case when it is tilted about x-axis at roll angle α = const, have the form (see Fig. 2a): mR yCM = ∑ i =1 Fiy − mR g sin( a) = 0 ,

(2.6)

mR zCM = ∑ i =1 Fiz − mR g cos( a) = 0 ,

(2.7)

I Rx a = ∑ i =1 Fiy h + ∑ i =1 Fiz yi = 0

(2.8)

where: yl = W/2, yr = –W/2, l = {1, 3} , r = {2, 4} , IRx – mass moment of inertia with respect to the axis parallel to x and passing through the robot’s mass centre.

3. Mobility analysis

The mobility analysis of the robot covers two characteristic cases in which it is tilted about x- or y-axis (see Fig. 2a-b). In both cases the robot’s body is in translational motion. In the investigations five types of ground are analysed. The interaction of the robot’s tyre with those types of ground is described with coefficients of friction, adhesion and rolling resistance. Values of those coefficients assumed in the work are summarized in Table 1 – for contact of dry surfaces. Coefficients in the table have the following meanings: μs – coefficient of static friction, μk – coefficient of kinetic friction, μp – coefficient of peak adhesion, fr – coefficient of rolling resistance. Because of specifics of interaction of the tyre with the ground, that is, existence within the contact area of regions of adhesion (where the coefficient μs is valid) and sliding (coefficient μk) in the literature usually the coefficient of peak adhesion μp is introduced, which reflects maximum value of adhesion for tyre-ground pair. The coefficient is defined as ratio of maximum value of longitudinal component of ground reaction force to value of normal component of ground reaction force at the area of tyre-ground contact. The coefficient of adhesion depends not only on types of contacting surfaces of the tyre and the ground, but also, for example, on tyre tread pattern, tyre pressure, etc. In turn, the coefficient of kinetic fricTab. 1. Coefficients of: sliding friction, adhesion and rolling resistance describing interaction of the tyre with selected ground types according to [1,2,3] other sources and author’s own estimations type of ground

μs

μp

μk

asphalt and concrete

1.00

0.85

0.75

0.015

unpaved road

0.80

0.68

0.65

0.050

rolled gravel

0.71

0.60

0.55

0.020

compressed snow

0.24

0.20

0.15

0.032

ice

0.12

0.10

0.07

0.010

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tion μk can be identified with the coefficient of sliding adhesion, frequently introduced in tyre modelling. Due to difficulties with finding in the literature the coefficient of static friction describing the interaction of a rubber tyre with all considered ground types, its value was estimated based on the known coefficients of peak adhesion, with the assumption that μp = 0.85 μs. This relationship is obtained on the basis of values of coefficients in the case of the rubber tyre interaction with dry asphalt, which are easily available.

3.1. Solution for the case of the robot tilted about x-axis

For the mobility analysis of the robot, in the case when it is tilted through angle α = const equations (2.6) – (2.8) are used and additional constraints are introduced. The first constraint follows from the fact, that robot cannot slide sideways, so side components of ground reaction forces are not allowed to exceed the value of developed friction. Taking in account the coefficient of static friction μs for the tyre-ground pair, this constraint can be written in the form: − µ s Fiz ≤ Fiy ≤ µ s Fiz .

(2.9)

One should underline that trivial solution for the range of robot roll angles α: α min1 = − arctan(µ s ) ≤ α ≤ arctan(µ s ) = α max1 . (2.10) in this case depends only on the value of coefficient µs. Limiting values of this angle (i.e., minimum and maximum) for particular types of ground are given in Tab. 2. Tab. 2. Limiting values of terrain inclination and robot’s roll angles for various types of the ground type of ground

αmax1/αmin1 [o]

asphalt and concrete

±45.00

unpaved road

±38.66

rolled gravel

±35.37

compressed snow

±13.50

ice

±6.843

From the fact that the robot cannot experience rollover follows the other constraint, according to which normal components of ground reaction forces must be positive, that is: (2.11) Fiz ≥ 0 . After taking into consideration critical cases, where Flz = 0 (so in consequence Fly = 0) and Frz = 0 (hence Fry=0), one obtains

α min 2 = − arctan(W / (2h) ≤ α ≤ ≤ arctan(W / (2h) = α max 2

(2.12)

It follows that in this case the range of angles of robot roll α depends only on position of the robot’s mass centre and the track width. After taking into account the assumed values of the robot’s parameters, the following solution is obtained: Articles

Journal of Automation, Mobile Robotics & Intelligent Systems

amin 2 = −67.96 [deg] ≤ a ≤ 67.96 [deg] = amax 2

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(2.13)

Eventually, the allowable range of robot’s roll angles depends on ground type as well as on position of the robot’s mass centre and robot’s track width, and is equal to: max(α min1 , α min 2 ) ≤ α ≤ min(α max1 , α max 2 ) ,

(2.14)

where angles αmin1 and αmin2 are negative, while angles αmax1 and αmax2 positive.

