Dual Quaternion Matrices in Multi-Agent Formation Control

TL;DR

This paper explores the properties of dual quaternion matrices related to multi-agent formation control, introducing new spectral analysis tools and discussing their applications in controlling agent formations.

Contribution

It introduces dual quaternion Laplacian matrices and proves a Gershgorin-type theorem for dual quaternion Hermitian matrices, advancing spectral analysis in formation control.

Findings

Relative configuration and logarithm adjacency matrices are Hermitian.

Dual quaternion Laplacian matrices are introduced and analyzed.

A Gershgorin-type theorem for dual quaternion Hermitian matrices is proved.

Abstract

Three kinds of dual quaternion matrices associated with the mutual visibility graph, namely the relative configuration adjacency matrix, the logarithm adjacency matrix and the relative twist adjacency matrix, play important roles in multi-agent formation control. In this paper, we study their properties and applications. We show that the relative configuration adjacency matrix and the logarithm adjacency matrix are all Hermitian matrices, and thus have very nice spectral properties. We introduce dual quaternion Laplacian matrices, and prove a Gershgorin-type theorem for square dual quaternion Hermitian matrices, for studying properties of dual quaternion Laplacian matrices. The role of the dual quaternion Laplacian matrices in formation control is discussed.