## 2 Vehicle and Wheel Dynamics Model
### Tire Model  
When a vehicle is driven on the road, the interaction forces between the vehicle and the road are determined by the interaction between the tires and the road. Therefore, tire forces are a key factor in the dynamic control of the vehicle, and tire force calculation models are necessary. It is well known that there are many tire models for calculating interaction forces, such as the Burckhardt tire model, Duggoff tire model, UniTire tire model, and Magic Formula tire model. Since the Magic Formula tire model has high accuracy in calculating longitudinal and lateral tire forces, especially when the tire has a large longitudinal slip rate and large lateral sideslip angle, it was chosen for calculating the tire forces in this paper. The Magic Formula tire model is a semi-empirical tire model that is widely used in vehicle engineering. It uses a uniform formula to calculate the longitudinal and lateral forces of tires. The formula of the model is shown in Equation (8) [34]. \[\left\{ \begin{array}{l}x=X+S_{x}\\ y(x)=D\sin\{C\text{arctan}[Bx-E(Bx-\text{arctan}(Bx))]\}\\ Y=y+S_{y}\end{array} \right., \tag{8}\]  
where \(X\) represents the longitudinal slip rate or lateral sideslip angle of the tire in a pure situation, and \(Y\) represents the longitudinal force or lateral force of the tire in a pure situation. In Equation (8), when \(X\) represents the longitudinal slip rate, \(Y\) represents the longitudinal force, and when \(X\) represents the lateral sideslip angle, \(Y\) represents the lateral force. \(S_{x}\) and \(S_{y}\) are correcting parameters. \(B\), \(C\), \(D\), and \(E\) are the performance parameters determined from the testing data. Figure 3 shows the longitudinal force and lateral force in pure slip circumstances, varying with the different vertical loads of the tire. (\(F_{2}=1000\), \(2000\), \(3000\), \(4000\), and \(5000\) N). Figure 3 presents the longitudinal force of the tire in pure slip circumstances, according to varying values of the tire vertical load from \(1000\) to \(5000\) N and varying values of the tire longitudinal slip rate from \(-0.3\) to \(0.3\). Figure 3 presents the lateral force of the tire in pure slip circumstances, according to varying values of the tire vertical load from \(1000\) to \(5000\) N and varying values of the tire lateral sideslip angle from \(-30^{\circ}\) to \(30^{\circ}\). Figure 2: Schematic diagram of the wheel dynamics equation. Furthermore, the tire works with longitudinal slip and lateral slip in many cases, which are called combined slip situations. In the case of combined slip situations, the calculation of the longitudinal and lateral forces of the tire can be performed as follows [34]:  
\[\left\{ \begin{array}{l}F_{x}^{*}=F_{x0}-\varepsilon\big{(}F_{x0}-F_{y0}\big{)} \Big{(}\frac{\sigma_{y}^{*}}{\sigma_{total}^{*}}\Big{)}^{2}\\ F_{y}^{*}=F_{y0}-\varepsilon\big{(}F_{y0}-F_{x0}\big{)}\Big{(}\frac{\sigma_{y}^ {*}}{\sigma_{total}^{*}}\Big{)}^{2}\end{array} \right., \tag{9}\]  
where the total normalized slip \(\sigma_{total}^{*}=\sqrt{{\sigma_{x}^{*}}^{2}+{\sigma_{y}^{*}}^{2}}\) and \(\sigma_{x}^{*}=\frac{\sigma_{x}}{\sigma_{max}}\), \(\sigma_{y}^{*}=\frac{\sigma_{y}}{\sigma_{max}}\). \(\varepsilon=min\big{(}\sigma_{total}^{*},1\big{)}\). The theoretical slips are normalized by the peak slip values, \(\sigma_{xmax}\) and \(\sigma_{ymax}\). \(\sigma_{x}\) and \(\sigma_{y}\) are calculated according to  
\[\left\{ \begin{array}{l}\sigma_{x}=-\frac{k}{1+k}\\ \sigma_{y}=\frac{\tan\alpha}{1+k}\end{array} \right., \tag{10}\]  
The equivalent longitudinal and lateral slips are calculated from the normalized total theoretical slip according to  
\[\left\{ \begin{array}{l}k^{\prime}=-\frac{\sigma_{total}^{*}\sigma_{max}sign (\sigma_{x})}{1+\sigma_{total}^{*}\sigma_{max}sign(\sigma_{x})}\\ \alpha^{\prime}=\tan^{-1}\left[\sigma_{total}^{*}\sigma_{ymax}sign(\sigma_{y} )\right]\end{array}