The bench mover's problem: Minimum-time trajectories, with cost for switching between controls

Abstract

Analytical results describing the optimal trajectories for general classes of robot systems have proven elusive, in part because the optimal trajectories for a complex system may not exist, or may be computed only numerically from differential equations. This paper studies a simpler optimization problem: finding an optimal sequence and optimal durations of motion primitives (simple preprogrammed actions) to reach a goal. By adding a fixed cost for each switch between primitives, we ensure that optimal trajectories exist and are well-behaved. To demonstrate this approach, we prove some general results that geometrically characterize time-optimal trajectories for rigid bodies in the plane with costly switches (allowing comparison with previous analysis of optimal motion using Pontryagin's Maximum Principle), and also present a complete analytical solution for a problem of moving a heavy park bench by rotating the bench around each end point in sequence.