On singular points and arcs in path constrained time optimal motions

Time optimal motions of articulated systems along specified paths have been efficiently solved by formulating the problem in terms of a scalar path coordinate and its time derivatives. This transforms the actuator constraints to state dependent constraints on the acceleration and on the velocity along the path. It was shown that the optimal control for this problem is bang-bang in the acceleration along the path. Points on the optimal trajectory that reach the velocity constraint are switching points from deceleration to acceleration. An important class of such switching points consists of singular points at which the optimal trajectory is determined by the slope of the velocity constraint rather than maximizing the acceleration along the path (Shiller and Lu 1992). In this paper, it is shown that singular points occur only if the set of admissible controls is not strictly convex. A strictly convex admissible set is therefore guaranteed to yield totally nonsingular trajectories. An efficient method for computing suboptimal but nonsingular trajectories is also presented and demonstrated for a two link planar manipulator.