# A Geometric Model for a Shield TBM Steering Simulator
## 4 Example of Curved Excavation Simulation
### Curved Excavation in the Top-Right Direction  
The planned route in the top-right direction was created by horizontal (\(a_{i}\)) and vertical (\(b_{i}\)) angles of 0.5\({}^{\circ}\). To travel along this route, \(\alpha\) and \(\beta\) were set to be the same as \(a_{i}\) and \(b_{i}\), respectively. The articulation jacks were then extended, as shown in Table 3 (Equations (5) and (6)). The flow chart of the curved excavation process is presented in Figure 10.  
The simulation progress in curved excavation in the top-right direction, without consideration of the two rotation angles (pitch and yaw), is depicted in Figure 11a,b, for 25 and 75 cycles. Blue lines (Figure 11) indicate the planned route, and red triangles indicate the three reference points. The red triangle travels along almost the entire route during the simulation. After 100 cycles, the TBM has travelled \(\sim\)150 m, with the final reference points being recorded in Table 4.  
The deviation between tracking points of the planned route (\(T_{x}\), \(T_{y}\), \(T_{z}\)) and the rear reference point (\(M_{x}\), \(M_{y}\), \(M_{z}\)) are shown in Figure 12, without consideration of rotation angles. The estimated rear reference points are almost the same as the tracking point, as the 15,000 calculated deviations overlap each other around zero (Table 5). If the articulation angles are adjusted to be almost identical to the direction angles of the planned route, the curved excavation can be successfully performed by the TBM without consideration of the rotation angles. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Location of**} & \multirow{2}{*}{**Coordinate**} & \multicolumn{2}{c}{**Position (mm)**} & \multirow{2}{*}{**Error (\%)**} \\ \cline{3-3} \cline{5-5}  & & **X-Type** & & **V-Type** \\ \hline \multirow{3}{*}{\(L_{r}\)} & \(L_{x^{\prime}}\) & \(-3986.7\) & \(-3986.3\) & \(0.0\) \\  & \(L_{y^{\prime}}\) & \(1534.9\) & \(1604.9\) & \(-4.6\) \\  & \(L_{z^{\prime}}\) & \(13.1\) & \(13.3\) & \(-1.2\) \\ \hline \multirow{3}{*}{\(R_{r}\)} & \(R_{x^{\prime}}\) & \(4013.0\) & \(4013.4\) & \(0.0\) \\  & \(R_{y^{\prime}}\) & \(1464.9\) & \(1534.9\) & \(-4.8\) \\  & \(R_{z^{\prime}}\) & \(13.1\) & \(13.3\) & \(-1.2\) \\ \hline \multirow{3}{*}{\(F_{r}\)} & \(F_{x^{\prime}}\) & \(39.4\) & \(39.8\) & \(-1.2\) \\  & \(F_{y^{\prime}}\) & \(4499.7\) & \(4569.7\) & \(-1.6\) \\ \cline{1-1}  & \(F_{z^{\prime}}\) & \(39.4\) & \(39.5\) & \(-0.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Differences in reference points between the X- and V-type systems. \begin{table}
\begin{tabular}{c c} \hline \hline
**Location of Jack** & **Required Length of Articulation Jack for \(\alpha\) = 0.5\({}^{\circ}\) and \(\beta\) = 0.5\({}^{\circ}\)** \\ \hline Upper & \(0\) mm \\ Lower & \(70\) mm \\ Left & \(70\) mm \\ Right & \(0\) mm \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Adjustment of articulation jacks for curved excavation in the top-right direction. Figure 11: Simulated curved excavation in top-right direction without consideration of rotation angles: (**a**) simulation with 25 cycles; (**b**) simulation with 75 cycles. Figure 10: Flowchart for the curved excavation process. The simulated progress of curved excavation in the top-right direction, with consideration of the rotation angles (pitch = yaw = 1.0\({}^{\circ}\)), is shown in Figure 13 for 25 and 75 cycles. Here, the red triangle does not fully follow the route during the simulation. Due to pitch and yaw, the TBM is inevitably liable to vibrate during the excavation, causing deviations. The total distance travelled by the TBM is again \(\sim\)150 m, with the final reference points being recorded in Table 4.  
The deviation between tracking points of the planned route (\(T_{x},T_{y},\ T_{z}\)) and the rear reference point (\(M_{xx},M_{yy},\ M_{zz}\)), with consideration of the rotation angles, is shown in Figure 13. The estimated rear reference points differ from the tracking point, as the 15,000 calculated deviations plot as an almost straight line.