Generalized Comprehensive Motion Theory for High-Order Differential Dynamics

We address the problem of calculating complex Jacobian matrices that can arise from optimization problems. An example is the inverse optimal control in human motion analysis which has a cost function that depends on the second order time-derivative of torque τ. Thus, its gradient decomposed to, among other, the Jacobian δ τ/δq. We propose a new concept called N-order Comprehensive Motion Transformation Matrix (N-CMTM) to provide an exact analytical solution of several Jacobians. The computational complexity of the basic Jacobian and its N-order time-derivatives computed from the N-CMTM is experimentally shown to be linear to the number of joints Nj. The N-CMTM is based on well-known spatial algebra which makes it available for any type of robots. Moreover, it can be used along classical algorithms. The computational complexity of the construction of the N-CMTM itself is experimentally shown to be N².