Application to the golf swing
Concepts
Dave Tutelman
 August 27, 2024
Coordinate systems
At the beginning of the book,
where the concept of vectors was first introduced, it became clear that
sometimes a proper choics of axes for the vectors would make the
problem easier to formulate or to solve. Any particular choice of axes
is called a "coordinate system". Before we talk about the coordinate
systems commonly used in golf biomechanics, let's review some general
principles for choosing a set of axes.
 An axis can be
either linear or angular. We have already seen examples of both.
For instance, spin is
twodimensional; it isn't just backspin. But there are two different
twodimensional coordinate systems we can use to describe it.
 Total spin and
axis tilt. These
two axes are, respectively, a magnitude (linear) and an angle.
 Backspin and sidespin. Both of these are linear
magnitudes.
Both are equally correct and equally "real". The choice depends on what
problem you are trying to solve. Whichever system makes the solution
easier or the explanation clearer is the right one.
 Linear axes
should be perpendicular to one another. If there are more than
two axes, they should all be mutually perpendicular. Note that I said
"should", not "must". It is possible to set up axes that are at any
nonzero angle. But it makes the math harder, and the whole point of a
coordinate system is to describe a problem in a way it can be
mathematically modeled. So why would you ever choose axes that make the
math harder?
There are three coordinate systems we
see over and over in research papers in golf science:
Let's understand each of them.
World coordinates  x,y,z
This
is the simplest coordinate system to visualize. It may not be the
system of choice for the computations, though it is for some problems.
But it is frequently what the results are converted to so the reader
can understand and appreciate them.
The diagram shows the world coordinates in the context of a golfer
practicing with algnment sticks. The exercise is very common, and the
sticks show the direction of the X and Y axes.
The axes are all linear. They are:
 Y
 the "downrange" direction. It reflects the golfer's alignment.
 Z
 Vertical
 X
 The axis perpendicular to both the other axes.
Another question about a coordinate system which is sometimes but not
always relevant is, "What is the origin?" That's the point that is
[0,0,0] for the three axes. It may vary with the problem being
addressed. For instance, the center of the ball is an easytodefine
origin. But it may be something completely different. For instance, it
could easily be the point centered between the shoulders. It depends on
the problem being studied. For instance:
 If the problem is an impact model or ball flight model, the
ball is a good origin.
 If the problem is a model of the arms swinging the club,
the midpoint between the shoulders is a good origin.
I have only seen examples of this coordinate system  or any other,
for that matter  demonstrated for righthanded golfers. I confess to
having no idea what happens to the positivenegative sense of the
vectors when you change to a lefthanded golfer..

Swing plane coordinates 
alpha,beta,gamma
The
swing plane coordinate system is seldom if ever used to mathematically
model a golf swing. But it is exceptionally useful in understanding the
workings of the swing, so many analyses done in another coordinate
system are transformed into swing plane coordinates for the discussion
phase of the report.
The three axes are all rotational, and their values would be measured
in degrees, radians, revolutions, etc. They are:
 Alpha
 the swing plane. Positive is the direction of the club during the
downswing.
 Beta
 the outofplane motion. Beta angles are perpendicular to the swing
plane, and measured above the plane. (That is, a negative angle
reflects a motion below the plane.)
 Gamma
 rotation about the axis of the shaft. This also happens to be
perpendicular to the swing plane, but is not typically thought of that
way. Positive gamma is a direction to close the clubface.
(This diagram has been
adapted from Barry Eisenzimmer's web site.)
Please do not ask detailed questions about this model yet. There are so
many that would be difficult to answer with what we know so far. We
will try to fill in any blanks as we go.

Right hand rule
As
soon as you start thinking constructively about the swing plane
coordinates, a bunch of difficult questions arise. Probably the most
difficult conceptually (not necessarily arithmetically) is the fact
that we don't have a way to add torques that are in different planes 
at least not yet. We also need to include forces along with the
torques, and we haven't covered how to resolve forces in a set of
angular coordinates. Let's resolve those issues now.
When
we draw a picture of a torque as a "circular arrow", it lies within a
plane. It is easy to visualize that way, but a circular arrow is not a vector. If we want to
treat torque as a vector, we need a different concept and a different
representation. And there is one.
A torque is applied around an axis. When the torque provides angular
acceleration, this axis is the axis of rotation. So let's represent
torque as a vector along that axis, with a magnitude that is the size
of the torque.
It turns out that this representation works really well. In fact, we
can add torques in 3D space by adding those vectors. The magnitude of
the resultant vector is the size of the net torque. The sumoftorques
vector direction is the axis that the net torque would be applied to.

