## 1 Introduction
### Dual Quaternion Algebra and Hand-Eye Calibration
[PERSON] presented an algorithm for calibrating the transformation parameters of robot-robot and tool-flange based on dual quaternions, simultaneously, and modified the accuracy of coordinate calibration in robot collaborative motion [35; 36]. [PERSON] used dual quaternion algebra to decompose the constraint equation of the hand-eye rotation matrix and obtained the displacement matrix by using the characteristics of screw axes as well. Then, the classical numerical solution method, which is a singular-value decomposition, was used to obtain the corresponding identification results [14]. Furthermore, the method based on dual quaternion algebra can not only express the rotation and translation changes in a rigid body, simultaneously, but can also obtain more accurate calibration results. The identification method proposed by [PERSON] uses the characteristics of screw axes and a \(6\times 1\) dimensional matrix, which represents the rigid body in three-dimensional space. However, the two elements ignored in this method contain some information on the transformation matrix of hand-eye coordination, which means all of the eight elements should be analyzed for the complete pose information of transformation. More specifically, the corresponding dual quaternion can be integrated into an \(8\times 1\) dimensional matrix, and the scalar parts of the real part and the dual part may not be equivalent to zero always. The dual quaternion contains eight elements, and all of them can be calculated by its own algebra. Furthermore, it can be seen from the kinematic model of the hand-eye robot system that the scalar part of the real part of a dual quaternion does not belong to the null space, but the real number space, and the dual part does as well. The eight elements of a dual quaternion contain all of the information on the position and orientation of the rigid body in space. Most hand-eye calibration methods based on dual quaternion ignore the scalar parts of the real part and the dual part of the dual quaternion. Therefore, research on the calibration of hand-eye robot systems using eight elements of a dual quaternion should be carried out. Furthermore, the extrinsic parameters of the camera are the main consideration during hand-eye calibration [35]. In addition, this paper mainly considers the modified solution of hand-eye calibration based on dual quaternion, which provides full information on the transformation between the robot coordinate and the camera coordinate. Thus, the improved hand-eye calibration algorithm based on dual quaternion includes full information on the pose transformation. It can include considerations of the uncertainty of the pose representation of the camera relative to the robot in the hand-eye system, and a complete form of hand-eye calibration using dual quaternion is formed. This is very useful for constituting a robot visual system as well as vision savoring control in further studies. In the rest of the paper, the full model for hand-eye calibration based on dual quaternion is presented in an explicit form. In addition, this paper is organized as follows: Section 2 briefly introduces the kinematic analysis of a dual-arm hand-eye robot based on dual quaternion and shows the derivation of the identification algorithm for each dual part and scalar part of the dual quaternion. Section 3 presents a computer simulation example of the corresponding algorithm. The hand-eye identification experiment is conducted on a dual-arm robot (the robot type is SDA5F) in Section 4. Section 5 concludes the paper.