## 4 Positioning Method of Supported Anchor Rod in Roadway
### Coordinate Transformation Model  
In the positioning process of the supported bolt, to determine the corresponding relationship between the three-dimensional coordinates of the target point of the supported anchor rod and its image at a particular time, it is necessary to establish a camera-pixel coordinate system conversion model. According to the position of the target point in the pixel coordinate system, the position coordinates relative to the RGB camera are obtained by the conversion model. The conversion relationship between the camera coordinate system and the pixel coordinate system is shown in Figure 8. In the figure, \(P\) is the real target point, \(P^{\prime}\) is the corresponding point of the target point on the image, \(uv\) is the pixel coordinate system, \(xoy\) is the image coordinate system, and \(O_{L}\), \(O\), \(O_{R}\) are the origin of the camera coordinate system of the left camera, RGB camera, and right camera, respectively. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & \multicolumn{2}{c}{**Left Imager**} & \multicolumn{2}{c}{**Right Imager**} & \multicolumn{2}{c}{**RGB**} \\ \hline Focal length & \(\left[381.5116\right.\) & \(383.3934\) & \(\left[381.2979\right.\) & \(383.5978\) & \(\left[378.7371\right.\) & \(380.0363\) \\ \hline Intrinsic matrix & \(\left[\begin{matrix}381.5116\end{matrix}\) & \(0\) & \(312.2119\) \\ \(0\) & \(383.3934\) & \(239.3964\) \\ \(0\) & \(0\) & \(1\) \\ \end{matrix}\) & \(\left[\begin{matrix}381.2979\end{matrix}\) & \(0\) & \(312.4911\) \\ \(0\) & \(383.5978\) & \(239.1390\) & \(0\) & \(380.0363\) & \(241.0257\) \\ \(1\) & \(0\) & \(0\) & \(1\) \\ \hline Radial distortion & \(\left[0.0018\right.\) & \(-0.0032\) & \(\left[0.0028\right.\) & \(-0.0033\) \\ \hline Tangential distortion & \(\left[-6.3533e^{-4}\right.\) & \(-0.0012\) & \(\left[9.4072e^{-4}\right.\) & \(-0.0016\) & \(\left[-0.0012\right.\) & \(2.7327e^{-4}\) \\ \hline Reprojection error & \(\left[3.5172e^{-5}\right.\) & \(5.8203e^{-5}\) & \(\left[2.1552e^{-5}\) & \(5.5460e^{-5}\right]\) & \(\left[3.3221e^{-5}\) & \(6.0154e^{-5}\right]\) \\ \hline Rotation matrix & & & \(\left[\begin{matrix}1.0000&0.0005&-0.0001\\ -0.0005&1.0000&0.0010\\ 0.0001&-0.0010&1.0000\end{matrix}\right]\) & \(\left[\begin{matrix}1.0000&-0.0005&0.0013\\ 0.0005&1.0000&0.0001\\ -0.0013&-0.0001&1.0000\end{matrix}\right]\) \\ \hline Translation matrix & & \(\left[-0.0951\right.\) & \(6.6814\) & \(0.0006\) & \(\left[0.0361\) & \(-4.4994\) & \(-0.0004\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sensor calibration parameters. Figure 8: Coordinate transformation relationship. Figure 7: Multivision sensor. The image and pixel coordinate systems are on the same imaging planes. The measurement unit and the origin position of the two are different. The coordinate transformation formula is:  
\[ \begin{bmatrix}u\\ v\\ 1\end{bmatrix} = \begin{bmatrix}\frac{1}{dx}&0&u_{0}\\ 0&\frac{1}{dy}&v_{0}\\ 0&0&1\end{bmatrix}\begin{bmatrix}x\\ y\\ 1\end{bmatrix}  \tag{5}\]  
where \((u,v)\) is the coordinate of the target point in the pixel coordinate system, and the unit is the pixel; \((x,y)\) is the coordinate of the target point in the image coordinate system, the unit is mm; \(dx\) and \(dy\) represent the actual size of a single pixel along the x and y axes, and the unit is mm; and \((u_{0},v_{0})\) is the coordinate value of the origin of the image coordinate system in the pixel coordinate system. The conversion relationship formula between the camera coordinate system and the image coordinate system is:  
\[Z \begin{bmatrix}x\\ y\\ 1\end{bmatrix} = \begin{bmatrix}f&0&0&0\\ 0&f&0&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}X\\ Y\\ Z\\ 1\end{bmatrix}  \tag{6}\]  
where \(f\) is the focal length of the camera and \((X,Y,Z)\) is the coordinates of the target point in the camera coordinate system. Combining Formulas (5) and (6), the conversion relationship between the camera coordinate system and the pixel coordinate system can be obtained as follows. \[Z \begin{bmatrix}u\\ v\\ 1\end{bmatrix} =