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Parametrized Topological Complexity for a Multi-Robot System with Variable Tasks

Gopal Chandra Dutta, Amit Kumar Paul, Subhankar Sau

TL;DR

This work extends sequential parametrized topological complexity to a heterogeneous multi-robot motion-planning setting with unknown obstacle positions, framing the problem through the Fadell–Neuwirth fibration on configuration spaces. It derives an explicit upper bound and leverages cup-length methods to obtain lower bounds, yielding exact results in odd dimensions and a constructive algorithm proving the even-dimensional case. The odd-dimensional case gives ${\sf TC}_{\bar{\mathbf{r}}}[p] = \sum_{i=1}^n r_i + m -1$, while the even-dimensional case yields ${\sf TC}_{\bar{\mathbf{r}}}[p] = \sum_{i=1}^n r_i + m -2$, with the latter established via a detailed motion-planning construction. The results provide precise, dimension-dependent characterizations of the minimal algorithmic instability required to plan collision-free trajectories under heterogeneous task demands and unknown obstacles, and they offer a practical, explicit planning algorithm for the even-dimensional setting.

Abstract

We study a generalized motion planning problem involving multiple autonomous robots navigating in a $d$-dimensional Euclidean space in the presence of a set of obstacles whose positions are unknown a priori. Each robot is required to visit sequentially a prescribed set of target states, with the number of targets varying between robots. This heterogeneous setting generalizes the framework considered in the prior works on sequential parametrized topological complexity by Farber and the second author of this article. To determine the topological complexity of our problem, we formulate it mathematically by constructing an appropriate fibration. Our main contribution is the determination of this invariant in the generalized setting, which captures the minimal algorithmic instability required for designing collision-free motion planning algorithms under parameter-dependent constraints. We provide a detailed analysis for both odd and even-dimensional ambient spaces, including the essential cohomological computations and explicit constructions of corresponding motion planning algorithms.

Parametrized Topological Complexity for a Multi-Robot System with Variable Tasks

TL;DR

This work extends sequential parametrized topological complexity to a heterogeneous multi-robot motion-planning setting with unknown obstacle positions, framing the problem through the Fadell–Neuwirth fibration on configuration spaces. It derives an explicit upper bound and leverages cup-length methods to obtain lower bounds, yielding exact results in odd dimensions and a constructive algorithm proving the even-dimensional case. The odd-dimensional case gives , while the even-dimensional case yields , with the latter established via a detailed motion-planning construction. The results provide precise, dimension-dependent characterizations of the minimal algorithmic instability required to plan collision-free trajectories under heterogeneous task demands and unknown obstacles, and they offer a practical, explicit planning algorithm for the even-dimensional setting.

Abstract

We study a generalized motion planning problem involving multiple autonomous robots navigating in a -dimensional Euclidean space in the presence of a set of obstacles whose positions are unknown a priori. Each robot is required to visit sequentially a prescribed set of target states, with the number of targets varying between robots. This heterogeneous setting generalizes the framework considered in the prior works on sequential parametrized topological complexity by Farber and the second author of this article. To determine the topological complexity of our problem, we formulate it mathematically by constructing an appropriate fibration. Our main contribution is the determination of this invariant in the generalized setting, which captures the minimal algorithmic instability required for designing collision-free motion planning algorithms under parameter-dependent constraints. We provide a detailed analysis for both odd and even-dimensional ambient spaces, including the essential cohomological computations and explicit constructions of corresponding motion planning algorithms.

Paper Structure

This paper contains 16 sections, 15 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A motivating problem
  4. Motion planning of multiple robots for variable target states
  5. Cohomology algebra of $H^{*}(E^{\bar{\mathbf{r}}}_B)$
  6. Topological complexity in odd-dimensional Euclidean space
  7. Topological complexity in even-dimensional Euclidean space
  8. An algorithm
  9. Partition of :
  10. Decomposition of the set :
  11. Algorithm over :
  12. Algorithm over :
  13. Appendix
  14. Appendix I-1
  15. Appendix I-2
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $p: E \to B$ be a fibration, and $R$ be a coefficient ring. If there exist cohomology classes $u_1, \cdots, u_k\in \ker[p^*: H^*(B; R)\to H^*(E; R)]$ such that their cup-product is nonzero, $u_1 \smile \cdots \smile u_k \neq 0 \in H^*(B; R)$, then ${\sf {secat}}(p) \geq k$.

Theorems & Definitions (29)

  • Lemma 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 4.1
  • ...and 19 more