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:

  1. 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.
  2. 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.
  3. 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.

Three Dimensionality
  1. The torso rotates about an axis 30 from vertical.
  2. The shoulder "hinge pin" is 50 from vertical, and stays in that orientation throughout the downswing. That's 20 more towards an upright swing plane than the torso rotation.
  3. In addition -- though #1 and #2 already constitute 3D -- the left arm is itself an axis of rotation

Four Rotational Axes

The basic motions in the model are all rotations. (Linear motions, like clubhead speed, can be calculated. But the model itself is one of rotations.) Here are the four axes about which rotation occurs:
  1. Q_Torso is the angle of body turn.
  2. Q_Shoulder is the angle between the torso and the left arm.
  3. Q_Arm is the angle of rotation of the left arm about its own axis.
  4. Q_Wrist is the angle between the left forearm and the club shaft at the grip.
To denote the muscular torque at the joint, the Q_ is changed to M_ (e.g., M_Shoulder).

The muscular torques are not the only torques acting in the model (though, appropriately, they are the torques presented in the graphs). There are "stops", similar to the wrist cock stop in the double-pendulum model. For instance:
  • The left arm lies across the chest for much of the early downswing. But two solid bodies cannot occupy the same space at the same time. This constraint means that the chest may exert a force on the arm that assists M_Shoulder, but isn't reported as part of muscular torque M_Shoulder.
  • There is a ligament that limits wrist motion at about zero degrees of wrist cock. This explains some of the shaft angle data reported here and in other papers.
Muscle Limitation

Naturally, the muscle group controlling each joint has a maximum torque it can deliver. As a simple-minded engineer, I view the joint as having a torque generator that can deliver to the joint the torque that the modeler wants, instant by instant, up to the maximum torque for that joint.

Biomechanics guys have a different perception of torque applied to a joint, and MacKenzie's model incorporates the biomechanics view. Realistically, there are limitations to how fast a muscle torque can ramp up (or, for that matter, ramp down or shut off). As shown in the diagram to the right, this is modeled as an exponential buildup and decay. An electrical engineer like me recognizes this as an RC circuit; a mechanical engineeer recognizes it as a dashpot.

The "switch on" and "switch off" time constants can be different. In MacKenzie's model, the time constant is 20msec for activation and 40msec for deactivation.
On top of that, a moving joint cannot exert as much torque as a stationary joint. Think of this analogy. You are pushing an automobile along the street. As long as the car is not moving, you can apply the maximum force your body is capable of. Once the car starts moving, it is harder to apply a force to it, since it is moving away from you. You have to maintain your velocity as well as apply a force. In fact, at some point the car may move faster than you can keep up. At that point, your ability to exert a force is down to zero. Alternatively, consider pedaling a bicycle going really fast downhill; you can't apply force to the pedals because your feet can't keep up.

The same principle applies to joints. The force becomes torque, and the velocity is angular velocity. MacKenzie uses a simple relationshop (see the curve at left) to describe the torque at the joints in his model. The wrist and elbow joints max out (lose all torque) at 60 radians per second, and the torso and shoulder at 30 rad/sec. But at even half the maximum angular velocity, the torque you can use is down to about 20% of its maximum value.

Here is an interesting consequence of this torque-velocity curve. At impact, with the clubhead moving at 100mph, the angular velocity of the wrist joint is in the vicinity of 40 rad/sec. Even if you wanted to apply wrist torque to the club, you would be limited to only 11% of the torque your wrists could have applied statically.

The model was validated by taking high-speed video (500fps) of a very consistent low-handicap golfer, and comparing model output to the golfer. The video captured the torso angle, the left arm angle, and the club angle (corresponding to Q_Torso, Q_Shoulder, and Q_Wrist) at 2msec intervals. The captured data was digitized for use in the computer. Then the model torques were adjusted by an optimization program, to make the fit to the human-golfer data as closely as possible -- which turned out to be very close indeed.

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.

Maximum Clubhead Speed

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.)

Here is a graph adapted from the paper, showing the muscular torques that were exerted during the optimized swing.

We can estimate the angles (the Q_ values) just looking at the diagrams above the graph. The angles release from the inside out. That is:
  • The torso turns first.
  • Before the torso has finished its rotation, the arm separates from the chest (torso) and starts to angle outward. That is the shoulder angle increasing.
  • It is not until quite late in the downswing that the wrist cock angle begins to release.
So, for anyone who isn't convinced by the double pendulum model that clubhead lag is a major contributor to clubhead speed, here's another, more detailed, model that says so.

As the torso and shoulder angles release, they are driven by the corresponding torques. But the same is not true of the wrist angle!There is only a small blip of wrist torque as the club begins to release, and it drops down again immediately. MacKenzie assures me that this does have a material effect on the clubhead speed. But, judging from the work done by the blip, it cannot be a large effect. Like the double pendulum model, the MacKenzie model gets maximum clubhead speed from centrifugal release, not from hitting with the hands.

Another point to note: the large-muscle torques, the shoulder and especially the torso, continue to provide accelerating torque right through impact. In fact, the torso torque is increasing until immediately before impact.[1] That is a lesson we could also learn from the double-pendulum model (see my article on accelerating through impact); but the fact that a speed-optimized model recommends it definitely reinforces the lesson.

In short, the four-rotation model refines where the power comes from to create clubhead speed, but it doesn't undo anything we learned from the double-pendulum model about achieving distance. M_Torso and M_Shoulder combined represent the shoulder torque of the double pendulum, along with the lateral movement of the shoulder pivot. Before, we lumped them together as part of the body's contribution. Now we can separate out the contribution of the left shoulder extending the arm away from the body. But "The Paradox" -- the most counterintuitive lesson from the double-pendulum model -- remains intact.

