# Multi-Objective Collaborative Control Method for Multi-Axle Distributed Vehicle Assisted Driving
## 3 Lower-Level Torque Distribution Strategy
The specific control allocation (CA) problem can be expressed as:  
\[J_{CA}=J_{virtual}+J_{tire}+J_{energy} \tag{27}\]  
where \(J_{CA}\) represents the overall optimization cost of the control allocation problem, \(J_{virtual}\) represents the virtual control cost, \(J_{tire}\) represents the tire load rate optimization cost, and \(J_{energy}\) represents the energy efficiency optimization cost. The following is a specific explanation of each cost function. 1. **Virtual control objectives** Define an expression for virtual control based on Formula (26). \[\mathbf{B}_{CA}\cdot\mathbf{u}_{T}=\mathbf{v}\] (28) where \(\mathbf{B}_{CA}\) is the coefficient matrix, \(\mathbf{u}_{T}\) is the control variable, and \(v\) is the virtual control objective variable, which can be obtained according to Equation (26). Meanwhile, the virtual control cost is expressed in the form of an approximation error, as follows: \[J_{virtual}=||(\mathbf{B}_{CA}\mathbf{u}_{T}-\mathbf{v})||_{\mathbf{P}_{1}}^{2}\] (29) where \(\mathbf{P}_{1}\) is the two-dimensional virtual control variable error weight matrix. 2. **Tire load rate optimization objectives** The tire load rate can be used to describe the utilization level of the tire under different working conditions. Considering that the lateral force of each tire in the vehicle is not controlled in longitudinal torque allocation, the tire load rate can be expressed as: \[\rho_{ij}=\frac{F_{x=ij}^{2}}{(\mu F_{x=ij})^{2}}=\frac{(T_{ij}i_{0})^{2}}{( \mu R_{w}F_{x=ij})^{2}}\] (30)When the overall tire load rate is low, it helps to improve vehicle stability. Therefore, the stability objective in control allocation is to minimize the tire load rate of each wheel, which can be expressed as:  
\[J_{tire}=||\mathbf{B}_{s}\mathbf{u}_{T}||_{\mathbf{P}_{2}}^{2} \tag{31}\]  
where the coefficient matrix \(\mathbf{B}_{s}\) is obtained by Equation (30). \(\mathbf{P}_{2}\) is the stability control objective weight matrix. (3) **Energy efficiency optimization objectives**  
As shown in Figure 10, there is a corresponding motor output torque \(T_{ij,me}\) that satisfies the highest energy efficiency at different speeds and satisfies the external characteristics of the motor, referred to as the optimal energy efficiency point. This study combines the total demand torque on the left and right side of the heavy vehicle and the reference torque point corresponding to the optimal energy efficiency point of each motor to develop the relevant control logic to achieve the goal of energy efficiency optimization, which can avoid the problem of real-time solution due to the non-linear energy loss function introduced by energy efficiency optimization. The energy efficiency optimization cost function can be expressed as:  
\[J_{energy}=||\mathbf{u}_{T}-\mathbf{u}_{ref}||_{\mathbf{P}_{3}}^{2} \tag{32}\]  
where \(\mathbf{P}_{3}\) is the weight matrix of energy efficiency optimization objectives, and \(\mathbf{u}_{ref}\) is the reference control matrix, meeting \(\mathbf{P}_{3}=p_{3}\cdot diag(c_{1i},c_{1r},\dots,c_{Sq},c_{Sr})\), \(\mathbf{u}_{ref}=[T_{1I,ref}\ T_{1r,ref}\ \dots\ T_{S,ref}\ T_{S,ref}]\). Here, \(p_{3}\) is a constant, and the coefficient \(c_{ij}\) and the reference torque \(T_{ij,ref}\) are varied according to the control logic. The left and right sides of the vehicle have the same logic for optimizing the energy efficiency of the torque distribution, so the left side of the vehicle is chosen to describe the control logic for energy efficiency optimization. Firstly, calculate the motor distribution coefficient \(\varsigma\) based on the target total torque \(T_{all,l}\) that needs to be allocated on the left side:  
\[\varsigma=\frac{T_{all,l}}{T_{il,me}} \tag{33}\]  
where \(T_{all,l}\) and \(T_{il,me}\) are of the same sign. When \(T_{all,l}\) is a negative braking torque, the value of \(T_{il,me}\) in the negative torque working area of the motor is opposite to that under driving conditions. The energy efficiency control logic is divided into two modes: priority allocation and average allocation.