## 1 Introduction
### Selection of Control Parameters for Yaw Stability Control of the Liquid Tank Semi-Trailer  
In terms of the selection of the control parameters, the yaw rate, the sideslip angle of the centroid, and the lateral acceleration are generally taken for stability control [16]. When these variables are used for the control, the vehicle's yaw stability will be greatly improved, but the driving path deviation would be produced compared with the path under steady-state driving. The magnitude of the articulated angle of the liquid tank \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(m_{1}\) & 5876 & kg & \(m_{1s}\) & 4457 & kg \\ \(m_{2}\) & \(25,876\) & kg & \(m_{2s}\) & \(20,000\) & kg \\ \(a\) & 2 & m & \(b\) & 2.478 & m \\ \(c\) & 2.189 & m & \(d\) & 4.693 & m \\ \(e\) & 5.4 & m & \(h_{1}\) & 1.175 & m \\ \(h_{2}\) & 2.125 & m & \(h_{1x}\) & 1.1 & m \\ \(h_{2c}\) & 1.1 & m & \(l_{1 xx}\) & 2283 & kg\(\cdot\)m\({}^{2}\) \\ \(l_{1 yy}\) & \(35,402\) & kg\(\cdot\)m\({}^{2}\) & \(l_{1 zz}\) & \(34,802\) & kg\(\cdot\)m\({}^{2}\) \\ \(l_{1 xx}\) & 1626 & kg\(\cdot\)m\({}^{2}\) & \(l_{2 xx}\) & \(22,330\) & kg\(\cdot\)m\({}^{2}\) \\ \(l_{2 zz}\) & \(250,416\) & kg\(\cdot\)m\({}^{2}\) & \(l_{2 xz}\) & 0 & kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Basic parameters of the liquid tank semi-trailer. semi-trailer directly reflects the steering characteristics and driving path of the liquid tank semi-trailer. The deviation of the driving path indicates the path-following ability of the semi-trailer to the tractor [17]. The greater the deviation is, the worse the path-following performance would be. Therefore, if the articulation angle can be adjusted to that of the steady state, while maintaining the stability of the vehicle's yaw rate, the vehicle would have both good yaw stability and good path-following performance. Therefore, the tractor yaw rate, the semi-trailer yaw rate, and the articulation angle are selected as the control parameters. Determination of Expected Values for Yaw Stability Control Parameters of the Liquid Tank Semi-Trailer  
When implementing the control method, it is necessary to obtain the vehicle state value when the vehicle is running stably, that is, the expected value of the vehicle state response. Take the derivative value of each state variable in the six degrees of freedom model of the liquid tank semi-trailer in Section 2.1 as zero, to obtain the vehicle linear reference model required by the control system, that is \(\hat{\beta_{1}}=\hat{\psi}_{1}=\hat{\psi}_{1}=\hat{\psi}_{1}=\hat{\psi}_{2}= \hat{\psi}_{2}=\hat{\theta}=0\). Then, the expected value of each state variable under a stable state can be determined as \(X_{d}\):  
\[\left\{ \begin{array}{l}X_{d}=\left[\beta_{1d}\;\dot{\psi}_{1d}\;\phi_{1d}\; \phi_{2d}\;\theta_{d}\right]^{T}\\ 0=N_{d}X_{d}+K_{d}U_{d}\end{array} \right. \tag{15}\]  
where \(U_{d}\) is the system input, \(U_{d}=\delta\); \(K_{d}\) is a \(5\times 1\) dimensional matrix, \(K_{d}=\left[-(a+c)k_{1}-h_{1c}k_{1}-k_{1}\;0\;0\;0\right]^{T}\); and \(N_{d}\) is a \(5\times 5\) dimensional matrix. \[N_{d}=\left[ \begin{array}{ccccc}(a+c)k_{1}+(c-b)k_{2}&N_{d12}&0&0&0\\ (k_{1}+k_{2})h_{1c}&N_{d22}&N_{d23}&k_{12}&0\\ k_{1}+k_{2}+k_{3}&N_{d32}&0&0&-k_{3}\\ -(d+e)k_{3}&N_{d42}&0&0&(d+e)k_{3}\\ h_{2c}k_{3}&N_{d52}&k_{12}&N_{d54}&-h_{2c}k_{3}\end{array} \right] \tag{16}\]  
\[N_{d12}=\frac{\left(a^{2}+ac\right)k_{1}}{u_{1}}+\frac{\left(b^{2}-bc\right) k_{2}}{u_{1}}-m_{1}u_{1}c;\;N_{d22}=(m_{1s}h_{1}-m_{1}h_{1c})u_{1}+\frac{(ak_{1}- bk_{2})h_{1c}}{u_{1}}\]  
\[N_{d32}=\frac{ak_{1}-bk_{2}-ck_{3}}{u_{1}}-m_{2}u_{2}-m_{1}u_{1}-\frac{(d+e)k_{ 3}}{u_{2}};\;N_{d42}=\frac{(d+e)^{2}k_{3}}{u_{2}}+m_{2}u_{2}e+\frac{(d+e)ck_{3} }{u_{1}}\]  
Figure 6: Comparison of vehicle state responses between simplified model and TruckSim model: (**a**) comparison of yaw rate of the tractor and (**b**) comparison of articulation angle.