## 3 Design of Trajectory Tracking Control System
### Torque Distributor Design
The controller's input is the deviation value between the expected and actual slip rates. In this paper, the expected slip rate is set to \(15\%\), and the controller's output is the wheel torque adjustment value. \[\Delta T_{i}=K_{p}e_{\lambda i}+K_{i}\int_{0}^{t}e_{\lambda i} \tag{41}\]  
The above design shows a drive anti-slip scheme for a single wheel, but for the whole vehicle if the torque of one side of the wheel is reduced. In contrast, if the torque of the other side is unchanged, an additional yaw moment will be generated so that the actual value of the additional yaw moment is not consistent with the value calculated by the yaw stability control algorithm, which will impact the vehicle's yaw motion. Based on the above reasons, in order to ensure that the additional yawing moment value is not affected, the driving anti-skid function needs to add a multi-wheel coordination mechanism, that is, the left and right wheels need to reduce the same driving anti-skid adjustment torque value; however, if the left and right wheels are sliding, the reduced torque value will be too large, which will affect the vehicle dynamic performance. Therefore, based on balancing power and coordination, this paper takes the maximum value of the driving anti-slip adjusting torque of the left and right wheels as the common driving anti-slip adjusting torque. \[\Delta T_{f}=\max(\Delta T_{fl},\Delta T_{fr}) \tag{42}\] \[\Delta T_{r}=\max(\Delta T_{rl},\Delta T_{rr})\]  
Finally, the output of the torque value of each motor is as follows:  
\[\left\{ \begin{array}{l}T_{ff}=\frac{F_{2 ff}}{(F_{2 ff}+F_{2 ff}+F_{2 ff}+F_{2 rr} )}*(T-\frac{\Delta\rm M*R}{2B_{f}})+\Delta T_{f}\\ T_{fr}=\frac{F_{2 ff}}{(F_{2 ff}+F_{2 ff}+F_{2 ff}+F_{2 rr})}*(T+\frac{\Delta\rm M*R }{2B_{f}})+\Delta T_{f}\\ T_{rf}=\frac{F_{2 ff}}{(F_{2 ff}+F_{2 ff}+F_{2 ff}+F_{2 rr})}*(T-\frac{\Delta\rm M* R}{2B_{f}})+\Delta T_{r}\\ T_{rr}=\frac{F_{2 rr}}{(F_{2 ff}+F_{2 ff}+F_{2 ff}+F_{2 rr})}*(T+\frac{\Delta\rm M* R}{2B_{r}})+\Delta T_{r}\end{array} \right. \tag{43}\]  
In the above equation, Fzf and Fzr are the front and rear axle loads, respectively; \(\rm h\) is the height of the centroid; \(\rm a_{x}\) and \(\rm a_{y}\) are the longitudinal acceleration and lateral acceleration, respectively; \(\rm B_{f}\) and \(\rm B_{r}\) are the front and rear wheel tracks; \(\rm T_{ij}\) is the driving torque of the in-wheel motor; \(\Delta T\) is the driving anti-slip adjustment torque of each wheel, where f, fr, rf, and rr represent the left front wheel, right front wheel, left rear wheel, and right rear wheel, respectively; and \(\rm R\) is the effective rolling radius of the wheel. In order to verify the effectiveness of the anti-skid strategy proposed in this section, the driving torque of each wheel in the case of not considering anti-skid and considering anti-skid is calculated, respectively, on the double-moving line road with a road adhesion coefficient of 0.3 and a vehicle speed of 60 km/h, as shown in Figure 7. The driving torque of each wheel after considering anti-skid is lower than that without considering anti-skid. It can ensure that the vehicle has stability on the low-adhesion road surface, indicating that the designed torque distributor--considering the anti-skid strategy--is compelling.