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 $\mathcal{E}$ consisting of polygonal obstacles, two feasible configurations, a length ℓ, 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 $\mathcal{E}$), m (the bit precision of the input), ε ^− 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 + 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.