Article Contents

Opening the loop  instrumented grips
Dave
Tutelman  February 26, 2021
Until now, kinetic analysis of the golf swing has been limited by the
"closed loop problem". There was no way to deduce mathematically which
hand, arm, or shoulder, right or left, was producing a given force or
torque on the golf club. Inverse
dynamics kinetic analyses are done by making a detailed
motion capture of a golf swing, and computing from that what the forces
and torques must have been in order to produce that motion. But there
is a continuum of forces and torques
the hands impart to the club that might cause the observed motion. That
continuum is based on how the forces
and torques are divided between left and right (or lead and trail)
hands and arms.
Over the years, several biomechanists have attempted to "open the loop"
by various methods. The purely analytical attempts seem to require
assumptions beyond what most researchers feel is acceptable; they
almost mandate the solution rather than discovering it. But there are
experimental methods that show a lot more promise. These add some
measurements to the standard inverse dynamics experiment. Specifically,
they instrument the grip to capture some data that can separate the
action of the two hands.
This article discusses the two leading attempts to open the loop by
instrumenting the grip.
 Choi and Park published a paper in 2020; their experiment added a
6DOF force and torque sensor between the hands, to measure the difference between what the hands are doing..
 Koike
published a paper in 2005 and another in 2016; his team's experiments
added strain gauges
to the grip to measure directly the forces being applied by each hand.
A few words about how my investigation came about:
 In late 2020, the ChoiPark paper got some discussion on the
Facebook group Golf
Biomechanists.
(Despite the fact it is Facebook, there is a lot of very serious
conversation by many of the leading members of the golf biomechanics
community.)
 Art Maffei (a systems engineer retired
from the DOD missile and NASA space programs, who feels that
biomechanics is "today's rocket science") started drawing conclusions
about what
the hands were doing based on the graphs in Choi & Park.
 On February 22, 2021, Mike Finney (a golf coach who focuses on
being cuttingedge with golf science) hosted a Zoom meeting to
understand the paper by Choi and Park.
The first two points got me investigating the subject. I was mostly
limiting myself to Choi and Park until the Zoom meeting. In trying to
consolidate what I learned from that, I went back and took a more
quantitative look at the Koike work. This article is an evaluation and
comparison of the two approaches to opening the loop: Choi & Park's
and
Koike's.
Executive summary
While both studies added
instrumentation to the grip, they approached the problem of getting the
hand forces rather differently.
 Choi and Park started with a
conventional inverse dynamics
computation to get the total bothhands forces and torques. Their
instrumented grip measured the differences between what the left hand
and the right hand were doing on the grip. Knowing the totals and the
hand differences, they were able to compute each hand separately.
 Koike placed strain gauges on the
grip to measure directly
each hand's
influence on its own part of the grip. Since forces were being measured
directly, there was no need for an inverse dynamics computation.
My contribution to the discussion is to apply a reasonableness test to
the results.I apply two tests here:
 Take the forces and torques applied separately by the right
and left hands,
and recombine them to see if they match
the wellknown closedloop
solutions. If you can't get the closedloop answer right, I must have
serious reservations about your openloop answer.
 Compare the two papers' results
to see if they separate
right from left to the same answer. If they do, then I have some
confidence in the answer, which would have been derived two different
ways. If they do not, then at least one is wrong and possibly both.
Neither method gave a strong enough showing, in my opinion, that we can
trust their results to provide us with enough knowledge to affect how
we teach the swing. Here are the most serious discrepancies I observed:
 Compared with many years of inversedynamics results, the
hand couple's magnitude is too low for both studies. ChoiPark was
worse than Koike in this regard, but nether was close to the numbers
found by MacKenzie, Kwon, or Nesbit..
 On the same basis, the moment of force is the wrong shape
as well as size for both studies, with Koike completely off the mark.
 There is considerable difference between the studies in how
the components of the hand couple divide up to provide the total hand
couple. So teasing apart the left and right hands is giving very
different
proportions.
 Similarly, how the hands divide up the axial force varies
markedly between the studies.
But
both are promising. I am not saying we should stop. Rather, we
should continue developing this sort of instrumentation; I am confident
that refinement will provide more robust answers. At the end of this
article, I suggest things that need to be improved
to arrive at useful
results.

The ChoiPark study
This is an interpretation of the paper "Three
Dimensional Upper Limb Joint Kinetics of a Golf Swing with Measured
Internal Grip Force"
by Hyeob Choi and Sukyung Park (The National Center for Biotechnology
Information, June 2020).
Choi & Park attacked a
classical problem in the biomechanics of the
golf swing, the "closed loop" problem. That is, an inverse dynamics
analysis of a golf swing can measure the combined contribution of
hands, arms, and shoulders, but not the right side or left side
individually. Choi & Park did this by instrumenting the grip of a
club to get
distinct measurements for the right and left hand. In particular, they
placed a 6D sensor in the grip between the right and left hands. The
sensor measures the difference between the hands, and allows teasing
apart the contributions of individual hands, arms, and shoulders.
The participants in the study were "nine healthy male professional
golfers registered by the Korea Golf Federation or the Korea
Professional Golf Association." Each took five full swings with the
instrumented driver. The swings were measured with respect to:
 Motion capture for the club, hands, arms, and torso, so
that
inverse dynamics could be performed.
 Differential force and torque from the sensor at the middle
of
the grip.
Here's a little more detail about what they did.
 They started with a conventional inverse dynamics analysis.
That was done with a motion capture in 3D using a camera system. Once
the 3D motion is completely known, it is possible to work backwards
using Newton's laws of motion, and infer what forces and torques must
have been needed to create the observed motion. There is a problem with
that, illustrated by the picture of the "Inverse dynamics golfer". It
is impossible to separate the forces and torques imposed by the right
hand from those of the left hand. The shoulders, arms, and hands are
assumed to be a rigid triangle, hinged at the red centerlines. Those
are the base of the neck which allows the triangle to turn, and the
hands which allow the club to turn. It is possible to use inverse
dynamics to derive one
force vs time function and one
torque vs time function describing what the hands are doing to the golf
club. But we can't derive
two functions, one for each hand.
 The conventional inverse dynamics gave Choi and Park the
total force and torque of the hands on the grip. But remember, this
grip has a sensor between
the hands. It can measure the difference in force between the
hands, likewise the difference in torque. So for each dimension, and
separately for force and torque, they have equations for:
 The sum of RH and LH, which is the inverse dynamics
output.
 The difference of RH and LH, which is the sensor output.
That is good enough to solve for each RH and LH force and
torque.
Here is a key result from their paper. Their Figure 5 is a collection
of
graphs of all the individual hand forces and torques (that is,
separated left hand from right). The forces and torques are projected
separately on the local x, y, and zaxes relative
to the club handle. (We will review what the local axes are in the next
subsection.) There are also graphs of the accelerations, and also
presented by axis. 
Figures 6 and 7 present similar curves for the wrists, elbows, and
shoulders. I'm not looking at those results in detail here, just the
forces and torques that the hands impose on the club.
The graphs are the averages of all 45 swings (9 golfers, 5 swings
each). The range over the 45 swings is shown as gray shading surounding
the curve of the graph.
Definition of local axes
Before we start, we must
understand the axes in the paper. Here is
part of Figure 4. It shows the X, Y, and Z axes used to
describe the club. The notion of "local axes" means that the axes are
part of
the club; when the club moves, so do the axes. Table 1 of the paper
describes the axes in words; here is the first line of that table,
describing the
axes of the club as acted on by the hands. (Other lines describe the
wrists, elbows, and shoulders; we won't do anything with those here.)
Table
1. Local axes of each segment. 
Segment 
Local
xAxis 
Local
yAxis 
Local
zAxis 
Both
club segment 
Cross
product of the local y and zaxis 
The
direction of the clubface 
Grip
direction of the longitudinal axis 
But the diagram and the table are both ambiguous. Does the definition
of the yaxis mean "a direction the clubface lies in" or "the direction
the clubface is pointing"? And the crude drawing does not help much; is
the club a driver at address or an iron laid open 90°? The key to this
ambiguity is in the hands
in the diagram. Given the hands gripping the club, I can only conclude
that the yaxis is the direction the clubface is pointing, and the club
in the diagram is intended to be a driver.
With that interpretation, here are the directions of the forces and
torques for each axis. I am assuming that Choi and Park are using the
conventional righthand rule for torques, common in engineering and
physics texts.
Forces
and torques


Local
xAxis 
Local
yAxis 
Local
zAxis 
Force

The
toeheel plane, perpendicular to the shaft axis. Positive is toward the
toe. 
The
direction the clubface is facing, perpendicular to the shaft axis.

The
shaft axis. Positive is toward the grip and away from the clubhead.

Torque

Rotation
in the direction the clubface is facing. 
Rotation
in the heeltoe plane. Positive rotates toward the heel. 
Rotation
about the shaft axis. Positive closes the clubface.


The Koike studies
Sekiya Koike (and his team, which seems to vary from study to study)
published
two studies that I have seen. Unfortunately, I no longer have the long
form of either study. I have a marginal photocopy of some key graphs
from the 2005 paper, and the short form of the 2016 paper
("Force and Moment Exerted by Each
Hand on an Instrumented Golf Club",
34th International Conference on Biomechanics in Sports, July 2016)
from the conference proceedings. The 2016 study is more interesting
because it includes forces along the shaft axis (not in the 2006
paper); that allows the computation of the moment of force, not just
the hand couple.
Koike's grip instrumentation is very different from Choi and Park's.
Koike embeds strain gauges in the grip, to measure the forces on the
shaft. The strain gauges are pressed by a rigid, segmented replacement
for the usual grip. There are four grip segments, by hand (lead and
trail) and by side of shaft. The gauges are placed so that not only
forces but individual hand torques can be inferred from the combination
of gauge readings.
The graphs showing the results are provided in a different set of axes
from Choi and Park. Instead of the local club axes, they present axes
related to the swing plane. (This will save us some computation, since
the Koike axes are what we need in order to compare results with the
wellestablished outputs from inverse dynamics studies.)
The graphs are:
 Inplane forces across the shaft.
 Forces along the shaft axis.
 Individual hand torques in the swing plane.
The
short form of Koike's paper does not address some things that gave me a
hint of what was wrong when the answers proved to be questionable. For
instance, it did not specify the data smoothing nor the weight of the
instrumented grip. Both of those were detailed in the ChoiPark paper,
and are plausible explanations of some of the problems with the
results. Wish I had them for Koike.

