Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Time-Varying Coverage Control via Singular Perturbations

Brandon Bao, Jorge Cortes, Sonia Martinez

TL;DR

The paper tackles dynamic coverage control for multi-robot systems under time-varying density, proposing a singular perturbation framework that decouples fast control updates from slow motion. It develops a two-time-scale, 2-hop distributed algorithm (TVD-SP_ε) that asymptotically tracks a centroidal Voronoi configuration, and provides discrete-time 1-hop variants with delayed information, along with rigorous convergence results. Convergence is established via Tikhonov’s theorem and generating-function analysis under appropriate invertibility and spectral conditions. Simulations and hardware experiments demonstrate that the proposed methods achieve near-centralized performance with limited communication and improved convergence when fresher information is available.

Abstract

This paper presents a novel dynamic coverage control algorithm allowing a group of robots to track an optimal-deployment configuration for arbitrary time-varying density functions. Building on singular perturbation theory, the proposed design employs a two-time scale separation approach, with a fast time scale corresponding to communication and a slow time scale corresponding to agent motion. The resulting algorithm is distributed over the 2-hop Delaunay graph and, for small enough values of the perturbation parameter, achieves the same performance as its centralized counterpart. We also introduce three discrete-time versions that rely only on 1-hop communication at the cost of having to use delayed information and formally establish their asymptotic convergence properties. Our technical approach combines computational geometry, singular perturbation theory, generating functions, and linear iterations with delayed updates. Various simulations illustrate the performance of the proposed algorithms.

Distributed Time-Varying Coverage Control via Singular Perturbations

TL;DR

The paper tackles dynamic coverage control for multi-robot systems under time-varying density, proposing a singular perturbation framework that decouples fast control updates from slow motion. It develops a two-time-scale, 2-hop distributed algorithm (TVD-SP_ε) that asymptotically tracks a centroidal Voronoi configuration, and provides discrete-time 1-hop variants with delayed information, along with rigorous convergence results. Convergence is established via Tikhonov’s theorem and generating-function analysis under appropriate invertibility and spectral conditions. Simulations and hardware experiments demonstrate that the proposed methods achieve near-centralized performance with limited communication and improved convergence when fresher information is available.

Abstract

This paper presents a novel dynamic coverage control algorithm allowing a group of robots to track an optimal-deployment configuration for arbitrary time-varying density functions. Building on singular perturbation theory, the proposed design employs a two-time scale separation approach, with a fast time scale corresponding to communication and a slow time scale corresponding to agent motion. The resulting algorithm is distributed over the 2-hop Delaunay graph and, for small enough values of the perturbation parameter, achieves the same performance as its centralized counterpart. We also introduce three discrete-time versions that rely only on 1-hop communication at the cost of having to use delayed information and formally establish their asymptotic convergence properties. Our technical approach combines computational geometry, singular perturbation theory, generating functions, and linear iterations with delayed updates. Various simulations illustrate the performance of the proposed algorithms.

Paper Structure

This paper contains 8 sections, 8 theorems, 52 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem Statement
  4. Time-varying Coverage Control via Singular Perturbations
  5. 1-Hop Distributed Discrete-Time Implementations with Delayed Information
  6. 2-Hop Distributed Discretization of TVD-SP$_{\epsilon}$
  7. 1-Hop Distributed Implementation with Delayed Updates
  8. Conclusions

Key Result

Theorem II.1

Assume that $u= h(p,t)$ is an isolated root of the equation eq: reduced model:b, and that the functions $f$, $g$, and $h$ are Lipschitz, and $g$ is bounded. Assume that the origin is a uniformly, asymptotically stable equilibrium of the dynamics eq:reduced-model-with-root, and that the boundary laye

Figures (2)

  • Figure 1: Evolution of the coverage control metric \ref{['eq:\n location cost']} with $n=10$ agents and dynamic density function $\phi_4$ under the (a) all-neighbors-delayed update \ref{['eq:homogeneous']}, with total cost 3687.5 (b) $2\setminus \! 1$-neighbors-delayed update \ref{['eq:discrete-u2']}, with total cost 2876.11, and (c) $2$-neighbors-delayed update \ref{['eq:discrete-u3']}, with total cost 3175.49.
  • Figure 2: Hardware experiments with $n=8$ differential drive robots performing time-varying coverage control in the Robotarium, a remotely accessible swarm robotics testbed SW-PG:20. The robots execute the TVD-SP$_{\epsilon}$ coordination algorithm with the $2\setminus \! 1$-neighbors-delayed update \ref{['eq:discrete-u2']}. (a) initialization, (b)-(d) snapshots during the execution.

Theorems & Definitions (15)

  • Theorem II.1: Tikhonov’s theorem VV:97
  • Theorem IV.1
  • proof
  • Example IV.2
  • Lemma V.1
  • proof
  • Theorem V.2
  • Theorem V.3
  • proof
  • Lemma V.4
  • ...and 5 more