# Research on the Attitude Control Strategy of TBM Digging with V-Shaped Propulsion System
## 2 V-TBM Propulsion System Analysis
### Mathematical Model of V-TBM Propulsion System
#### 2.2.1 Valve-Controlled Cylinder Model Construction  
Taking into account the pressure compensation mechanism of the constant differential pressure reducing valve within the proportional speed control valve structure, as well as valve body leakage [13], the control cylinder model of the proportional speed control valve is formulated as:  
\[y=C_{1}e^{\frac{-B+\sqrt{2^{*}-4 mK_{\mathrm{s}}}}{2m}}+C_{2}e^{\frac{-B-\sqrt {2^{*}-4 mK_{\mathrm{s}}}}{2m}}+\frac{1}{K_{\mathrm{s}}}\Bigg{[}2A_{3}p_{1}- \rho A_{3}\bigg{(}\frac{A_{4}}{C_{\mathrm{d2}}\pi D_{2}\sin\alpha_{2}x_{ \mathrm{v}}}\bigg{)}^{2}-p_{4}A_{4}-F_{\mathrm{L}}\Bigg{]}, \tag{1}\]  
where \(C_{1},\ C_{2}\) is the constant of integration; \(m\) is the mass of the hydraulic cylinder piston rod, kg; \(B\) is the load viscous damping coefficient, N/(m/s); \(K_{\mathrm{s}}\) is the spring stiffness, N/m; \(A_{3}\), \(A_{4}\) is the area of the hydraulic cylinder rodless chamber, rod chamber, mm\({}^{2}\); \(p_{1},\ p_{4}\) is the speed control valve inlet pressure, rod chamber pressure, MPa; \(C_{\mathrm{d2}}\) is the throttle valve orifice flow coefficient; \(D_{2}\) is the throttle valve orifice circumferential diameter of the opening, mm; \(\alpha_{2}\) is the throttle valve spool semi-cone angle, \({}^{\circ}\); \(x_{\mathrm{v}}\) for the throttle spool displacement; \(F_{\mathrm{L}}\) is the hydraulic cylinder piston rod external load force, N. It can be seen that the propulsion cylinder displacement \(y\) and spool displacement \(x_{\mathrm{v}}\) are related by a quadratic system of equations with load disturbances.  
Figure 2: Hydraulic schematic diagram of V-TBM propulsion system.  
\[\mathbf{A}=\begin{bmatrix}\frac{\partial f_{1}}{\partial m_{1}}&\frac{\partial f_{1}}{ \partial m_{2}}&\cdots&\frac{\partial f_{1}}{\partial m_{6}}\\ \frac{\partial f_{2}}{\partial m_{1}}&\frac{\partial f_{2}}{\partial m_{2}}& \cdots&\frac{\partial f_{2}}{\partial m_{6}}\\ \vdots&\vdots&\vdots&\vdots\\ \frac{\partial f_{6}}{\partial m_{1}}&\frac{\partial f_{6}}{\partial m_{2}}& \cdots&\frac{\partial f_{6}}{\partial m_{6}}\end{bmatrix}. \tag{5}\]  
If \(\mathbf{A}\
eq 0\), then the system of Equation (4) has a solution that solves for \(\Delta m_{\text{i}}(i=1,2,\ldots,6)\).  
Let the cyclic matrix \(\mathbf{B}_{n}\) be:  
\[\begin{cases}\mathbf{B}_{0}=\begin{bmatrix}\mathbf{m}_{10},\mathbf{m}_{20},\cdots,\mathbf{m}_ {60}\end{bmatrix}^{\mathsf{T}}\\ \mathbf{B}_{1}=\mathbf{B}_{0}+\begin{bmatrix}\Delta m_{1},\Delta m_{2},\cdots,\Delta m _{6}\end{bmatrix}^{\mathsf{T}}\\ \mathbf{B}_{2}=\mathbf{B}_{1}+\begin{bmatrix}\Delta m_{1},\Delta m_{2},\cdots,\Delta m _{6}\end{bmatrix}^{\mathsf{T}}\\ \vdots\end{cases}. \tag{6}\]  
After solving for \(\Delta m_{\text{i}}=(m_{\text{i}}-m_{\text{i}0})(i=1,2,\ldots,6)\), the matrix loop of Equation (6) is carried out up to \(\max(\Delta m_{1},\Delta m_{2},\cdots,\Delta m_{6})<\varepsilon\), and the resulting matrix \(\mathbf{B}_{n}\) is the required front shield platform positional solution.  
Thus, the Newton-Raphson method can be employed to perform the forward solution analysis of the positional attitude. Based on Equation (3), a multivariate quadratic nonlinear relationship is identified between the elongation of propulsion hydraulic cylinder \(\Delta l_{\text{i}}\) and the positional attitude \(\mathbf{m}_{\text{i}}(i=1,2,\ldots,6)\) of the front shield platform.