## 2 Nonlinear Dynamic Vehicle Model
### Seven Degrees of Freedom Vehicle Model  
The nonlinear vehicle model described in this paper makes the following assumptions [26]: (1) the vehicle is only considered to move parallel to the ground; (2) the origin of the vehicle coordinate system coincides with the centroid point of the vehicle; and (3) we ignore the influence of longitudinal rolling resistance on the vehicle state. The nonlinear vehicle model studied in this paper is shown in Figure 2; \(a\) is the distance from the center of mass to the front axle; \(b\) is the distance from the center of mass to the rear axle; \(l=a+b\) is the distance from the front axle to the rear axle; \(B_{f}\) is the front wheel track; \(B_{r}\) is the rear wheel track; \(\alpha_{ij}\) is the tire sideslip angle; \(F_{xij}\) is the longitudinal force of the tire; \(F_{yij}\) is the lateral force of the tire; \(\delta_{ij}\) is the front wheel steering angle; \(v_{x}\) is the velocity of the center of mass along the \(x\)-axis; and \(v_{y}\) is the velocity of the center of mass along the \(y\)-axis. Among them, \(ij\) is \(fl\), \(fr\), \(rl\), \(rr\), which denote front left, front right, back left, and back right, and \(ij\) denotes this in the following model. The longitudinal force balance equation:  
\[m(\vec{v}_{x}-\omega_{r}\cdot v_{y}) =F_{xfl}\cos\delta_{fl}+F_{xfr}\cos\delta_{fr}-F_{yfl}\sin\delta_{fl} \tag{1}\] \[-F_{yfl}\sin\delta_{fr}+F_{xfl}+F_{xrr}-F_{w}\]  
Figure 2: The nonlinear dynamic vehicle model. The lateral force balance equation:  
\[m(\overset{\bullet}{v_{y}}+\omega_{r}\cdot v_{x})=F_{xfl}\sin\delta_{fl}+F_{xfr} \sin\delta_{fr}+F_{yfl}\cos\delta_{fl}+F_{yfr}\cos\delta_{fr}+F_{yrl}+F_{yrr} \tag{2}\]  
The moment balance equation around the \(z\)-axis:  
\[ \begin{array}{rl}I_{z}\cdot\overset{\bullet}{\omega_{r}}&=a(F_{xfl}\sin \delta_{fl}+F_{xfr}\sin\delta_{fr}+F_{yfl}\cos\delta_{fl}+F_{yfr}\cos\delta_{ fr})\\ &+\frac{B_{f}}{2}(F_{xfr}\cos\delta_{fr}-F_{xfl}\cos\delta_{fl})+\frac{B_{f}}{2 }(F_{yfl}\sin\delta_{fl}-F_{yfr}\sin\delta_{fr})\\ &+(F_{xrr}-F_{xrl})\frac{B_{f}}{2}-(F_{yrl}+F_{yrr})b\end{array}  \tag{3}\]  
The absolute acceleration of the vehicle's center of mass on the \(x\)-axis of the vehicle coordinate system:  
\[a_{x}=\overset{\bullet}{v_{x}}-\omega_{r}\cdot v_{y} \tag{4}\]  
The absolute acceleration of the vehicle's center of mass on the \(y\)-axis of the vehicle coordinate system:  
\[a_{y}=\overset{\bullet}{v_{y}}-\omega_{r}\cdot v_{x} \tag{5}\]  
The vehicle sideslip angle of the center of mass:  
\[\beta=\arctan\bigl{(}v_{y}/v_{x}\bigr{)} \tag{6}\]  
The vertical load equation for each tire:  
\[ \begin{array}{rl}F_{zfl(r)}=mg\frac{b}{2l}-ma_{x}\frac{h_{x}}{2l}\mp ma_{y} \frac{h_{y}}{B_{f}}\cdot\frac{b}{l}\\ F_{zrl(r)}=mg\frac{a}{2l}+ma_{x}\frac{h_{x}}{2l}\mp ma_{y}\frac{h_{x}}{B_{r}} \cdot\frac{a}{l}\end{array}  \tag{7}\]  
The vehicle sideslip angle equation for each tire:  
\[ \begin{array}{rl}\alpha_{fl(r)}=\delta_{fl(r)}-\arctan(\frac{v_{y}-b\cdot \omega_{r}}{v_{x}-\frac{b}{2}\cdot\omega_{r}})\\ \alpha_{rl(r)}=\ -\arctan(\frac{v_{y}-b\cdot\omega_{r}}{v_{x}-\frac{b}{2}\cdot \omega_{r}})\end{array}  \tag{8}\]  
The longitudinal velocity of each wheel center in the wheel coordinate system:  
\[ \begin{array}{rl}v_{fl(r)}=(v_{x}\mp\frac{B_{f}}{2}\cdot\omega_{r})\cos \delta_{fl(r)}+(v_{y}+a\cdot\omega_{r})\sin\delta_{fl(r)}\\ v_{rl(r)}=v_{x}\mp\frac{B_{r}}{2}\cdot\omega_{r}\end{array}  \tag{9}\]  
The slip rate equation for each wheel:  
\[\lambda_{ij}=\frac{\Omega_{ij}\cdot R-v_{ij}}{\Omega_{ij}\cdot R}=1-\frac{v_{ ij}}{\Omega_{ij}\cdot R} \tag{10}\]  
In the above formulas, \(F_{w}\) is the vehicle air resistance.