# Evolving the Controller of Automated Steering of a Car in Slippery Road Conditions
## 2 Materials and Methods
### Evolving the Steering Controller via GP
#### 2.5.2 Fitness Evaluation  
The fitness value is intended to estimate the quality of the steering produced by the evolved SAF. We defined the criteria of such a quality from the desired characteristics of the trajectory of the car during the trial. It is implemented on a given test track (as shown in Figure 2) featuring a given friction coefficient, \(\mu\), as follows: first, the simulated car is initially positioned at the starting position of the track. Similarly to [3], in order to include in the evaluation of the evolved SAF its response to a step disturbance, the starting position is set to be parallel, 5 m off the center of the lane. Then, the car accelerates slowly to a given constant target speed. Then, in order to abstract the dynamics of the car from the effect of eventual excessive traction or braking forces, the car is maintained at a constant speed by simulated cruise control. To render the task of controlling the car challenging, but solvable, the target speed is set to 0.85 of the critical speed \(V_{CR}\). The critical speed is the speed at which the car theoretically could not negotiate the corners of the test track with the given coefficient of friction without losing control, running off the track, and eventually crashing. This speed is approximated as the speed at which the centrifugal forces during steady-state cornering \(F_{C}=mV_{CR}^{2}/R\) (where \(m\) is the mass of the car, and \(R\) is the radius of the turn) become theoretically equal to the friction force \(F_{F}=\mu\ m\ g\) (where \(\mu\) is the overall coefficient of friction and \(g\) is the gravitational acceleration). At the traveling speed of 0.85 of \(V_{CR}\), the car inherently suffers from intermittent instability due to the yaw inertia both in the entry and exit of corners and due to the dynamics of lateral weight transfer in corners [3; 14] that we intend to counter by the evolved SAF. The car, traveling at \(V_{CR}\) (and above), is theoretically uncontrollable. Consequently, there would be no existing SAF that results in a steerable car. Similarly, a car traveling considerably slower than \(V_{CR}\) does not suffer from any instability and its steering could be accomplished adequately by the canonical servo-control models. The speeds of the car during the trials on the track featuring different friction coefficients are shown in Table 3.  
The speed of the car is kept constant during the trial by a simple, handcrafted cruise control mechanism that maps the difference between the desired speed (e.g., as shown in Table 3, 13.3 m/s for the trial on a track with friction coefficient \(\mu\) = 0.5) and the actual one into an increment (or decrement) of the position of the accelerator pedal. As the car reaches the desired speed, the steering of the car is assumed by the evolved SAF. Then, the SAF starts to continuously (with a sampling frequency \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**\#Road** & **Friction of** & **Friction of Road** & **Overall Friction,** & **Critical Speed,** & **Speed of the Car** \\
**Condition** & **Tires, \(\mu_{t}\)** & **Surface, \(\mu_{s}\)** & \(\mu\) = \(\mu_{t}\times\mu_{s}\)** & \(V_{CR}\), \(\mu\)/s** & **(0.85 \(V_{CR}\)), \(\mu\)/s** \\ \hline \(1\) & 1.0 & 1 (dry) & 1 & 22.13 & 18.82 \\ \(2\) & 1.0 & 0.8 (dry) & 0.8 & 19.79 & 16.8 \\ \(3\) & 1.0 & 0.6 (rainy) & 0.6 & 17.15 & 14.5 \\ \(4\) & 1.0 & 0.5 (rainy) & 0.5 & 15.65 & 13.3 \\ \(5\) & 1.0 & 0.4 (snowy) & 0.4 & 14 & 11.9 \\ \(6\) & 1.0 & 0.3 (icy) & 0.3 & 12.12 & 10.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Speed of the car during the fitness trial on the test track with different friction coefficients. of 40 Hz) produce the desired steering angle \(\delta\) calculated for the currently perceived values of the parameters pertinent to the state of the car. The intended, optimal, trajectory of the car steered by the evolved SAF should feature both a quick and smooth return to the center of the lane followed by a precise drive along it (Figure 6, Case 3).