# Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving
## 3 Lower-Level Torque Distribution Strategy  
This section is based on the virtual control variable composed of the generalized additional yaw moment obtained by the upper layer MPC controller and the generalized longitudinal force required for longitudinal motion tracking, combined with the minimum tire load rate and minimum energy consumption for multi-objective torque allocation. The longitudinal motion control adopts the sliding mode control algorithm, which will not be repeated here. The generalized longitudinal force and additional yaw moment required by the vehicle are generated by the longitudinal force of the tire, which can be expressed as:  
\[\left\{ \begin{aligned} F_{xc}=\sum\limits_{i=1}^{2}\left(F_{xwii} cos\delta_{il}+F_{xwir}cos\delta_{ir}\right)+\sum\limits_{i=3}^{5}\left(F_{xwil}+F_{ xwir}\right)\\ M_{d}=\frac{\beta}{2}\left(\sum\limits_{i=1}^{2}\left(-F_{xwil} cos\delta_{il}+F_{xwir}cos\delta_{ir}\right)+\sum\limits_{i=3}^{5}\left(-F_{xwil}+F_{ xwir}\right)\right)+\sum\limits_{i=1}^{2}\sum\limits_{j=l}^{r}\left(L_{i}F_{xwij}sin \delta_{ij}\right)\end{aligned} \right. \tag{23}\]  
If the rolling resistance moment is ignored and it is believed that the slip rate/speed change of each tire is not significant within a finite time [19], then the longitudinal force of each wheel can be expressed as:  
\[F_{xwij}=\frac{T_{wij}}{R_{w}}=\frac{\left(T_{ij}\cdot i_{0}\right)}{R_{w}} \tag{24}\]  
In the formula, \(i_{0}\) is the reduction ratio and \(T_{wij}\) is the driving torque of each wheel. \(T_{ij}\) is the torque output by the wheel motor. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{\(\delta_{1}\)} & \multicolumn{5}{c}{\(\dot{\delta_{1}}\)} \\ \cline{2-6}  & **NL** & **NS** & **ZO** & **PS** & **PL** \\ \hline NL & HSC & HSC & HSC & MHSC & MSC \\ NS & HSC & HSC & MHSC & MSC & MLSC \\ ZO & MLSC & MLSC & LSC & MLSC & MLSC \\ PS & MLSC & MSC & MHSC & HSC & HSC \\ PL & MSC & MHSC & HSC & HSC & HSC \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Operational stability collaborative fuzzy rule library. Figure 9: The fuzzy logic MAP diagram. (**a**) Human–machine collaboration. (**b**) Operational stability collaboration. Substitute Equation (24) into Equation (23), and based on the small angle assumption of steering angle, the formula can be simplified as:  
\[\left\{ \begin{aligned} F_{xc}=\frac{i_{0}}{R_{w}}\cdot\frac{5}{i}(T_{il}+ T_{ir})=\frac{i_{0}}{R_{w}}\cdot(T_{all,l}+T_{all,r})\\ M_{d}=\frac{i_{0}\cdot B}{2R_{w}}\sum\limits_{i=1}^{5}(-T_{il}+T_{ir})= \frac{i_{0}\cdot B}{2R_{w}}\cdot(-T_{all,l}+T_{all,r})\end{aligned} \right. \tag{25}\]  
where \(T_{all,l}\) and \(T_{all,r}\) are the sums of the target torque output by the left and right wheel motors of the vehicle, \(B\) is the wheelbase. Once the generalized longitudinal force and additional yaw moment are determined, the total target torque of the left and right motors can be expressed as:  
\[\left\{ \begin{aligned} T_{all,l}=\frac{1}{2}\left(\frac{F_{xc}R_{w}} {i_{0}}-\frac{2M_{c}R_{w}}{i_{0}B}\right)\\ T_{all,r}=\frac{1}{2}\left(\frac{F_{xc}R_{w}}{i_{0}}+\frac{2M_{c}R_{w}}{ i_{0}B}\right)\end{aligned} \right. \tag{26}\]  
The torque distribution will be described as a multi-objective optimization problem consisting of three optimization objectives, the first of which is to ensure that the torque corresponding to the drive motor meets the requirements of the generalized virtual control variables, which can be represented by the tracking effect of the total torque of the left and right wheel motors on the target torque \(T_{all,l}\) and \(T_{all,r}\), respectively. The second optimization objective is the minimum tire load rate. Finally, it is necessary to consider the optimization of the overall energy efficiency of the vehicle, that is, to maximize the operation of each wheel motor in high efficiency areas and reduce energy loss.