Decidability of Motion Planning with Differential Constraints

Classical path planning does not address many of the challenges of robotic systems subject to differential constraints. While there have been many recent efforts to develop motion planning algorithms for systems with differential constraints (MPD), very little has been said about the existence of exact algorithms. In other words, the decidability of MPD problems is still an open question. In this paper, we propose a partial answer to this question limiting ourselves to special cases where the trajectory functions of the systems under the finite-dimensional piecewise-continuous controls have a closed-form polynomial formulation. We define an abstract formulation for the MPD problem based on the concept of a control space. We provide an incremental decision algorithm to answer the decidability question and present sufficient conditions for problems to which this algorithm can be applied. Decidability results for several non trivial MPD problems are presented. For example, we show that the question of existence of a trajectory for a Dubin's car with a polygonal rigid body between two specified positions and orientations in a polygonal environment with a fixed and finite number of discontinuities in curvature is decidable.