## 2 Methodology
### Modeling and Problem Formulation  
The main goal of this work is to enable the vehicle to track the target path quickly and steadily. The vehicle suspension characteristics are relatively small in relation to the research content, so the single-track model is selected. At the same time, the dynamic model established in this paper is mainly used to design the predictive model in the model predictive controller. It needs to be simplified as much as possible while accurately describing the dynamic characteristics of the vehicle. The following idealized assumptions are first proposed when performing dynamic modeling: (1) road fluctuations are ignored, assuming that the vehicle is always driving without vertical motion; (2) suspension motion is ignored; (3) the load movement of the front and rear axles is not considered, and the left and right transfer of the load is ignored; (4) the tire cornering characteristics are considered, but the vertical and horizontal coupling relationships are ignored; and (5) the mechanical effects of steering system are ignored. In this paper, a three-degree-of-freedom vehicle dynamics model is constructed, including longitudinal motion, lateral motion, and yaw (see Figure 2). The longitudinal force, lateral force, and yaw moment received by the vehicle can be written as:  
\[\left\{ \begin{array}{l}m\Big{(}\ddot{x}-\dot{y}\dot{\varphi}\Big{)}=\sum F_{ x}=2F_{lf}\cos\delta_{f}-2F_{cf}\sin\delta_{f}+2F_{lr}\\ m\Big{(}\ddot{y}+\dot{x}\dot{\varphi}\Big{)}=\sum F_{y}=2F_{lf}\sin\delta_{f}-2 F_{cf}\cos\delta_{f}+2F_{cr}\\ I_{z}\ddot{\varphi}=\sum M_{z}=2L_{f}\Big{(}F_{lf}\sin\delta_{f}+F_{cf}\cos \delta_{f}\Big{)}-2L_{r}F_{cr}\end{array} \right. \tag{1}\]  
where \(m\) is the vehicle mass; \(x\) is the longitudinal position; \(\varphi\) is the yaw angle; \(\delta_{f}\) is the front wheel rotation angle; \(y\) is the lateral position; \(I_{z}\) is the \(z\)-axis moment of inertia; \(F_{x}\) is the total longitudinal force;\(F_{y}\) is the total lateral force; \(M_{z}\) is the total yaw moment; \(F_{cf}\) and \(F_{cr}\) are the lateral forces on the front and rear tires of the vehicle and are related to the corner stiffness and corner angle of tires; \(F_{lf}\) and \(F_{lr}\) are the longitudinal force on the front and rear tires of vehicle, which are related to the longitudinal stiffness and slip rate of the tire; \(F_{xf}\) and \(F_{xr}\) are the forces on the front and rear tires of the vehicle in the \(x\) direction; \(F_{yf}\) and \(F_{yr}\) are the forces on the front and rear tires of the vehicle in the \(y\) direction; and \(L_{f}\) and \(L_{r}\) are the distances from the front and rear axis to the center of mass. Generally, the stable driving vehicle has small variation angle and slip rate. The tire forces involved in this paper are all approximated by a linear function. In addition, there are many trigonometric functions in the vehicle dynamics model described by formulas. Since each angle involved in the dynamics model is in a small angle interval, each trigonometric function can satisfy the following approximate conditions: \(\cos\theta\approx 1\), \(\sin\theta=0\), \(\tan\theta=\theta\). Hence, the tire force of the vehicle can be expressed as:  
\[\left\{ \begin{array}{l}F_{lf}=C_{lf}s_{f}\\ F_{lr}=C_{lr}s_{r}\\ F_{cf}=C_{cf}\Big{[}\delta_{f}-\Big{(}\dot{y}+L_{f}\dot{\varphi}\Big{)}/\dot{ x}\\ F_{cr}=C_{cr}\Big{(}L_{r}\dot{\varphi}-\dot{y}\Big{)}/\dot{x}\end{array} \right. \tag{2}\]  
Figure 2: Schematic of the single-track model. where \(C_{cf}\) and \(C_{cr}\) are the lateral stiffness of the front and rear tires; \(C_{lf}\) and \(C_{lr}\) are the longitudinal stiffness of the front and rear tires; and \(s_{f}\) and \(s_{r}\) are the slip ratio of the front and rear tires. The non-linear dynamic vehicle model can be written as:  
\[\left\{