## 2 Methodology
### Kinematic Analysis of the Dual-Arm Hand-Eye Robot Based on Dual Quaternion
\(\hat{\mathbf{q}}_{\alpha-1}^{\alpha-2}\) is the dual quaternion representation of the coordinate \(\{ca\_1\}\) of camera 1 with respect to the coordinate \(\{ca\_2\}\) of camera 2, \(\hat{q}^{ca,1}_{robot}\) is the dual quaternion representation of the robot coordinate \(\{robot\}\) with respect to the coordinate \(\{ca\_1\}\) of camera 1, \(\hat{q}^{robot}_{ca\_2}\) is the dual quaternion representation of the coordinate \(\{ca\_2\}\) of camera 2 with respect to the robot coordinate \(\{robot\}\), \(\hat{q}^{c-1}_{robot}\) is the dual quaternion representation of the robot coordinate \(\{robot\}\) with respect to the end-effector coordinate \(\{e\_1\}\) of arm 1, \(\hat{q}^{c-2}_{robot}\) is the dual quaternion representation of the robot coordinate \(\{robot\}\) with respect to the end-effector coordinate \(\{e\_2\}\) of arm 2, \(\hat{q}^{c-1}_{tr-1}\) is the dual quaternion representation of the calibration plate \(\{tr\_1\}\) of arm 1 with respect to the end-effector coordinate \(\{e\_1\}\) of arm 1, and \(\hat{q}^{c-2}_{tr-2}\) is the dual quaternion representation of the calibration plate \(\{tr\_2\}\) of arm 2 with respect to the end-effector coordinate \(\{e\_2\}\) of arm 2. These coordinates construct the hand-eye robot coordinate system. There are two cameras in Figure 1 in order to show the example of the relative transformation between the robot and different cameras, which capture pictures used by the dual-arm robot system. Two or more cameras can offer the robot system a greater range of visual perception, which can extend the functionality of the dual-arm robot in further usages. The procedure of hand-eye calibration not only requires the optical information, which is determined by the pictures, but also obtains the external parameters of the camera. Furthermore, it was necessary to determine the transformation model of the output from the nonlinear model, which was formed by the pixel coordinate of the picture in the two-dimensional sensor space and the world coordinate in the three-dimensional scene state. The pictures of the calibration board contain the information of that with respect to the system coordinates as the robot moved to different poses so that the constraints could be constructed from the data of multiple two-dimensional pictures, which were captured by the cameras, and the external parameters of the cameras could be obtained. As Figure 1 is a dual-arm hand-eye robot system, it was necessary to perform a hand-eye calibration procedure on each of the two cameras and to construct the relative transformation between the two cameras. The equation of the two camera coordinates could be formed as:  
\[\hat{q}^{ca-1}_{ca\_2}=\hat{q}^{ca,1}_{robot}*\hat{q}^{robot}_{ca\_2} \tag{1}\]  
Nevertheless, the hand-eye calibration of one end-effector and one camera should be carried out first. In general, one arm in the example shown in Figure 1 is chosen for further derivation. The relative pose constraint between the end-effector coordinate of the one arm and the other coordinate should mainly be considered. Two-dimensional hand-eye calibration boards with known geometric parameters were fixed to the end-effectors of the robot in Figure 1, and the cameras captured the different poses of the target as the robot's arm pose changed. Figure 1: Example setup of the coordinate systems in the dual-arm hand–eye calibration.