The
remaining concern is the sign of the torque vector. Physicists the
world over have adopted a convention called "the right hand rule".
Here's how it works.
You "grab" the axis of rotation or torque, with your right hand. Be
sure it's the right hand; it comes out backwards if you use your left
hand. The fingers have to wrap around the axis in the direction of the
rotation or torque. If you do that, the thumb will point along the axis
in the positive direction.
BTW, it doesn't make any technical
difference whether you use your right or left hand; it is merely a convention.
As long as you are completely consistent, you will get an answer that
will serve physics well. But you may have trouble explaining your work
to someone using the right hand  so everybody agrees to use the right
hand.

Club coordinates  x,y,z
If you look back at the swing plane
coordinate system,
you will notice that the axes do not stay in the same place throughout
the swing. This is easiest to see for the gamma axis. It is the
centerline of the shaft. Of course, that moves as the shaft moves,
changing direction quickly and drastically as the club swings. The beta
axis changes as well, rotating around during the swing. If the swing
were perfectly planar, then at least the alpha axis would be fixed. In
fact, it is not a perfect plane, but it is close enough (especially, as
we will see soon, the "functional swing
plane") that we can get very useful answers from considering the
alpha axis fixed.
When
we think critically about the moving axes, we realize that they are
close to being fixed to the golf club itself, rather than fixed in
space. And indeed, an important coordinate system in many biomechanics
problems is a set of axes referred to the club itself. In this system:
 Z
 The axis of the shaft of the club.
 Y
 The direction the face is pointing, but perpendicular to the Zaxis.
More formally, the direction perpendicular to both the Zaxis and the
grooves in the face.
 X
 The perpendicular to both the Xaxis and Yaxis. It is not
parallel to the grooves, though you might expect it to be. If the club
had a perfectly upright lie angle of 90°, then it would be along the
grooves and very easy to visualize. But the typical lie angle of a golf
club at impact is 50°65°, so we need to make a choice. And the choice
is to keep the Xaxis perpendicular to both the Yaxis and the Zaxis;
any other choice would give very difficult arithmetic when we analyze
the model.
There is an important class of problems where this coordinate system is
the preferred measurement. Many golf experiments are done using
measurement instruments attached to the golf club itself. A few
examples:
It stands to reason that any data recorded from such instruments needs
to be recorded in club data coordinates.
Let's reinforce the workings of the righthand rule by applying it to
the clubreferenced coordinates. Here is a table I originally put
together for a previous
study that used club coordinates.
Forces and
torques


Local
xAxis 
Local
yAxis 
Local
zAxis 
Force

The
toeheel plane, perpendicular to the shaft axis. Positive is toward the
toe. 
The
direction the clubface is facing, perpendicular to the shaft axis.

The
shaft axis. Positive is toward the grip and away from the clubhead.

Torque

Rotation
in the direction the clubface is facing. 
Rotation
in the heeltoe plane. Positive rotates toward the heel. 
Rotation
about the shaft axis. Positive closes the clubface.


Other coordinate systems
Those
are the three main coordinate systems you will encounter in discussions
of golf biomechanics. But there are others you will see less
frequently. Here are a few; let's just gloss over them to get the
flavor of what is possible.
The
first comes from a paper we mentioned above, the attempt by Choi
and
Park to distinguish the forces of the right and left hand. (That
turns
out to be a fairly difficult problem with conventional modeling, as we will see later.)
They took the idea of clubreferenced coordinates and extended it to
each individual body part in their analysis. They have an [X,Y,Z]
coordinate system on each of:
 The lower part of the club.
 The upper part of the club; they separated upper from lower
at the grip, between the two hands.
 Each hand.
 Each forearm.
 Each upper arm.
 The torso.
This concept is not original to Choi and Park, but their paper
contained the clearest diagram I could find of it. If you are doing a
fullbody analysis, you might easily have 1520 body parts, each with
its own coordinate system. The analysis would treat each body part as a
separate free body diagram, and come up with the net forces and torques
applied to it in order to produce a golf swing.

