## 3 Design of Trajectory Tracking Control System
### Vehicle Instability Judgment and Proportional Controller Allocation Based on the Phase Plane Method
\tag{33}\]  
where \(\text{B}_{i}=\overset{\bullet}{\beta}+\text{A}_{i}\beta(i=1,2,3,4)\) are the four boundary equations of diamond shape; \(\text{A}_{i}\) is the slope of the equation; and \(\text{B}_{i}\) is the constant term of the equation. According to the stability theory of the vehicle, the stability of the vehicle mainly depends on the yaw rate when the sideslip angle of the center of mass is negligible. Referring to the results found in the literature, the critical value of instability U for the yaw rate deviation is shown in Table 4. According to the \(\beta\)-method theory, the yaw rate cannot effectively characterize the vehicle's stability when the mass center lateral angle is large, and the influence of the mass center lateral angle on the vehicle's stability is dominant. Therefore, the car's instability judgment is carried out according to Figure 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Variables** & **Range of Value** & **Step Size** \\ \hline \(v_{x}\)(km/h) & 60–120 & 10 \\ \(\mathfrak{u}\) & 0–1.0 & 0.1 \\ \(\delta_{f}\) (\({}^{\circ}\)) & 0–10 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Value ranges and step sizes of the three variables. Figure 5: Five-parameter diamond phase plan. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \(v_{x}\) (km/h) & \(\mu\) & \(\delta_{f}\) (\({}^{\circ}\)) & \(\dot{\beta}_{lim}^{\star}\) & \(\dot{\beta}_{lim}^{-}\) & \(\dot{\beta}_{lim}^{\star}\) & \(\dot{\beta}_{lim}^{-}\) & \(\dot{\beta}_{re}\) \\ \hline
60 & 0.8 & 0 & 0.835 & \(-\)0.835 & 0.079 & \(-\)0.079 & 0 \\
60 & 0.3 & 0 & 0.265 & \(-\)0.265 & 0.036 & \(-\)0.036 & 0 \\
80 & 0.8 & 0 & 0.672 & \(-\)0.672 & 0.0685 & \(-\)0.0685 & 0 \\
80 & 0.3 & 0 & 0.303 & \(-\)0.303 & 0.034 & \(-\)0.034 & 0 \\
120 & 0.8 & 0 & 0.48 & \(-\)0.48 & 0.055 & \(-\)0.055 & 0 \\
120 & 0.3 & 0 & 0.104 & \(-\)0.104 & 0.018 & \(-\)0.018 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of some database parameters. In the stability control, the controller's weight should be increased when the center of mass lateral deflection angle is large. The stability affected by the longitudinal velocity, the road adhesion coefficient, and the front wheel angle can effectively characterize this situation, as follows: near the diamond boundary, that is, when the stability is small, one situation is that the center of mass lateral deflection angle is large, then, the weight of the controller should be significant. The other case is that the centroid side angle is slight. However, in this case, the velocity of the centroid side angle is large, and the vehicle is about to enter the state with a large lateral angle of the center of mass, and so the weight of the centroid side angle controller should be significant now. Therefore, the stability degree can be used to reasonably allocate the controller weight P and be substituted into Equation (34) to obtain the weighted yaw moment \(\Delta\)M, as follows:  
\[\left\{ \begin{array}{ll}\Delta\text{M}=\text{P}\Delta\text{M}_{\beta}+(1-P) \Delta\text{M}_{\omega}\\ P=\frac{H}{H+S_{\beta}}\end{array} \right. \tag{34}\]  
where H is the distance from the equilibrium point to the vehicle state point on the phase diagram. \begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{v_{x}}\) (km/h) & **U (rad/s)** & \(\mathbf{v_{x}}\) (km/h) & **U (rad/s)** \\ \hline
60 & 0.025 & 90 & 0.028 \\
70 & 0.026 & 100 & 0.030 \\
80 & 0.027 & 120 & 0.030 \\ \hline \end{tabular}
\end{table} 
Table 4: Critical value of yaw rate deviation instability. Figure 6: Process of instability judgment.