A THREE DIMENSIONAL KINEMATIC AND KINETIC STUDY OF THE GOLF SWING

Department of Mechanical Engineering, Lafayette College, Easton, PA, USA
Received 

20 May 2005 
Accepted 

02
September 2005 
Published 

01
December 2005 
©
Journal of Sports Science and Medicine (2005) 4, 499
 519
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ABSTRACT 
This
paper discusses the threedimensional kinematics and kinetics of a
golf swing as performed by 84 male and one female amateur subjects
of various skill levels. The analysis was performed using a variable
fullbody computer model of a human coupled with a flexible model
of a golf club. Data to drive the model was obtained from subject
swings recorded using a multicamera motion analysis system. Model
output included club trajectories, golfer/club interaction forces
and torques, work and power, and club deflections. These data formed
the basis for a statistical analysis of all subjects, and a detailed
analysis and comparison of the swing characteristics of four of the
subjects. The analysis generated much new data concerning the mechanics
of the golf swing. It revealed that a golf swing is a highly coordinated
and individual motion and subjecttosubject variations were significant.
The study highlighted the importance of the wrists in generating club
head velocity and orienting the club face. The trajectory of the hands
and the ability to do work were the factors most closely related to
skill level.
KEY
WORDS: Golf biomechanics, computer modeling, kinematics, kinetics.

INTRODUCTION 
Since
the golf shot is one of the most difficult biomechanical motions
in sport to execute, a detailed understanding of the mechanics of
the swing would be beneficial to the golfer and teacher (Vaughn,
1979).
It would also provide equipment manufacturers with useful data for
club analysis and design (Thomas, 1994).
Most biomechanical studies of golf swings have employed models of
varying degrees of sophistication (Budney and Bellow, 1979;
1982;
Jorgensen, 1970;
Lampsa, 1975; Neal and Wilson, 1985;
Vaughn, 1979;
Williams, 1967). Generally, these models were limited to one or two rigid
link (double pendulum) systems and constrained the motion to two
dimensions. The double pendulum models were further limited by fixing
the pivot point of the upper link. Notable exceptions are Vaughn
(1979)
who analyzed the threedimensional (3D) mechanics of a swing using
a rigid onelink model and Milne and Davis (1992)
who utilized a twolink planar system with a flexible lower link
to study shaft behavior. These models have been applied to only
a single (male) subject each with the exception of Neal and Wilson
(1985)
who applied their model to six male subjects. Unfortunately, the
only comparative information presented is the linear velocity of
the club mass center, all other information is given for one subject.
These modeling endeavors have yielded important information on various
mechanical quantities of the golf swing. However, these findings
represent only a beginning to the full understanding of the entire
mechanics of the golf swing. One method of obtaining a more complete
understanding of the golf swing is the development of a threedimensional
biomechanical model of the golfer (Dillman, 1994).
What has limited previous attempts at developing this type of model
is the high degree of difficultly in deriving and solving the resulting
equations of motion. The addition of links, the inclusion of the
third dimension, and the use of nonrigid elements represent major
increases in system and thus equation complexity. Fortunately, multibody
analysis software has become available that aides in the development
of analytical models for highly complex dynamic systems.
This paper presents a comprehensive study of the 3D kinematics and
kinetics of a golf swing using a model created with the aid of multibody
analysis software. The golf swing model combines a variable fullbody
multilink threedimensional representation of a human with a flexible
parametric model of a golf club, a ground surface model, and an
impact force model. This model is applied to a large sampling of
subjects for statistical and comparative information. By analyzing
a variety of subjects, the study attempted to discover where differences
in swing style, skill level, body type, and experience reveal themselves
in the kinematic and kinetic quantities. In summary, the purposes
of this study are the following:
 Advance golf swing computer modelling.
 Completely characterize the 3D kinetics and kinematics of the
club swing.
 Analyze several diverse subjects for statistical information of
all quantities.
 Highlight similarities and differences in swing mechanics among
select golfers.
 Attempt to describe the golf swing from a mechanics perspective.

