# Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors
## 2 Vehicle Modelling and Maneuvers
### Model and Torque Allocation of the Hybrid Powertrain  
The components of the hybrid powertrain (PWT), as well as the control logic that defines the distribution between ICE and EMs, are shown in Figure 4, where each block represents a key element of the control system. This layout is presented in detail in [15], \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property** & **Symbol** & **Value** \\ \hline Total mass & \(M\) & 1006 kg \\ Wheelbase & \(l\) & 2.3 m \\ Distance between CoG and front axle & \(a\) & 0.805 m \\ Distance between CoG and rear axle & \(b\) & 1.495 m \\ Track & \(t\) & 1.413 m \\ CoG height & \(h_{cg}\) & 0.537 m \\ Yaw inertia & \(I_{z}\) & 965.6 kg\(\cdot\)m\({}^{2}\) \\ Tire radius (no load) & \(r\) & 0.291 m \\ Lateral stiffness (front tires) & \(C_{f}\) & 21,094 N/rad \\ Lateral stiffness (near tires) & \(C_{r}\) & 14,556 N/rad \\ Motor peak torque & \(T_{EM}\) & 103 N-m \\ Motor peak power & \(P_{EM}\) & 25 kW \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the vehicle [15]. Figure 3: Schematic representation of the 2 DoF bicycle model. where a similar model is used for a preliminary analysis of the parallel hybrid P4 layout in the Fiat 500. The central idea of the system is that the imbalance between torques applied in the left and right wheel will create a yaw moment \(M_{z}\) on the vehicle, which will be the main control action. The interface chosen is a co-simulation environment between MATLAB/Simulink and VI-CRT. As displayed in Figure 4, the control system receives the torque request and EM limitations from the plant and, with the block \"hybrid logic\", (1) decides on the best split between the front ICE and the rear EMs. The ICE torque request is directly sent to the plant again, while the overall EM torque request passes through the block \"torque vectoring\" (2), which will determine the split between left and right commands. Block (2) is then composed of three internal logic subsystems: the \"bicycle model\" block (3) calculates the _ideal yaw rate_ using the 2 DoF handling model to estimate the yaw rate that a neutral vehicle would present in a given status of steering angle and velocity; then the block \"control system\" (4) obtains the reference yaw rate and compares it to the real yaw rate coming from the plant observation and, through the various proposed logics, defines the desired control action in terms of _torque bias_; finally, in the final \"torque allocation\" block (5), the total EM torque and the torque bias requests are combined to calculate the effective left and right torque requests for the EMs [15]. The ideal yaw rate is calculated by targeting the neutral understeer behavior characterized by an understeer coefficient \(K_{u}\) with zero value. The ideal yaw rate can then be determined as follows:  
\[\frac{\gamma}{\delta}=\frac{\frac{v_{x}}{l}}{1+K_{u}v_{x}}\rightarrow\gamma_{ i}=\delta\cdot\frac{v_{x}}{l} \tag{2}\]  
The EM model is energetic, based on the maximum torque and power curves supplied by the manufacturer, as well as the efficiency map. Regarding the battery, a simple saturation strategy is put in place to restrain the current based on the series/parallel configuration of the cells and their maximum discharge rates and nominal voltage. The characteristics of the E-PWT are considered fixed for all values of state of charge and no thermal features are considered. Some implementations of TV systems are based on the application of braking action. However, these strategies are less efficient because the braking power is dissipated (rather than regenerated). Being able to control each EM individually, this paper does not consider the actuation of the braking system as a control parameter. After the definition of the individual torque requests, they are sent to CRT each with a timestep of 0.001 s, depicting all significant dynamic quantities.