## 2 Methodology  
### Kinematic Analysis of the Dual-Arm Hand-Eye Robot Based on Dual Quaternion  
Generally speaking, there are two ways to set up the vision system of the hand-eye robot. More importantly, different hand-eye systems can be selected according to different operation tasks and control objectives. The first one is the eye-to-hand system, in which the camera is fixed near the workspace of the robot. In this kind of system, the relative pose between the camera and the end-effector of the robot is not fixed, so that the system can observe local image details. The second one is the eye-in-hand system, in which the camera is also fixed on the robot body. In this kind of system, the camera can follow the movement of the end-effector of the robot in the world coordinate. Even though different installation ways have different features, all of them have to determine the pose constraints among the world coordinate, the robot coordinate, the end-effector coordinate, the camera coordinate, and the picture coordinate, etc., so that the vision system can describe and express the target information using pictures in the three-dimensional space accurately. Generally, the pictures captured by commercial cameras are corrected or compensated according to the internal parameters of the camera itself provided by the manufacturers. However, usually it is necessary to identify the external parameters of the camera in order to construct the transformation model between the corresponding coordinates in the hand-eye system. This paper mainly considers the identification of the external parameters of the camera, which is important in the hand-eye calibration procedure. The dual-arm robot contains two normal manipulators, and both of them share some common workspace, while the system has more flexible operation ability. The dual-arm robot can also use the vision system to acquire a picture of the external environment in order to expand the system function. For example, the vision system can acquire data on the moving object in order to derive the relative pose of that expressed in the robot coordinate for further control commands. As mentioned above, many papers have proposed effective methods to calibrate the constraint model of hand-eye systems so that the premise of the robot vision ability can be obtained for further development. In addition, among the classical hand-eye calibration solutions up to now, [PERSON] and [PERSON] have summarized the methods and the procedure of that in order to obtain the equation AX = XB and also have studied the corresponding solutions earlier. In addition, [PERSON] presented an online identification algorithm for the extrinsic parameters of the camera based on dual quaternion and solved the corresponding numerical equation by using the globally fast optimization method [34]. [PERSON] presented a solution for the trocar limitation which may cause ill-conditioned constraints during the procedure of hand-eye calibration based on dual quaternion and solved the problem by using the Levenberg-Marquardt algorithm [7]. These subtle methods use the operational derivation of dual quaternion algebra but are also not the same as that presented by [PERSON] [14]. The main purpose of this section is to deduce the procedure of identifying the external parameters of the hand-eye equation by using the eight elements of the dual quaternion. Assuming that, first, in the example of eye-to-hand systems, the camera is mounted near the base of the dual-arm robot, so that it mimics the human view, the relative position and orientation of the object or the end-effectors of the robot can be obtained from the image data. For the general analysis, Figure 1 shows the example setup of the coordinate systems for a dual-arm robot equipped with two cameras.