## 4 Experimental Verification and Analysis
### Rigid Transformation from Calibration Plate Coordinate System to Robot Coordinate System  
The calibration plate selected in this experimental step is a rectangular plate of \(2000\text{ mm}\times 1200\text{ mm}\). The smallest enclosing rectangle of the original image is shown in Figure 8a. Due to the effects of lens distortion, the entire image is barrel-shaped [30]. The image effect after distortion correction is shown in Figure 8b. The image exactly matches the standard rectangular frame, indicating that the image distortion has been significantly corrected. \begin{table}
\begin{tabular}{c c} \hline
**Parameter Type** & **Calibration Results** \\ \hline
**Equivalent focal length** & [5010.81789826726, 5011.24572785725] \\ \hline
**Principal point coordinates** & [2768.70959640510, 1806.34169717229] \\ \hline
**Radial distortion** & k1 = -0.0625967952090845, \\  & k2 = 0.13398419477852 \\ \hline
**Tangential distortion** & p1 = -0.000122713140590104, \\  & p2 = 0.00160031845139996 \\ \hline
**Scaling factor** & \(\Gamma\) = 0.168076608984161 \\ \hline
**Mean reprojection error** & Mean reprojection error = 0.08 \\ \hline \end{tabular}
\end{table} 
Table 2: Calibration results of camera internal parameters. Figure 7: Calibration plate attitude and calibration average projection error. The calibration plate selected in this paper is a rectangular calibration plate. We calibrated the four vertices on the upper surface of the rectangular calibration plate, moved the calibration plate to different heights, calibrated the four-point coordinates in turn, obtained the four-point coordinates of the workpiece in the image coordinate system and the machine coordinate system at different heights, and calculated the different \(\prescript{R}{C}M\) matrix parameters. For workpieces of different sizes and heights, we know that three non-collinear points can determine a plane. If the accuracy of the four-point calibration results is enough, there is no need for more point calibration. For a quadrilateral workpiece, you can take the four vertices of the upper surface of the workpiece, move the workpiece to different heights, calibrate the four-point coordinates of the upper surface in turn, obtain the four-point coordinates of the workpiece in the image coordinate system and the machine coordinate system at different heights, system at different heights, and calculate the \(\prescript{R}{C}M\) matrix parameters at different heights. For irregular workpieces, we can take the non-collinear key four-point positions on the upper surface of the workpieces to fit into a quadrilateral, determine the key quadrilateral shape of the workpiece, move the workpiece to different heights, calibrate the four-point coordinates of the upper surface in turn, obtain the four-point coordinates of the workpiece in the image coordinate system and the machine coordinate system at different heights, and calculate the \(\prescript{C}M\) matrix parameters at different heights. In the actual grasping work, the transformation from the image coordinate system to the robot coordinate system is realized by the \(\prescript{R}{C}M\) matrix parameters at different heights, the four-point positions obtained by the camera and the \(\prescript{R}{C}M\) matrix so that the shape and position of the workpiece can be determined, and then the target grasping work of the robot can be realized. We used the robot's tip tool (TCP) to establish a tool coordinate system, sequentially aligned the four corner points of the calibration board, and read the corresponding coordinate values on the teaching pendant, as shown in Figure 9.  
Due to the large variety of products and the different sizes of products at different heights, a hand-eye calibration is required at each product height position. However, in order to adapt to new height products, hand-eye calibration is generally required at a sufficiently small height step.