On shortest Dubins path via a circular boundary

The paper characterizes the shortest bounded-curvature paths from an initial configuration (a location and a heading orientation), via the boundary of an intermediate circle, to a target configuration. Such paths are fundamentally required in motion planning of a Dubins vehicle that has to avoid entering certain forbidden regions and when addressing the Dubins Traveling Salesman Problem with Neighborhoods. By using Pontryagin's maximum principle and analyzing the necessary conditions for state inequality constraints, the geometric properties of the shortest bounded-curvature paths are established. These geometric properties not only allow restricting the shortest bounded-curvature path within a sufficient family of 26 candidates but also enable us to devise a completely analytic solution for finding the candidate path. As a consequence, the shortest bounded-curvature path can be computed in a constant time by checking the path lengths of these 26 candidates. Finally, numerical examples validate the developments in the paper and highlight its importance in addressing some Dubins Traveling Salesman Problems with Neighborhoods.