Biomechanics of Golf: Inertia Characteristics of the Golfer During the Swing

Biomechanics of Golf:

Inertia Characteristics of the Golfer During the Swing

Steven M. Nesbit, PhD Professor Department of Mechanical Engineering Lafayette College Easton, PA nesbits@lafayette.edu

Michael D. Jacobs PGA Golf Professional Jacobs 3D Golf Rock Hill Golf Club Long Island, NY mj@jacobs3d.com

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1. ABSTRACT 2. Background An analyses of the global tensor of the golfer/club system may reveal important quantities and characteristics that affect the golf shot. This type of analyses has been performed for acrobatic (torque-free) movements. No such analyses have been done for the golfer/club system (non-torque free) movement.

2. Objective Quantify and discuss the global inertia tensor, and associated principal inertia values and directions of the golfer/club system during the swing as performed by 14 male and one female professional and amateur subjects of various skill levels and body types. Correlate the results to various measures of swing performance to determine the relative importance of the global inertia quantities.

3. Methods This study was performed using a 17-segment model of the body combined with a rigid model of the club. Data to drive the model was obtained from subject swings recorded using an 8-camera motion capture system. Body segment mass and inertia data were specified using anthropomorphic data based upon subject height, weight, and gender, and segment lengths were determined from maker data. The global inertia tensor of the golfer/club system (and principal values and directions) was determined relative to the global centre-of-mass (COM) using methods of tensor mechanics.

4. Results The analyses determined that variations of the principal values and directions of the golfer/club system were substantial during the swing motion, that the principal values generally aligned with two stability and one motion axes, and that skill level was a factor in the manipulations of the principal inertias and directions. Comparisons among subjects revealed that while subject-to-subject variations were significant, patterns and correlations related to skill level and swing performance measures were discovered. The minimum value of the 1st principal inertia (absolute, ratio, normalized, and timing) was strongly correlated to several of the performance measures.

5. Conclusions The results support the hypothesis that the global inertia tensor is an important mechanical quantity of the golfer/club system during the swing motion and is related to skill level and swing performance. It appears that the golfer senses and controls the relative principal values and directions to positively influence the golf shot, however the degree to which this occurs merits further investigation.

Keywords: Golf biomechanics, computer modeling, inertia tensor, golf swing performance, golf swing stability, principal inertia values, principal inertia directions

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2.0 INTRODUCTION Since the golf shot is one of the most difficult biomechanical motions in sport to execute, a detailed understanding of the comprehensive 3D mechanics of the swing would be beneficial to the golfer and teacher (Dillman and Lange, 1994). The execution of the golf swing requires carefully timed movements of all the segments of the body which are coordinated with the purposeful movement of the club to achieve the ultimate and elusive goal of producing an accurate and high-speed golf shot (Thomas, 1994). These complex and coordinated movements may result in changes to the global inertia tensor of the golfer/club system during the complete swing motion (Hinrichs, 1978).

This global inertia tensor may be relevant to the overall resistance the golfer must overcome in moving the body and swinging the club, the manner in which the golfer interacts with the ground, and the relative stability of the golfer/club system. The study of the global inertial tensor characteristics of the golf/club system may offer an informative and beneficial measure of swing mechanics to quantify, visualize, and summarize the dynamic effects of the movements of all the body segments and club during the entire motion, and as such may be a factor in golf swing performance.

Studies of the global inertia tensor characteristics of the human body performing athletic motions are limited in scope and context. The tensor characteristics of the human body during acrobatic movements have been studied by Jenkins (2018), Yeardon and Mikulcik (1996), and Hinrichs (1978). These studies were performed in the limited context of conservation of angular momentum situations only since the athlete is not in contact with the ground thus experiencing non-torque motion. Jenkins quantified the 3D full-body inertia tensor values at various points during a backwards flip movement. He found that the inertia tensor was diagonally dominant and ranged from a maximum value of 11.09 kg- m2 about an axis going through the body COM in the lateral direction while in the take-off position, to a minimum of 2.8 kg-m2 about the flipping rotation axis while in the tuck position. He did not determine the principal values or directions as part of his analyses. Theoretical analyses have shown that rotations of a rigid body about the principal axis corresponding to the intermediate principal moment of inertia are unstable (Haug, 1992), and may affect humans

performing rotational dominant acrobatic motions (Hinrichs, 1978).

Yeadon and Mikulcik (1996)

determined that this effect was a potential motivation for some movements of acrobatic athletes by causing the build up of twist about an axis other than the desired rotation axis. Purposeful body movements by the athlete including arm abduction and body flexion are often employed by the athlete to prevent and/or mitigate this issue. Thus, Yeadon and Mikulcik (1996) and Hinrichs (1978) have shown that for torque-free body motions the orientation of the full-body principal axes, and the relative values of the principal inertias play a role in the relative stability and uniformity of the motion.

These studies have shown that quantifying the global inertia tensor of the body has value in describing and understanding the associated human movement and actions of the athlete. However, such analyses has yet to be applied to the full-body inertia tensor characteristics of non-torque-free athletic motion, or athletic motions that involves an implement, and more specifically to the golfer/club system during the entirety of the swing. Thus the global inertia tensor of the golfer/club system has not been quantified, described, and its function and importance relative the golf swing are not known.

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The goals of this study are to quantify, analyze, and determine the relative importance of the global inertia tensor characteristics of the golfer/club system during the full swing motion. The study employs a full-body model of the golfer linked to a model of the club. This model is applied to a large sampling of subjects in an attempt to discover where differences in the global inertial tensor of the body/club system reveal themselves in swing performance measures. In summary, the purposes of this study are the following: •

Quantify the global inertial tensor of the golfer/club system during the entire swing motion

•

Determine the principal inertia values and directions from the global inertia tensor

•

Correlate the principal values to several established swing performance measures

•

Analyze several diverse subjects for statistical information of all quantities

•

Highlight similarities and differences among selected subjects

•

Offer practical advice to golf practitioners to improve performance based upon implementing significant findings

1. METHODS 2. Golfer/Club Model A seventeen-segment full-body model of the golfer interfaced to a rigid model of a golf club (Figure 1) was developed using methods described in Nesbit (2005a, 2007), Huston (2013), Kane et al (1983), and Craig (1986) in order to determine the global 3D inertia tensor of the golfer/club system, and the relevant performance measures during the golf swing. The basic model development is summarized herein with new analyses presented in detail. The individual segment inertia properties (mass, local COM locations, and principal inertia values) of the body portion of the model were specified from anthropomorphic data based upon subject height, weight, and gender (Huston, 2013). The relative segment mass (size of local sphere) and local COM location (origin of sphere) can be seen in Figure 1 on the left-hand side stick-figure. The properties of the club were measured experimentally using methods described in Nesbit et al.

