## 1 Introduction  
When we want to use the manipulator combined with vision to grasp, we usually obtain the pose information of the object in space through the camera. However, the pose information is based on the camera coordinate system and cannot be used directly. If we want the robot end effector to reach the target position, we need to know the coordinate information of the target position in the robot coordinate system. Robot hand-eye calibration is the key technology to solve this problem. It refers to the establishment of the relationship between the manipulator coordinate system and the image coordinate system so that the manipulator can accurately reach the specified position under the guidance of the camera. At present, many experts and scholars have proposed many hand-eye calibration methods, which are mainly divided into the linear calibration method and the iterative calibration method. The iterative method is mainly used to improve accuracy and robustness. The linear method is an efficient on-line hand-eye calibration method [1]. [PERSON] and [PERSON] [2] first introduced the dynamic equation AX = XB into hand-eye calibration and provided the minimum configuration for the unique solution. [PERSON] and [PERSON] [3] proposed another highly linear method. Firstly, the translation of hand-eye transformation is estimated, and then its rotation is estimated. This method has been widely used in many systems. The traditional hand-eye calibration method is mainly based on [4,5] the two-step method. Firstly, the rotation matrix of hand-eye parameters is calculated, and then the translation vector is solved by using the rotation matrix, and the parameters of hand-eye matrix are obtained by combining the rotation and translation parameters. This method is simple and easy to understand, but its inherent defect is error transmission [6]. [PERSON] et al. [7] summarized the hand-eye calibration method into four steps: camera pose, mechanical claw pose, mathematical model, and error measurement. [PERSON] et al. [8] proposed a new analytical solution based on cost function to estimate the hand-eye matrix with measurement error. In order to obtain the nonsingular analytical solution based on the modified Rodrigues parameters, a new additional rotation theory must be introduced. [PERSON] et al. [9] solved the rotation relationship by twice transforming the motion of the target carried by the robot and then solved the transformation relationship through multiple rotating motions in the robot tool coordinate system. To solve this problem, some scholars have proposed a synchronous calibration method of rotation and translation parameters. [PERSON] et al. [10] proposed [10] a synchronous optimization method for the calibration and measurement of a typical hand-eye positioning system. By establishing the measurement calibration model under nonlinear constraints, an iterative method and closed solution method were proposed to effectively suppress the influence of calibration error on measurement accuracy. [PERSON] et al. [11] proposed a calculation method of nonlinear robot vision guidance calibration parameters based on maximum likelihood estimation on the basis of analyzing the transformation relationship of coordinate systems in robot vision guidance. This method constructs the matrix transformation measure function in the calculation process according to the maximum likelihood estimation and uses the Levenberg Marquardt algorithm to solve the nonlinear least squares problem. The camera calibration parameters and hand-eye calibration parameters can be solved at one time. In addition, for the establishment of the model, many scholars have discussed the modeling and solution methods of hand-eye calibration. Common modeling methods include quaternion, dual quaternion, and Denavit-Hartenberg (D-H). [PERSON] et al. [12] proposed a collaborative motion coordinate calibration method based on dual quaternion, which effectively solved the problem of synchronous calibration.