METHODS 
A fullbody
computer model of a golf swing (Figure
1) was developed under the direction of the United States Golf
Association (USGA) to study the biomechanics of the golfer, the
interactions between the golfer and his equipment, and the behavior
of the clubs. The model was built, analyzed and postprocessed with
the aid of the commercial software package ADAMS (Mechanical Dynamics,
Inc.). An ADAMS model is built from rigid segments connected with
flexible elements and/or a variety of joints. Forces and motions
can be superimposed on the model. ADAMS derives the differential
equations of motion for the model employing methods of Lagrangian
dynamics. The resulting equations of motion are integrated using
one of several backward differentiation formula (BDF) integrators.
The results are output and the model simulated using the ADAMS postprocessor.
Golfer model
The golfer was modeled as a variable fullbody, multilink, threedimensional
humanoid mechanism made up of fifteen rigid segments interconnected
with spherical joints. The segment size, mass and inertia properties
were determined from gender and overall body height and weight using
the GeBod data base accessible through the ADAMS ANDROID module
(Mechanical Dynamics, Inc.). The standard available joints are ankles,
knees, hips, lumbar, thoracic, neck (2), shoulders, and elbows.
Wrist joints were added. A notable generality of the model is the
simplified representation of the back and spine joints. The model
divided the entire torso and spine into two segments and joints
(lumbar and thoracic). A finer division was attempted, however severe
marker crowding resulted, and tracking was compromised. All joints
were spherical yielding a maximum of three relative angular degreesoffreedom
(DOF's) with the exceptions of the knees, elbows, and wrists which
were modeled as two degreeoffreedom joints (bending and twisting
for the knees and elbows, bending and yawing for the wrists). The
motions superimposed upon the joints were specified in terms of
Bryant angles (see below) and their time dependent derivatives.
Club model
The golf club was modeled as a flexible shaft joined to a rigid
club head. The shaft was made up of 15 rigid subsegments each with
representative mass and inertia properties. The subsegments were
connected by massless 3D beam elements with the appropriate flexibility
and damping characteristics. The mass and flexibility properties
for the shaft subsegments were calculated using standard analytical
methods. Global shaft damping was determined experimentally by fixing
the grip end of a club in a cantilever manner, deflecting the club
head, and measuring the rate of amplitude decay. This value was
assumed to apply to all shaft subsegments. The rigid club head
segment with hosel contains the representative mass, CG location,
and 3x3 inertia tensor which were determined using solid modeling
techniques described by Oglesby et al. (1992).
The club and golfer models were interconnected with sphericaltype
joints placed at the ends of the lower arms and attached to the
grip point of the shaft to simulate the motions of the wrists and
hands. The model does not explicitly contain hands. However since
the hands experience the same kinematic trajectories as the club
handle grip point, the mass and inertia properties of the hands
were combined with the properties of the handle of the club model.
The angular motions of the wrist joints were driven kinematically
while the three linear DOF's were designated as flexible for both
wrists. This designation avoided a closed loop (indeterminate) configuration
which can cause the simulation to fail.
Swing data and joint motions
Data to kinematically drive the joints of the golfer model were
obtained from subject golf swings. A multicamera Motion Analysis
System (Motion Analysis, Inc.) tracked passivereflective markers
(13 and 19 mm) that were strategically placed on the golfer and
the club. There were 23 markers placed on the golfer and three on
the club. On the golfer the markers were placed at the wrists, forearms,
elbows, shoulders, cervical and lumbar vertebra, head, hips, knees,
mid lower leg, ankles, and feet. All markers were located relative
to bony landmarks for subjecttosubject consistency, and securely
attached with twosided tape (skin) or Velcro (clothing). Markers
were attached directly to the skin wherever possible. The subjects
wore snugfitting clothing (tanktop and bicyclestyle shorts),
a baseball hat (head marker), and shoes of their choice. Marker/joint
offsets were measured, and virtual jointcenter markers were located
from these data using features provided by the data collection software.
The three markers on the club were arranged in a rigid triad that
was attached to the shaft just below the handgrip.
The system was calibrated until the combined 3D residual for all
cameras was under 1.00mm. Test/retest of static marker locations
varied by less than 0.20mm for a given calibration. The threedimensional
marker paths were recorded at 180 Hz then smoothed and processed
to yield global body 123 angular motions of each body segment
and the club. The global angular motions were transformed into local
relative joint motions (Bryant angles) by comparing the motions
of adjacent body segments. The motion of the club relative to the
lower arm segments represented the wrist motions. The relative angular
motions were used to kinematically drive the joints and wrists of
the golfer model. This process is described in Appendix.
Impact model
A springdamper impact function was included to model the ballclub
head collision at impact. The impact force is calculated from the
expression:
where X is the impact deformation, V is the impact deformation velocity,
K is the spring stiffness, e is the stiffening exponent, and C is
the damping factor. The values for K (K = 912,975 Nm), and e (e
= 1.5265) were obtained from static compression tests performed
on a variety of golf balls (Johnson, 1995).
The damping factor C was set to 5% as no experimental or analytical
data were available. This value was selected as it reflects the
underdamped impact phenomena, and it results in a rapid removal
of impact energy without noticeably increasing the impact force.
The impact force calculated from Eqn (1) gave results consistent
with impact forces reported by Gobush (1990)
and Ujihashi (1994).
More sophisticated impact models may be developed from the work
of Lieberman and Johnson (1994).
Ground surface model
A ground surface model was added to support the golfer. A linear
springdamper system was used to represent the contact between the
feet and the ground, and frictional forces provided traction. The
initial contact parameters were obtained from Scott et al. (1993)
and were adjusted at solution time to prevent overstiffening the
model. The golfer model was balanced by kinematically driving the
angular DOF's of the lower torso segment (hips) relative to the
global coordinate system. To avoid overconstraining the model,
the linear DOF's were set free.
Individual force plates were used to measure the vertical reaction
forces between the golfer's feet and the ground. The data provided
kinetic verification of the model since ground reaction forces are
one of the outputs of the model. The data was also used to cause
the android to keep both feet on the ground. A kinematically driven
model is infinitely stiff, therefore small joint angle errors can
cause one of the feet to leave the ground surface. To solve this
problem, the Beta motion (up and down) of one of the ankle joints
was dynamically driven to give the model compliance. A torque control
function [Eqn (2)] that incorporated the force plate data was applied
to the beta motion of the ankle joint to force the foot down.
where T_{Beta} is the applied torque, Ci and Pi are the
function constants, F_{MEAS} and F_{CALC} are the
measured and calculated ground reaction forces respectively, and
T_{WEIGHT} is the torque in the ankle joint imposed by the
weight of the golfer on that foot. The form of the function assured
that the model results would agree with the measured ground reaction
forces. The function constants are adjusted through trial solutions.
Once an acceptable set of torque control function constants was
found, the solution was iterated until the individual ground reaction
forces from the analysis matched the force plate data.
Work
and power
Traditional kinetic analyses of the golfer have focused on determining
the forces and torques generated during the downswing (Dillman and
Lange, 1994). However, this information provides insight to instantaneous
accelerations, not overall changes in velocity thus yielding a snapshot
image of the swing dynamics. An energy analysis has the following
advantages: Only the forces/torques that change the velocity of
the club are taken into account, i.e., forces/torques that do no
work are ignored; the cumulative effects of forces/torques applied
over a distance are determinable which introduces factors such as
range of motion, timing, and sustainability of forces/torques; the
collective effect of various body motions can be summarized by looking
at the output (i.e., the energy transferred to the club and the
resulting club velocity) (Nesbit, 2003).
The work and power expressions were developed from the analytical
equation for the work on a rigid body in threedimensional motion:
Where
_{i} is external
force vectors, _{i}
is the linear velocity vector, _{i}
is the angular velocity vector, and _{i}
is the external moment vector. Power was determined by numerical
differentiation of the results of Eqn (3).
Solution
and model output
Once all the elements of the model were assembled, the resulting
dynamic equations of motion were solved using a Wielenga Stiff Integrator
(Mechanical Dynamics Inc.). This integrator is the most stable and
accurate, however occasional local errors do occur as evident by
small spikes in some of the figures (see Figures
6 and 11). These discontinuities
quickly damped out, and the errors did not propagate. Solution of
the model yielded the threedimensional club trajectories, club
kinematics, golfer/club interaction forces and torques, club work
and power, club deflections, joint kinematic and kinetic quantities,
and ground reaction forces. From the club trajectories, the quantity
"swing radius ratio" is calculated as the ratio of the
radius of the path of the club head through impact to the radius
at the beginning of the downswing as measured in the swing plane.
Model verification
Verification of the model was done in several phases. The first
phase compared the simulated swing of the model with the motion
analysis data taken for each subject. The joint angles for the model
were calculated from the marker data using the analytical methods
described in Appendix. The joint angular velocities and accelerations
were subsequently determined by numerical differentiation of the
joint angle information. These kinematic quantities were used to
drive the joints of the model. The model simulations exactly reproduced
the subjects' motions in terms of joint and club angles, velocities,
and accelerations. This comparison provided kinematic verification
of the model.
To verify the internal loads predicted by the model, several carefully
configured inverse static and dynamic test cases and simulations
were applied to the model. The static analysis consisted of posing
the humanoid model in a variety of stationary positions (such as
the arms straight out to the side) and having the model solve for
the static torques and forces in the joints to support the segments
against gravitational loads. The model results and analytically
determined results were identical. Next, harmonic motions were applied
to individual segments (inverse dynamic simulation) and the model
determined joint torques was compared to analytically predicted
joint torques. Both methods gave identical results. This verification
gave confidence in the internal loads predicted by the model. How
well these loads represent actual subject joint loads is not known.
The one kinetic output of the model that could be directly and accurately
measured was ground reaction forces. Force plate data compared well
with model calculated vertical ground reaction forces with less
than 7% difference after local smoothing (Nesbit et al., 1994).
Finally, model output is compared to available published data. These
data are limited to the kinematic and kinetic quantities of the
club. These comparisons are presented in the Results section.
Subjects
A total of 84 male and one female amateur golfers of various skill
levels, experience, age, height, weight, and
competitive rounds played per year were analyzed using the computer
model. All subjects were righthanded. A summary of the data for
the male subjects is given in Table
1. All subjects used the same driver for the study. The only
study of this type that analyzed multiple subjects used the same
club for all subjects (Neal and Wilson, 1985).
A subset of four subjects (three males and the one female) was selected
for a detailed comparison of their swing mechanics (their data are
given in Table 2). The three
males were selected from the aggregate group. A diversity of skill
levels, body types, gender, and swing styles were the criteria for
selecting these four subjects. The detailed comparison is intended
to present a crosssection of time histories of these quantities,
and to illustrate similarities and differences in swing mechanics
among select golfers. No effort at completeness is attempted here
as every golfer has unique kinematic and kinetic swing signatures.
Informed consent for the following procedure was obtained from all
of the subjects. Each subject had reflective markers placed upon
his/her body. After practicing for several minutes to acclimate
to the markers and testing environment, the subjects were asked
to execute a series of swings which included striking a golf ball.
A swing from each subject was selfselected then analyzed.

