## 2 Methodology
### Identification Algorithm of Each Dual Part and Scalar Part of the Dual Quaternion  
Using Equation (2) and the differences among different pictures, the following equations were obtained:  
\[\hat{\mathbf{q}}_{tr,k}^{c,1}=\hat{\mathbf{q}}_{tr,k}^{c,2} \tag{3}\]  
\[\hat{\mathbf{q}}_{robot}^{c,2*}*\hat{\mathbf{q}}_{robot}^{c,1}*\hat{\mathbf{q}}_{ca,j}^{robot }=\hat{\mathbf{q}}_{ca,j}^{robot}*\hat{\mathbf{q}}_{tr,2}^{ca}*\hat{\mathbf{q}}_{tr,1}^{ca *} \tag{4}\]  
On the other hand, the homogeneous transformation matrixes \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{X}\) were directly converted into the corresponding dual quaternion forms by using the basic equation of the hand-eye calibration, and the similar form of the basic equation was obtained as:  
\[\hat{\mathbf{A}}*\hat{\mathbf{q}}_{ca,j}^{robot}=\hat{\mathbf{q}}_{ca,j}^{robot}*\hat{\mathbf{B}} \tag{5}\]  
where \(\hat{\mathbf{A}}\) is the dual quaternion form of \(\mathbf{A}\), \(\hat{\mathbf{B}}\) is the dual quaternion form of \(\mathbf{B}\), and \(\hat{\mathbf{q}}_{ca,j}^{robot}\) is the dual quaternion form of \(\mathbf{X}\). Furthermore, there is the equation:  
\[\left(\left[\hat{\mathbf{A}}^{+}\right]-\left[\hat{\mathbf{B}}^{-}\right]\right)\hat {\mathbf{q}}_{ca,j}^{robot}=\mathbf{Q}_{AB}\hat{\mathbf{q}}_{ca,j}^{robot}=0 \tag{6}\]  
where \(\left[\hat{\mathbf{A}}^{+}\right]\) is the matrix form of \(\hat{\mathbf{A}}\) when it used left multiplication, \(\left[\hat{\mathbf{B}}^{-}\right]\) is the matrix form of \(\hat{\mathbf{B}}\) when it used right multiplication, \(\mathbf{Q}_{AB}\) is an \(8\times 8\) dimensional matrix, and \(\hat{\mathbf{q}}_{ca,j}^{robot}\) is an \(8\times 1\) dimensional matrix. The matrix \(\mathbf{Q}_{AB}\) can be written in a partitioned matrix form as:  
\[\mathbf{Q}_{AB}= \begin{bmatrix}\mathbf{Q}_{11}&\mathbf{0}_{4\times 4}\\ \mathbf{Q}_{21}&\mathbf{Q}_{22}\end{bmatrix}  \tag{7}\]  
The matrix \(\hat{\mathbf{q}}_{ca,j}^{robot}\) can also be written in a partitioned matrix form as:  
\[\hat{\mathbf{q}}_{ca,j}^{robot}= \begin{bmatrix}\mathbf{q}_{11}^{rob,ca,j}\\ \mathbf{q}_{rbb,ca,j}^{rob}\end{bmatrix}  \tag{8}\]  
The first rows of \(\mathbf{Q}_{11}\), \(\mathbf{Q}_{21}\), and \(\mathbf{Q}_{22}\) in Equation (7) do not belong to the zero vectors, and the scalar parts of the real and dual parts of \(\hat{\mathbf{q}}_{ca,j}^{robot}\) in Equation (8) do not belong to the null space either. In addition, the matrix \(\mathbf{Q}_{AB}\) is not fully ranked so that the block matrices of \(\mathbf{Q}_{AB}\) and \(\hat{\mathbf{q}}_{ca,j}^{robot}\) could be identified, respectively. The formulas that were obtained from Equations (6)-(8) are:  
\[\mathbf{Q}_{11}\mathbf{q}_{11}^{rob,ca,j}=\mathbf{0}_{4\times 1} \tag{9}\]  
\[\mathbf{Q}_{21}\mathbf{q}_{11}^{rob,ca,j}+\mathbf{Q}_{22}\mathbf{q}_{21}^{rob,ca,j}=\mathbf{0}_{4 \times 1} \tag{10}\]It was necessary and convenient to solve Equation (9) using the singular-value decomposition (SVD) method first and then to obtain the real part result of the rotation information contained in the dual quaternion \(\hat{\mathbf{q}}^{robot}_{ca,j}\). The dual part of the dual quaternion \(\hat{\mathbf{q}}^{robot}_{ca,j}\) includes the combination information of the rotation and the translation, and it is not an intuitive representation form of the position vector. Therefore, it was necessary to substitute the result of Equation (9) into Equation (10), and the dual result of \(\hat{\mathbf{q}}^{robot}_{ca,j}\) was obtained by using the SVD method so that a complete representation of the identification result was formed finally.