# Research on the Vehicle Steering and Braking Stability Region
## 2 Stability Analysis of 5 DOF Steering and Braking Vehicle Model
### Vehicle Model
\tag{1}\]  
where \(m\) is the mass of the vehicle, \(I_{z}\) is the yaw moment of inertia of the vehicle, \(v_{x}\) is the longitudinal velocity, \(v_{y}\) is the lateral velocity, \(\omega\) is the yaw rate, \(\omega_{f}\) is the angular velocity of the front wheels, \(\omega_{r}\) is the angular velocity of the rear wheels, \(l_{f}\) is the distance from the front axle to the mass center, \(l_{r}\) is the distance from the rear axle to the mass center, \(J\) is the yaw moment of inertia of the wheels, \(\delta_{f}\) is the front wheels' steering angle, \(C_{\text{air},x}\) is the longitudinal air resistance coefficient, \(C_{\text{air},y}\) is the lateral air resistance coefficient, \(A_{\text{L},x}\) is the longitudinal area of the vehicle, \(A_{\text{L},y}\) is the lateral area of the vehicle, \(\rho\) is the density of air, \(T_{bf}\) is the braking torque of the front wheels, \(T_{bf}\) is the braking torque of the rear wheels, \(R_{\text{e}}\) is the rolling radius of the wheels, \(F_{lf}\) and \(F_{lr}\) are the longitudinal tire forces of the front wheel and the rear wheel, respectively, and \(F_{sf}\) and \(F_{sr}\) are the lateral tire forces of the front wheel and the rear wheel, respectively. For the convenience of description, the braking torque \(T_{b}\) is regarded as a positive scalar in this paper. It is mainly applied to the front wheel to prevent the rear wheel from locking and slipping first. The braking torque of the front and rear wheels is distributed according to the following formula:  
\[\left\{ \begin{array}{l}T_{bf}=\eta T_{b}\\ T_{br}=(1-\eta)T_{b}\end{array} \right. \tag{2}\]  
where \(\eta\) is the braking torque distribution coefficient, and \(\eta\) = 0.7. To improve the stability of braking and make the best use of the road adhesion conditions, a braking mode with both the front and rear wheels locked simultaneously is adopted. On any road surface with an adhesion coefficient \(\mu\) (\(\mu\) = 0.3 in this paper), the  
Figure 1: The 5 DOF model: (**a**) Single-track model with longitudinal force added; (**b**) The front and rear wheel braking model. condition for both the front and rear wheels to lock at the same time is that the sum of the braking forces of the front and rear wheels are equal to the adhesion force, and the braking forces of the front and rear wheels are equal to their respective adhesion forces [27], that is:  
\[\left\{ \begin{array}{l}F_{\mu f}+F_{\mu r}=\mu F_{z}\\ F_{\mu f}=\mu F_{zf}\\ F_{\mu r}=\mu F_{zr}\end{array} \right. \tag{3}\]  
where \(F_{\mu f}\) is the braking force of the front wheel brake, \(F_{\mu r}\) is the braking force of the rear wheel brake, and \(F_{z}\) is the total load of the front and rear wheels of the vehicle. Only plane motion is considered, therefore the front wheel load \(F_{\vec{z}f}\) and the rear wheel load \(F_{zr}\) are fixed values, and they are distributed as follows:  
\[\left\{ \begin{array}{l}F_{zf}=\frac{mgI_{f}}{I_{f}+I_{c}}\\ F_{zr}=\frac{mgI_{c}}{I_{f}+I_{r}}\end{array} \right. \tag{4}\]  
Without loss of generality, the simulation in this paper is carried out on a low adhesion coefficient road. The model parameters of the 5 DOF system are selected according to [3; 28], as shown in Table 1.