Approximation algorithms for curvature-constrained shortest paths

Let B be a point robot in the plane, whose path is constrained to have curvature ofat most 1, and let Ω be a set ofpolygonal obstacles with n vertices. We study the collision-free, optimal path-planning problem for B. Given a parameter e, we present an O((n2/e4) log n)-time algorithm for computing a collision-free, curvature-constrained path between two given positions, whose length is at most (1+e) times the length ofan optimal path, provided it is robust. (Roughly speaking, a path is robust ifit remains collision-free even ifcertain positions on the path are perturbed). Our algorithm thus runs significantly faster than the previously best known algorithm by Jacobs and Canny whose running time is O(( n+L e2 ) 2 + n 2 ( n+L e2 ) log n), where L is the total edge length ofthe obstacles. More importantly, the running time ofour algorithm does not depend on the size ofobstacles. The path returned by this algorithm is not necessarily robust. We present an O((n 2.5 /e 4 ) log n)-time algorithm that returns a robust path whose length is at most (1 + e) times the length ofan optimal path, provided it is robust. We also give a stronger characterization ofcurvature-constrained shortest paths, which, apart from being crucial for our algorithm, is interesting in its own right. Roughly speaking, we prove that, except in some special cases, a shortest path touches obstacles at points that have a visible vertex nearby.