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### Methods

Passive protective linear-elastic torque elements were installed at each joint and were only activated if the anatomical limits of a particular joint were in danger of being exceeded during the simulation. Parameter values for segment length, moment of inertia, and mass for a representative golfer with a body mass of 80 kg and a standing height of 1.83 m were calculated using the values of de Leva (1996). Parameter values for a standard driver, 43.5 in. in length, were taken from the work of Cochran and Stobbs (1968).

The equations of motion for the three-segment system were written using a Newtonian formulation in combination with the known equations of constraint for a system linked with pin joints (Sprigings, Lanovaz, Watson, & Russell, 1998). The torso segment was treated as a special case in that gravitational force was prevented from supplying rotational motion to the segment. We reasoned that during the golf swing, the torso segment rotates about the spine, and since the spine is a line of symmetry for this segment, gravitational torque cannot assist rotation about this axis. The value used for the moment of inertia of the model’s torso segment about its proximal end was that of the anatomical torso’s moment of inertia about its longitudinal axis. A fifth-order Runge-Kutta-Fehlberg algorithm (Burden, Faires, & Reynolds, 1981) with variable step size was programmed and used to drive the simulation model.

The simulation process commenced with the assumption that the golfer had just completed his back swing and was just about to commence his down swing. It was assumed that at time zero the golfer’s torso segment was rotated 90° clockwise (top view) from the address position, with the arm and club segments positioned 60° and 10°, respectively, above a horizontal line through their proximal end, which is a typical configuration for an elite golfer (Yun, 1996; Figure 1). The acute 70° of wrist-cock angle that corresponds to this starting configuration takes into account the club’s inertial effects that are observed for a real player during the dynamic transition from the backward to forward swing.

The optimization scheme employed a single activation muscular control strategy where the onset of voluntary torque at each joint was controlled separately. The time of onset, as well as the length of time that the joint torques acted, provided six control variables for the optimization. The optimization search engine was based on Powell’s algorithm (Press, Teukolsky, Vetterling, & Flannery, 1992). The objective function was composed of the clubhead speed at impact, along with penalty variables that reflected inappropriate behavior by the model during the simulated golf swing. For example the position of the club’s shaft at impact was constrained using a penalty variable to be within +0.5% of the vertical position. This impact position constraint is consistent with the observation made by internationally acclaimed professional golf instructor, Jim McLean, that the greatest drivers of the modern era (Nicklaus, Norman, Hogan, Nelson, Snead, Price, Woods, Lietzke, Peete, Sutton) when viewed face on, all had their clubshaft vertical at impact (McLean, 1999). In the optimization scheme, the clubhead speed was expressed as a negative valued penalty variable so that its minimization in the optimization scheme would actually reflect a maximum. The sum of the accrued penalty variables served as the objective function, which was minimized by varying the values of the six control parameters that regulated the onset and duration of the muscular torque activation strategy.

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