# A Theoretical Method and Model for TBM Tunnelling Trajectory Adjustment
## 3 Case Study
### Trajectory Adjustment Model  
In order to apply the proposed method for TBM trajectory adjustment, we need to know the values of several constraints introduced in Section 3.2. These constraint values vary depending on the specific TBM used. By combining the design parameters of the TBM  
Figure 5: Flowchart of the relationship between the rotated angle of the cutterhead and TBM penetration. Figure 6: Part of the sixth section of the Qingdao Metro Project geological profile. used in the sixth section of the Qingdao Metro Project with the actual project requirements, we obtain the values of these constraints and establish the TBM trajectory adjustment model. These constraints and their values are summarized in Table 1. Among them, the main beam rotated angle is calculated according to the limitation of the penetration difference between adjacent cutters (5 mm) and the actual cutter spacing (73 mm), and the total tunneling distance is set to one ring. When the values of these constraints and the initial bias angle and distance are known, reasonable tunnelling trajectories can be calculated using the method proposed in Section 2. The main beam rotated angles (\(\alpha_{1}\) and \(\alpha_{2}\)) are used to express the tunnelling trajectory. As an example, assuming an initial bias angle of 9 mm/m and a bias distance of 40 mm, multiple trajectories meeting the constraints can be obtained through calculation. Some of these trajectories are listed in Table 2.  
Table 2 shows five combinations of \(\alpha_{1}\) and \(\alpha_{2}\). In the first combination, \(\alpha_{1}\) and \(\alpha_{2}\) reach their permitted maximum value of 6.82 \(\times\) 10\({}^{-2}\), resulting in a rapidly corrected trajectory with a total tunnelling distance of 1582.03 mm. However, the trajectory adjustment process could be slower and steadier. In the second and third combinations, the total tunnelling distance reaches its maximum of 1800 mm. In the second combination, \(\alpha_{1}\) is fixed at its maximum, while in the third combination, \(\alpha_{2}\) is fixed at its maximum. The values of \(\alpha_{2}\) in the second combination and \(\alpha_{1}\) in the third combination are calculated to be 5.16 \(\times\) 10\({}^{-2}\) and 2.41 \(\times\) 10\({}^{-2}\), respectively. Figure 7 illustrates the relationship between \(\alpha_{1}\) and \(\alpha_{2}\), representing different trajectory adjustment schemes. Each point on the figure represents a combination of \(\alpha_{1}\) and \(\alpha_{2}\) as well as a trajectory adjustment scheme. The five black points with numbers correspond to the five schemes listed in Table 2. The dotted lines and curves represent the constraints proposed in this paper. The orange area is formed by the overlapping of the red and yellow areas, indicating that the schemes in this area do not satisfy both the main beam rotated angle and total tunnelling distance constraints. Only the schemes located in the green area satisfy all of the constraints and are permitted. It is important to note that these results are based on the assumed condition with an initial bias angle of 9 mm/m and a bias distance of 40 mm. The results will differ for different conditions, including the location of each constraint and the area. In other words, the model can calculate the reasonable area of the trajectory adjustment scheme based on the initial bias angle and distance, thereby providing the TBM main driver with a reference. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(d_{\mathbf{0 min}}\) & \(d_{\mathbf{1 max}}\) & \(\alpha_{i}\) & \(s_{\mathbf{max}}\) \\ \hline
40 mm & 60 mm & \(6.82\times 10^{-2}\) & 1800 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The constraint values suitable for the TBM used in the sixth section of the Qingdao Metro Project.