RESULTS 
The
following data were determined for each subject: the trajectory
of the club, the magnitude of the linear velocity and acceleration
of the hands and club head, the magnitude of the golfer/club interaction
force, the three components of the angular velocity and acceleration
of the club, the three components of the golfer/club interaction
torque, the total, linear, and angular components of work and power,
and the club head deflection patterns. These data for the four selected
subjects are given in Figures 2 through 17. Table
3 presents statistical information for the maximum (M) and impact
(I) values for the aggregate group. The maximum values are reported
for the portion of the swing from the top of the backswing to impact.
The linear quantities are reported in resultant form since in each
case (velocity, acceleration, and force) the dominant component
was centrifugal and the magnitudes of and differences among the
subjects for the other linear components were negligible. The angular
quantities are resolved according to the relative body (Euler angle)
123 Bryant angle convention where alpha motion (α) is about
the Xaxis, beta motion (ß) is about the Y'axis, and gamma motion
(γ) is about the Z''axis (Kane et al., 1983).
The reference coordinate system, established when the subject addresses
the ball, places the Xaxis (alpha) perpendicular to the club shaft
and aligned with the bottom edge of the club face as viewed down
the club shaft, the Zaxis (gamma) pointing down the club shaft,
and the Yaxis (beta) completing a righthanded coordinate system.
The alpha component coincides with the swing angular motion, the
beta component is a measure of the pitch motion of the club relative
to the swing, and the gamma component is the roll angular motion
about the long axis of the shaft.
While the majority of the data in Table
3 have not been previously reported, some data does exist. Differences
in Table 3 values versus the
reported values can be attributed to differences in subjects as
well as analysis methodologies, and the clubs used. In all cases
the reported data is for a few subjects only. For example, the magnitude
of the grip velocity agrees well with Vaughn (1979),
however there was not the significant reduction in hand speed prior
to impact as reported and which is also discussed by Cochran and
Stobbs (1969).
The maximum club head velocity values and velocity profiles agree
with previous studies (Budney and Bellow, 1979; Cochran and Stobbs, 1969;
Milburn, 1982;
Vaughn, 1979).
The magnitude of the linear force at the grip and the shape of these
curves generally agree with that previously reported (Budney and
Bellow, 1979;
Vaughn, 1979;
Williams 1983). Alpha torque magnitudes generally agree with previous
data (Budney and Bellow, 1979;
Vaughn, 1979),
however the torque profiles are quite different. In addition, the
beta torque values agree in magnitude with Vaughn (1979),
however these curve profiles are also quite different. The gamma
torque magnitudes and profiles obtained are considerably different
than those reported by Vaughn (1979).
The club deflection magnitudes generally agree with those obtained
by Milne and Davis (1992)
who report deflections for a driver based upon a twodimensional
model. However, there are small differences in the deflection profiles.
Significant relationships
The data in Tables 1 and 3
were correlated to determine which data are most indicative of performance
as opposed to just being characteristic of the individual and their
swing style. The most significant linear relationships found are
reported in Table 4. This table
lists the slope and Yintercept values for the linear curve fits
(trend lines), and the R^{2} values (coefficient of determination)
which are a measure of how well the data fit the trend lines. These
relationships are discussed in the following section.
The data in Table 3, the correlations
in Table 4, and the graphical
information in Figures 2 through 17 completely characterize the
3D kinetics and kinematics of the club during the downswing. This
information is used to identify important swing characteristics,
describe the swing mechanics, and compare the selected subjects
in the subsections that follow. The statistical information in Table
3 reveals an unexpectedly large range of values for the majority
of kinematic and kinetic quantities. In addition, the correlations
given in Table 4 are at best
relatively low. These two findings expose the high degree of individuality
of the golf swing.
Club trajectories
A front superimposed view of the trajectory of the club for selected
subjects' swing is shown in Figure
2 with the golfer graphics removed for clarity. The swing is
shown starting from the top of the backswing. Each frame represents
0.01 seconds. The separation between the shaft and the club head
is an indication of club deflections. Individual swing characteristics
are evident by differences in the amount of backswing, the path
of the club head, the shape and size of the inner hub, the spacing
between the frames, club deflection patterns, and the action of
the wrists.
The figure clearly shows that the inner hub has a constantly changing
radius which is necessary for delaying the outward motion of the
club (discussed later). This subtle action is negated by the fixed
pivot of the of the upper link of double pendulum models and may
explain why there was much contradictory discussion as to the exact
mechanics involved in executing delayed wrist uncocking. Table
4 illustrates the relatively strong correlation between a reducing
inner hub radius and skill level for all the subjects.
Figure 3 illustrates each swing
from a side view showing the paths of the grip point and club head
mass center. The figure clearly shows that the swing does not take
place in a fixed plane and that there is significant pitch (beta)
motion of the club during the swing. There appears to be two planes;
one traced out by the club head, and the other by the path of the
hands. The angle between these planes ranges from 9 to 12 degrees.
Linear quantities
Figure 4 illustrates the grip
and club head mass center linear velocities for the four subjects.
Impact occurs at 0.0 seconds. The grip velocity curves for the subjects
are surprisingly similar (less than 0.4 m/sec separates the fastest
from slowest), and the degree of difference in the curve profiles
is very small especially when compared to the differences in the
club head velocity curves. This similarity is true for the aggregate
group as well. Generally, the maximum grip velocity was reached
just before impact and remained constant or decreased slightly through
impact. There was a small correlation between grip velocity and
skill level for all subjects (Table
4).
There are large differences in both the shape and magnitude of the
club head velocity curves. The figure illustrates the relationship
between maximum club head velocity and skill level as indicated
in Table 4 for all subjects.
Maximum club head velocity occurred at impact for the scratch golfer
and on both sides of impact for the other subjects with the degree
of spread related to the skill level of the four subjects. This
finding is not unexpected noting the precise timing required to
simultaneously coordinate the swing motion, wrist uncocking, wrist
roll, swing plane stabilization, and shaft unflexing to cause the
peak velocity to occur at impact. The large differences between
grip and club head velocity highlights the importance of the wrists
in generating club head velocity. The slope of the velocity curve
during the downswing is an indication of both the delay and the
magnitude of wrist swing motion. It also shows when the wrist motion
occurs and its relationship to skill level of the four subjects.
Figure 5 illustrates the magnitude
of the grip and club head linear accelerations. These curves reflect
the dominant centrifugal acceleration component, hence they are
similar in shape and subject trends to the velocity curves. It is
interesting that while the grip velocity remains relatively constant
or slightly decreases near impact, the grip acceleration curves
increase slightly. This indicates a shortening of the hub radius
near impact and is seen in all 84 subjects.
Figure 6 shows the magnitude
of the golfer/club interaction force at the grip. The force is directed
along the shaft through the entire downswing. Initially this force
does work to accelerate the club, then gradually changes function
as the downswing progresses to reacting to the centrifugal acceleration
at the time of impact. The force curves have the same general shape
and subject trends as the club head velocity and acceleration curves.
Comparison of this data with Figure
4 highlights how subject differences in club head velocity magnify
the differences in the interaction forces.
Angular motion: Alpha component
Figures 7, 8,
and 9 illustrate the alpha
component of the angular velocity, angular acceleration, and torque
applied to the grip respectively for the four subjects. The alpha
components indicate the swinging action of the club and are the
most significant angular motions. Referring to Figure
7, the relationship between skill level and maximum swing angular
velocity and the slope of the curve prior to impact can be seen.
The alpha angular velocity of the club reflects the summation of
the rotation of the upper body with the motion of the wrists. It
was shown that the grip point linear velocity was similar for the
subjects (Figure 4). This fact
reveals that the rotational velocity of the upper body did not differ
much among the subjects which is surprising, noting the significant
differences in body type and skill level. Thus the curves are a
direct indication of differences in the wrist motion among the subjects.
There is a weak correlation between subject height and alpha velocity
(Table 4).
The maximum swing angular velocity consistently occurred 0.025 seconds
prior to impact for the scratch golfer and generally occurred in
the range of 0.020 to +0.010 seconds relative to impact for the
