# Research on the Vehicle Steering and Braking Stability Region
## 3 Solution of Equivalent Equilibrium Point
### Introducing the D'Alembert Principle into the 5 DOF Model
\tag{13}\]  
The equilibrium point solution of a SDOF system can be regarded as a programming problem constrained by Equation (13):  
\[\text{min}f\Big{(}\dot{v}_{x},\dot{v}_{y},\dot{\omega},\dot{\omega_{f}},\dot{ \omega_{r}}\Big{)}=\left|\dot{v}_{x}\right|+\left|\dot{v}_{y}\right|+\left| \dot{\omega}\right|+\left|\dot{\omega_{f}}\right|+\left|\dot{\omega_{r}}\right| \tag{14}\]  
The optimal solution \(\left(\dot{v}_{x}^{*},\dot{\omega}_{y}^{*},\dot{\omega}_{f}^{*},\dot{\omega}_ {r}^{*}\right)\) must satisfy \(f\Big{(}\dot{v}_{x}^{*},\dot{v}_{y}^{*},\dot{\omega}_{f}^{*},\dot{\omega}_{r }^{*}\Big{)}=0\), and the fitness value of the 5 DOF system is as follows:  
\[\text{Fitness}=\left|\dot{v}_{x}\right|+\left|\dot{v}_{y}\right|+\left|\dot{ \omega}\right|+\left|\dot{\omega_{f}}\right|+\left|\dot{\omega_{r}}\right| \tag{15}\]  
To verify the equivalent equilibrium state of the nonlinear system after adding the virtual force, Figure 4 shows the comparison of multiple initial point phase trajectory and single initial point state variables time series of the system without or with the virtual force. Figure 4: _Cont._  
The initial conditions of the multi-initial points simulation are: longitudinal velocity \(v_{x}=30\) m/s, front wheel steering angle \(\delta_{f}=0.001\) rad, braking torque \(T_{b}=500\) N\(\cdot\)m, lateral velocity \(v_{y}\) ranging from \(-10\) m/s to \(10\) m/s, with intervals of \(1\) m/s, and the value range of the yaw rate \(\omega\) is \(-0.5\) rad/s to \(0.5\) rad/s, with value intervals of \(0.1\) rad/s. The single initial  
Figure 4: Comparison of the 5 DOF model with and without virtual force: (**a**) Phase space without adding virtual force; (**b**) Phase space with virtual force; (**c**) Local view of (**a**); (**d**) Local view of (**b**); (**e**) Phase plane without virtual force; (**f**) Phase plane with virtual force; (**g**) Time domain without virtual force; (**h**) Time domain with virtual force. point simulation conditions are: longitudinal velocity \(v_{x}\) = 30 m/s, front wheel steering angle \(\delta_{f}\) = 0.001 rad, and braking torque \(T_{b}\) = 500 N\(\cdot\)m.  
Figure 4a,b show the global dynamic characteristics of the system, and it can be seen that the global characteristics of the system have not changed obviously after adding the virtual force. However, by comparing the local diagrams of Figure 4c,d, it can be seen that the downward spiral motion trend of the phase trajectory (phase trajectory in the green rectangle in Figure 4a) disappears and is replaced by the spiral motion in the two-dimensional plane in Figure 4d. This means that the longitudinal velocity of these phase trajectories remains constant. The attraction domains of Figure 4e,f are almost the same. It is noted that the two green phase trajectories at the edge of the attraction domain in Figure 4f can return to the original point, while they cannot become stable in Figure 4e. It is indicated that the stability of the system is slightly improved. Figure 4g,h are time series of phase trajectory state variables under stable initial conditions. It can be seen that the longitudinal velocity changes from linear decreasing to basically constant, and the lateral velocity and yaw rate also remain constant during the simulation time. It shows that the system under these initial conditions can achieve the equivalent equilibrium state. Figure 4 confirms the equivalent equilibrium state of the system from the global and local perspectives, so that the static bifurcation analysis of the nonlinear system can be carried out.