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Elongation of Curvature-Bounded Path

Zheng Chen, Kun Wang, Yang Lu

TL;DR

This work addresses finding curvature-bounded (Dubins) paths between two oriented points with a prescribed length, a problem central to time-window-constrained and multi-vehicle coordination. It develops explicit, numerically verifiable conditions based on reachability sets to determine when a path of length $s$ exists, distinguishing CCC and CSC shortest-path patterns and, for CSC, partitioning boundary conditions into two subspaces with a potential nonexistence interval $(\ell_1,\ell_2)$. When existence holds, the authors introduce elongation strategies to transform the shortest path into one of the desired length, with detailed results showing CCC paths can be elongated to any $s\geq\ell_m$, while CSC paths depend on the endpoint region. Theoretical results are complemented by numerical examples with UAVs forming a triangle formation, demonstrating how to compute $\ell_m^i$, $\ell_1^i$, $\ell_2^i$ and apply elongation to achieve synchronized arrivals, confirming the method's practical impact for coordinated nonholonomic systems.

Abstract

The paper is concerned with elongating the shortest curvature-bounded path between two oriented points to an expected length. The elongation of curvature-bounded paths to an expected length is fundamentally important to plan missions for nonholonomic-constrained vehicles in many practical applications, such as coordinating multiple nonholonomic-constrained vehicles to reach a destination simultaneously or performing a mission with a strict time window. In the paper, the explicit conditions for the existence of curvature-bounded paths joining two oriented points with an expected length are established by applying the properties of the reachability set of curvature-bounded paths. These existence conditions are numerically verifiable, allowing readily checking the existence of curvature-bounded paths between two prescribed oriented points with a desired length. In addition, once the existence conditions are met, elongation strategies are provided in the paper to get curvature-bounded paths with expected lengths. Finally, some examples of minimum-time path planning for multiple fixed-wing aerial vehicles to cooperatively achieve a triangle-shaped flight formation are presented, illustrating and verifying the developments of the paper.

Elongation of Curvature-Bounded Path

TL;DR

This work addresses finding curvature-bounded (Dubins) paths between two oriented points with a prescribed length, a problem central to time-window-constrained and multi-vehicle coordination. It develops explicit, numerically verifiable conditions based on reachability sets to determine when a path of length exists, distinguishing CCC and CSC shortest-path patterns and, for CSC, partitioning boundary conditions into two subspaces with a potential nonexistence interval . When existence holds, the authors introduce elongation strategies to transform the shortest path into one of the desired length, with detailed results showing CCC paths can be elongated to any , while CSC paths depend on the endpoint region. Theoretical results are complemented by numerical examples with UAVs forming a triangle formation, demonstrating how to compute , , and apply elongation to achieve synchronized arrivals, confirming the method's practical impact for coordinated nonholonomic systems.

Abstract

The paper is concerned with elongating the shortest curvature-bounded path between two oriented points to an expected length. The elongation of curvature-bounded paths to an expected length is fundamentally important to plan missions for nonholonomic-constrained vehicles in many practical applications, such as coordinating multiple nonholonomic-constrained vehicles to reach a destination simultaneously or performing a mission with a strict time window. In the paper, the explicit conditions for the existence of curvature-bounded paths joining two oriented points with an expected length are established by applying the properties of the reachability set of curvature-bounded paths. These existence conditions are numerically verifiable, allowing readily checking the existence of curvature-bounded paths between two prescribed oriented points with a desired length. In addition, once the existence conditions are met, elongation strategies are provided in the paper to get curvature-bounded paths with expected lengths. Finally, some examples of minimum-time path planning for multiple fixed-wing aerial vehicles to cooperatively achieve a triangle-shaped flight formation are presented, illustrating and verifying the developments of the paper.

Paper Structure

This paper contains 7 sections, 11 theorems, 11 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Definitions and Notations
  3. Elongation of Curvature-Bounded Path
  4. Elongation of CCC-Path
  5. Elongation of CSC-Path
  6. Numerical Examples
  7. Conclusions

Key Result

Lemma 1

Given any curvature-bounded path $\gamma \in \Gamma(X,Y)$, if it has parallel tangents, then for any $s\geq \ell(\gamma)$ there exists a curvature-bounded path $\bar{\gamma}\in \Gamma(X,Y)$ so that $s = \ell(\bar{\gamma})$.

Figures (13)

  • Figure 1: The geometry of $C_{X}^r$, $C_X^l$, $C_{Y}^r$, and $C_Y^l$ with respect to $X$ and $Y$.
  • Figure 2: Elongation of curvature-bounded path with parallel tangents.
  • Figure 3: An elongation strategy for the shortest path of type CCC.
  • Figure 4: Elongation of a path in $\Gamma(X,Y)$ with a straight line segment and the length of the straight line segment is no less than $4/\kappa$.
  • Figure 5: An elongation strategy for $\gamma_m\in \Gamma(X,Y)$ with $(X,Y)\in \mathcal{O}_4$.
  • ...and 8 more figures

Theorems & Definitions (17)

  • Definition 1: Curvature-Bounded Path Ayala:2016
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Remark 1: Dubins path Dubins:1957
  • Lemma 1: J. Ayala Ayala+2017+283+292
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • ...and 7 more