## 2 2-DOF Vehicle Dynamics Model and Ideal Reference Model  
Due to the simplification of the vehicle structure, the 2-DOF dynamics model is widely used to analyze vehicle-related problems. In the 2-DOF model, the left and right wheels of the vehicle are considered as a whole, as shown in Figure 1 below. The distributed drive vehicle can independently control the torque of the four steering wheels to achieve four-wheel steering control. In addition, controlling the direct yaw moment is necessary to ensure stability. Therefore, the 2-DOF model can characterize the lateral motion of the vehicle and is suitable for this study [27]. \begin{table}
\begin{tabular}{c c} \hline
**Ref.** & **Approach** & **Limitations** \\ \hline [PERSON] [8] & The kinematic MPC is used to deal with the road curvature disturbance, the yaw rate PID feedback control is used to eliminate the uncertainty and modeling error, and the vehicle sideslip angle compensator is used to correct the kinematic model prediction. This strategy improves the control accuracy and ensures vehicle stability. \\ \multirow{4}{*}{[PERSON] [11]} & A force-driven switched MPC path following the control strategy is proposed to coordinate the active front wheel steering and external yaw torques. This strategy improves the control accuracy and ensures the stability of the vehicle. \\ \multirow{4}{*}{[PERSON] [18]} & Quasi-linearization techniques are used to simplify the vehicle model, which preserves the inherent coupling effects between the longitudinal acceleration/braking force, steering angle, and vehicle state. Based on this model, an input-output decoupling controller is proposed. \\ \multirow{4}{*}{[PERSON] [20]} & In this paper, the decoupling design method of the longitudinal and lateral controllers is used. For the longitudinal controller, the proportion containing the feedforward term is used. On the other hand, an adaptive backstepping method is used in the lateral case to deal with the nonlinearity and parameter uncertainty of the model. \\ \hline \end{tabular}
\end{table} 
Table 1: Literature review and analysis. Based on the 2-DOF model in the figure, the longitudinal velocity of the car along the \(x\)-axis direction is assumed to be constant, and only the lateral motion of the vehicle on the \(y\)-axis and the yaw motion around the \(z\)-axis are considered. In order to obtain the ideal front wheel angle, the influence of air resistance, suspension, and rear wheel angle is not considered. The lateral motion of the vehicle is shown in Equation (1). \[\mathrm{m}(\dot{v_{y}}+v_{x}\omega)=F_{yf}\cos\delta_{f}+F_{yr} \tag{1}\]  
The yaw motion of the vehicle is shown in Equation (2). \[I_{z}\dot{\omega}=aF_{yf}\cos\delta_{f}-bF_{yr} \tag{2}\]  
where \(\mathrm{F_{yf}}\) and \(\mathrm{F_{yr}}\) are the lateral reaction forces of the ground to the front and rear wheels, respectively. a and b are the distances between the center of the mass of the vehicle and the front and rear axles, respectively. \(\delta_{f}\) is the front wheel angle; \(\omega\) is the car's yaw rate; \(\mathrm{I_{z}}\) is the moment of inertia of the vehicle around the z-axis; and \(\mathrm{v_{x}}\) and \(\mathrm{v_{y}}\) are the forward and lateral speeds of the vehicle, respectively. In the classic 2-DOF vehicle model, the longitudinal force of the tire is usually ignored when analyzing the lateral dynamics. However, in the research of MPC trajectory tracking control, the front wheel angle is assumed to be small for calculating the speed, and the influence of the longitudinal force of the tire is also ignored to simplify the model. Therefore, in this paper, the front wheel angle is assumed to be a slight angle, and Equations (1) and (2) can be deduced. \[\left\{ \begin{array}{l}\sum F=F_{yf}\cos\delta_{f}+F_{yr}=F_{yf}+F_{yr}=k_{1 }\alpha_{f}+k_{2}\alpha_{r}\\ \sum M=aF_{yf}\cos\delta_{f}-bF_{yr}=ak_{1}\alpha_{f}-bk_{2}\alpha_{r}\end{array} \right.