3.2. Solution for the case of the robot tilted about y-axis

(2.15)

(2.16)

• for independent drive of each axle Ffx / Fbx = F fz / Fbz , Tt = 0 ,

(2.17)

t p = 0, t a = (2 IWy / rg + mR rg / 2)v R + + mR g rg ( f r cβ − sβ ) / 2

(2.21)

Tt = ± IWy v R / rg +

(

)

− mR v R rg ( f r h ± la )cβ − h(2 sβ − v R / g ) /(2 L cβ ) + − mR g rg ( f r cβ − sβ )(± la cβ − h sβ ) /(2 L cβ )

(2.23)

− mR g tβ (l p cβ ± h sβ ) /(2 L)

(

)

Fpx = − mR v R −la cβ ± h(2 sβ − v R / g ) /(2 Lcβ ) +

(2.24)

+ mR g tβ (−la cβ ± h sβ ) /(2 L)

where: tβ = tan(β) = sβ / cβ, and „+” sign is valid for the case of driving the front wheels, and „–” for the back wheels. 3.2.3. Solution for the case of independent driving of two axles Solution for the case of independent driving of two axles (Tt = 0) has the form: τ f = IWy v R / rg +

(

)

− mR v R rg − ( f r h − lb )cβ + h(2 sβ − v R / g ) /(2 Lcβ ) +

τ b = IWy v R / rg +

3.2.1. Solution for the case of single-axle drive Solution for the case of single-axle drive (a – active axle, p – passive axle, a = f and p = b or a = b and p = f) has the form:

Ffx = mR v R lb cβ + h(2 sβ − v R / g ) /(2 Lcβ ) +

(2.18)

+ mR g rg ( f r c b − s b ) / 2, Tt = 0

(

)

Fax = 2 IWy L / r + mR ( L ± f r h) v R /(2 L) + 2 g

(

)

− mR g Lsβ + f r (−la cβ ± h sβ ) /(2 L)

(

)

(2.19)

Fpx = − 2 IWy L / rg2 ± mR f r h v R /(2 L) + − mR g f r (la cβ ± h sβ ) /(2 L)

(2.20)

where: sβ = sin(β), cβ = cos(β), lf = L/2 ‒ xCM, lb = L/2 + xCM, h = rg + zCM, and the „+” sign is valid for the case of driving the front wheels, and „–” for the back wheels. 3.2.2. Solution for the case of drive transmission from one axle to another by means of the toothed belt Solution for the case of driving one axle, and drive transmission to another axle by means of the toothed belt Articles

(2.22)

)

where: p – passive axle, p = b (for front wheels driven) or p = f (for back wheels driven). Based on the above assumptions, the following solutions for driving torques and components of ground reaction forces are obtained.

t p = 0, t a = (2 IWy / rg + mR rg / 2)v R +

(a – active axle, p – passive axle, a = f and p = b or a = b and p = f ):

(

• for single-axle drive and transmission of drive to another axle by means of the toothed belt Ffx / Fbx = Ffz / Fbz , τ p = 0 ,

2012

Fax = mR v R l p cβ ± h(2 sβ − v R / g ) /(2 Lcβ ) +

In order to obtain solutions of dynamic equations of motion for the case of robot motion on inclined ground, that is, to determine components of ground reaction forces and driving torques, the following assumptions are introduced: • for single-axle driver τ p = 0 , Tt = 0 ,

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+ mR g rg ( f r cβ − sβ )(lb cβ + h sβ ) /(2 L cβ )

(

)

(2.25)

− mv R rg − ( f r h + l f )cβ + h(2 sβ − v R / g ) /(2 Lcβ ) + + mR g rg ( f r cβ − sβ )(l f cβ − h sβ ) /(2 L cβ )

