# Torque Allocation of Hybrid Electric Trucks for Drivability and Transient Emissions Reduction
## 2 Hybrid Vehicle Powertrain Layout
### Dynamic Equations
#### 2.2.3 Vehicle and Wheels  
The vehicle model considers the longitudinal dynamics and load transfer caused by the vehicle acceleration \(a_{x}\), which is computed from the equilibrium equation--reported in Equation (7)--of the forces acting on the vehicle, i.e., the longitudinal forces on the tyres \(F_{x,FL}\), \(F_{x,FR}\), \(F_{x,RL}\), \(F_{x,RR}\), where the subscripts \(FL\), \(FR\), \(RL\), \(RR\) refer to the front-left,  
Figure 5: Elastic torque and viscous damping of the clutch damper.  
front-right, rear-left and rear-right wheels, respectively, the aerodynamic force \(F_{a}\) and the resistance force \(F_{i}\) due to the road slope angle \(\alpha\).  
\[a_{x}=\frac{\sum_{t}^{4}F_{x,n}-F_{a}-F_{i}}{M_{\wp}} \tag{7}\]  
The term \(\sum_{t}^{4}F_{x,n}\)--where the subscript \(n\) refers to the \(FL\), \(FR\), \(RL\), \(RR\) wheels--sums all the longitudinal forces on the four wheels, while the two terms \(F_{a}\) and \(F_{i}\) are as described by Equation (8):  
\[\left\{\begin{aligned} F_{a}&=0.5\,\rho\;S_{\wp}\;C_{d} \;V^{2}\\ F_{i}&=M_{\wp}\;g\;\sin\alpha\end{aligned}\right. \tag{8}\]  
Regarding the aerodynamic terms, \(\rho\) is the air density, \(S_{\wp}\) is the vehicle front area, \(C_{d}\) is the aerodynamic drag coefficient, and \(V\) is the vehicle speed. Moreover, each tyre is described by an additional degree of freedom, and since vehicle lateral dynamics are not taken into account, the angular accelerations \(\ddot{\theta}_{F}\) and \(\ddot{\theta}_{R}\) are the same for the two wheels at the front and rear axles, and are computed according to Equation (9):  
\[\left\{\begin{aligned} \ddot{\theta}_{F}&=\frac{-T_{ \wp,F}-F_{x,R}R_{w}-T_{r,F}}{I_{w,F}}\\ \ddot{\theta}_{R}&=\frac{T_{w}-T_{\wp,R}-F_{x,R}R_{w }-T_{r,R}}{I_{w,R}}\end{aligned}\right. \tag{9}\]  
where \(T_{w}=T_{ds,L/R}\) is the wheel torque coming from the half shaft--which participates only for the rear wheels--\(T_{b,F}\) and \(T_{b,R}\) are the braking torques, and \(T_{r,n}\) (\(T_{r,F}\) for the front and \(T_{r,R}\) for the rear) is the rolling resistance torque, which has a quadratic relationship to the wheels' rotational speed \(\theta_{n}\), according to Equation (10):  
\[T_{r,n}=F_{z,n}\;R_{w}\;(f+\dot{K}\dot{\theta}_{n}^{2}) \tag{10}\]  
The terms \(f\) and \(K\) are the constant and quadratic coefficients of the rolling resistance and \(F_{z,n}\) is the vertical load on each wheel, which is computed considering the static load distribution between the two axles--by means of parameter \(\gamma\)--and the dynamic load transfer due to vehicle longitudinal acceleration, using Equation (11):  
\[\left\{\begin{aligned} F_{z,F}&=\frac{m\;g\;\gamma}{2 }-\frac{m\;h_{G}\;a_{x}}{2\;L}\\ F_{z,R}&=\frac{m\;g\;(1-\gamma)}{2}+\frac{m\;h_{G}\;a_{x}}{ 2\;L}\end{aligned}\right. \tag{11}\]  
where \(L\) is the vehicle wheelbase and \(h_{G}\) is the height of the vehicle's centre of mass, whereas \(F_{z,F}\) and \(F_{z,R}\) indicate the vertical load on the front and rear wheels, respectively. For each wheel, the longitudinal tyre slip is computed using Equation (12):  
\[\sigma=\frac{\omega R_{w}-V}{\omega R_{w}} \tag{12}\]  
For sake of simplicity, the tyre model assumes a constant relaxation length \(L_{t}\), even if--for more accurate results--it should be considered dependent on the vertical load and longitudinal slip [26]. Introducing the relaxation length, it is possible to correct the tyre slip, obtaining a transient slip for each tyre. Finally, the Pacejka formulation was adopted in the tyre model to evaluate the longitudinal force \(F_{x,n}\) on each tyre, as a function of the different constant Pacejka parameters and three main dynamic quantities:  
\[F_{x,n}=f(F_{z,n},\mu,\sigma) \tag{13}\]  
where \(\mu\) is the tyre-road friction coefficient.