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 ill-understood. 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, ill-understood 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 real-world 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 human-golfer 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 near-impossibility -- 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 in-plane 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, out-of-plane 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 double-pendulum 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 jack-knifing 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 different-looking 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