The second picture I was referring to.

Not sure you can actually solve the problem, but let's see if I can give you some different point of view. Three Euler angles uniquely identify the orientation of an object in space, so the the 3 markings can uniquely identify the positioning of the ball (two angles define an axis, e.g. that passing in the center of the triangle of the 3 dots, and the third defines the rotation around it). The impact point of the putter defines a point on the surface and thus identifies just 2 Euler angles (i.e. an axis). What you can do is to calculate the relative orientation of the ball with respect to the impact axis (it's a relative orientation between two directions in space with respect to a common frame, something we do often in crystallography) and this should be unique. To have some fun you can also try to see what's going on in the Euler space of the relative orientation (i.e. setting the zero on the axis of the impact point)

Hi Matteo,

Do excuse me, as this problem is somewhat outside of my current comfort zone being a sports scientist by trade!

For your method to work would you need to know the impact point in relation to balls position in the three pictures as the impact point was identified post trial (picture attached below)? However there may be potential to calculate a figure to offset each trial (would essentially zero the data for the first trial and adjust from there) by taking some other pictures of the golf ball and calculating the relative position of the black markings in relation to something visible on picture I took to locate the impact point. Could this potentially work or have I totally missed the point you were raising?

Thanks

Ashley

oh well it's not my field either but it's nice to think sideways from time to time.

So consider the golf ball being the Earth. You have defined 3 points on the surface and you can unequivocally identify them (the three dots). Actually you know also a 4th point.. the center of the Earth! And this is the origin of your reference frames for axes and rotations..

Now consider your have parked your car somewhere. What is your current position with respect to the car and what's the orientation of the car with respect to you?

You can define the orientation of the car using the axis passing through the center of the Earth and its rotation on the surface. In practice, take the pic:

http://en.wikipedia.org/wiki/File:ECEF_ENU_Longitude_Latitude_relationships.svg

put the car originally oriented north in the green reference and rotate it around its axis. The latitude and longitude are two Euler angles. is The third is the spin angle (around its axis). Now take your latitude and longitude. You defined another set of lambda and phi angles. Now take a map (of the Earth) and draw the two things: you have parallels and meridians to see the difference in position and you can draw an arrow to identify the orientation of the car. Your problem reduces to the rotation of the map in such a way that your car is always in the origin of the coordinates and parked with the arrow pointing e.g. north (i.e. the dots always oriented in the same way) and you want to find your position in this map (i.e. the position of the hit point with respect to a situation where the three dots have the same orientation).

So, practically, you identify your initial reference A, based on the three dots. You might set the origin in one of them i.e. the bottom left one. From figure 2 you see that in each trial you have a rotated reference B, totally similar to A, just a bit rotated with respect to it. You calculate the rotation needed to go from B to A (3 rotation angles defining 3 rotation matrices. There is probably a bit more math involved as you see just the projection). You then identify the hit point with respect to the reference B (easy as it means calculating it with respect to the three dots, irrespectively of the ball positioning) and you call it XB. To calculate XA (the position of the hit point with respect to the reference A, i.e. the "perfect" positioning, you apply to XB the rotation to go from B to A. It is a set of 3 matrix multiplications.

So you need the position of the impact point with respect to the dots and the position of the dots with respect to the one you assume as "reference"

Ashley,

I think Matteo has provided a nice mathematical solution to solve for the small differences in positioning you are referring too. However, just to provide yet another perspective, I am curious as to why you are attempting to determine your system with such precision when you are trying to measure a system that I think could be considered chaotic? I mean, it sounds like you are talking about very small variations in a complex system that affect the outcome in highly non-linear ways. Even though the outcome is determinate, I wonder if it will be highly challenging to relate specific outcomes back to initial conditions. Although, perhaps I am completely misunderstanding the situation.

Matteo, thank you for explaining this method in more detail, I am certainly reading off the same page now!

Sean, thanks for your interest. I currently contemplating analysing the impact point both corrected and uncorrected due to the following reasons. I want to see whether golfers with a higher handicap have an increased amount of variation of the impact point on the golf ball, now I would like to ensure that this is true variation and not due to the minor differences in the original ball placement. I am also looking to look for associations between my impact variables (X, Y coordinates and length of the impact point from the centroid, and surface area of the impact zone) to kinematic variables the software measures including sidespin, initial ball roll, forward roll, horizontal and vertical launch angle. I am unsure as to whether to test this adjusted, or unadjusted, I plan to look at the variability of ball placement between subjects and decide from there.

I looked at this impact mechanism in previous work of mine that I am looking to publish in the near future. Via multiple regression analysis I found that associations exist between the impact variables and a number of the kinematic variables. For this protocol however it was me and a mechanical putting robot, so I was able to control for the placement of the ball in a much stricter fashion, whereby ball placement did not become a problem. I found with actual subjects and time constraints this was near on impossible during data collection. With this work being for my PhD the next logical step was to see whether associations exist with a human subject cohort, hence I find myself where I am now.

