# Research on the Vehicle Steering and Braking Stability Region
## 6 Conclusions  
In this paper, the stability region of the steering and braking condition in vehicle planar motion is solved by the equilibrium point bifurcation analysis method. In addition, it is verified using the energy dissipation method. The main work and conclusions are as follows:  
1. A 5 DOF vehicle dynamic model has been established that can analyze the stability of vehicles under steering and braking conditions effectively. The application conditions of braking torque, the locking of front and rear wheels, and the limitation of road adhesion are considered in the model. The validation results indicate that this model can describe the dynamic characteristics of vehicle braking and steering conditions accurately. It can serve as the basic model for braking and steering stability analysis. 2. The original system has been transformed into the equivalent system by using the D'Alembert principle. The equilibrium points of the equivalent system are solved by the hybrid algorithm combining the genetic algorithm and sequential quadratic programming method. The bifurcation diagram of the equilibrium point with the braking torque is calculated. The results indicate that as the braking torque increases, the number of equilibrium points increases to three from one, and the system state changes to stable from unstable. The bifurcation characteristics have been confirmed by the phase portrait. 3. The process of solving the stability region under steering and braking conditions has been determined, and the two-dimensional bifurcation parameter set (front wheel steering angle and braking torque) has been obtained for different longitudinal velocities. The three-dimensional stability region (longitudinal velocity, front wheel steering angle and braking torque) was fitted and verified. 4. Stable regions of the system with and without inertia force have been solved using the energy dissipation method, and compared with the stability region obtained by the equivalent static bifurcation method. The results indicate that the equilibrium point bifurcation method proposed in this paper for steering and braking conditions can effectively solve the stability region of the equivalent system. The stability region of the equivalent system is almost the same as that of the original system without applying virtual force. When the limited braking torque is 500 N-m and the initial longitudinal velocity increases from 30 m/s to 50 m/s, the range of the stable region significantly decreases, and the absolute value of the front wheel steering angle at the boundary point changes from less than 0.02 rad to more than 0.02 rad. Due to the danger of high-speed steering and braking conditions, as well as the extensive experiments required to explore the unstable region of the vehicle system, it has been difficult to achieve actual vehicle experimental verification so far. In the future, we will verify and analyze the results of steering and braking stability regions with a driving simulator. This paper mainly studies vehicle steering and braking stability in planar motion, without considering the impact of the dynamic characteristics of the pitch and roll. In future research, factors such as pitch and roll will be considered to explore the effects of tire load changes and different braking modes on the system equilibrium point bifurcation and stability region. **Author Contributions:** Conceptualization, X.W. and W.L. ; methodology, X.W. ; software, W.L. ; validation, F.Z., Z.L. and W.B. ; writing--original draft preparation, W.L. ; writing--review and editing, W.L. and X.W. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the Key R&D Plan of Heilongjiang Province, grant number JD22A014 and by National Natural Science Foundation of China, grant number 52175497. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest.