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 Choi-Park 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 cutting-edge 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 both-hands 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:
  1. Take the forces and torques applied separately by the right and left hands, and recombine them to see if they match the well-known closed-loop solutions. If you can't get the closed-loop answer right, I must have serious reservations about your open-loop answer.
  2. 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 inverse-dynamics results, the hand couple's magnitude is too low for both studies. Choi-Park 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 Choi-Park 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 z-axes 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 x-Axis Local y-Axis Local z-Axis
Both club segment Cross product of the local y- and z-axis 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 y-axis 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 y-axis 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 right-hand rule for torques, common in engineering and physics texts.

Forces and torques

Local x-Axis Local y-Axis Local z-Axis
Force
The toe-heel 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 heel-toe 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 well-established outputs from inverse dynamics studies.)

The graphs are:
  1. In-plane forces across the shaft.
  2. Forces along the shaft axis.
  3. 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 Choi-Park 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 time-series graph from the study, to compare with previous (necessarily closed-loop) inverse dynamics studies from researchers Sasho MacKenzie, Young-Hoo Kwon, and Steven Nesbit. That involves finding the in-plane 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 swing-plane 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 axis-by-axis forces and torques in whatever manner necessary to keep the reference in-plane. 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 point-for-point 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 open-source 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 Choi-Park paper, the graphs also contain some of the backswing and much of the follow-through. 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 Newton-meters -- 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:
  1. Take the zero reference height in pixels (the height of the x-axis).
  2. Subtract the height of the value pixel (the pixel that is part of the curve), to get the height above the x-axis in pixels.
  3. 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 Newton-meters), 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 Choi-Park 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 Choi-Park 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 z-axis 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 in-plane axis is the local (club-referenced) y-axis for forces and x-axis for torques.
  • At transition, the clubface lies approximately within the swing plane. Numerically, it would be at a z-axis 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 in-plane 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 z-rotation r
-90
0
In-plane force F
-Fx
Fy
In-plane torque T
Ty
Tx


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 in-plane results in the table. The formulas for in-plane force and torque become:

F  =  Fx sin(r) + Fy cos(r)

T =  Tx cos(r) - Ty 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 in-plane forces and torques.

The problem is, we only know rotation r at impact (very close to zero, or the shot is off-line) 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 z-axis angular acceleration in Figure 5 of Choi-Park. 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.
  1. 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 ith table entry, and Δt is the time between the ith and (i+1)th table entry. This is the easiest to implement, but usually the least accurate.
  2. 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.
  3. Runge-Kutta method, usually the fourth-order 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 Runge-Kutta is the usual standard for research-quality 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 Runge-Kutta 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 Runge-Kutta, 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 Choi-Park 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
Runge-Kutta
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 Runge-Kutta.

But even having chosen Runge-Kutta, the integration was not straightforward to implement. It requires three values for every iteration: at time sample i, i+, and i+1. But we don't have a fomula we can use to evaluate at i+. 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 Runge-Kutta process:
  1. Overlap: Use modified Euler to get a value at i=1. Then use the three-sample Runge-Kutta method, grouping together: (0-1-2), (1-2-3), (2-3-4), (3-4-5), etc. You always have a value for all but the last, which is the one you are trying to find.
  2. Alternating: Use modified Euler to get a value at i=1. Then use a single Runge-Kutta iteration (0-1-2) to get the value at i=2. Then start over; use modified Euler to get a value at i=3. A single Runge-Kutta iteration (2-3-4) will give a value at i=4. Continue in this manner, computing odd-numbered samples with modified Euler and even-numbered samples with Runge-Kutta.
  3. 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 in-plane.
  • The Choi-Park 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 in-plane total hand couple, to relate it to all the inverse dynamics studies that form the basis of our hand-club knowledge.

Torques of interest to answer this question are y-axis torques at the beginning of the downswing and x-axis torques near impact. In between, it is a mix of the two. In particular, it is a weighted sum of x-axis and y-axis torques, where the weighting coefficient is a function of the angle of rotation.

