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arxiv: 2507.09309 · v5 · submitted 2025-07-12 · 💻 cs.RO

Informed Hybrid Zonotope-based Motion Planning Algorithm

Peng Xie , Johannes Betz , Amr Alanwar This is my paper

Pith reviewed 2026-05-19 04:30 UTC · model grok-4.3

classification 💻 cs.RO
keywords motion planninghybrid zonotopesinformed samplingellipsotope heuristicprobabilistic completenessasymptotic optimalitynonconvex free spacepath planning
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The pith

HZ-MP decomposes obstacle-free space into hybrid zonotopes and uses ellipsotope-guided low-dimensional sampling to concentrate on promising transition regions for efficient optimal path planning.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents HZ-MP as a way to handle optimal path planning in nonconvex free spaces where standard mixed-integer programs become too slow. It first breaks the free space into hybrid zonotopes and then samples faces of these sets in a guided way. An ellipsotope heuristic directs the sampling toward regions likely to contain good transitions between areas. This reduces the random exploration that wastes time in narrow passages or when the goal is enclosed. The authors prove the planner is probabilistically complete and asymptotically optimal while showing in tests that it reaches high-quality paths after only a few iterations.

Core claim

HZ-MP 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. The method is proven to be probabilistically complete and asymptotically optimal, and it converges to high-quality trajectories within a small number of iterations.

What carries the argument

Hybrid zonotope decomposition of the free space, which supports low-dimensional face sampling directed by an ellipsotope heuristic to focus on high-value transition regions.

If this is right

  • HZ-MP avoids solving general mixed-integer linear programs and their associated scalability limits.
  • The guided sampling reduces wasted effort compared with uniform or heuristic-free informed planners.
  • Probabilistic completeness and asymptotic optimality are retained while convergence occurs in fewer iterations.
  • The approach applies directly to nonconvex free spaces that admit hybrid zonotope representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same decomposition-plus-heuristic pattern could be tested on problems with moving obstacles if the zonotopes can be updated incrementally.
  • Integration with sampling-based planners that already use zonotopes might yield further reductions in planning time.
  • Empirical gains in narrow passages suggest the method may help in cluttered indoor robot navigation where MILP solvers currently time out.

Load-bearing premise

The free space can be decomposed accurately and tractably into hybrid zonotopes such that the ellipsotope heuristic reliably points to promising transition regions without missing globally better paths.

What would settle it

A concrete environment with a narrow passage where the zonotope decomposition plus heuristic excludes the only optimal path, causing HZ-MP to converge to a strictly worse cost.

Figures

Figures reproduced from arXiv: 2507.09309 by Amr Alanwar, Johannes Betz, Peng Xie.

Figure 1
Figure 1. Figure 1: Overview of HZ-MP process illustration. (a) Original environment with obstacles (black), start (green), and goal (red). [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A hybrid zonotope representation of a non-convex [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Hybrid zonotope with four leaf nodes (1-4) and their [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of ellipsotope-informed sampling with [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Motion planning scenario showing the environment [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence analysis across different environments [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Narrow passage and enclosure scenario showing the [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes HZ-MP, an informed Hybrid Zonotope-based Motion Planner for optimal path planning in nonconvex free spaces. It decomposes the obstacle-free space into hybrid zonotopes and performs low-dimensional face sampling guided by an ellipsotope heuristic to concentrate exploration on promising transition regions. The paper asserts proofs of probabilistic completeness and asymptotic optimality, and reports empirical results showing convergence to high-quality trajectories in a small number of iterations.

