# 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 steering system model can be represented as:  
\[J_{s}\bar{\theta}_{s}+b_{s}\dot{\theta}_{s}=T_{h}+T_{a1}+T_{a2}-\frac{F_{yi} \cdot R_{w}\cdot r+M_{a1}}{k_{s}}-\frac{F_{yi}\cdot R_{w}\cdot r+M_{a1}}{p\cdot k _{s}} \tag{3}\]  
The 2-DOF vehicle dynamics model is chosen for the control model, and the force analysis of the 2-DOF vehicle dynamics model is shown in Figure 3.  
The model considers the influence of generalized additional yaw moment \(M_{d}\) on yaw motion, and the 2-DOF vehicle model is represented as follows:  
\[\left\{ \begin{aligned} m\Big{(}v_{x}\dot{\beta}+v_{x}\omega_{z} \Big{)}=\sum\limits_{i=1}^{5}F_{yi}\\ l_{z}\dot{\omega}_{z}=\sum\limits_{i=1}^{2}L_{i}F_{yi}-\sum\limits_{i=3 }^{5}L_{i}F_{yi}+M_{d}\end{aligned} \right. \tag{4}\]  
where \(m\) is the total vehicle mass, \(v_{x}\) is the longitudinal speed, \(I_{z}\) is the yaw inertia of the entire vehicle. \(L_{i}\) represents the distance from each axle to the center of gravity (CG). Vehicle sideslip angle and yaw rate can be expressed as \(\beta\) and \(\omega_{z}\). The path tracking model can be represented as  
\[\left\{ \begin{aligned} \dot{y}_{e}\approx v_{x}\psi_{e}+v_{x} \beta\\ \dot{\psi}_{e}=\omega_{z}-v_{x}\rho\end{aligned} \right. \tag{5}\]  
where \(y_{e}\) is the lateral distance error, \(\psi_{e}\) is the heading angle error, and \(\rho\) is the path curvature. Figure 3: The 2-DOF vehicle model. Figure 2: The assisted steering system model. Organize (3)-(5) to obtain the control system model as shown below:  
\[\dot{\mathbf{x}}=f(\mathbf{x},\mathbf{u},\mathbf{w})= \begin{bmatrix}\dot{\theta}_{s}\\ \frac{\left(-\mathrm{b}_{s}\dot{\theta}_{s}+\mathrm{T}_{h}+T_{a1}+T_{a2}^{m} \right)}{I_{s}}-\frac{\left(F_{yi}R_{w}+M_{a1}\right)}{(\delta_{s}I_{s})}- \frac{\left(F_{yi}R_{w}+M_{a2}\right)}{(p\delta_{s}I_{s})}\\ \frac{\sum_{i=1}^{5}F_{ii}}{(mv_{x})_{x}}-\omega_{z}\\ \frac{\left(\sum_{i=1}^{2}\left(L_{i}F_{yi}\right)-\sum_{j=3}^{5}\left(L_{i}F_ {yi}\right)\right)}{I_{z}}+\frac{M_{d}}{I_{z}}\\ \upsilon_{x}\psi_{e}+\upsilon_{x}\beta\\ \omega_{z}-\upsilon_{x}\rho\end{bmatrix}  \tag{6}\]  
where the state variable \(\mathbf{x}=\left[\theta_{s}\quad\dot{\theta}_{s}\quad\beta\quad\omega_{z}\quad y_ {e}\quad\psi_{e}\right]^{\mathsf{T}}\), the control variable \(\mathbf{u}=\left[T_{a2}\quad M_{d}\right]^{\mathsf{T}}\), and the system disturbance \(\mathbf{w}=\left[T_{h}\quad\rho\right]^{\mathsf{T}}\) Due to the calculation of nonlinear tire lateral force and the aligning torque involved in Formula (6), in order to maintain model accuracy and improve control solution speed, the magic tire model under pure side slip conditions is selected to undergo first-order Taylor expansion approximation at the current tire side slip angle \(\overline{\alpha}_{i}\) to obtain the corresponding tire force, as follows:  
\[ \begin{cases}F_{yi}=\overline{F}_{yi}+\overline{C}_{i}(\alpha_{i}-\overline{ \alpha}_{i})\\ M_{zi}=\overline{M}_{zi}+\overline{K}_{i}(\alpha_{i}-\overline{\alpha}_{i}) \end{cases}  \tag{7}\]  
where \(\overline{F}_{yi}\) and \(\overline{M}_{zi}\) are the current lateral force and the aligning torque of the tire, which meet the following requirements \(\overline{F}_{yi}=f_{yi}(\overline{\alpha}_{i},F_{zi})\), \(\overline{M}_{zi}=m(\overline{\alpha}_{i},F_{zi})\). At the same time, the local lateral stiffness meets \(\overline{C}_{i}=\left(\frac{\partial f_{yi}(\alpha_{i},F_{zi})}{\partial \alpha_{i}}\right)\Big{|}_{\overline{\alpha}_{i}}\). The slope of the local aligning torque satisfies \(\overline{K}_{i}=\left(\frac{\partial m(\alpha_{i},F_{zi})}{\partial\alpha_{i }}\right)\Big{|}_{\overline{\alpha}_{i}}\), where \(F_{zi}\) is the vertical load of each axis under static load conditions. The tire sideslip angle can be expressed as  
\[\alpha_{i}=