Here's
another system, though it isn't a fullfledged coordinate system. But
it looks like one because it orients everything in the body according
to three mutually perpendicular planes. It is the system the science of
anatomy uses to talk about the direction of anything the
body is or does.
I mention it because you will occasionally run across the terms
"sagittal", "frontal" (AKA "coronal"), and "transverse" in golf
biomechanics papers. It
isn't usual, but it happens. This diagram should be all you need to
understand what the author is saying.
There are other, specialpurpose coordinate systems that you might
encounter. Now that you have seen a few, you should be able to figure
them out when you run into them.

Functional swing plane
One of the coordinate systems above is referenced
to the swing plane.
But there are numerous studies that show the swing is not really
planar; it doesn't restrict itself to a single plane. Some golfers have
a more planar swing than others, but nobody is really close enough to
model the entire swing with any accuracy using coordinates tied to some
specification of the plane of the actual swing. Except for one such
spec: the "functional swing plane".
"Functional swing
plane" is a term originated by biomechanics professor Dr YoungHoo Kwon
in a paper
coauthored with a number of his students (including Chris Como). It
refers to a portion of the late downswing and early followthrough
where the motion of the club is governed in large part by the momentum
of the club itself. The clubhorizontal position in the downswing to
the clubhorizontal position in the followthrough, shown in yellow, is
the functional swing plane. During this part of the swing, the angular
velocity of the club is such that it pulls outward very strongly.
In fact, the functional swing plane is characterized by its stability.
A more precise term is "stable equilibrium". That means that any
attempt to move the club out of plane will itself create a force or
torque to move it back into the functional swing plane. It is stable
because any disturbing force is met automatically with a restoring
force. Let's look at swing plane stability more closely.

Stability
of the functional swing plane
The
stability comes from the centripetal force (a pull) the hands need to
exert on the grip in order to keep the club from flying off in a
straight line. The reaction to this force is an outward pull that the
clubhead exerts on the shaft tip. These forces are so large in a full
golf swing that outofplane forces trying to pull the club out of the
functional swing plane can't get it very far off plane.
I examine swing
plane stability in detail in another
article,
but let's look at a quick example. Elite golfers have enough driver
clubhead speed so that the yellow forces approach 100 pounds. Let's see
how much beta torque would be
applied to the driver just 1° off plane. (Beta torque? .That is
outorplane torque, as named in the swing
plane coordinate system we discussed.) Here is how we would
compute it.

The
diagram shows the club angled away from the functional swing plane. The
beta angle is exaggerated so we can visualize it better; it is more
like 10° than 1°, but the calculations will proceed as if it is just 1°,
 1° of outofplane rotation at the butt of the club moves
the clubhead 0.8" out of the swing plane. The forces are still parallel
to the functional swing plane, so they are no longer on the same
straight line; they are parallel to one another. More precisely, they
are parallel and 0.8" apart.
 That 0.8" is a moment arm. A couple has been formed of the
two forces and the moment arm. The torque of that couple is in a
direction to restore the club back to the functional swing plane. (A
clockwise torque.)
As we said, the functional swing plane is stable,
in that moving the club offplane creates a beta torque that tries to
restore the club to the original plane.

