Coverage and Field Estimation on Bounded Domains by Diffusive Swarms

TL;DR

This paper presents control strategies for robot swarms to efficiently cover bounded areas and estimate scalar fields using diffusive motion, with proven convergence and field reconstruction capabilities.

Contribution

It introduces diffusion, advection, and reaction control methods for swarm coverage, with a focus on diffusion-based control ensuring convergence to the scalar field.

Findings

Diffusion coefficient depends only on local measurements for convergence.

Swarm density converges exponentially to a function proportional to the scalar field.

Field reconstruction is possible from robots' observations over a small domain subset.

Abstract

In this paper, we consider stochastic coverage of bounded domains by a diffusing swarm of robots that take local measurements of an underlying scalar field. We introduce three control methodologies with diffusion, advection, and reaction as independent control inputs. We analyze the diffusion-based control strategy using standard operator semigroup-theoretic arguments. We show that the diffusion coefficient can be chosen to be dependent only on the robots' local measurements to ensure that the swarm density converges to a function proportional to the scalar field. The boundedness of the domain precludes the need to impose assumptions on decaying properties of the scalar field at infinity. Moreover, exponential convergence of the swarm density to the equilibrium follows from properties of the spectrum of the semigroup generator. In addition, we use the proposed coverage method to construct a time-inhomogenous diffusion process and apply the observability of the heat equation to reconstruct the scalar field over the entire domain from observations of the robots' random motion over a small subset of the domain. We verify our results through simulations of the coverage scenario on a 2D domain and the field estimation scenario on a 1D domain.