other subjects. Figure 8 (alpha
angular acceleration) further illustrates the timing of the maximum
alpha angular velocity relative to impact, as well as the smoothness
of the wrist swing plane motion. The large spike after impact for
the scratch golfer was seen for all his swings and may reflect the
sudden rolling over of the right wrist. (This view was deduced by
how quickly the club face turns in after impact as seen in Figure
2).
Alpha torque (Figure 9) is
the dominant torque component. Again, there is a relationship between
peak positive torque and skill level as reported in Table
4. The maximum values occurred well before impact and generally
came close to zero near impact. The scratch golfer consistently
exhibited a negative alpha torque 0.01 seconds prior to impact,
however it is too late in the swing to suggest that it was related
to the purposeful delaying of wrist motion. The figure clearly shows
that delayed wrist motion is not achieved by applying a hindrance
torque as suggested by Jorgensen (1970)
and Milburn (1982).
Also, it does not appear that the wrist behaves as a free hinge
until impact (Jorgensen 1970;
Milburn, 1982),
but supports both Lampsa (1975) and Budney and Bellow (1979)
that this torque should be positive up to impact to achieve maximum
club head velocity. There is a weak correlation between subject
weight and alpha torque (Table
4).
The subjects exhibited alpha torque profiles that were both unique
and consistent among trails revealing a alpha torque "signature"
for each subject. Two distinct swing styles were revealed however.
The scratch and 5 handicap subjects were "hitters" appearing
to exert considerable effort in swinging the club. Their alpha torques
increased significantly during the downswing and reached large maximum
values at the midpoint of the downswing. These maximum values were
maintained until close to impact. The other two subjects were "swingers"
with a swing style that was smooth and appeared almost effortless.
Their maximum torques were much lower and the curves had smaller
variations during the downswing. While there were significant subject
differences in maximum alpha torques and the shape of the curves
during the downswing, these differences did not seem to affect the
maximum alpha angular velocity or the curve profiles to the degree
suggested by the torque data.
Angular motion: Beta component
Figures 10, 11,
and 12 illustrate the beta
component of the angular velocity, angular acceleration, and torque
respectively for the selected subjects. The beta angular components
indicate the pitch motion of the club. While the beta motion is
the smallest of the angular motions, Figures
10 and 11 show that it
is still significant. Since the path of the grip and club head define
different planes (Figure 3),
pitch motion of the club must take place. As the speed of the club
increases, so must the beta motion as is indicated in Figure
10. The large variations in beta velocity curves among subjects
further emphasize the relative difference in hand and club head
paths taken to impact. The scratch golfer had the least pitch motion
up to impact and the lowest beta angular velocity at impact. All
four subject exhibited a stabilization of the pitch motion as indicated
by the low beta angular accelerations at impact (Figure
11).
The beta torque curves exhibit large subjecttosubject variations.
In general, the torques increased towards impact which coincides
with the rapid pitching of the club, then tend toward zero near
impact as the pitch accelerations approach zero. The large negative
torques after impact result mainly from the mechanical rolling over
of the wrists which attempts to pitch down the club. All beta actions
exhibited large ranges (Table 3).
These motions and torques reflected characteristics of individual
swing style, and were not related to skill level or club head velocity.
Angular motion: Gamma component
Figures 13, 14,
and 15 illustrate the gamma
component of the angular velocity, angular acceleration, and torque
respectively for the four subjects. The gamma angular components
indicate the rolling motion about the long axis of the club shaft
and are important in squaring up the club face for impact. Figures
13 and 14 reveal that
the gamma motion is significant yielding angular velocity values
that are approximately half of that for the alpha component, plus
the largest angular acceleration component. While the most important
function of the gamma motion is to square up the club face for impact,
it does contribute to the overall club head velocity. For example,
the scratch golfer generates approximately 1.5 m·s1 club head mass
center velocity at impact from the gamma angular velocity.
The subjects exhibit two distinct styles as illustrated by the shape
of these curves. The 5 and 18 handicaps initiate the gamma motion
with the start of the downswing, have a linear increase in speed,
peak prior to impact, and generate significantly lower angular velocities.
The scratch golfer and 13 handicap exhibit delay in initiating this
motion relative to the start of the downswing. In addition, they
have a nearly uniform increases in acceleration, peak near impact,
and generate significantly higher angular velocities.
Figure 15 reveals that the
gamma torque is the smallest torque component. This finding is expected
noting that the inertia of the club relative to the grip point is
significantly smaller about the gamma axis than the alpha and beta
axes. Also evident is the two swing styles described above; the
5 and 18 handicap exhibit a double dip curve up to impact, and scratch
golfer and 13 handicap have a single dip curve. The single dip curves
reached maximum values approximately 0.06 seconds prior to impact
and the double dip curves reached maximum values at 0.10 seconds
(5 handicap) and 0.12 seconds (18 handicap) prior to impact with
their second smaller peak occurring just before impact. All curves
nearly passed through zero torque at impact with the scratch golfer
about 0.01 seconds early and the 18 handicap about 0.01 seconds
late. It appears that delaying the initiation of this motion aides
in the generation of speed. It is interesting that the ability to
generate alpha and gamma angular velocities are not necessarily
related. Most gamma actions exhibited large ranges (Table
3). These motions and torques also reflected individual swing
style, and
were not related to skill level or club head velocity.
Club deflection
Figure 16 illustrates the
club deflection patterns for each subject. The deflection includes
bending in, and perpendicular to the swing plane, bending down of
the club head, and twisting and elongation of the shaft. The deflection
in the swing plane was by far the largest component. The curves
clearly demonstrate that individual swing mechanics greatly effect
shaft deflection patterns, with the patterns loosely following each
individual's alpha torque curve (Figure
9). There is a delay of approximately 0.015 to 0.020 seconds
from the time an alpha torque curve passes through zero, and the
club shaft inplane deflection goes through zero. The scratch golfer
was superior in coordinating his alpha torque so to release the
maximum stored strain energy in the shaft at impact. This timing
is important since the unflexing of the shaft can contribute to
the club head velocity. The magnitude and timing of the club shaft
deflections varied greatly among subjects in the aggregate group
(Table 3), and no significant
correlations were found.
Work
and power
The ability to apply forces and torques in the direction of motion
during the downswing is indicated by the total work, and the ability
to apply forces and torques as the swing increases in velocity is
indicated by the total power. Figure
17 illustrates total work curves and reveals differences among
the four
subjects in magnitude, shape, and timing. It is interesting that
all subjects had the same total work at time 0.085 seconds which
corresponds to the club position shown in Figure
1 for all subjects. The better golfers initially do work at
a slower rate, then do work more rapidly through impact. The better
golfers also had higher club head velocities, higher total work
done, and were able to peak total work closer to impact. Referring
to Table 4, the strongest correlation
found was between total work and club head velocity. This correlation
is expected since the total work is the primary factor in generating
club head velocity as predicted by Newton's Laws. Table
4 also points out a strong correlation between total work and
handicap.
Total work is a combination of angular work (torques x angular motions)
and linear work (forces x linear translations). The linear force,
work, and power are primarily transferred from the golfer to the
club via pulling on the club by and through the arms. The angular
torque, work, and power are transferred by and through the wrists.
The ability to develop high peak forces and torques reflects the
strength of the arms and wrists respectively. Table
3 shows a large range in values for both quantities among the
subjects. An analysis of the ratio of linear work to angular work
seems to indicate that better golfers use their arms more relative
to their wrists to do work (by a 1.41:1 ratio for the scratch golfer).
Table 4 shows a strong correlation
between this ratio and handicap.
Figure 18 reveals differences
among the subjects in the magnitude, shape, and timing of the total
power profiles. Total power is approximately the
same until 0.12 seconds which roughly corresponds to the vertical
position of the club. The power then peaks at different times prior
to impact for each subject. More importantly, the scratch golfer
was able to zero his power output at impact resulting in maximum
work output. The differences in total power are quite significant
as is the balance between angular and linear power components. The
arms are more important for generating power than the wrists. The
angular power peaks prior to the linear power for each subject.
Table 4 shows a correlation
between total power and handicap

