An Efficient Snell's Law Method for Optimal-Path Planning across Multiple Two-Dimensional, Irregular, Homogeneous-Cost Regions

Abstract

We are exploring a new approach to high-level optimal-path

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

“…Thus by picking A and B sufficiently close to the region boundary, it is always possible to find a locally-optimal path obeying Snell's Law and departing at that connects A and B on thepath. Since Snell's-Law paths are the single local optimum among isotropicweighted-region paths as proved in [11] and [15] the path must be optimal if thepath is totally nonbraking. Otherwise, if either segment of the latter path is braking, then its cost can only be worse by Lemma 2.1, so the path is still optimal.…”

“…To facilitate mathematical analysis, we assume as in the Army Mobility Model [18] and our own earlier work based on it [12,15] http://faculty.nps.edu/ncrowe/elevation2.htm that the terrain is partitioned into polygonal regions within which the coefficient of friction can be assumed constant; any terrain can be sufficiently closely approximated by some such polygons [10]. For overland terrain, often this can be done from map overlays or classification of surface covering in aerial photography.…”

“…We will show there are only four ways an optimal path can traverse a homogeneous-friction-coefficient planar anisotropic region (henceforth called just a "region"): type I, straight across at a permissible nonbraking non-critical heading, turning on boundaries according to standard weighted-region constraints [15]; type II, straight across at a critical-impermissibility heading to follow as close as possible to a desired direction; type III, "switchbacks" or "slaloming" alternating arbitrarily between a matched pair of critical-impermissibility headings; and type IV, straight across http://faculty.nps.edu/ncrowe/elevation2.htm with braking at a non-critical heading, with the associated braking cost function. Other key results of this section are summarized in Figures 3, 4, and 5.…”

“…As in our previous work [14,15], we will specify local criteria that an optimal path must obey. First consider isotropic regions.…”

“…But they can have several serious disadvantages: (1) potentially much wasted computation, since search for the goal is mostly blind; (2) potentially much wasted space, since homogeneous areas of the terrain require as much storage per unit area as heterogeneous areas; (3) directional biases in the propagation grid giving path-cost errors; (4) accumulation of roundoff errors; (5) digitalization effects causing unreasonable jagged paths where the optimal path should be straight (while jagged paths can be smoothed, there are many kinds of smoothing to choose among, and optimality or even feasibility of the smoothed path cannot be guaranteed); and (6) inability to converge to the optimal solution as grid size increases. [15,6] discuss these disadvantages in more detail for a related problem. Note that disadvantage (4) is particularly serious for grid algorithms on anisotropic terrain, because repeated rounding to the same side of the recommended path direction, as in near-homogeneous terrain, will force the path further and further afield; on the other hand, increasing the number of angles of…”