## 4 Discussion  
Data from Table 8 demonstrate that improvements of a new model affected both target quality function addends--deviation from the trajectory and lateral acceleration. An increase of one of the parameters may lead to the decrease of another. The closer the vehicle is to the center of the lane, the stronger the forces affecting it during the turn. However, in this case, the new model demonstrated improvements in both parameters, so called non-zero sum, which indicates qualitative model enhancement in contrast to parameter tuning. Another issue that was not covered previously is the value of prediction time distance. In other words, it is the task of finding the optimal parameter for Equations (7)-(9). Previously, the value was chosen manually for each configuration and was in the range [0.8... 1.8] seconds. Too short of a prediction time does not allow achieving the effect described in Section 2.4 and leads to trajectory Type (1) in Figure 14. In contrast, too long of a period provides changes to the trajectory, which makes it too far from that desired, as shown on Trajectory (3), Figure 14.  
Figure 14: Different types of the car trajectory behavior depending on the prediction time. From (1) to (3), this duration becomes longer. Figure 13: Distance error (left picture) and yaw angle error (right picture) with 0.95 \(V_{cr}\) and 0.3\(\mu\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**\#Road**} & \multirow{2}{*}{**Overall Friction**} & \multicolumn{2}{c}{**0.85 of Critical Speed**} & \multicolumn{2}{c}{**0.9 of Critical Speed**} & \multicolumn{2}{c}{**0.95 of Critical Speed**} \\ \cline{3-8}  & & **PD** & **PD** & **PD** & **PD** & **PD** & **PD** \\ \hline
1 & 0.5 & 239 + 446 & 79 + 219 & 255 + 456 & 103 + 231 & 288 + 555 & 110 + 254 \\
2 & 0.3 & 779 + 914 & 186 + 196 & 823 + 978 & 215 + 201 & 870 + 984 & 252 + 206 \\
3 & 0.1 & 1241 + 418 & 346 + 62 & 1291 + 426 & 367 + 65 & 1305 + 477 & 405 + 66 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Steering controllers target quality function for each parameter combination split by addends corresponding to deviation from the center of the lane and lateral acceleration, respectively. Therefore, we ran a series of experiments in order to find the optimal value of this parameter. Figure 15 shows the results. We started the search from the small prediction times provided trajectory (1) and increased the time value until the result become equal to 1000 (which means the car crashed). The optimal values of prediction time could not be less than what we start from because they will turn into the PD controller results, and they could not be more than those that lead to the car crash already because of its remoteness from reality. We also note that an increase of target speed along with a decrease of friction coefficient increases the optimal prediction time. In other words, under less stable conditions, the prediction should be farther. Another thought that we mentioned before in the Introduction Section is a special form of the safe trajectory that has a linear curvature profile, which is comfortable for the turning car with high speed. In order to compare the obtained by PPD controller trajectory shape with clothoid, we combined them on a single plot. Clothoid could be constructed by solving the system of differential equations below, called the \"reconstruction equation\". A clothoid is uniquely defined by:  
* Coordinates and heading at which it starts: \((x_{0},\ y_{0},\ \theta_{0})\);
* Its length \(L\);
* Its linear curvature function, which is determined by the two coefficients \((k_{0},\ k_{1})\). With these five values \((x_{0},\ y_{0},\ \theta_{0},\ L,k_{0},\ k_{1})\), we can evaluate the clothoid's position and heading \((x(s),\ y(s),\ \theta(s))\) at any point \(s\) in area \([0;L]\).