APPENDIX
(Work of the Body Joints)

The mechanical work of the body joints done during the tennis swing is one of the quantities being assessed in this study.  The humanoid model does not evaluate the work of the body joints directly, however it does provides the necessary data to calculate joint work as a post-processing function. Driving the joints of the model kinematically yields the internal reaction forces and driving torques at each joint.  Since there is no relative linear motion at the joints, the linear forces do no work, thus from the joint angular kinematics and driving torques, the work done at a joint can be determined from Eqn (A1):

(A1)

where ω_i is the relative angular velocity vector, T_i is the joint torque vector, and i is the alpha (medial/lateral), beta (anterior/posterior), and gamma (long-axis twisting) motion of the joint.

Using the body 1-2-3 Euler angle representation, the work of a joint can be determined by summing each separate angular movement over time as:

(A2)

where n is the number of numerical time steps, T_α, T_β and T_γ are the torque components, t is time, and is the time interval of the numerical integrator.

Since work is a scalar quantity, the work of individual joints can be summed directly to determine the work of a group of joints, or the work of the entire body. These methods have been used to evaluate the work of the body joints of the golfer (Nesbit and Serrano, 2005).