# Lateral Oscillation Characteristics of Vehicle Trajectories on the Straight Sections of Freeways
## 5 Discussion
For extra-large trucks, there is a massive correlation between the eigenvalues of the lateral amplitude and the polynomial function, and the regression efficiency of the third-order polynomial is optimal. The goodness-of-fit for extra-large trucks is 0.927 and 0.900, and the model expressions are shown in Equations (12)-(13). \[LA_{PC,Mean}=0.435+1.003\exp(-V_{PC}/41.706) \tag{8}\]  
\[LA_{PC,85 th}=15.534-0.545V_{PC}+(7.480e-3)V_{PC}^{2}-(4.494e-5)V_{PC}^{3}+(9.963e -8)V_{PC}^{4} \tag{9}\]  
\[LA_{MLC,Mean}=0.515+0.032\sin(\pi\frac{V_{MLC}+3.085}{22.130}) \tag{10}\]  
\[LA_{MLC,85 th}=0.758+0.038\sin(\pi\frac{V_{MLC}-16.483}{11.752}) \tag{11}\]  
\[LA_{ELT,Mean}=40.880-1.433V_{ELT}+0.017V_{ELT}^{2}-(6.663e-5)V_{ELT}^{3} \tag{12}\]  
\[LA_{ELT,85 th}=55.608-1.950V_{ELT}+0.023V_{ELT}^{2}-(9.067e-5)V_{ELT}^{3} \tag{13}\]  
where \(LA_{PC,Mean}\) and \(LA_{PC,85 th}\) are the mean and 85 th percentile values of the lateral amplitude of passenger cars, \(V_{PC}\) is the speed of passenger cars. Similarly, \(MLC\) stands for medium-large cars and \(ELT\) stands for extra-large trucks. The relationship between the bounds of \(LA\) and speed for all vehicles is shown in Figure 12. Meanwhile, it reflects the regression model's parameters and the fitting efficiency. After establishing the prediction model of lateral amplitude, the validity of the models needs to be verified. Mean relative error (\(MRE\)), and root mean square error (\(RMSE\)) were picked as model assessment indexes. The calculation formulas are as follows:  
\[MRE=\frac{100}{n}\sum_{k=1}^{n}\left|\frac{y_{k}-x_{k}}{y_{k}}\right| \tag{14}\]  
\[RMSE=\sqrt{\frac{\sum\limits_{k=1}^{n}{(y_{k}-x_{k})}^{2}}{n}} \tag{15}\]  
where \(n\) represents the number of sample points, \(y_{k}\) stands for the predicted value of the \(k\)th sample point, and \(x_{k}\) is the actual value. The specific calculation results are shown in Table 6. From Table 6, the prediction performance of all models was accurate, with the \(MRE\) staying within 3% and the \(RMSE\) controlled within 0.03 m. Naturally, the models have certain conditions of applicability. For passenger cars, the models apply to calculate the lateral amplitude at vehicle speeds between 80 and 150 km/h; the applicable speed range for medium-large cars is 80 to 125 km/h; the suitable content for extra-large trucks is 80-95 km/h. Following the prediction of the lateral amplitude, the vehicle width and the left and right clearance can be added to calculate the theoretical value of the lane width. The specific calculation expression is shown in Equation (16). \[LA_{PC,85 th}=15.534-0.545V_{PC}+(7.480e-3)V_{PC}^{2}-(4.494e-5)V_{PC}^{3}+(9.963e -8)V_{PC}^{4} \tag{9}\]0  
where \(LW\) is the lane width, \(LC\) is the left clearance, and \(RC\) is the right clearance. Experience dictates that each clearance is taken as 0.3 m. \(VW\) is the vehicle width, which is 1.9 m for passenger cars, 2.4 m for medium-large cars, and 2.5 m for extra-large trucks. Then, the lane width's theoretical minimum and optimal value can be derived from the mean and 85 th percentile values of lateral amplitude, as shown in \[LA_{PC,85 th}=15.534-0.545V_{PC}+(7.480e-3)V_{PC}^{2}-(4.494e-5)V_{PC}^{3}+(9.963e -8)V_{PC}^{4} \tag{9}\]1 \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Vehicle Types**} & \multicolumn{2}{c}{**MRE (\%)**} & \multicolumn{2}{c}{**RMSE (m)**} & \multicolumn{2}{c}{**Adj. R\({}^{2}\)**} \\ \cline{2-7}  & **Mean** & **85 th** & **Mean** & **85 th** & **Mean** & **85 th** \\ \hline Passenger Cars & 2.614 & 2.069 & 0.015 & 0.022 & 0.827 & 0.755 \\ Medium-large Cars & 2.070 & 1.594 & 0.013 & 0.015 & 0.609 & 0.582 \\ Extra-large Trucks & 0.680 & 0.587 & 0.004 & 0.005 & 0.927 & 0.900 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Verification of the lateral amplitude prediction model. Figure 12: The relationship between lateral amplitude and speed. (**a**) All types of vehicles; (**b**) Passenger Cars; (**c**) Medium-large cars; (**d**) Extra-large Trucks.