2D and 3D Coordinates for Golf Swing Analysis

Introduction
This article is meant as an introduction to the basics of measuring position in both two dimensions and three-dimensions; specifically to the golf swing.  I wrote it a while ago but would like to bring it back and then continue the idea.

Position Measurement in Two-Dimensions
When we want to measure the position of something on a flat plane, say a piece of paper or the screen of a computer, we can use two perpendicular axes, one along the bottom and one along the left side of the paper. Lets give each of these lines a name, lets call the bottom one the X-axis and the one on the left side the Y-axis. These two lines join at the bottom left corner of the paper and we call this point the origin or (0, 0). It is zero feet along the X-axis and zero feet along the Y-axis. To place a point on the paper we can simply measure a specific distance along the X-axis and a specific distance along the Y-axis and write is as: (x, y). These two numbers completely define a point anywhere on our piece of paper. Now if we chose to move our two lines to the center of the page, that is, we draw a cross with one line up and down and the other left to right, we would now have added a negative X-axis and negative Y-axis. Anything to the left of the center will get a negative X value and anything below the center would get a negative Y value. Further, any point in the left bottom corner would have both X and Y values negative. The combination of the origin, the X-axis and the Y-axis is called a two-dimensional (2D) coordinate system or reference frame. The coordinate system below is labeled with several coordinate pairs as examples.

 

The Golf Mat as a Two-Dimensional Coordinate System
Now let’s take this piece of paper and lay it down on the floor, and let’s enlarge the paper to about a 5-foot square, paint it green and call it a golf mat coordinate system. If our golfer stood on this mat in the address position we could move the center (or origin) of our two lines between his feet. We can also make the line in the direction of the pin, red and call it the X-axis and make the line that is forward and backward with respect to the golfer, green (a different green than the mat so we can see it) and call it the Y-axis, maybe even position it through the ball for convenience.

Now since we have the golfer standing on a two-dimensional coordinate system we can measure certain things about his stance, for example, we can find the coordinates of the ball are (0, 3), that is, 0 feet along the X-axis and 3 feet along the Y-axis. We can also measure the positions of his right heel (0.8, 0), right toe (0.9, 0.8), left heel (-0.6, 0) and left toe (-0.7, 0.8). Note that all these values are in feet. Now we have some simple two-dimensional information about his stance. From these points we can calculate things like: width of stance, distance away from the ball, position of the ball between the feet, how much each foot is open or closed to the target in degrees, and if we draw a line connecting each heel we can calculate the angle that his stance is open or closed to the target. Note that the Y-coordinate of each heel is zero, which means he is square to the target in this example.

Measurement in Three-Dimensional Space
Now that we have the golf mat as our two-dimensional X-Y plane, lets add another axis and extend our discussion to the world we live in; three-dimensional space. Lets place an imaginary vertical line starting from the origin and going all the way to the ceiling. Lets color this line blue and call it the Z-axis. Now we have three axes, the X, Y and Z-axes and an origin. The origin is now defined with three number as (0, 0, 0), one each for its X, Y and Z values. The combination of the origin and the X, Y and Z-axes is called a three-dimensional (3D) global coordinate system or a global reference frame. We use the word “global” because it is the reference frame from which everything is measured. If we measure a distance along each of these three axes we can define any point in the 3D space around the golfer. Now we have the power to define the position of any point on the golfer’s body or club at the address position, not just the position of his feet and the ball as we did earlier.

Lets not worry how we do it for now but lets just assume we can measure the 3D coordinates of a point on his pelvis at the right thigh and at the left thigh. Lets also measure the position of a point on his left shoulder and a point on his right shoulder. Now we can join the left and right pelvis points with an imaginary line and the left and right shoulder points an imaginary line. From these lines we can measure some more interesting things about his stance. With the pelvis line we can measure whether his hips are open or closed at address, we call this the pelvis rotation angle, and how he is tilted to the right or left, we call this pelvis side bend. We can do the same for his upper body using the line from the left to the right shoulder. Notice that we can get rotation and side bend from this line but we cannot get forward-backward bend. We don’t know how much he is bending forward with his upper body since we only have a line from left shoulder to right shoulder. We need another line or point to figure that out. In fact we need to define a three-dimensional “local reference frame” in his upper body and all the other body parts in order to calculate everything we need for a true 3D analysis. I will explain this in my next article and also explain the term “six-degrees-of-freedom”. Here’s a hint; with a local reference frame on each body segment we can measure three positions - sway, thrust and lift (x, y, z) and three angles, one around each of the axes - forward bend, side bend and axial rotation (θ_x, θ_y, θ_z). Remember, positions are measured in feet (or meters) and angles are measured in degrees.

Changing the Kinematic Sequence with Biofeedback Drills

I am currently working with an LPGA player to improve her efficiency in the kinematic sequence and get her more swing speed.  Remember that the kinematic sequence is the graph of turning speed of pelvis (red), thorax (green), arm (blue) and club (brown).  It is measured in degrees/second along the vertical axis and shows time through the swing along the horizontal axis.  (Check my earlier blog articles for more detail or amm3d.com).  After capturing her swing with the AMM3D golf motion capture system, I analyzed it with the TPI 3D biomechanics report.  Below is the graph of her kinematic sequence with a driver.  There are many “expert” characteristics in the graph; transition order is good, peaking order is good, accelerations and decelerations generally look good too.

