Minimum-Time Paths for a Small Aircraft in the Presence of Regionally-Varying Strong Winds

We consider the minimum-time path-planning problem for a small aircraft flying horizontally in the presence of obstacles and regionally-varying strong winds. The aircraft speed is not necessarily larger than the wind speed, a fact that has major implications in terms of the existence of feasible paths. First, it is possible that there exist configurations in close proximity to an obstacle from which a collision may be inevitable. Second, it is likely that points inside the obstacle-free space may not be connectable by means of an admissible bidirectional path. The assumption of a regionally-varying wind field has also implications on the optimality properties of the minimum-time paths between reachable configurations. In particular, the minimum-time-to-go and minimum-time-to-come between two points are not necessarily equal. To solve this problem, we consider a convex subdivision of the plane into polygonal regions that are either free of obstacles or they are occupied with obstacles, and such that the vehicle motion within each obstacle-free region is governed by a separate set of equations. The equations of motion inside each obstacle-free region are significantly simpler when compared with the original system dynamics. This approximation simplifies both the reachability/accesibility analysis, as well as the characterization of the locally minimum-time paths. Furthermore, it is shown that the minimum-time paths consist of concatenations of locally optimal paths with the concatenations occurring along the common boundary of neighboring regions, similarly to Snell’s law of refraction in optics. Armed with this representation, the problem is subsequently reduced to a directed graph search problem, which can be solved by employing standard algorithms.