## 3 Steady-State Models of Vehicle Dynamics for Lateral Torque Vectoring  
To analyze the vehicle dynamics with lateral torque vectoring, some definitions of the vehicle geometry for front-wheel drive (FWD) are illustrated in Figure 6, where \(R_{w}\) is tire radius, \(B\) is track or tread, \(L\) is wheel base, \(\delta\) is steering angle, and the location marked by G is the center of gravity. Accordingly, the total yaw moment \(M_{g}\) for FWD is written as  
\[M_{g}=\frac{B}{2}\cos\delta\cdot(F_{xr}-F_{xl}) \tag{21}\]  
where \(F_{xl}\) and \(F_{xr}\) are the left and right driving forces and are equal to \(T_{\text{nl}}^{\prime}/R_{w}\) and \(T_{\text{zw}}^{\prime}/R_{w}\), respectively. The left-right torque vectoring generates two main effects during driving. First, the total maximum cornering force varies as difference between the left and right driving forces changes. Second, a yaw moment generated by left-right torque vectoring results in a variation in front and rear cornering forces. As a consequence, it is expected that a maximum lateral acceleration with an optimal distribution between front and rear driving forces can be attained. For the purpose of approximating lateral acceleration, the two-axle model, or called bicycle model, as shown in Figure 7, is utilized. Accordingly, the equations of motion are written as  
\[F_{yf}l_{f}-F_{yr}l_{r}+M_{g}=0 \tag{22}\]  
\[F_{yf}+F_{yr}=ma_{y} \tag{23}\]  
where \(F_{yf}\) and \(F_{yr}\) are the cornering forces of front and rear axles, respectively, \(m\) is the mass of vehicle, \(a_{y}\) is lateral acceleration, \(l_{f}\) is the distance between the front axle and G, and \(l_{r}\) is the distance between the rear axle and G, i.e., \(l_{f}+l_{r}=L\). Figure 5: Operational conditions of DM for left and right turn. Figure 6: Some definitions of vehicle geometry with lateral torque vectoring for FWD. Most forces acting upon the vehicle during driving are not through the center of gravity G. Therefore, the resulting net pitch moment about G will redistribute the normal loads between front and rear tires and thus generate a so-called front-rear weight transfer. The driving force of a vehicle is to overcome the tractive resistance, \(F_{trac}\), expressed below:  
\[F_{trac}=F_{air}+F_{roll}+F_{grd}+ma_{x} \tag{24}\]  
where \(F_{air}\) is the air resistance in terms of \(\frac{1}{2}\rho AC_{a}v^{2}\) with \(\rho\), \(A\), \(C_{a}\), and \(v\) being air density in kg/m\({}^{3}\), frontal area of vehicle in m\({}^{2}\), drag coefficient, and vehicle speed in m/s, respectively; \(F_{roll}\) is the rolling resistance in terms of \(C_{r}mg\cos\theta\) with \(C_{r}\) and \(\theta\) being coefficient of rolling resistance and slope angle of climbing in degree, respectively; \(F_{grd}\) is the grade resistance in terms of \(mg\sin\theta\) with \(C_{r}\) and \(\theta\) being the coefficient of rolling resistance and slope angle of climbing in degree; and \(a_{x}\) is the longitudinal acceleration. In addition, the gradeability of vehicle is defined as Grade \(=\tan\theta\cdot\)100%. Considering the typical tractive resistance during driving on a horizontal road, i.e., \(\theta=0^{\circ}\), the front-rear weight transfer, \(\Delta F_{zfr}\), can be approximated as  
\[\Delta F_{zfr}=\left[F_{trac}-F_{roll}\left(1-\frac{R_{w}}{h_{g}}\right)\right] \frac{h_{g}}{L} \tag{25}\]  
where \(h_{g}\) is the height of G.  
In order to investigate the effect of roll moment on the weight transfer between left and right tires, a four-wheel model rather than two-axle model is required as shown in Figure 8, where the subscripts, \(x\), \(y\), z, \(fl\), \(fr\), \(rl\), \(rr\), and \(\_lim\), denote longitudinal, lateral, vertical, front-left, front-rear, rear-left, rear-rear, and adhesive limit of the tire, respectively. It is noted that the adhesive limit of each tire \(F_{lim}\) is expressed as  
\[F_{lim}=\mu_{p}F_{z} \tag{26}\]  
where \(\mu_{p}\) is the adhesive coefficient of tire, and \(F_{z}\) is the normal load of each tire.