Shortest paths for the Dubins´ vehicle in heterogeneous environments

In this paper, the problem of finding minimum length paths for a Dubins´ vehicle that can only move forward in an heterogeneous environment is considered. An hybrid version of the Pontryagin´s maximum principle is used to derive necessary conditions for optimality. Unlike in the case of homogeneous environments, it is deduced that heterogeneity of the environment implies that optimal paths can contain reflections. A subclass of environments is analyzed more specifically in order to obtain additional necessary conditions. Based on these results, two concrete application cases are detailed to demonstrate the usefulness of the approach in practice. The first example concerns a mobile robot and the second example concerns a glider.