# A Theoretical Method and Model for TBM Tunnelling Trajectory Adjustment
## 2 Method  
### Overview of the Method System  
In actual projects, different types of TBMs use different trajectory adjustment methods. Double-shield TBMs typically adjust the main beam direction by changing the stroke of the thrust cylinder systems, which in turn adjusts the tunnelling direction, while open-type TBMs adjust the main beam direction and tunnelling direction by changing the stroke of the left and right gripper cylinders. Regardless of the TBM type, the common goal of trajectory adjustment is to change the direction of the main beam based on the existing tunnelling bias condition, as shown in Figure 1[21]. This process is conducted based on human subjective experience, making it challenging to control the accuracy and efficiency of trajectory adjustment. This paper proposes a theoretical method to quantify this task. The tunnelling bias condition of the TBM, including the bias angle and bias distance in both the horizontal and vertical directions, is quantitatively described using four physical quantities, as shown in Figure 1. Correspondingly, a suitable rotated angle of the main beam will be obtained by the proposed method, which is directly related to the tunnelling trajectory.  
Ideal TBM trajectory adjustment should have two key characteristics. First, it should be infrequent. The resultant force and tunnelling direction should not require frequent adjustments. For TBM tunnelling, there is a certain distance permitted between the actual and expected measured point. Tunnelling within the permitted deviation can be achieved without human intervention. Additionally, the direction of the main beam should be considered during trajectory adjustment. If the deviation directions of the key point and main beam are opposite, meaning that the TBM can naturally and gradually return to the expected trajectory, then the tunnelling state can be maintained. For example, if the measured point deviates to the left of the expected position, while the main beam slightly deviates to the right, the measured point will naturally move to the right, reducing its deviation without the need for human control. The second key characteristic is being smooth and progressive. The angle deviation of the TBM main beam and cutterhead should be limited. Excessive angle deviation results in a large bias moment acting on the cutterhead. This can be harmful to the cutters, especially the cutter located on the edge of the cutterhead.  
According to the above considerations, a TBM trajectory adjustment method with constraints is proposed. The method consists of three steps:  
1. Judging the current tunnelling state. In this step, the horizontal and vertical deviation of the measured point and main beam are used to judge the necessity of trajectory adjustment. There are two criteria for judgement; one is if the actual deviation from the measured point meets the requirements of the permitted value, and the other is if the deviation direction of the measured point and main beam are opposite. Only when neither of these criteria are met does the tunnelling need human controlling and the method continue. This step is described in Section 2.2.
2. Building and conducting the TBM trajectory adjustment model. Based on the field-measured data, the theoretical relationship between the TBM controlling parameter (bias angle of the main beam) and the trajectory parameters (bias angle and distance) are researched. This step is described in Section 2.3.
3. Constructing the constraints of the main beam bias angle. In this step, the bias angle of the main beam and cutterhead is controlled in order to protect the cutters from abnormal wear. Specifically, a permitted difference of the penetration of adjacent cutters is determined. Combining this difference value with the fixed cutter spacing, the permitted bias angle of the main beam and cutterhead can be calculated, which is used as a constraint in the process of calculating the bias angle of the main beam. This step is described in Section 2.4.