Modeling the Swing  Cochran & Stobbs
Dave Tutelman
 January 16, 2012
Why We Need a Model of the Swing
I was intending at some point to write an article
on the value of physics to golf. Cochran & Stobbs' chapter 2 does a
fine job saying most of what needs saying. I have added just a bit here,
giving a 2012 view of scientific modeling on top of C&S's original
1968 treatise.
Let me begin by excerpting/paraphrasing a few words from their book.
In science a model is a simple
representation of something complicated or illunderstood. A scientist
can actually build such a model of wood, plastic, metal, etc, then test
it in detail to see how it works. More commonly, he can just describe
his model mathematically, and work out on paper all he wants to find
out about its behavior.
What a wonderful, simple, elegant statement! It captures the essence of
a physical/mathematical model in so few words. The model's value is
that it can be analyzed and characterized, even if the complicated,
illunderstood real world cannot. But, in order for the analysis to
give useful results, a few more things must be true:
 The model must actually faithfully represent the real world in
its behavior. This usually means that each part of the model must
correspond to a part of the reality it represents. "Correspond to" does
not have to mean "look like"; they may look very different and still be
a valid model.^{[1]}
 The result of analyzing the model must correspond to the
realworld system's measured outputs. If it can't correctly predict
what we can measure, the model cannot be trusted to predict anything
else.
If the model can be trusted to give reasonable predictions, then its
value is obvious. We can tweak the parameters (inputs) of the model,
and see what happens to the outputs. For instance, if we had a valid
model of the swing that includes the wrist cock at the top of the
backswing, we can try different wrist cocks in the model to investigate how wrist cock affects
things like ball speed and trajectory. It might be difficult to
investigate the effect of wrist cock without a model. Go ahead and try
to design a humangolfer experiment to answer that question. It is
almost impossible, given that any one golfer might go to a less
comfortable swing (and therefore less effective) if asked to try a
different wrist cock than usual  and that two different golfers with
different natural wrist cocks have a lot of other differences between
their swings, which would preclude attributing different results to
wrist cock alone.
Note that the model does not have to represent perfectly all things
about the real system. But it must represent the things that are
important to the output. And therein lies the art and the difficulty of
modeling the golf swing. The debates in current research center around
things like whether both arms must be modeled in all their joints, or
is a straight left arm sufficient? How about the left arm plus some
additional motion attributable to a bulk summary of the right arm's
effect? So far, nobody is criticizing any of the models for not
representing individual fingers gripping the club, nor what color shirt
the golfer is wearing. It is generally agreed that these attributes of
the real system do not have an important effect on the interesting
outputs  so they do not need to be accounted for in the model.
But why don't we include them anyway, just to make sure we are not leaving out anything important?
As we add features to the model, we put additional burdens on analyzing
it. A few decades ago, this absolutely mandated keeping the model as
simple as possible. But the abundance of computing power today weakens
that argument. True, every variable added to the model is going to make
it harder to program, and is going to make the program run longer.
However, adding that variable is not likely to make the model difficult
to the point of nearimpossibility  as was the case when Cochran
& Stobbs first wrote their book.
But there is another reason to make every feature of the model "earn
its keep". That isn't the mechanics of analysis, but the difficulty of
understanding what the model is telling you. Every additional variable
is something you have to take into account, tweak, and see what it
does. And what it does may be affected by where the other variables are
set. So you have yet another dimension, and you have to
mentally assimilate and understand what that dimension is telling you.
Consequence: In order for the model to provide understanding of the
underlying reality, it has to be kept simple enough so the model itself
can be understood, and manipulated in a sensible way.
One last observation about models, before we look at almost 50 years of
modeling the swing. What the model needs to include depends on what
questions you want to ask about the real system. Consider:
 If the important output is clubhead speed, then you will get very
close to reality even if you only consider inplane forces. (That is,
forces and torques in the swing plane.)
 If the important output is directional accuracy, then you need to
know the angle of the swing plane, the axial rotation of the club's
shaft, outofplane forces, and perhaps other things as well  in
addition to just about everything you needed for clubhead speed.
So you can have a perfectly good and worthwhile model of the golf
swing, but there may still be questions you can't ask it. That doesn't
mean the model is worthless; it just means you need a different model
for the particular question you want to ask.
The Double Pendulum Model
In addition to a good
explanation of why we need a swing model, Cochran & Stobbs gave us
a simple first model, one that served the golf community for 30 years.
It was the doublependulum model of the golf swing.
Shoulder turn

Wrist hinge

(Photos from Cochran & Stobbs, "Search for the Perfect Swing")
Cochran & Stobbs
observed that the most prominent motions in a golf swing were a turn of
the shoulders and a hinging of the wrists. (Each motion is demonstrated
separately in the photos above.) Then they made the scientist/modeler's
leap: Suppose I constructed
a model of just these two motions. That would be manageable for
analysis. Does it match what actually goes on in a golf swing well
enough to be a useful model?
History has proven it very useful, even if missing some of the details of the swing. Here then are the elements of that model:
 A
double pendulum, consisting of two levers, an upper and a lower. The
upper corresponds to the arms, and the lower corresponds to the club.
 A fixed pivot at the inner end of the upper lever. This
corresponds to a point between the shoulders, about which the body
turns and rotates the arms.
 A hinge between the upper and lower levers. This corresponds to the arms and hands; the angle at this hinge is wrist cock.
 A pair of "torque generators", one at each hinge,
that allow the application of an arbitrary torque at the hinges. (More
explanation of this below.)
 A "stop" at the hinge, limiting the angle of wrist cock.
Quoting C&S, "You can think of the stop as a wedge of slightly
yielding material which prevents the two levers from jackknifing on
each other. It represents the golfer's inability to cock his wrists by
more than 90º or so."

