Modeling the Swing - The Future

Dave Tutelman  --  January 16, 2012

So where do we go from here?

That is a reasonable question to ask. Modeling the golf swing is not likely to stop after 2011 (unless the Mayans were right). Where do I see this science going in the next few years, and why? I have spent part of my engineering career as a technology forecaster, and will try to apply those methods to the question, What's next for mathematical models of the golf swing.

Demand Pull or Supply Push

In my article on technology forecasting, I started by insisting that we must identify the force driving the evolution we are investigating. It can be supply-driven or demand-driven. Let me repeat something I said in that article:

Think about the two adages:
  • Build a better mousetrap and the world will beat a path to your door.
  • Necessity is the mother of invention.
These are diametrically opposed views of what drives the evolution of technology. The views might be expressed as technology-driven vs market driven. Another way of expressing the dichotomy is supply-driven vs demand-driven.
  • The first says that evolution is technology driven. If you come up with a truly better technology, then people will buy it.
  • The second says that evolution is market-driven. If there is a big enough need, the technology will (or  must) be invented to fill the need.

There are success stories for both views; they are not mutually exclusive. Let's try to identify the supply and demand considerations driving the modeling of the golf swing.

Demand (Let's hit this first, because it is where the need is. If the modeling is purely supply-driven, then it probably is neither useful nor interesting except to the modelers themselves.)
  • Golf instructors want to know what science says they should be teaching. (In my experience, most golf instructors don't want to know this, but they should. Unfortunately, it can take over a decade for established scientific fact to become accepted -- along with its implications -- by the instruction community. This is especially true if science contradicts the traditional myths of golf. Sorry, PGA, but that's what I've seen. OTOH, there are certainly a significant minority of instructors who really do want to understand the science behind what they teach. The best of them will ask a lot of challenging questions, and they will accept answers that can be backed up.)
  • Golfers have the same needs -- and often the same questions -- as the instructors. This is true at all levels, from the technically-minded beginner to the tour pro looking for an edge.
  • Golf club manufacturers want to know the science behind the swing, so they can design and build better-performing clubs. (Again, the community contains both the scientifically curious and the Luddites and myth-perpetuators. Club companies are sales-driven. Some actually improve performance and then convince the customers of the fact. Unfortunately, many of them find it easier to sell their wares by pandering to the myths the customers already believe.)
  • Technology is improving all the time, including the technology of mathematically modeling the golf swing. Computers are bigger, faster, and cheaper than last year, and will be more so this year -- so computing-intensive models that are infeasible now might well be routine with the faster computers of the future. And the instrumentation to do the kinematic models has been proliferating for the past few years. The precursors are in a lot of hands now. It should only be a matter of time before it produces some new, interesting models.
  • The skilled labor supply is there. Our institutions of higher learning have a bumper crop of technically savvy graduate students, many looking for thesis topics. In departments of kinesiology, biomechanics, and even engineering, mathematical models of the golf swing have been a ticket to a cap and gown. I see this trend continuing, and even growing.
We will look at the trends in more detail starting with supply, so we know what is possible. Then we will examine demand, in the form of interesting questions that a model might help to answer.

Supply Push

Most of our supply side discussion deals with technology. But not all. We mentioned "skilled labor" as a supply factor. I'd like to say a few things relating the skill level to the sort of model you might expect as a result.


Since it was postulated around 1970, Moore's Law has described the cost of computing remarkably accurately. Roughly paraphrased, it says that computer bang for the buck doubles every two years. This has tracked reality for four decades now, and through a trillion-fold increase in computing power per dollar. That is not just a difference in degree; it is a difference in kind.

What do I mean by that? In 1970, it was important for the programmer to be clever. Organize the program to compute only what really needs to be computed. Go to considerable effort to write bug-free programs, because you didn't have a computer right at hand; you submitted your job to a big comp center machine (well, big by the standards of the day), and didn't see the output for half a working day. Today, by contrast, there is so much storage and compute power right on your desktop (and even in your pocket) that it's OK to throw computing at a problem without thinking much about it. Compute everything first and select the important results after you see the output. Unthinkable in 1970!