Calculations
My goal for the
calculations was to get a timeseries graph from the study, to compare
with previous (necessarily closedloop) inverse dynamics studies from
researchers Sasho MacKenzie, YoungHoo Kwon, and Steven Nesbit. That
involves finding the
inplane torques acting on the shaft. The graph will separate the hand
couple from the moment of force, but combine the hands. It was
important that these torque graphs be the torques in the swing plane like the classic
studies' results,
not those of
Choi and Park's local axes. Koike's results were presented as
swingplane forces and torques already; the step of mapping them from
local club axes to the swing plane was unnecessary.
To do the mapping from local club coordinates to swing plane coordinates, I
combine the axisbyaxis
forces and torques in whatever manner necessary to keep the reference
inplane. Such combination turns out to be different from
step to step, but is always dependent on the angle of rotation between
the clubface and the swing plane.
If you are not interested in the calculations but just the results, you
can go now to the discussion section and
skip the details.
Digitize the graphs
The
first step was to enter into a spreadsheet the force and torque graphs
(Figures 4 and 5
from Choi and Park, and the three graphs from Koike). I know of no way
to do it except the painstaking work to
transcribe it pointforpoint by hand, which is what I did. To save
anybody else the trouble, I have made the spreadsheet available for download. The file is in ODS format. My
spreadsheet was created in
LibreOffice Calc, an opensource spreadsheet program. Excel claims to
be able to import .ODS files, but I have not tested this one in Excel.
Step
1
 Digitizing graphs by hand is tedious work. The only bright spot is
that the graphs are already in the form of pixels in my computer, and I
have an image editing program to help me deal with it. The first
step was to zoom in on the image until I could see some pixel
granularity or artifacts on my 24" monitor screen. I took screenshots
of thse enlarged graphs, and saved them in JPEG form. These I
loaded into a photo editor program with an important
property: it has a box that gives the [x,y]
coordinates of the current cursor position, in pixels. (The program was
ULead PhotoImpact, but most serious image editors have this feature.)
Then I placed
the cursor on a desired point on the graph, and recorded the
appropriate coordinate[s] on the spreadsheet. After transcribing the
first graph  with the hand tenseness you would expect  I got the
idea of zooming in another factor of 3 so I had some slack at finding
the
point on the graph. A few details of the digitization of the pixels:
 I only digitized from transition to impact. In the
ChoiPark paper, the graphs also
contain some of the backswing and much of the followthrough. I had no
intention of using that for anything, so I didn't bother to digitize it.
 In Choi and Park, the
length of the downswing, transition to impact, was 108
pixels on all the graphs. (Choi and Park conveniently placed vertical
lines to show transition and impact.) I broke that into 18 segments of
6 pixels each. When I take into account the horizontal scale, that is
18.5 milliseconds per segment, and a pixel is 3.1 milliseconds.
 In
Koike, the length of the downswing was 288 pixels on all three graphs.
That gave 16 segments of 18 pixels each. A segment is 14
milliseconds, and a pixel is 0.784 milliseconds.
Step
2  But that's only part of the job of digitizing the graphs. So
far, we have a collection of pixel positions. But the graphs are about
physical
quantities  milliseconds, Newtons, and Newtonmeters  not pixels. We
have
to take all those pixels and convert them to the quantities they
represent. I used the first page of the spreadsheet for
the pixel representation, and the second page
for the quantities. The typical operation to convert a cell with a pixel
position to a
cell with real data is:
 Take the zero reference height in pixels (the height of the
xaxis).
 Subtract the height of the value pixel (the pixel that is part of the curve), to get the height
above the xaxis in pixels.
 Multiply that by the unit scale factor (units per pixel).
Step
3
 The final step of digitization is a sanity check. Draw a graph of
each
of the waveforms using the spreadsheet "chart" function. By eyeball,
compare the graph to the original in the paper. If it is the same
shape,
and the peaks look about the same height (in units like Newtonmeters),
then the digitization was probably done correctly. Those curves are on the
"graphs" page for the relevant figures in Choi & Park and Koike. An
eyeball
check suggests that we have a reasonable digitization
of the graphs in the papers.

Rotation function
The data in the Koike paper is given as the components of force and
torque in
the swing plane.
That is what we need, because the inverse dynamics studies we are
comparing it with uses the swing plane frame of reference. But the
ChoiPark data is given in terms of axes local to the club itself. So
in order to project the force and torque data into the swing plane, we
need to know the angle between the club and the swing plane. And it
varies over the downswing. In this section, we look at what we need to
do for the ChoiPark data to convert it to forces and torques in the
swing plane  as the Koike data already is.
The club rotates roughly
90° about its zaxis
during the downswing. Consider:
 At impact, the clubface is pointing toward the target. More
precisely, it is pointing in the direction given by the swing plane. So
at impact,
the inplane axis is the local (clubreferenced) yaxis for
forces and xaxis
for torques.
 At transition, the clubface lies approximately within the
swing plane. Numerically, it would be at a zaxis rotation
of about
90°. Depending on the wrist flexion, it might be more negative (cupped
wrist)
or less (bowed wrist), but not a long way from 90°. If it were exactly
90°, the inplane axis is x for forces
and +y for
torques.
 From transition much of the way down, the clubface stays
close to the swing plane. It is only in the last half or less of the
downswing that the club begins to rotate to clubface square, the impact
position.
Let's take this word description and convert it to a table.