(1994). The model was driven kinematically via 3D joint trajectories determined from motion capture data using experimental and analytical methods discussed below.

This information provides the basic quantities needed to

determined the global inertia tensor of the golfer/club system as described in the following sections. The global reference is shown in Figure 1. Relative linear directions of the golfer with respect to this reference are: forward(X)/backward(+X), lead(-Y)/trail(+Y), and upward(+Z)/downward(-Z). Angular reference directions from Jacobs (2019) are yaw (Y-Z plane), pitch (X-Z plane), and roll (X-Y plane).

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Figure 1 – Seventeen-Segment Human Model Plus Rigid Club Model Showing Segment (LHS) and Composite (RHS) COM Locations – Global Coordinate Reference is Identified

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3.2 Linear and Angular Kinematic Analyses, and Joint Trajectory Motions The data needed to determine local joint positions, global linear positions of the COM’s of the body segments and club, and the global angular positions of the body segments and club were obtained experimentally using a motion capture system to record subject swing trials (see below). All body segments were tracked with at least three markers as demonstrated for the lower arm and club in Figure 2, with the exception of the upper legs and lower arms. Thus, it was possible to directly determine the global 3D orientation matrix for each body segment (Craig, 1986). The missing information for the upper legs and arms were determined from the lower legs and arms due to the hinge-joint constraint of the knees and elbows (Nesbit, 2007).

Figure 2 – Generic Body Segment and Club Model Highlighting Marker Configurations and Local References

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From marker information, the rotation matrix of the body segment (seg) with respect to the global coordinate system (G) is described as:

and for the club (club):

𝑅𝑅𝐺𝐺𝑠𝑠𝑒𝑒𝑔𝑔

(2)

𝑅𝑅𝐺𝐺𝑐𝑐𝑙𝑙𝑢𝑢𝑏𝑏

(3)

The global body segment and club angular positions (body 1-2-3: αG, βG, and γG: Kane et al., 1983) are extracted from the rotation matrices of Equations (2) and (3) using methods described in Nesbit (2005) and Craig (1986). The local joint angles of adjacent body segments (including the club with respect to the lower arm) are found from:

[ R] G P

−1 G D

R= DPR

(4)

where D is the distal segment, and P is the proximal segment. The relative (local) joint angles (αL, βL, and γL) are contained inside the P R matrix and are extracted from the P R matrix using well-known methods from Craig (1986).

D D Basic interpolation methods using the global marker position data, and the local COM data for each body segment and the club are used to determine the global COM position for each body segment and the club. Numerical differentiation is applied to the local joint angles, the global segment angular positions, and the global segment linear positions. A combination of central-difference methods of order ∆t4, and forward-difference and backward difference methods of order ∆t2 was most effective (Dean and Nesbit, 1988). From this information, local and global angular velocities and accelerations of the body segments, club, and body joints, and the global linear velocities and accelerations of the segment mass center locations were determined using methods described in Nesbit (2007), Craig (1986), and Kane et al. (1983).

3.3 Dynamic Equations of Motion The Iterative Newton-Euler Dynamic Formulation was used to derive the dynamic equations of motion for the model. This method is often used for serial chain robot manipulator arms (Craig, 1986), and was adopted to the golfer/club biomechanical model in Nesbit (2007). Referring to Figure 3 which shows free-body diagrams of three adjacent body segments i-1 (proximal), i, and i +1 (distal), the force and moment balance on body segment i yield Newton’s Equation (5) and Euler’s Equation (6):

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Figure 3 – Free-Body Diagrams of Adjacent Body Segments in a Serial Chain Configuration

i i 1

R i 1f i

i

f i=

i

ni = i N i

i i 1

1

i

Fi

Ri 1 ni

(5) i

1

Pc i

i

Fi

i

Pi

1

i i 1

R i 1f i

1

(6)

where the inertial forces and moments are determined from: i 1

Fi

i 1

N i 1 =i 1Ic i

1

= mi

1

i 1

Ac i

i 1 • 1

(7)

1

ωi 1

ωi 1

i 1

i 1

Ic i

1

i1

ωi 1

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(8)

where for all variables, the subscript refers to the body segment, the superscript to the body segment frame of reference and: R is the 3x3 rotation matrix f is the 3x1 vector of link interaction forces F is the 3x1 vector of inertia forces at the link COM n is the 3x1 vector of link interaction torques N is the 3x1 vector of inertia moments Pc is the 3x1 position vector of the link COM P is the 3x1 position vector of the link Ac is the 3x1 linear COM acceleration vector m is the link mass Ic is the 3x3 inertia tensor about the link COM

ω is the 3x1 angular velocity vector •

ω

is the 3x1 angular acceleration vector

Equations (5) through (8), the rotation matrices, and the free-body diagrams of Figure 3 are applicable to the humanoid model because of the serial arrangement of the rigid links and joints in configuring the arms, legs, torso, and head and neck. The indicial form of these equations facilitates the programming of these expressions to either solve them directly as a numerical algorithm, or to derive the symbolic equations of motion in the following manner (Craig, 1986; Nesbit, 2007). The pelvis/lumbar body segment is designated as segment 1. Outward iterations through the torso and legs from this segment compute the inertial forces and torques acting on each segment. Once the ends of each serial chain are reached (club, feet, and head), the iterations reverse and work inward to determine joint forces and torques.

Adding a club to the model complicates this process due to the closed-looped configuration. In addition, an appropriately configured ground surface model is necessary to support and balance the model, and to yield a properly constrained and solvable system (Nesbit, 2007). The entire model including the equations and graphics that follow was programmed, and solved with MATLAB. 3.4 Global Center-of-Mass of Golfer/Club System From the global positions of the segment and club COM’s, the net COM relative to the global coordinate system shown in Figure 1 was calculated from (Choi et al., 2016):

𝐶𝐶𝑂𝑂𝑀𝑀𝑁𝑁𝑒𝑒𝑡𝑡 =

1 𝑀𝑀

∑𝑖𝑖𝑛𝑛=0 𝐶𝐶𝑂𝑂𝑀𝑀𝑖𝑖 x 𝑚𝑚 𝑖𝑖

(9)

where M is the whole-body mass plus club, mi is the mass of the ith segment, COMi is the COM of the ith segment, and n is total number of segments. The global inertia tensor of the golfer/club system is determined about this point.

