## 3 Mathematical Solution of the Problem
### Regression Model Substituting Numerical Solution of Transcedental Equation
This means that the maximum error allowed or root mean square error (RMSE), when predicting the value of the substitution \(t\) by using the tested regression model is as follows:  
\[RMSE_{t}=tg\bigg{(}\frac{\beta_{d}}{2}\bigg{)}=tg\bigg{(}\frac{0.10^{\circ}}{2} \bigg{)}=0.000873 \tag{29}\]  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & \\ \hline
**No \({}^{\star}\)** & \(\theta_{\mathrm{S}}\) [\({}^{\circ}\)] & \(R\) **[m]** & \(d\) **[m]** & \(R\)/\(d\) & \(K\) & \(e^{\mathrm{K}\theta s}\) & \(c\) & \(\beta_{max}\) [\({}^{\circ}\)] & \(t\) & \(\beta_{d}\) [\({}^{\circ}\)] \\ \hline \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) \\ \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) \\ \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) & \(\bullet\) \\
26,384 & 180.00 & 20.00 & 16.60 & 1.204819 & 0.672004 & 8.257840 & 0.484965 & 51.7435 & 0.357094 & 39.3026 \\
26,385 & 180.00 & 20.00 & 16.80 & 1.190476 & 0.645936 & 7.608518 & 0.492151 & 52.4083 & 0.359695 & 39.5668 \\
26,386 & 180.00 & 20.00 & 17.00 & 1.176471 & 0.619744 & 7.007523 & 0.499366 & 53.0720 & 0.362259 & 39.8268 \\
26,387 & 180.00 & 20.00 & 17.20 & 1.162791 & 0.593365 & 6.450203 & 0.506612 & 53.7347 & 0.364787 & 40.0826 \\
26,388 & 180.00 & 20.00 & 17.40 & 1.149425 & 0.566726 & 5.932363 & 0.513888 & 54.3961 & 0.367279 & 40.3345 \\
26,389 & 180.00 & 20.00 & 17.60 & 1.136364 & 0.539743 & 5.450199 & 0.521193 & 55.0564 & 0.369736 & 40.5823 \\
26,390 & 180.00 & 20.00 & 17.80 & 1.123596 & 0.512315 & 5.000239 & 0.528526 & 55.7153 & 0.372158 & 40.8263 \\
26,391 & 180.00 & 20.00 & 18.00 & 1.111111 & 0.484322 & 4.579285 & 0.535898 & 56.3727 & 0.374546 & 41.0665 \\
26,392 & 180.00 & 20.00 & 18.20 & 1.098901 & 0.455613 & 4.184352 & 0.543280 & 57.0287 & 0.376901 & 41.3030 \\
26,393 & 180.00 & 20.00 & 18.40 & 1.086957 & 0.425998 & 3.812606 & 0.550699 & 57.6830 & 0.379224 & 41.5358 \\
26,394 & 180.00 & 20.00 & 18.60 & 1.075269 & 0.395225 & 3.461274 & 0.558145 & 58.3357 & 0.381514 & 41.7651 \\
26,395 & 180.00 & 20.00 & 18.80 & 1.063830 & 0.362952 & 3.127537 & 0.565619 & 58.9866 & 0.383773 & 41.9909 \\
26,396 & 180.00 & 20.00 & 19.00 & 1.052632 & 0.328684 & 2.808334 & 0.573119 & 59.6357 & 0.386001 & 42.2132 \\
26,397 & 180.00 & 20.00 & 19.20 & 1.041667 & 0.291667 & 2.500108 & 0.580647 & 60.2829 & 0.388198 & 42.4322 \\
26,398 & 180.00 & 20.00 & 19.40 & 1.030928 & 0.250624 & 2.197581 & 0.588200 & 60.9281 & 0.390366 & 42.6480 \\
26,399 & 180.00 & 20.00 & 19.60 & 1.020408 & 0.20 But, for engineers and practitioners it is probably more important what the previously set benchmark for the maximum RMSE value represents in a geometrical sense. Bearing in mind that the maximum allowed single-bus length is around 14.5 m, a 0.1\({}^{\circ}\) error in the predicting steering angle \(\beta_{d}\), with a fourth-degree polynomial regression function, guarantees a 0.025 m lateral accuracy (tan (0.1\({}^{\circ}\))\(\cdot\)14.5 m = 0.025 m). Polynomial regression of the third degree produces a 0.05 m lateral accuracy, while second-degree polynomial regression results in errors exceeding 0.10 m. The single bus was selected as an appropriate design vehicle to calculate the lateral positioning error due to its long wheelbase. In Table 2, the results of testing various nonlinear regression models and their corresponding RMSE metrics are listed. Evaluation of the compared regression models was not performed by using the whole set of input data (26,400 combinations in total)--80% of the combinations from the input dataset were used to train the models, while the remaining 20% were used for testing. This specific ratio for the input dataset split used later for the model training and testing is recommended by advanced Python users [34].