## 5 Conclusions  
Through the introduction, this paper introduces the specific working mode of the hand-eye calibration method and the current research status of the hand-eye calibration method. It also discusses the defect in hand-eye calibration: due to the small phenomenon of camera imaging, it cannot be applied to machine grasping at any height. Based on the imaging model of a standard pinhole camera, the rigid transformation matrix is introduced to establish the rigid transformation mathematical model from image coordinate system to robot coordinate system at any height h, and the hand-eye transformation is simplified into plane rigid transformation. In the process of the experiment, the internal parameters and distortion parameters of the camera were obtained according to the Zhang Zhengyou calibration method, which eliminates the influence of the camera barrel distortion on the imaging quality and ignores the influence of the calibration height. Then, the machine coordinate system and image coordinate system of the calibration plate at different heights were calibrated using the four-point calibration method, and the \(\lx@converttounder{R}_{C}M\) parameter matrix at different heights h was obtained. From the obvious linear relationship between the distribution of the \(\lx@converttounder{R}_{C}M\) parameter matrix in the three-dimensional coordinate system, the linear relationship between the parameters of the rigid transformation matrix from the image coordinate system to the robot coordinate system and the calibration height was further fitted. The random error of the experimental results was analyzed, and the linear relationship between the calibration height and the pixel density of the image was further studied. Through the experimental results, the systematic errors in the experimental process were deeply analyzed. The experimental results show that the hand-eye relationship of objects at any height in the camera's field of view is calculated according to this linear relationship, which is accurate and suitable for the positioning of products at any height, and the positioning error is less than 0.08%.  
According to this conclusion, the eye-to-hand type hand-eye calibration system only needs to calibrate and calculate the rigid transformation matrix of several sampling heights to calculate the hand-eye rigid transformation matrix of the system at any height, and the result is accurate and effective. The calibration of the coordination system brings great, convenience,  
Resources, S.S.; Writing--original draft, D.Z. and W.W.; Writing--review & editing, S.G., D.Z. and W.W. All authors have read and agreed to the published version of the manuscript.  
This paper is supported by the National Natural Science Foundation of China (grant no. 51475129, 51675148, 51405117). This research was funded by the application and demonstration of \"intelligent generation\" technology for small- and medium-sized enterprises--R & D and demonstration application of the chair of the industry Internet innovation service platform based on artificial intelligence (grant no. 2020C01061).  
The author of this article declares that there is no conflict of interest related to this manuscript.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Height** **(mm)**} & \multicolumn{3}{c}{**Results of Testing (mm)**} & \multicolumn{3}{c}{**Calculation Results (mm)**} \\ \cline{2-7}  & \(\mathbf{X}\) & \(\mathbf{Y}\) & \(\mathbf{X}^{\prime}\) & **Error** & \(\mathbf{Y}^{\prime}\) & **Error** \\ \hline
20 & 691.07 & 1983.25 & 691.2215 & 0.0219\% & 1982.9732 & 0.0139\% \\ \hline
55 & \(-1487.95\) & 1901.81 & \(-1488.0752\) & 0.0084\% & 1901.3576 & 0.0238\% \\ \hline
88 & \(-1419.21\) & 2504.49 & \(-1419.7325\) & 0.0368\% & 2503.8941 & 0.0238\% \\ \hline
105 & 537.27 & 2542.82 & 537.6739 & 0.0752\% & 2543.1803 & 0.0142\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Verification of the applicability of the rigid transformation matrix.