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Efficient Path Planning with Soft Homology Constraints

Carlos A. Taveras, Santiago Segarra, César A. Uribe

TL;DR

The paper tackles efficient path planning under soft homology constraints on disk-with-holes surfaces by leveraging discrete Hodge theory. It introduces the H$^*$ algorithm, which uses a harmonic-projection based heuristic $\gamma(\tau)$ and a cost $C^\alpha(\tau,\bar{\tau})=W(\tau)+\alpha\Delta\gamma(\tau,\bar{\tau})$ to approximate shortest paths within homology classes similar to a reference path $\bar{\tau}$; Fortified Rollout variants (RH$^*$ and PRH$^*$) further improve results. The method avoids exhaustively enumerating all homology classes, achieving substantial reductions in node visits while providing interpretable control via the parameter $\alpha$ that trades off path length and topological similarity. Numerical experiments on synthetic SCs show that H$^*$ often matches or closely approximates optimal homologous paths with far fewer expansions, and scales better as the number of holes increases, compared with prior methods like BLK. The work thus offers a practical, topology-aware planning tool for real-time robotics in cluttered environments where homology plays a crucial role.

Abstract

We study the problem of path planning with soft homology constraints on a surface topologically equivalent to a disk with punctures. Specifically, we propose an algorithm, named $\Hstar$, for the efficient computation of a path homologous to a user-provided reference path. We show that the algorithm can generate a suite of paths in distinct homology classes, from the overall shortest path to the shortest path homologous to the reference path, ordered both by path length and similarity to the reference path. Rollout is shown to improve the results produced by the algorithm. Experiments demonstrate that $\Hstar$ can be an efficient alternative to optimal methods, especially for configuration spaces with many obstacles.

Efficient Path Planning with Soft Homology Constraints

TL;DR

The paper tackles efficient path planning under soft homology constraints on disk-with-holes surfaces by leveraging discrete Hodge theory. It introduces the H algorithm, which uses a harmonic-projection based heuristic and a cost to approximate shortest paths within homology classes similar to a reference path ; Fortified Rollout variants (RH and PRH) further improve results. The method avoids exhaustively enumerating all homology classes, achieving substantial reductions in node visits while providing interpretable control via the parameter that trades off path length and topological similarity. Numerical experiments on synthetic SCs show that H often matches or closely approximates optimal homologous paths with far fewer expansions, and scales better as the number of holes increases, compared with prior methods like BLK. The work thus offers a practical, topology-aware planning tool for real-time robotics in cluttered environments where homology plays a crucial role.

Abstract

We study the problem of path planning with soft homology constraints on a surface topologically equivalent to a disk with punctures. Specifically, we propose an algorithm, named , for the efficient computation of a path homologous to a user-provided reference path. We show that the algorithm can generate a suite of paths in distinct homology classes, from the overall shortest path to the shortest path homologous to the reference path, ordered both by path length and similarity to the reference path. Rollout is shown to improve the results produced by the algorithm. Experiments demonstrate that can be an efficient alternative to optimal methods, especially for configuration spaces with many obstacles.
Paper Structure (14 sections, 2 theorems, 15 equations, 4 figures)

This paper contains 14 sections, 2 theorems, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Simplicial Complexes
  4. Paths
  5. Hodge Theory, Homology and Harmonics
  6. Problem Statement
  7. Methods
  8. The $\mathcal{H}^*$ Algorithm
  9. Fortified Rollout
  10. Numerical Analysis
  11. Illustrative Example
  12. Characterizing the Effect of
  13. Characterizing the Effect of the Number of Holes
  14. Conclusion

Key Result

Proposition 1

Two paths $\tau_1$ and $\tau_2$ connecting the same points are homologous if and only if $\gamma(\tau_1) = \gamma(\tau_2)$.

Figures (4)

  • Figure 1: (a) Simplicial Complex with two holes, (b) four distinct homotopy/homology classes with three paths each, (c) harmonic projection basis vectors, and (d) the progression of the harmonic projection for each path in (b).
  • Figure 2: (a) Paths produced by $\mathcal{H}^*$ for various $\alpha$ and (b) the projection difference of all visited nodes after a path has been found.
  • Figure 3: (a) Path length, (b) projection difference, and (c) number of nodes visited for $\mathcal{H}^*, \mathcal{RH}^*, \text{ and } \mathcal{PRH}^*$ as a function of $\alpha$. The color-coded (and labeled) horizontal lines each correspond to a distinct homology class in (a) denoting the classes' shortest path length and (b) the projection difference. The horizontal line in (c) shows the number of nodes visited by BLK.
  • Figure 4: (a) The length of paths produced by algorithms for each SC, (b) number of nodes visited to produce the paths, and (c) example paths.

Theorems & Definitions (5)

  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof