## 2 Vehicle and Tire Model
### Brush Tire Model  
According to the above formula, the calculation of vehicle state requires known longitudinal and lateral tire forces, which are generally calculated using a tire model in vehicle dynamics research, and the accuracy of the tire model directly affects the accuracy of state estimation. Compared with the currently widely used Magic Formula (MF) tire model, the brush tire model requires fewer parameters to be fitted and can accurately represent the longitudinal and lateral forces of the tire, so the brush tire model is used for simulation in this paper [24].  
Figure 1: Planar vehicle model.  
As can be seen in Figure 2, the brush tire model has a high fitting accuracy under the small lateral deflection angle condition, but the peak lateral force of the brush tire model is slightly larger than that of the MF tire model under the large slip angle condition. In this paper, the simulation conditions are low vertical force and lateral slip angle, and the tries are not in the strong nonlinear region, so the brush tire model is chosen to ensure the simulation accuracy. The brush tire model calculation is shown below.  
The joint longitudinal force and lateral force model is defined as:  
\[f=\left\{\begin{array}{rl}\rho s&-\frac{(\rho s)^{2}}{3\mu r_{z}^{2}}+\frac{ (s)^{3}}{27(\mu r_{z})^{2}}&|s|<s_{m}\\ \mu F_{z}&|s|\geq s_{m}\\ \rho=4\epsilon_{a}{}^{2}\epsilon_{b}k_{c}\\ s_{m}=\frac{3\mu F_{z}}{\rho}\\ s=\sqrt{s_{x}^{2}+s_{y}^{2}}\end{array}\right. \tag{7}\]  
The longitudinal force and lateral force of the tire are expressed as:  
\[F_{x}=\frac{s_{x}}{s}f,F_{y}=\frac{s_{y}}{s}f \tag{8}\]  
where \(\epsilon_{a}\) is half of the contact patch length, \(\epsilon_{b}\) is half of the contact patch width, \(k_{c}\) is the tire tread stiffness coefficient in unit length, \(s\) is the total slip rate, \(F_{z}\) is the tire vertical force, and \(\mu\) is the friction coefficient. The longitudinal and lateral slip rates, denoted by \(s_{x}\) and \(s_{y}\), are expressed as follows:  
\[s_{x,ij}=\frac{R\omega_{ij}-v_{ij}}{\max(R\omega_{ij},v_{ij})},s_{y,ij}=\frac {v_{ij}}{R\omega_{ij}}\tan\alpha_{ij} \tag{9}\]  
where \(R\) is the wheel effective radius, \(\omega_{ij}\) is the angular speed of wheel rotation, \(v_{ij}\) is the wheel center speed, \(\alpha_{ij}\) is the tire side slip angle, and the calculation formula is as follows:  
\[\left\{\begin{array}{l}v_{1j}=(v_{x}\mp w\dot{\psi})\cos\delta+(v_{y}+a\dot{ \psi})\sin\delta\\ v_{2j}=v_{x}\mp w\dot{\psi}\end{array}\right. \tag{10}\]  
\[\left\{\begin{array}{l}\alpha_{1j}=\delta-\arctan(\frac{V_{y}+a\dot{\psi}} {V_{x}\mp w\dot{\psi}/2})\\ \alpha_{2j}=-\arctan(\frac{V_{y}-b\dot{\psi}}{V_{x}\mp w\dot{\psi}/2})\end{array}\right. \tag{11}\]  
Figure 2: Lateral force calculation of brush tire model and MF tire model.  
when the vehicle is steering, the longitudinal force of the tire as well as the lateral one, is influenced by the transfer of the vertical load, which is calculated by the following equations:  
\[\left\{\begin{array}{l}F_{\mathrm{z1}j}=\frac{b}{2L}mg-\frac{h_{\mathrm{cog}}}{ 2L}ma_{x}\mp\frac{bh_{\mathrm{cog}}}{2w}ma_{y}\\ F_{\mathrm{z2}j}=\frac{a}{2L}mg+\frac{h_{\mathrm{cog}}}{2L}ma_{x}\mp\frac{ah_{ \mathrm{cog}}}{Lw}ma_{y}\end{array}\right. \tag{12}\]  
where \(h_{cog}\) is the height of the center of gravity.