Modeling the Swing - MacKenzie
Three Dimensions, Three Pendulum Levers
-- 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
University of Saskatchewan, with an advance that was a considerable
breakthrough. They made several significant extensions
to the 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.
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.
increased the complexity of muscles and
joints a bit. He still assumes a rotating torso and a one-armed model.
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.
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.
- The torso rotates about an axis 30º from vertical.
- The shoulder "hinge pin" is 50º from vertical, and stays in that orientation throughout the downswing. That's 20º
more towards an upright swing plane than the torso rotation.
- In addition -- though #1 and #2 already constitute 3D --
the left arm is itself an axis of rotation
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:
To denote the muscular torque
at the joint, the Q_ is
changed to M_ (e.g., M_Shoulder).
- Q_Torso is the
angle of body turn.
- Q_Shoulder is the
angle between the torso and the left arm.
- Q_Arm is the angle
of rotation of the left arm about its own axis.
- Q_Wrist is the
angle between the left forearm and the club shaft at the grip.
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.
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.
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:
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.
- 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.
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:
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.
- The torso turns first.
the torso has finished its rotation, the arm separates from the chest
(torso) and starts to angle outward. That is the shoulder angle
- It is not until quite late in the downswing that the wrist
cock angle begins to release.
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. 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.
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
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. 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.
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 25º, the way we usually think of the plane -- that is the angle
from horizontal. It gets even flatter (under 20º) early in the
downswing, then steepens considerably. By the time of impact, it is up
to 50º, 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".
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. 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?
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 30º to 50º.
golfer in the Coleman-Rankin study was at least 20º steeper at impact
than he was at transition (the beginning of the downswing). The
highest-handicapper in the study showed a 50º variation of left arm
plane during the downswing, and even the scratch golfers had a 30º
variation. The MacKenzie model shows just over a 30º variation,
certainly consistent with Coleman-Rankin.
left arm angle at impact was much steeper for Coleman & Rankin
(70º-80º) than MacKenzie (50º). A few degrees (certainly less than 10º,
and probably more like 5º)
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.
- 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.
|Coleman & Rankin
also ascertained that the club shaft
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
issue; one of the scratch players started about 20º above and the other
the same amount below. And these were not the extremes either; those
ranged from 30º above to 23º below the left arm plane. (Independent of
left arm plane for that golfer, there was about 30º of variation
It is worth noting that the Coleman & Rankin data shows a much
tighter grouping at impact.
The shaft plane at impact was centered around 15º, with only the
15-handicapper more than 3º
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 12º and 18º.
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
15º 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
(wrist cock or technically, ulnar deviation) of about 10º 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
We have a sort of negative lesson here. We learned that the swing is
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?
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
the more interesting properties of a model -- in fact, any model in any
science -- are its extensibility
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.
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:
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
- 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.
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.
- 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.
- 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 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.
- The chosen low-handicap golfer was close enough to a good
release that the release angles didn't change much during the
- 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.
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.
modified -- March 1, 2012