# Filtering Process to Optimize the Technical Data of Prototype Race Cars
## 4 Description of the Filtering Process
### Determination of the Uncertainties of the Simulated Values  
As a preliminary step, we evaluated the sensitivity of the simulation results to uncertainties in the input parameters. Specifically, we analyzed how the uncertainty of each input propagates to the uncertainty of the vehicle speed calculated by the simulation using the  
Figure 5: The covered distance–time functions obtained by simulation, applying the optimal (3.871) and two other gear ratios [11].  
Gaussian Error Propagation Law [47] (3). To illustrate this approach, we selected a scenario in which the vehicle accelerates from rest to 40 [km/h]. The estimated uncertainties of relevant input parameters--based on prior experimental data and the literature values--are listed in Table 1. Using these data, the propagated uncertainties in the simulated vehicle speed were computed and are presented in Figure 6.  
\[\Delta v_{\mathrm{s}}=\sqrt{\sum_{i=1}^{n}\left(\frac{\partial v_{\mathrm{s}}}{ \partial x_{i}}\right)^{2}\cdot\Delta x_{i}^{2}} \tag{3}\]  
In Equation (3), \(\Delta x_{i}\) and \(\Delta v_{\mathrm{s}}\) denote the uncertainties of the input data \((x_{i})\) and the simulated vehicle speed \((v_{\mathrm{s}})\), respectively. Partial derivative \(\frac{\partial v_{\mathrm{s}}}{\partial x_{i}}\) was approximated by the ratio \(\frac{\Delta v_{i}}{\Delta x_{i}}\), where \(\Delta v_{\mathrm{s}i}\) and \(\Delta x_{i}\) are the uncertainties that belong to the 5% relative uncertainty of variable \(x_{i}\) if the uncertainties of the other variables are zero.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \(I_{f}[\mathbf{m}]\) & \(I_{b}[\mathbf{m}]\) & \(w[\mathbf{m}]\) & \(m_{0}[\mathbf{kg}]\) & \(m_{f}[\mathbf{kg}]\) & \(m_{b}[\mathbf{kg}]\) \\ \(0.735\pm 0.007\) & \(0.615\pm 0.030\) & \(0.240\pm 0.012\) & \(247.7\pm 0.500\) & \(8.000\pm 0.010\) & \(16.90\pm 0.017\) \\ \hline \(J_{f}\big{[}\mathbf{kg}\cdot\mathbf{m}^{2}\big{]}\) & \(J_{b}\big{[}\mathbf{kg}\cdot\mathbf{m}^{2}\big{]}\) & \(J_{r}\big{[}\mathbf{kg}\cdot\mathbf{m}^{2}\big{]}\) & \(i_{\mathbf{i_{2}}}\) & \(\eta\) & \(R\) \\ \(0.182\pm 0.010\) & \(0.230\pm 0.011\) & \(0.020\pm 0.001\) & \(4.000\pm 0.004\) & \(0.970\pm 0.020\) & \(0.220\pm 0.011\) \\ \hline \(C\) & \(A\big{[}\mathbf{m}^{2}\big{]}\) & \(\mu_{roll_{f}}\) & \(\mu_{roll_{b}}\) & \(Hbearing\) & \(d[\mathbf{m}]\) \\ \(0.300\pm 0.060\) & \(1.000\pm 0.100\) & \(0.014\pm 0.001\) & \(0.018\pm 0.002\) & \(0.002\pm 0.0002\) & \(0.025\pm 0.00002\) \\ \hline \(U[\mathbf{V}]\) & \(U_{brush}[\mathbf{V}]\) & \(R_{r}[\mathbf{\Omega}]\) & \(R_{s}[\mathbf{\Omega}]\) & \(R_{view}[\mathbf{\Omega}]\) & \(M_{res}[\mathbf{Nm}]\) \\ \(48.00\pm 2.400\) & \(1.200\pm 0.120\) & \(0.01\pm 0.0001\) & \(0.016\pm 0.0002\) & \(0.006\pm 0.0001\) & \(0.600\pm 0.030\) \\ \hline \(\rho_{air}\big{[}\mathbf{kg}/\mathbf{m}^{3}\big{]}\) & \(L_{sr}[\mathbf{H}]\) & \(L_{s}[\mathbf{H}]\) & \(L_{r}[\mathbf{H}]\) & \(\mu_{s}\) **and \(\mu_{c}\)** & \\ \(1.200\pm 0.024\) & \(0.00120\pm 0.00012\) & \(0.000340\pm 0.000034\) & \(0.00010\pm 0.00001\) & \(0.74\pm 0.074\) & \\ \hline \end{tabular}
\end{table}
Table 1: Estimated uncertainties of the input data.  
Figure 6: Simulated velocities and their uncertainties.  
The definitions of the symbols used in Table 1 are provided in the Nomenclature section. The uncertainty values were estimated based on previous experimental measurements and relevant literature sources. The simulated vehicle speeds and their associated uncertainties, corresponding to the input data and uncertainty estimates from Table 1, are depicted in Figure 6.  
The results indicate that the relative uncertainty of the simulated speed ranges from 5% to 9% within the analyzed velocity interval (1-40 [km/h]), confirming the model's robustness while highlighting the variable sensitivity of the output to different inputs.