# Evolving the Controller of Automated Steering of a Car in Slippery Road Conditions
## 3 Experimental Results  
For each of the six road conditions (as described in Table 3) we employed GP and performed 20 independent evolutionary runs to obtain the best SAF. The fitness convergence characteristics of these independent runs are shown in Figure 7.  
As Figure 7 illustrates, in all road conditions the fitness of the best-evolved SAF converges to values that are better (i.e., lower) than those of the best PD and PID controllers. We obtained the optimal values of parameters of the best PD and PID controllers by their complete enumeration (i.e., a \"brute-force\" search). For the PD controller, we evaluated 25 discrete values of each of the two parameters \(k_{1}\) and \(k^{\prime}\)2 (resulting in a size of the search space equal to \(25^{2}=625\)). For the PID, \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Evolved individuals & SAF \(\delta\) \\ \hline Genetic representation & Parse tree \\ \hline Set of non-terminals (functions) & [+,-,*,/] \\ \hline \multirow{3}{*}{Set of terminals} & Variables pertinent to the state of the car, and their derivatives: lateral deviation (\(e\), \(e^{\prime}\), \(\int e\)), speed (\(V\)), steering angle (\(\delta\)), lateral acceleration (\(u\), \(a^{\prime}\)) angular deviation (\(\theta\), \(\theta^{\prime}\)), \(\quad\) and a random constant (C) \\ \hline Population size & 200 individuals \\ \hline Selection & Binary tournament, ratio 0.1 \\ \hline Elitism & Best 4 individuals \\ \hline Crossover & Single point, random, ratio 0.9 \\ \hline Mutation & Single point, random, ratio 0.05 \\ \hline Fitness value & Sum of (i) the area under the trajectory of the car around the center of the lane and (ii) the average of its lateral velocity. \\ \hline Termination criteria & (\#Generations >200) _or_ (no improvement of fitness during 16 consecutive generations) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of genetic programming (GP). we borrowed the optimal values of the two parameters of the PD controller, and used only 10 discrete values in the vicinity of these values and 25 values for the third parameter \(k_{3}\) (i.e., the size of the search space is 2500). The best fitness of the three proposed controllers--PD, PID, and GP-RMEP--and the optimal values of the coefficients of the PD and PID controllers are shown in Table 5.  
As the results shown in Table 5 demonstrate, the PID controller outperforms the PD controller in that its fitness is lower in all road conditions. However, the quality of steering of the best-evolved GP-RMEP controller is even better than that of PID, and the difference between them widens with the decrease of the friction coefficient, reaching a maximum of about 4 times (1532 versus 374) on icy roads (\(\mu=0.3\)). Conversely, on grippy, dry roads (\(\mu=0.8\) and \(\mu=1.0\)), this difference is not very significant (587 versus 545 and 613 versus 498, respectively), implying that the PID model provides good enough steering of the car in these road conditions. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & & & & \multicolumn{2}{c}{**Controller**} \\ \cline{4-6}
**\#Road** & **Overall** & & & & & **GP-RMEP, an** \\
**Condition** & **Friction**\(\mu\) & & & & & **Arbitrary SAF** \(\delta\) (\(e\), \(e^{\prime}\), \(\int e\), \(\int e\), \(\int e\), \(  
The lack of generality is one of the well-documented drawbacks of solutions obtained via GP [4; 5; 26] that hinders the applicability of this algorithm to real-world problems. Indeed, we could not be sure about how well the SAF that was evolved in a single car driven at a fixed speed on a fixed track featuring a fixed coefficient of friction would perform in different situation(s).