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3.5 Global Inertia Tensor of Golfer/Club System The general (local and global) 3D inertia tensor of a rigid body is represented in the form: 𝐼𝐼𝑥𝑥𝑥𝑥 𝐼𝐼𝑥𝑥𝑦𝑦 IGENERAL = [ 𝐼𝐼𝑥𝑥𝑦𝑦 𝐼𝐼𝑦𝑦𝑦𝑦 𝐼𝐼𝑥𝑥𝑧𝑧 𝐼𝐼𝑦𝑦𝑧𝑧

𝐼𝐼𝑥𝑥𝑧𝑧 𝐼𝐼𝑦𝑦𝑧𝑧 ] 𝐼𝐼𝑧𝑧𝑧𝑧

(10)

𝑠𝑠𝑒𝑒𝑔𝑔 The local body segment and club inertia tensors (𝐼𝐼𝑙𝑙𝑜𝑜𝑐𝑐𝑎𝑎𝑙𝑙 ) are relative to their respective local segment COM, and are

aligned with their local coordinate systems (Figure 2). The global inertia tensor of the golfer/club system is

determined by transforming the inertia tensors of the individual body segments and club to the global coordinate system at the composite COM location of the golfer/club system using (Haug, 1992): 𝐼𝐼𝑠𝑠𝑒𝑒𝑔𝑔 = 𝑖𝑖𝑛𝑛𝑣𝑣(𝑅𝑅𝑠𝑠𝑒𝑒𝑔𝑔) ∗ 𝐼𝐼𝑠𝑠𝑒𝑒𝑔𝑔 ∗ 𝑖𝑖𝑛𝑛𝑣𝑣(𝑅𝑅𝑠𝑠𝑒𝑒𝑔𝑔)𝑇𝑇 + 𝑚𝑚𝑎𝑎𝑠𝑠𝑠𝑠 ∗ (𝐶𝐶𝑂𝑂𝑀𝑀 𝑠𝑠𝑒𝑒𝑔𝑔 ∗ (𝐶𝐶𝑂𝑂𝑀𝑀 𝑠𝑠𝑒𝑒𝑔𝑔)𝑇𝑇 ∗ 𝐸𝐸 − 𝑜𝑜𝑢𝑢𝑡𝑡𝑒𝑒𝑟𝑟 𝑝𝑝𝑟𝑟𝑜𝑜𝑑𝑑) (11) 𝑠𝑠𝑒𝑒𝑔𝑔 𝐺𝐺

𝐺𝐺

𝑙𝑙𝑜𝑜𝑐𝑐𝑎𝑎𝑙𝑙

𝐺𝐺

𝑠𝑠𝑦𝑦𝑠𝑠

𝑠𝑠𝑦𝑦𝑠𝑠

where 𝐼𝐼𝑠𝑠𝑒𝑒𝑔𝑔 is the general 3D inertia tensor of a body segment, 𝑚𝑚𝑎𝑎𝑠𝑠𝑠𝑠𝑠𝑠𝑒𝑒𝑔𝑔 is the mass of the body segment or club, E is 𝐺𝐺

𝑠𝑠𝑒𝑒𝑔𝑔 the 3x3 identity tensor, 𝐶𝐶𝑂𝑂𝑀𝑀𝑠𝑠𝑦𝑦 is the global position vector from the composite COM of the golfer/club system to

the local COM of a body segment or the club, and outer prod is the outer product of 𝐶𝐶𝑂𝑂𝑀𝑀𝑠𝑠𝑒𝑒𝑔𝑔 𝑠𝑠𝑦𝑦𝑠𝑠 . T and inv are the

standard matrix transpose and inverse functions respectively. The total global inertia tensor of the golfer/club system

at the golfer/club COM can be found by element-by-element addition of the segment and club inertia tensor at each point in time of the digitized full swing motion. 𝑔𝑔𝑜𝑜𝑙𝑙𝑓𝑓𝑒𝑒𝑟𝑟/𝑐𝑐𝑙𝑙𝑢𝑢𝑏𝑏 𝐼𝐼𝑔𝑔𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 = ∑ 𝐼𝐼𝑠𝑠𝑒𝑒𝑔𝑔 + 𝐼𝐼𝑐𝑐𝑙𝑙𝑢𝑢𝑏𝑏 𝐺𝐺 𝐺𝐺

(12)

Once the time history of the global inertia tensor of the total golfer/club system is quantified for the entire swing, the principal inertia values and associated principal directions were determined at every point in the digitized swing of the subject using standard methods of tensor analyses (see Haug, 1992; note that the principal values will be ordered as I1 < I2 < I3, and the principal directions associated accordingly).

6. Club and Golfer Performance Measures The following club and golfer performance measures were determined from the golfer/club model to help assess the relative importance of the global inertia tensor characteristics and behaviors during the execution of the golf shot.

These include: 1) Club head speed at impact 2) The overall maximum work done by the club, and joints of the body 3) The overall maximum kinetic energy of the club, and the golfer/club system 4) The magnitude and direction of the maximum overall angular momentum of the golfer/club system

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The methods used to calculate the work and kinetic energy of the club, the work of the body joints, and the kinetic energy of the body are found in Nesbit (2005a, 2005b, 2007, 2009). An analyses of the angular momentum of the golfer/club system has not been reported in the literature. Previous work concerning the analyses of the full-body inertia tensor characteristics during an athletic (torque-free) movement included much qualitative discussions of the global angular momentum about the global COM of the system (Hinrichs, 1978; Jenkins, 2018; Yeardon and Mikulcik, 1996), thus it is included here. For the golfer/club system, the importance of the relationship between the global inertia tensor and the angular momentum vector is not clear since the system is not torque-free. Classical mechanics would indicate that the change in the direction and/or the magnitude of the angular momentum vector would result in a moment about the COM of the golfer/club system that would be reacted in the feet according to: ̇ 𝐻𝐻𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 = 𝑀𝑀 𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙

(13)

where 𝐻𝐻̇𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 is the time rate of change of the global angular momentum vector, and 𝑀𝑀𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 is the global moment vector about the COM of the golfer/club system. The angular momentum of a body segment (and club) about the

𝑠𝑠𝑒𝑒𝑔𝑔 composite COM of the golfer/club system in the global reference systems (𝐻𝐻𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 ) is found from: 𝑠𝑠𝑒𝑒𝑔𝑔 𝑠𝑠𝑒𝑒𝑔𝑔 𝑠𝑠𝑒𝑒𝑔𝑔 𝑠𝑠𝑒𝑒𝑔𝑔 𝑠𝑠𝑒𝑒𝑔𝑔 𝐻𝐻𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 = 𝑅𝑅𝐺𝐺𝑠𝑠𝑒𝑒𝑔𝑔 ∗ (𝐼𝐼𝑙𝑙𝑜𝑜𝑐𝑐𝑎𝑎𝑙𝑙 ∗ 𝜔𝜔𝑙𝑙𝑜𝑜𝑐𝑐𝑎𝑎𝑙𝑙 ) + 𝐶𝐶𝑂𝑂𝑀𝑀𝑠𝑠𝑦𝑦𝑠𝑠 𝑐𝑐𝑟𝑟𝑜𝑜𝑠𝑠𝑠𝑠 (𝑚𝑚 𝑖𝑖 ∗ 𝑉𝑉𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 )