What does this mean to golf swing modeling? Consider a statement in MacKenzie's paper: "Due to the length of the closed form solution of the dynamical equations representing the model, the number of shaft segments was limited to four. The inclusion of additional shaft segments resulted in the computer program becoming too large to compile into an executable file." This suggests that, even in 2009, there were problems too large to be conveniently handled by the existing computer solutions (both the computers themselves and the program packages that support modeling). As the modelers get access to increased computing power, the complexity of the models can increase. More segments to a flexible shaft. More elements in a model. More iterative convergence algorithms applied to larger multivariate models.

This has special impact on inverse dynamics kinetic models, because they require convergence algorithms (to match the kinetics to the kinematics) for a large number of variables (over 40 time-varying functions had to be synthesized in Nesbit's work). The march of ever-cheaper computing will encourage studies with:
  • More complex models, meaning more variables in the model.
  • Models of more golfers (because it takes less time and money to model each golfer).
  • Computer comparison of golfers. The eyeball is still the way most lessons are learned from a swing model. Graphs and the researchers' insight picks up the important differences between golfers. But cheap computing and storage has made possible "data mining" -- programs looking for interesting trends in complex data. I foresee data mining programs making suggestions to the researchers on what to examine in the model output.
  • More portable data capture. Computers the size of cell phones (perhaps the computers in cell phones), and even computers in golf clubs, will allow data capture to leave the laboratory and go out on the golf course. I don't know where that will lead, but the possibilties are exciting.
I'm confident there are other things that swing modelers will figure out to do with all their computing power, but I'm pretty sure the above list will happen.

Motion Capture Systems

Every model needs to be validated by measurements on real golfers. This is especially true with kinematic studies, which are all about capturing and correlating the motions of real golfers. (And, by extension, inverse dynamics kinetic studies, whose first step is a kinematic model.) Fortunately, the technology is moving rapidly in motion capture systems.

Most such systems are camera-based. That is not surprising, given how technology has put digital photography on a trend curve that rivals computing. (I have been serious about photography since 1950. I had a Rolleiflex as a teenager, and several good 35mm SLRs since. None of them can match the performance nor features of my current camera, a Canon for which I paid just over $100 two years ago. The successor to that camera has a few features I covet, and it sells for under $100.) So the ability to do 3D motion capture with multiple high-speed digital cameras is definitely "there", and is getting more capable and cheaper as you read this.

A newer technology with fewer entrants so far is motion capture by sensors. (Think accelerometers, gyroscopes, strain gauges, etc.) The one that pops immediately to mind is the K-Vest. The existing product is being used only for teaching, and does not instrument nearly enough for research and modeling. But it is a credible forerunner for implementations that could support modeling.

So far, these motion capture systems -- photographic and sensor based -- are being used mostly for instruction. At least one is being used for shaft fitting (the Fujikura/Vicon Enso). Not many are there for research, much less for the specialized research of swing modeling.

Let me expand a bit on the "creative tension" between teaching and swing modeling. Instructors teach what they believe to be true. In most cases, they believe because of what they have been taught (tradition) and their own personal experience (observation). Fairly little of their "knowledge" comes from understanding swing models. That means that what they are teaching might be right -- or not -- but it is not based on science.

But teaching is a high-numbers game. There are thousands of golf teachers[1] and millions of golf students (or potential students). If my business were technology applicable to sports, it is worth competing in a market that sells to golf instructors. The market for researching swing models is smaller by several orders of magnitude; I would be much less motivated to invent a product for that market. Fortunately, the needs are not all that different. Much of the hardware is basically the same for swing instruction and swing modeling. Clearly, some of the software is different. But our technological society has a history of graduate students writing sophisticated software for which the market was not clearly large. (Those familiar with the history of the Internet know exactly what I'm talking about.) And graduate schools are the center of activity in swing modeling. So we have the technology, or we almost certainly will soon.

