A Dual Quaternion Control Law for Formation Control of Multiple 3-D Rigid Bodies

TL;DR

This paper introduces a dual quaternion control law for multi-agent 3D rigid body formation, enabling flexible relative configurations and directed interactions, with proven convergence and verified through numerical experiments.

Contribution

It develops a novel dual-quaternion framework with directed graph representation for formation control, extending prior methods to more general configurations.

Findings

Global asymptotic convergence to desired formations

R-linear convergence rate based on spectral properties

Numerical experiments validate the control law

Abstract

This paper studies the integrated position and attitude control problem for multi-agent systems of 3D rigid bodies. While the state-of-the-art method in [Olfati-Saber and Murray, 2004] established the theoretical foundation for rigid-body formation control, it requires all agents to asymptotically converge to identical positions and attitudes, limiting its applicability in scenarios where distinct desired relative configurations must be maintained. In this paper, we develop a novel dual-quaternion-based framework that generalizes this paradigm. By introducing a unit dual quaternion directed graph (UDQDG) representation, we derive a new control law through the corresponding Laplacian matrix, enabling simultaneous position and attitude coordination while naturally accommodating directed interaction topologies. Leveraging the recent advances in UDQDG spectral theory, we prove global asymptotic convergence to desired relative configurations modulo a right-multiplicative constant and establish an R-linear convergence rate determined by the second smallest eigenvalue of the UDQDG Laplacian. A projected iteration method is proposed to compute the iterative states. Finally, the proposed solution is verified by several numerical experiments.