# Vehicle Motion Control for Overactuated Vehicles to Enhance Controllability and Path Tracking
## 6 Results and Discussion  
The effectiveness of the proposed control architecture is studied concerning the following aspects: (1) the adherence of the actual vehicle handling characteristics to the reference vehicle behavior under quasi-steady-state conditions; (2) the impact of MF on the vehicle model predictability in the MPC and the subsequent path and speed tracking capabilities; (3) the impact of overactuation and MF on the stable handling envelope of the vehicle in the phase plane and in transient driving conditions. Therefore, two respective maneuvers were selected to investigate the characteristic properties of the proposed approach, as depicted in Figure 6.  
* Euler Spiral (ES). The main aims of this quasi-steady-state maneuver are to examine the effectiveness of MF in following the reference vehicle behavior and to study the respective enhancement of the MPC in path tracking at a constant \(V_{\text{ref}}\). Figure 6a (top) illustrates the reference path of the ES. At the bottom, the handling behavior, \(\delta_{f}\) and \(\beta\)  
Figure 5: Left plot shows the normalized lateral tyre force \(\frac{F_{x}}{\kappa_{i}}\) over the normalized longitudinal tyre force \(\frac{F_{x}}{\kappa_{i}}\) for various tyre sideslip angles \(\alpha\). Right plot shows the normalized longitudinal force \(\frac{F_{x}}{\kappa_{i}}\) over the longitudinal tyre slip \(\kappa\) for various tyre side slip angles \(\alpha\), with the gray area defining the slip constraint for the CA.  
over normal acceleration \(a_{n}\) of the uncontrolled vehicle and the respective reference vehicle behavior are displayed.
* U-Turn. The main aims of this transient maneuver are to investigate the MF and the MPC performance up to the limits of handling under pure longitudinal, pure lateral, and combined driving conditions. Figure 5(b) (top) shows the reference path with a left turn; at the bottom, the desired trajectory in the gg diagram and the respective velocity profile are depicted.  
Two vehicle configurations are compared using a detailed multi-body vehicle model set up in _Simpack_; see Figure 7. Both configurations utilize the control architecture depicted in Figure 3 and differ only in the available actuator set.  
The first configuration features 4 WS and wheel-independent 4 WD and is denoted as \"overactuated\" in the subsequent discussion. The second overactuated configuration features a reduced actuator set with FWS and dual-motors enabling axle-wise independent drive and is denoted as \"dual-motor\". Considering the reduced actuator set, additional constraints are introduced in the CA (11) of the dual-motor configuration: \(\delta_{r}=0\), \(T_{w,1}=T_{w,2}\) and \(T_{w,3}=T_{w,A}\).  
The parameters for the LTV-MPC are given Table  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Name** & **Value** & **Unit** \\ \hline \(T_{\text{CA}}\) & 0.01 & s \\ \(q^{2}\) & \(\frac{C_{f}l_{f}^{2}+C_{f}l_{g}^{2}}{C_{f}+C_{r}}\) & m\({}^{2}\) \\ \hline \(\kappa_{\text{max}}\) & 0.25 & - \\ \(k_{\text{max}}\) & 0.25 & 1/s \\ \(\delta_{r,\text{max}}\) & 10 & deg \\ \(\delta_{r,\text{max}}\) & 10 & deg/s \\ \hline \(q_{x}\) & \((\
icefrac{{1}}{{m_{\text{g}}}})^{2}\) & N\({}^{-2}\) \\ \(q_{y}\) & \((\
icefrac{{1}}{{m_{\text{g}}}})^{2}\) & N\({}^{-2}\) \\ \(q_{z}\) & \((\
icefrac{{2}}{{4}}/l_{z,\text{g}})^{2}\) & \((\text{Nm})^{-2}\) \\ \(q_{\delta_{z}}\) & \((\
icefrac{{0.03}}{{10}})^{2}\) & \((\text{deg}/\text{s})^{-2}\) \\ \(q_{\kappa}\) & \((\
icefrac{{0.1}}{{0.3}})^{2}\) & - \\ \hline \(w_{f}\) & 0.829 & mm \\ \(w_{r}\) & 0.826 & mm \\ \(h\) & 0.507 & m \\ \(h_{1}\) & \(h\) & m \\ \(h_{2}\) & 1.08\(h\) & m \\ \(r_{l}\) & 0.361 & m \\ \hline \hline \end{tabular}
\end{table}
Table 2: CA parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Value** & **Unit** \\ \hline \(m\) & 1310 & kg \\ \(l_{z}\) & 2006 & kgm\({}^{2}\) \\ \(l_{f}\) & 1.387 & m \\ \(l_{r}\) & 1.107 & m \\ \(C_{f}\) & 140.86 & kN/rad \\ \(C_{r}\) & 176.86 & kN/rad \\ \hline \hline \end{tabular}
\end{table}
Table 3: Two-wheel vehicle model parameters.