## 4 Yaw Stability Control Method for the Liquid Tank Semi-Trailer and Its Implementation
### General Architecture of Yaw Stability Control Method for the Liquid Tank Semi-Trailer
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(\Delta K_{I}\) & \multicolumn{8}{c}{\(\mathbf{\omega_{c}}\)} \\ \cline{3-10}  & \(\Delta K_{I}\) & & **NB** & **NM** & **NS** & **ZO** & **PS** & **PM** & **PB** \\ \hline \multirow{6}{*}{\(e\)} & NB & NB & NB & NM & NM & NS & ZO & ZO \\  & NM & NB & NB & NM & NS & NS & ZO & ZO \\  & NS & NB & NM & NS & NS & ZO & PS & PS \\  & ZO & NM & NM & NS & ZO & PS & PM & PM \\  & PS & NM & NS & ZO & PS & PS & PM & PB \\  & PM & ZO & ZO & PS & PS & PM & PB & PB \\  & PB & ZO & ZO & PS & PM & PM & PB & PB \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fuzzy rule table of \(\Delta K_{I}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \(\Delta K_{p}\) & \multicolumn{8}{c}{\(\mathbf{\omega_{c}}\)} \\ \cline{3-10}  & & **NB** & **NM** & **NS** & **ZO** & **PS** & **PM** & **PB** \\ \hline \multirow{6}{*}{\(e\)} & NB & PB & PB & PM & PM & PS & ZO & ZO \\  & NM & PB & PB & PM & PS & PS & ZO & NS \\  & NS & PM & PM & PM & PS & ZO & NS & NS \\  & ZO & PM & PM & PS & ZO & NS & NM & NM \\  & PS & PS & PS & ZO & NS & NM & NM & NM \\  & PM & PS & ZO & NS & NM & NM & NM & NB \\  & PB & ZO & ZO & NM & NM & NM & NB & NB \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fuzzy rule table of \(\Delta K_{p}\). Figure 10: Fuzzy PID output surface relationships: (**a**) \(\Delta K_{p}\) output surface, (**b**) \(\Delta K_{I}\) output surface, and (**c**) \(\Delta K_{D}\) output surface. \[\left\{ \begin{array}{l}M_{\omega_{1}}=K_{P}e_{\omega_{1}}+K_{1}\int_{0}^{t}e_{ \omega_{1}}dt+K_{D}\frac{d\omega_{1}}{dt}\\ M_{\omega_{2}}=K_{P}e_{\omega_{2}}+K_{1}\int_{0}^{t}e_{\omega_{2}}dt+K_{D}\frac {d\omega_{2}}{dt}\\ M_{\theta}=K_{P}e_{\theta}+K_{I}\int_{0}^{t}e_{\theta}dt+K_{D}\frac{d\omega_{ 2}}{dt}\end{array} \right. \tag{21}\]  
Through the weighted control of the three, the additional yaw moment \(\Delta M_{1}\) required by the tractor and the additional yaw moment \(\Delta M_{2}\) required by the semi-trailer can be obtained by Equation (22). \[\left\{ \begin{array}{l}\Delta M_{1}=M_{\omega_{1}}\\ \Delta M_{2}=w_{1}M_{\omega_{2}}+w_{2}M_{\theta}\end{array} \right. \tag{22}\]  
where \(w_{1},w_{2}\) are the weight coefficients of the yaw rate controller and the articulation angle controller of the semi-trailer, respectively. After several simulation comparisons, the weight coefficient \(w_{1}\) is taken as 0.6, and \(w_{2}\) is taken as 0.4. Calculation and Distribution of Braking Torque of Yaw Stability Control Method for the Liquid Tank Semi-Trailer  
When the additional yaw moment required for differential braking of the liquid tank semi-trailer is calculated based on the multi-object PID controller, it is necessary to calculate the braking moment generated by the wheels of the tractor and semi-trailer \(T_{1}\), \(T_{2}\). The additional yaw moment of the vehicle is provided by the braking force applied to the wheels. Therefore, only by determining the relationship between the additional yaw moment and the braking force of different wheels can the braking torque be accurately calculated. Take the direction in which the vehicle turns left as the positive one and the other as the negative one. The relationship between the additional yaw moment and the braking moment of the liquid tank semi-trailer under the left-turning condition is shown in Figure 11.  
In Figure 12, \(t_{w1},t_{w2},\ t_{w3}\) is the track width of the front axle of the tractor, the second and third axles of the tractor and the fourth, fifth, and sixth axles of the semi-trailer; \(\Delta M_{1},\Delta M_{2}\) is the additional yaw moment of the tractor and semi-trailer. When the vehicle performs differential braking, supposing that the braking force is applied to the left wheel  
Figure 11: Relationship between additional yaw moment and braking torque of liquid tank semi-trailer. of the first axle of the tractor and the left wheels of the fourth, fifth, and sixth axles of the semi-trailer, it can be obtained from the mechanical knowledge that  
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