# Modeling the Swing - MacKenzie

## Three Dimensions, Three Pendulum Levers

Dave Tutelman  --  January 16, 2012

In the early 2000s, there were quite a few attempts to add "realism" to the Double Pendulum model. But I credit Sasho MacKenzie, working under Eric Sprigings at the University of Saskatchewan, with an advance that was a considerable breakthrough. They made several significant extensions to the model:

 The double-pendulum model is two-dimensional; it behaves in a single plane. Actually the usual analysis tacitly assumes a vertical plane, but it needn't. If you assume that the pivots only act about their axes, and their axes are parallel, you can assume a tilted plane (the "swing plane") and leave it to the pivots to absorb the forces outside the plane. But quite a few researchers have verified that there is activity outside a single plane which might be of interest. MacKenzie accounts for this in his model. Mackenzie increased the complexity of muscles and joints a bit. He still assumes a rotating torso and a one-armed model. But his model has that arm as the left arm explicitly, and it is connected to a left shoulder with its own motion. The picture at the right shows the elements of MackKenzie's model. It is essentially a triple pendulum, but he has also removed the restriction that the joints only operate in one plane. In addition, the arm itself may rotate about its own axis -- a fourth pivot. Finally, MacKenzie models the shaft of the club itself as having segments connected by joints with springs in them. In that way, the shaft can approximate a real shaft in that it is flexible. His shaft results are very interesting, but a discussion will have to wait for a later article; this one is about the swing model and not the shaft flex model. In this article, we concern ourselves with points #1 and #2: out-of-plane motion (if indeed there is a "swing plane"), and four axes of rotation instead of just two.

### Some Details of the Model

Here are a couple of pictures adapted from MacKenzie's paper, showing the 3D aspect of the model and the four ways motion can occur.

### Lessons from the Model

Let's remember why we want a mathematical model in the first place. We want to see what swing changes do to performance -- ideally. It is impossible to do such tests with real golfers, because:
• It is very difficult to produce an isolated change in a swing. Any change from the swing they have practiced will drag along a lot of other changes as well.
• No golfer actually makes the changes that they feel they do. It is always a little different; at the very least, it is probably smaller than the intended change. You have to exaggerate a change to make it effective at all.
So we turn to a mathematical model, where we can make such changes by plugging in a different value for a torque or a shaft weight and see what the model (usually a computer program) says about what happens.

The most telling change that MacKenzie made was to run the optimization program again. But this time, instead of making the closest match to the original measured golfer, the program was trying to achieve the greatest clubhead speed. (Of course, constraints were added so that a real, human golfer of the same size and strength would be able to make the optimized swing.)

### What we can and can't learn from the model

Among the more interesting properties of a model -- in fact, any model in any science -- are its extensibility and its limitations. What questions can it answer about the golf swing that the modeler didn't think to ask -- and what questions do we know it can't answer? Here is an example of each.

 Right arm  The most common criticism of the old double-pendulum model is that it doesn't account for the right arm. The same accusation can be leveled at MacKenzie's model; it is still a left-arm-only swing. But the accusation would be premature; the model can indeed be used to investigate several interesting scenarios involving the right arm. For instance, consider the torque M_Shoulder. The tacit assumption is that it is generated by muscles in the shoulder. But the important thing for the model is that it is the torque that motivates the separation of the left arm from the torso, the torque that changes Q_Shoulder. That torque can come from either or both of: A contraction of the muscles behind the left shoulder, shown as a blue arrow in the diagram. An extension of the right arm, shown as a hydraulic actuator in the diagram. But, in order for this to be a pure M_Shoulder play, this extension must be accomplished without contributing to M_Wrist. That can be a bit tricky to perform, because a pulling left hand and pushing right hand can constitute a torque at the grip of the club if you're not careful. This, of course, is no big surprise to MacKenzie. At one point in his paper, he says, "M_Shoulder peaked at 85Nm which is slightly greater than maximum shoulder abduction torques (~80Nm) previously reported. However, this was as expected since [the maximum allowable torque] for M_Shoulder was doubled to compensate for the [model's] lack of a trailing arm." He fully expected right arm extension to help T_Shoulder if needed. Body motion Here's one the model can't help with. The torso torque M_Torso is an indivisible quantity. The model sheds no light on how much comes from the legs, how much from the obliques, etc. It has no way to tell whether you do better letting the legs turn the hips in the backswing, or keeping the lower body still and making the entire backswing with upper body "X-factor" rotation. The only thing that matters to the model is the total amount of muscular torque that turns the top of the torso.

### Notes:

1. But note, however, that an accelerating torque does not assure acceleration. For instance, when M_Shoulder kicks in it actually retards the torso rotation. This is basic Newtonian physics; every action has an equal and opposite reaction. In order for the torso to exert torque M_Shoulder to release the left arm, the left arm exerts an equal and opposite torque back on the torso slowing its release. This is analogous to an effect often noted with the double pendulum model; the hands are slowed by the release of the club by a very similar mechanism occurring here. (For animations explaining the latter effect, see Rod White's article on the physics of the swing.) In fact, we can see the torso slowing its rotation in the last tenth of a second of the graph; while Q_Torso is still increasing, its slope (which is velocity) is decreasing. So the net effect is deceleration.
2. There is an assumption built into this assertion. The graphs are from the model optimized for maximum clubhead speed; the paper does not present the optimum match to the real golfer. I am assuming that these two curves did not change much during the process of optimizing for maximum speed. I believe this is a good assumption, but that belief is based on my guesses that
• The chosen low-handicap golfer was close enough to a good release that the release angles didn't change much during the optimization.
• Right arm rotation does not have much to do with clubhead speed -- all that motion is in a plane perpendicular to linear clubhead acceleration -- so that curve didn't change much either.
3. In the graphs for swing plane angle, I have transformed the G-coordinate angles from both papers into the sort of angle we generally associate with swing plane -- simply related to the lie angle of the club. The transformation is very simple: just subtract the papers' angle from 180º. In addition, the Coleman-Rankin data showed swings of different durations; I normalized them all into the same 0.3sec time line for comparison purposes. Finally, Sasho MacKenzie has pointed out to me that the "left arm plane" is defined differently in the two papers. Looking over the difference, it is possible that this accounts for part of the difference in the base angles -- the amount that MacKenzie's "plane" is flatter than Coleman-Rankin's.
4. For a really good example of something that is good instruction, even though the physics doesn't support what the instructor thinks it does, look at my article on accelerating through impact.