# Vehicle Control Strategy Evaluation Based on the Driving Stability Region
## 3 The Driving Stability Region for Control Strategy Evaluation
### Solving the Driving Stability Region of the Vehicle with DYC  
Using the solution process shown in Section 3.1, the driving stability region of the 5 DOF vehicle system was solved with DYC. The vehicle system model (Formula (11)) with an all-wheel drive mode was used for the simulation. The initial simulation conditions were as follows. The value range of the longitudinal velocity \(v_{x}\) was from 10 m/s to 60 m/s and the value interval was 5 m/s; the value range of the front wheel angle, \(\delta_{f}\), was from \(-0.08\) rad/s to 0.08 rad/s and the value interval was 0.01 rad/s. To ensure that the system maintains the dynamic equilibrium state, the magnitude of the driving torque was  
Figure 3: A flow chart for solving the driving stability region of the vehicle.  
calculated according to the Formula (17), with a value range of driving torque, \(T_{d}\), from 7.0018 N\(\cdot\)m to 252.0637 N\(\cdot\)m:  
\[T_{d}=\sqrt{\left(\frac{1}{2}C_{\text{air}\_x}A_{\text{L}\_x}p(v_{x0})^{2} \right)^{2}+\left(\frac{1}{2}C_{\text{air}\_y}A_{\text{L}\_y}\rho(v_{y0})^{2} \right)^{2}}\cdot R_{e} \tag{17}\]  
where \(v_{x0}\) is the initial longitudinal velocity of the vehicle, \(v_{y0}\) is the initial lateral velocity of the vehicle, \(C_{\text{air}\_x}\) is the longitudinal air resistance coefficient, \(C_{\text{air}\_y}\) is the lateral air resistance coefficient, \(A_{\text{L}\_x}\) is the longitudinal windward area of the vehicle, and \(A_{\text{L}\_y}\) is the lateral windward area of the vehicle.  
The bifurcation points of the 5 DOF vehicle system model (an all-wheel drive mode system) with DYC are shown in Table 3.  
Figure 4 shows the driving stability region of the 5 DOF vehicle system model (an all-wheel drive mode) after the introduction of DYC. The region surrounded by the two-dimensional parameter bifurcation set composed of the initial longitudinal velocity and the driving torque, and the coordinate axis, is the driving stability region (the equilibrium points are drawn in Table 3).  
Figure 5 compares the differences between the driving stability regions with and without DYC. When the initial longitudinal velocity is the same, and the longitudinal velocity is 10 m/s, the range of the front wheel steering angle with DYC control is from \(-\)0.0737 rad to 0.0737 rad, which is larger than the range without DYC control (from \(-\)0.0603 rad to 0.0603 rad). When the initial longitudinal speed is 15 m/s, the difference between the extreme values of the front wheel steering angle is the largest, and the DYC control effect is the best. When the initial longitudinal velocity is between 10 m/s and 30 m/s, the expansion of the driving stability region is particularly significant. At this time, the vehicle has a larger steering range, which means it is more stable. When the initial longitudinal velocity is larger than 30 m/s, the driving stability region of the vehicle with DYC still shows an expanding trend, which also means that DYC control still has an effect on improving vehicle stability when driving at high speeds.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{\
\begin{tabular}{c} **Initial Value of** \\ **Longitudinal** \\ **Velocity (m/s)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Driving Torque** \\ **(N\(\cdot\)m)** \\ \end{tabular} } & \multicolumn{4}{c}{**Front Wheel Steering Angle**} \\ \cline{3-6}  & & **Bifurcation** & **Bifurcation** & **Bifurcation** & **Bifurcation** \\  & & **Point 1** & **Point 2** & **Point 2** \\  & & **(with DYC)** & **(without DYC)** & **(without DYC)** & **(with DYC)** \\ \hline
10 & 7.0018 & 0.0737 & 0.0603 & \(-\)0.0603 & \(-\)0.0737 \\
15 & 15.754 & 0.0407 & 0.0271 & \(-\)0.0271 & \(-\)0.0407 \\
20 & 28.0071 & 0.0262 & 0.0163 & \(-\)0.0163 & \(-\)0.0262 \\
25 & 43.7611 & 0.0188