Optimal grid-free path planning across arbitrarily-contoured terrain with anisotropic friction and gravity effects

Abstract

Anisotropic (heading-dependent) phenomena arise in the "two-and-one-half-dimensional" path-planning problem of finding minimum-energy routes for a mobile agent across some hilly terrain. We address anisotropic friction and gravity effects, and ranges of impermissible-traversal headings due to overturn danger or power limitations. Our method does not require imposition of a uniform grid, nor average effects in different directions, but reasons about a polyhedral approximation of terrain. It reduces the problem to a finite but provably-optimal set of possibilities, and then uses A* search to find the cost-optimal path. However, the possibilities are not physical locations but path subspaces. Our method exploits the surprising insight that there are only four ways, mathematically simple, to optimally traverse an anisotropic homogeneous region: (1) straight across without braking, the standard isotropic-weighted-region traversal; (2) straight across without braking but as close as possible to a desired impermissible heading; (3) making impermissibility-avoiding switchbacks on the path across a region; and (4) straight across with braking. We prove specific optimality criteria for transitions on the boundaries of regions for each combination of traversal types. These criteria subsume previously published criteria for traversal of isotropic weighted-region terrain. Our method can take considerably less computer time and space than previous methods on some terrain; at the same time, we believe it is the first algorithm to provide truly optimal paths on anisotropic terrain.