Transition

Impact

Angle
of zrotation r

90°

0°

Inplane
force F

F_{x}

F_{y}

Inplane
torque T

T_{y}

T_{x}

In the [x,y]
plane, the forces and torques in the x and y axes combine
trigonometrically. Let's use the sine and cosine of rotation r to come up
with a function that gives the inplane results in the table. The
formulas for inplane force
and torque become:
F = F_{x} sin(r) + F_{y} cos(r)
T = T_{x} cos(r)  T_{y} sin(r)
So now we know how to combine the forces and torques in the local axes
(what we have in the waveforms) to get the inplane forces and torques.
The problem is, we only know rotation r at impact
(very close to zero,
or the shot is offline) and approximately at transition (approximate
because different golfers have different wrist flexion at transition).
What can we do to get a useful function of rotation vs time?
Well, we do have the angular acceleration of the club shaft for the
whole downswing. (In calculus terms, that is the second derivative of r, denoted r''.) It is the zaxis angular
acceleration in Figure 5 of ChoiPark. If the waveform, which we have
also
digitized, is the second derivative of r, then we can find r by two
integrations. The first integration gives the angular velocity r' and the
second the angle r
itself.
Numerical integration
This is not about golf nor physics nor biomechanics. It is about how to
best do
integration on a table of purely empirical numbers. If you wish to skip
it, I will not be insulted, nor even surprised. It is here mostly to
document the choice of method.
We
don't have an analytical function
for the angular acceleration, just
an empirical waveform with numerical values. So we will have to use
numerical integration to get the rotation function. The resulting
rotation
function itself will be a numerical waveform.
There are
several possible approaches commonly used for numerical integration.
 Euler
integration. Mathematically, this is characterized by
the formula:
r[i+1]
=
r[i] + Δt r'[i]
where r'[i]
is the derivative (the value in the table) for the i^{th} table
entry, and Δt
is the time between the i^{th} and (i+1)^{th}
table entry.
This is the easiest to implement, but usually the
least accurate.
 Modified
Eulier integration. Here the next value depends on
both the current and next acceleration.
r[i+1]
=
r[i] + (Δt/2) (r'[i] + r'[i+1])
This is considerably more
accurate for only a small increase in computational complexity. Instead
of using the slope at point i to compute the
value at i+1,
it uses the average slope between i and i+1.
 RungeKutta
method, usually the fourthorder version. This
is more complex, and usually does not lend itself well to the case
where the accelerations are just a table of values. In general, it
requires a formula to compute the function. But our simple
case of just integration and not a rich differential equation causes
the iteration formula to be much simpler.
r[i+1]
=
r[i] + (Δt/6) (r'[i] + 4 r'[i+1/2] + r'[i+1])
Fourth order RungeKutta
is the usual standard for researchquality numerical integration,
because the added accuracy is worth all the extra computation.
So which method
did I use? I started with
just an Euler computation to keep it simple, but it gave results too
far off to be taken
seriously. At impact, the face was still open 40°. Alternatively, we
could postulate that the downswing started with the face closed 40°, so
it would get to square at impact. Sorry, but that's a stretch even for
Dustin Johnson  much less the average professional. So I
implemented all three on the spreadsheet. Here is how
they looked when I was done. The Modified Euler and RungeKutta are
indistinguishable at this scale. I added the green curve, an
idealization
of the rotation function. It is implemented as a sine curve from 90°
to +90°, and the spreadsheet user can define both the impact point
(rotation=0) and the duration of rotation.
For my first cut at choosing an integration method, I wanted one that
would give close to a square face at impact. (We will get more
sophisticated below.) I had
two degrees of freedom
to play with: the initial values for (i) the rotation and (ii) the
angular
velocity. I started out with rotation=90° (the face in the swing
plane)
and angular velocity=0 (the face is neither opening nor closing at the
beginning of the downswing). I did play with those values to see if I
could make any of the methods come out with a square face at impact.
None of my tries gave reasonable curves, given what I have seen of
existing swings. The only ones that made any sense at all were to
assume a rather bowed wrist in transition, like Dustin Johnson. (I
already referred to this above.) Remember that we are talking about an
average of nine professional golfers. For Euler to work, it would have
required a bowed wrist of 40°; for modified Euler or RungeKutta, it
would be down to
16°, but that is still more than I was willing to assume for an average
over the study subjects.
It is very reasonable to make the argument that we know
the clubface is close to square at impact. If not, the ball would fly
off in a completely wrong direction. So that is probably more important
than the clubface lying in the swing plane at transition. In fact, we
know exceptions to the latter, golfers who bow or perhaps even cup
their wrist at the top of the backswing.
To
test whether it really matters, I tried playing with the initial
position, choosing a value so the clubface was square at impact
(rotation=0). This gave curves of the same shape but higher or lower,
as the accompanying graph shows.
When this rotation function was applied, the resulting torque and force
curves moved, but not enough to make any significant difference in (a)
how well the ChoiPark study agreed with others, nor (b) what to teach
about the swing.
Finally, I tested the three methods of integration for accuracy. I applied them to an analytical
function similar to what we have here, but where we know the correct
answer. I used all
three methods to integrate a sine function, which will give a raised cosine,
both of which we know exactly. The percentage errors accumulated are
shown in the following table.
Angle
(rad)

Percentage
error

Euler

Mod.
Euler

RungeKutta

0 
0.0 
0.0000 
0.0000 
0.2 
2.0 
0.0066 
0.0066 
0.4 
3.9 
0.0263 
0.0001 
0.6 
5.7 
0.0583 
0.0065 
0.8 
7.3 
0.1012 
0.0003 
1 
8.6 
0.1533 
0.0063 
1.2 
9.5 
0.2127 
0.0006 
1.4 
10.1 
0.2769 
0.0059 
1.6 
10.3 
0.3433 
0.0009 
This strongly recommends using RungeKutta.
But even having chosen RungeKutta, the integration was not
straightforward to implement. It
requires three values for every iteration: at time sample i, i+½, and i+1. But we
don't have a fomula we can use to evaluate at i+½. So we have
to use three values from our table of samples. Each iteration will
involve i,
i+1,
and i+2,
and Δt is
the time interval for two samples,
not one. I tried
three different variants to compute the RungeKutta process:
 Overlap:
Use modified Euler to get a value at i=1. Then use the threesample
RungeKutta method, grouping together: (012), (123), (234),
(345), etc. You always have a value for all but the last, which is
the one you are trying to find.
 Alternating:
Use modified Euler to get a value at i=1. Then use a single RungeKutta
iteration (012) to get the value at i=2. Then start over; use
modified Euler to get a value at i=3. A single RungeKutta iteration
(234) will give a value at i=4. Continue in this manner, computing
oddnumbered samples with modified Euler and evennumbered samples with
RungeKutta.
 Slipped
alternating:
We have to perform two integrations, one to get angular velocity and
the next to get angle. Use the alternating method with modified Euler
with odd numberered samples for the first integration, and even numbers
for the
second integration.
All three gave very similar results. The Overlap method gave the most
accurate results for the sine function problem, so I used that.

Hand couple
At this point, we have the force exerted by each hand individually:
 For the Koike data, the forces and torques are inplane.
 The ChoiPark data is separated into x, y,
and z
components. We also have the couple exerted by each hand individually,
separated into x,
y,
and z
components. And, from the previous section, we know how to project each
of those forces and torques into the swing plane.
What we need is the inplane total
hand couple, to relate it to all the inverse dynamics studies that form
the basis of our handclub knowledge.
Torques of interest to answer this question are yaxis torques at the
beginning of the downswing and xaxis torques near impact. In
between, it is a mix of the two. In particular, it is a weighted sum of
xaxis and yaxis torques, where the weighting coefficient is a
function of the angle of rotation.
We are trying to compute the
inplane couple exerted by both hands together. It's obvious how to
handle the
individual hand torques; just add them.
But we also need to account for the inplane crossshaft forces,
because they
are exerted
at different places on the grip. As we can see from the diagram
the L and
R
forces are applied a distance d apart,
creating a torque. (The distance d is about 8.5cm for a golfer's grip.)
We can represent the blue quantities with an
equivalent force F
and couple T
shown in red. The point of the equivalent force and couple is the
midhands point shown in green, which is where most biomechanics
studies apply the hand forces. The equivalent for F is clearly L+R. The couple
is a little more work to find, but not much more.
When we move the point of application of L and R to the
midhands point:
T_{L} =  ½ Ld
and T_{R} = ½ Rd
Therefore:
T = ½ (R 
L)d
(If you need to know how I converted forces into a force and a couple,
there is an excellent tutorial
video on YouTube.)
For the ChoiPark
calculations, where we are given clubreferenced axes, it is
important to note that these are y forces
creating an xaxis
torque. The direction of the torque can be seen by applying the
righthand rule to the xaxis. When we
make the same observation about x forces
creating a yaxis
torque, the sense is the opposite and the forces' factor is (LR) instead.
But we're not done. This is the way to find x and y forces and
their couple. But our goal is inplane
forces and torques. This requires using the angle of rotation r in the
projection formulas from the section on Rotation.
Let's use C
to designate hand couple, and use subscripts r and l for right and
left and x
and y for
the respective axes.
C_{x} = T_{rx} + T_{lx} + ½d(F_{ry}  F_{ly})
C_{y} = T_{ry} + T_{ly} + ½d(F_{lx}  F_{rx})
C_{inplane} = C_{p} = C_{x} cos(r) + C_{y} sin(r)
This
allows us to compute the inplane couple, so we can compare it with
this heavilystudied question for both hands together. If it doesn't
match previous results, that puts the ChoiPark and/or Koike results
under suspicion.