DISCUSSION 
The
ability to completely describe the threedimensional kinematics
and kinetics of the golf swing utilizing a computer model has numerous
practical implications for practitioners and researchers. The information
obtained from the computer model allows one to precisely explain
a subject's golf swing from a mechanics perspective by explicitly
detailing the time history of the motions, forces, and torques.
Doing so for several subjects revealed a number of important characteristics
of the golf swing, and similarities/differences among subjects.
The following observations and practical implications are offered:
 An important component in generating club head velocity is the
reducing radius path of the hands during the downswing. The study
revealed a relatively strong correlation between a reducing radius
path of the hands and skill level.
 The torques and range of motion of the wrists are important factors
in generating club head velocity, more so than the speed of the
hands. In addition, the actions of the wrists identified the better
golfer more so than the speed of the hands.
 The notion of delayed wrist motion to generate club head velocity
is valid, however the mechanism to achieve it is based upon the
path of the hands and the initial wrist angle, not a retarding wrist
torque.
 Shaft flexibility plays a part in generating club head velocity.
The straightening of the shaft continues to accelerate the club
head through impact even after the work by the wrist on the club
is done. Approximately half of the shaft stored strain energy is
released by impact and converted to higher club head velocities.
 Work and power analysis is a valuable method for evaluating a
golf swing since this approach considers the cumulative effects
of forces/torques applied over a distance thus including factors
such as range of motion, timing, and sustainability of forces/torques.
 Work and power were well correlated to skill level, and were essential
factors for generating club head velocity. Range of motion was important
for generating maximum positive work.
 Swinging harder does little to generate additional club head velocity.
Swinging further (expanded range of motion) has the potential to
generate additional club head velocity if the subject possess sufficient
muscular power. Exercise programs thus should promote flexibility,
and strength training for power as opposed to just strength development.
 Subject differences in work, power, force, and torque do translate
to differences in club velocity, however not to the degree one would
expect. Factor in the greater losses associated with impact and
aerodynamic drag at higher club speeds and the results are driving
distances that are not that different. This observation is especially
important for the individual golfer to realize as swinging the club
"harder" may do little to improve driving distance. In
fact, it may be more difficult to do useful work with tight muscles,
and the cost associated with increased effort is often a reduction
in accuracy.
Description
of the swing mechanics
The following description the golf swing is offered as an aide to
understanding the fundamental mechanics involved. The description
is from the top of the backswing through impact based upon data
from the scratch golfer.
The downswing is initiated with a pulling along the shaft while
simultaneously applying a positive alpha (swing) torque resulting
in positive linear and angular work being done. As the club head
moves away from the body, the action of the linear force becomes
less directed at speeding up the club and more toward controlling
the path of the grip point. About the time the club becomes vertical
in the downswing, the alpha torque increases in magnitude as it
takes over the acceleration of the club from the linear force. Simultaneously,
the gamma (rolling) torque is initiated to square up the club head
for impact, and a beta torque is applied to pitch the club forward.
From this position up until the club shaft is roughly parallel with
the ground, all the torque components increase smoothly and reach
their maximum values. From the parallel position to impact, which
coincides with the increase in swing motion of the wrists, the torque
components rapidly decrease. All the torque components pass through
zero at or near impact resulting in maximum angular work just before
impact. By the time impact is reached, the linear force is maximized
and perpendicular to the path of the club head in the plane of the
swing. At this time the linear force is reacting to the centrifugal
loading of the club thus maximizing the linear work at impact.
Just
before impact the wrists momentarily approximate a "free hinge"
configuration as the golfer merely holds on to the club as its momentum
carries it to impact. By the time impact is reached, all torque
components are in opposite directions because the wrists cannot
keep up with the rotational speed of the club at this time in the
downswing. The club head does not slow down however, as the straightening
of the shaft continues to accelerate the club head. The club head
swing plane deflection component passes through zero at impact releasing
about half of the shaft stored strain energy, and resulting in the
club head velocity peaking exactly at impact.
This subject exhibited a swing hub curve with a large initial radius
of curvature that decreased continuously during the downswing. He
also had a highest degree of initial wrist cocking. Together, these
served to reduce the initial centrifugal acceleration which in turn
diminished the tendency of the club to move outward even though
a positive alpha torque was applied from the initiation of the downswing.
This large radius path was carried through most of the downswing
as the hand speed was increased by the linear force. Approaching
impact, the hub radius was quickly reduced by a redirection of the
linear force, which in turned caused a rapid increase in the centrifugal
acceleration. This action which was coordinated with a large increase
in alpha torque, pulled the club outward and through impact. These
coordinated actions give the impression of a consciously delayed
wrist motion. It is believed that this sequence of events are necessary
to yield the optimum segmental addition, thus the largest possible
club head velocities.

