Hybrid Geodesics as Optimal Solutions to the Collision-Free Motion Planning Problem

In this paper we address the problem of designing energy minimizing collision-free maneuvers for multiple agents moving on a plane. We show that the problem is equivalent to that of finding the shortest geodesic in a certain manifold with nonsmooth boundary. This allows us to prove that the optimal maneuvers are C1 by introducing the concept of u-convex manifolds. Moreover, due to the nature of the optimal maneuvers, the problem can be formulated as an optimal control problem for a certain hybrid system whose discrete states consist of different "contact graphs". We determine the analytic expression for the optimal maneuvers in the two agents case. For the three agents case, we derive the dynamics of the optimal maneuvers within each discrete state. This together with the fact that an optimal maneuver is a C1 concatenation of segments associated with different discrete states gives a characterization of the optimal solutions in the three agents case.