(14)

where 𝜔𝜔𝑠𝑠𝑒𝑒𝑔𝑔 is the angular velocity vector of the body segment about the local coordinate system, 𝑉𝑉𝑠𝑠𝑒𝑒𝑔𝑔 𝑙𝑙𝑜𝑜𝑐𝑐𝑎𝑎𝑙𝑙

𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙

is the linear

velocity of the COM of the segment in the global coordinate system. The global angular momentum of the golfer/club

system about the composite COM, (𝐻𝐻 𝑏𝑏𝑜𝑜𝑑𝑑𝑦𝑦/𝑐𝑐𝑙𝑙𝑢𝑢𝑏𝑏 ) is the sum of the global angular momentums of all the body 𝐺𝐺𝑙𝑙𝑜𝑜𝑏𝑏𝑎𝑎𝑙𝑙 segments and club as determined by Equation (14).

4. 0 SUBJECTS AND TESTING PROTOCOL 1. Subjects The subjects of this experiment were 14 male golfers of height (1.801 ± 0.052 m), weight (78.342 ± 10.928 kg), age (30.647 ± 11.921 years), and skill level (4 professional subjects and 5.615 ± 7.478 handicap for the 10 amateur subjects), and 1 female golfer (data withheld to protect subject). The subjects were all right-handed.

2. Experimental Setup, and Data Collection and Processing Procedures The experimental setup, and data collection and processing procedures follow standard methods associated with camera-based motion capture systems utilized in the recording of subjects executing golf swing trials (Nesbit et al.

1994, 2005a, 2005b, 2007, 2009). An overview of the methods will be presented here with any differences and/or additions relative to basic procedures highlighted and explained in more detail. An eight-camera motion capture system (Gears, Inc.) tracked passive-reflective markers that were strategically placed on the golfer and the club. There were 27 markers placed on the golfer and six on the club. On the golfer the markers were placed at the wrists/hands (2), elbows (2), shoulders (2), upper arm (2), upper back (2), mid-back (2), lower back (2), head (3), hips (2), knees

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(2), and feet (6). Two sets of three markers were placed on the club, three (3) just below the handle, and three (3) on the clubhead, with each set of three defining a plane.

All subjects used a driver of their choice for their respective trials. The mass properties were measured, and other relevant information collected for the club used by each subject. The subjects stretched and warmed up in accordance with normal protocols. The Gears Motion Capture system was calibrated until the combined 3D residual for all cameras was less than 1.00 mm. The subjects were asked to execute a series of “competitive effort” swings that consisted of hitting a real ball placed on a tee into a simulator. Algorithms within the motion capture system detected the impact with the ball. Data were collected at 360 Hz then smoothed using a single application of five-point rectangular sliding-average methods. Eight trials were recorded for each subject. Poor trials as reported by the subject (uncomfortable swing, poor flight of the ball, etc.) were disregarded, and the trial repeated. At the conclusion of the trials, the subjects assessed (approved/rejected) each swing trial based upon an overall visual assessment of the motion capture data. A single representative trial from each subject was used for this study.

1. RESULTS 2. Global Principal Inertia Values The global principal inertia values for the aggregate group of all male subjects, the female subject, and professional subjects from the aggregate group are given in Table 2. The table presents the maximum, minimum, average, range, and standard deviation of the three principal inertia values. The average time relative to impact (time = 0 seconds is impact) when the average maximum and minimum principal values occurred is given in parentheses where relevant. Additionally, the following inertia-based data are presented: The ratio of the 1st principal inertia value at impact over the mass of the subject (1st normal.); and the ratio of the 1st principal inertia at impact over the 1st principal inertia at address (1st ratio). The 1st principal value (and associated quantities) was assessed from address to impact only since this value aligns with the swinging motion, and the relevant swing the performance data occurs during this portion of the swing as can be seen in Table 3 (Nesbit, 2005). The data for the 2nd and 3rd principal values were assessed for the entire motion as these values align with the stability directions which are relevant for the entire swing motion.

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1st Principal (kg-m2) 3.451/2.108

2nd Principal (kg-m2) 12.731/10.721

3rd Principal (kg-m2) 13.421/11.682

(-0.611/-0.192)

(0.049/-0.053)

(0.114/-0.470)

Maximum

4.680/2.747

15.780/13.270

Minimum

3.442/2.083

Range

Average

Std. Dev. Female Ave Pros

1st Normal. (m2) 0.043/0.027

1st Ratio

16.550/14.402

0.053/0.032

1.040

9.690/7.663

10.120/8.670

0.039/0.022

0.803

1.238/0.664

6.090/5.607

6.430/5.770

0.014/0.010

0.237

(0.230/0.303)

(0.869/0.794)

(0.817/1.247)

0.419/0.221

2.295/1.927

2.438/2.022

0.004/0.003

0.071

(0.078/0.098)

(0.332/0.322)

(0.277/0.322)

2.447/1.843

8.233/6.904

8.757/7.661

0.043/0.026

0.961

(-0.611/-0.326)

(-0.356/-0.058)

(0.331/0.011)

3.601/2.393

13.115/10.749

13.668/11.625

0.045/0.030

1.005

(-0.528/-0.255)

(-0.153/-0.018)

(0.232/0.000)

0.931

Table 2 – Principal Inertia Values for All Subjects for the Golfer/Club System Figures 4a-f present plots of the absolute and normalized principal inertia values of the golfer/club system for the four professional subjects, the female subject, and three male amateur subjects (0 HDC, 8 HDC, and 20 HDC). The LHS plots are the absolute 1st, 2nd, and 3rd principal inertia values, and the RHS is the associated normalized 1st, 2nd, and 3rd principal values. Graphically, the professional subjects are indicated with solid lines, the amateurs with line/symbol combinations, and the female subject with symbols only to facilitate visualization of these groupings. Note that including all subjects in these graphs rendered them indecipherable. The subjects that were included were deemed representative of their respective sub-groups. Animations of a professional subject, 0 HDC, 8 HDC, and 20 HDC are included with a view point from every angle of the golfers’ swing.

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Animations of each of four players from all different viewing angles.

Principal Inertia Values (LHS Graphs) vs Normalize Principal Inertia Values (RHS Graphs) for selected subjects and groupings.

