A geometric view of formation control with application to directed sensing

TL;DR

This paper introduces a geometric framework for formation control using Riemannian gradient descent, providing new controllers for directed sensing graphs and practical convergence tests.

Contribution

It presents a unified geometric approach to formation control, including directed variants and a simple local convergence test applicable to any directed graph.

Findings

Proposes a geometric formation control method based on Riemannian gradient descent.

Introduces a directed controller with a practical local convergence test.

Shows that persistence is not necessary or sufficient for convergence in directed graphs.

Abstract

We propose a geometric approach to distance-based formation control modeled on a minimum-norm lifting of Riemannian gradient descent in edge-space to node-space. This yields a unified family of controllers, including the classical gradient controller and its directed variant. For the directed case, we give a simple numerical test for local convergence that applies to any directed graph and target. We show that persistence is neither necessary nor sufficient for local convergence of our directed controller and propose an alternative that is necessary and more easily checked.