Dual Complex Adjoint Matrix: Applications in Dual Quaternion Research

TL;DR

This paper introduces dual complex adjoint matrices for dual quaternion matrices, enabling new solutions for eigenvalue problems and practical applications like Hand-Eye calibration in robotics.

Contribution

The paper presents the novel concept of dual complex adjoint matrices and applies it to eigenvalue analysis and calibration problems in dual quaternion research.

Findings

Validated the method for Hand-Eye calibration

Improved Rayleigh quotient iteration efficiency

Confirmed correctness through numerical experiments

Abstract

Dual quaternions and dual quaternion matrices have garnered widespread applications in robotic research, and its spectral theory has been extensively studied in recent years. This paper introduces the novel concept of the dual complex adjoint matrices for dual quaternion matrices. We delve into exploring the properties of this matrix, utilizing it to study eigenvalues of dual quaternion matrices and defining the concept of standard right eigenvalues. Notably, we leverage the properties of the dual complex adjoint matrix to devise a direct solution to the Hand-Eye calibration problem. Additionally, we apply this matrix to solve dual quaternion linear equations systems, thereby advancing the Rayleigh quotient iteration method for computing eigenvalues of dual quaternion Hermitian matrices, enhancing its computational efficiency. Numerical experiments have validated the correctness of our proposed method in solving the Hand-Eye calibration problem and demonstrated the effectiveness in improving the Rayleigh quotient iteration method, underscoring the promising potential of dual complex adjoint matrices in tackling dual quaternion-related challenges.