# A Theoretical Method and Model for TBM Tunnelling Trajectory Adjustment
## 4 Model Modification
### Analysis of the Trajectory Adjustment of the Target Samples  
The proposed model is analyzed using nine target samples listed in Table 3, each with a different bias angle and distance. According to their bias angle and distance, corresponding possible combinations of \(\alpha_{1}\) and \(\alpha_{2}\) are calculated by the proposed model. Considering that the TBM does not have the function of recording the actual rotated angle of the main beam and the cutterhead \(\alpha_{1}\) and \(\alpha_{2}\), in this project, the actual \(\alpha_{1}\) and \(\alpha_{2}\) are approximately calculated according to the propulsion stroke of the thrust cylinder. In detail, the TBM is equipped with four groups of thrust cylinders located at the top, bottom, left, and right of the main beam, as Figure 10 shows. In general, the stroke of cylinders in the same group is consistent, while the difference from the stroke of different groups of cylinders results in the included angle between the cutterhead and the tunnel face. Therefore, the rotated angle can be calculated according to the stroke of each group of cylinders and their geometric center, as Equation (10) shows [27]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of Rings** & **Direction** & **Bias Angle (mm/m)** & **Bias Distance (mm)** \\ \hline
136 & Horizontal & 4 & 50 \\
142 & Horizontal & 1 & 45 \\
189 & Horizontal & 6 & 44 \\
232 & Horizontal & 7 & 44 \\
626 & Horizontal & 4 & 53 \\
286 & Vertical & \(-3\) & \(-43\) \\
334 & Vertical & 4 & 42 \\
356 & Vertical & \(-5\) & \(-53\) \\
363 & Vertical & \(-1\) & \(-57\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Basic information about the samples needing human invention using the proposed method. Figure 9: The bias angle and distance in the vertical direction. As Equation (10) shows, \(\alpha_{H}\) and \(\alpha_{Y}\) represent the rotated angle of the main beam in the horizontal and vertical direction; \(s_{A}\), \(s_{B}\), \(s_{C}\), and \(s_{D}\) represent the cylinder stroke of each group; \(d_{AC}\) means the distance between the geometric centers of group A and C; and \(d_{BD}\) means the distance between the geometric centers of group B and D. Usually, constant rotated angles of the main beam are used during the excavation process in a single ring. Therefore, \(\alpha_{1}\) and \(\alpha_{2}\) are recorded as the same value, i.e., the calculated results of Equation (10). The comparison results of the actual rotated angle of the nine target samples and the corresponding trajectory adjustment feasible region calculated by the proposed model are shown in Figure 11.  
As Figure 11a-i shows, the main beam rotated angles used in the actual project of the nine samples are located below their feasible region. According to the analysis from Section 3 of this paper, these samples are located in the yellow area, which means that the tunnelling distances used to correct their tunnelling trajectory are longer than 1800 mm (one ring). In order to verify this conclusion, the changing rules between the bias distance and the tunnelling distance of these target samples are calculated by the proposed method. The results are shown in Figures 12 and 13. Specifically, the bias distances shown in Figures 12 and 13 only represent their absolute value, while their direction information can be found in Table 3.  
As Figures 12 and 13 show, all of the tunnelling distances used to correct the bias distance of the target samples are greater than 1800 mm, and this conclusion is consistent with the comparison results of the actual main beam rotated angles and their feasible regions, which are shown in Figure 7. In addition, the tunnelling distances of rings 142# and 232# in the horizontal direction and rings 286# and 334# in the vertical direction are relatively small--less than 2000 mm. Meanwhile, their actual main beam rotated angles are also relatively close to the corresponding feasible regions, which are shown in Figure 11b,d,f,g, respectively.