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Constraint-Aware Mesh Refinement Method by Reachability Set Envelope of Curvature Bounded Paths

Juho Bae, Ji Hoon Bai, Byung-Yoon Lee, Jun-Yong Lee

TL;DR

This work tackles inter-sample constraint violations in real-time direct-method optimal control by constructing an envelope of reachability sets for curvature-bounded planar paths (Dubins-style) and integrating a patch-based mesh refinement that guarantees obstacle-avoidance between sample points. The envelope is built via a sequence of augmented optimal-control problems and PMP analysis, yielding a finite set of trajectory classes (CSC/CCC) that bound feasible interpolants. A Rectangular Patch Cover framework tests envelope intersections with forbidden regions, and a Definiteness Theorem proves finite mesh refinement under a tolerance, enabling reliable inter-sample collision avoidance within a Sequential Convex Programming setting. Numerical demonstrations on 2D fixed-wing UAVs with circular no-fly zones show improved convergence and maintained real-time feasibility, suggesting practical impact for onboard trajectory optimization and potential extension to higher dimensions.

Abstract

This paper presents an enhanced direct-method-based approach for the real-time solution of optimal control problems to handle path constraints, such as obstacles. The principal contributions of this work are twofold: first, the existing methods for constructing reachability sets in the literature are extended to derive the envelope of these sets, which determines the region swept by all feasible trajectories between adjacent sample points. Second, we propose a novel method to guarantee constraint violation-free between discrete states in two dimensions through mesh refinement approach. To illustrate the effectiveness of the proposed methodology, numerical simulations are conducted on real-time path planning for fixed-wing unmanned aerial vehicles.

Constraint-Aware Mesh Refinement Method by Reachability Set Envelope of Curvature Bounded Paths

TL;DR

This work tackles inter-sample constraint violations in real-time direct-method optimal control by constructing an envelope of reachability sets for curvature-bounded planar paths (Dubins-style) and integrating a patch-based mesh refinement that guarantees obstacle-avoidance between sample points. The envelope is built via a sequence of augmented optimal-control problems and PMP analysis, yielding a finite set of trajectory classes (CSC/CCC) that bound feasible interpolants. A Rectangular Patch Cover framework tests envelope intersections with forbidden regions, and a Definiteness Theorem proves finite mesh refinement under a tolerance, enabling reliable inter-sample collision avoidance within a Sequential Convex Programming setting. Numerical demonstrations on 2D fixed-wing UAVs with circular no-fly zones show improved convergence and maintained real-time feasibility, suggesting practical impact for onboard trajectory optimization and potential extension to higher dimensions.

Abstract

This paper presents an enhanced direct-method-based approach for the real-time solution of optimal control problems to handle path constraints, such as obstacles. The principal contributions of this work are twofold: first, the existing methods for constructing reachability sets in the literature are extended to derive the envelope of these sets, which determines the region swept by all feasible trajectories between adjacent sample points. Second, we propose a novel method to guarantee constraint violation-free between discrete states in two dimensions through mesh refinement approach. To illustrate the effectiveness of the proposed methodology, numerical simulations are conducted on real-time path planning for fixed-wing unmanned aerial vehicles.
Paper Structure (21 sections, 39 equations, 15 figures, 2 tables, 2 algorithms)

This paper contains 21 sections, 39 equations, 15 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Direct Formulation of the Optimal Control Problem
  3. Reachability Set Envelope of Curvature Bounded Paths
  4. The Reachability Set of Curvature Bounded Paths
  5. The Original Optimal Control Problem
  6. Augmentation of Two Optimal Control Problems
  7. Necessary Conditions of the Augmented Problem
  8. Schematic Examples
  9. Rectangular Patch Cover and the Mesh Refinement Algorithm
  10. Upmost Bound of CSCSC Curves
  11. Winding direction of the middle C is opposite
  12. Winding directions of all three C's are the same
  13. Horizontal Bound of CSCSC Curves
  14. Bottommost Bound and the Non--CSCSC Curves
  15. Mesh Refinement Algorithm and the Definiteness Theorem
  16. ...and 6 more sections

Figures (15)

  • Figure 1: Example of a curve $(z_{1}(\tau), z_{2}(\tau))$ and a forbidden region
  • Figure 2: Schematic examples of the curves of fundamental classes
  • Figure 3: Example of rectangular patches covering $\mathcal{B}'(\mathbb{X}, \kappa_m)$
  • Figure 4: Four scenarios of reaching the upmost boundary of $\mathcal{B}'(\mathbb{X}, \kappa_m)$
  • Figure 5: Illustration of the CSCSC curves that attain the maximum $y$ coordinate
  • ...and 10 more figures