# Exploiting Lunar Navigation Constellation for GNC Enhancement in Landing Missions
## 4 Results
### Monte Carlo Analysis
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline  & Position \(\varepsilon_{r}\) (x, y, z): \(15\,\mathrm{m}\) \\ LCNS SISE (\(1\sigma\)) & Velocity \(\varepsilon_{v}\) (x, y, z): \(0.15\,\mathrm{m}/\mathrm{s}\) \\  & Clock bias \(\varepsilon_{\theta}\): \(10\,\mathrm{m}\) \\  & Clock drift \(\varepsilon_{\theta}\): \(0.1\,\mathrm{m}/\mathrm{s}\) \\ User clock Allan variance & \(h_{0}\): \(2\times 10^{-25}\) \\  & \(h_{-2}\): \(6\times 10^{-25}\) \\ \hline Altimeter noise (\(1\sigma\)) & \(1\%\) of \(\zeta\) \\ IMU noise (\(1\sigma\)) & \(1\,\mathrm{\SIUnitSymbolMicro g}\) \\ \hline  & Position \(\sigma_{r}\) (x, y, z): \(1\,\mathrm{km}\) \\ Initial filter uncertainty (\(1\sigma\)) & Velocity \(\sigma_{v}\) (x, y, z): \(10\,\mathrm{m}/\mathrm{s}\) \\  & Clock bias \(\sigma_{b}\): \(100\,\mathrm{m}\) \\  & Clock drift \(\sigma_{d_{c}}\): \(1\,\mathrm{m}/\mathrm{s}\) \\ Filter rate & \(10\,\mathrm{Hz}\) \\ \hline Landing start time \(t_{S}\) & Uniform distribution in (\(3000\,\mathrm{s}\), \(5000\,\mathrm{s}\)) \\ Initial state perturbation (\(1\sigma\)) & Position (x, y, z): \(100\,\mathrm{m}\) \\  & Velocity (x, y, z): \(0.1\,\mathrm{m}/\mathrm{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of the navigation simulation settings. As we can see by looking first at the line colour distribution, in both cases there is not a specific correlation between the landing start time, \(t_{S}\), and the navigation and control errors, \(\delta r_{\text{NAV}}\) and \(\delta r_{\text{CTRL}}\). Overall, the control error in particular is quite uniform once convergence is reached. \begin{table}
\begin{tabular}{c c c c c c} \hline
**CTRL** & **Success Rate** & \(\mu_{\delta r_{\text{NAV}}}\) **(m)** & \(\mathbf{3\sigma_{\delta r_{\text{NAV}}}}\) **(m)** & \(\mu_{\delta r_{\text{CTRL}}}\) **(m)** & \(\mathbf{3\sigma_{\delta r_{\text{CTRL}}}}\) **(m)** \\ \hline A & 51.2\% & 1.8 & 2.8 & 56.5 & 20.1 \\ B & 86.0\% & 1.7 & 2.8 & 388.9 & 22.8 \\ \hline \end{tabular}
\end{table} 
Table 3: Collection of performance for the two proposed controllers in the Monte Carlo analysis. Statistical distribution of navigation and control errors are reported for successful samples, alongside the success rate. Figure 11: Case A. Navigation (**top**) and control (**bottom**) errors of the successful landing trajectories, with average and upper \(3\sigma\) bound in logarithmic scale. Figure 12: Case B. Navigation (**top**) and control (**bottom**) errors of the successful landing trajectories, with average and upper \(3\sigma\) bound in logarithmic scale. \(\delta r_{\text{NAV}}\) is, as expected, comparable for the two cases, providing final errors below 5 m, also in the worst cases. The control error instead varies in accordance with the nominal performance. As one can see in Table 3, the average values are very close to the nominal values for both case A and case B. Both cases also show a similar value for the standard deviation, around \(\sim\)20 m.  
Regarding case A (Figure 11), a quite unsatisfactory success rate of 51.2% is reached, while a much larger value of 86.0% is obtained with the milder controller of case B. This reflects how the larger gains of the more aggressive tuning are less reliable and robust when a dispersion from the nominal scenario is introduced. An interesting visualisation of the successful trajectories is obtained by looking at the final points observed on the horizontal plane of the Moon-fixed IAU_MOON frame at 1 km altitude. This is what the scatter plots in Figure 13 represent, together with two sets of covariance ellipses. The upper plot reports the distributions of the two cases together, with the aim point of the South Pole visible. The two lower plots instead present the distribution of each case individually. In all representations, the grey ellipses represent the 3\(\sigma\) covariance of the estimated trajectory in the nominal scenario, while the remaining three ellipses represent the three sigma levels of the successful Monte Carlo samples.