Insight, imagination, and forward dynamics

This is a supply problem! Forward dynamics kinetic models, especially simple ones, are a great way to learn what matters. Granted, more parameters in the model allow a more faithful representation of the swing. But they also tend to obscure the essential points of what is happening, what we want to accomplish. It takes insight to figure out what matters when looking at all the data from a full-body model -- or any model with a lot of parameters. And it takes both insight and imagination to devise a model with a small number of parameters, that is still complex enough to deal accurately with interesting questions about the golf swing. Once again, let me invoke Einstein: "Everything should be as simple as possible, but not simpler."

The real problem is that it takes insight and imagination to do useful modeling. The army of grad students getting into the picture have the training and the skills, but only the rare individual has the insight and imagination to come up with a simple model that works. That is why we still only have two relatively simple models: the double pendulum and MacKenzie's model.

What path will swing modeling follow for the next few years? Absent someone with great talent or a flash of imagination and insight, we can still make progress. It will be the progress of looking for patterns in a sea of multivariate data from too-complex models. Given the advances in technology, we can still learn new and interesing things from the models. But there may be a way to get a kinetic model with more detail than MacKenzie's but without the multivariate clutter of a 40-plus-parameter full-body model. Some approaches that may bear fruit include:
  • We learned from Nesbit & Serrano's work that the big providers of power are the hips, lumbar and thoracic regions, and the right elbow. The most advanced "simple" model we have currently is MacKenzie's. That model accounts for the right elbow as left shoulder torque, and lumps the other major providers into torso torque. The addition of three to six parameters (roughly double the complexity) might handle all these known-to-be-important factors. That is still far less than a full-body model, and manageable for what-if questions.
  • Examination of the full-body model may show that the behavior of some joints are not independent; that changes in one parameter are always accompanied by a predictable corresponding change in another parameter -- or perhaps more than one other parameter. Each such strong correlation is an opportunity to combine two or more parameters into one. In that way, we reduce the dimensionality of the model.[2] Do that enough, and even a full-body model may become a manageable what-if tool.
  • So far, my assumption about what-if questions assumed that they are questions that involve tweaking parameters of a forward-dynamics kinetic model. (E.g.- change the time profile of wrist torque.) But suppose the what-if question is a kinematic question? (E.g.- what is the effect of more hip slide -- or more hip rotation -- early in the downswing?) If that can be expressed by tweaking the kinematic measurements, then a full-body inverse dynamics model can be exactly the way to answer the question. But be careful; expressing those changes may be harder than you think. Changing the way the hips behave is very likely to affect other kinematics of the swing: the knees, the shoulders, etc. So it may be very difficult to set up a kinematic what-if situation that reflects what a real human can do.

Demand Pull

Here are some kinds of questions that could be answered by the proper model, if it were available -- and rather difficult to answer any other way.

Validate Existing Swing "Folklore"

There is a lot of existing "folklore" about the golf swing. Undoubtedly much of it is true. Almost equally undoubtedly, some of it is not. And then there is contradictory folklore, assertions that have both pro and con adherents. It would be very nice if we had golf swing models that could address such questions.

As an example, take the matter of centrifugal release vs wrist torque. The folklore was to hit with the hands. No less an expert than Ben Hogan said so. But the double pendulum model says otherwise, emphatically. And each successive model has confirmed it. So not all of the common wisdom about the golf swing is true wisdom.

