# A Geometric Model for a Shield TBM Steering Simulator
## 3 Geometrical Model for Curved Excavation
### (b) X-type (vertical steering)
#### 3.2.2. Articulation Jack Operation in the V-Type System  
The V-type articulation system is depicted in Figure 7. Here, only the articulation jack in the target direction is unchanged, while the others are extended. To steer the TBM in an upward direction, only the upper articulation jack is unchanged, while the others are retracted. The length of each articulation jack is determined as follows:  
\[l= \begin{pmatrix}l_{u}\\ l_{d}\\ l_{t}\\ l_{r}\end{pmatrix} = \begin{pmatrix}t_{d}v_{d}+t_{t}v_{l}+t_{r}v_{r}\\ t_{u}v_{u}+t_{l}v_{l}+t_{r}v_{r}\\ t_{u}v_{u}+t_{d}v_{d}+t_{r}v_{r}\\ t_{u}v_{u}+t_{d}v_{d}+t_{l}v_{l}\end{pmatrix}  \tag{10}\]  
For horizontal steering, two reference points change together, with that at the front moving to new coordinates, regardless of the direction of horizontal steering. However, as the articulation point is located within the articulation jack, the reference points at rear right and left either change or are unchanged, depending on the target direction. To determine the new reference points in horizontal steering, auxiliary variables are calculated first (Equations (11) and (12); Figure 7a):  
\[d_{FR}=\sqrt{{l_{F}}^{2}+\left(\frac{D}{2}\right)^{2}} \tag{11}\]  
\[d_{Fa}=\sqrt{2{(d_{FR})}^{2}(1-\cos\alpha)} \tag{12}\]  
where \(d_{FR}\) is the distance between \(R_{r}\) (\(R^{0}\)) and \(F_{r}\) (\(F^{0}\)), and \(d_{Fa}\) is the distance between \(F^{0}\) and \(F_{r}\), which depends on the articulation angle in the horizontal direction. Based on these auxiliary variables, reference points are recalculated depending on the rotation direction of the articulation angle and the angle of the planned route. If the articulation angle in the horizontal direction (\(\alpha\)) is negative, the reference points are determined as follows (Figure 7a):  
\[L_{r}= \begin{pmatrix}L_{x^{\prime}}\\ L_{y^{\prime}}\\ L_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}L_{x}^{0}\\ L_{y}^{0}\\ L_{z}^{0}\end{pmatrix}  \tag{13}\]  
\[R_{r}= \begin{pmatrix}R_{x^{\prime}}\\ R_{y^{\prime}}\\ R_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}R_{x}^{0}+|l_{Ir}|\cos(b_{0}+ \beta)\sin(a_{0}+\frac{\pi}{2})\\ R_{y}^{0}+|l_{Ir}|\cos(b_{0}+\beta)\cos(a_{0}+\frac{\pi}{2})\\ R_{z}^{0}+|l_{Ir}|\sin(b_{0}+\beta)\end{pmatrix}  \tag{14}\]  
\[F_{r}= \begin{pmatrix}F_{x^{\prime}}\\ F_{y^{\prime}}\\ F_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}F_{x}^{0}+d_{Fa}\cos(b_{0}+\beta) \sin(-c+a_{0}+\frac{\pi}{2})\\ F_{y}^{0}+d_{Fa}\cos(b_{0}+\beta)\sin(-c+a_{0}+\frac{\pi}{2})\\ F_{z}^{0}+d_{Fa}\sin(b_{0}+\beta)\end{pmatrix}  \tag{15}\]  
where \(c\) is the base angle of an isosceles triangle comprising three reference points based on the reference line at the rear (\(\angle F^{0}L^{0}R^{0}=\angle F_{r}R_{r}L_{r}=\angle F^{0}R^{0}L^{0}=\angle F_{ r}L_{r}R_{r}\)) (Figure 7a). If \(\alpha\) is positive, the reference points are determined as follows:  
\[L_{r}= \begin{pmatrix}L_{x^{\prime}}\\ L_{y^{\prime}}\\ L_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}L_{x}^{0}+|l_{Ir}|\cos(b_{0}+\beta) \sin(a_{0}+\frac{\pi}{2})\\ L_{y}^{0}+|l_{Ir}|\cos(b_{0}+\beta)\cos(a_{0}+\frac{\pi}{2})\\ L_{z}^{0}+|l_{Ir}|\sin(b_{0}+\beta)\end{pmatrix}  \tag{16}\]  
\[R_{r}= \begin{pmatrix}R_{x^{\prime}}\\ R_{y^{\prime}}\\ R_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}R_{x}^{0}\\ R_{y}^{0}\\ R_{z}^{0}\end{pmatrix}  \tag{17}\]\[F_{r}= \begin{pmatrix}F_{x^{\prime}}\\ F_{y^{\prime}}\\ F_{z^{\prime}}\end{pmatrix} = \begin{pmatrix}F_{x}^{0}+d_{Fa}\text{cos}(b_{0}+\beta) \text{sin}\big{(}c+a_{0}+\frac{a}{2}\big{)}\\ F_{y}^{0}+d_{Fa}\text{cos}(b_{0}+\beta)\text{cos}\big{(}c+a_{0}+\frac{a}{2} \big{)}\\ F_{z}^{0}+d_{Fa}\text{sin}(b_{0}+\beta)\end{pmatrix}  \tag{18}\]  
**(a)**: **V-type (horizontal steering, when _a<0_)**  
For vertical steering, the three reference points change together, with those at the rear moving to new coordinates by the same distance and those at the front moving more than those at the rear (Figure 6b).