# Research on the Vehicle Steering and Braking Stability Region
## 1 Introduction
used the Lyapunov exponents method and energy dissipation method to quantitatively analyze the stability region of the system under steering and driving conditions and obtained the stability region of state variables (lateral velocity and yaw rate) [15] and the stability region of system control parameters (front wheel angle and driving torque) [16], respectively. The different characteristics of dynamics between the high-degree-of-freedom model and the 2 DOF model were revealed and explained. It was also proved that the maximum deviation of the energy dissipation method from the equilibrium point bifurcation method is less than 1.2%. Braking is a common condition in vehicle driving, and many scholars have also studied its stability through nonlinear analysis methods. Based on considering the wheel slip dynamics, [PERSON] et al. studied the longitudinal braking dynamics of a two-wheeled vehicle model coupled with the vehicle body and studied the variety of wheel slip rate when the braking torque changes through a bifurcation diagram, showing the whole process of how the system transits from stable braking to locking in one or two wheels, and then locking in two wheels [17]. [PERSON] et al. proposed an analysis method based on constrained bifurcation and continuity methods with the D'Alembert principle introduced, established a vehicle dynamics model including longitudinal velocity, lateral velocity, yaw rate and roll, and analyzed the bifurcation phenomenon caused by braking deceleration [18]. [PERSON] et al. studied the 3 DOF dynamic model combined with the Burckhardt tire model, studied the change of system equilibrium point under front/rear/four-wheel braking by the phase plane method, and obtained the stable region corresponding to the slip rate of the front and rear wheels when the vehicle turns to brake [19]. [PERSON] et al. considered engine braking in the 5 DOF model and obtained the bifurcation diagram of the equilibrium point with the change of front wheel angle and longitudinal velocity through the phase plane method and static bifurcation theory [20]. [PERSON] et al. constructed an 18-degree-of-freedom unified dynamic model of a vehicle chassis including three subsystems: a steering subsystem, braking subsystem and suspension subsystem, analyzed the influence of front and rear wheel braking modes on vehicle driving stability by the phase plane analysis method, and found that front wheel braking is more stable than rear wheel braking [21]. A new model based on factors of the driver, vehicle and environment was developed to determine the'safe stopping distance' of vehicles, and tested on real road conditions by using equipment placed on a vehicle [22]. It can be stated that, as a nonlinear dynamic system with multiple degrees of freedom, the steering instability mechanism of a vehicle has been basically recognized as a bifurcation phenomenon in a nonlinear system, and the instability mechanism under driving and steering conditions can also be explained by bifurcation theory. From the above research, it can be found that due to the particularity of braking dynamics, many scholars have focused on the design of controllers and the analysis of braking mechanisms based on simple vehicle models. When the system dimensions increase, it is necessary to simplify the model or improve the nonlinear analysis methods for analysis. However, there is still a lack of achievements in terms of analyzing or calculating the braking stability region. By systematically studying the vehicle steering and braking stability region, the integrated control theory of vehicle handling stability will be further improved, and the theoretical fundamentals of nonlinear vehicle dynamics will be provided for the longitudinal and lateral coupled control of intelligent vehicles under extreme conditions (mostly steering and braking working conditions) [23; 24; 25].