# Vehicle Control Strategy Evaluation Based on the Driving Stability Region
## 2 The Vehicle Dynamics Model with the Control Strategy
### The Vehicle Model with DYC  
In this section, a mature linear feedback control method [18] is selected, and the direct yaw moment calculation expression is:  
\[M=h_{r}(\omega-\omega_{e}) \tag{1}\]  
where \(\omega\) is the actual body yaw rate, \(\omega_{e}\) is the expected body yaw rate, \(h_{r}\) is a negative real constant, and \(h_{r}=-10,000\) is taken for better control effect. The expected yaw rate [19] model selected in the model is:  
\[\omega_{e}=\frac{v_{x}/L}{1+Kv_{x}^{2}}\delta_{f} \tag{2}\]  
where \(L=l_{f}+l_{r}\) is the sum of the front and rear wheelbases and \(K\) is the vehicle stability factor [20]:  
\[K=\frac{m}{L^{2}}\Bigg{(}\frac{l_{f}}{k_{r}}-\frac{l_{r}}{k_{f}}\Bigg{)} \tag{3}\]  
where \(k_{f}\) and \(k_{r}\)[21] are the cornering stiffness of the front and rear wheels, respectively, so that \(k_{f,r}=\mathrm{B}_{f,r}\times\mathrm{C}_{f,r}\times\mathrm{D}_{f,r}\). \(\mathrm{B}_{f,r}\) are the stiffness factors of the front and rear wheels, \(\mathrm{C}_{f,r}\) are the shape factors of the front and rear wheels, and \(\mathrm{D}_{f,r}\) are the peak factors of front and rear wheels. The 5 DOF vehicle system model with an all-wheel drive mode is selected as the analysis object [22], and the direct yaw moment control is introduced into the wheel rotation equation, including driving torque. The 5 DOF vehicle system model with DYC is obtained and is shown as follows:  
\[\left\{ \begin{array}{l}\dot{v}_{y}=-v_{x}\omega+\frac{\Gamma_{ff}\sin\delta_{f}+ \Gamma_{sf}\cos\delta_{f}+\Gamma_{br}-\text{sgn}(v_{y})\cdot C_{air,y}A_{L,y} \frac{\xi}{\tau_{y}^{2}}}{m}\\ \dot{v}_{x}=v_{y}\omega+\frac{\Gamma_{ff}\cos\delta_{f}-F_{sf}\sin\delta_{f}+ \Gamma_{br}-\text{sgn}(v_{x})\cdot C_{air,x}A_{L,x}\frac{\xi}{\tau_{y}^{2}}\sigma _{x}^{2}}{m}\\ \dot{\omega}=\frac{(\frac{\Gamma_{ff}\sin\delta_{f}+F_{sf}\cos\delta_{f}}{m}) \frac{\Gamma_{br}-F_{sf}\sin\delta_{f}}{l_{x}}+M}{l_{x}}\\ \dot{\omega_{f}}=\frac{\Gamma_{ff}-\text{sgn}(\omega_{f})\cdot T_{bf}-R_{e} \cdot F_{lf}}{l_{x}}\\ \dot{\omega_{r}}=\frac{T_{bf}-\text{sgn}(\omega_{f})\cdot T_{br}-R_{e}\cdot F _{lf}}{l_{x}}\end{array} \right. \tag{4}\]  
where \(v_{y}\) is lateral velocity, \(\omega\) is yaw rate, \(v_{x}\) is longitudinal velocity, \(\omega_{f}\) is the angular velocity of front wheels, \(\omega_{r}\) is the angular velocity of rear wheels, \(T_{bf}\) is the braking torque of front wheels, \(T_{br}\) is the braking torque of rear wheels, \(F_{lf}\) is the longitudinal tire force of front wheels, \(F_{lr}\) is the longitudinal tire force of rear wheels, \(F_{sf}\) is the lateral tire force of front wheels, \(F_{sr}\) is the lateral tire force of rear wheels, \(l_{f}\) is the distance from the front wheels to the center of mass, \(l_{r}\) is the distance from the rear wheels to the center of mass, \(R_{e}\) is the wheel rolling radius, \(J_{\omega}\) is the moment of inertia of the wheel, \(l_{z}\) is the moment of inertia of the vehicle around the \(Z\) axis, \(m\) is the vehicle mass, \(\delta_{r}\) is the rear wheel steering angle, \(\delta_{f}\) is the front wheel steering angle, \(C_{air\_x}\) is the longitudinal air drag coefficient, \(A_{L\_y}\) is the lateral windward area of the vehicle, \(C_{air\_y}\) is the lateral air resistance coefficient, \(A_{L\_x}\) is the longitudinal windward area of the vehicle, \(\rho\) is air density, \(T_{df}\) is the driving torque of the front wheels, and \(T_{dr}\) is the driving torque of the rear wheels. The calculation method of tire force adopts [PERSON]'s classic Magic Formula [23]:  
\[F=\text{Dsin}(\text{Carctan}(\text{B}x-\text{E}(\text{B}x-\text{arctan}\ \text{B}x))) \tag{5}\]  
where \(\text{B},\text{C},\text{D},\text{and}\ \text{E}\) are tire parameters (B is the stiffness factor, C is the shape factor, D is thepeak factor, and E is the curvature factor), \(F\) is the steady longitudinal force or lateral force of the tire, and \(x\) is the longitudinal slip rate or sideslip angle.