## 1 Introduction
[PERSON] and [PERSON] put forward the hand-eye calibration algorithm of octet [13] on the basis of dual quaternion and realized the synchronous optimal solution of rotation and translation matrix through the model of AX = XB. [PERSON] et al. [14] proposed a new model based on a pair of dual equations. When the homogeneous matrix is decoupled or not, the simultaneous solutions and separable solutions of the dual equations are given. Simulation and experiments verify the feasibility and superiority of the proposed method in terms of accuracy and robustness. [PERSON] and [PERSON] [15] proposed a hand-eye calibration technology based on minimizing the re-projection error. This method directly obtains the images of the calibration mode without explicitly estimating the camera attitude for each input image, effectively solves the estimation problem, and is easy to extend to different projection models. It can deal with different camera models by changing the projection model. Some scholars have also combined big data with deep learning [16,17] to explore a new method of hand-eye system calibration. This method does not need to establish an accurate system mathematical model. Based on a large number of experimental data and neural network learning modes, the parameters of the hand-eye system are obtained through training. However, there are some disadvantages. In order to ensure the accuracy of training, tens of thousands of data samples are usually needed, and it takes a lot of time and energy to collect samples. In hand-eye calibration, the vision system in which the camera is installed in a fixed position outside the manipulator body and the camera does not move with the manipulator is called an eye-to-hand hand-eye system [18]. This installation method has the advantages of convenient installation, simple calculation, and a low rate of measurement errors and is the preferred scheme in machine vision projects [19], as shown in Figure 1[20]. The defect is that when the camera is imaging, there will be a phenomenon of near--large and far--small. Therefore, one hand-eye calibration is only suitable for grasping objects of the same height. If you want to grab objects of different heights, you need to calibrate several times, and the calibration results are only applicable to grabbing objects with the calibrated height, which is discrete. Once a new product is involved in the system, you need to recalibrate the hand-eye transformation matrix at the height of the product. Moreover, each recalibration requires the disassembly of the robot end effector. Considering the mechanical installation error, the previously calibrated height also needs to be recalibrated. In this way, with the increase in product models, the calibration work becomes more and more arduous. In order to solve the above problems, this paper first calibrates the camera parameters to eliminate the effect of camera distortion on imaging quality. It ignores the effect of the calibration height, simplifies the hand-eye transformation into a plane-rigid transformation, and then introduces the calibration height parameters to establish a mathematical model that has a highly linear relationship between the parameters of the hand-eye rigid transformation matrix and the calibration and verifies the correctness of the model through experiments. According to the hand-eye calibration model, in any eye-to-hand-type hand-eye system, only a few key height hand-eye relationships need to be calibrated, and the hand-eye relationships of objects of any height in the camera field of view can be calculated based on this linear relationship. The number of hand-eye calibrations at different heights is reduced, and the efficiency of hand-eye calibration is improved. The rest of this paper is organized as follows: in the second section, the pinhole camera model is introduced. The third section introduces the rigid transformation of the coordinate system.