Which brings us to the sort of questions that we would want physics to answer. Which means, of course, we want a mathematical model that answers them. For instance:
  • It has become generally accepted that the "X-factor" is associated with clubhead speed.[3] The X-factor is the establishment of a big angle between the hips and the shoulders, and holding that angle through the downswing. It would be nice to have a model where you can play with that angle, and quantify the effect.[4] Such research would either go a long way towards telling us why the X-factor provides more distance -- or perhaps let us know that the X-factor was not the primary factor at work in distinguishing the long-hitting from short-hitting golfers.
  • The notion of an actual swing plane has been demolished by data from Coleman and Rankin, and the modeling work of Mackenzie and Sprigings. But the MacKenzie model could be used to study questions like, "What does a flatter/more upright 'plane' actually do?" and, "What would be the result of actually keeping the club on a plane?" Such questions are actually being addressed as this is written.[4]
This barely scratches the surface of interesting questions. You can make a game of it while you watch golf on TV. Any time Johnny Miller or Peter Kostis or one of the other swing commentators analyzes a swing, just ask yourself, "Is what he is saying actually true?" It's shocking how often the answer is sufficiently non-obvious (or even blatantly wrong) that a look at the physics is appropriate.


Almost every modeling study to date has focused on clubhead speed at impact. There are a few reasons for this:
  • Distance sells! It is dramatic, and most people who play golf want to hit the ball farther.
  • It is measurable. It's temptingly easy to look at the output of a model as a single number. And clubhead speed is a very easily interpreted number.
But accuracy is just as important in scoring well, and it's hard to find a single physical model (either mathematical or mechanical) that addresses accuracy. The only one that comes to mind involves putting; Dave Pelz and the "Perfy" robot have given rise to a number of mechanical putting simulators and putter testers that might be considered mechanical models of the putting stroke. But there are lots of questions of accuracy that could and should be addressed by a swing model. A few examples:
  • What are the muscle moves that lead to impact with a closed face, an open face, and a square face? That probably means closed or open compared with the position of the clubface at address, since the absolute position can be dealt with by strengthening or weakening the grip.[4]
  • What are the muscle moves that lead to an out-to-in clubhead path, and in-to-out path, and a down-the-line path? With the loss of the swing plane myth, we need something to replace the concept in a practical, teachable way.
  • Consider things we do to increase distance, such as increasing wrist cock, holding the lag, and increasing the X-factor[3]. What does each of these do to directional accuracy?
In addition to control of direction, "accuracy" can be taken to mean repeatability of result. Repeatability is closely related to the notion of a solid impact. Both are definitely desirable. Is there any way to quantify that in the output of a model, with that sort of criterion for accuracy? I can think of a few.
  1. The most direct way to determine repeatability is to run the model a statistically significant number of times, with random variation of the parameters around their design value. That is called by simulation people a "Monte Carlo study".
  2. Less direct, but undoubtedly more efficient and probably almost as effective, is to look at how fast the outputs (clubhead direction, clubface position, clubhead speed) are changing in the vicinity of impact. The first derivatives of these quantities are good indicators of how repeatable they are; the larger the absolute value of the derivative, the harder it is to replicate on the golf course.
  3. The problem with looking at output derivatives is that things producing a high clubhead speed are likely to produce high derivatives of most measures in the vicinity of impact. (For instance, look at the total power curves compared to total work in the Nesbit-Serrano paper. Or look at the various graphs that can be output from SwingPerfect.) That implies the not-necessarily-correct conclusion that clubhead speed and accuracy work against one another. So let us look at the sensitivity of the outputs to changes in input. This can be quantified as the partial derivatives of the output at impact, with respect to the various inputs.
Two of these approaches (#1 and #3) require a model that can be run repeatedly with slightly changed inputs, to perform numerical differentiation or to get a statistically significant sample. Any of the models we have looked at fills that bill, because we are not running iterative convergence on them. Even a model with 40-plus parameters should be able to run pretty quickly on today's desktop computers. The real problem with more parameters means there are more partial derivatives to consider, or more inputs to Monte Carlo.

The approaches where we vary the inputs probably should have the input variations scaled, according to which muscles are involved and how much control we have in repeating that force or torque. That requires a lot of input from biomechanics experts. That's not a bad thing; I'm just pointing out that it is more than a physics or engineering problem.