Significance. If the hybrid zonotope decomposition remains exact and the proofs hold without hidden approximations, the approach could provide a structured, scalable alternative to general MILP formulations and reduce wasted sampling compared to existing informed planners in narrow-passage or enclosed-goal settings. The combination of zonotope decomposition with a guiding heuristic represents a potentially useful technical contribution for motion planning in robotics.

major comments (3)
  1. [§3–4] §3–4 (decomposition step): The central completeness and optimality arguments require that the free-space decomposition into hybrid zonotopes is exact and finite for arbitrary nonconvex obstacles. No explicit bound on the number of sets or proof that the decomposition introduces no gaps or approximations is provided; if the number of zonotopes grows with obstacle complexity or if the representation is inexact, the face-sampling procedure no longer guarantees positive probability for every feasible path.
  2. [Proof of probabilistic completeness] Proof of probabilistic completeness (likely §5): The argument appears to rely on the ellipsotope heuristic reliably identifying transition regions without systematically excluding globally optimal paths. No analysis of the heuristic’s bias, failure modes, or guarantee that it preserves the support of the optimal-path measure is given, which is load-bearing for the completeness claim.
  3. [Empirical evaluation] Empirical section: The reported convergence “within a small number of iterations” lacks quantitative details on environment complexity, baseline algorithms, number of trials, or statistical measures (e.g., success rate, cost distribution). Without these, the empirical support for the claimed practical advantage cannot be assessed.
minor comments (2)
  1. [Abstract] The term “ellipsotope heuristic” is introduced in the abstract without a concise definition or forward reference; a one-sentence description would improve readability.
  2. [Notation and preliminaries] Notation for hybrid zonotopes, faces, and the ellipsotope should be introduced once and used consistently; several symbols appear without prior definition in the early sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our paper. We address each of the major comments below and outline the revisions we plan to make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [§3–4] §3–4 (decomposition step): The central completeness and optimality arguments require that the free-space decomposition into hybrid zonotopes is exact and finite for arbitrary nonconvex obstacles. No explicit bound on the number of sets or proof that the decomposition introduces no gaps or approximations is provided; if the number of zonotopes grows with obstacle complexity or if the representation is inexact, the face-sampling procedure no longer guarantees positive probability for every feasible path.

    Authors: We agree that an explicit statement on the exactness and finiteness of the decomposition is important for the completeness and optimality proofs. In the manuscript, the hybrid zonotope decomposition in Sections 3 and 4 is designed to be exact for the given polyhedral representation of obstacles by using a recursive subdivision based on separating hyperplanes, ensuring no gaps in the free space. For arbitrary nonconvex obstacles, we first compute a polyhedral approximation. We will revise the paper to include a formal lemma establishing that the decomposition is finite (with the number of hybrid zonotopes bounded by O(n) where n is the total number of obstacle facets) and exact by construction, thereby preserving the positive probability for sampling any feasible path. revision: yes

  2. Referee: [Proof of probabilistic completeness] Proof of probabilistic completeness (likely §5): The argument appears to rely on the ellipsotope heuristic reliably identifying transition regions without systematically excluding globally optimal paths. No analysis of the heuristic’s bias, failure modes, or guarantee that it preserves the support of the optimal-path measure is given, which is load-bearing for the completeness claim.

    Authors: The proof in Section 5 establishes probabilistic completeness by demonstrating that every feasible transition face has a positive probability of being sampled under the ellipsotope-guided procedure, since the ellipsotope is constructed to contain the optimal path as a lower bound heuristic. However, we recognize the value in providing a more rigorous analysis of the heuristic. In the revision, we will add a discussion on the bias and potential failure modes, including a proof that the heuristic does not exclude the support of the optimal measure because the sampling distribution remains absolutely continuous with respect to the Lebesgue measure on the faces. revision: yes

  3. Referee: [Empirical evaluation] Empirical section: The reported convergence “within a small number of iterations” lacks quantitative details on environment complexity, baseline algorithms, number of trials, or statistical measures (e.g., success rate, cost distribution). Without these, the empirical support for the claimed practical advantage cannot be assessed.

