# Tube-Based Robust Model Predictive Control for Autonomous Vehicle with Complex Road Scenarios
## 4 Simulation Result and Analysis
### Co-Simulation Results  
In this study, a specific subset of experimental conditions was analyzed to evaluate the system's performance at varying road adhesion levels. The analysis focused on a constant vehicle speed of 20 m/s, whereas the road adhesion coefficient (\(\upmu\)) was varied across three levels: 0.4, 0.6, and 0.8. These conditions thoroughly represent a wide spectrum of road surfaces, ranging from low-adhesion to high-adhesion surfaces.  
The parameters used in this study were \(m=1723\) kg, \(a=1.232\) m, \(b=1.468\) m, \(I_{z}=4175\) kg, \(T=0.02\) s, \(C_{cf}=66.9\) kN/rad, and \(C_{cr}=62.7\) kN/rad.  
According to Theorem 1, the feedback gain is given by  
\[K=\left[\begin{array}{cccc}0.112&0&-18.0&0&-0.0297&-0.0297\end{array}\right]\]  
To facilitate a comprehensive comparison of the experimental datasets, the analysis focused on two key metrics: the maximum value Y and yaw angle error, with the mean squared error (RMSE) compared to the original reference estimation.  
As shown in Figure 3 and Table 1, reduction in maximum tracking error in S-curve: Tube-RMPC achieves a 9.17% reduction in the peak deviation compared to MPC and a 20.5% improvement over PID; Lower mean squared error (RMSE): Tube-RMPC reduces MSE by 14.0% compared to MPC and leads to a 36.2% improvement over PID. Moreover, Reduction in Yaw angle error in S-curve: Tube-RMPC achieves a 1.77% reduction in the peak deviation compared to MPC and a peak deviation compared to MPC and a 5.66% improvement over PID; Lower Yaw angle RMSE: Tube-RMPC achieves a 7.00% reduction in peak deviation compared to MPC and an 9.44% improvement over PID.  
Similarly, as shown in Figure 4 and Table 2, the following results were obtained for the double lane change. Reduction in maximum tracking error in double lane change: Tube-RMPC achieves a 2.25% reduction in the peak deviation compared to MPC and a 4.58% improvement over PID; Lower mean squared error (RMSE): Tube-RMPC reduces MSE by 29.38% compared to MPC and leads to a 71.17% improvement over PID. Moreover, Reduction in Yaw angle error in double lane change: Tube-RMPC achieves a 23.06% reduction in the peak deviation compared to MPC and a 28.10% improvement over PID;Lower Yaw angle RMSE: Tube-RMPC achieves a 0.03% reduction in the peak deviation compared to MPC and a 6.83% improvement over PID.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mean Tracking Error** & **Y** & **RMSE** \\ \hline Tube-RMPC & 7.6390 & 0.2482 \\ MPC & 8.4103 & 0.2885 \\ PID & 9.6074 & 0.3890 \\ \hline
**Yaw Angle Error** & \(\Phi\) & **RMSE** \\ \hline Tube-RMPC & 17.5866 & 4.4729 \\ MPC & 17.8978 & 4.7853 \\ PID & 18.5831 & 4.8950 \\ \hline \hline \end{tabular}
\end{table}
Table 1: S-curve result analysis.  
Figure 3: S-curve of Y position tracking result and yaw angle result. **(a–c)** Positions with 0.4, 0.6, and 0.8, respectively; **(d–f)** yaw angles with 0.4, 0.6, and 0.8, respectively.  
However, asymptotic tracking cannot be strictly guaranteed due to model simplifications and external disturbances, which aligns with the bounded tracking errors observed and discussed in the simulation analysis.