# Research on the Vehicle Steering and Braking Stability Region
## 2 Stability Analysis of 5 DOF Steering and Braking Vehicle Model
### Tire Model  
To reflect the nonlinear characteristics of tire force, the expression of tire force adopts the classic magic formula [29]:  
\[F=\mathrm{D}\sin(\mathrm{C}\mathrm{arctan}(\mathrm{B}x-\mathrm{E}(\mathrm{B}x- \mathrm{arctanB}x))) \tag{5}\]  
where B, C, D, and E are tire parameters, \(F\) is the tire steady-state longitudinal force or lateral force, and x is the longitudinal slip rate or sideslip angle.  
\begin{table}
\begin{tabular}{c c} \hline
**Component Names and** & **Values** \\
**Parameters** & **Values** \\ \hline vehicle mass & 1500 \\ yaw moment of inertia & 3000 \\ \(\mathrm{l}_{z}/\mathrm{kg}\cdot\mathrm{m}^{2}\) & 1.2 \\ the distance from the front axle to the mass center & 1.3 \\ the distance from the rear axle to the mass center & 2.0 \\ yaw moment of inertia of the wheels & 1.3 \\ \(\mathrm{J}/\mathrm{kg}\cdot\mathrm{m}^{2}\) & 2.0 \\ longitudinal air resistance & 1.3 \\ coefficient & 2.0 \\ \(\mathrm{C}_{\mathrm{air},x}\) & 1.3 \\ the lateral air resistance & 1.3 \\ \(\mathrm{C}_{\mathrm{air},y}\) & 1.3 \\ the longitudinal area of the vehicle & 1.3 \\ \(\mathrm{A}_{\mathrm{L},x}/\mathrm{m}^{2}\) & 1.3 \\ the lateral area of the vehicle & 3.5 \\ \(\mathrm{A}_{\mathrm{L},y}/\mathrm{m}^{2}\) & 1.3 \\ the density of air & 1.2258 \\ \(\rho/\mathrm{kg}/\mathrm{m}^{3}\) & 1.2258 \\ the rolling radius of the wheels & 0.224 \\ \(R_{\mathrm{e}}/\mathrm{m}\) & \\ \hline \end{tabular}
\end{table}
Table 1: Vehicle model parameters.  
The calculation of the slip rate is based on the unified formula considering all tire operating conditions in [13].  
\[k=\frac{\omega_{w}R_{e}-v_{wx}}{|v_{wx}|} \tag{6}\]  
The tire velocity in the tire coordinate system is calculated by coordinate transformation.  
\[\left\{\begin{array}{l}v_{xf}=v_{x}\cos\delta_{f}+(v_{y}+\omega l_{f})\sin \delta_{f}\\ v_{xr}=v_{x}\cos\delta_{r}+(v_{y}-\omega l_{r})\sin\delta_{r}\end{array}\right. \tag{7}\]  
\[\left\{\begin{array}{l}v_{yf}=-v_{x}\sin\delta_{f}+(v_{y}+\omega l_{f})\cos \delta_{f}\\ v_{yr}=-v_{x}\sin\delta_{r}+(v_{y}-\omega l_{r})\cos\delta_{r}\end{array}\right. \tag{8}\]  
where \(v_{xf}\) is the longitudinal velocity of the front wheel in the tire coordinate system, and \(v_{xr}\) is the longitudinal velocity of the rear wheel in the tire coordinate system.  
The sideslip angle is calculated by using the unified formula which is suitable for the conditions of a large sideslip angle and reverse wheel rotation [1]:  
\[\left\{\begin{array}{l}\alpha_{f}=\arctan\frac{v_{yf}}{v_{xf}}\cdot\text{ sgn}\Big{(}v_{xf}\Big{)}\\ \alpha_{r}=\arctan\frac{v_{yr}}{v_{xr}}\cdot\text{sgn}(v_{xr})\end{array}\right. \tag{9}\]  
The combined slip of the tire is analyzed and calculated according to the model proposed by [PERSON] [30].  
\[\left\{\begin{array}{l}F_{ff}=F_{ff0}\cdot G_{x}\\ F_{lr}=F_{ff0}\cdot G_{x}\\ G_{x}=\cos[\arctan\big{\{}B_{g,x}(\alpha)\cdot\alpha\big{\}}]\\ B_{g,x}(\alpha)=r_{x,1}\cos[\arctan(r_{x,2}\cdot k)]\\ F_{sf}=F_{sf0}\cdot G_{y}\\ F_{sr}=F_{sr0}\cdot G_{y}\\ G_{y}=\cos[\arctan\big{\{}B_{g,x}(k)\cdot k\big{\}}]\\ B_{g,g}(k)=r_{y,1}\cos[\arctan(r_{y,2}\cdot\alpha)]\end{array}\right. \tag{10}\]  
where \(F_{ff0}\), \(F_{ff0}\), \(F_{sf0}\), \(F_{sr0}\) are the longitudinal forces and lateral forces of the front and rear wheels under a steady state, \(G_{x}\) and \(G_{y}\) are the tire force combined slip correction function, and \(r_{x,1}\), \(r_{x,2}\), \(r_{y,1}\), \(r_{y,2}\) are the tire force combined slip correction coefficient. The values of the tire model parameters [30] are shown in Table 2.