Moment of force
Next, we calculate the other torque giving the club angular velocity 
the moment of the force on the handle. The quantities we need are shown
in the diagram below.
F_{z}

The zaxis (axial)
force applied by the hands.

F_{xy}

The force in the swing plane
perpendicular to the shaft. For the Koike data, it is given to us. For
the ChoiPark data, it is a combination of x and y forces
applied by the hands, a
weighted sum where the weighting factor is based on the value r of the
rotation
function.

F

The total force the hands
exert on the handle. It is the resultant force of Fz and Fxy.

a

The angle between the
shaft and the line of action of the total force.

L

The distance between the
midhands point and the CoM of the club. I measured a half dozen
drivers, and this was between .65 and .75 meters. We don't know
anything about L
for the driver in the ChoiPark experiments, so we will use 0.7m for
our calculation.

D

The distance from the CoM
to the line of action of the the force, perpendicular to that line of
action.

For Choi & Park, we need to start by projecting separate x and y forces into the swing plane, to combine to form F_{xy}.
F_{xy}
=  F_{x} sin(r)  F_{y} cos(r)
where r is
the angle of rotation of the club about the shaft axis. The
sign of the inplane force was chosen to denote a force away from
the target. We need that direction because we want a moment of
force that helps the release of the club to be positive. Helping
release means
the line of action of the force is in front of the CoM of the club.
Just from the geometry in the diagram, it is easy to conclude that:
F
= sqrt (F_{z}^{2} + F_{xy}^{2})
a = arcsin
(F_{xy} / F)
D
= L sin(a)
Moment of force = F D
If we apply these formulas in the order shown, we will get the moment
of force. Note that this does not include the effect of gravity. But
gravity contributes less than 3Nm of torque at any point of the swing,
and considerably less for most of the swing.

Summary of computed torques
The end goal is to get the three
inplane torques that are interesting:
 The hand couple.
 The moment of force.
 The total torque, which is the sum of the previous two.
We need them for two big reasons:
 We would like a sanity check; we will compare with studies.
If the ChoiPark results are too different, they are suspect. So are
any new conclusions we draw from ChoiPark.
 We would like to learn things that were not in the prior
studies, specifically how the forces and torques applied to the club
differ between the hands. The novelty of the instrumented shaft studies is that
the hands' forces and torques can be separated into RH and LH.

Sanity tests
My first goal, before
"What can we learn about the golf swing?" has been "How much can we
trust the results of these studies?" To this end, I looked at the
ChoiPark and Koike results with two sanity tests in mind:
 We have a wealth of inversedynamics studies that don't try to
open the loop. They show what both hands together do to the club. The
most familiar results are the hand couple
and the moment of force in the swing plane. Most importantly, those studies all agree with
one another in several important features. (We'll look at those
features below.) That begs us to ask the important question, "How far from
existing, trusted studies are the ChoiPark and Koike results?"
We already saw how to calculate the bothhandstogether hand couple and
moment of force. Let's compare them with the classic results and see if
they agree.
 The other sanity test is, "How well do
the ChoiPark and Koike results compare with one another?"
If they agree, then we might be able to learn something about how the
hands' role differs. But if they disagree, then we don't know which to
believe, if either.
Spoiler: Unfortunately, the studies fail both tests.
Comparison with previous studies
Let's
look at three sets of curves that are rather representative of what
inverse dynamics tells us about the handstoclub torque. That's "hands"
 plural! Remember that inverse dynamics cannot distinguish which hand
does what.

Dr.
Sasho MacKenzie has posted multiple instructional videos to the
web. Here is a screenshot from one of his videos,
where he makes the point that this picture is a very common pattern.
The video showed this pattern of hand couple, moment of force, and
their sum (the total torque applied to the club) for several golfers,
including a multiple major winner. They all looked pretty much like
this, and Sasho says all decent swings look more or less like this .

Dr.
YoungHoo Kwon posted a tutorial
on Facebook about the mathematical model of the golf swing. (The
introductory installment was on November 4, 2015.) He included this
graph as generally representative of the swing of an elite competitor.
He works with many scholastic and professional golfers on their
biomechanics, so he would be in a position to know what is typical.
On the graph:
 T (blue) is the hand couple.
 Tf (red) is the moment of force.
 Ttotal (green) is the sum of the two,
the total torque applied to the club.

Dr Steven Nesbit and Monika Serrano
published a paper in the Dec 2005 issue of the Journal of Sports Science and
Medicine on work and power in the golf swing. They did a
multiplebodypart inverse dynamics study of the driver swings of four
golfers of very different skill levels.
This is Figure 11 of that paper, with a few of my own notations in
color. It shows the inplane component of the hand couple for the four
golfers. Since all the other studies (MacKenzie, Kwon, Choi & Park,
and Koike) were based on highly skilled golfers, we will use Nesbit's
scratch golfer as the model here. (Note: The Nesbit paper did not
include moment of force, nor the data from which moment of force could
be readily calculated.)
Wait! Nesbit? Wasn't there a lot of controversy
about Nesbit's results vs the rest of the world of biomechanics? Yes,
indeed! But in the overall scheme of things, Nesbit's results look a
lot like the MacKenzie and Kwon results we have already seen. And that
is even more true for Nesbit's scratch golfer. The controversy was
whether the hand couple is positive or negative at impact; for the
scratch player, there is no dispute on this point.