CONCLUSIONS 
The
overall goal of this study was to create a computer model of a golfer,
then use the model to analyze the 3D mechanics of a golf swing for
several subjects. Novel components included completely characterizing
the 3D kinetics and kinematics of the downswing, performing an energy
analysis of the swing, analyzing a large group of subjects for statistical
information, searching for significant correlations, and highlighting
similarities and differences in swing mechanics among select subjects.
An important advance over previous studies of this type was the development
of the fullbody golfer model and a flexible steppedshaft club model.
This modeling effort consciously avoided applying the simplifying
assumptions that limited previous modeling attempts. The model generated
considerable valuable 3D data which were used to describe the golf
swing from a mechanics perspective, and to identify important swing
characteristics.
This analysis revealed the true complexity and individuality of the
golf swing motion. While some data were similar among subjects, most
data illustrated vast differences both in terms of magnitude and profile.
For example, the kinetic quantities consisting of the work, power,
linear interaction force and the three components of torque illustrated
how differently each subject drives and controls the golf club. These
differences have important implications for golf instruction, equipment
design, and injury assessment. Also revealed were the quantities that
were related to skill level such as hand trajectory, work ratio, work,
club head and grip velocity, alpha torque and angular velocity, and
power. The other quantities seemed to reflect swing style and not
skill level. The study discovered little correlation between body
type and swing characteristics or skill level. 
ACKNOWLEDGEMENT 
Funding
for this project was provided by grants from the United States Golf
Association and the National Science Foundation. 
KEY
POINTS 
 Fullbody
model of the golf swing.
 Mechanical
description of the golf swing.
 Statistical
analysis of golf swing mechanics.
 Comparisons
of subject swing mechanics