A word about forces  and torque
The swing models we discuss are generally composed of limbs and joints.
Muscular "forces" are applied at the joints. They can't just move a
limb arbitrarily. The only way they can move a limb is for the muscles
that span the joint to cause one limb to exert a force to move the
other.
But
how do joints move? They are pivots or hinges. They allow the limbs to
turn, one with respect to another. Therefore, the only forces the
muscles can exert at the joint is a turning force  a torque. (See the physics primer if this is at all mysterious to you. Understanding torque is absolutely essential to understanding the models.)
The picture shows how a joint turns a muscular contraction into a
torque. The joint in this case is the elbow. The biceps muscle
contracts (shortens forcefully  that is what muscles do), pulling on
the inside of the joint. At the same time, the triceps muscle is
relaxed, exerting no force on the outside of the joint. The result is a
net torque (turning force) exerted on the forearm by the upper arm  the blue arrow in the picture. The
torque "flexes" the arm; that is, creates a more acute angle at the
elbow joint.
To exert a torque in the opposite direction  to "extend" the arm
rather than "flex" it  the triceps would contract and the biceps
would relax. The pull of the triceps is exerted on the outside of the
elbow joint, so the torque would be the reverse of the blue arrow shown in
the picture.
All the muscular "forces" of interest in all the swing models here are
of this type: a torque at a joint, rather than a pure pushing or
pulling force.

There are exactly two torques that can be applied in the Double Pendulum model:
 At the fixed pivot. This is the body turn, the shoulders turning the "triangle" of the arms. In modeling the swing, we refer to this as "shoulder torque".
 At the hinge between the levers. This is the hands, wrists and forearms changing the angle of the club. In modeling the swing, we refer to this as "wrist torque".
With only two moving parts, the model is very simple, making it
relatively easy to analyze. Another advantage of simplicity is that it
makes it easier to learn lessons from the model; cause and effect are
more readily seen if the possible causes are few. The weakness of such
a simple model is that it is easy to question its relevance to reality.
We will deal with that in the next few pages.

Analyzing the model
A
reader will observe that the equations for the model are nowhere in
Cochran & Stobbs' book. Neither are any diagrams annotated with the
sort of variables one could use in analysis. So why do we call it a
model? (Especially since we declined to so designate Kelley's TGM, for
lack of quantification.)
Well, the book is full of illustrations like the one at the right, and
plenty of others that indicate C&S did the math homework, and
compared the results to real swings in detail. There are several reason
why that detail is not in the book:
 This is not an engineering or math text. C&S did a
remarkably good job of not including equations (except in the Appendix,
for techies like me). So they showed a lot of the results of working
with the model, without giving the reader the math involved in the
workings.
 In the 1960s, computing was expensive and somewhat rare.
There was no computer on everybody's desk. Even terminal access to
computers was very rare. In order to analyze the model, you not only
had to set up the equations (true for any model, even today); you also
had to get the computation done in a world where computation was not
easy to come by. Computers were in big computer rooms, usually behind
locked doors; if
you wanted computing done, you submitted a deck of punch cards or
magnetic tapes over a counter to the computer operators, then came back
in an hour or three to get your printout. No, not diagram; printout. No
spreadsheets either; you probably did it in Fortran. Or... You did the
computation by hand, with an adding machine or slide rule to help.
(Hey, I was a working engineer at the time. Putting a model on a
computer was much more of an investment of time back then. No
spreadsheets, almost no graphing programs, and I already detailed the
clumsy way of submitting the job to the computer.)
So I'm convinced that they did the analysis, and presented the results
in their book, without showing the details of the analysis. They might
have also published technical papers with those details, but I haven't
seen them. For most folks interested in swing models, the details of
the analysis were revealed by Theodore Jorgensen in 1994 or so. Which
brings us to...

Notes:
 In fact, two very differentlooking
physical systems can have the same mathematical model. Science is full
of phenomena that are completely different but are governed by the same
equations. When that happens, one of the systems can be used as a
"computer" to act as a working model for the other system. I ran across
a few such pairs in engineering school, back in 1960 give or take a
year.
 I took a course in structural materials. One of the topics was
the way stress flows in an irregular structure, such as stress
concentration at a notch. That was very hard to measure directly. (More
modern and more expensive instruments are available today.) But the
lines of stress follow the same equations as lines of field flow in an
electric field. So we could copy the structural shape, making it out of
an electrically resistive material, then measure and plot the voltage
in the resistive material. And we could be confident that the lines of
stress in the real structural material would be perpendicular to the
lines of constant voltage in the resistive material. We were using the
resistive material as a "computer model" of the structural material.
 I also took some military training as part of the ROTC program
(Reserve Officers Training Corps). Part of the training was using an
artillery plotting board as a forward observer would, to transform the
observer's point of view to the gun's point of view. Being an engineer
more than a forward observer, I mused about the algebra and
trigonometry of the conversion, and recognized it as being identical to
what I did every night in my AC circuit analysis homework. I borrowed a
forward observer board from the military science department for the
semester, and recalibrated "yards" to "volts" (or ohms or amperes,
depending on the problem). Then I used it as a "graphical computer" for
my circuit analysis homework. It halved the time I spent on AC
circuits, and I still got the right answers.
Last
modified  March 3, 2012