Hand Couple
Here
is a graph showing the hand couple for each of the five studies: Choi
& Park, Koike, MacKenzie, Kwon, and Nesbit. The "classic" studies
are shown as dashed lines and the instrumented grip studies as heavy
solid lines.
They have a few features in common:
 They are significantly positive for most of the downswing.
 They drop later in the downswing and are negative by the
time of impact.
But there are two very important characteristics of the classic studies
that are quite different from the instrumented grip studies:
 The classic studies have positive peaks between 37 and 50
Newtonmeters of torque, while Koike barely gets above 20 and ChoiPark
never makes it to 15. They are woefully lacking in torque.
 While the classic studies dive negative, crossing the axis
in the last 40msec or less, ChoiPark crosses the axis 80msec before
impact. Koike is much more conventional in the time it crosses zero,
but it limps rather than dives; it is barely negative at impact.
On balance, graphs computed from ChoiPark and Koike do not match what
we should expect, either from previous inverse dynamics or other
evidence like TrueTemper's studies
of shaft bend.

Moment of force
Here
is a graph of moment of force for four of the studies. (Nesbit is missing; that study does not
provide the moment of force, nor the forces necessary to compute it.)
Neither ChoiPark nor Koike look anything like what we are used to.
 The ChoiPark curve does start negative and go positive,
but:
 It is only slightly negative, not nearly enough.
 It goes positive 180msec before impact, not 50msec or
less like the classic studies.
 It is already running out of enthusiasm (turning back
toward zero) before impact.
 The Koike curve has the right size negative peak, but
nothing else about it looks like the moment of force. In particular, it
contributes nothing to clubhead speed, which we know not to be true for
any decent swing.

Neither ChoiPark nor Koike is
close enough to
the timetested inverse dynamics studies that we can trust it to be
true. Let's move on to...
Comparison with one another
Choi & Park and Koike teased apart the left and right hand forces
and torques, but in two different ways. Let's look at how the hands
differ right from left, and specifically whether the two approaches
find them differing
in the same way. If not, ChoiPark and Koike have a consistency problem. We would not
know which of them to trust to answer our questions  if either.
Hand couple
Here are graphs of the hands' contribution to the hand couple. The
color coding for both graphs is:
 Red
 right hand torque.
 Green
 left hand torque
 Blue
 the torque couple due to a "pushpull" of forces
from both hands.
 Black  the total
hand couple torque, the sum of the three
components above.
The total hand couple torque is only roughly the same shape. More
significant, the way the hands contribute to the couple is completely
different between ChoiPark and Koike. Specifically:
 For Choi & Park, the first hundred milliseconds of the
downswing (and especially the first 50msec) shows all three components
have roughly the same contribution to the total couple, each of them
36 Newtonmeters.
 For Koike, that same time period has negligble right hand
contribution, about 7Nm of torque from the left hand, and 1316Nm from
the pushpull couple. The ratio of the contributions is completely
different and the total is about twice as large as was Choi & Park.
 The left hand and the pushpull are much the same shape for
both, though very different magnitudes. The right hand torque is not at
all similar between the two studies.
Axial force
The moment of force of both studies is far from reality. The
reason is that the magnitude and even the sign of the inaxis
crossshaft force is far from reality. But the axial force is the right
shape for both (if not necessarily the right size), so let's compare
that. The
color code is:
 Red
 right hand contribution to axial force.
 Green
 left hand contribution to axial force.
The distribution of right vs left is completely different.
 Koike has the left hand doing essentially all the work,
with the right hand just "along for the ride". There are certainly
players on tour with this use of the hands; we know that because they
allow the right
hand to come completely off the club at impact. Vijay Singh is a good
example, but hardly the only example.
 Choi & Park have the left hand doing all the work for
the first third of the downswing, then briefly "dying" and the right
hand taking over as the total axial force curves upwards. Finally the
left kicks in again when the right hand peaks, and they are sharing the
load about equally just before impact.
 Now let's look at the magnitudes. We know the axial force
in the vicinity of impact is about 100 pounds, or 450N. The Koike graph
has just about that total (both hands), while Choi & Park have a
bit more than half of what we would expect.
The results simply do not agree! Koike's results are more credible in
this test than in the test against the inverse dynamics results. Choi
& Park's results do not provide credible answers.
Conclusions
I think that both teams are
onto something
important!
Instrumenting a grip should
be a good way to determine the answers to those questions. But I am
convinced we don't have answers yet. We still
have
to refine the technique, get rid of the things that made the current
studies so inaccurate.
My bottom line is that neither the Choi & Park results nor the
Koike results are ready for prime time. We cannot use them yet to
decide what torques and forces better players apply to the club, much
less what to teach golfers to do.

What needs to be done better?
I know it is unconventional to have a major
section after the "Conclusions", but I'd like to itemize things I see
as needing fixing in order to get useful results from an instrumented
grip.
Instrumenting a grip is a promising approach, but neither
implementation gave trustworthy results  yet. That says
something
needs to be done better. Here is a list of things I would suggest for
the next try.
But before we do that, I'd like to review possible ways the fault could be mine. I don't
think I have mistakes here, but if I do...
 My digitization
of their data might be faulty. I doubt this is the case. The
graphs produced by my digitized data look very similar to those in the
published papers.
 My calculations
might be faulty.
There are lots of places where it is too easy to invert plus and minus
in the calculations. I have gone over the spreadsheet several times,
but errors may still be there. I'm willing to share my spreadsheet with
anyone who wants to go over them in detail. Contact
me if you're interested. Or, better yet, build your own spreadsheet
starting with my digitized graphs from Choi & Park and Koike. If
there is an error, I hope someone can
find it and tell me about it. I'll be glad to rewrite this article in
light of the corrected information, or withdraw it altogether if that
is appropriate.
 The previous
inverse dynamics
studies used for comparison might be faulty.
I am not inclined to entertain this possibility. There have been some
disputes about those results, leading to a very close inspection of the
studies. I don't think there have been many stones left unturned. And
even the opponents in those disputes had results that looked much more
like one another's than any look like the ChoiPark or Koike results.
Let us look at the ChoiPark and Koike studies as a good start, and try
to identify the
problems that most need solution. I have listed what I see as the
problems, in the order of most important first. Each issue is flagged
for either Choi
& Park, Koike, or both.

