# The Development of the Pipe Jacking Guidance Technology
## 4 Prism Method
### Measuring Principle
#### 4.1.2 Angular Measurement  
According to the center coordinate values O\({}_{1}\) (\(X_{\rm O1}\), \(Y_{\rm O1}\), \(Z_{\rm O1}\)) and O\({}_{2}\) (\(X_{\rm O2}\), \(Y_{\rm O2}\), \(Z_{\rm O2}\)) of the pipe jacking machine's head and tail solved in Section 4.1.1, the angles between the coordinate axes can be calculated. For example, the angle \(\delta\) with the \(x\)-axis is calculated and solved by Equation (15). \[\delta=\arctan\left(\frac{\sqrt{\left(Y_{\rm O2}-Y_{\rm O1}\right)^{2}+\left( Z_{\rm O2}-Z_{\rm O1}\right)^{2}}}{X_{\rm O2}-X_{\rm O1}}\right) \tag{15}\]  
The calculation principle of the space vector of the azimuth, pitch and roll angle of the pipe jacking machine is shown in Figure 9. Vector a is the direction vector of the designed trajectory route, the red dotted line is the projection of vector a on the horizontal plane and the angle between vector a and the horizontal plane is \(\theta\). The yellow local plane is defined as the plane coplanar with vector a and at an angle \(\theta\) to the horizontal, denoted as plane m. Vector b is defined as the normal vector to plane m.  
Define the known vector a (\(X_{\rm a}\), \(Y_{\rm a}\), \(Z_{\rm a}\)) and the normal vector b (\(X_{\rm b}\), \(Y_{\rm b}\), \(Z_{\rm b}\)) to be solved of the plane m, where the vector a (\(X_{\rm a}\), \(Y_{\rm a}\), \(Z_{\rm a}\)) is calculated before construction. Because vector a is in plane m, the normal vector b is perpendicular to vector a; since the angle between plane m and the horizontal plane is \(\theta\), we can know that the angle between the normal vector b of plane m and the normal vector (0, 0, 1) of the horizontal plane is \(\theta\). Therefore, according to the cosine theorem, the following Equation (16) can be obtained. \[\left\{ \begin{array}{l}X_{a}X_{b}+Y_{a}Y_{b}+Z_{a}Z_{b}=0\\ \frac{Z_{b}}{\sqrt{X_{b}^{2}+Y_{b}^{2}+Z_{b}^{2}}}=\cos\theta\end{array} \right. \tag{16}\]  
Normalize the vector b (\(X_{\rm b}\), \(Y_{\rm b}\), \(Z_{\rm b}\)) to obtain Equation (17). \[X_{b}^{2}+Y_{b}^{2}+Z_{b}^{2}=1 \tag{17}\]  
The resolution of the joint Equations (16) and (17) results in the determination of the normal vector b (\(X_{\rm b}\), \(Y_{\rm b}\), \(Z_{\rm b}\)) of the plane m.  
Figure 9: Schematic diagram of the Angle calculation. In Figure 9, A\({}_{1}\)A\({}_{2}\) is the projection of the jacking machine axis O\({}_{1}\)O\({}_{2}\) on the plane m. The vector O\({}_{1}\)O\({}_{2}\) is calculated according to Equation (18). \[\left(X_{O1O2}\quad Y_{O1O2}\quad Z_{O1O2}\right)=\left(X_{O1}-X_{O2}\quad Y_{O1 }-Y_{O2}\quad Z_{O1}-Z_{O2}\right) \tag{18}\]  
(X\({}_{O1O2}\), Y\({}_{O1O2}\), Z\({}_{O1O2}\)) are the coordinates of the vector O\({}_{1}\)O\({}_{2}\). In Figure 9, vector c (X\({}_{C}\), Y\({}_{C}\), Z\({}_{C}\)) is in plane m and perpendicular to O\({}_{1}\)O\({}_{2}\). According to the vector product formula, Equation (19) can be obtained:  
\[\overrightarrow{c}=\overrightarrow{O_{1}O_{2}}\times\overrightarrow{b}= \left(X_{c}\quad Y_{c}\quad Z_{c}\right)=\left( \begin{vmatrix}Y_{O1O2}&Z_{O1O2 }\quad\left|\begin{matrix}Z_{O1O2}&X_{O1O2}\\ Y_{b}&Z_{b}\end{vmatrix} \end{vmatrix}\right. \begin{vmatrix}Z_{O1O2}&Y_{O1O2 }\\ Z_{b}&X_{b}\end{vmatrix} \end{vmatrix} \tag{19}\]  
According to the vector product formula (Equation (20)), the coordinates of vector A\({}_{1}\)A\({}_{2}\) can be solved:  
\[\overrightarrow{A_{1}A_{2}}=\left(X_{A1A2}\quad Y_{A1A2}\quad Z_{A1A2}\right) =\left( \begin{vmatrix}Y_{c}\quad Z_{c}\\ Y_{b}\quad Z_{b}\end{vmatrix}\begin{vmatrix}Z_{c}\quad X_{c}\\ Z_{b}\quad X_{b}\end{vmatrix}\begin{vmatrix}X_{c}\quad Y_{c}\\ X_{b}\quad Y_{b}\end{vmatrix} \right) \tag{20}\]  
(1) Pitch angle \(\beta\)  
As shown in Figure 9, O\({}_{2}\)O\({}_{3}\) is parallel to A\({}_{1}\)A\({}_{2}\). Therefore, the angle between vectors O\({}_{1}\)O\({}_{2}\) and A\({}_{1}\)A\({}_{2}\) is the pitch angle \(\beta\).