# Research on the Vehicle Steering and Braking Stability Region
## 2 Stability Analysis of 5 DOF Steering and Braking Vehicle Model
### Model Validation  
In this section, the effectiveness of the model is verified by the phase space trajectory (\(v_{y}\)-\(\omega\)-\(v_{x}\)) of different single initial conditions, the time series of state variables, and the change of body attitude. See Table 3 for the initial conditions of specific simulations.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Tire**} & \multicolumn{3}{c}{**Longitudinal** Tire Parameters} & \multicolumn{3}{c}{**Lateral Tire Parameters**} & \multicolumn{3}{c}{**Longitudinal**} & \multicolumn{3}{c}{**Lateral**} \\  & \multicolumn{3}{c}{**Longitudinal** Tire Parameters} & \multicolumn{3}{c}{**Lateral Tire Parameters**} & \multicolumn{3}{c}{**Combined Slip Coefficients**} & \multicolumn{3}{c}{**Combined Slip Coefficients**} \\ \cline{2-11}  & **B** & **C** & **D** & **E** & **B** & **C** & **D** & **E** & \(r_{x,1}\) & \(r_{x,2}\) & \(r_{y,1}\) & \(r_{y,2}\) \\ \hline Front & 11.275 & 1.56 & 2574.8 & 0.4109 & 11.275 & 1.56 & 2574.7 & \(-\)1.999 & & & & & \\ Rear & 18.631 & 1.56 & 1749.6 & 0.4108 & 18.631 & 1.56 & 1749.7 & \(-\)1.7908 & & 35 & 40 & 40 & 35 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Tire model parameters.  
Figure 2 shows the simulation results, with a simulation time of 30 s. The blue line, red line, and black line in Figure 2a represent the dynamic behavior of the system under the input of initial conditions 1, 2, and 3, respectively. Under the input of initial condition 1, the phase space trajectory first appears a wide range of semi-circular motion, corresponding to the fluctuation of lateral velocity and yaw rate in Figure 2c for the first 5 s. After that, the phase trajectory inclines downward, which also corresponds to the rapid convergence of lateral velocity and yaw rate in Figure 2c and the linear decrease of longitudinal velocity. It can be seen in Figure 2d from the change of body attitude that the vehicle is in a steady-state steering state and the distance between the two vehicles is decreasing continuously, indicating that the vehicle will eventually be stationary.  
The initial condition 2 increases the front wheel steering angle input compared with the initial condition 1. It can be seen that the motion range of the phase trajectory in the  
Figure 2: Time-domain responses of single initial conditions and vehicle body attitude: (**a**) Phase space trajectories; (**b**) Top view of a; (**c**) Time-domain responses; (**d**) Vehicle body attitude.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Initial Conditions** & \(\begin{array}{c}\mathbf{v_{x}}\\ \mathbf{(m/s)}\end{array}\) & \(\mathbf{v_{y}}\) & \(\mathbf{\omega}\) & \(\delta_{f}\) & \(\mathbf{T_{b}}\) \\ \hline
1 & 30 & 0.1 & 0.1 & 0.010 & 300 \\
2 & 30 & 0.1 & 0.1 & 0.015 & 300 \\
3 & 30 & 0.1 & 0.1 & 0.015 & 600 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Initial conditions.  
early stage is larger, and the state variables fluctuate more violently in the early stage as well, but it can still be stabilized. We also can find that the turning radius of the vehicle is smaller than that in the initial condition 1.  
The initial condition 3 increases the braking torque input compared with the initial condition 2. It can be seen that the motion range of the phase trajectory in the early stage is smaller, and the fluctuation amplitude of state variables is smaller in the early stage too. The longitudinal velocity linearly decreases to 0 when the simulation time is about 16 s. The vehicle can also be observed to be stationary during the body attitude change, and the overall braking distance is significantly reduced compared with the previous two groups of simulations.  
From the above analysis, it can be found that the 5 DOF model can reflect the motion process of the vehicle under the combined steering and braking conditions, thus providing a basic analysis model for studying the dynamic bifurcation characteristics of the vehicle under the combined steering and braking conditions.