## 4 Numerical Simulation  
To verify the effectiveness of the proposed ISPFCE algorithm, a numerical simulation was conducted with CarSim and Simulink. The vehicle model in CarSim is a 27-degree-of-freedom full-vehicle model that is equipped with nonlinear tire models. The measurement noise characteristics of realistic sensors (such as those in the RT4000 navigation system from Oxford Technical Solutions Ltd., Bicester, U.K.) were reproduced in the simulation. The proposed algorithm was implemented using Simulink. The simulation parameters are listed in Table 1.  
In the simulation, the initial speed of the vehicle at the origin point, which was also considered to be the desired speed, was set to 23 m/s. The shape and curvature of the road path are shown in Figures 20 and 21, respectively. The minimum radius of the path was 187.5 m (\(1/R=0.0053(1/\)m)), as shown in Figure 21. The friction coefficient was initially 0.85; subsequently, it decreased to 0.2 and then returned to 0.85 in a stepwise manner, as shown in Figure 21. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(m\) & 1412 kg & \(R_{g}\) & 0.3 m \\ \(l_{f}\) & 1.016 m & \(l_{r}\) & 1.564 m \\ \(l_{s}\) & 0.77 m & \(h_{c}\) & 0.54 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 19: A path segment. Figure 20: Road path. As discussed previously, the proposed ISPFCE algorithm can be applied in combination with previously-investigated motion controllers. In the simulation, two motion controllers were designed for comparison, one based on fuzzy model predictive control [12] (Controller A) and one based on a linear quadratic regulator [40] (Controller B). The performances of these motion controllers with and without the proposed ISPFCE algorithm are presented for comparison. The lane-keeping performance and speeds of vehicles controlled by Controllers A and B with and without ISPFCE are illustrated in Figures 22-24. As shown in Figure 23, without the proposed ISPFCE algorithm, Controllers A and B attempt to control the corresponding vehicles such that they travel at the desired speed along the entire road. However, as discussed above, due to the limited road friction at a tight bend, it is impossible for the vehicles to pass through the bend at the desired speed. Figure 22 indicates that the vehicles controlled by Controllers A and B cannot remain on the path at the bend,  
By contrast, with the proposed ISPFCE algorithm, the speeds of the vehicles controlled by Controllers A and B are reduced to low values before the tight bend with the corresponding low friction coefficient, as indicated in Figure 23. Thus, these vehicles can move along the reference path with small tracking errors, as shown in Figures 22 and 24. This comparison demonstrates that the proposed ISPFCE algorithm can be applied with different motion controllers to handle roads with changes in their friction coefficients. Notably, traveling at a low speed at all times would lead to increased time consumption and decreased traffic flow. Thus, once a high friction coefficient is again detected, the vehicle speed should be increased. As shown in Figure 23, the vehicle speeds return to the desired value once a high friction coefficient is again detected, demonstrating that the proposed ISPFCE algorithm can handle various changes in the friction coefficient on a curved road. The control inputs to the vehicles controlled by Controllers A and B with ISPFCE are shown in Figures 25 and 26. The details of the friction estimation and speed planning are further plotted in Figures 27-34.