A complete approximation algorithm for shortest bounded-curvature Paths

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 consisting of polygonal obstacles, two feasible configurations, a length ε, and an approximation factor ε, 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∈+∈ε) 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 ), m (the bit precision of the input), ε -ε1, and ε. 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∈+∈ε) approximations to the shortest ε-robust bounded-curvature path [11,3] in time that is polynomial in n and ε -∈1. (Intuitively, a path is ε-robust if it remains feasible when simultaneously twisted by some small amount at each of its environment contacts.) Unfortunately, ε-robust solutions do not exist for all problem instances that admit bounded-curvature paths. Furthermore, even if a ε-robust path exists, the shortest bounded-curvature path may be arbitrarily shorter than the shortest ε-robust path. In effect, these earlier results confound two distinct sources of problem difficulty, measured by ε -∈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. © 2008 Springer Berlin Heidelberg.