Discrete Dubins Paths

TL;DR

This paper introduces a discrete analogue of Dubins paths, demonstrating that shortest bounded-curvature polygonal paths share the same structure as classical Dubins paths, providing a new discrete proof of the original result.

Contribution

The paper develops a discrete model of curvature-constrained paths and proves they have the same structure as continuous Dubins paths, offering a novel proof approach.

Findings

Shortest discrete paths have the same structure as Dubins paths

Discrete model provides a new proof of Dubins' classical result

Properties of continuous paths are derived as a limiting case

Abstract

A Dubins path is a shortest path with bounded curvature. The seminal result in non-holonomic motion planning is that (in the absence of obstacles) a Dubins path consists either from a circular arc followed by a segment followed by another arc, or from three circular arcs [Dubins, 1957]. Dubins original proof uses advanced calculus; later, Dubins result was reproved using control theory techniques [Reeds and Shepp, 1990], [Sussmann and Tang, 1991], [Boissonnat, C\'er\'ezo, and Leblond, 1994]. We introduce and study a discrete analogue of curvature-constrained motion. We show that shortest "bounded-curvature" polygonal paths have the same structure as Dubins paths. The properties of Dubins paths follow from our results as a limiting case---this gives a new, "discrete" proof of Dubins result.