5.2 Global Principal Directions In order to visualize and facilitate subject comparisons of the principal inertia values, and especially the principal directions, an inertially equivalent ellipsoid is superimposed over the golfer/club system at different points in the swing (Figure 5). This ellipsoid, which is sized to have the same absolute principal inertia values as the golfer/club

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system at every instant during the swing, is centered on, and moves with the COM of the golfer/club system, and is oriented according to the principal directions (the principal axes are included with the ellipsoid, and are color coded accordingly). The principal values are further illustrated in the upper portion of the figure by color-coded inertially equivalent spheres and include the instantaneous numerical values of the golfer/club system (top and larger font), and golfer body alone (bottom and smaller font). The ellipsoid, and spheres will change size, and in the case of the ellipsoid, orientation to reflect their respective values and directions during the swing. The global trajectories of the golfer/club COM (body+club), and golfer alone COM (body only) are also shown although these quantities are beyond the scope of this study. An additional graphic (black vector) that can be seen in some frames of this graphic is the angular momentum vector.

Figures 5a-d, Inertia Ellipse Relative Values at Impact (note angular momentum vector) (Note that subject is a male professional golfer of medium build)

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Figures 6 through 11 utilize these graphical means to compare the principal inertia values and directions of selected subjects at important points in the swing progression (address, top of the backswing, club in vertical position during downswing, club in horizontal position during downswing, impact, and near the end of the follow-thru). These figures can be seen for a professional subject (Figures a - top LHS), a zero-handicap amateur (Figures b - top RHS), an eight- handicap (Figures c - bottom LHS), and a twenty-handicap subject (Figures d - bottom RHS). The vertical and horizontal positions of the club during the downswing are considered significant since the vertical position corresponds to the maximum total body/club work, and the horizontal position corresponds to the maximum total body/club angular momentum. Other important events happen at these club positions as well (Nesbit, 2005a; Jacobs, 2019).

Figures 6a-d – Address Figures 7a-d – Top of the Backswing Figures 8a-d – Downswing to Vertical Club Figures 9a-d – Last Parallel Club Position Figures 10a-d – Impact

Figures 11a-d – End of Follow-Thru

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The data of Table 2, Figures 4 and Figures 6 thru 11 have not been reported in the literature for the golfer/club system. Jenkins (2018) reports the complete global inertia tensor value in the takeoff position of an aerobic movement for a male subject of 1.75 m in height, and 65.3 kg in weight. The principal values were determined from this information and correspond to [2.115, 11.090, 11.385] kg-m2 for the absolute 1st, 2nd, and 3rd principal inertia respectively. These values are slightly lower than what is presented here, however the subject in Jenkins was of average height, and below average weight, analyzed for a different posture, and did not have an implement when performing the movement. Data on the principal directions of the global inertia tensor of the golfer/club system has not been previously reported.

5.4 Swing Performance Measures Table 3 presents the data for the performance measures for the aggregate group of male subjects. These data are based upon the maximum values that occur during the entire swing motion except club head speed which is reported at impact. The table presents the average, maximum, minimum, range, and standard deviations of all quantities. In addition, the associated time for each of the above data where relevant is in parentheses below the quantity. These data agree well with the literature (Nesbit et al., 2005a, 2005b, 2007, 2009) with the exception of maximum angular momentum which has not been previously reported. These data represent a broad range of performance measures that form the basis of correlation analyses which follows.

Measure

Average

Max Angular

(N-m)

Body/Club

Momentum

(N-m)

(kg-m2/sec)

174.12

292.15

22.12

(-0.109)

(-0.018)

(-0.033)

(-0.038)

533.00

1216.00

270

441.0

29.22

(-0.008)

(-0.002)

(-0.001)

(-0.013)

(0.050)

143.00

371

132.7

220.0

14.79

(-0.036)

(-0.119)

(-0.033)

(-0.052)

(-0.083)

390

845

137.3

221.0

14.43

(0.028)

(0.117)

(0.033)

(0.039)

(0.133)

95.58

202.56

28.69

54.84

4.38

(0.008)

(0.046)

(0.009)

(0.010)

(0.032)

Max Work Body/Club

(m/sec)

(N-m)

(N-m)

45.10

277.59

533.42

(-0.016) 49.70

Minimum

39.35

Std. Dev.

Max KE

Max Work Club

Maximum

Range

Max KE Club

Club Head Speed Impact

10.35 2.71

Table 3 – Performance Measures of Aggregate Male Subjects

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5.5 Significant Correlations Linear regression analysis was performed to test the relationships between the independent variables of the principal inertia values (maximums, minimums, ranges, and the timing of occurrence), and dependent variables of the performance measures. Table 4 provides the most significant correlations found. This table presents the independent variable, dependent variable, R2 values, and the slope and Y-intercept values for the linear curve fits that satisfies a threshold of R2 ≥ 0.500. Where indicated and warranted, some entries include both maximum and minimum independent variables. Note that skill level was not included in the regression analysis since handicap does not indicate a quantifiable difference for professional subjects, and subjects with a zero handicap.

Independent Var

Dependent Var

R2

Slope

Intercept

Min Normalized 1st

Club Head Speed

0.6354

793.86

24.303

Club Head Speed

0.5590

-27.233

57.647

Club Head Speed

0.630

24.28

22.848

Max Total Body/Club

0.5711

301.01

-124.78

Principal Value Time of 1st Min Principal Value 1st Ratio Min 1st

Principal

Value

Work

Min Normalized 1st

Max Club Work

0.5071

9440.8

-60.317

1st Ratio

Max Club Work

0.564

339.33

-124.72

Min Normalized 1st

Max KE Body/Club

0.8073

153.71

-31.851

Max/Min 1st Principal

Max Angular

0.7558/0.8151

6.1472/11.48

0.9029/-2.0783

Value

Momentum 0.8151/0.8134

1.7232/2.0504

0.1798/0.1359

0.8284/0.8603

11.48/2.0096

0.1767/-1.3559

Principal Value

Principal Value

Max/Min 2nd

Principal

Max Angular

Value

Momentum

Max/Min 3rd Principal

Max Angular

Value

Momentum

Table 4 – Most Significant Correlations Identified

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1. DISCUSSIONS The following discussions are based upon the results for all subjects. Discussions of specific subjects are intended to be exemplary of the composite group unless noted otherwise.

