Reachability by Paths of Bounded Curvature in a Convex Polygon

TL;DR

This paper characterizes the reachable region for a curvature-constrained point robot inside a convex polygon, providing an efficient algorithm to compute it and identifying the path structure necessary for reachability.

Contribution

It offers a complete characterization of the reachability region, an $O(n^2)$ algorithm for computation, and a path structure description for curvature-constrained motion.

Findings

Reachability region has complexity $O(n)$.

An $O(n^2)$ algorithm computes the region.

Reachability paths are of type CCSCS.

Abstract

Let $B$ be a point robot moving in the plane, whose path is constrained to forward motions with curvature at most one, and let $P$ be a convex polygon with $n$ vertices. Given a starting configuration (a location and a direction of travel) for $B$ inside $P$, we characterize the region of all points of $P$ that can be reached by $B$, and show that it has complexity $O(n)$. We give an $O(n^2)$ time algorithm to compute this region. We show that a point is reachable only if it can be reached by a path of type CCSCS, where C denotes a unit circle arc and S denotes a line segment.