## 3 Mathematical Solution of the Problem
### Regression Model Substituting Numerical Solution of Transcedental Equation
As can be seen in Table 2, only polynomial fourth-degree regression fulfills the set benchmark for the maximum RMSE value. Even when the whole set of the input data was considered, the obtained RMSE value for the polynomial fourth-degree regression is 0.000508, which is still the best in comparison to the other tested regression models. In addition to the desired value of RMSE, the coefficient of determination of the R-squared value (R\({}^{2}\)) for the selected polynomial regression model is R\({}^{2}\) = 0.99988. Despite the R\({}^{2}\) not considering overfitting, the achieved R\({}^{2}\) value is very close to 1. Before the fourth-degree polynomial regression model was adopted as an optimal solution, second- and third-degree polynomial regression models were also tested, but their RMSE values, as well as their lateral positioning errors, were considerably higher than the previously established benchmark. In addition, as displayed in the Table 3, the obtained R\({}^{2}\) values for the second and third polynomial regression models were also very close to 1, but their RMSE metrics could not meet the desired level of accuracy. The relationship between the dependent variable, i.e., the substitution \(t\), and the independent variables representing the turn angles \(\theta_{S}\) and \(R/d\) ratio in the modeled polynomial regression function looks as follows: \begin{table}
\begin{tabular}{c c} \hline \hline
**Nonlinear Regression Model Type** & **RMSE** \\ \hline Polynomial Regression (4 th degree) & 0.000496 \\ Ridge Regression & 0.002261 \\ Lasso Regression & 0.073637 \\ Elastic Net Regression & 0.073637 \\ Decision Tree Regression & 0.002843 \\ Random Forest Regression & 0.000995 \\ Support Vector Regression—SVR & 0.051933 \\ Gradient Boosting Regression & 0.001854 \\ K Neighbors Regression & 0.001480 \\ AdaBoost Regression & 0.009943 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Tested nonlinear regression model for the prediction of substitution \(t\) values. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Degree of Polynomial Regression Model** & **Lateral Positioning Error [m]** & **RMSE** & **R\({}^{2}\)** \\ \hline
2 nd-degree polynomial regression & \(>\)0.100 & 0.002453 & 0.99904 \\
3 rd-degree polynomial regression & 0.050 & 0.001754 & 0.99944 \\
4 th-degree polynomial regression & 0.025 & 0.000496 & 0.99988 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Lateral positioning error, RMSE, and R\({}^{2}\) values for polynomial regression of various orders. \[t= -0.044417+0.044078\cdot\left(\frac{R}{d}\right)+0.004485\cdot\theta_{ \mathrm{g}}+(-0.010769)\cdot\left(\frac{R}{d}\right)^{2}+(-0.001131)\cdot\left( \frac{R}{d}\right)\cdot\theta_{\mathrm{s}}+ \tag{30}\] \[\left(-6.53\cdot 10^{-6}\right)\cdot\theta_{\mathrm{s}}^{2}+\left(-6.78 \cdot 10^{-5}\right)\cdot\left(\frac{R}{d}\right)^{3}+0.000310\cdot\left( \frac{R}{d}\right)^{2}\cdot\theta_{\mathrm{s}}+\left(-6.35\cdot 10^{-6}\right) \cdot\left(\frac{R}{d}\right)\cdot\theta_{\mathrm{s}}^{2}+\] \[2.61\cdot 10^{-8}\cdot\theta_{\mathrm{s}}^{3}+0.000215\cdot\left( \frac{R}{d}\right)^{4}+\left(-3.68\cdot 10^{-5}\right)\cdot\left(\frac{R}{d} \right)^{3}\cdot\theta_{\mathrm{s}}+\left(7.34\cdot 10^{-7}\right)\cdot\left( \frac{R}{d}\right)^{2}\cdot\theta_{\mathrm{s}}^{2}+\] \[9.36\cdot 10^{-9}\cdot\left(\frac{R}{d}\right)\cdot\theta_{\mathrm{s }}^{3}+\left(-5.80\cdot 10^{-11}\right)\cdot\theta_{\mathrm{s}}^{4}\]  
The formula is rather a complex one, but it could be processed easily on a handheld computer, even the one installed in an autonomous vehicle. Though perhaps impractical for common use and not concurrent with vehicle movement simulation software, the mathematical solution developed herein could easily be used to guide the vehicle safely through the sharp turn. With the substitution \(t\) finally retrieved, the critical angle \(\beta_{d}\) and critical distance \(F_{d}\) could be calculated easily (\(\beta_{d}=2\cdot arctg(t)\) and \(F_{d}=(d-R\cdot sin\beta_{d})/cos\beta_{d}\)).