 How big is the restoring
torque. Very simple calculation: the force times the moment arm. That is
100# * 0.8 = 80 inchpounds of torque = 6.7 footpounds.
Here's a way to visualize 80 inchpounds. Take a 2½ pound dumbbell
plate and hang it on a dowel, broomstick, or narrow PVC pipe 32 inches
from the end. (Why? 2½*32=80) Now grasp it near the end of the
stick and try to hold it
level. If the club gets one degree out of plane, that effort is the
torque you will have to apply at impact in order to keep it out of
plane. That is how much stability the functional swing plane has!
Yes, I know this is not very precise. Here are a few of the
approximations we made.
 The force is through the center of mass of the club, not
the tip of the shaft. But the functional swing plane is also defined
that way, so we don't have an error there.
 The center of mass is not as far out on the club as the
clubhead, so the moment arm from a 1° angle will not be quite as large.
That is an approximation.
 I'm playing a little fast and loose with the difference
between the butt of the club and the midhands point. But they are not
far enough apart to make a large difference for this visualization; you
can get the idea just fine.
But
even if the numbers are not spoton, the trend is obvious. The
functional swing plane is very stable; it requires a lot of effort to
move the club offplane or keep it offplane. One important thing that
means for golfers and instructors is: it is extremely important to have the club move from transition into
the downswing on the proper plane. Once the downswing is
substantially under way, it is between difficult and impossible to
"save" the swing by rerouting it.

Implications for 2dimensional models
I feel I have learned a lot about the swing from the double pendulum model. A detailed
analysis based on the double pendulum was published by Theodore Jorgensen in the
early 1990s, but it dates back at least to Alastair Cochran and John
Stobbs
in the 1960s. But any time I cite the double pendulum, someone is sure
to criticize it as, "Oh, that's no good. It's 2D. You can only get
worthwhile information from 3D models." While it is true that there is
a lot that a 3D model might teach that you can't get from a 2D model,
there is very little that the double pendulum teaches that is
overturned by a 3D model.
So
why this kneejerk prejudice against 3D models, and an immediate
willingness to reject its teachings? The most obvious answer (obvious
to me, anyway) is the common  and incorrect  assumption that the 2D
plane in question is vertical, a faceon view of the swing. It can't
be, especially if there is an extremely stable functional swing plane
that is not
vertical. Everything going on with alpha torques and their
inplane accelerations and motion is happening in the functional swing plane.
The only legitimate conclusion is that Cochran and Stobbs and Jorgensen
were talking about a pendulum in the swing plane. The valid view of
their 2D model is the green arrow in the image, not the red one.
Here are some reasons that everybody assumes the the doublependulum
swing is vertical  even though clear thinking indicates it should be
on the plane of the actual swing.
 As soon as we hear "pendulum", we tend to think of a
mechanism operated by gravity, which would make its motion vertical.
Except for a golf swing, every pendulum I have ever been exposed to has
been vertical, and I am sure my experience in this regard is more
diverse than average. So it is natural to visualize it as vertical.
 The most common pictures and videos of the golf swing is
frontal, faceon. There are other we see (e.g. down the line), but I
have never seen a YouTube video (or any other, for that matter) taken
perpendicular to the swing plane. So we are not used to visualizing the
swing that way.
 Teachers, coaches, mathematicians, and others working on
the doublependulum model do not help matters. I seldom see gravity
done right in such a model. It is usually included as a full
32ft/sec/sec. But that is only correct if the swing is vertical. If the
plane of the swing is slanted, The gravitational force has to be
multiplied by the cosine of the plane angle. For instance, the swing
plane is about 30° from vertical, so the inplane force of gravity is
only 87% of the full value of gravity.

The next question to ask about
2D models is what we lose by not
making it 3D.
The first and most obvious thing we lose is any motion, force, or
torque perpendicular to the swing plane. Those are completely lost;
there is no way to even represent them in a 2D model, much less
calculate or assess them. The most obvious such consideration is
squaring of the clubface. That occurs at a substantial angle to the
swing plane, and most of what causes it (rotation around the shaft
axis) is completely normal to the swing plane. So we really have no way
of talking about a square clubface in a 2D model that lives in the
swing plane.
But there are things that critics throw out that apply to the double
pendulum model, but not to 2D models in general. Yes, the double
pendulum is the first and simplest 2D model, but hardly the only one.
For instance, I have heard the assertion that "parametric acceleration"
can only be studied in 3D. Parametric acceleration is the angular
acceleration of the golf club by pulling up and in on the handle during
the last part of the downswing. It was introduced in a 2001 technical paper by K. Miura, and talked about in Book 1
without calling it parametric acceleration. (Let me urge you to review the latter reference. It will
be essential to understanding how the golf swing is really powered.)
But how can that be? Up and in
is 2dimensional. It's up and
it's in.
Well, remember that "up" and "in" are concepts  even primary axes 
in the rectangular "world" coordinate system.
But we are not talking about that system! Remember, that is the fallacy
of thinking the 2D model lives only in a vertical plane  so it can
handle "up" but it doesn't know anything about "in". And we debunked
that notion already; our 2D model is in the tilted swing plane, not a
vertical plane. It works wherever that plane is for the swing we are
looking at. So  appropriately enough  let's look at the swing plane coordinate system.
In that system, we can talk about "up and in" and still stay in the
swing plane. If the "up" and the "in" stay in the same ratio as the
tilt of the swing plane, then "up and in" fits this 2D
model very well.
In fact, any force, torque, or motion that is wholly within (or even
parallel to) the swing plane fits the 2D swing plane model. It is only
things that happen outside those two dimensions that would require a 3D
model to handle them.