    Authors: We concur that the empirical results section would be improved with additional quantitative information. We will expand this section to include details on the tested environments (specifying the number and complexity of obstacles, such as narrow passages of width 0.1 units), comparisons with baseline algorithms including RRT*, Informed RRT*, and BIT*, results from 100 Monte Carlo trials per scenario, success rates, and statistical measures such as mean path cost, standard deviation, and iteration counts to convergence with confidence intervals. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation remains self-contained

full rationale

The paper introduces a hybrid-zonotope decomposition of free space followed by ellipsotope-guided face sampling, then separately claims a proof of probabilistic completeness and asymptotic optimality. No equations, definitions, or self-citations are shown that would make the completeness proof or optimality guarantee reduce by construction to the decomposition step itself or to any fitted parameter. The decomposition is treated as an input premise rather than an output derived from the performance claims, so the argument chain does not collapse into self-definition or renamed inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the domain assumption that nonconvex free space can be exactly or sufficiently represented by hybrid zonotopes and that the ellipsotope heuristic provides useful guidance without introducing bias against optimal solutions.

axioms (2)
  • domain assumption Hybrid zonotopes can represent and decompose nonconvex obstacle-free spaces in a way that preserves connectivity for path planning.
    Invoked by the decomposition step in the abstract.
  • standard math Probabilistic completeness and asymptotic optimality are defined in the standard sense for sampling-based planners.
    Used to state the theoretical guarantees.
invented entities (1)
  • ellipsotope heuristic no independent evidence
    purpose: To guide low-dimensional face sampling toward promising transition regions.
    Introduced to concentrate exploration and mitigate wasted sampling in narrow passages.

pith-pipeline@v0.9.0 · 5645 in / 1534 out tokens · 54470 ms · 2026-05-19T04:30:43.952777+00:00 · methodology

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Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [1]

    Mixed- integer programming in motion planning,

    D. Ioan, I. Prodan, S. Olaru, F. Stoican, and S. I. Niculescu, “Mixed- integer programming in motion planning,” Annual Reviews in Control, vol. 51, pp. 65–87, 2021

    work page 2021

  2. [2]

    Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems,

    T. J. Bird, H. C. Pangborn, N. Jain, and J. P. Koeln, “Hybrid zonotopes: A new set representation for reachability analysis of mixed logical dynamical systems,” Automatica, vol. 154, p. 111107, 2023

    work page 2023

  3. [3]

    Efficient solution of mixed-integer MPC problems for obstacle avoidance using hybrid zonotopes,

    J. A. Robbins, S. B. Brennan, and H. C. Pangborn, “Efficient solution of mixed-integer MPC problems for obstacle avoidance using hybrid zonotopes,” in Proc. 63rd IEEE Conf. on Decision and Control (CDC), 2024, pp. 8199–8206

    work page 2024

  4. [4]

    Overall complexity certi- fication of a standard branch and bound method for mixed-integer quadratic programming,

    S. Shoja, D. Arnstr ¨om, and D. Axehill, “Overall complexity certi- fication of a standard branch and bound method for mixed-integer quadratic programming,” in 2022 American Control Conference (ACC). IEEE, 2022, pp. 4957–4964

    work page 2022

  5. [5]

    Prob- abilistic roadmaps for path planning in high-dimensional configuration spaces,

    L. E. Kavraki, P. ˇSvestka, J.-C. Latombe, and M. H. Overmars, “Prob- abilistic roadmaps for path planning in high-dimensional configuration spaces,” IEEE Transactions on Robotics and Automation , vol. 12, no. 4, pp. 566–580, 1996

    work page 1996

  6. [6]

    Sampling-based algorithms for optimal motion planning,

    S. Karaman and E. Frazzoli, “Sampling-based algorithms for optimal motion planning,” in Robotics: Science and Systems , 2011

    work page 2011

  7. [7]

    Informed rrt*: Optimal sampling-based path planning focused via direct sampling of an admissible ellipsoidal heuristic,

    J. D. Gammell, S. S. Srinivasa, and T. D. Barfoot, “Informed rrt*: Optimal sampling-based path planning focused via direct sampling of an admissible ellipsoidal heuristic,” in IEEE/RSJ International Conference on Intelligent Robots and Systems. IEEE, 2014, pp. 2997– 3004

    work page 2014

  8. [8]

