Characterization of Shortest Paths on Directional Frictional Polyhedral Surfaces

In this paper, we address a shortest path problem where an autonomous vehicle moves on a polyhedral surface according to a distance function that depends on the direction of the movement (directional) and on the friction of the space (frictional). This shortest path problem generalizes a hierarchy of problems and finds geometric structure to solve several proximity problems. We perform the characterization of shortest paths for a directional frictional geodesic (DFG) distance function on polyhedral surfaces. We derive the local optimality criterion necessary to solve the corresponding shortest path problem using the continuous Dijkstra algorithm [11]. The derivation of this optimality criterion essentially involves demonstrating the strict convexity of the DFG distance function. This contribution is the most fundamental result that enables all constructions of the continuous Dijkstra algorithm to solve the corresponding DFG shortest path problem.