## 2 System Modeling
### Tire Model  
As the vehicle lateral dynamics possesses inherent nonlinearities, the linear tire model cannot accurately represent nonlinear characteristics of tire dynamics. Several tire models have been used in the literature to describe nonlinear characteristics of the tire force. Pacejka tire models are one of the most commonly used tire models. Pacejka tire model, also called Magic Formula, contains the combination formula of trigonometric functions to describe the tire characteristics under different  
Figure 2: Schematic of tire force. (**a**) Schematic of front left wheel tire force; and (**b**) Schematic of rear left wheel tire force.  
working conditions [26]. Should the tire operates under the assumption of pure cornering and pure longitudinal slip condition, the Pacejka tire model can be expressed as:  
\[F_{x0}=D_{x}\sin\{C_{x}\text{arctan}[B_{x}s-E_{x}(B_{x}s-\text{arctan}(B_{x}s) )]\} \tag{10}\]  
\[F_{y0}=D_{y}\sin\{C_{y}\text{arctan}\big{[}B_{y}\alpha-E_{y}(B_{y}\alpha-\text {arctan}(B_{y}\alpha))\big{]}\} \tag{11}\]  
Since the form and definition of each group of parameters are similar, the parameters \(B_{x}\), \(C_{x}\), \(D_{x}\), \(E_{x}\)\(B_{y}\), \(C_{y}\), \(D_{y}\), \(E_{y}\) can be abbreviated as \(B\), \(C\), \(D\), \(E\). Where parameters \(B\), \(C\), \(D\), \(E\) are stiffness factors, curve shape factors, peak factors, and curvature factors, respectively. These parameters variables are the functions of tire vertical load, camber angle and tire characteristics; \(s\) is the longitudinal slip ratio; \(\alpha\) is the tire slip angle; and \(F_{x0}\) and \(F_{y0}\) are the longitudinal and lateral tire force.  
The tire model on the combined condition of lateral and longitudinal force can be expressed as follows:  
\[F_{x}=\frac{\sigma_{x}}{\sigma}F_{x0} \tag{12}\]  
\[F_{y}=\frac{\sigma_{y}}{\sigma}F_{y0} \tag{13}\]  
\[\sigma_{x}=-\frac{s}{1+s} \tag{14}\]  
\[\sigma_{y}=\frac{-\tan\alpha}{1+s} \tag{15}\]  
\[\sigma=\sqrt{\sigma_{x}{}^{2}+\sigma_{y}{}^{2}} \tag{16}\]  
The tire slip angle, defined as the angle between the orientation of wheel longitudinal velocity vector and the wheel resultant velocity vector, can be written as:  
\[\alpha_{i}=arc\tan(\frac{v_{ni}}{v_{mi}})\qquad(i=1,2,3,4) \tag{17}\]  
where \(v_{mi}\) and \(v_{ni}\) are the longitudinal and lateral velocities at the center of i-th wheel, respectively. \(v_{mi}\) and \(v_{ni}\) can be calculated by the \(v_{xi}\) and \(v_{yi}\), which are wheel velocities of each wheel in the x direction and y direction. As shown in Figure 2, the matrix form is obtained as:  
\[\left[\begin{array}{c}v_{mi}\\ v_{ni}\end{array}\right]=\left[\begin{array}{cc}\sin\delta_{i}&\cos\delta_{ i}\\ \cos\delta_{i}&-\sin\delta_{i}\end{array}\right]\left[\begin{array}{c}v_{yi} \\ v_{xi}\end{array}\right]\qquad(i=1,2,3,4) \tag{18}\]  
The wheel velocities of each wheel in the \(x\) direction and \(y\) direction can be calculated by vehicle longitudinal and lateral velocities:  
\[\begin{array}{c}v_{y1}=v_{y2}=v_{y}+a\phi\\ v_{y3}=v_{y4}=v_{y}-b\phi\\ v_{x1}=v_{x2}=v_{x3}=v_{x4}=v_{x}\end{array} \tag{19}\]  
Since the effects of load transfer are caused by lateral and longitudinal accelerations, the tire normal forces are defined as:  
\[\left\{\begin{array}{l}F_{21}=\frac{bmg}{2(a+b)}-\frac{m\dot{v}_{y}h_{CG}}{2 B_{1}}-\frac{m\dot{v}_{x}h_{CC}}{2(a+b)}\\ F_{22}=\frac{bmg}{2(a+b)}+\frac{m\dot{v}_{y}h_{CG}}{2B_{1}}-\frac{m\dot{v}_{x}h_ {CG}}{2(a+b)}\\ F_{23}=\frac{amg}{2(a+b)}-\frac{m\dot{v}_{x}h_{CC}}{2B_{1}}+\frac{m\dot{v}_{x}h_ {CC}}{2(a+b)}\\ F_{24}=\frac{amg}{2(a+b)}+\frac{m\dot{v}_{y}h_{CC}}{2B_{1}}+\frac{m\dot{v}_{x}h_ {CC}}{2(a+b)}\end{array}\right. \tag{20}\]where \(h_{CG}\) is the height of mass center. The longitudinal slip ratio \(s\) describes as the proportion of sliding motion in wheel motion is defined as:  
\[s_{i}=\frac{R_{w}\omega_{i}-v_{mi}}{v_{mi}} \tag{21}\]  
where \(R_{w}\) is wheel rolling radius.