# Vehicle Motion Control for Overactuated Vehicles to Enhance Controllability and Path Tracking
## 6 Results and Discussion
### Euler Spiral Maneuver  
The reference velocity \(V_{\text{ref}}\) is set to \(25\,\text{m/s}\). The reference path features a linear increase in curvature over a distance of \(2250\,\text{m}\), starting from a straight line and culminating in a circle with a radius of \(62.8\,\text{m}\). By traveling at the constant velocity \(V_{\text{ref}}\), the controlled vehicle is expected to follow the path in a quasi-steady-state manner, with a linear increase in the normal acceleration \(a_{n}\) over both the traveled distance \(s\) and time \(t\). Consequently, the handling diagram of the vehicle is derived by tracking the reference path. The parameters in Table 3 refer to the (linear) uncontrolled vehicle and parameterize the prediction/reference vehicle behavior for the MPC and MF in (7) and in (9), respectively. In Figure 6a (bottom), the corresponding handling diagram of the uncontrolled vehicle (dotted) and desired reference vehicle behavior (black line) is shown. Black dots relate the reference vehicle behavior to the corresponding position at the reference path. The ES results are depicted in Figure 8. The response of the overactuated configuration is represented by solid lines with square markers, while the dual-motor configuration is represented by dotted lines with circle markers. The reference vehicle response is depicted in gray. The top row displays the steering angle \(\delta_{f}\), the sideslip angle \(\beta\) and the path tracking error \(e\) over the normal acceleration \(a_{n}\). In the bottom row, the actuator utilization for the wheel torques \(T_{w,i}\) and the rear-wheel steering angle \(\delta_{r}\) are shown. The vehicle response will only follow the reference vehicle behavior when both MF and CA can compensate for the error terms \(e_{x}\), \(e_{y}\) and \(e_{z}\) in (11). This results in a distribution of \(T_{w,i}\) and \(\delta_{r}\) according to (11a) that minimizes tyre slips \(\kappa_{i}\). Below \(a_{n}\approx 4.5\,\text{m/s}^{2}\), the CA of both the overactuated and the dual-motor configuration allows for the accurate MF of the reference vehicle behavior. Since the chosen reference vehicle behavior closely matches the linear domain of the uncontrolled vehicle in this range of normal acceleration \(a_{n}\)--see Figure 6a (bottom)--minimal torque vectoring is needed to compensate for deviations. While only very small drive torques are required by the CA of the overactuated configuration (see \(T_{w,i}\) in Figure 8) to match the required additional yaw  
Figure 8: Euler spiral maneuver simulated with the overactuated and dual-motor configurations. Linear (reference) vehicle behavior given in gray. Top row shows front steering angle \(\delta_{f}\), vehicle sideslip angle \(\beta\) and path tracking error \(e\) characteristics over \(a_{n}\). Bottom row shows actuator usage for wheel torques \(T_{w,i}\) and rear-wheel steering angle \(\delta_{r}\). moment, in the dual-motor configuration, a large drive torque split is needed, driving the rear axle and recuperating at the front axle. Consequently, the MPC tracks the reference trajectory; see \(\delta_{f}\) and \(\beta\) and \(e\) in Figure 8. A closer inspection of the steering angle \(\delta_{f}\) and the path tracking error \(e\) reveals that the CA of the dual-motor configuration more accurately follows the reference vehicle behavior, and the MPC more precisely tracks the reference path, compared to the overactuated configuration. Prohibiting opposite signs of the front and rear drive torques in the CA (11) for the dual-motor configuration results in a slight decrease in the tracking performance and a considerable decrease in the drive torque split.