# Vehicle Motion Control for Overactuated Vehicles to Enhance Controllability and Path Tracking
## 6 Results and Discussion
### U-Turn Maneuver  
The U-turn incorporates trail-braking and out-of-corner accelerating to study the stability and path tracking properties of the controlled vehicle at combined lateral and longitudinal acceleration and transient driving conditions. The reference path is calculated using the minimum curvature method [26], while the velocity profile is optimized by maximizing the combined lateral and longitudinal acceleration subject to an upper limit, defined by a reference acceleration level \(|a_{\text{ref}}|\). By design, the U-turn is symmetrical with respect to the apex of the corner in both the reference path and velocity profile. Considering the linear reference vehicle behavior--see Figure 6a (bottom)--the trajectory traced in the \(\beta\)-\(\psi\) phase plane during the trail-braking and out-of-corner accelerating phases should be similar. Consequently, no hysteresis should appear in the phase plane. In Figure 10, both the dual-motor and the overactuated vehicle configuration are compared. Each row in the figure corresponds to a different combined reference acceleration level \(|a_{\text{ref}}|\) and therefore velocity profile: \(|a_{\text{ref}}|=$5\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}$\) in the first row, \(6\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}$\) in the second, \(7\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}$\) in the third and \(8\,\mathrm{m}\mathrm{/}\mathrm{s}^{2}$\) in the fourth. This approach aims to explore the limits of handling and to investigate the MF error and consequently the path tracking error of the MPC for both configurations. Within each row, the first plot shows the path tracking error \(e\), the second plot shows the gg-diagram and the third plot shows the phase plane \(\beta\)-\(\psi\)  
Figure 9: Phase planes at varying FWS angles corresponding to the dual-motor and overactuated configurations at \(V=$25\,\mathrm{m}\mathrm{/}\mathrm{s}$\). Stable and unstable regions are shown in white and red, respectively. The \(I_{\text{MF}}\) measure is overlaid in grayscale. Red dots indicate stable equilibria corresponding to Figure 8 (top). The solid line represents the overactuated configuration, the dashed line represents the dual-motor configuration and the reference line is shown in red with circle markers. As \(|a_{\text{ref}}|\) increases, the hysteresis of the dual-motor configuration appears; see the phase plane \(\beta\)-\(\psi\) at the right of Figure 10. This leads to skidding under braking. The noticeable hysteresis in the phase plane of the dual-motor configuration suggests higher MF errors compared to the overactuated configuration. At lower levels of acceleration, the path tracking errors \(e\) are comparable for both configurations. However, differences become apparent for \(|a_{\text{ref}}|=7\,\text{m}/\text{s}^{2}\) during corner entry (\(175\,\text{m}\) to \(225\,\text{m}\)) under trail-braking, where the dual-motor configuration saturates the front types, followed by skidding and steering corrections by the path tracking controller (not depicted). The increased path error \(e\) under trail-braking correlates well with the loss of model following in the trail-braking branch in the phase plane. For both configurations, starting at \(|a_{\text{ref}}|>5\,\text{m}/\text{s}^{2}\), a vertical line in the gg diagram at \(a_{x}=5\,\text{m}/\text{s}^{2}\) is visible due to the \(F_{d,\text{max}}\) constraint in (8e), considering the drive force limitations of the powertrain in the MPC. At \(|a_{\text{ref}}|=8\,\text{m}/\text{s}^{2}\), large path tracking errors \(e\) in the MPC become apparent in the dual-motor configuration due to insufficient MF. In contrast, the CA of the overactuated configuration utilizes both TV and RWS to minimize and maintain consistent hysteresis while increasing the maximum acceleration levels. The trajectories in the phase plane for this configuration remain entirely within the black areas at the apex of the curve in the phase planes depicted in Figure 9; hence, reasonable MF is provided to the MPC.