The Cost of Bounded Curvature

TL;DR

This paper analyzes the additional length required for a car-like robot with bounded turning radius to travel between two points, characterizing the worst-case scenarios and how this extra length varies with distance.

Contribution

It provides a detailed analysis of the function measuring the difference between shortest bounded-curvature paths and Euclidean distance, including its monotonicity and worst-case configurations.

Findings

The function ub(d) decreases from 7/3 to 2

ub(d) is constant for d _s  approximately 1.5874

Worst-case configuration pairs are characterized for all distances

Abstract

We study the motion-planning problem for a car-like robot whose turning radius is bounded from below by one and which is allowed to move in the forward direction only (Dubins car). For two robot configurations $\sigma, \sigma'$, let $\ell(\sigma, \sigma')$ be the shortest bounded-curvature path from $\sigma$ to $\sigma'$. For $d \geq 0$, let $\ell(d)$ be the supremum of $\ell(\sigma, \sigma')$, over all pairs $(\sigma, \sigma')$ that are at Euclidean distance $d$. We study the function $\dub(d) = \ell(d) - d$, which expresses the difference between the bounded-curvature path length and the Euclidean distance of its endpoints. We show that $\dub(d)$ decreases monotonically from $\dub(0) = 7\pi/3$ to $\dub(\ds) = 2\pi$, and is constant for $d \geq \ds$. Here $\ds \approx 1.5874$. We describe pairs of configurations that exhibit the worst-case of $\dub(d)$ for every distance $d$.