Communications in Mathematical Sciences

Volume 21 (2023)

Number 7

Dual quaternion matrices in multi-agent formation control

Pages: 1865 – 1874



Liqun Qi (Department of Mathematics, School of Science, Hangzhou Dianzi University, Hangzhou, China)

Xiangke Wang (College of Mechatronics and Automation, National University of Defence Technology, Changsha, China)

Ziyan Luo (School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, China)


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 both 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.


unit dual quaternions, formation control, dual quaternion matrices, dual quaternion Hermitian matrices, eigenvalues

2010 Mathematics Subject Classification

15A18, 15A66, 70E60, 93C85

The work of Z. Luo was supported by the National Natural Science Foundation of China (Grant No. 12271022).

Received 11 April 2022

Received revised 20 December 2022

Accepted 20 January 2023

Published 9 October 2023