Solving dual quaternion matrix equation $AX-YB=C$

Abstract

In robotics research, hand-eye calibration is a challenging problem, often represented by the matrix equation $AX=YB$. In this paper, by using the vector operators of dual quaternion matrices, the properties of the real representation of dual quaternion matrices and the properties of semi-tensor product (STP) of dual quaternion matrices, we aim to propose the general solution and the Hermitian solution of the dual quaternion matrix equation $AX-YB=C$, which is the general case of $AX=YB$, where $X$ and $Y$ are unknown dual quaternion matrices. Firstly, we can vectorize the matrix equation $AX-YB=C$ and combine it with STP and the real representation of dual quaternion matrices, to transform the dual quaternion matrix $AX-YB=C$ into a real linear system, thus we can get the necessary and sufficient condition for the solvability and the general solution expression of the dual quaternion matrix equation $AX-YB=C$. Based on this, we also get the Hermitian solution of the dual quaternion matrix equation $AX-YB=C$ by simplifying the complexity of computation with $\mathbb {GH}$-representation of special dual quaternion matrix. Additionally, we propose corresponding algorithms and provide the numerical examples to verify the effectiveness of the corresponding method.