AUTHOR
BIOGRAPHY 
Steven M. NESBIT
Employment: Associate Professor and Head, Department of
Mechanical Engineering, Lafayette College, Easton, PA, USA.
Degree: PhD, PE, MS, BS.
Research interests: Sports Biomechanics, mechanisms,
computer modelling.
Email: nesbits@lafayette.edu


APPENDIX 
This
section describes the methods used to locate local body segment
coordinate systems, determine global orientations of the body
segment coordinate systems, and extract the relative orientations
of adjacent body segments which are the angles that drive the
joints of the humanoid and golf club models. The methods were
developed from the works of Kane et al. (1983)
and Craig (1986).
Other methods for determining relative body segment orientations
and velocities are given by Teu et al. (2005).
Figure 1app illustrates
the club and right forearm marker locations and coordinate systems.
The markers at the ends of the right forearm segment are virtual
makers located at the joint centers. Their positions are determined
by the Motion Analysis System software. Also shown is the global
coordinate system attached to the ground. The configuration
of markers, local segment coordinate systems, and relative and
global angles is typical of all adjacent segments in the body,
thus the figure is relevant to the general discussions that
follow.
Local coordinate systems were defined for each body segment
(and the club) from groups of three adjacent marker locations.
Generally, markers were placed at the distal and proximal ends
of each segment, and are represented as marker i+1 and marker
i, respectively. In addition, a third noncollinear marker is
placed between marker i and i+1, and is designated marker i+2.
Taken together, the three markers form a plane from which the
local coordinate systems are established. The local Zaxis is
coincident to the long axis of the segment and is determined
from the following vector difference:
An intermediate vector Q is determined from markers i and i+2:
Using cross products, the local X and Y axes can be determined
as follows:
The local coordinate system is then represented in matrix form
as follows:
Where the first column X_{x}, X_{y}, X_{z}
is the Xaxis unit vector components, the second column is the
Yaxis unit vector, and the third column is the Zaxis unit
vector components. This process is repeated for all body segments
and the club. All of the terms in the above matrix are known.
The android model is driven kinematically by specifying the
relative body 123 Euler angles (Bryant angles alpha (α),
beta (β), and gamma (γ))
for each joint. The Bryant angle transformation matrix is as
follows:
Where

