# Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving
## 2 Fuzzy Cooperative Control Strategy Based on LTV-MPC
### Vehicle Steady State Analysis Based on Phase Plane  
The dynamic response of a vehicle depends on the characteristics of nonlinear tires. Under extreme driving conditions, the dynamic characteristics of vehicle dynamics will change rapidly, and even lead to instability [17; 18]. The phase plane method, as a time-domain analysis method, is widely used in the study of stability and dynamic characteristics of nonlinear differential systems because it can reveal the equilibrium positions and types in the system, as well as the attractive regions and stable regions related to these equilibria. This section analyzes the dynamic performance of multi-axle heavy vehicles based on the phase plane \(\beta-\omega_{z}\), obtaining their critical stable steering angle and critical linear steering angle, and dividing the vehicle stability region based on this. On the basis of Equation (4), considering the magic tire model with pure side slip, and without considering the additional yaw moment \(M_{d}\), the nonlinear system differential equation about \(\beta\) and \(\omega_{z}\) can be recorded as  
\[ \begin{cases}\dot{\beta}=f_{1}(\beta,\omega_{z})\\ \dot{\omega}_{z}=f_{2}(\beta,\omega_{z})\end{cases}  \tag{10}\]  
By giving different initial values \((\beta,\omega_{z})\) to Equation (10), the phase trajectories of the differential equation and the phase plane diagram composed of the phase trajectories can be obtained. With the increase of vehicle steering angle input in the phase plane, the vehicle will undergo a state change from stable to unstable, which is also known as the phase plane bifurcation phenomenon. In order to study the vehicle steering angle under this critical state, two-axle passenger cars usually introduce the lateral slip angle \(\alpha_{imax}\) corresponding to the lateral force saturation of the front and rear tires and analyze the phase plane in detail by combining the yaw rate boundary. For five-axle vehicles, the side slip angles of the front two axles are relatively close, while in the rear three axles, there is a relationship between the side slip angles: \(\alpha_{5}>\alpha_{4}>\alpha_{3}\). Therefore, the mean of the first two axles' sideslip angles and the fifth axle sideslip angle are chosen as the boundary. The specific boundary conditions can be expressed as:  
\[ \begin{cases}|\omega_{z}|\leq\left|\frac{\mu\overline{\alpha}}{\overline{ \alpha}_{z}}\right|\\ |\alpha_{f}|\leq\left|\alpha_{fmax}\right|\\ |\alpha_{5}|\leq|\alpha_{5 max}|\end{cases}  \tag{11}\]  
Figure 4 shows the phase plane corresponding to different steering angle inputs under the conditions of road adhesion \(\mu=0.8\) and vehicle speed \(v_{x}=20\) m/s. The solid circle, solid triangle, and solid diamond in the figure represent the equilibrium point, saddle point, and focal point, respectively. The black, blue, and red lines represent the yaw rate boundary, \(\alpha_{f}\) boundary, and \(\alpha_{5}\) boundary, respectively. It can be seen that when \(\delta_{1}=0.1\) rad, the vehicle equilibrium points are all within the three corresponding boundaries. As the steering angle increases, the equilibrium point gradually approaches the left saddle point. When \(\delta_{1}=0.188\) rad, the left saddle point coincides with the equilibrium point, and the vehicle reaches a critical stable state. At this point, the equilibrium point precisely reaches the saturation limit of \(\alpha_{5}\). Note that \(\delta_{1c}=0.188\) rad is the critical stable steering angle. When \(\delta_{1}>\delta_{1c}\), the system will undergo bifurcation and the equilibrium point of the system will shift from a stable node to a focal point. Comparing Figure 4b,c, it can be seen that the stable node near the equilibrium point converges faster than the stable focal point. When \(\delta_{1}=0.3\) rad, the vehicle state oscillates around the focal point and finally stabilizes.