# A Geometric Model for a Shield TBM Steering Simulator
## 3 Geometrical Model for Curved Excavation
### Calculation of Reference Point Change Owing to Shield Jack Operation  
After articulation, a group of shield jacks is extended. The mechanism of shield jack operation is similar to that of the V-type articulation system, with only selected jacks being extended while the other shield jacks remain unchanged. As explained in Section 3.2, shield jack differences caused by one-sided shield jack extension are required for successful curved excavation during articulation. However, as we assumed that the shield jacks are extended equally and simultaneously at the same jack speed, the length of shield jacks is calculated as follows:  
\[ll=\begin{pmatrix}ll_{u}\\ ll_{d}\\ ll_{l}\\ ll_{r}\end{pmatrix}=tt\cdot V \tag{24}\]  
where \(tt\) is the operation time for extension of the shield jack, and \(V\) is the extension speed of the jack.  
During TBM advance, three reference points are changed by the same distance during the extension of the shield jack. Therefore, the new reference points are calculated after articulation work as follows:  
\[L=\begin{pmatrix}L_{x}\\ L_{y}\\ L_{z}\end{pmatrix}=\begin{pmatrix}L_{x^{\prime}}+ll\cos(b_{0}+\beta)\sin(a_{0}+ \alpha)\\ L_{y^{\prime}}+ll\cos(b_{0}+\beta)\cos(a_{0}+\alpha)\\ L_{z^{\prime}}+ll\sin(b_{0}+\beta)\end{pmatrix} \tag{25}\]\[R=\begin{pmatrix}R_{x}\\ R_{y}\\ R_{z}\end{pmatrix}=\begin{pmatrix}R_{x^{\prime}}+ll\cos(b_{0}+\beta)\sin(a_{0}+ \alpha)\\ R_{y^{\prime}}+ll\cos(b_{0}+\beta)\cos(a_{0}+\alpha)\\ R_{z^{\prime}}+ll\sin(b_{0}+\beta)\end{pmatrix} \tag{26}\]  
\[F=\begin{pmatrix}F_{x}\\ F_{y}\\ F_{z}\end{pmatrix}=\begin{pmatrix}F_{x^{\prime}}+ll\cos(b_{0}+\beta)\sin(a_{0}+ \alpha)\\ F_{y^{\prime}}+ll\cos(b_{0}+\beta)\cos(a_{0}+\alpha)\\ F_{z^{\prime}}+ll\sin(b_{0}+\beta)\end{pmatrix} \tag{27}\]  
where \(F\) is the front reference point shifted by the TBM advance \(\left(F_{x},F_{y},F_{z}\right)\); \(R\) is the rear-right point shifted by the TBM advance \(\left(R_{x},R_{y},R_{z}\right)\); and \(L\) is the rear-left point shifted by TBM advance \(\left(L_{x},L_{y},L_{z}\right)\).  
During TBM advance, deviations in its position caused by pitch, yaw, and roll may occur, so the reference points are recalculated depending on these rotations according to [PERSON] et al. [2] and [PERSON] et al. [3] as follows:  
\[T=\begin{pmatrix}\cos\mathcal{O}\!\cos\varphi-\sin\mathcal{O}\!\sin\theta\!\sin \varphi&\cos\mathcal{O}\!\sin\varphi+\sin\mathcal{O}\!\sin\theta\!\cos\varphi& -\sin\mathcal{O}\!\cos\theta\\ -\cos\theta\!\sin\varphi&\cos\theta\!\cos\varphi&\sin\theta\\ \sin\mathcal{O}\!\cos\varphi+\cos\mathcal{O}\!\sin\theta\!\sin\varphi&\sin \mathcal{O}\!\sin\varphi-\cos\mathcal{O}\!\sin\theta\!\cos\varphi&\cos \mathcal{O}\!\cos\theta\end{pmatrix} \tag{28}\]  
\[LL=\begin{pmatrix}LL_{x}\\ LL_{y}\\ LL_{z}\end{pmatrix}=T\cdot L \tag{29}\]  
\[RR=\begin{pmatrix}RR_{x}\\ RR_{y}\\ RR_{z}\end{pmatrix}=T\cdot R \tag{30}\]  
\[FF=\begin{pmatrix}FF_{x}\\ FF_{y}\\ FF_{z}\end{pmatrix}=T\cdot F \tag{31}\]  
where \(T\) is the transformation matrix of three rotation angles; \(\theta\) is the pitch angle; \(\varphi\) is the yaw angle; \(\mathcal{O}\) is the roll angle; \(FF\) is the front reference point affected by the three rotation angles \(\left(FF_{x},FF_{y},FF_{z}\right)\); \(RR\) is the rear-right reference point affected by the three rotation angles \(\left(RR_{x},RR_{y},RR_{z}\right)\); and \(LL\) is the rear-left reference point affected by the three rotation angles \(\left(LL_{x},LL_{y},LL_{z}\right)\).  
Based on the work of [PERSON] et al. [6], horizontal and vertical deviations caused by pitch and yaw can be calculated as follows, assuming that the TBM moves along the planned route well (Figure 8):  
\[d_{h}=\left(\frac{\left(MM_{x}-M_{x}\right)-\left(FF_{x}-F_{x}\right)}{l_{F}} \right)\cdot 100 \tag{32}\]  
\[d_{v}=\left(\frac{\left(MM_{z}-M_{z}\right)-\left(FF_{z}-F_{z}\right)}{l_{F}} \right)\cdot 100 \tag{33}\]  
Figure 8: Deviation tendencies: (**a**) horizontal tendency; (**b**) vertical tendency.