## 4 Experimental Verification and Analysis
### Analysis of Experimental Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Height (mm)** & \(\mathbf{A}_{11}\) & \(\mathbf{A}_{12}\) & \(\mathbf{T}_{x}\) & \(\mathbf{A}_{21}\) & \(\mathbf{A}_{22}\) & \(\mathbf{T}_{y}\) \\ \hline
15 & \(-0.00574\) & \(0.697815001\) & \(-1743.855103\) & \(0.695863008\) & \(0.005118\) & \(689.4714355\) \\ \hline
30 & \(-0.00531\) & \(0.694389999\) & \(-1738.144653\) & \(0.692277014\) & \(0.004842\) & \(699.4268799\) \\ \hline
45 & \(-0.00515\) & \(0.69036603\) & \(-1732.568604\) & \(0.689655006\) & \(0.004555\) & \(706.3405762\) \\ \hline
55 & \(-0.00428\) & \(0.688233972\) & \(-1729.682251\) & \(0.687260985\) & \(0.003596\) & \(713.5334473\) \\ \hline
75 & \(-0.00471\) & \(0.684557021\) & \(-1722.456787\) & \(0.683767021\) & \(0.00411\) & \(723.1809692\) \\ \hline
85 & \(-0.00460\) & \(0.682471991\) & \(-1718.754883\) & \(0.681801021\) & \(0.003916\) & \(728.6638184\) \\ \hline
95 & \(-0.00443\) & \(0.680664003\) & \(-1716.002808\) & \(0.679901004\) & \(0.003839\) & \(733.9542236\) \\ \hline
105 & \(-0.00475\) & \(0.678767979\) & \(-1711.740601\) & \(0.677829027\) & \(0.004294\) & \(739.3514404\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of rigid transformation matrix \(\overset{R}{\underset{C}{\text{C}}{M}}\) at each height. We further analyze the relationship between the calibration height and various parameters of \(\frac{R}{C}M\), as shown in Table 5.  
As can be seen from Table 5, except for the A11 and A22 two parameters, the other parameters have a strict linear relationship with the calibration height, and the correlation coefficient R\({}^{2}\) > 0.995, \(|\) Pearsons's R\(|\) > 0.997, adj R-Square > 0.995. Therefore, the proposed linear result is shown in Equation (11). Figure 11: Relationship between rigid transformation parameters and calibration height. Figure 10: Broken line diagram of rigid transformation matrix parameters and calibrated height parameters: (**a**) The Z\(-\)axis is the parameter value, and (**b**) Z\(-\)axis is the calibration height. \[\left\{ \begin{array}{l}\text{A}_{11}=10^{-5}H-0.0056\\ \text{A}_{12}=-2\times 10^{-4}H+0.7004\\ T_{x}=0.3511H-1748.8\\ \text{A}_{21}=-2\times 10^{-4}H+0.6984\\ \text{A}_{22}=-10^{-5}H+0.005\\ T_{y}=0.5465H+682.24\end{array} \right. \tag{11}\]  
Considering the error caused by the experiment, we further analyzed the experimental results. We can see from the calibration results in Table 3 that the calibration coordinates are in the camera and robot coordinate systems at different heights, as shown in Figure 12.  
It can be found that when H is 105 mm, XeB' has great errors, and errors will inevitably occur in reading data in the actual calibration process. Considering A11 and A22, the two parameters are relatively small, and the approximate linear processing will not have a great impact on the results. In practical application, one-step data preprocessing can be carried out first, and then the points of matrix \(\overset{R}{C}M\) parameters with high dispersion can be eliminated for refitting. In order to further study the experimental results, we found that there is also a linear relationship between pixel density and camera calibration height \(H\). The research is as follows:  
Pixels per inch (PPI) indicates the number of pixels per unit length of content. The larger the PPI, the higher the resolution and fidelity of the representative image. The size of the research object in this paper is in mm, so the pixel density is defined as the number of pixels contained per mm, using \(Pr\) as an indicator. Figure 12: Coordinate analysis diagram: (**a**) three\(-\)dimensional display of calibration plate coordinates and (**b**) actual camera robot coordinates.