## 3 Constructing the Pose Detection Model  
Given the coordinates \(P_{i}\big{(}x_{P_{i}},y_{P_{i}},z_{P_{i}}\big{)}^{T}\) and \(Q_{i}\big{(}x_{Q_{i}},y_{Q_{i}},z_{Q_{i}}\big{)}^{T}\), \((i=1,2,3)\), the direction vector of the line \(L_{i}\) can be expressed as follows:  
\[s_{L_{i}}=\big{(}X_{L_{i}},Y_{L_{i}},Z_{L_{i}}\big{)}^{T}=\big{(}x_{Q_{i}}-x_{ P_{i}},y_{Q_{i}}-y_{P_{i}},z_{Q_{i}}-z_{P_{i}}\big{)} \tag{1}\]  
Therefore, the equation of the line \(L_{i}\) can be expressed as follows:  
\[\frac{x-x_{P_{i}}}{X_{L_{i}}}=\frac{y-y_{P_{i}}}{Y_{L_{i}}}=\frac{z-z_{P_{i}}} {Z_{L_{i}}} \tag{2}\]  
Similarly, given the coordinates \(R_{i}^{{}^{\prime}}\Big{(}x_{R_{i}^{{}^{\prime}}},y_{R_{i}^{{}^{\prime}}},z_{ R_{i}^{{}^{\prime}}}\Big{)}^{T}\), \((i=1,2,3)\), where \(z_{R_{i}^{{}^{\prime}}}\) is equal to the focal length \(f\) of the camera, the direction vector of the line \(l_{i}\) can be expressed as follows:  
\[s_{l_{i}}=\big{(}X_{l_{i}},Y_{l_{i}},Z_{l_{i}}\big{)}^{T}=\Big{(}x_{R_{i}^{{}^ {\prime}}},y_{R_{i}^{{}^{\prime}}},z_{R_{i}^{{}^{\prime}}}\Big{)}^{T} \tag{3}\]  
As a result, the linear equation of the line \(l_{i}\) can be expressed as follows:  
\[\frac{x}{X_{l_{i}}}=\frac{y}{Y_{l_{i}}}=\frac{z}{Z_{l_{i}}} \tag{4}\]  
The straight-line equations of \(L_{i}\) and \(l_{i}\) can be written as the following four equations:  
\[\left\{ \begin{aligned} Y_{L_{i}}x-X_{L_{i}}y& =Y_{L_{i}}x_{P_{i}}-X_{L_{i}}y_{P_{i}}\\ Z_{L_{i}}y-Y_{L_{i}}z&=Z_{L_{i}}y_{P_{i}}-Y_{L_{i}}z_{P_{i}}\\ Y_{l_{i}}x-X_{l_{i}}y&=0\\ Z_{l_{i}}y-Y_{l_{i}}z&=0\\ \end{aligned} \right. \tag{5}\]  
Figure 2: Principle of the visual pose measurement system for anchor holes on the steel belt. Equation (5) is expressed in matrix form as follows:  
\[A_{i}X_{R_{i}}=B_{i} \tag{6}\]  
where  
\[A_{i}= \begin{pmatrix}Y_{L_{i}}&-X_{L_{i}}&0\\ 0&Z_{L_{i}}&-Y_{L_{i}}\\ Y_{l_{i}}&-X_{l_{i}}&0\\ 0&Z_{l_{i}}&-Y_{l_{i}}\end{pmatrix} \]  
\[B_{i}= \begin{pmatrix}Y_{L_{i}}x_{R_{i}}-X_{L_{i}}y_{R_{i}}\\ Z_{L_{i}}y_{R_{i}}-Y_{L_{i}}z_{R_{i}}\\ 0\end{pmatrix} \]  
\[X_{R_{i}}= \begin{pmatrix}x_{R_{i}}\\ y_{R_{i}}\\ z_{R_{i}}\end{pmatrix} \]  
The least squares solution to the equations is as follows:  
\[\overline{X}_{R_{i}}=\left(A_{i}^{T}A_{i}\right)^{-1}A_{i}^{T}B_{i} \tag{7}\]  
That is, the coordinate \(R_{i}\left(x_{R_{i}},y_{R_{i}},z_{R_{i}}\right)^{T}\) of the laser pointer's spot in the camera's coordinate system is \(\left(A_{i}^{T}A_{i}\right)^{-1}A_{i}^{T}B_{i}\), \(\left(i=1,2,3\right)\). From the coordinates \(\left(x_{R_{i}},y_{R_{i}},z_{R_{i}}\right)^{T}\) of \(R_{i}\) solved by Formula (7), it can be obtained that the normal vector of the coal wall plane \(S_{W}\) is as follows:  
\[n_{W}=R_{1}^{\rightarrow}R_{2}^{\rightarrow}\times R_{1}^{\rightarrow}R_{3 }=\left(x_{R_{2}}-x_{R_{1}},y_{R_{2}}-y_{R_{1}},z_{R_{2}}-z_{R_{1}}\right)^{T} \times\left(x_{R_{3}}-x_{R_{1}},y_{R_{3}}-y_{R_{1}},z_{R_{3}}-z_{R_{1}}\right) ^{T} \tag{8}\]  
Normalizing Equation (8) yields the unit normal vector of the coal wall plane as follows:  
\[e_{W}=\left(X_{W},Y_{W},Z_{W}\right)^{T}=\frac{n_{W}}{\left|n_{W}\right|} \tag{9}\]  
In the camera's coordinate system \(O_{C}x_{C}y_{C}z_{C}\), the unit normal vector of the camera's imaging plane can be set to the following:  
\[e_{C}=\left(X_{C},Y_{C},Z_{C}\right)^{T}=\left(0,0,1\right)^{T} \tag{10}\]  
From Equations (9) and (10), the rotation angle of the vector \(e_{C}\) to \(e_{W}\) is expressed as follows:  
\[\theta=arccos(e_{C},e_{W}) \tag{11}\]  
Its rotation axis can be expressed as the following:  
\[\omega=\left(\omega_{X},\omega_{Y},\omega_{Z}\right)^{T}=e_{C}\times e_{W} \tag{12}\]  
At this point, without considering the rotation around the \(Z\)-axis when transforming from the camera's coordinate system to the steel belt anchor hole's coordinate system, the rotation matrix of the steel belt anchor hole's coordinate system relative to the camera's coordinate system can be obtained using Rodrigues' rotation formula, as follows:  
\[R_{1}=