## 1 Introduction
### Dual Quaternion Algebra and Hand-Eye Calibration  
From a certain point of view, there are two forms of hand-eye robot systems, such as the eye-in-hand systems and the eye-to-hand systems. The difference between the two is whether the relative pose of the camera changes with the movement of the end-effector. Normally, the hand-eye calibration procedure contains the steps of extracting the pose information of image data from the camera coordinate to the robot coordinate. After hand-eye calibration, the robot system can transform image information extracted from the camera into the representation of that with respect to the base coordinate so that the robot operation can be processed into the motion control instruction of the system, which is expressed in a unified coordinate [24]. Moreover, based on the classification of the camera mode, the hand-eye system can also be divided into three types, such as monocular, binocular, and the multi-ocular. From another point of view, the main work of hand-eye calibration is to find the transformation matrix between the camera coordinate and the robot coordinate. Furthermore, the equivalence relationship of the basic transformation matrix can be deduced by analyzing the robot vision system as well. [PERSON] studied dual quaternion algebra and proposed an effective regularization-patching optimization method for hand-eye calibration based on dual quaternion [25]. [PERSON] and [PERSON] studied the camera projection matrix by analyzing the constraints step by step in order to identify the internal and external parameters of the camera, and then, the corresponding transformation matrix could be obtained by using the quaternion algebra [26; 27]. [PERSON] proposed an algorithm for analyzing the hand-eye equation by using the properties of the Euclidean groups; then, the equation could be decomposed based on the special orthogonal group SO(3) and the special Euclidean group SE(3) in Lie algebra so that the transformation matrix could also be solved by using multiple groups of the measured data and the least square method [28]. Furthermore, the nonlinear calibration equation could be established by collecting the information of the feature points among multiple images. The corresponding parametric results could be obtained by the optimization algorithm or the intelligent algorithm [29]. As mentioned above, early researchers such as [PERSON] and [PERSON] have summarized the methods of hand-eye calibration to obtain the hand-eye equation which can be solved by traditional algorithms. More importantly. this can elicit more researchers to study a variety of specific ways that can also solve this problem [30; 31]. Furthermore, most of these methods need the intrinsic parameters of the target in space and its image data in different directions in the camera view. [PERSON] and [PERSON] used quaternions to represent the rotation variables and vectors to describe the displacement of an object in space, and then, the calibration results could be obtained by nominal nonlinear algorithms [28]. [PERSON] and [PERSON] used quaternions to represent the pose transformation of the rigid body motion in space so that the rotation matrix and the displacement vector could be separated. Then, the corresponding identification model could be established and used for obtaining the calibration results [32]. However, the combination concept of quaternion and dual numbers, that is called dual quaternion, has been used repeatedly in robotics recently. Furthermore, dual quaternion algebra has its own calculating characteristics, and it is not similar to the homogeneous transformation, but also combines the characteristics of screw axes. This concept has been used as a mathematical tool to deal with robotic problems and is considered to be the most efficient and compact way of representing the transformations of a rigid body in coordinates [33; 34]. This could be the main factor that the calibration problem in robotics can also be solved by using dual quaternion algebra.