## 3 Experimental Results  
For each road condition described in Table 4, we developed a new analytic equation and tested them with different levels of target speed. We pick optimal parameters for a PD controller according to algorithms in our previous work [6] to optimize its quality function and controller perception. So, we ran simulations for predictive PD, best PD, and, constructed via genetic programming [7], GP-RMEP controlling models. Algorithms were designed for the same track with the same conditions and the same estimation quality function \(F\), so we could compare them. Resulting car trajectories are presented in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\#Road** & **Friction of** & **Friction of Road** & **Overall Friction,** & **Critical Speed,** & **Speed of the** & **Speed of the** & **Speed of the** & **Car** \\
**Condition** & **Tires, \(\mu\)** & **Surface, \(\mu\)s** & \(\mu=\mu\times\mu\)s & \(V_{CR}\), m/s & **(0.85 \(V_{CR}\)), m/s** & **m/s** & **(0.95 \(V_{CR}\)), m/s** \\ \hline \(1\) & \(1.0\) & 0.5 (rainy) & 0.5 & 15.65 & 13.3 & 14.2 & 15 \\ \(2\) & \(1.0\) & 0.3 (icy and snowy) & 0.3 & 14 & 10.4 & 11 & 11.6 \\  & \(1.0\) & 0.1 (icy) & 0.1 & 12.12 & 6 & 6.3 & 6.7 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Speed of the car during the trial on the test track with different friction coefficients.  
Figure 9: _Cont._  
As Figure 9 illustrates, under each road condition, the trajectory provided by the PD controller version with prediction was better (i.e., closer to the center of the lane). In Figure 10, the changes of steering angle for parameters \(\mu=0.3\), \(0.95V_{cr}\) is shown. It could be noted that the new controller provides smooth steering, similar to the PD controller but with lower amplitude. This characteristic also corresponds to driving stability. We analyzed the smoothness of each function as a number of sign changes in the approximate first derivative: \(d=\frac{\Delta{\sigma}}{\Delta t}\). The results were as follows: GP-RMEP--349 changes, PD controller--218 changes, PPD controller--98 changes. This PPD controller is more stable than PD and has lower angle change amplitudes. As for comparison with the GP method, the high frequency of oscillations makes them less influencing but exhausts the tires and mechanical parts of the steering. We compared their target quality function values, and the results are shown in Table 5.  
Figure 10: Dynamics of the steering angle for different types of controllers—the controller input is \(\mu=0.3,0.95V_{cr}\).  
Figure 9: Car trajectories on the track tuned with prediction SAF of standard PD controller for friction coefficients. The purple curves correspond to the GP-RMEP controller, blue curves to the trajectories controlled by SAF with prediction, red curves to the original SAF. (**a**) \(\mu\) 0.5, 0.85 \(V_{cr}\); (**b**) \(\mu\) 0.5, 0.9 \(V_{cr}\); (**c**) \(\mu\) 0.5, 0.95 \(V_{cr}\); (**d**) \(\mu\) 0.3, 0.85 \(V_{cr}\); (**e**) \(\mu\) 0.3, 0.9 \(V_{cr}\); (**f**) \(\mu\) 0.3, 0.95 \(V_{cr}\); (**g**) \(\mu\) 0.1, 0.85 \(V_{cr}\); (**h**) \(\mu\) 0.1, 0.9 \(V_{cr}\); (**i**) \(\mu\) =0.1, 0.95 \(V_{cr}\).  
As the results shown in Table 5 demonstrate, the PPD controller outperforms the PD and is comparable with the GP-RMEP controller in terms of target quality function. Thus, the new controller has a trajectory close to the best trajectory obtained with the GP controller but does not have oscillations that could lead to mechanical damage [15]. Since, according to our earlier studies [6; 13], oscillations in the GP-RMEP method are part of its tactics aimed at increasing and maintaining the largest slip angle on tires and, accordingly, cannot be removed from the method without reducing the speed and safety of the car, the result we obtained, which is close in quality to GP-RMEP and free from oscillations, is an important result for us.  
Also during the analysis of the proposed model, we discovered several features, as described below.