Sean could you explain why you think the system is chaotic? It is the first I've heard the use of this term within sports biomechanics, or suggest some reading for me? Essentially do you mean it cannot be controlled for and I therefore can consider it a limitation?

I hope I've explained my situation and research a bit clearer for you. I look forward to a good discussion!

Ashley

Hi Ashley,

First, thanks for clarifying your project. It sounds very interesting to me to find out about how impact dynamics may correlate with handicap.

To address your question about chaos... now that I understand your project more clearly, I am less inclined to call your system of interest chaotic. It seems to me that you are trying to measure some pre-impact/impact variables (impact location, ball and putter attitude and motion) and then trying to correlate these with kinematics of the ball afterwards. I would say that the closer to impact you measure post-impact kinematics, the better the relationships will be. As the ball moves farther away from impact, there will be many minor variations in the system that will produce determinate, but highly non-linear outcomes (ball movements). This is why I would lean towards calling the system chaotic.

I am, however, not quite clear on how this relates to your question. Meaning that your variable of interest seems to be the impact point on the putter between the putter and the ball. If you could measure this impact point very reliably, you would be able to answer your question without measuring ball kinematics at all. The ball kinematics could become a secondary analysis or even a simulation model question. Have you thought about measuring impact point directly somehow? That is, without relying on the ball as a reference frame (which as you said may not work if you can't place it consistently).

@Ashley: thanks for the extra info. I was suspecting you were looking for something similar. As a further thought I might then suggest you to definitely think about representing your impact points (and in general the whole problem) e.g. on stereographic maps or directly as volume rendering in Euler space.

You might consider a backwards map with the projection point in the direction of the hit to map the hit points,

It could be also worth marking the dimples on the same map.

I suspect it makes a huge difference to hit a dimple (in terms of impulse) versus hitting the region in between dimples (difference in impact areas and therefore efficiency in the hit).

Many thanks to both of you again for the advice. I think for what both of you are suggesting (Sean, in measuring the impact point directly) and (Matteo, looking at the impact point in terms of impulse) I would need to have recorded the impact between the putter face and golf ball at a frame rate of >2000 Hz upwards, unfortunately this equipment was not available to me and therefore I needed to take the reference frame of the pigmented emollient markings after each trial. And obviously Matteo without knowing the time of the impact between the putter and ball I cannot measure the impulse, although I wholeheartedly agree with what you are saying.

I am looking to meet with the technician at my University to ensure I do the mathematics correctly. So will make sure to update you both with the outcome, if you are interested?

Ashley, I would be very interested to hear more about the project. I think if you do have the opportunity to revisit your setup, a putter face instrumented with a force plate could be quite useful in this instance to accurately determine impact location and timing of the event. I understand that we don't always get the equipment that we want! Best of luck with your project!

@Ashley: sure I'd like to know the outcome. I am pretty sure you can have an idea of the impact area already from the position of the marking: I suspect a golfer with a low handicap has quite a calibrated swing and therefore impact time is the same.

Of course the more you want to be on the scientific side i.e. trying to use mathematics and physics to model some phenomena, the more you need to design your experiment and measurement in the most convenient way (which is usually not the most readily available one)

I am really happy to read this communication between scholars in the biomechanical field. Although it is not the exact interest of precise measurement to the golf, I learn from the discussion of the matter. Hope that Ashley got a nice feedback from Sean Osis & Matteo Leoni. Thanks for all contribution thoughts and may simulation in different approach will aid in helping this matter.

@Hashem: well, biomechanics is fascinating and something we experience daily, but my background and action field are completely out of here (as you can see from my cv)... but for a scientist the most important thing is not the field of action, but being in the right place at the right time to use his know-how for the solution of some problem! great works and great collaborations start this way, by chance

You would like to know changes in the position of the ball in the cartesian frame?

This is fairly easy to do but will require some work on your part:

Two web cams setup in the x-y plane , one looking down the x- axis and the other looking down the y-axis give two images that can be used to determine the position of the ball. the centroid is easily found by observing the diameter of the ball image in pixels. MATLAB may be used to acquire the orthogonal images from the web cam and the position of the centroid may be determined to within 0.25 pixel.

A third web cam on the z-axis, with a suitable mark on the ball, may be used to determine the rotation.

Jerry.

Thank you Jeremy for this answer. Unfortunately the data has already been collected, so I will continue with Mattteo's solution, although in the future I intend to adopt the approach you suggested. I did not as I did not incur problems of ball position when I did the testing with a putting robot where I placed the ball.

Ashley