## 3 Constructing the Pose Detection Model
\begin{pmatrix}cos\theta+\omega_{X}^{2}(1-cos\theta)&-\omega_{Z}sin \theta+\omega_{X}\omega_{Y}(1-cos\theta)&\omega_{Y}sin\theta+\omega_{X}\omega _{Z}(1-cos\theta)\\ \omega_{Z}sin\theta+\omega_{X}\omega_{Y}(1-cos\theta)&cos\theta+\omega_{Y}^{2} (1-cos\theta)&-\omega_{X}sin\theta+\omega_{Y}\omega_{Z}(1-cos\theta)\\ -\omega_{Y}sin\theta+\omega_{X}\omega_{Z}(1-cos\theta)&\omega_{X}sin\theta+ \omega_{Y}\omega_{Z}(1-cos\theta)&cos\theta+\omega_{Z}^{2}(1-cos\theta)\\ \end{pmatrix}  \tag{13}\]  
As shown in Figure 2, the slopes of the fitted lines of the upper and lower edges of the steel belt in the image's coordinate system are \(k_{1}\) and \(k_{2}\), respectively, and the angle of rotation around the \(Z\)-axis when transforming from the camera's coordinate system to the steel belt anchor hole's coordinate system can be expressed as follows:  
\[\gamma=\arctan\frac{k_{1}+k_{2}}{2} \tag{14}\]  
Then, the rotation matrix rotating around the \(Z\)-axis can be expressed as follows:  
\[R_{2}= \begin{bmatrix}cos\gamma&sin\gamma&0\\ -sin\gamma&cos\gamma&0\\ 0&0&1\end{bmatrix}  \tag{15}\]  
Then, the rotation matrix of the steel belt anchor hole's coordinate system in the camera's coordinate system can be expressed as follows:  
\[R=R_{2}\bullet R_{1} \tag{16}\]  
It is known that \(H_{j}^{{}^{\prime}}\bigg{(}x_{H_{j}^{{}^{\prime}}}y_{H_{j}^{{}^{\prime}}}z_{ H_{j}^{{}^{\prime}}}\bigg{)}^{T}\); then, the direction vector of the line \(l_{H_{j}}\) can be expressed as follows:  
\[s_{l_{H_{j}}}=\bigg{(}X_{l_{H_{j}^{{}^{\prime}}}}Y_{l_{H_{j}^{{}^{\prime}}}}Z_ {l_{H_{j}^{{}^{\prime}}}}\bigg{)}^{T}=\bigg{(}x_{H_{j}^{{}^{\prime}}}y_{H_{j} ^{{}^{\prime}}}z_{H_{j}^{{}^{\prime}}}\bigg{)}^{T} \tag{17}\]  
The linear equation of the straight line \(l_{H_{j}}\) can be expressed as follows:  
\[\frac{x}{X_{l_{H_{j}^{{}^{\prime}}}}}=\frac{y}{Y_{l_{H_{j}^{{}^{\prime}}}}}= \frac{z}{Z_{l_{H_{j}^{{}^{\prime}}}}} \tag{18}\]  
From Equation (9) and the coordinates of \(R_{1}\), the plane equation of the coal wall plane \(S_{W}\) can be obtained as follows:  
\[X_{W}\big{(}x-x_{R_{1}}\big{)}+Y_{W}\big{(}y-y_{R_{1}}\big{)}+Z_{W}\big{(}z-z_ {R_{1}}\big{)}=0 \tag{19}\]  
Formula (19) can be rewritten in the following form:  
\[X_{W}x+Y_{W}y+Z_{W}z=X_{W}x_{R_{1}}+Y_{W}y_{R_{1}}+Z_{W}z_{R_{1}} \tag{20}\]  
The linear equation of the straight line \(l_{H_{j}}\) is divided into two linear equations and combined with the plane equation of the coal wall plane \(S_{W}\) to obtain the following:  
\[\left\{ \begin{aligned} X_{W}x+Y_{W}y+Z_{W}z=X_{W}x_{R_{1}}+Y_{W}y_{ R_{1}}+Z_{W}z_{R_{1}}\\ Y_{l_{H_{j}^{{}^{\prime}}}}x-X_{l_{H_{j}^{{}^{\prime}}}}y=0\\ Z_{l_{H_{j}^{{}^{\prime}}}}y-Y_{l_{H_{j}^{{}^{\prime}}}}z=0\end{aligned} \right. \tag{21}\]  
Formula (21) is expressed in matrix form as follows:  
\[C_{j}X_{H_{j}}=D_{j} \tag{22}\]  
where  
\[C_{j}= \begin{pmatrix}X_{w}&Y_{w}&Z_{w}\\ Y_{l_{H_{j}^{{}^{\prime}}}}&-X_{l_{H_{j}^{{}^{\prime}}}}&0\\ 0&Z_{l_{H_{j}^{{}^{\prime}}}}&-Y_{l_{H_{j}^{{}^{\prime}}}}\end{pmatrix} ,\]\[D_{j}= \begin{pmatrix}X_{W}x_{R_{1}}+Y_{W}y_{R_{1}}+Z_{W}z_{R_{1}}\\ 0\\ 0\end{pmatrix} ,\]  
\[X_{H_{j}}= \begin{pmatrix}x_{H_{j}}\\ y_{H_{j}}\\ z_{H_{j}}\end{pmatrix} .\]  
Therefore, from Formula (19), we obtain the following:  
\[X_{H_{j}}=C_{j}^{-1}D_{j} \tag{23}\]  
That is, the translation vector \(t\) of the steel belt anchor hole's coordinate system in the camera's coordinate system is \(C_{j}^{-1}D_{j}\).