## 5 Simulations and Discussions
### Performance Evaluation Considering Road Topography  
To analyze the effect of considering road topography, four comparative MPC controllers are utilized for trajectory tracking of the test road, as shown in Figure 8. The desired velocity of AGV is set to be a constant of 72 km/h. Since the first 200 m of the test road is just a segment of a straight lane, we initialized the location of the AGV as 200 m behind the starting point of the test road.  
* The MPC controller considers both road curvature and bank angle, as proposed in this work, denoted as Controller I;
* The MPC controller without consideration of road topography, i.e., setting \(\mathbf{u}_{2}=[0,0]^{T}\), denoted as Controller II;
* The MPC controller only considers the effect of road bank angle, i.e., setting the second term of \(\mathbf{u}_{2}\) as zero, denoted as Controller III;
* The MPC controller only considers the effect of road curvature, i.e., setting the first term of \(\mathbf{u}_{2}\) as, denoted as Controller IV.  
The comparison of the simulation results is shown in Figure 9. It can be seen from Figure 9a that the lateral track errors when considering the road curvature (Controller I and Controller IV) are significantly smaller than those not considering the road curvature (Controller II and Controller III). Thus, we can conclude that the road curvature has more influence on the lateral track error than that of road bank angle. As shown in Figure 9b, the regularized ZMPs when considering road bank angle (Controller I and Controller III) coincide nicely with the LTR. On the other hand, the regularized ZMPs when not considering road bank angle (Controller II and Controller IV) diverge significantly from the LTR. Thus, the road bank angle is validated to be an important factor for rollover prevention. Figure 9c illustrates the comparison of front wheel angle (\(\delta_{f}\)) during the simulations. We can see that the maximum of front wheel angle is less than 0.04 (rad), satisfying the actuator saturation limits. Figure 9d shows the comparison of sideslips in the plane of lateral velocity \(v_{y}\) and yaw rate \(\omega_{z}\). It can be seen that the traces of Controller I, considering both road curvature and bank angle, are more compact than those of the other three controllers. This validates that the proposed MPC scheme considering the effect of road topography can be used to ensure stabilization of high-speed AGVs.  
Note that Controller I, considering the effects of both road curvature and bank angle, can provide better trajectory tracking performance than the other three controllers by limiting the lateral track error and the \(\bar{y}_{ZMP}\) within 0.15 m and 0.3 m, respectively. Thus, it validates the necessity of considering road topography in the stabilization of high-speed AGVs.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Values (Unit)** & **Parameters** & **Values (Unit)** & **Parameters** & **Values (Unit)** \\ \hline \(m\) & 1600 (kg) & \(\mathcal{C}_{af}\) & \(-\)110,000 (N/rad) & \(t_{s}\) & 0.05 (s) \\ \(m_{s}\) & 1430 (kg) & \(\mathcal{C}_{ar}\) & \(-\)92,000 (N/rad) & \(t_{l}\) & 0.5 (s) \\ \(I_{x}\) & 700. 7 (kg\(\cdot\) m\({}^{2}\)) & \(K_{\phi}\) & 145,330 (N\(\cdot\) m\(/\)rad) & \(N_{s}\) & 10 \\ \(I_{z}\) & 2059.2 (kg\(\cdot\) m\({}^{2}\)) & \(D_{\phi}\) & 4500 (N\(\cdot\) m\(\cdot\) s\(/\)rad) & \(N_{p}\) & 20 \\ \(l_{f}\) & 1.12 (m) & \(g\) & 9.81 (m/s\({}^{2}\)) & \(W_{cg}\), \(W_{ep\phi}\) & 500 \\ \(l_{r}\) & 1.48 (m) & \(\delta_{f,max}\) & 0.4 (rad) & \(W_{sh}\) & 50 \\ \(T_{r}\) & 1.565 (m) & \(\Delta\delta_{f,max}\) & 0.08 (rad\(/\)s) & \(W_{\delta f}\) & 5 \\ \(h_{sr}\) & 0.68 (m) & \(\alpha_{r,lim}\) & 0.1 (rad) & \(k_{1}\) & 0.5 \\ \(d_{s}\) & 0.5 (m) & \(\bar{y}_{ZMP,max}\) & 0.7 & \(k_{2}\) & 0.6 \\ \hline \hline \end{tabular}
\end{table}
Table 1: The parameter of the vehicle and weights of the MPC controller.