# Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving
## 2 Fuzzy Cooperative Control Strategy Based on LTV-MPC  
### Assisted Driving Control Model  
The controlled object is a five-axle heavy vehicle with an assisted steering system, which acts on the first two axles. The assisted steering system is a coupled assisted steering system, which works by outputting assisted steering torque through the assisted steering motor, coupling with the driver's input torque and the steering system's aligning torque, and ultimately forming a steering angle response to assist in controlling the vehicle. Meanwhile, each wheel can be controlled independently and precisely by torque. This section builds a path tracking model for assisted driving, including a steering system model, the 2-DOF vehicle model, and the path-tracking model. The structure of the assisted steering system model is shown in Figure 2. The assisted steering system includes the first and second axles, with the steering column between the steering wheel and the axle, and the steering dynamics are analyzed with the steering column as the object. The transmission between the steering column and the axle is carried out through gear and rack, and the transmission ratios between the gear and rack and the first and second axles are \(k_{s}\) and \(p\cdot k_{s}\), respectively. Here, \(k_{s}>1,p=1.25\). The active force received by the assisted steering system includes the torque \(T_{h}\) applied by the driver through the steering wheel and the assisted steering motor's assisted torque \(T_{a}\), which includes the driver assistance component \(T_{a1}\) and the auxiliary component \(T_{a2}\). \(T_{a1}\) and \(T_{a2}\) all act directly on the steering column. \(T_{a1}\) achieves fixed proportion torque assistance for the driver, meeting \(T_{a1}=k_{h}T_{h}\). Meanwhile, the system is also subjected to differential torque \(M_{di}\) and aligning torque \(M_{si}\) acting on the tires of the first and second axles, where differential torque is generated due to inconsistent longitudinal force allocation between the left and right tires. The dynamic equation of the assisted steering system can be expressed as:  
\[f_{s}\ddot{\theta}_{s}+b_{s}\dot{\theta}_{s}=T_{h}+T_{a1}+T_{a2}+\frac{M_{a1}} {k_{s}}+\frac{M_{a2}}{p\cdot k_{s}}-\frac{M_{a1}}{k_{s}}-\frac{M_{a2}}{p\cdot k _{s}} \tag{1}\]  
In the formula, \(f_{s}\) is the equivalent moment of inertia of the steering column, \(b_{s}\) is the equivalent damping, \(\theta_{s}\) is the steering column angle. The aligning torque \(M_{si}\) mainly consists of the lateral force \(M_{syi}\) of the tire and the component of the aligning torque of the tire \(M_{zzi}\), which can be expressed as:  
\[M_{si}=M_{syi}+M_{zzi}\approx F_{yi}\cdot R_{w}\cdot\tau+M_{zi} \tag{2}\]  
where \(F_{yi}\) and \(M_{zi}\) are the total lateral force and total aligning torque of the corresponding axle, \(R_{w}\) is the wheel radius, and \(\tau\) is the caster angle. Due to the fact that differential torque \(M_{di}\) is obtained from the lower layer torque allocation results, the output torque of the auxiliary steering motor serves as another controllable active force, which can balance the impact of differential torque when necessary to achieve precise control of the steering system. Therefore, the consideration of differential torque \(M_{di}\) is temporarily ignored in the control model, and the actual output torque of the final auxiliary motor can be calculated by the driver assistance component \(T_{a1}\), the auxiliary component \(T_{a2}\), and the differential torque \(M_{di}\).