A Complete Approximation Algorithm for Shortest Bounded-Curvature Paths

TLDR: This work addresses the problem of finding a polynomial-time approximation scheme for shortest bounded-curvature paths in the presence of obstacles and clarifies the critical factors contributing to the complexity of bounded-Curvature motion planning.

Abstract: We address the problem of finding a polynomial-time approximation scheme for shortest bounded-curvature paths in the presence of obstacles. Given an arbitrary environment $\mathcal{E}$ consisting of polygonal obstacles, two feasible configurations, a length l, and an approximation factor e, our algorithm either (i) verifies that every feasible bounded-curvature path joining the two configurations is longer than l or (ii) constructs such a path Π whose length is at most (1 + e) times the length of the shortest such path. The run time of our algorithm is polynomial in n (the total number of obstacle vertices and edges in $\mathcal{E}$), m (the bit precision of the input), e -1, and l. For general polygonal environments, there is no known upper bound on the length, or description, of a shortest feasible bounded-curvature path as a function of n and m. Furthermore, even if the length and description of a shortest path are known to be linear in n and m, finding such a path is known to be NP-hard [14]. Previous results construct (1 + e) approximations to the shortest e-robust bounded-curvature path [11,3] in time that is polynomial in n and e -1. (Intuitively, a path is e-robust if it remains feasible when simultaneously twisted by some small amount at each of its environment contacts.) Unfortunately, e-robust solutions do not exist for all problem instances that admit bounded-curvature paths. Furthermore, even if a e-robust path exists, the shortest bounded-curvature path may be arbitrarily shorter than the shortest e-robust path. In effect, these earlier results confound two distinct sources of problem difficulty, measured by e -1 and l. Our result is not only more general, but it also clarifies the critical factors contributing to the complexity of bounded-curvature motion planning.