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

TL;DR

This work analyzes reachability for curvature-bounded curves in $\mathbb{R}^3$ using the Pontryagin Maximum Principle, building on the Markov-Dubins minimum-time problem to characterize boundary points and the full position reachability set. The authors show boundary points (including directional information) are reachable by trajectories from the classes H, CSC, CCC, and their subsegments, with precise curvature-length constraints, and that the position reachability set is a solid of revolution generated by rotating the 2D Dubins boundary about the initial axis. By connecting reachability to time-optimality, the paper leverages existing 2D results and 3D PMP analysis to extend Dubins car theory to spatial curves, providing an implicit boundary description and a closed-form geometric construction of the 3D reachability set. The findings offer analytic tools for planning curvature-bounded spatial trajectories and pave the way for higher-dimensional extensions and Lie-group formulations in future work.

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 $\mathbb{R}^3$.

Figures (5)

• Figure 1: Evolution of $\pi_2\left(\mathcal{G}_2(t_f)\right)$ over multiple values of $t_f$.
• Figure 2: Structure of the case study analysis
• Figure 3: Illustration of auxiliary trajectories that are generated from controllers that are not maximal. Among the state variables, only the $\boldsymbol{x}$ component is depicted.
• Figure 4: Boundaries of $\pi_2\left(\mathcal{G}_2(t_f)\right)$ in $\mathbb{R}^2$ and in $P$. ($t_f = 1.4\pi$)
• Figure 5: Cross-sectional view of $\pi_3\left(\mathcal{G}_3(t_f)\right)$ over multiple $t_f$.