Both  Pay attention to inverse
dynamicsThere
have been a lot of inverse dynamics of the golf swing published
over the past decade or more. Together they constitute what we know
about the golf swing. And together they constitute a fairly consistent
picture of the forces and torques the hands apply to the club. The
biggest missing component, in the opinion of many, is failure to
distinguish between right and left arms/hands  the closed loop.
So it is imperative that any attempt to separate right from left  to
open the loop  needs to be consistent with what we already know.
Before you present your results to the world, do what I did: add the
hands back together and make sure the closed loop solution is right. If
your solution does is not consistent with the inverse dynamics
literature, it is not credible. Let's face it:
 The closed loop hand couple for Koike is not very good, and
ChoiPark is worse.
 The closed loop moment of force for ChoiPark is dismal,
and Koike is worse.
I am particularly surprised that the results in Choi & Park are not
consistent with our existing knowledge. Unlilke Koike, who measured
each hand's forces individually, Choi & Park started with an inverse dynamics solution
and separated the hands using a difference signal between what the
hands did to the grip. If their results don't meet this criterion then
they were starting with a suspect closedloop inverse dynamics solution.

Koike  More golfers
Choi & Park  Separate golfers
Both studies have the problem of only "one swing" graphed. The
explanation for the Koike study is easy; there was only one golfer. In
the case of Choi & Park, the graphs are the average of 45 swings from 9
golfers. To their credit, Choi & Park also included a gray area
indicating variation in the sample. That might be enough. Or it might
not, if there were several different ways of swinging the club
represented.
During the Zoom conference, someone (I think Michael Finney) said,
"There will always be criticism. If you average the data, someone will
need to see separate graphs. If you keep them separate, someone will
want to see an average." Too true. But let's look at a single issue 
how the hands divide up the axial force  and
see if that might guide us.
Here is a pair of graphs, one from Koike and the other from Choi &
Park. They show the axial force (pull along the shaft axis) vs time.
The red vertical line on each graph is the impact.
Again, remember that ChoiPark is an average of five swings for each of
nine golfers, while Koike is a single golfer. Choi & Park chose to
show the swingtoswing variation by a gray area surrounding the
average. They never indicate whether that is a total variation or one
standard deviation. Now to see what variation needs to tell us. Note
that the Koike data shows the right hand
to be exerting negligible force, while ChoiPark has a substantial
force (over 100N). That difference might be a real disagreement, or it
might mean that Koike's one golfer is something of an outlier. But...
The gray area showing variation does not show any data close to zero
righthand force at impact. Even a graph along the bottom edge of the
gray area attributes a lot more force to the right hand than Koike's
graph. And that is for the whole final third of the downswing.
What are we to make of this? Was Koike's one golfer such an outlier
than none of the ChoiPark golfers had that pattern? I find it hard to
believe, since we can point to tour players who obviously have the
Koike pattern at impact.
My takeaway is:
 Koike needs to do the study for a collection of golfers,
not just one.
 ChoiPark and Koike both need to find a good way to
represent variation in the results. That includes noting if there are
classes of swings with different graphical "signatures", and
characterizing their results rather than just averaging everybody.

Both  Different club, different swing
If the club is too different from a regular club, it will affect the
swing. In other words, it will affect the forces and torques the golfer
uses, and that the instrumented grip records. And both ChoiPark and
Koike had something very different from the regular club. The
instrumented grip added a lot of weight to the buttend 10 or 11 inches of
the club.
 Reading between the lines in Choi & Park, I would guess
that the top 10" of their club is 280 grams heavier than a normal club.
 Koike doesn't say anything about weight. But, based on my
own measurements of their diagram and their description of the
instrumented grip, I believe they added at least 200 grams to that top
12 inches.
Since most drivers are not much more
than 300g in the first place, ChoiPark comes close to doubling the
weight
of the club, and Koike adds about 70%. I have
done computer simulations of golf swings where up to 100g is added at
the butt; the effect on impact was negligible. However, the
changes in these studies are more likely to have an effect because:
 The added weight is two to three times what my simulations
investigated.
 The
added weight is not concentrated in the butt itself. Instead, it
appears to be close to uniform for the top 1011 inches of
the club.
The changes will increase the moment of inertia of the club, as well
as giving a different "feel" of the club in the hands. Weight changes
like this might well cause a change in the swing. Of course, if the
club changes the swing significantly, the conclusions drawn from any
measurement will be affected as well.
So how big were the changes in the measured forces and torques due to
the instrumentation of the club? Choi and Park anticipated this
question. Their Figure 3 (reproduced below) shows kinematic
graphs that compare the position
of a regular club with the instrumented club. (Not the torques, not the
speeds, but the position.)
The
"difference" curves are visually small, which may or may not be
intentional. I don't have any intuition for
what that says about the torques, so I'm not sure whether this explains
the difference from previous studies. But they do state in Section 2.4:
The trajectory of the left hand and the
clubhead and the main positions of the swing, showed relatively small
difference with an RMS of 0.06 and 0.16 m, respectively.
This is supposed to be a small difference. But 0.06m is 6cm or more
than 2 inches, and 0.16m is 16cm or more than 6 inches. That does not
sound small to me. Moreover, this is an RMS
value, not a peak value. The error graph looks to have a variability
similar to a sine wave, so we could expect the peak errors to be about 3 and
9 inches respectively. That is not a small effect, in my opinion.
There is one difference that is calculable. Adding that much weight at
the grip moves the club's center of mass much closer to the midhands
point. In doing so, it cuts the moment arm for the moment of force by
about 40% for the ChoiPark club, somewhat less for the Koike club. So the moment of force will probably be 40% lower than I
calculated. That doesn't help much. The big problem with
the moment
of force is its shape not its size, particularly how early it goes
positive. This
change of balance point will not affect the zerocrossing at all.