An up and in pull may indeed be
a force in the swing plane, but how could we represent it in a 2D
model? To understand, let's first look at the double pendulum as a 2D
model. Before we go any further, let me remind you that this is not a
frontal view of the golf swing; it is a view perpendicular to the swing plane.
Here
is a simplified diagram of the model. The inner arm of the pendulum
(representing the lead arm of the golfer) is in yellow, and the outer
arm of the pendulum (representing the golf club) is in gray. The two
crossed black lines are pivots. The pivots can be driven by active
motor torques: red for the shoulder torque and green for the hand
couple. Otherwise they are frictionless. The brown shaded area
represents the fixed framework the inner arm is attached to.
The diagram is simplified in that only the active, prescribed torques
are shown. Are there other torques at work, and maybe forces as well.
Not just yes, but hell yes.
If we were doing a proper analysis, we would need to account for:
 Gravity, whose force affects both the inner and outer arms.
 The hands are dragging the club along, as well as exerting
a centripetal force preventing the club from just flying off on its
own. So, in addition to the green torque, there would be a green force
at the wrist hinge. It is not shown here because it is a necessary,
computable consequence of the torques we do show.
 That hand force is not necessarily along the shaft axis. If
it does not act through the center of mass of the club, then there is a
torque (the moment of the force)
acting on the coub in addition to the hand couple.
 The golfer's body is accelerating the mass of the arms and
club forward (to the right). In order to keep the shoulder pivot in
place, it must be able to exert a force to the right, to make that
acceleration happen. Again, we can compute the magnitude and direction
of the force the shoulder pivot exerts.

Now that we understand what the
double pendulum looks like when modeled, let's generalize the model so
it can analyze an upandin swing. For that swing the shoulder joint
has to move up and in, in order to pull up and in on the club at the
wrist joint; the power comes from the body (and ultimately the legs);
the hands merely transmit the force to the club.
Here are two different ways of modeling the up and in move.
The
first adds an up and in force at the shoulder pivot. We no longer have
a fixed pivot for the shoulder; we have a force, and that force has to
account for the proper movement
of the shoulder pivot itself. Remember all those forces we had to
calculate to analyze the double pendulum? In particular, remember that
there were forces at the shoulder pivot just to keep the pivot pinned
in place? Without the given of the pin, you need to apply those forces
to the pivot first, then add the up
and in force to move the pivot the way you want it to move  up
and in. All this works, but it is a lot of computation.
The second way to model up and in is more like the double pendulum we
started with. We assume a path the shoulder pivot takes (the brown
heavy line in the diagram) and a "schedule" of how it moves along the
path during the downswing. From that, we can compute the
instanttoinstant magnitude and direction of the force needed to move
it along the path.
If we apply the force we computed from the second method to the first
model, we should get theshoulder pivot moving along the same path as
the brown line. (That is not a constant force at all; it varies during
the downswing. That is also true for the simple double pendulum.)

A 2D model can be very instructive, and isn't incorrect at all. It can
tell us an awful lot about how the club moves in the swing plane.
It cannot tell us about motion outside the swing plane, like clubface
squaring or the effect of shallowing in transition (a change of plane).
But it certainly has its place; it is far from useless.
Last
modified  Sept 28, 2024