(5)


(6)


(7)


(8)


(9)


(10)


(11)


(12)


(13)

The local coordinate system matrix and the Bryant angle transformation
matrix are set equal to each other for each segment. Thus the
left hand side of Eqns (5) through (13) are known. From these
equations, the global Bryant angles are extracted. For example,
solving for the angle α, note the
following:
Dividing Eqn (14) by Eqn (15) yields the formula for α:
Using a similar procedure, the expressions for Beta and Gamma
are found:
Thus Eqns (16), (17), and (18) yield the global Bryant angles
for each body segment and the club. Relative angles of the distal
segment with respect to the proximal segment are needed to drive
the joints of the model. Determination of the relative Bryant
angles is done the following way: The relationship between the
Bryant matrices of adjacent segments is given by:
where
G is ground (global reference system). D is the distal segment,
and P is the proximal segment. The relative Bryant angles
are contained inside the
matrix. In order to isolate this matrix, both sides of Eqn
(19) are multiplied by the inverse of the
matrix yielding:
The
global Bryant angles are substituted into the
and matrices yielding
all known elements of the
matrix. The relative Bryant angles are then extracted from
the matrix in a
manner similar to that used for the global Bryant angles.
Application of Eqn (20) to the digitized motion analysis data
yields tabular 3D relative motions for all the joints of
the model including the wrists which drive the club. Cubic
splines are used to create continuous functions from the tabular
data to kinematically drive each joint.



