# A Theoretical Method and Model for TBM Tunnelling Trajectory Adjustment
## 2 Method
### Constraints of the Main Beam Bias Angle  
In this section, constraints for the main beam bias angle are introduced when the two criteria discussed in Section 2.2 are not satisfied. Equation (6) represents the expression of the optimized tunnelling trajectory. However, it is evident that this equation has multiple solutions. Therefore, three types of constraints are established in this section to limit the range of solutions and address this issue.  
1. Constraints of the initial and maximum bias distance  
Generally, manual trajectory adjustment is only required when the bias distance reaches a certain threshold, represented by \(d_{\text{min}}\). Additionally, throughout the entire tunnelling process, the bias distance is limited and cannot exceed \(d_{\text{1 max}}\). The purpose of \(d_{\text{0 min}}\) is to limit the frequency of trajectory adjustments, while \(d_{\text{1 max}}\) ensures the quality of the tunnelling trajectory. In the proposed method, these constraints can be expressed using the following equations. The values of \(d_{\text{0 min}}\) and \(d_{\text{1 max}}\) are generally determined by the construction enterprise based on the characteristics of the TBM and tunnel [25,26].  
\[\begin{cases}d_{0}\geq d_{\text{0 min}}\\ d_{1}\geq d_{\text{1 max}}\end{cases} \tag{7}\]
2. Constraints of the main beam rotated angle  
The protection of TBM cutters should be considered during trajectory adjustment. To address this, a constraint is proposed in this paper. The TBM cutterhead is equipped with dozens of cutters arranged at equal intervals in the radius direction. To ensure the even distribution of tunnelling load among different cutters and protect them from abnormal damage, the penetration differences between adjacent cutters are limited, which is also a key reason for frequently updating cutters. During trajectory adjustment, the cutterhead will rotate slightly with the main beam, forming an included angle between the cutterhead and tunnel face, which leads to penetration differences between cutters in different positions. The relationship between the rotated angle of the cutterhead and the penetration of each cutter is illustrated in Figure 5. The permitted penetration differences between adjacent cutters are recorded as \(p^{\prime}\), and the cutter spacing is recorded as \(s\). The permitted rotated angle of the main beam, \(\theta^{\prime}\), which is also the cutterhead, can be calculated using Equation (8).  
\[\alpha_{i}=\tan^{-1}\frac{p^{\prime}}{s} \tag{8}\]  
3. Constraint of the total tunnelling distance  
The total length of tunnelling deviation from the expected trajectory is always limited, as the TBM needs to return to the expected trajectory as quickly as possible. The route length is measured as a straight line. In this method, the bias angle of the TBM is assumed to be small enough to be ignored, and the tunnelling length deviating from the expected trajectory is considered to be approximately the sum of \(s_{1}\), \(s_{2}\), and \(s_{3}\). The total length of tunnelling deviation from the expected trajectory, \(s_{\text{max}}\), can be expressed using the following equation.  
\[s_{1}+s_{2}+s_{3}\leq s_{\text{max}} \tag{9}\]