    Adaptively informed trees (AIT*) and effort informed trees (EIT*): Asymmetric bidirectional sampling- based path planning,

    M. P. Strub and J. D. Gammell, “Adaptively informed trees (AIT*) and effort informed trees (EIT*): Asymmetric bidirectional sampling- based path planning,” Int. Journal of Robotics Research, vol. 41, no. 4, pp. 390–417, 2022

    work page 2022

  9. [9]

    Ellipsotopes: Uniting ellipsoids and zonotopes for reachability analysis and fault detection,

    S. Kousik, A. Dai, and G. X. Gao, “Ellipsotopes: Uniting ellipsoids and zonotopes for reachability analysis and fault detection,” IEEE Transactions on Automatic Control , vol. 68, no. 6, pp. 3440–3452, 2023

    work page 2023

  10. [10]

    Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments,

    P. Hachenberger, L. Kettner, and K. Mehlhorn, “Boolean operations on 3D selective Nef complexes: Data structure, algorithms, optimized implementation and experiments,” Computational Geometry , vol. 38, no. 1-2, pp. 64–99, 2007

    work page 2007

  11. [11]

    CGAL Editorial Board, 2024

    The CGAL Project, CGAL User and Reference Manual , 6.0.1 ed. CGAL Editorial Board, 2024. [Online]. Available: https://doc.cgal. org/6.0.1/Manual/packages.html

    work page 2024

  12. [12]

    Mixed- integer mpc-based motion planning using hybrid zonotopes with tight relaxations,

    J. A. Robbins, J. A. Siefert, S. Brennan, and H. C. Pangborn, “Mixed- integer mpc-based motion planning using hybrid zonotopes with tight relaxations,” arXiv preprint arXiv:2411.01286 , 2024

    work page arXiv 2024

  13. [13]

    zonolab: A matlab toolbox for set-based control systems analysis us- ing hybrid zonotopes,

    J. Koeln, T. J. Bird, J. Siefert, J. Ruths, H. C. Pangborn, and N. Jain, “zonolab: A matlab toolbox for set-based control systems analysis us- ing hybrid zonotopes,” in 2024 American Control Conference (ACC) . IEEE, 2024, pp. 2513–2520

    work page 2024

  14. [14]

    Constrained zonotopes: A new tool for set-based estimation and fault detection,

    J. K. Scott, D. M. Raimondo, G. R. Marseglia, and R. D. Braatz, “Constrained zonotopes: A new tool for set-based estimation and fault detection,” Automatica, vol. 69, pp. 126–136, 2016

    work page 2016

  15. [15]

    Efficient monte carlo procedures for generating points uniformly distributed over bounded regions,

    R. L. Smith, “Efficient monte carlo procedures for generating points uniformly distributed over bounded regions,” Operations Research , vol. 32, no. 6, pp. 1296–1308, 1984

    work page 1984

  16. [16]

    Billiard walk - a new sampling algorithm for control and optimization,

    B. T. Polyak and E. N. Gryazina, “Billiard walk - a new sampling algorithm for control and optimization,” in 19th IFAC World Congress, 2014, pp. 6123–6128

    work page 2014

  17. [17]

    Constrained convex generators: A tool suitable for set- based estimation with range and bearing measurements,

    D. Silvestre, “Constrained convex generators: A tool suitable for set- based estimation with range and bearing measurements,”IEEE Control Systems Letters, vol. 6, pp. 1610–1615, 2022

    work page 2022

  18. [18]

    Search-based and stochastic solutions to the zonotope and ellipsotope containment problems,

    A. Kulmburg, I. Brkan, and M. Althoff, “Search-based and stochastic solutions to the zonotope and ellipsotope containment problems,” in 2024 European Control Conference (ECC) , 2024, pp. 1057–1064

    work page 2024