# Robotics: Science and Systems XVII

### Generalized Comprehensive Motion Theory for High-Order Differential Dynamics

*Vincent Samy, Ko Ayusawa, Eiichi Yoshida*

**Abstract:**

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.

**Bibtex:**

@INPROCEEDINGS{Samy-RSS-21, AUTHOR = {Vincent Samy AND Ko Ayusawa AND Eiichi Yoshida}, TITLE = {{Generalized Comprehensive Motion Theory for High-Order Differential Dynamics}}, BOOKTITLE = {Proceedings of Robotics: Science and Systems}, YEAR = {2021}, ADDRESS = {Virtual}, MONTH = {July}, DOI = {10.15607/RSS.2021.XVII.032} }