## 1 Introduction
### Simplified Six Degrees of Freedom Model of the Liquid Tank Semi-Trailer for Yaw Stability Control
where: \(m_{2}\) is the mass of the semi-trailer, kg; \(m_{2s}\) is the sprung mass of the semi-trailer, kg; \(u_{2}\) is the longitudinal velocity of the semi-trailer, m/s; \(\dot{\beta}_{2}\) is the sideslip angle of the semi-trailer, rad; \(\dot{\psi}_{2}\) is the yaw rate of the semi-trailer, rad/s; \(\dot{\phi}_{2}\) is the roll angular acceleration of semi-trailer, rad/s\({}^{2}\); \(F_{yT}\) is the lateral force at the articulated point, N; and \(h_{2}\) is the distance from the centroid of the semi-trailer to its roll axis, m.  
For the yaw motion of the semi-trailer:  
\[I_{2 zz}\ddot{\psi}_{2}-I_{2 xz}\ddot{\phi}_{2}=-F_{yT}e-F_{y3}d \tag{10}\]  
where \(I_{2 zz}\) is the moment of inertia of the semi-trailer around the Z axis, kg\(\cdot\)m\({}^{2}\); \(I_{2 xz}\) is the centroid yaw-roll inertia product of the semi-trailer, kg\(\cdot\)m\({}^{2}\); \(d\) is the distance from the axle of the semi-trailer to its center of mass, m; and \(e\) is the distance from the centroid of the semi-trailer to the articulated point, m.  
For the semi-trailer roll movement:  
\[ \begin{array}{c}\left(I_{2\mathrm{xx}}+m_{2s}h_{2}^{2}\right)\ddot{\phi}_{2 }-I_{2 xz}\ddot{\psi}_{2}=m_{2s}gh_{2}\sin\phi_{2}\\ +m_{2s}\left(u_{2}\left(\dot{\beta}_{2}+\dot{\psi}_{2}\right)-h_{2}\ddot{\phi }_{2}\right)h_{2}+F_{yT}h_{2c}-c_{2}\dot{\phi}_{2}-k_{r2}\phi_{2}-k_{12}(\phi _{2}-\phi_{1})\end{array}  \tag{11}\]  
where \(I_{2 xx}\) is the moment of inertia of the semi-trailer around the X-axis, kg\(\cdot\)m\({}^{2}\); \(h_{2c}\) is the distance from the articulated point to the roll axis of the semi-trailer, m; \(k_{r2}\) is the semi-trailer roll stiffness, N\(\cdot\)m/rad; and \(c_{2}\) is semi-trailer roll damping, N\(\cdot\)m\(\cdot\)s/rad. The constraint equation at the fifth wheel is as follows:  
\[\left\{ \begin{array}{c}F_{yT}=F_{yH}\cos\theta\\ \dot{\theta}=\dot{\psi}_{2}-\dot{\psi}_{1}\\ \beta_{2}=\beta_{1}-\frac{h_{1c}}{u_{1}}\dot{\phi}_{1}+\frac{h_{2c}}{u_{2}} \dot{\phi}_{2}-\frac{c}{u_{1}}\dot{\psi}_{1}-\frac{c}{u_{2}}\dot{\psi}_{2}- \theta\end{array} \right. \tag{12}\]  
The linear tire model is adopted for vehicle tires as follows:  
\[\left\{ \begin{array}{c}F_{y1}=k_{1}\alpha_{1}=k_{1}\left(\beta_{1}+\frac{a \dot{\psi}_{1}}{u}-\delta\right)\\ F_{y2}=k_{2}\alpha_{2}=k_{2}\left(\beta_{1}-\frac{b\dot{\psi}_{1}}{u}\right)\\ F_{y3}=k_{3}\alpha_{3}=k_{3}\left(\beta_{2}-\frac{d\dot{\psi}_{2}}{u}\right) \end{array} \right. \tag{13}\]  
where \(\theta\) is the articulated angle, rad; \(k_{i}\) is the lateral stiffness of wheel \(i(i=1,\ 2,\ 3)\), N\(/\)rad; and \(\alpha_{i}\) is the side slip angle of wheel i, rad. Combining Equations (6)-(13) and eliminating the fifth wheel force \(F_{yH}\), a simplified dynamic model of the liquid tank semi-trailer is obtained as follows:  
\[M\dot{X}=NX+KU \tag{14}\]  
where \(X\) is the state variable, and \(X=\left[\beta_{1}\ \dot{\psi}_{1}\ \dot{\phi}_{1}\ \dot{\phi}_{1}\ \dot{\phi}_{2}\ \dot{\phi}_{2}\ \dot{\theta}\ \theta\ \right]^{T}\); \(U\) is the input, and \(U=\delta\);\[\text{Matrix }M=\]  
\[ \begin{pmatrix}cm_{1}n_{1}&l_{1 xx}&0&-l_{1 xx}-ch_{1}m_{1x}&0&0&0&0\\ (h_{1x}m_{1}-h_{1}m_{1x})u_{1}&-l_{1 xx}&0&m_{1x}(2h^{2}-h_{1x}h_{1})+l_{1 xx}&0&0&0 &0\\ m_{1}u_{1}+m_{2}u_{2}&-m_{2}-\frac{m_{22}u_{2}}{u_{1}}&0&-h_{1}m_{1x}-\frac{h_{1 }m_{2}m_{2}}{u_{1}}&0&h_{2}m_{2}-h_{2}m_{2}&-em_{2}&0\\ -em_{2}u_{2}&l_{2 xx}+e^{2}m_{2}+\frac{em_{22}u_{2}}{u_{2}}&0&\frac{dh_{22}u_{2} }{u_{2}}&0&e(h_{2}m_{2}-h_{2}m_{2})-l_{2 xx}&m_{2}e^{2}+l_{2 xx}&0\\ (h_{2x}m_{2}-h_{2}m_{2})u_{2}&e(h_{2}m_{2}-h_{2x}-h_{2x})-l_{2 xx}+\frac{(3h_{2 }m_{2}-h_{2x}m_{2})u_{2}}{u_{1}}&0&\frac{h_{1x}(h_{2}m_{2}-h_{2x})+m_{2}h_{2}}{ u_{1}}+h_{2x}&e(h_{2}m_{2}-h_{2x}m_{2})-l_{2 xx}&0\\ 0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0\\ \end{pmatrix} \]  
\[\text{Matrix }N=\]  
\[