# Evolving the Controller of Automated Steering of a Car in Slippery Road Conditions
## 2 Materials and Methods
### Servo Control as a PD Controller  
The first of the three steering controllers that we investigated is based on the servo-control model of steering. It represents the canonical approach of defining the instant value of the steering angle in autonomous cars [23; 24]. According to this model, the SAF that defines the steering angle \(\delta\) is assumed to be the sum of the following two scaled parameters: the linear \(e\) and angular \(\theta\) deviations of the car from its desired trajectory (e.g., the center of the lane) as illustrated in Figure 3. The SAF of the servo control model could be expressed as the following Equation (1):  
\[\delta=k_{1}\,e+k_{2}\,\theta \tag{1}\]  
where the scaling coefficients (gains) \(k_{1}\) and \(k_{2}\) are calculated as follows:  
\[k_{1}=\text{Steering\_Factor}\text{Steering lock angle} \tag{2}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Feature** & **Value** \\ \hline Total length, m & 300 \\ Lane width, m & 20 \\ Length of sector 1, m & 90 \\ Radius of turn 1, R\({}_{1}\) m & 50 \\ Length of sector 2, m & 210 \\ Radius of turn 2, R\({}_{2}\), m & 50 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Main Features of the Test Track.  
Figure 2: Hook-type test track.  
\[k_{2}=\mathcal{I}\text{Steering lock angle}. \tag{3}\]  
The steering lock angle is the maximum angle of turning of the front wheels of the car, and its value, for the car used in our experiments, is 0.62 rad (35\({}^{\circ}\)). Alternation of the coefficients \(k_{1}\) and \(k_{2}\) affects the tradeoff between the main requirements of steering [25]: smoothness, fast response, and stability in the way the car returns to the center of the lane if it occasionally deviates from it. The servo control model is intended to steer the car in such a way that results in minimization of the mean value of the absolute deviation (the error) \(e\) of the geometrical center of the car from the center of the lane during a given trial.  
Assuming that, for small angles \(\theta\) and very short periods of time \(dt\),  
\[\theta\approx de\dot{\theta}x=de(V\ dt) \tag{4}\]  
where \(V\) is the speed of the car. For a constant speed \(V\), Equation (4) could be rewritten as  
\[\theta\approx k_{v}\ de\dot{\theta}t=k_{v}\ e^{\prime} \tag{5}\]  
where \(e^{\prime}\) is the first derivative of the lateral deviation of the car from the center of the lane. Substituting the value of \(\theta\) from Equation (5) in Equation (1), we obtain a different form of the servo-control model:  
\[\delta=k_{1}\ e+k_{2}\ (k_{v}\ e^{\prime})=k_{1}\ e+k^{*}_{2}\ e^{\prime}. \tag{6}\]  
Equation (6) expresses the servo-control model of steering as a PD controller. Its output--the _control variable_--the steering angle \(\delta\), is a sum of the proportional (P) and derivative (D) terms of the _error_: the deviation of the car from the center of the lane \(e\). Because the _desired process value_ (i.e., the desired deviation from the center of the lane) is 0, the absolute value of the error \(e\) is equal to the _measured process value_. The controller attempts to minimize the value of the _error_ by constantly adjusting the steering angle \(\delta\), which, presumably, would yield a trajectory as close as possible to the center of the lane (Figure 4). We will explain the mechanism of tuning the parameters \(k_{1}\) and \(k^{*}_{2}\) in Section 3.  
Figure 3: Most relevant variables pertinent to the state of the car.