Choi & Park  Smoothing
From Choi and Park, Section 2.4:
All collected data were filtered with a
fifthorder bidirectional
Butterworth lowpass filter with a cutoff frequency of 10 Hz.
A
fifthorder filter is pretty sharp, so we can assume any frequency
components over 10Hz would be negligible. That may be a problem if
you are looking for rapidly varying behaviors, for instance in the vicinity of
impact. I come from an EE background, and worked with sampled data
systems in the 1960s. For those projects, I lived by the Sampling Theorem. This was originally stated by
Nyquist in 1928, and formalized as part of information theory by
Shannon in 1948.
The sampling theorem says that, given a signal with no frequencies in
it higher than F,
you can take samples of the signal at 2F or
anything above that, and reconstruct the original signal with perfect
accuracy based only on the samples. The reconstruction is conceptually
pretty simple; just pass the samples
as pulses through another lowpass filter
of cutoff frequency F. That's what the sampling theorem says you can do. It also tells you what not to do. If
you sample at less than 2F, any
reconstruction is going to have some artifacts that keep it from being
accurate.
Let's look at what the sampling theorem implies for this system.
Because of the 10Hz filter, any sampling frequency above
20Hz should be sufficient. A 20Hz sampling rate means a sample every
50msec.
Underlying that calculation of sampling frequency is an interesting
correlary. Any feature of the waveform that lasted less than 50msec was
probably filtered out
by the smoothing filter. It may also have been replaced by distracting
artifacts of the smoothing filter.
Let's
look at what such a filter does to a fast transition in its input. The
figure shows what a step function input (the red line) looks like
after
going through a fifthorder Butterworth filter (the white oscilloscope
trace). The
filter does several things to the sudden input change:
 It creates a smooth, continuous function. That is what we
want to happen when our purpose is smoothing the data.
 Because of the slope of the resulting change, the "middle"
of the step is effectively delayed. The delay is about 35% of a
cycle at the cutoff frequency of the filter. For a 10Hz filter the
delay will be about 35msec. So what we see in the graphs was delayed
about 35msec from when it occurred.
 There is a "ringing" effect that dies out after a few
cycles. The frequency of the ringing is 10Hz. It can deform the curve,
obscuring or at least distracting
from effects you are looking for in the data.
Remember, in a golf swing a lot of the interesting activity occurs in
the last
40msec before impact. That is where most of the inertial release of the
club occurs. So it is likely that the most interesting
information about the downswing has been removed  smoothed right out
of existence  by the smoothing
filter, or distorted by ringing.
What this tells me, at the very least, is that I cannot depend on any
of the details during the last 40msec of the downswing. They constitute
the release of the club, and occur on too fast a time scale for the
smoothed data to represent. It also tells me that the graphs may be
delayed from when they actually occurred. Or perhaps not; I didn't see
anything in the ChoiPark paper that said they corrected for this, but
they might have.
If they wanted, could they have used a higher filter cutoff than 10Hz?
Yes, unless they were
removing so much noise they needed such heavy smoothing. They sampled
their
motion capture system at 200Hz and the
sensor in the grip at 400Hz. So why use a smoothing filter suitable for
a 20Hz sampling rate? If they had used a smoothing filter of
3050Hz, that would have allowed capture of timedomain features of the
order of 17msec to 10msec, fast enough to capture most of the
interesting features of the downswing. And it still leaves a factor of
23 to reduce noise and quantization errors, assuming that would be
enough. I don't know what noise they were dealing with apart from
quantization noise, so I don't know if it is enough smoothing for what
they encountered. But the smoothing they used smothers anything that
happens during release.
While we are talking about smoothing the data at the Zoom metting, Rob
Neal made an interesting comment. He pointed out that the curves
published by Choi and Park are the average of the 45 swings in the
experiment. Averaging is itself a smoothing operation. So the smoothing
may well be even heavier than the 10Hz cutoff frequency would suggest.

Both  Overlapping hands
Both ChoiPark and Koike
instrumented the grip by dividing it into a lower part (toward the
clubhead) and an upper part (toward the butt). They assumed that
anything done to the upper part was done by the lead (left) hand and anything
done to the lower part was done by the trail (right) hand. But that is not
perfectly true; the hands overlap, and therefore necessarily impinge on
the "wrong" part of the grip.
In the picture, the red dashed line is the best possible split if the
instrumented grip is simply split into an upper and lower portion. We
can see that the lead thumb and forefinger live on the trail part of
the grip, and the trail fifth finger lives on the lead side of the
grip. And this is probably the best
split; move the red line and the overlap errors might become greater.
But that is not the only problem with overlap. The hands actually
interact. The green arrow represents a force exerted by the base of the
trail thumb on the lead thumb, which in turn increases the force the
lead thumb exerts on the grip. That's an added complication.

Choi and Park do not simply
dismiss this problem. They write:
The hand grip posture by covering the left
thumb by the thenar eminence of the right hand leads to the coupling of
the grip forces of both hands. Then, the embedded force sensor
measurement could possibly underestimate the difference in the
handgrip
forces. To examine this overlap effect on grip force measurement,
subjects also performed swings by grasping the club with two hands
completely separated.
They then measured the "internal" forces and torques from normal swings
and handsseparated swings. Here is their result.
Their "internal grip force and torque" is a processing of the sensor
output. It represents in some way the difference between the hands, but
it isn't exactly that difference. (I did a spreadsheet's worth of
arithmetic to verify that, but I don't know exactly what it is.) Choi
& Park seem to believe it says the thumb interactions can be
neglected without interfering too much with the results. It may also
say something about the finger overlap from one part of the grip to
another, but I'm not sure.
In any event, I think there are pretty substantial differences between
the regular grip and the split grip, in the order of 50% of the total
force for much of the downswing. The torque differences were less, but
still more than 10% of the total torque. So I am not prepared to
dismiss the difference out of hand.
The Koike conference paper is only four pages, and doesn't get to that
level of detail. So I don't know how carefully Koike considered this
problem.
The problem needs to be solved. The solution needs to be both correct
and convincing. I must admit that I have no idea how to do it with an
instrumented grip. All I can think of is instrumented gloves, which
have their own problems.

There are other things I can think of that might be improved. But they are considerably less
important than these five. Solve these, and we can all learn something
useful about the golf swing while we do the next round of refinements.
Last
modified  3/21/2021


