# Vehicle Motion Control for Overactuated Vehicles to Enhance Controllability and Path Tracking
## 6 Results and Discussion
### Euler Spiral Maneuver
The CA in the overactuated configuration improves the MF of the reference vehicle behavior up to \(a_{n}\approx$8.5\,\mathrm{m}\,\mathrm{/}\mathrm{s}^{2}$\) by creating an additional positive yaw moment (braking inner (left) and driving outer (right) wheels) to compensate for terminal understeer in the uncontrolled vehicle and steering the rear wheels to compensate for the otherwise increased sideslip angle \(\beta\). The larger torque split for the overactuated configuration starting at around \(a_{n}\approx$5\,\mathrm{m}\,\mathrm{/}\mathrm{s}^{2}$\) is used in combination with RWS to regulate the increase in the vehicle sideslip angle as well. The maximum normal acceleration \(a_{n}\) is raised from \($9.2\,\mathrm{m}\,\mathrm{/}\mathrm{s}^{2}$\) to \($9.4\,\mathrm{m}\,\mathrm{/}\mathrm{s}^{2}$\), resulting in smaller possible quasi-steady-state cornering radii. As a result, the path tracking error \(e\) is reduced from \(-$1\,\mathrm{m}$\) for the dual-motor configuration to \(-$0.25\,\mathrm{m}$\) for the overactuated configuration at the maximum \(a_{n}\). Consequently, the enhanced MF enlarges the applicable range of the MPC's prediction model, improving the model predictability and resulting in lower path tracking errors \(e\). To improve the understanding of the root causes behind the varying MF performance and path tracking accuracy of the overactuated configuration and the dual-motor configuration, and to examine how MF facilitates the stable domain of handling under disturbances in these configurations, phase planes for both configurations are drawn at three equilibria along the ES, indicated by the red dots in Figure 8. The phase planes are calculated by evaluating the CA response at discrete points in the phase plane and assessing the resulting vehicle response. This allows for the identification and comparison of stable and unstable domains between vehicle configurations. Moreover, the RMSE metric is introduced to quantify how accurately the reference model is followed:  
Figure 8:0  
This metric provides an assessment of the MF error in the phase plane, highlighting the deviation between the reference vehicle behavior and the actually achieved response. In the top row of Figure 9, phase planes for constant \(\delta_{f}=\{\)0, 1.5, 2\(\}\) deg and \(T_{w,i}\) according to CA (11) are shown for the dual-motor configuration. The bottom row displays phase planes for the overactuated configuration at the same \(\delta_{f}\) as in the top row and \(T_{w,i}\) and \(\delta_{r}\) according to CA (11). A separatrix separates the unstable domain (red) from the stable domain (white), where the trajectories converge to an equilibrium (red dots). Evident from the phase planes is the enlarged stable domain for the overactuated configuration compared to the dual-motor configuration. These findings closely align with the enlarged controllability envelope described by [19] for TV and RWS. Moreover, the distance between the equilibrium and the separatrix remains large even in cornering conditions (\(\delta_{f}\
eq 0\)) at higher lateral accelerations for the overactuated configuration. Based on these findings, the values for the SE for the overactuated vehicle in Table 1 are chosen. The \(I_{\text{MF}}\) measure is overlaid in grayscale with the indicator bar to the left of Figure 8:1 Darker shades indicate a close match from the controlled vehicle to the reference vehicle behavior. Conversely, lighter shades indicate an increasing MF error, where the cut-off value of \(I_{\text{MF}}\) is set to 5%.