## Kinetic Analysis

The graphical musculoskeletal models also allow quantification of the joint reaction forces and moments during the applied motions (Fig. 3). Each joint of the model is treated as a ball joint and each limb is treated as a rigid link. A mass, center of mass, and moment of inertia are assumed for each body segment, and the acceleration of each link is quantified from the kinematic data. The joint reaction force and moment are quantified using an inverse dynamics approach [7]. The equations of motion for each limb segment about an arbitrary point are:

F = m(rA + a xrC + ax (ax rC)) y^ M = la + ax la, where ^ M are the resultant force and moment vectors of the external forces and moments acting on the body segment; m is the mass of the segment; rA is the position of the local coordinate system in the global frame; rc is the position vector of the segment mass center relative to the local coordinate system; a is the angular velocity of the local coordinate system; and I is the moment of inertia of the segment about the center of mass. The external forces and moments include terms for the joint reaction force and moment acting on each end of the segment and the gravitational force acting at the center of mass.

A distal limb, such as the lower leg or forearm, is analyzed first. The link should be unconstrained at the distal end or have known loading conditions at that end. The equilibrium equations are used to quantify the joint reaction force and moment acting at the proximal joint. The joint reaction force and moment acting on the proximal joint of the most distal segment are used to apply the same analysis to the next link, such as the upper arm or upper leg.

The joint reaction forces and moments can be distributed among the muscles that cross each joint. The muscles acting about the proximal joint are typically modeled as vectors acting along straight lines between the point of origin and the point of insertion. The point of origin and the point of insertion of each muscle on the bones being modeled are identified on the graphic model, and followed throughout the applied motion. The equations of motion for each degree of freedom are expressed as:

FIGURE 3 A model of the skeleton during baseball pitching. The model is shown from the side (a) and from the front (b) prior to the ball release. A collegiate baseball pitcher was videotaped in three dimensions while wearing reflective markers to obtain the pitching kinematics. The inverse dynamics technique was used to quantify the joint reaction force (single arrow head) and the joint reaction moment (double arrow head) at the shoulder and the elbow during the pitching motion. See also Plate 34.

FIGURE 3 A model of the skeleton during baseball pitching. The model is shown from the side (a) and from the front (b) prior to the ball release. A collegiate baseball pitcher was videotaped in three dimensions while wearing reflective markers to obtain the pitching kinematics. The inverse dynamics technique was used to quantify the joint reaction force (single arrow head) and the joint reaction moment (double arrow head) at the shoulder and the elbow during the pitching motion. See also Plate 34.

where m represents the number of muscles; F^ represents the magnitude of the ¿th muscle force; r,- represents a force unit vector of the ¿th muscle; FJ and T represent the joint reaction force and moment vectors; Fc and Te represent the external force and moment vectors; and r, represents the location of the ¿th muscle insertion [23]. The graphical model may show that some muscles wrap around the bone during the motion. In these cases, the centroidal line of the muscle should be plotted along the muscle length to determine the appropriate moment arm.

Quantifying the individual muscle forces is an indeterminate problem, since there are more unknowns than equations. The number of unknowns can be reduced by combining muscles with similar orientations or removing the least active muscles from the analysis based on EMG studies [11]. For most joints, however, reduction alone can not remove enough muscles to make the analysis determinate. Therefore, optimization techniques are combined with the equations of motion to solve for the muscle forces. The underlying assumption behind the optimization method is that the central nervous system controls muscle action by minimizing some performance criteria or cost function, such as muscle activity [23,24], muscle stress [1], or the square of the muscle stress [21,36]. The system of equations are also subjected to the constraint that the muscle stresses, expressed as the muscle force divided by the physiological cross-sectional area, are nonnegative and less than a maximum value [1]. The optimization procedure allows quantification of the individual muscle forces and the joint contact force. Ligament forces are treated as part of the joint contact force.

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