# A Geometric Model for a Shield TBM Steering Simulator
## 3 Geometrical Model for Curved Excavation
### Calculation of TBM Position Change by Articulation Jack Operation
#### 3.2.1 Articulation Jack Operation in the X-Type System  
The X-type articulation system is shown in Figure 6. The articulation jack in the target direction is retracted, while the others are extended simultaneously. If the intention is to  
Figure 4: Example of a planned route in the vertical direction.  
steer the TBM forward in an upward direction, only the upper articulation jack is retracted while the other jacks are automatically extended. Based on this principle, the length of each articulation jack in the X-type system is determined by the following:  
\[l=\begin{pmatrix}l_{u}\\ l_{d}\\ l_{l}\\ l_{r}\end{pmatrix}=\begin{pmatrix}-t_{u}v_{u}+t_{d}v_{d}+t_{l}v_{l}+t_{r}v_{r}\\ t_{u}v_{u}-t_{d}v_{d}+t_{l}v_{l}+t_{r}v_{r}\\ t_{u}v_{u}+t_{d}v_{d}-t_{l}v_{l}+t_{r}v_{r}\\ t_{u}v_{u}+t_{d}v_{d}+t_{l}v_{l}-t_{r}v_{r}\end{pmatrix} \tag{4}\]  
where \(l_{u}\), \(l_{d}\), \(l_{l}\), and \(l_{r}\) are the length variations of upper, lower, left, and right articulation jacks, respectively; \(t_{u}\), \(t_{d}\), \(t_{l}\), and \(t_{r}\) are the operation times for the setting of articulation jacks in the upper, lower, left, and right directions, respectively; and \(v_{u}\), \(v_{d}\), \(v_{l}\), and \(v_{r}\) are the extension speeds of upper, lower, left, and right articulation jacks, respectively.  
As a result of the changing length of articulation jacks, the articulation angle is determined as follows:  
\[\alpha=\tan^{-1}\left(\frac{l_{l}-l_{r}}{D}\right)=\tan^{-1}\left(\frac{l_{lr} }{D}\right) \tag{5}\]  
\[\beta=\tan^{-1}\left(\frac{l_{d}-l_{u}}{D}\right)=\tan^{-1}\left(\frac{l_{du}} {D}\right) \tag{6}\]  
where \(\alpha\) is the articulation angle in the horizontal direction on the \(x\)-\(y\) plane; \(\beta\) is the angle in the vertical direction on the \(y\)-\(z\) plane; \(l_{lr}\) is the length difference between left and right articulation jacks; \(l_{du}\) is the length difference between lower and upper articulation jacks; and \(D\) is the TBM shield diameter (Figure 6).  
During articulation for horizontal steering, three reference points change together. As the articulation point is located at the center, both rear reference points change symmetrically relative to the origin, while the front point moves to a new coordinate (Figure 6a). However, to steer the TBM in the vertical direction, both rear reference points remain unchanged while the front point changes during the extension of the articulation jack (Figure 6b). The reference points are recalculated for the articulation angle and the angle of the planned route as follows:  
\[L_{r}=\begin{pmatrix}L_{x^{\prime}}\\ L_{y^{\prime}}\\ L_{z^{\prime}}\end{pmatrix}=\begin{pmatrix}L_{x}^{0}+\frac{D}{2}\left(\cos a _{0}-\cos(a_{0}+\alpha)\right)\\ L_{y}^{0}+\frac{D}{2}\left(\sin(a_{0}+\alpha)-\sin a_{0}\right)\\ L_{z}^{0}\end{pmatrix} \tag{7}\]  
Figure 5: Cross section of the arrangement of articulation jacks.  
\[R_{r}=\begin{pmatrix}R_{x^{\prime}}\\ R_{y^{\prime}}\\ R_{z^{\prime}}\end{pmatrix}=\begin{pmatrix}R_{x}^{0}-\frac{D}{D}(\cos a_{0}-\cos(a_ {0}+\alpha))\\ R_{y}^{0}-\frac{D}{2}(\sin(a_{0}+\alpha)-\sin a_{0})\\ R_{z}^{0}\end{pmatrix} \tag{8}\]  
\[F_{r}=\begin{pmatrix}F_{x^{\prime}}\\ F_{y^{\prime}}\\ F_{z^{\prime}}\end{pmatrix}=\begin{pmatrix}\frac{(L_{x^{\prime}}+R_{x^{\prime }})}{2}+l_{F}\cos(b_{0}+\beta)\sin(a_{0}+\alpha)\\ \frac{(L_{y^{\prime}}+R_{y^{\prime}})}{2}+l_{F}\cos(b_{0}+\beta)\cos(a_{0}+ \alpha)\\ \frac{(L_{x^{\prime}}+R_{x^{\prime}})}{2}+l_{F}\sin(b_{0}+\beta)\end{pmatrix} \tag{9}\]