Time-optimal controls of kinematically redundant manipulators with geometric constraints

Theoretical investigations of time-optimal control of kinematically redundant manipulators subject to control and state constraints are presented. The task is to move the end-effector along a prescribed geometric path (state equality constraints). In order to address a structure of time-optimal control, the concept of a regular trajectory derived in Pontryagin et al. (1961) and the extended state space introduced herein are used. Next, it is proved that if the dynamics of a manipulator are defined by n actuators and m path-constrained equations, where m<n, then at most n-m+1 actuators are saturated, provided that the time-optimal manipulator trajectory is regular with respect to a prescribed geometric path given in the work space. Besides, it is shown that these results are also consistent for a point-to-point time-optimal control problem. A computer example involving a planar redundant manipulator of three revolute kinematic pairs is included which confirms the obtained theoretical results.