# Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors
## 2 Vehicle Modelling and Maneuvers  
### Vehicle Dynamics  
The model used to represent the dynamic response of the vehicle is a 14 DoF model widely exploited and presented in the literature [19; 20], an application of which can be found in [21].  
The derivation of the 14 DoF model and its complete equations is beyond the scope of this paper. The 14 DoF of the model are briefly introduced below, corresponding to the scheme in Figure 2.  
* 6 DoF describe the free body motion of the vehicle sprung mass;
* 2 DoF for each of the four wheels represent the generalized position with respect to the body of the vehicle \(\zeta\) (derived from the suspension kinematics) and the longitudinal slip of the tire.  
The actual implementation of the model is performed using the CarRealTime software (CRT) from VI-Grade. CRT's interface allows a complete description of the studied vehicle, including the characteristics of its many subsystems (steering, brakes, suspensions, body, powertrain, and so on). Being so, it is possible to analyze a fully non-linear response, especially when it comes to the limit driving conditions, with superior precision when compared with lower DoF mathematical models. The studied vehicle is based on the physical features of the Fiat 500, as shown in Table 1.  
The vehicle model relies on custom damper, spring, and suspension curves; as far as tires are concerned, the appropriate Pacejka models are implemented, widely used in professional vehicle dynamics simulations [22]. The 14 DoF model, while suitable for the complete description of the vehicle's motion and dynamics, is rather inconvenient during the implementation and theoretical calculations related to the control systems, owing to its complexity and inherent non-linearity. For that purpose, a simpler model is used in the preliminary phases of system setup, namely a 2 DoF bicycle model [19; 20].  
Figure 2: Scheme and variables of the 14 DoF vehicle model.  
This widely known model considers the automobile as a single-track vehicle (collapsing the contribution of left-right wheels); Figure 3. It is valid under certain conditions such as small steering angles and linear range of the tires and it neglects some contributions such aerodynamic forces, suspension compliance, and load transfer.  
In the bicycle model, the 2 DoF used to describe the behavior of the vehicle are the side slip angle \(\beta\) and yaw rate \(\gamma\). By mild computations, which are not reported for the sake of brevity, the equations of motion can be written in a state-space representation as follows:  
\[x=Ax+Bu\] \[x=(\beta),\ A=\left(\begin{matrix}\frac{C_{f}+C_{r}}{m\
u_{x}}&\frac{aC _{f}-bC_{r}}{m\
u_{2}}-1\\ \frac{aC_{f}-bC_{r}}{I_{z}}&\frac{a^{2}C_{f}+b^{2}C_{r}}{I_{z}\
u_{x}}\end{matrix} \right),\ u=(\delta_{M_{z}}),\ B=\left(\begin{matrix}-\frac{C_{f}}{m\
u_{x}}&0 \\ -\frac{aC_{f}}{I_{z}}&\frac{1}{I_{z}}\end{matrix}\right) \tag{1}\]  
where the state vector \(x\) consists of the variables \(\beta\) and \(\gamma\), while the input vector \(\mathrm{u}\) is made up of the steering angle at the wheels \(\delta\) and the external yaw moment \(M_{z}\). The elements of the matrices \(A\) and \(B\) are calculated as functions of the physical parameters of the vehicle (as in Table 1) and of \(v_{x}\), which is the longitudinal velocity of the vehicle.  
Notice that the used 2 DoF bicycle model has \(M_{z}\) as one of the two inputs. This variable is not commonly present in the standard representation of the model. Its addition is particularly well suited for describing the effects of the TV, in which a yaw moment is generated as a consequence of left/right torque imbalance (dimension not present in the \"mono-track\" approach of the bicycle model).