# Evolving the Controller of Automated Steering of a Car in Slippery Road Conditions
## 3 Experimental Results
Ultimately, we should have considered an evolving SAF that performs (nearly) equally well on several fitness  
Figure 7: Fitness convergence characteristics for 20 independent runs of GP evolving the SAF of the GP-RMEP controller for friction coefficients \(\mu\) = 0.3 (**a**), \(\mu\) = 0.4 (**b**), \(\mu\) = 0.5 (**c**), \(\mu\) = 0.6 (**d**), \(\mu\) = 0.8 (**e**), and \(\mu\) = 1.0 (**f**), respectively. The bold curves correspond to the mean, while the envelope shows the minimum and maximum values in each generation. The best fitness of the PD and PID controllers (obtained via a complete enumeration of the values of their parameters) are shown as horizontal lines. cases that correspond to these different conditions. Moreover, in order to bridge the inevitable reality gap, we should have implemented an evolutionary adaptation of a set of the best SAFs, evolved on the simulated car, to a real one driven on a real track. However, in our current, seminal work, we will report the results of testing the best SAF evolved via GP with a single fitness case (fixed track, fixed coefficient of friction, and driven at fixed speed \(V=0.85\)\(V_{CR}\)) in different road conditions. As shown in Figure 8, all GP-RMEP controllers, evolved for a particular slippery condition (e.g., for \(\mu\) equal to 0.3, 0.4, 0.5, or 0.6, respectively) feature a reasonably small degradation (if any) when tested in different (unforeseen during the evolution) road conditions. For example, the GP-RMEP controller, evolved on snowy roads with \(\mu=0.5\), offers a comparatively good quality of steering (i.e., lower fitness values) compared to the alternative PD and PID controllers when tested on roads with friction \(\mu\) equal to 0.3 (icy), 0.4 (snowy), and 0.6 (rainy). The analytical expression of the SAF of this GP-RMEP controller is shown in Equation (8). \[\delta=(V-\epsilon^{\prime}+a+8e+14)+\frac{0.571(7-V)}{\delta\left(-\epsilon^ {\prime}+\frac{42}{\epsilon^{\prime}}\right)}[\theta^{\prime}\left(-\frac{e}{ 108}+8\right)(\epsilon^{\prime}+\frac{\theta\left(\frac{30}{\sigma}-e\right)} {7-V}+5-\theta-\frac{30}{\theta a^{\prime}})+\frac{3\alpha(1-0.2\epsilon)}{7 \epsilon V\left(3a+1.125+\frac{7}{8\sigma}\right)}] \tag{8}\]  
It is not uncommon that the solutions obtained via GP are way too complex to be easily comprehended by a human [5]. The presented best-evolved SAF is not an exception to this trend, and we cannot explain precisely either why or how the SAF, shown in Equation (8), works. We could only confirm that the SAF implements a proportional-derivative (PD) control of the steering in that both (i) the direct values of parameters and (ii) their derivatives are incorporated in its code. The only integral term included in the terminal set of the GP--the integral of the lateral deviation--is not incorporated by the GP in the best-evolved SAF, which is consonant with the findings that the integral term could be obsolete in some nonlinear PID controllers [27]. The dynamics of the steering angle and the deviation from the center of the lane of the car steered by the sample best-evolved SAF shown in Equation (8) in snowy (\(\mu=0.5\)) road conditions are illustrated in Figures 9 and 10, respectively. The same figures also show the behavior of the car steered by the best PD and PID controllers with values of parameters optimized for this particular road condition (as shown in Table 5). As these two figures illustrate, the lateral deviation of the  
Figure 8: Generality of the best-evolved steering functions manifested by a reasonable change in their respective fitness value when tested on different (unforeseen during the evolution) friction coefficients. The fitness of the steering functions evolved for a particular friction coefficient is indicated by larger circular markers. car steered by the sample best-evolved steering function is significantly lower than that of the best solution of the servo-control model, especially during steady-state cornering in the left Turn 1 and right Turn 2.