We are trying to compute the in-plane 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 in-plane cross-shaft 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 mid-hands 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 mid-hands point:

TL = - Ld    and    TR = Rd

Therefore:

T  =  (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 Choi-Park calculations, where we are given club-referenced axes, it is important to note that these are y forces creating an x-axis torque. The direction of the torque can be seen by applying the right-hand rule to the x-axis. When we make the same observation about x forces creating a y-axis torque, the sense is the opposite and the forces' factor is (L-R) instead.

But we're not done. This is the way to find x and y forces and their couple. But our goal is in-plane 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.

Cx = Trx + Tlx + d(Fry - Fly)

Cy = Try + Tly + d(Flx - Frx)

Cin-plane  =  Cp  =  Cx cos(r) + Cy sin(r)

This allows us to compute the in-plane couple, so we can compare it with this heavily-studied question for both hands together. If it doesn't match previous results, that puts the Choi-Park 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.

Fz
The z-axis (axial) force applied by the hands.
Fxy
The force in the swing plane perpendicular to the shaft. For the Koike data, it is given to us. For the Choi-Park 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 mid-hands 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 Choi-Park 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 Fxy.

Fxy  =  - Fx sin(r)  -  Fy cos(r)

where r is the angle of rotation of the club about the shaft axis. The sign of the in-plane 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 (Fz2Fxy2)

a  =  arcsin (Fxy / 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 in-plane 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:
  1. We would like a sanity check; we will compare with studies. If the Choi-Park results are too different, they are suspect. So are any new conclusions we draw from Choi-Park.
  2. 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 Choi-Park and Koike results with two sanity tests in mind:
  1. We have a wealth of inverse-dynamics 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 Choi-Park and Koike results?" We already saw how to calculate the both-hands-together hand couple and moment of force. Let's compare them with the classic results and see if they agree.
  2. The other sanity test is, "How well do the Choi-Park 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 hands-to-club 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. Young-Hoo 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 multiple-body-part 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 in-plane 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:
  1. The classic studies have positive peaks between 37 and 50 Newton-meters of torque, while Koike barely gets above 20 and Choi-Park never makes it to 15. They are woefully lacking in torque.
  2. While the classic studies dive negative, crossing the axis in the last 40msec or less, Choi-Park 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 Choi-Park 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 Choi-Park nor Koike look anything like what we are used to.
  • The Choi-Park 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 Choi-Park nor Koike is close enough to the time-tested 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, Choi-Park 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 "push-pull" 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 Choi-Park 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 3-6 Newton-meters.
  • For Koike, that same time period has negligble right hand contribution, about 7Nm of torque from the left hand, and 13-16Nm from the push-pull 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 push-pull 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 in-axis cross-shaft 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...
  1. 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.
  2. 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.
  3. 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 Choi-Park or Koike results.
Let us look at the Choi-Park 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 dynamics

There 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 Choi-Park is worse.
  • The closed loop moment of force for Choi-Park 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 closed-loop 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 Choi-Park is an average of five swings for each of nine golfers, while Koike is a single golfer. Choi & Park chose to show the swing-to-swing 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 Choi-Park 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 right-hand 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 Choi-Park 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.
  • Choi-Park 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 Choi-Park and Koike had something very different from the regular club. The instrumented grip added a lot of weight to the butt-end 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, Choi-Park 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 10-11 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 mid-hands point. In doing so, it cuts the moment arm for the moment of force by about 40% for the Choi-Park 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 zero-crossing at all.
 

Choi & Park - Smoothing

From Choi and Park, Section 2.4:
All collected data were filtered with a fifth-order bi-directional Butterworth low-pass filter with a cut-off frequency of 10 Hz.
A fifth-order 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 fifth-order 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 Choi-Park 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 30-50Hz, that would have allowed capture of time-domain 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 2-3 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 Choi-Park 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 hand-grip 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 hands-separated 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