Dynamic Optimization Of The Golf Swing (P3)
where ω was the joint’s instantaneous angular velocity, ω(max) was the joint’s maximum angular velocity , and Γ was a shape factor affecting the curvature. With the incorporation of the backswing into this new model, some of the joints may have a negative angular velocity at the time of activation (i.e., ω< 0), potentially creating a singularity as ω approaches −ω/Γ . The pioneering experiments of Katz  suggest that the force-velocity relationship diverges from Hill’s model when a force greater than isometric tension is applied. The divergence was illustrated by a discontinuity in velocity at the onset of lengthening and an exponential increase in the rate of lengthening with increasing applied force. Van Soest et al. provide a generic equation for the eccentric forcevelocity relationship observed in the experiments of Katz. The rotational equivalent of this equation was manipulated to use the same variables as Sprigings and Neal:
where S is the slope-factor, defined by van Soest et al. as the ratio between the eccentric and concentric derivatives of force with respect to velocity at isometric force, and T is the ratio between the maximum eccentric and isometric force. A slope-factor of 2 and a force ratio of 1.5 was used in this model, matching the values used by van Soest et al. Using a piecewise function, an example torque-velocity scaling profile is shown in Figure 2 for a nominal isometric torque of 50 Nm, a maximum angular velocity of 10 rad/s, and a shape factor of 3. Besides the pelvis, all the torque generators use the same parameters as MacKenzie and Sprigings. The pelvis is more proximal than the torso, so it was given a maximum isometric torque of 250 Nm, 25% greater than the torso, and a maximum angular velocity of 30 rad/s, equal to that of the torso.
Golf Drive Simulation
The address position (position and orientation of a golfer’s joints before the swing commences) was predetermined. The pelvis and torso angles were taken from mean values in literature, and the remaining joint angles were determined by matching the golfer model’s grip position and orientation to the mean grip position and orientation from the motion capture experiment in the associated work of McNally et al.. The golf swing is driven by 12 muscle torque generators—two for each DOF (backswing and downswing). In the preceding models, the golf swing simulation began at the top of the backswing and assumed the shaft initially had no deflection. In real golf swings, the shaft has significant toe-up deflection caused by the quick transition from backswing to downswing. The shaft initialization issue is solved by modelling the backswing. The set of backswing torque generators use the same parameters as those used for the downswing except with negative maximum isometric torques and scaled maximum angular velocities. The maximum angular velocities for the backswing were scaled by a factor of 0.07, and the maximum isometric torques were scaled by a factor of 0.65 to create a realistic 3:1 tempo (duration of backswing to duration of downswing) and swing duration (address to impact) of approximately 1 second. To simulate the shaft deflection and impact, the validated flexible club and impact models described in the work of McNally et al. are used. The simulated launch conditions of the golf ball are recorded shortly after impact, and used to initialize an aerodynamic model that simulates the ball flight.