# A Theoretical Method and Model for TBM Tunnelling Trajectory Adjustment
## 2 Method
### Excavation Trajectory Design  
This step aims to solve this problem and design a scientific and safe route to adjust the TBM so as to excavate along the expect trajectory. With this target, a three-stage trajectory adjustment strategy is proposed and depicted in Figure 3.  
Figure 2: Flowchart of the consistency between the bias angle and distance; (**a**) same direction, (**b**) opposite. In this method, the known variables, i.e., the basis of the trajectory adjustment, are the initial bias angle \(\theta_{1}\) and distance \(d_{0}\). The controlled index is the rotated angle of the main beam in the three stages and is recorded as \(\alpha_{1}\), \(\alpha_{2}\), and \(\alpha_{3}\). On this basis, the curvature and the curvature radius of the tunnelling trajectory should be determined. The TBM tunnelling can be divided into several sections with the same length \(l\) which are commonly referred to as \"rings\" in the actual project. Due to the slightly rotated angle of the main beam, the TBM's turning trajectory can be approximated as a broken line composed of multiple short lines (Figure 4), and the included angle difference of the adjacent line is \(\alpha\). On this basis, the curvature \(K\) and curvature radius \(\rho\) can be calculated by the following equation. \[\sin\alpha_{i}=\frac{l}{2\cdot\rho_{i}} \tag{1}\]  
\[K_{i}=\frac{1}{\rho_{i}}=\frac{2\cdot\sin\alpha_{i}}{l} \tag{2}\]  
The curvature of the first-stage tunnelling trajectory \(K_{1}\) can be calculated using the rotated angle of the main beam \(\alpha_{1}\). Further, the length of the first-stage tunnelling trajectory \(s_{1}\) and the bias distance \(d_{1}\) can be calculated as  
\[s_{1}=\frac{\theta_{1}}{K_{1}} \tag{3}\]  
\[d_{1}=d_{0}+d_{1}^{\prime}=d_{0}+\frac{(1-\cos\theta_{1})}{K_{1}} \tag{4}\]  
Similarly, in the second and third stages, a fixed curvature is also used to adjust the bias distance. In detail, the values of the curvature of the second and third stages are the same, but their directions are opposite. Therefore, the second and third stages can be regarded as a pair of inverse processes; in other words, when the measurement point of the  
Figure 4: Flowchart of calculating the tunnelling curvature. Figure 3: The flow chart of the trajectory adjustment strategy with three stages. TBM reaches half of the bias distance \(d_{1}\), the TBM tunnelling curvature turns to the opposite direction, and the third stage starts, while the bias angle is recorded as \(\theta_{2}\). The rotated angle of the main beam and the tunnelling curvature in the second stage are recorded as \(\alpha_{2}\) and \(K_{2}\). The second and third stages are a pair of reciprocal process, so their reductions of the TBM bias distance are the same and are recorded as \(d_{2}^{\prime}\) and \(d_{3}^{\prime}\). Ideally, when the bias angle of the TBM reaches 0, the bias distance reaches 0 exactly. With this target, the relationship between the increment of bias distance in each stage \(d_{1}^{\prime}\), \(d_{2}^{\prime}\), and \(d_{3}^{\prime}\) and the initial bias distance \(d_{0}\) can be expressed by the following equations. \[d_{2}^{\prime}=d_{3}^{\prime}=\rho_{2}\cdot(1-\cos\theta_{2})=\frac{(1-\cos \theta_{2})}{K_{2}} \tag{5}\]  
\[0=d_{0}+d_{1}^{\prime}-d_{2}^{\prime}-d_{3}^{\prime}=d_{0}+\frac{l\cdot(1-\cos \theta_{1})}{2\cdot\sin\alpha_{1}}-\frac{l\cdot(1-\cos\theta_{2})}{\sin\alpha_ {2}} \tag{6}\]  
Ideal trajectory adjustment should satisfy Equation (6), where \(d_{0}\) and \(\theta_{1}\) are the known variables, namely the mentioned bias distance and bias angle. The selection of \(\alpha_{1}\), \(\alpha_{2}\), and \(\theta_{2}\) requires manual intervention. This equation is the expression of the optimized tunnelling trajectory. Specifically, the rotated angles of the main beam used in the second and third stages are equal, but their direction is opposite. It is necessary to adjust the rotated angle to zero at the transition between the second and third stages.