Dynamic Optimization Of The Golf Swing (End)
The optimization variables are the activation and deactivation times of the muscle torque generators, the shoulder torque ratio r and a parameter h, permitting slight adjustment of the height of the pelvis. h is attributed to small variances in knee flexion among golfers. To reduce the size of the solution space, some assumptions were made:
1. the torso and shoulder activate and deactivate simultaneously,
2. the pelvis, torso, shoulder and arm all activate at =0 to initiate the backswing, and
3. at the transition from backswing to downswing, the pelvis, torso and shoulder activate at a time τ after their respective backswing deactivation times (τ=40 ms [2,5]).
The objective of the biomechanical optimization is to maximize carry distance, measured as the point-to-point distance the golf ball travels in the air. Shots that travel offline are not penalized because it is assumed that the golfer may adjust their alignment such that the same shot lands in the center of the fairway. Rather than defining the initial ball position and forcing the model to make contact, the initial ball position is determined as follows: for each simulated swing, the impact and carry distance are simulated assuming a center-face impact at several points in the swing where the clubhead is within a height achievable using a maximum length tee of 4 inches (101.6 mm). If the model swings above the maximum achievable tee height or too close to the body, the swing is discarded. Moreover, if the clubhead touches the ground, or the clubface angle is such that the resulting impact would produce too much sidespin, the swing is also discarded. A genetic algorithm (MATLAB, 2017a) was used in a preliminary optimization to find the general location of the optimal solution, and then the patternsearch algorithm was used to refine the solution.
Results, Discussion and Conclusions
The optimization variable boundaries and optimal solution are provided in Table 1, where P, TS, A, and W denote the pelvis, torso/shoulder, arm and wrist, respectively. Some timings were omitted from Table 1 as they can be inferred from the foregoing assumptions. All values are in seconds, except for dimensionless r and hp. The optimized biomechanical timings indicate that the torso activated before the pelvis to commence the downswing. This result is inconsistent with real golf swings, where the rotation of the pelvis typically initiates the downswing. The model is not exploiting the extra power that could be generated by creating a large X-factor. It is possible that biomechanical constraints render the model’s X-factor ineffective at generating extra clubhead speed with a favorable clubhead delivery. One of the major limitations of the biomechanical model is the rigid body representation of the torso. In a real golf swing, the spine bends and contorts, causing a noticeable displacement of the thorax during the downswing. The flexibility of the spine could be what permits the sequential driving of the pelvis followed by the torso during the downswing.
Other than the torso activating before the pelvis, the optimized timings are reasonable. The late deactivation of the wrist during the backswing suggests the optimizer discovered the benefits of a large wrist cock and corresponding delayed wrist release. Furthermore, power was maximized during the downswing by deactivating the active muscle torques of the torso/shoulder, arm and wrist after impact, which occurred at t = 0.968 s. The golf ball was teed far forward in the stance and as high as possible to allow for an ideal clubhead delivery with a large angle of attack (11.9°) and dynamic loft (21.9°), thus minimizing spin rate (1620 rpm) and maximizing launch angle (20.5°). The clubhead and ball speed were 164 km/h and 241 km/h, respectively; the golf ball carried 257 yards.
To validate the biomechanical model, the simulated grip kinematics are compared to those of elite golfers in Figure 3. The motion capture data was taken from the associated work of McNally et al. . The grip orientation is defined using a Y-X-Z Euler rotations, and the kinematics are plotted against the normalized progression of the swing, from address to impact. The shaded bands represent the standard deviation of the measured kinematics for the ten elite golfers (100 golf swings total). At some points in the swing, the simulated grip kinematics fall outside the standard deviation of the experimental measurements. The discrepancies in the final quarter of the swing arise from the forward ball position that was required for the ideal clubhead delivery. It is atypical for a golfer to place the ball so far forward in the stance, but based on the results of this optimization it may be advisable to do so to create a large angle of attack and increase dynamic loft. As a result of the forward ball position, the downswing of the golfer model is effectively longer than that of a real golfer, which explains the offsets in the kinematics for the downswing portion of the swing. Despite these discrepancies, the model effectively recreated the motion of an elite golf swing with no prior knowledge of what such a swing should look like. The authenticity of the simulated golf swing lends credibility to the results of simulation experiments involving the use of this golfer model.