Dynamic controllers for column synchronization of rotation matrices: a QR-factorization approach

TL;DR

This paper presents a QR-factorization based method for distributed synchronization of specific columns of rotation matrices in multi-agent systems, achieving almost global convergence.

Contribution

It introduces a novel QR-factorization approach to decouple column dynamics, enabling effective distributed control for rotation matrix synchronization.

Findings

Achieves almost global convergence under certain graph topologies.

Decouples column dynamics from remaining matrix parts.

Provides a scalable control law for multi-agent rotation synchronization.

Abstract

In the multi-agent systems setting, this paper addresses continuous-time distributed synchronization of columns of rotation matrices. More precisely, k specific columns shall be synchronized and only the corresponding k columns of the relative rotations between the agents are assumed to be available for the control design. When one specific column is considered, the problem is equivalent to synchronization on the (d-1)-dimensional unit sphere and when all the columns are considered, the problem is equivalent to synchronization on SO(d). We design dynamic control laws for these synchronization problems. The control laws are based on the introduction of auxiliary variables in combination with a QR-factorization approach. The benefit of this QR-factorization approach is that we can decouple the dynamics for the k columns from the remaining d-k ones. Under the control scheme, the closed loop system achieves almost global convergence to synchronization for quasi-strong interaction graph topologies.