Fekete points, formation control, and the balancing problem

TL;DR

This paper introduces a formation control method that maximizes pairwise distances and stabilizes systems on a submanifold, combining distributed and decentralized strategies for stable formation shaping.

Contribution

It proposes a novel control algorithm for formation shaping that is distributed, decentralized, and applicable to systems on the Euclidean group, with theoretical and practical insights.

Findings

Stable convergence to desired formations demonstrated.

Graph-theoretical interpretation of equilibria provided.

Extension to systems on the special Euclidean group achieved.

Abstract

We study formation control problems. Our approach is to let a group of systems maximize their pairwise distances whilst bringing them all to a given submanifold, determining the shape of the formation. The algorithm we propose allows to initialize the positions of the individual systems in the ambient space of the given submanifold but brings them to the desired formation asymptotically in a stable fashion. Our control inherently consists of a distributed component, maximizing the pairwise distances, and a decentralized component, asymptotically stabilizing the submanifold. We establish a graph-theoretical interpretation of the equilibria that our control enforces and extend our approach to systems living on the special Euclidean group. Throughout the paper, we illustrate our approach on different examples.