1. Golfer/Club System Global Inertia Tensor The global inertia tensor of the golfer/club system represents a quantifiable summary of the mass, inertia, COM location, and movements of the various body segments and club relative to the global mass center of the golfer/club system. Referring to Figure 5, this global inertia tensor reveals principal values and associated principal directions that align with fundamental directions of the golf swing. The 1st principal value and associated gamma/green direction aligns with the long axis of the golfer/club system, or the primary swing axis for all subjects. This inertia value indicates the overall swing resistance that the golfer must manipulate and overcome through interactions with the ground to effectively swing the club. The 2nd principal value and associated alpha/red direction generally aligns with the lateral side-to-side bending motion of the torso, and this axis rotates with the torso as the subject moves through the swing phases. This inertia quantity has a stabilizing effect to counter the wide arc accelerated motion of the arms and club during the swing. The 3rd principal value and associated beta/blue direction generally aligns with the relative anterior-posterior or forward-to-back bending motion of the torso, and this axis also rotates with the torso as the subject moves through the swing phases. This inertia quantity also has a stabilizing effect due to its large value that acts to counter the significant centrifugal loading of the club, especially at impact. Taken together, the alpha and beta axes create a stability plane of high relative inertia with respect to the orientation of the golfer/club system. This stability plane contributes to the overall vertical balance of the golfer who remains in a nearly fixed stance during a high speed and complex 3D motion of all the body segments and the club. These directional alignment trends apply to most subject, however some lesser skilled subjects exhibited switched alpha and beta principal directions (see Figures 6c and 7c).

The global inertia tensor of the golfer/club system is found to be a dynamic quantity that undergoes significant variation which is evident from both the changing magnitudes of the principal inertia values (Figures 4), and the changing orientations of the principal directions (Figures 6 through 11). The global inertia tensor is established at a baseline level at address by the initial stance of the subject. Both the principal values and directions oscillate above and below relative to this baseline value for most subjects (all the professional subjects demonstrated this behavior).

Applying Equations (11) and (12) to individual components of the golfer/club system reveals that the movement of the club has a large effect on the tensor values as expected, and is evident by the differences in principal inertia values between the golfer/club system and the golfer body alone (Figures 6 through 11). From a relative perspective, the motion of the club has the most significant effect on the 1st principal value (approximately 10% of total for most subjects). Another contributor to the dynamic nature of the inertia tensor is the movements of the arms (upper and lower) and hands. As will be discussed further, the position of the arms at various points in the swing and the

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associated effect on the 1st principal value is an important factor in swing performance. The third contribution comes from the movement of the legs (upper and lower) and feet (which move little). It was found that the width of the leg stance affects the relative difference between the 2nd and 3rd principal values, and can cause a switching of the alpha and beta principal axes for wide stances. Lastly, the body segments of the torso (lumbar, thoracic, chest, neck, and head) contribute greatly to the absolute tensor value, however little to the dynamic variation since these body segments primarily rotate thus do not move linearly a significantly amount relative to the COM of the golfer/club system about any axes. Isolating contributions of the club and body segment movements to the changes in principal directions is more difficult due to the indirect relationship between Equations (11) and (12) and the principal directions, thus these will be analyzed and discussed relative to the full body movements of Figures 6 through 11 in Section 6.3.

6.2 Golfer/Club System Global Principal Inertia Values The data of Table 2 and the graphs of Figures 4a, 4c, and 4e show that all three principal inertia values vary considerably among the subjects, between the three principal values for a given subject, and during the swing motion itself. These differences include the maximum and minimum values, the timing of occurrences, and the overall profiles. These become more consistent among subjects when considering the normalized principal values (Figures 4b, 4d, and 4f). The 1st principal inertia value varies the most among subjects, and for a given subject from a relative perspective. It is established at approach, reaches a local maximum for the aggregate group (3.451 kg-m2) when the club is in the horizontal position in the backswing (-0.611 sec), then a local minimum (2.108 kg-m2) near the top of the backswing (-0.292 seconds). For the subgroup of professional golfers the maximum and minimum values were higher, and occurred later in the swing, and for the female subject they were considerably lower and exhibited timing differences. The 1st principal value reaches a local maximum during the downswing when the club was near the vertical position. From this point until impact, the 1st principal inertia dropped rapidly. The value at impact was lower compared to address (1st ratio) for the aggregate group (0.931), near equal for the subgroup of professional golfers (1.005), and 0.961 for the female subject. Analyses of the normalized 1st principal inertia profile (Figure 4b) reveals patterns and grouping of the higher skilled subjects with the exception of one professional subject. The profiles of all subjects are fairly similar from address to the top of the backswing. At this point to impact, the lesser skilled subjects demonstrated smaller ranges and lower local minimums during the downswing, and tended toward lower values and declining profiles overall compared to higher skilled subjects. The normalize profile for the female subject was similar to that of the higher skilled subjects. The ability of a subject to manipulate the 1st principal value from address to impact is an important predictor in swing performance (see below).

Data for the 2nd principal inertia shows that for the aggregate group the average maximum value (12.731 kg-m2) occurred during the follow-thru (0.049 sec), and the average minimum value (10.721 kg-m2) occurred late in the downswing (-0.053 sec – club in horizontal position), however the range and standard deviation on the timing of the maximums and minimums were considerable. The sub-group of the professional subject presented slightly higher maximums and similar minimums as the aggregate group. The most significant difference was the timing of the

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maximum value which occurred during the downswing. The female subject presented maximum and minimum values much lower than the males subjects of the aggregate group. The timing of these was similar to the professional golfer sub-group where both extremes occurred during the downswing, however earlier in the downswing. Figures 4c and 4d present the profiles of the absolute and normalized 2nd principal inertia respectively. The normalized profiles indicate that the sub-group of professional subjects present uniform profiles with considerable ranges and vertical spread. The remaining lower skilled subjects and the female subject were more closely grouped together and presented smaller ranges. The relative values at impact were generally lower or equal compared to address. Note the considerable difference in magnitude between the 2nd principal inertia and the 1st principal inertia. This is an advantage since the 2nd principal value is one of the stabilizing inertias.

Data for the 3rd principal inertia shows that for the aggregate group the average maximum value (13.421 kg-m2) occurred during the follow-thru (0.114 sec), and the average minimum value (11.682 kg-m2) occurred during the backswing (-0.470 sec – club in vertical position), however the range and standard deviation on the timing of the maximums and minimums was considerable. The sub-group of the professional golfers presented slightly higher maximums and similar minimums as the aggregate group. The most significant difference was the timing of the minimum value which occurred at impact (0 sec). The female subject presented maximum and minimum values much lower than the males subjects of the aggregate group. The timing of these was similar to the professional sub-group.

Figures 4e and 4f present the profiles of the absolute and normalized 3rd principal inertia respectively. The normalized profiles reveal groupings and trends similar to that found for the normalized 2nd principal value. Generally, the subjects had lower or nearly equal values at impact relative to address. Again, note the considerable difference in magnitude between the 3rd principal inertia and the 1st principal inertia, and the similarity in magnitude the 2nd principal value.

This is also an advantage to the golfer since the 3rd principal value is the other stabilizing inertia.