Clubface Squaring

Here is another graph with another lesson. It shows the actual angles over the course of the downswing. I have highlighted the wrist cock angle Q_Wrist (in yellow) and the arm rotation angle Q_Arm (in blue). The wrist cock angle, we should certainly know, is the angle between the club shaft and the left forearm at any time during the downswing. The arm rotation angle is closely related to how much the face of the club is open or closed, especially in the vicinity of impact.

The important thing to note here is that the shape of the two curves is very similar, almost identical. Therefore, you can assume that shaft rotation and clubface closure angles track the club's release angle during the downswing, for purposes of back-of-the-envelope estimates at the very least.[2] I have used this in the past, to estimate things like how much shaft torque it takes to square the clubface given the moment of inertia of a driver head. It is useful because there are tools to calculate or estimate the club's angular velocity during release. So we can also use those methods to estimate clubface closure rates during release.

MacKenzie's results say that this gives a very reasonable estimate. To the extent that the curves don't match exactly, the estimate will not be exactly correct. But it will give a  good enough approximation for most purposes.
Swing Plane

The model of the swing shows that there is no constant "swing plane", whether you are talking about the left arm's plane or the club shaft's plane. MacKenzie plots the left arm plane, and it starts very flat: about 25, the way we usually think of the plane -- that is the angle from horizontal. It gets even flatter (under 20) early in the downswing, then steepens considerably. By the time of impact, it is up to 50, a more reasonable number for a driver. (But still not as steep as we would expect from a driver's lie angle.)

MacKenzie references a paper by Coleman & Rankin, in which they photographically measured the swing plane of a variety of golfers. (Seven right-handed male golfers with handicaps ranging from zero to fifteen.) Let's see what lessons are to be learned from both papers about the "swing plane".

The picture at the right shows the plane of the lead (left) arm over the time of the downswing, with the transition on the left and impact on the right.[3] I have combined all seven curves from the Coleman-Rankin study into an "envelope" that contains all the data points.

How consistent is Coleman-Rankin's data with MacKenzie's model?
  • Every golfer in the Coleman-Rankin study was at least 20 steeper at impact than he was at transition (the beginning of the downswing). The highest-handicapper in the study showed a 50 variation of left arm plane during the downswing, and even the scratch golfers had a 30 variation. The MacKenzie model shows just over a 30 variation, certainly consistent with Coleman-Rankin.
  • The left arm angle at impact was much steeper for Coleman & Rankin (70-80) than MacKenzie (50). A few degrees (certainly less than 10, and probably more like 5) might be attributed to the fact that Coleman-Rankin's golfers were swinging a 5-iron, and MacKenzie's a driver. But the differences are too great for the choice of club to be the only difference.[3]
  • Most of the golfers also showed flattening early in the downswing increasing and even peaking, like MacKenzie's model. But this last point does not seem to be a quality-of-golf issue; Coleman & Rankin's data includes two scratch golfers, one with the further flattening and one without. So early flattening does not seem to be something that needs to be either taught or discouraged -- unless... If MacKenzie's arm plane changed significantly in going from the best golfer match to the best clubhead speed, then maybe there is something here worth looking at. There is not enough information presented in the paper to tell.
Bottom line: Even though numbers and details disagree, both papers show that the left arm does not stay in a plane during the downswing. The left arm plane's variation during the downswing ranged from 30 to 50.

Coleman & Rankin also ascertained that the club shaft plane is not the same as the left arm plane, and often not even close. Some were above the left arm plane and some below. Again, it's not a quality-of-golf issue; one of the scratch players started about 20 above and the other almost the same amount below. And these were not the extremes either; those ranged from 30 above to 23 below the left arm plane. (Independent of the left arm plane for that golfer, there was about 30 of variation there, too.)

It is worth noting that the Coleman & Rankin data shows a much tighter grouping at impact. The shaft plane at impact was centered around 15, with only the 15-handicapper more than 3 away from this number. (And both scratch golfers were spot on it.) But even this final angle is not zero, as you might expect. Centrifugal force is trying to straighten out the line from shoulder to clubhead, via the tension it puts on the shaft. But all this tension seems to produce not a straight line, but an upward angle between 12 and 18.

I verified this by looking at a sampling of about a dozen down-the-line shots of good golfers at impact. (Most of them are celebrated pros; the golf magazines love to do full sequence shots of them.) In every case, the plane of the club was above the plane of the left arm in the impact photo. The right forearm was frequently on the shaft plane, but the left arm was above it. I did not measure the angular difference, but 15 is not a bad estimate by eye.

Could it be that the minimum wrist cock (at least the minimum without straining) is in this vicinity? That would explain the Coleman & Rankin results. When we look at MacKenzie's results, that shows a residual Q_Wrist (wrist cock or technically, ulnar deviation) of about 10 at impact. While not the exact same number, it is in the right direction and the right ball park.

What can we take away from all this? We have a sort of negative lesson here. We learned that the swing is not on a single plane. Moreover, every golfer departs from plane in a different way, and almost all of those deviations are pretty significant, not trivial. Finally, the best golfers in the sample differed in opposite directions. My conclusion is that we have identified another case of something that may be good instruction even if it isn't essential to the swing. How can that be?[4]

The data says fairly clearly, there is no one swing plane (nor one pattern of planes, since the swing plane varies during the downswing) that is superior to the others. On the other hand, working on a "classic" swing plane (a la Hogan) may be the surest way to a good clubhead path at impact, not the beginner's usual outside-in path. But obviously one or both of the scratch players in the sample were able to get a good clubhead path with very different approaches to the swing plane. So it isn't essential to a good clubhead path, but it may be the best way to teach a good clubhead path.

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.


  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 180. 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.

Last modified -- March 1, 2012