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Informed Hybrid Zonotope-based Motion Planning Algorithm

Peng Xie, Johannes Betz, Amr Alanwar

Abstract

Optimal path planning in nonconvex free spaces poses substantial computational challenges. A common approach formulates such problems as mixed-integer linear programs (MILPs); however, solving general MILPs is computationally intractable and severely limits scalability. To address these limitations, we propose HZ-MP, an informed Hybrid Zonotope-based Motion Planner, which decomposes the obstacle-free space and performs low-dimensional face sampling guided by an ellipsotope heuristic, thereby concentrating exploration on promising transition regions. This structured exploration mitigates the excessive wasted sampling that degrades existing informed planners in narrow-passage or enclosed-goal scenarios. We prove that HZ-MP is probabilistically complete and asymptotically optimal, and demonstrate empirically that it converges to high-quality trajectories within a small number of iterations.

Informed Hybrid Zonotope-based Motion Planning Algorithm

Abstract

Optimal path planning in nonconvex free spaces poses substantial computational challenges. A common approach formulates such problems as mixed-integer linear programs (MILPs); however, solving general MILPs is computationally intractable and severely limits scalability. To address these limitations, we propose HZ-MP, an informed Hybrid Zonotope-based Motion Planner, which decomposes the obstacle-free space and performs low-dimensional face sampling guided by an ellipsotope heuristic, thereby concentrating exploration on promising transition regions. This structured exploration mitigates the excessive wasted sampling that degrades existing informed planners in narrow-passage or enclosed-goal scenarios. We prove that HZ-MP is probabilistically complete and asymptotically optimal, and demonstrate empirically that it converges to high-quality trajectories within a small number of iterations.

Paper Structure

This paper contains 21 sections, 7 theorems, 22 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Notations
  4. Hybrid zonotope
  5. Informed sampling
  6. Ellipsoids and ellipsotopes
  7. Informed hybrid zonotope-based motion planning with space decomposition
  8. Problem Formulation and Overview
  9. Free space decomposition via hybrid zonotopes
  10. Adjacency computation and path identification
  11. Informed sampling and ellipsotope refinement
  12. Sampling on shared faces
  13. Ellipsotope-based reachable set refinement
  14. Path Cost Structure
  15. Practical path enumeration.
  16. ...and 6 more sections

Key Result

Theorem 2.2

With uniform sampling of the informed subset, $x \sim \text{Uniform}(X_{\text{informed}})$, the cost of the best solution, $c_{\text{best}}$, converges linearly to the theoretical minimum, $c_{\text{min}}$, in the absence of obstacles. $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Overview of HZ-MP process illustration. (a) Original environment with obstacles (black), start (green), and goal (red). (b) Hybrid zonotope representation of obstacle-free space decomposed into convex leaves. (c) Three possible connected paths identified through adjacency computation, with the brown path being the shortest. (d) Ellipsotope reachable set constructed based on the optimal path cost from (c). (e) Updated reachable set after pruning using ellipsotope-based bounds, which refines the feasible space for subsequent planning iterations.
  • Figure 2: A hybrid zonotope representation of a non-convex feasible region with an obstacle (white area).
  • Figure 3: Hybrid zonotope with four leaf nodes (1-4) and their adjacency.
  • Figure 4: Illustration of ellipsotope-informed sampling with path cost.
  • Figure 5: Motion planning scenario depicting the environment with obstacles (gray), the start state (green), the goal (red), and the solution path obtained by the proposed algorithm.
  • ...and 4 more figures

Theorems & Definitions (12)

  • Definition 2.1: Bird2023
  • Example 1
  • Theorem 2.2: gammell_informed_2014
  • Definition 2.3: Ellipsoids and Ellipsotopes kousik2023
  • Remark 3.1
  • Proposition 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Example 2
  • Lemma 3.5: Safe ellipsoidal pruning
  • ...and 2 more