(

)

(2.27)

− mR g tβ (lb cβ + h sβ ) /(2 L)

(

(2.26)

)

Fbx = mR v R l f cβ − h(2 sβ − v R / g ) /(2 Lcβ ) +

(2.28)

− mR g tβ (l f cβ − h sβ ) /(2 L)

3.2.4. Relationships common for all cases The characteristic of the presented dynamic equations of motion is that for each discussed configuration of the robot one obtains the same solution for the normal components of ground reaction forces, that is of the form:

(

)

(2.29)

(

)

(2.30)

Ffz = mR g (lb cβ + h sβ ) − v R h /(2 L) Fbz = mR g (l f cβ − h sβ ) + v R h /(2 L)

It should be also noted that this solution is independent of the type of terrain on which the robot moves. However, the type of terrain will affect values of the tangent components of ground reaction forces, and as a result, the values of driving torques.

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3.3. Results of mobility analysis for the case of robot tilted about y-axis and moving with constant velocity

The analysis of robot’s mobility, for the case in which it is tilted at the pitch angle β, will cover all possible options of drive system configuration. For each option, considered are equations (2.1) – (2.5) and the constraints: Fiz ≥ 0 ,

− µ p Fiz ≤ Fix ≤ µ p Fiz ,

(2.31)

that is all robot wheels must be in contact with the ground at all times and the value of longitudinal component of ground reaction force cannot exceed the value of developed friction force. Limiting values of pitch angle β are determined after considering dynamic equations of motion and particular constraints. In case when for the given constraint is obtained a solution which violates any other constraint, the solution is discarded. Also, presented are values of driving torques and ground reaction forces corresponding to those angles, which may be important from the point of view of required drives’ capabilities and mechanical strength of the robot’s structure. In particular, the values of driving torques can become a decisive factor at the choice of specific robot’s drive system configuration. Because of very complex form of general solution, the following analysis will be conducted for specific robot parameters. In the present work discussed are results of mobility analysis of the robot for the case of motion with constant velocity, and: • front-wheel drive only (Tab. 3), • transmission of drive from back axle to front axle by means of toothed belt (Tab. 4), • independent two-axle drive (Tab. 5). In tables 3-5 are given the limiting values of terrain longitudinal inclination, driving torques and ground reaction forces for given constraints. Because of the fact that the robot’s mass centre is located at the geometric centre of the body, robot’s mobility is affected mainly by the coefficients of peak adhesion μp. With increasing pitch angle, robot can at first undergo downward slide (in case when constraint − µ p Fiz ≤ Fix ≤ µ p Fiz is not satisfied), and only then experience overturn (when constraint Fiz ≥ 0 is not satisfied). Analogous conclusions also pertain to the other considered robot’s drive system configurations.

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Conclusions and future work

From the conducted analysis of robot’s mobility for the case of its motion with constant velocity can be drawn the following conclusions. The worst of the analysed options with respect to robot’s mobility is driving of only one axle, that is, only front or back wheels. In this case, range of angles of longitudinal terrain inclination, at which motion is possible, is the narrowest. Moreover, the range is non-symmetric, so for instance in the case of front-wheel-drive when going up the hill the angle is smaller, than while going down. The allowable range of longitudinal terrain inclination angles is the same for the case of drive transmission from back to front axle and for independent two-axle drive. Additionally, in the case of symmetric robot’s mass distribution between axles, the range is symmetric. After assuming the same total robot’s mass in all analysed cases, it is evident that for single-axle drive and for back-to-front drive transmission, driving torque necessary for motion is equal to the sum of driving torques required for independent driving of both axles. The advantage of independent two-axle drive configuration as compared to the other considered solutions are the smaller required values of driving torques per single drive. The advantage of solution with back-to-front-axletransmission drive is the analogous robot’s mobility with respect to the independent two-axle drive, but with smaller number of applied drives. This solution is associated with minor increase in complexity of design, because of application of additional toothed belts. On the other hand, the belts can enhance robot mobility in case of motion on uneven terrain if used as a caterpillar track – example of this is the PIAP Scout robot of PIAP Poland. After taking this into consideration, design solution like that can be optimal in the analysed case and become a reasonable trade-off between single-axle drive and independent two-axle drive. However, it should be emphasized that in order to point-out the true optimal solution for the robot’s drive system configuration, a more comprehensive research is required. In accordance, during future research will be analysed robot’s mobility including: • more advanced cases of motion (e.g., negotiation of a curved path),

