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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 two-dimensional; it isn't just backspin. But there are two different two-dimensional 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:

A "world coordinate system".
Coordinates taken with respect to the swing plane.
Coordinates taken with respect to the golf club.

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 easy-to-define 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 right-handed golfers. I confess to having no idea what happens to the positive-negative sense of the vectors when you change to a left-handed 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 out-of-plane 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 sum-of-torques 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 Z-axis. More formally, the direction perpendicular to both the Z-axis and the grooves in the face.
X - The perpendicular to both the X-axis and Y-axis. 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 X-axis perpendicular to both the Y-axis and the Z-axis; 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:

The TrueTemper ShaftLab, with strain gauges embedded in the shaft.
The SkyPro golf swing analyzer, with accelerometer and gyro clamped below the grip.
Various experiments with an instrumented grip, to determine grip pressure or distinguish force applied by each hand.

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 right-hand rule by applying it to the club-referenced coordinates. Here is a table I originally put together for a previous study that used club coordinates.

Forces and torques
	Local x-Axis	Local y-Axis	Local z-Axis
Force	The toe-heel 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 heel-toe 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 club-referenced 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 full-body analysis, you might easily have 15-20 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 full-fledged 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, special-purpose 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 Young-Hoo Kwon in a paper co-authored with a number of his students (including Chris Como). It refers to a portion of the late downswing and early follow-through where the motion of the club is governed in large part by the momentum of the club itself. The club-horizontal position in the downswing to the club-horizontal position in the follow-through, 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 out-of-plane 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 out-or-plane 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 out-of-plane 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 off-plane 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 inch-pounds of torque = 6.7 foot-pounds.

Here's a way to visualize 80 inch-pounds. 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 mid-hands 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 spot-on, the trend is obvious. The functional swing plane is very stable; it requires a lot of effort to move the club off-plane or keep it off-plane. 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 re-routing it.

Implications for 2-dimensional 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 knee-jerk 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 face-on 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 in-plane 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 double-pendulum 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, face-on. 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 double-pendulum 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 in-plane 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 2-dimensional. 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 up-and-in 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 instant-to-instant 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