## 2 Methodology
### Identification Algorithm of Each Dual Part and Scalar Part of the Dual Quaternion
In order to improve the accuracy and the precision of the calibration results, it was necessary to acquire data of multiple pictures in order to form a combined identification algorithm during the specific implementation procedure of the hand-eye calibration so that there would be equations expressed as follows:  
\[\mathbf{T}_{11}=\left(\mathbf{G}_{1}^{T}\;\mathbf{G}_{2}^{T}\;\cdots\;\mathbf{G}_{n}^{T}\right) ^{T} \tag{11}\]  
\[\mathbf{T}_{21}=\left(\mathbf{H}_{1}^{T}\;\mathbf{H}_{2}^{T}\;\cdots\;\mathbf{H}_{n}^{T}\right) ^{T} \tag{12}\]  
\[\mathbf{G}_{n}^{T}=\left(\mathbf{Q}_{11}\mathbf{q}_{11}^{rob,ca}\right)_{k}^{T} \tag{13}\]  
\[\mathbf{H}_{n}^{T}=\left(\mathbf{Q}_{21}\mathbf{q}_{11}^{rob,ca}+\mathbf{Q}_{22}\mathbf{q}_{21}^{rob,ca}\right)_{k}^{T} \tag{14}\]  
where \(n\) is the number of the measurement, and \(k\) is the serial number of the \(k\)-th picture. Using the SVD method to identify Equations (11) and (12), successively, can obtain more reasonable results than that identified by using Equations (9) and (10). Furthermore, based on the dual quaternion \(\hat{\mathbf{q}}^{robot}_{ca,j}\), which is the camera coordinate \(\{ca\}\) with respect to the robot coordinate \(\{robot\}\), the dual quaternion \(\hat{\mathbf{q}}^{robot}_{ca,j}\) contains the same transformation information as the homogeneous transformation matrix of the actual robot system. Otherwise, the rotation angle and the translation value contained in the dual quaternion can represent the intuitive numerical meaning in the corresponding coordinate. From the above analysis and derivation, the steps of the modified hand-eye calibration algorithm based on dual quaternion algebra can be concluded as follows: First, fix the two-dimensional calibration board at the end-effector of one arm, and use the camera to capture different poses of the board. Second, compute the real part of \(\hat{\mathbf{q}}^{robot}_{ca,j}\) by using the SVD method to solve the combination of Equations (6) and (11). Third, compute the dual part of \(\hat{\mathbf{q}}^{robot}_{ca,j}\) by using Equations (6) and (12) and the result of the second step. Therefore, complete the overall result of \(\hat{\mathbf{q}}^{robot}_{ca,j}\). After following these steps of the procedure, the relative transformation between the two cameras can be obtained by using Equation (1). Furthermore, the dual-arm hand-eye robot system discussed in Figure 1 has two arms, and the relative position and orientation between the two end-effectors and the two cameras should be obtained in order to complete the procedure so that the kinematic equation of these two end-effectors and the corresponding two cameras can be written as:  
\[\hat{\mathbf{q}}^{c,2}_{ca,1}=\hat{\mathbf{q}}^{c,2}_{robot}*\hat{\mathbf{q}}^{robot}_{ca,1} \tag{15}\]  
\[\hat{\mathbf{q}}^{c,1}_{ca,2}=\hat{\mathbf{q}}^{c,1}_{robot}*\hat{\mathbf{q}}^{robot}_{ca,2} \tag{16}\]  
From the discussion above, the modified hand-eye calibration for the dual-arm robot system based on dual quaternion can be summarized in Algorithm 1. In addition, an abbreviation is used to express the corresponding algorithm as MDQ, which means the modified hand-eye calibration using dual quaternion. ```
1:Input: The pictures captured by cameras; the poses data of end-effectors; the corresponding dual quaternions \(\hat{A},\hat{B}\). 2:1: Compute the real part and the dual part of \(\hat{\mathbf{q}}^{robot}_{ca,j}\) based on Equations (6), (11), and (12). 3: initial \(\hat{\mathbf{q}}^{robot^{(i)}}_{ca,j}\leftarrow\) the first solution in step 1; \(n\gets 1\)
4:while\(\left\|\left(\left[\hat{\mathbf{\lambda}}^{+}\right]-\left[\hat{\mathbf{B}}^{-}\right] \right)\hat{\mathbf{q}}^{robot^{(i)}}_{ca,j}\right\|>\varepsilon\),do  Form solution based on Equations (11) and (12); \(n\gets n+1\);  Form kinematic equation of robot system based on Equations (15) and (16); endwhile Output:\(\hat{\mathbf{q}}^{robot}_{ca,j}\leftarrow\hat{\mathbf{q}}^{robot^{(i)}}_{ca,j}\)
```  
**Algorithm 1**Modified hand-eye calibration for the dual-arm robot system using dual quaternion