6.3 Golfer/Club System Global Principal Inertia Directions Figures 6 through 11 demonstrate that the three principal directions change orientations during the swing motion, and the degree to which they angularly displace during the swing is subject dependent. These figures present the manner in which the golfer controls the swing axis and stability plane through the swing phases. Also, the relative orientation of the three principal directions at important points in the swing reveal the effects of golfer stance and club position at these instances.

For all subjects, the gamma (green) direction or swing axis which is associated with the 1st principal value, was pitched slightly forward, and simultaneously yawed in the trail direction at address (Figures 6). During the backswing (Figures 7), this axis either yawed further in the trail direction (Figure 7a and 7d), remained constant in this direction (Figure 7b), or yawed in the lead direction (Figure 7c). For all subjects this axis pitched upward during the backswing. The swing axis yawed in the trail direction for all subjects as the swing progressed to the vertical club position (Figures 8) with the amount of yaw inversely related to skill level. From this position to the horizontal club position (Figures 9),

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all subjects yawed further in the trail direction while generally maintaining their forward/backward pitch orientation as established at the top of the backswing (Figure 9d shows an exception). From this point until impact (Figure 10), the swing axis pitched forward for most subjects (Figures 10a,b, and c), and yaws further in the trail direction (Figure 10d illustrates an extreme shift in the trail direction). After impact, the swing axis returns to the top of the backswing position for most higher skilled golfers by the end of the follow-thru (Figure 11) with much subject-to-subject variation on the trajectory of this axis during this phase of the swing. Comparing impact to address, the orientation of the swing axis is pitched forward a greater amount at impact with respect to the original orientation at address with the higher skilled golfers exhibiting less relative forward pitch.

The 2nd and 3rd principal directions (alpha/red and beta/blue respectively) that define the stability plane, primarily align with the side-to-side yawing and forward/back pitching of the subject. These two axes contribute to golfer stability due to their respective orientations, and associated large inertia values. The similarity of the 2nd and 3rd principal values provides a uniformity of feel relative to the alpha and beta principal axes as they continuously reorient relative to the feet of the golfer. While these two principal values change during the swing, the relative change for each is around 15% which is much less than the relative change for the 1st principal value (40%). These stability directions generally orient with the beta axis pointing forward and slightly upward at address (Figures 6), and the alpha axis pointing downward and in the trail direction. Together, these axes orient the stability plane as pitched downward in the forward direction, and yawed upward in the lead direction. These orientations of the alpha and beta axes, and associated stability plane were consistent among the higher skilled subjects (see Figures 6a and 6b), and more variable for the lesser skilled subjects (Figures 6c and 6d). For some lesser skilled subjects, the alpha and beta axis were switched at address (Figure 6c) due to an initial wide stance of the legs. As the swing progresses to the top of the backswing (Figures 7), the alpha and beta axis roll about the swing axis roughly equal to the angular movement of the club in the swing plane (Jacobs, 2019). This roll angular movement has the effect of reorienting the stability axes such that the beta axis is more aligned to front-back stability, and the alpha axis is more aligned with side-to-side stability. There is the most variability of the alpha and beta principal axes among the subjects at the top of the backswing. From the top of the backswing to the horizontal position of the club in the downswing (Figures 8), the higher skill subjects maintained their orientation of the alpha and beta axes (Figure 8a and 8b), with the lesser skilled subjects presenting considerable and variable movements of these axes (Figures 8c and 8d). From the vertical to horizontal club positions (Figures 9), most subjects present little movement of the beta axis, and some upward yawing of the alpha axis. As the swing progresses toward impact (Figures 10), both axes roll considerably about the swing axis, the alpha axis pitches downward, and the beta axis becomes parallel with the horizontal plane. The stability plane once established at the top of the backswing, yaws slightly upward during the first part of the downswing, and pitches downward later in the downswing. All subjects experience these same relative movements, however the degree and the timing are subject-to-subject dependent with the higher skilled subjects experiencing less movement in both directions. During the follow-thru phase, the alpha and beta axes, and the stability plane generally return to their orientations established at address although there was considerable trajectory variation among subjects during

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this phase. Similar to the other phases, the higher skilled subjects experience less movement of the stability plane compared to less skilled subjects. Through all swing phases, the higher skilled players present less yaw and pitch movement of the stability plane, and greater roll, with the higher skilled subjects presenting equal roll displacements from address to impact, and impact to the end of the follow-thru. It appears that a wide leg stance at address contributes to greater movements of the principal directions as it has a constraining effect on the movement on the lower body segments which may cause the golfer to linearly displace the torso body elements a greater amount resulting in more yaw and pitch of the stability plane. Minimal movement of the stability plane would serve to reduce the variability of feel to the feet of the golfer which may be an advantage in the effort to maintain balance during the swing (Choi et al., 2016).

6.5 Correlations to Performance Measures Referring to Table 4, several of the principal inertia values and one of the timing values correlated to some degree with various performance measures. Club head speed correlated strongly with the 1st ratio, minimum normalized 1st principal inertia, and less strongly with the timing of this quantity. These correlations suggest that it is beneficial for increased club head speed to manipulate one’s 1st principal value through movement and posture means, to reach a higher minimum value later in the swing, specifically just after initiating the downswing. The normalized principal inertia indicates that relative body segment positions rather than body segment masses are important here. In addition, it is also beneficial to increase, or at least equate the comparative value of the 1st principal inertia at impact relative to address. The ratio also indicates that it is the relative positions of body segments and the club and not their relative masses that are significant. Note that the professional subjects nearly matched 1st principal values at impact and address (see Table 2) which is interesting considering that the body postures and club positions were different at these two points in the swing which is evident in the different 1st principal directions (see Figures 10). This would imply at least an awareness of this inertia value by these subjects.

The maximum total body/club work correlated somewhat with the minimum absolute 1st principal inertia indicating that a subject’s ability to limit the minimum value is beneficial. One possible explanation is that limiting the minimum 1st principal value requires that the body segments are either heavier, and/or further from the swing axis of the golfer/club system. To do this may necessitate either higher joint forces and torques, or higher ranges of linear displacement at the joint centers, both of which result in higher joint work. The cumulative nature of joint work during the downswing explains why the maximum total body/club works are higher (Nesbit and Serrano, 2005b).

