Sampling Extremal Trajectories for Planar Rigid Bodies

This paper presents an approach to finding the time-optimal trajectories for a simple rigid-body model of a mobile robot in an obstacle-free plane. Previous work has used Pontryagin’s Principle to find strong necessary conditions on time-optimal trajectories of the rigid body; trajectories satisfying these conditions are called extremal trajectories. The main contribution of this paper is a method for sampling the extremal trajectories sufficiently densely to guarantee that for any pair of start and goal configurations, a trajectory can be found that (provably) approximately reaches the goal, approximately optimally; the quality of the approximation is only limited by the availability of computational resources.

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