Tab. 3. Results of robot’s mobility analysis for the case of motion with constant velocity and the front-wheel driv type of ground

constraint

β [o]

τf [Nm]

Ffx [N]

Fbx [N]

Ffz [N]

Fbz [N]

asphalt and concrete

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–19.0 28.5

1.95 –2.65

22.52 –31.79

–0.56 –0.33

26.50 37.40

37.17 21.80

unpaved road

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–15.0 23.3

1.76 –2.00

19.24 –25.43

–1.84 –1.22

28.30 37.40

36.76 24.45

rolled gravel

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–14.1 19.8

1.51 –1.84

17.19 –22.34

–0.73 –0.52

28.65 37.23

36.65 26.12

compressed snow

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–4.5 6.9

0.64 –0.51

6.45 –7.08

–1.15 –1.01

32.27 35.39

34.86 31.47

ice

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–2.5 3.2

0.31 –0.26

3.29 –3.45

–0.34 –0.33

32.92 34.54

34.36 32.70 Articles

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Tab. 4. Results of robot’s mobility analysis for the case of motion with constant velocity and transmission of drivefrom back axle to the front axle via toothed belt type of ground

constraint

β [o]

τb [Nm]

Tt [Nm]

Ffx [N]

Fbx [N]

Ffz [N]

Fbz [N]

asphalt and concrete

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–40.4 40.4

3.77 –3.64

1.11 –2.57

12.80 –30.81

30.81 –12.80

15.06 36.25

36.25 15.06

unpaved road

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–34.2 34.2

3.46 –2.98

1.16 –1.98

12.68 –25.19

25.19 –12.68

18.65 37.04

37.04 18.65

rolled gravel

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–31.0 31.0

3.04 –2.85

1.08 –1.84

12.28 –22.37

22.37 –12.28

20.46 37.29

37.29 20.56

compressed snow

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–11.3 11.3

1.30 –0.94

0.59 –0.52

5.96 –7.25

7.25 –5.96

29.81 36.23

36.23 29.81

ice

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–5.7 5.7

0.63 –0.51

0.30 –0.27

3.19 –3.51

3.51 –3.19

31.88 35.13

35.13 31.88

Tab. 5. Results of robot’s mobility analysis for the case of motion with constant velocity and the independent two-axle drive type of ground

constraint

β [o]

τf [Nm]

τb [Nm]

Ffx [N]

Fbx [N]

Ffz [N]

Fbz [N]

asphalt and concrete

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–40.4 40.4

1.11 –2.57

2.67 –1.07

12.80 –30.81

30.81 –12.80

15.06 36.25

36.25 15.06

unpaved road

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–34.2 34.2

1.11 –2.57

2.30 –1.00

12.68 –25.19

25.19 –12.68

18.65 37.04

37.04 18.65

rolled gravel

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–31.0 31.0

1.08 –1.84

1.97 –1.01

12.28 –22.37

22.37 –12.28

20.46 37.29

37.29 20.46

compressed snow

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–11.3 11.3

0.59 –0.52

0.71 –0.43

5.96 –7.25

7.25 –5.96

29.81 36.23

36.23 29.81

ice

Ffx ≤ μp Ffz Ffx ≥ –μp Ffz

–5.7 5.7

0.30 –0.27

0.33 –0.24

3.19 –3.51

3.51 –3.19

31.88 35.13

35.13 31.88

• motion with variable speed (accelerating, braking), • motion with occurrence of wheel sliding, • traversing various obstacles of environment, e.g. kerbs. Also planned are investigations which will consist in analysis of various robot’s drive system configurations from the point of view of ensuring the best accuracy of realisation of motion in the presence of wheels’ slip.

Acknowledgements

The work has been realised as a part of the project entitled “Dynamics modeling of four-wheeled mobile robot and tracking control of its motion with limitation of wheels slip”. The project is financed from the means of National Science Centre of Poland granted on the basis of decision number DEC-2011/03/B/ST7/02532.

AUTHOR

Maciej Trojnacki* – Industrial Research Institute for Automation and Measurements PIAP, Aleje Jerozolimskie 202, 02-486 Warsaw, POLAND, mtrojnacki@piap.pl *Corresponding author

Articles

References

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