Science frequently uses models for studying complex problems. The scientists developing such models strive for the simplest model possible yet still giving useful results. The simplest model possible for a golf swing consists of two segments. It still is surprisingly useful considering its utmost simplicity and the great complexity inherent in a golf swing.

We will look presently at an interesting and very specific way to use mathematical models. A model can be considered as a black box, where we define inputs and observe the resulting outputs. Normally this would be torques as inputs and angular position//velocity/acceleration information as outputs. However with mathematical models we can also inverse the role of input and output. We can define angular position/velocity/acceleration as inputs and find the torques required, to produce such motion, as outputs. This we can define as inverse dynamics.

A scientists wanting to study a golf swing faces a difficult problem of having to measure the torques developed by the various muscle groups. For instance, where and how to apply gauges to measure the shoulder torque? Nasty task. Here is where inverse dynamics comes into the picture as an very elegant and practical measuring approach. It is relatively easy to measure the angular position/velocity/acceleration of various parts and segments of the body. If this information is inputted into an adequate mathematical model we can solve this tricky problem of measuring various torques simply by using the model in an inverse way.

With a simple 2 segment model we can, using inverse dynamics, determine the torques at the center (spine/shoulder) and the wrists. We require as input either angular position, velocity or acceleration. I thought it a good idea to show, for an change, not just results but also the operations involved. This might be of particular interest to some who seem to assume that I have secret assumptions not to be revealed. So, more details than usually are shown, hopefully making it very obvious that posts like this are tedious and very time consuming to put together. Not only are there these mathematical operations/calculations to perform but also the various housekeeping functions such as transfer to proper HTML format, uploading etc.

As input we will use information accessible on the Skill Technologies site. I have chosen the 4 sensor swing of the unskilled amateur for being a greater challenge and to make feel some perhaps more comfortable if comparing their swing. I like to stress strongly that this is primarily to illustrate the method since I will be using 3D measurements as input for a 2D model. The results obtained are still quite interesting and rather convincing.

The first step is to isolate the curve with the angular velocities for lead arm and club, shown above, and paste it into suitable software such as CorelDRAW to measure the xy-coords of a sufficient number of points to allow a mathematical expression to be developed. Once a mathematical expression is developed we can use the angular velocity information as input to a mathematical model. Below is shown the matching obtained between the measured curves and those calculated from the derived equivalent mathematical expressions.

**Pointsarm**** = **{{0,0},{9.18,3.06},{20.9,6.63},{30.08,9.18},{40.8,13.77},{50.5,17.8},{60.1,23.5},{70.9,28.5},{80.5,34.7},{90.2,40.3},{100.4,42.8},{110.1,45.9},
{119.8,48.9},{130.5,54},{139.2,60.1},{150.4,67.8},{160.6,72.9},{170.2,76},{179.9,77.5},{190.6,79.5},{199.3,84.6},{210,93.3},{220.2,102.5},
{230.9,115.2},{240.1,125.4},{250.8,138.6},{260,152.4},{270.7,167.7},{280.3,185.0},{290.5,199.8},{299.2,208.5},{310.4,217.1},{319.6,224.3},
{329.8,234.5},{340,245.7},{349.7,252.3},{359.9,254.3},{367,252.8},{378.2,255.9},{387.4,263},{395,270.7},{404.2,270.1},{412.9,270.7}}

The three curves above from left to right show the experimental points measured for the arm, (Skill Technologies), the mathematically derived approximation and the two curves merged to show the degree of matching obtained.

**Pointsclub** = {{0,0},{10.2,.5},{19.9,2.55},{29.6,6.1},{40.3,7.7},{50,10.2},{60.7,14.3},{70.3,18.4},{80,22.4},{90.2,25},{100.4,27.5},{109.6,29.6},{120.8,31.6},{131,34.7},{139.2,38.2},{150.4,41.8},{160.6,43.3},{170.8,46.4},{179.9,47.4},{190.1,50.5},{199.8,53},{210.5,59.1},{219.2,63.2},{229.9,70.9},{241,78},{250.3,85.1},{260.5,91.2},{269.6,95.8},{279.8,98.9},{289.5,98.9},{300.7,98.4},{310.4,96.3},{320.1,93.3},{330.8,92.3},{341,89.2},{350.2,85.1},{360.4,80},{370.6,76.5},{380.8,69.3},{389.9,55.6},{396,44.3},{400.1,37.2},{404.2,29.6},{408.8,29.6},{413.9,29.6}}

The three curves above from left to right show the experimental points measured for the club, (Skill Technologies), the mathematically derived approximation and the two curves merged to show degree of matching obtained.

Once the analytical expressions derived we can obtain angular positions and accelerations through integration and differentiation of this velocity information. The angular positional information obtained this way allows us to use the 2D display of the mathematical model as shown below. It is a surprising good match to the Skills Technologies' swing display. Notice that this golfer does not try to retain the angle, immediately releasing from the top, and that the inner segment is steadily increasing its speed up to impact.

We are finally ready to have some fun and to perform the inverse dynamics operation with the mathematical model. The torques derived with the model are subsequently for arm and club:

Substituting the information derived for angular displacement, velocity and acceleration into the equations above gives as result the curves show below. I have assumed a swing using a driver. The swing as displayed is that of a very strong individual, the torque levels are very high. The wrist torque is immediately active from the top.

Conclusions:

-1- We have demonstrated above an elegant and practical use of mathematical models in determining torques developed by a golfer - hence the usefulness of Inverse Dynamics.

-2- The result of our inverse dynamics analysis shows the unskilled amateur indeed to be unskilled but very strong. Very large torques throughout the swing and a considerable wrist torque immediately applied from the top and yet relatively low club head speed of only about 90 mph at impact. A very inefficient swing indeed. These magnitudes of torques, if used by a pro, in a proper sequential fashion, would probably give closer to 140 mph as club head speed.

-3- The swing being very inefficient can also be noted from the speed of 'hands', as shown below. There is no reduction of the hand speed prior to impact. In an efficient golf swing hand speed reduces indicating that the arm's kinetic energy is used to boost club head speed. The unskilled amateur's swing is probably not unlike many on the driving range, almost falling on their face due to strenuous efforts and yet getting the ball barely out of their shadow.

mandrin