## 2 Materials and Methods
### Servo Control as a PD Controller  
In our earlier studies, we showed that the canonical well-known servo control (Figure 4) is a variation of the PD controller. To avoid repetition, we give here only the main statements; the details can be found in our paper [6].  
\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 3: Rotation curve. The transition between the straight part (blue) and circular part (green) is a spiral curve (red). Picture is taken from [[https://cifrasyteclas.com/clotoide-la-curva-que-vela-por-tu-seguridad-en-carreteras-y-ferrocarriles/](https://cifrasyteclas.com/clotoide-la-curva-que-vela-por-tu-seguridad-en-carreteras-y-ferrocarriles/)]([https://cifrasyteclas.com/clotoide-la-curva-que-vela-por-tu-seguridad-en-carreteras-y-ferrocarriles/](https://cifrasyteclas.com/clotoide-la-curva-que-vela-por-tu-seguridad-en-carreteras-y-ferrocarriles/)).  
The steering angle defined by this model is a linear combination of scalable deviation from the desired trajectory parameters--the distance \(e\) and angle \(\theta\) shown in Figure 5. The SAF (steering angle function) of the servo control model could be expressed as the following Equation (1):  
\[\delta=k_{1}\,e+k_{2}\,\,\theta \tag{1}\]  
where \(\delta\) is the steering angle, and the optimal values of the scaling coefficients (gains) \(k_{1}\) and \(k_{2}\) have an impact to the main requirements to the steering [11]--smoothness, fast response, and stability in the way the car returns to the center of the lane after it deviates from it. These parameters could be chosen from the steering lock angle restrictions in \(30^{0}\) in different ways depending on the specific conditions and features of the road [4].  
For small angles \(\theta\) and very short periods of time \(dt\):  
\[\theta\approx d\theta dx=d\sigma(V\,dt) \tag{2}\]  
Figure 4: Servo-control model of steering as a PD steering controller. The steering angle function (SAF), defining the steering angle \(\delta\) is implemented as a sum of the proportional-(P) and derivative (D) terms of the _error_—the deviation \(e\) from the center of the lane.  
Figure 5: Most relevant variables pertinent to the state of the car. Variables \(\theta\) and \(e\) are directly involved in the calculation of the steering angle \(\delta\) according to the servo-control model.  
For constant speed \(V\), Equation (2) could be rewritten as:  
\[\theta\approx k_{v}\;de\dot{dt}=k_{v}\;e^{\prime} \tag{3}\]  
Applying simple mathematical transformations, we can represent Equation (1) in the following shape expressed the servo-control model of steering as a PD controller:  
\[\delta=k_{1}\;e+k_{2}\;(k_{v}\;e^{\prime})=k_{1}\;e+k^{*}{}_{2}\;e^{\prime} \tag{4}\]  
where \(e^{\prime}\) is the first derivative of the lateral deviation of the car from the center of the lane. From the PD controller point of view, servo control model represents by itself a closed-loop system. The input--measured _process value_ is equal to the absolute value of the error \(e\)--the deviation of the car from the center of the lane. Its output--the _control variable_--the steering angle \(\delta\), is a sum of the proportional (P) and derivative (D) terms of the _error_. 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.