A Multiscale Analysis of Multi-Agent Coverage Control Algorithms

TL;DR

This paper develops a multiscale theoretical framework for gradient descent algorithms in multi-agent coverage control, ensuring large-scale consistency and connecting with Lloyd-based methods, supported by convergence proofs and numerical experiments.

Contribution

It introduces a multiscale analysis framework for coverage algorithms, linking macroscopic probabilistic models with finite-agent implementations and convergence guarantees.

Findings

Proves convergence of the proposed gradient descent algorithms in Wasserstein space.

Establishes a connection between the framework and Lloyd-based algorithms.

Demonstrates the effectiveness of the approach through numerical experiments.

Abstract

This paper presents a theoretical framework for the design and analysis of gradient descent-based algorithms for coverage control tasks involving robot swarms. We adopt a multiscale approach to analysis and design to ensure consistency of the algorithms in the large-scale limit. First, we represent the macroscopic configuration of the swarm as a probability measure and formulate the macroscopic coverage task as the minimization of a convex objective function over probability measures. We then construct a macroscopic dynamics for swarm coverage, which takes the form of a proximal descent scheme in the $L^2$-Wasserstein space. Our analysis exploits the generalized geodesic convexity of the coverage objective function, proving convergence in the $L^2$-Wasserstein sense to the target probability measure. We then obtain a consistent gradient descent algorithm in the Euclidean space that is implementable by a finite collection of agents, via a "variational" discretization of the macroscopic coverage objective function. We establish the convergence properties of the gradient descent and its behavior in the continuous-time and large-scale limits. Furthermore, we establish a connection with well-known Lloyd-based algorithms, seen as a particular class of algorithms within our framework, and demonstrate our results via numerical experiments.