Although she transitions in the correct order; pelvis, thorax, arm, club; you can see that she pauses with her hips before transition (the red curve flattens out), also her transition period is too long, there is minimal downswing loading of any joint and her turning speeds are fanning early in the downswing.  This pause really limits the speed contribution of the “stretch-shorten cycle” of the muscles.  In and ideal stretch-shorten cycle there should be virtually no time between the eccentric/concentric contraction phases of the muscle; this “amortization” phase or pause should be as short as possible.  The lack of downswing joint loading means she is leaving a lot of power on the table and the fanning of the curves means that the relative speed between the body segments is increasing, reducing the amount of force that can be produced in each muscle; (see my article on amm3d.com regarding Riding, Stretching, Fanning, I’ll post it soon on this blog too).

So while she was still in the AMM3D system we began using real-time audio biofeedback to see if we could change the sequence and teach her not to pause at the top; hence allowing the smooth acceleration into and out of transition and improving the stretch-shorten cycle.  We set the audio tone to sound if she achieved a good core stretch at the beginning of downswing and worked on half backswing drills.  Very quickly she was able to coordinate her downswing loading (aka “X-Factor Stretch”) and the curves became smoother before and after transition.  We also saw that thorax, arm and shaft “rode” up together (good energy transfer); however, her total transition time was still very long.  This means that the drill was definitely successful but that we still need to work on lessening the transition time from about 0.17 seconds to about 0.8 seconds.  This will require a combination of half swing drills and strengthening of the core muscles so they can support fast pelvis firing without leaving the thorax behind.  Check out the kinematic sequence graph below, this is from one of the half swing drills. 

Drills can definitely change the kinematic sequence very quickly, but conditioning exercises will also need to be done in the gym to support the change.  The trick will be to transfer this to the full swing.

Where is Peak Rotation Speed in the Kinematic Sequence of the Golf Swing?

An article I did earlier this year on where in the downswing the pelvis rotation speed peaks and then decelerates.  The interesting this is how early it happens.

The Kinematic Sequence is an indicator of swing efficiency and speed. If you look at just the downswing phase of the swing (the center section) the graph above displays how fast you turn or swing each part of your body for about 0.25 seconds before impact. During the downswing each part accelerates then decelerates in a smooth and sequential order. In an efficient swing the red (pelvis) curve should peak first followed by the green (thorax) curve then the blue (lead upper arm) curve and finally the brown (shaft) curve hits maximum swing speed at impact. Also each curve should peak higher than the previous one.

One thing that is not obvious from the graph: What is the typical body position at each of these peaks? In this article we look at where the pelvis speed peaks. The strange thing from the graph is that it looks like it is about half way down in the swing. Well, in terms of time it may be; but in terms of body position it is not. In fact peak pelvis turning speed occurs surprisingly early in the downswing in terms of body and club position; look at the first image in the diagram above. That’s because at the beginning of the downswing you are moving slow and at the end you are moving very fast.

Let’s look at where 45 touring pros are at this point in the downswing. They are taken from the TPI 3D database. It is very instructive to look at their positions. Look at the angles of the shaft, lead arm, thorax and pelvis. I tried to group them with respect to shaft angle. I couldn’t group them with respect to thorax and pelvis angle but you can study that yourself.

One very important observation is that in most cases the club is still very close to, or before vertical, the pelvis looks pretty close to square, and as I said, peak pelvis speed occurs pretty early in the downswing. That’s where the pelvis is turning the fastest, after this point the pelvis turning speed slows down. Notice that at this point there is still a good wrist set and in many cases also a good shoulder load.

The last four images in this article are definitely the odd ones out.

In a subsequent article I will look at the positions of high handicappers also at peak pelvis speed and you definitely won’t see this consistency and “strength” of position; quite often will see a very late pelvis peak.

So when doing a 3D analysis, in my opinion, it is good to visualize what the golfer should look like at the specific points in the kinematic sequence, the first one being shown here; the point at which the pelvis reaches its peak turning speed.

This was not meant to be a rigorous scientific analysis and you can probably argue some of my groupings, however, it is just meant to give you a feel of where these top golfers are when their pelvis speed peaks and then begins to decelerate as the rest of the segments continue to accelerate. Enjoy studying the pictures on the next few pages.

The first group has the club shaft well before vertical and the lead arm close to horizontal.

Next group is on the next page.
This next group has the club still before vertical but the arm is a little below horizontal.

Next group is on the next page.
This next group has the club shaft a little past vertical and the arm is around 30 to 45 degrees below parallel

These last four have the latest point of peak pelvis speed and hence a larger more released wrist angle. These four guys are not the longest hitters on the tour.

Copyright © Phil Cheetham “The 3D Guy” November 2009