Time-optimal trajectories with bounded curvature in anisotropic media

This paper characterizes time-optimal trajectories in anisotropic (direction-dependent) environments where path curvatures are bounded by the minimum turning radius of a mobile agent. Such problems are often faced in the navigation of aerial, ground and naval vehicles when a mobile agent cannot instantaneously change its heading angle. The presented work is a generalization of the Dubins car problem, which considers fastest paths with bounded curvature while assuming constant speed and minimum turning radius. We relax this assumption and discuss fastest-path finding problems for the generalized direction-dependent speed and minimum turning radius functions, to account for the effects of waves, winds and slope of the terrain on the agent’s motions. We establish that there exists an optimal path such that it is a portion of a path of the type CSCSC where C denotes a sharpest turn curve and S a straight line segment. Further analysis of a special case (wherein the speed polar plot is convex) delivers a more detailed characterization of an optimal path and presents an algorithm that implements our results.