Maximum club work correlated to 1st principal ratio, and minimum normalized 1st principal inertia. The relationship to the minimum normalized 1st principal inertia may result from a greater amount of total body work as described above, which creates the potential for greater work transfer to the club. The energy transfer from the body to the club is enhanced by maximizing the 1st principal ratio. The maximum club work for the aggregate group occurs at a swing angle of about 30 degrees prior to impact (-0.016 sec). From this point in the downswing to impact the first principal

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inertia value decreases rapidly and by a significant amount for most subjects. This action corresponds to a reduction of the hub path radius (Nesbit and McGinnis, 2009) which is known to enhance energy transfer to the club (Nesbit and Serrano, 2005b). Even with this reduction, the higher skilled subjects had a larger hub path radius at impact (Jacobs, 2019) which effectively moves the arm segments and club further from the gamma axis thus increasing the value of the local minimum 1st principal inertia compared to lesser skilled subjects. These two correlations indicate that body and club positioning are most important to limiting the minimum value of the 1st principal inertia early in the downswing, and maximizing the relative value at impact with respect to address. The strong correlation between club work and club head speed (Nesbit and Serrano, 2005b) makes this result expected.

Maximum kinetic energy of the golfer/club system correlated quite strongly to the minimum normalized 1st principal inertia. Limiting the minimum normalized 1st principal value requires that the body segments are further from the global COM of the golfer/club system about the swing axis. To do this results in higher linear velocities at the body segment and club COM’s, which results in higher kinetic energies at the initiation of the downswing, and higher maximum values overall.

Maximum angular momentum correlated to all absolute values (max and min) of the three principal inertia values.

The maximum angular momentum occurred at an average time of -0.038 sec for the aggregate group of male subjects which corresponds to the horizontal position of the club (see Figures 9). This result is not surprising noting the similarity in how the quantities are calculated (see Equations (11) and (14)). An interesting finding is the angular momentum vector (Figures 8, 9, and 10) keeps a fairly consistent orientation relative to the inertia ellipse during the downswing for the higher skilled subjects, and points more often in the lead vs trail direction. The role of total body/club angular momentum on golf swing performance has yet to be determined.

5. Practical Applications to Golf Instruction and Golfer Performance The results and discussions above support the hypothesis that the global inertia tensor of the golfer/club system is important to the golf shot. However, the degree to which it can be sensed, controlled, and manipulated by the golfer is unknown at this point. Since humans have an inherent control mechanism that allows them to maintain stability during complex motions with the aid of neuromuscular activation that utilize continuous feedback between visual, vestibular, and somatosensory inputs (Choi et al., 2016), then it is quite reasonable to assume that the golfer is at least aware of their inertia tensor characteristics implying control and manipulation may be possible. Based upon this premise, then the logical question becomes, “How can this knowledge be utilized to improve player performance?”

The results strongly support the following modifications to the global inertia tensor and associated quantities to enhance golf shot performance and stability: 1) Limit the minimum value of the 1st principal inertia value (absolute and normalized) as the golfer nears the top of the backswing.

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2) Delay the timing of reaching the minimum value of the 1st principal value to just after initiation of the downswing.

3) Match and/or exceed the value of the 1st principal value at impact with respect to address.

4) Minimize the relative yaw and pitch movement of the swing axis and stability plane.

Physically, these recommended modifications to the golfer’s global inertia tensor must be described in terms of specific body postures, movements, and visualizations in order for a practitioner to implement and/or teach them. The following are suggested: 1) A straighter alignment of torso, neck, and head body segments (torso alignment axis).

2) The torso alignment axis should be yawed in the trail direction and pitched in the forward direction from address to impact.

3) A higher position of the hands at the top of the backswing and during the initiation of the downswing.

4) A greater distance between COM of club and torso alignment axis at the top of the backswing.

5) Limit the inward movement of the hands and arms at the initiation of the downswing.

6) A larger hub path radius trajectory during the downswing.

7) A narrower stance that promotes greater linear and angular displacement of the knees.

8) Visualization of the swing axis as intersecting the head and left foot at address with its movement generally tracking with the torso alignment axis.

9) Visualization of the stability plane as aligning with the elbow joints and navel at address with its movement generally perpendicular to, and rotating about the swing axis with the movement of the club in the swing plane.

6.6 Areas of Future Research This study was successful in describing the global inertia tensor of the golfer/club system of several subjects of various skill levels and body types during the full swing motion. However, the fundamental role of the inertia tensor in the swing production of the golfer is only inferred from comparisons, correlations, and laws of mechanics, and not directly proven through the results of this study. For example, the role of the inertia tensor to the stability of the golfer/club system, the momentum and energy transfers from the body to the club, the reactions to such in the feet, the mechanisms in which the golfer controls his/her own inertia tensor characteristics, and the awareness of one’s inertia tensor during the swing merit further study. Such questions have been partially addressed for some torque-free acrobatic movements, however the same is not the case for this specific torque-applied motion. Follow-up studies that build upon the results of this work should consider these important questions.

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7.0 CONCLUSIONS This paper quantifies and discusses the global inertia tensor, and principal inertia values and directions of the of the golfer/club system during the swing as performed by 14 male and one female professional and amateur subjects of various skill levels and body types. The analyses determined that variations of the principal values and directions of the golfer/club system were substantial during the swing motion, that the principal values generally aligned with two stability and one motion axes, and that skill level was a factor in the manipulations of the principal inertias and directions. Comparisons among subjects revealed that subject-to-subject variations were significant, however patterns and correlations related to skill level and swing performance measures were discovered. The minimum value of the 1st principal inertia (absolute, normalized, and ratio) and the timing of occurrence in the downswing were strongly correlated to several of the performance measures of the swing. Practical advice to golf practitioners is presented to improve performance based upon implementing significant findings. It appears that the golfer senses and controls the relative principal values and directions to positively influence the golf shot, however the degree to which this occurs merits further investigation.

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LIST OF APPREVIATIONS COM

Center-of-mass

HDC

Handicap

RHS

Right hand side

LHS

Left hand side

ETHICS APPROVAL AND CONSENT TO PARTICIPATE All subjects in this study were informed of the purposes of the study, and gave written consent for the use of their data for research purposes, in accordance with local IRB requirements. Local IRB approval for this study was obtained prior to recruiting and testing of subjects.

HUMAN AND ANIMAL RIGHTS All research procedures followed were in accordance with the ethical standards mandated by the local IRB committee responsible for human experimentation. Local IRB approval for this study was obtained prior to recruiting and testing of subjects.

CONSENT FOR PUBLICATION Not applicable.

AVAILABILITY OF DATA AND MATERIALS The data supporting the findings of the article is available upon request to the corresponding author (Steven M. Nesbit).

FUNDING None.

CONFLICT OF INTERESTS The authors declare no conflict of interest, financial or otherwise.

ACKNOWLEDGEMENTS We are gratefully with all the participants that volunteered in this study.

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REFERENCES Choi, A., Kang, T.G. and Mun, J.H. (2016). Biomechanical evaluation of dynamic balance control ability during golf swing. J. Med. Biol. Eng. 36, 430-439.

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