# The Development of the Pipe Jacking Guidance Technology
## 5 Inertial Guidance Method  
### Principle  
The inertial guidance method, which is predicated on inertial navigation technology, is of particular significance in the context of underground space development [20]. The inertial guidance method utilizes equipment such as inertial measurement units (gyroscopes and accelerometers), odometers and static leveling, in addition to control terminals, as illustrated in Figure 11. The inertial measurement unit (IMU) is responsible for the collection of attitude and velocity information, while the odometer quantifies the distance increment of the pipe jacking process. Additionally, the static leveling provides elevation data of the pipe jacking machine, as illustrated in Figure 12. The data presented herein has been aggregated and processed to calculate the deviation distance between the pipe jacking route and the designed route. This information will be used to provide further feedback to control the jacking direction of the pipe jacking machine and reduce the deviation.  
Figure 11: System composition of the inertial guidance method.  
Figure 12: Sketch of the data fusion.  
The horizontal position measurement of the pipe jacking machine can derive the position coordinates in the horizontal plane through the attitude angle information output from the inertial measurement unit and the distance increment information from the odometer, using the Dead Reckoning (DR) [21]. As shown in Figure 13, the equation for calculating the position coordinates of Q is shown in Formula (24):  
\[\left\{\begin{array}{l}X_{n}=X_{0}+\sum_{i=1}^{n}\Delta L_{i}\cdot\cos\alpha _{i}\\ Y_{n}=Y_{0}+\sum_{i=1}^{n}\Delta L_{i}\cdot\sin\alpha_{i}\end{array}\right. \tag{24}\]  
where the initial horizontal position coordinate of Q is set to be Q\({}_{0}\) (X\({}_{0}\), Y\({}_{0}\)) and \(\Delta\)Li is the distance increment of Q in the direction of horizontal azimuth angle \(\alpha_{i}\) (i = 1, 2, 3).  
The position is updated through the application of Dead Reckoning, with the eastward offset \(\Delta\)L\({}_{\text{E}}\) and northward offset \(\Delta\)L\({}_{\text{N}}\), both before and after the update, being related to the displacement increment, azimuth \(\alpha\), and pitch angle \(\beta\). The eastward offset \(\Delta\)L\({}_{\text{E}}\) and northward offset \(\Delta\)L\({}_{\text{N}}\) are calculated as shown in Equation (25):  
\[\left\{\begin{array}{l}\Delta L_{E}=\Delta L_{i}\cos\beta\cos\alpha\\ \Delta L_{N}=\Delta L_{i}\cos\beta\sin\alpha\end{array}\right. \tag{25}\]  
According to Equation (26), the latitude and longitude information can be calculated.  
\[\left\{\begin{array}{l}L=L_{0}+\sum_{i=1}^{j}\Delta L_{Ej}/(R_{M}+h)\\ \lambda=\lambda_{0}+\sum_{j=1}^{i}\Delta L_{Nj}/\{(R_{N}+h)\cdot\cos L\} \end{array}\right. \tag{26}\]  
where L and \(\lambda\) are longitude and latitude, L\({}_{0}\) and \(\lambda_{0}\) are the longitude and latitude information of the initial position, R\({}_{\text{M}}\) is the radius of the local earth's meridian circle and R\({}_{\text{N}}\) is the radius of the local earth's prime unitary circle. h is the current height.  
By obtaining the water plane coordinate information above, combined with the height measurement of the static leveling, we can obtain the pipe jacking position information under the spatial coordinate system and calculate the deviation compared with the ideal design pipeline route.