Spectral Control of Mobile Robot Networks

TL;DR

This paper introduces a control method for mobile robot networks that adjusts their structure to achieve specific spectral properties, enhancing distributed coordination capabilities.

Contribution

A novel spectral control scheme using spectral moments and gradient descent to shape the eigenvalue spectrum of robot network adjacency matrices.

Findings

Successfully controls network eigenvalues via spectral moments.

Ensures convergence of network topology to desired spectral properties.

Validated through computer simulations.

Abstract

The eigenvalue spectrum of the adjacency matrix of a network is closely related to the behavior of many dynamical processes run over the network. In the field of robotics, this spectrum has important implications in many problems that require some form of distributed coordination within a team of robots. In this paper, we propose a continuous-time control scheme that modifies the structure of a position-dependent network of mobile robots so that it achieves a desired set of adjacency eigenvalues. For this, we employ a novel abstraction of the eigenvalue spectrum by means of the adjacency matrix spectral moments. Since the eigenvalue spectrum is uniquely determined by its spectral moments, this abstraction provides a way to indirectly control the eigenvalues of the network. Our construction is based on artificial potentials that capture the distance of the network's spectral moments to their desired values. Minimization of these potentials is via a gradient descent closed-loop system that, under certain convexity assumptions, ensures convergence of the network topology to one with the desired set of moments and, therefore, eigenvalues. We illustrate our approach in nontrivial computer simulations.