# The Development of the Pipe Jacking Guidance Technology
## 4 Prism Method
### Measuring Principle  
#### 4.1.1 Coordinate Measurement  
Coordinate measurement uses the principle of three points to determine a plane to calculate the center coordinates of the pipe jacking machine's head and tail. As shown in Figure 8, target prisms are placed at target points A, B and C at the tail of the pipe jacking machine. The coordinates of the target prisms are obtained by the total station and are marked as (X\({}_{i}\), Y\({}_{i}\), Z\({}_{i}\)) (i = A, B, C). The center coordinates of the pipe jacking machine head are set (X\({}_{\text{OI}}\), Y\({}_{\text{OI}}\), Z\({}_{\text{OI}}\)) and the center coordinates of the pipe jacking machine tail are set (X\({}_{\text{O2}}\), Y\({}_{\text{O2}}\), Z\({}_{\text{O2}}\)). The calculation of the head center coordinate of the pipe jacking machine is based on the same principles and formulae as the calculation of the tail center coordinate of the pipe jacking machine. Consequently, the following discussion will focus exclusively on the head center coordinate of the pipe jacking machine. \begin{table}
\begin{tabular}{c c} \hline
**Product Advantages** & **Contents** \\ \hline Measurement accuracy & Guidance system with an accuracy of 2 mm \\ Measurement range & Total station and laser target measuring \\ Laser angle of incidence range & distance up to 300 m \\ \multirow{2}{*}{Convenient communication} & \multirow{2}{*}{
\begin{tabular}{c} Wireless communication, communication \\ distance up to 500 m \\ \end{tabular} } \\  & & 2–15 m \\ \hline \end{tabular}
\end{table} 
Table 1: RMS-D automatic guidance system. The spatial positions of the three target prisms from the center of the pipe jacking machine head remain unchanged. Before the pipe jacking construction, the distances from prisms A, B and C to the center of the pipe jacking machine head O\({}_{1}\) can be measured based on the initial measurement, which are recorded as D\({}_{\text{AO1}}\), D\({}_{\text{BO1}}\) and D\({}_{\text{CO1}}\). In the pipe jacking construction process, once the target prism coordinates have been obtained through the total station, the distance Equation (10) can be utilized. \[\left\{ \begin{array}{l}(X_{A}-X_{\text{O1}})^{2}+(Y_{A}-Y_{\text{O1}})^{2}+ (Z_{A}-Z_{\text{O1}})^{2}=D_{\text{AO1}}^{2}\\ (X_{B}-X_{\text{O1}})^{2}+(Y_{B}-Y_{\text{O1}})^{2}+(Z_{B}-Z_{\text{O1}})^{2}=D _{\text{BO1}}^{2}\\ (X_{\text{C}}-X_{\text{O1}})^{2}+(Y_{\text{C}}-Y_{\text{O1}})^{2}+(Z_{\text{C} }-Z_{\text{O1}})^{2}=D_{\text{CO1}}^{2}\end{array} \right. \tag{10}\]  
In Equation (10), (Xi, Yi, Zi) (i = A, B, C) denote the target prism coordinates, (X\({}_{\text{O1}}\), YO1, ZO1) represent the head center coordinates of the pipe jacking head and D\({}_{\text{IO1}}\) (i = A, B, C) are the known distance parameters. Assume that the normal vector coordinates of the plane where the target prism is located are (XABC, YABC, ZABC), and the calculation Equation (11) is as follows based on the target prisms A, B and C:  
\[\left(X_{ABC}\quad Y_{ABC}\quad Z_{ABC}\right)=\left( \begin{array}{c|c}Y_{A} -Y_{B}\quad Z_{A}-Z_{B}\\ Y_{A}-Y_{\text{C}}\quad Z_{A}-Z_{\text{C}}\end{array} \right|\quad \begin{array} []{c|c}Z_{A}-Z_{B}\quad X_{A}-X_{B}\\ Z_{A}-Z_{\text{C}}\quad X_{A}-X_{\text{C}}\end{array} \right|\quad \begin{array} []{c|c}X_{A}-X_{B}\quad Y_{A}-Y_{B}\\ X_{A}-X_{\text{C}}\quad Y_{A}-Y_{\text{C}}\end{array} \right) \tag{11}\]  
According to the plane normal vector coordinates (XABC, YABC, ZABC), the equation for the plane is:  
\[X_{ABC}x+Y_{ABC}y+Z_{ABC}z+D=0 \tag{12}\]  
In Equation (12), (x, y, z) are the coordinates of any point on the plane where the target prism is located.