Popular Science

No, I don't mean the magazine. Let's reword that as "participatory science". An important characteristic of science is the reproducibility of results, and the invitation to others to reproduce the results. This calls for programs that simulate the models, so other researchers can verify the lessons for themselves, then go on to learn more lessons from the model.

I can cite an admirable example for the double pendulum: Max Dupilka's SwingPerfect program. I said admirable, not perfect. There are some controls that I and other researchers wish had been incorporated. But such shortcomings are few and far between. The program does a truly remarkable job of answering questions that the model can deal with. And it's free.

What we need now is a similar program for other models. The MacKenzie-Sprigings model just cries out for such a program. Does anybody know if one is available? It would seem that an open-source project for this would be appropriate.
  • It would leave the model itself open to inspection. SwingPerfect is closed source. I am pretty sure that it is a correct representation of the double pendulum, but I am not 100% certain. There might have been errors in implementation, errors so small and subtle that the answers are close enough to be reasonable -- but are wrong. (In fact, I have worked with such a program for driver trajectories: plausible but wrong. Such programs can teach incorrect lessons. I had to rewrite one of my web articles after realizing it and switching to a correct program.)
  • It would allow "anybody" to make enhancements to the program as researchers desire them. SwingPerfect can only be enhanced if Max decides he wants to make the enhancement.
It would be difficult to open-source Nesbit's full-body model for several reasons. One is legal; the GeBOD model and the ADAMS software are proprietary and substantial. Another is complexity; today's typical desktop machine may not have enough horsepower to run both the model and the matcher (optimizer).


We have reviewed a few important models of the golf swing, specifically:

Type of model
Double pendulum
Cochran, Stobbs,
& others
Simple forward dynamics
3D multiple pendulum
Simple forward dynamics
Planarity of swing
Full-body model
Complex inverse dynamics

Each has its own lessons to teach. And each type of model answers questions in a different way. How we use and extend these models going forward will be interesting to watch.


I'd like to thank Jim McLean, Sasho MacKenzie, and Paul Wilson for constructive comments that resulted in a significant improvement in this article.


  1. According to the information I could glean in February 2012, the PGA of America has 28,000 members and the US Golf Teachers Federation has 25,000. While not all the PGA professionals are active teachers and while there may be some overlap of membership, I'm pretty sure that those numbers include 30,000-40,000 golf instructors.
  2. For an example of reducing the dimensionality of a model based on a correlation discovered in the data, see my article on gear effect. I observed a strong, nearly linear correlation between I and c, and was able to combine them into a straight line (so the single parameter was the position along the straight line). In fact, I was able to reduce dimensionality even further. The straight line passed through the origin with a slope I/c. Everywhere I or c appeared in my model, it was always the ratio I/c. That ratio is the constant slope of the straight line, not a parameter at all. We managed to lose two parameters, I and c, and replace both with a known constant. Even though I and c did vary, their ratio did not, and the ratio was always the same in the real world.
  3. The X-Factor is an observation made by Jim McLean and first described in Golf Magazine in 1992. It holds that a prime determinant of clubhead speed, and thus distance, is the size of the angle between the hips and the shoulders from the top of the backswing through impact. In his book "The X-Factor Swing" (Collins, 1997), McLean provides more details on how he came to that conclusion, and how to incorporate it into your swing.
  4. Some of the questions listed as "future work" above are already being addressed. For instance:
    • Sasho MackKenzie has made me aware of a recent paper of his that looks at the effect of swing plane and the club's position above or below it. In the paper, he investigates how this affects clubhead speed, clubface squaring, and other results of the swing.
    • MacKenzie has also experimented with incrementally more complex models. In private communications, he discussed his model with the added parameter of a pelvis section, or a bending left elbow. As questions come up that need such models, it is clearly not prohibitive to build and use those models.

Last modified -- March 7, 2012