# Integrated Adaptive Steering Stability Control for Ground Vehicle with Actuator Saturations
## 4 Simulation Verification
### Scenario 1: Square-Wave Front Wheel Input with Flat Road Surface  
In this scenario, the vehicle runs on a flat road under a square-wave front road wheel input with 0.5 Hz frequency and 0.01 rad amplitude. The simulation results are exhibited in Figures 3-10.  
The vehicle body vertical displacement and acceleration are illustrated in Figures 3 and 4, respectively, from which it can be seen that the proposed controller can render the heave motion of the sprung mass as zero, asymptotically stable, and decrease the vertical acceleration dramatically, hence, improving vehicle ride comfort. Although the vertical displacement under the proposed controller converges slower than under the comparative controller \(\Psi\), the vertical acceleration converges much faster, effectively improving ride comfort, as the vertical acceleration, rather than the displacement, is the main comfort index.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m_{s}\) & 1110 kg & \(m_{w}\) & \(2\times 30\) kg \\ \hline \(l_{x}\) & 440.6 kg\(\cdot\)m\({}^{2}\) & \(l_{z}\) & 1343.1 kg\(\cdot\)m\({}^{2}\) \\ \hline \(c_{s}\) & \(2\times 4000\)Ns\(/\)m & \(k_{s}\) & \(2\times 28\),000 N\(/\)m \\ \hline \(c_{w}\) & \(2\times 1000\)Ns\(/\)m & \(k_{w}\) & \(2\times 232\),000 N\(/\)m \\ \hline \(c_{f}\) & 22,010 N\(/\)rad & \(c_{r}\) & 22,010 N\(/\)rad \\ \hline \(h_{g}\) & 0.54 m & \(d\) & 0.74 m \\ \hline \(a\) & 1.04 m & \(b\) & 1.56 m \\ \hline \(\theta_{1 max}\) & \(1/400\) kg\({}^{-1}\cdot\)m\({}^{-2}\) & \(\theta_{1 min}\) & \(1/600\) kg\({}^{-1}\cdot\)m\({}^{-2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the vehicle model.  
Figure 3: Vehicle body vertical displacement.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(k_{1}\) & \(1\) & \(k_{2}\) & 10,000 & \(k_{3}\) & 10 \\ \hline \(k_{4}\) & \(1\) & \(k_{10}\) & 100 & \(k_{\zeta}\) & 10 \\ \hline \(r_{0}\) & \(10\) & \(\sigma\) & 0.01 & \(r_{1}\) & 5000 \\ \hline \(r_{2}\) & 0.001 & \(k_{11}\) & 0.1 & \(k_{\gamma},k_{\theta},k_{z}\) & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameters of the proposed controller and the compared controller.  
The vehicle sideslip angle and yaw rate are illustrated in Figures 8 and 9, supposing that the maximum yaw control effort satisfies \(u_{max}=1000\) N. Figure 9 indicates that lower performance degradation can be guaranteed with the proposed controller when saturation of the yaw control actuator occurs. In addition, Figure 8 shows that the vehicle under the proposed controller possesses a lower sideslip angle compared with respect to the compared controller, making for more stable lateral motion with the proposed controller.  
The yaw motion control input is illustrated in Figure 10, and the corresponding state of the anti-windup compensator \(\zeta\) is illustrated in Figure 11. From these two figures, it can be seen that when the control input reaches the actuator limitation \(u_{max}\), the state of the anti-windup compensator \(\zeta\) accordingly increases its value such that the control input decreases by feeding \(\zeta\) back into the control input \(u_{y}\), consequently reducing the impact of actuator saturation on control performance.  
Figure 8: Vehicle sideslip angle.  
Figure 7: Active suspension input.  
Figure 9: Vehicle yaw rate.