An Orthogonal Basis Approach to Formation Shape Control (Extended Version)

TL;DR

This paper introduces an orthogonal basis method to enhance 3D formation control, ensuring unique tetrahedral formations with global stability without restrictive conditions.

Contribution

It presents a novel orthogonal basis approach that guarantees unambiguous 3D formations with decentralized control and stability guarantees, surpassing prior limitations.

Findings

Ensures global asymptotic stability of formations.

Provides a decentralized control method for 3D formations.

Achieves stability without restrictions on tetrahedron configurations.

Abstract

In this paper, we propose a novel approach to the problem of augmenting distance-based formation controllers with a secondary constraint for the purpose of preventing 3D formation ambiguities. Specifically, we introduce three controlled variables that form an orthogonal space and uniquely characterize a tetrahedron formation in 3D. This orthogonal space incorporates constraints on the inter-agent distances and the signed volume of tetrahedron substructures. The formation is modeled using a directed graph with a leader-follower type configuration and single-integrator dynamics. We show that the proposed decentralized formation controller ensures the \textit{global} asymptotic stability and the local exponential stability of the desired formation for an \textit{n}-agent system with no ambiguities. Unlike previous work, this result is achieved without conditions on the tetrahedrons that form the desired formation or on the control gains.