On the Reachability of 3-Dimensional Paths with a Prescribed Curvature Bound

Abstract

This paper presents the reachability analysis of curves in $\mathbb{R}^3$ with a prescribed curvature bound. Based on Pontryagin Maximum Principle, we leverage the existing knowledge on the structure of solutions to minimum-time problems, or Markov-Dubins problem, to reachability considerations. Based on this development, two types of reachability are discussed. First, we prove that any boundary point of the reachability set, with the directional component taken into account as well as geometric coordinates, can be reached via curves of H, CSC, CCC, or their respective subsegments, where H denotes a helicoidal arc, C a circular arc with maximum curvature, and S a straight segment. Second, we show that the reachability set when directional component is not considered\textemdash{}the position reachability set\textemdash{}is simply a solid of revolution of its two-dimensional counterpart, the Dubins car. These findings extend the developments presented in literature on Dubins car into spatial curves in $\mathbb{R}^3$.