# Torque Vectoring Control Strategies Comparison for Hybrid Vehicles with Two Rear Electric Motors
## 4 Simulation Results
### Ramp Steering  
The last maneuver studied is ramp steering, highlighting the steady-state response of the vehicle under a wide range of lateral accelerations. As shown in Figure 12, the control systems present a first overshooting and slight vibrations (due to the initial threshold) followed by a linear ascending phase and a saturation around 12.5 s. Once again, the FOSM lowpass controller behaves poorly regarding the actuation smoothness, presenting quite accentuated vibrations before saturation. Figure 11: Yaw rate (**a**) and yaw rate error (**b**) in the 80\({}^{\circ}\) step steering. Figure 13 displays the yaw rate and its error, while Figure 14 shows a zoomed-in section of the error plot between 0 and 14 s.  
The controlled vehicles clearly change their behavior and follow quite well the yaw goal for the first part of the maneuver. After the saturation of the actuators, the controlled vehicles see their behavior move closer to the baseline version, as it is no longer possible to supply the required yaw moment to create a perfectly neutral vehicle. At around 16 s, the baseline vehicle achieves its maximum lateral performance, as shown by the yaw rate peaking, corresponding to the reduction in the requested torque  
Figure 12: Torque in the right-side EM during ramp steering. Figure 13: Yaw rate (**a**) and yaw rate error (**b**) in ramp steering. presented before. The same peak effect is displayed at around 17 s for all controlled systems, at a yaw rate of 40.3 rad/s vs. the 35.5 rad/s baseline. After this peak, a highly non-linear response is observed, as expected, as the tires are no longer in their preferred working zone. It is hard to draw meaningful conclusions from the results after the tire saturation because the mechanical models tend to be much less precise. A closer look at the first section of the maneuver allows for a more interesting discussion of the controller. The plots confirm some previously stated conclusions, such as the high sensitivity and vibrational response of the FOSM lowpass controller; the quick settling time and prompt response of SOSM suboptimal; the PID controller with good transient behavior but finding it difficult to eliminate small steady-state error; the slower but smooth response of the SOSM twisting algorithm; and the similar smooth response of LQR and FOSM continuous. Lastly, Figure 15 presents the lateral acceleration and steering angle. In Figures 9 and 15, the advantage of all control systems over the baseline vehicle is clear. Higher maximum lateral acceleration and more responsive behavior can be observed in the plot. It is interesting to observe that the linearity goal is not completely achieved; in fact after the saturation of the EMs at 0.8 g, the slope of the plot changes significantly. As can be noted in Figure 9, SOSM suboptimal exhibited more responsive behavior. In this case, the SOSM twisting and LQR controllers are much closer and have a higher lateral acceleration in several zones of the plot. This suggests that the SOSM suboptimal controller performs better in transient situations but gives no significant advantage in less dynamic conditions. Figure 14: Yaw rate error detail: zoomed-in region before saturation. Figure 15: Steering angle vs. lateral acceleration during ramp steering. In [15], the constant radius corner condition was studied. In this paper, both of the presented maneuvers are open loop. Finally, Table 5 shows, for comparison purposes, the numerical results obtained by the control systems considered in the three conditions studied and with reference to the proposed PF. The time considered for the step steering maneuvers is the total simulation time, while for ramp steering, the PF was calculated until 17 s of simulation, because, after that, the systems behave similarly, and the simulation precision is lower.