ISAAC 2008: Algorithms and Computation pp 628-643

# A Complete Approximation Algorithm for Shortest Bounded-Curvature Paths

• Jonathan Backer
• David Kirkpatrick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

## 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.

## Keywords

Short Path Curvature Path Feasible Path Interesting Open Problem Internal Contact
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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