# Exploiting Lunar Navigation Constellation for GNC Enhancement in Landing Missions
## 4 Results
### Monte Carlo Analysis
The aspect ratio and the orientation of nominal scenario covariance ellipses (the grey ones) are comparable with those of the samples, thus providing a sensible expectation of the landing mission dispersion ratios among the various components. The additional increment of the area is associated with the introduced control error. The 3\(\sigma\) uncertainty ellipses of these sample is characterised by a semi-major and semi-minor axes of 11.7 m and 4.2 m for case A, and 14.0 m and 1.9 m for case B. In both cases, the semi-major axis is mostly aligned with the \(X_{\text{IAU\_MOON}}\) axis. This elongation is consistent with the nominal scenario errors presented in the previous section, where most of the errors and dispersion were in the along-track component. Figure 13: Covariance ellipses of the successful Monte Carlo samples, taken on the 1 km altitude horizontal plane. Both case A (red) and case B (blue dots) are displayed: together in the upper plot, individually in the two lower plots. Such effects, reflected in the navigation error as well, are also caused by the geometry involving the user and some of the servicers that are visible in this specific time-frame. Their LoS with respect to the South Pole (where the lander is practically placed) during the few minutes of thrusted arc have a relatively large variation along the \(Y_{\text{IAU\_MOON}}\) axis and a practically null one along the \(X_{\text{IAU\_MOON}}\) axis, causing the \(y\) component to be much more observable, thus with a reduced error. Analysing instead the different failed samples, it is possible to say that the vast majority of them failed by crossing the 1 km altitude prematurely. Even though such an outcome is not desirable, it is possible to say that the failure is due to very high state discrepancies between the desired and actual conditions at the beginning of the thrusted phase. Indeed, by looking at Figure 14, it is possible to see the distribution of such an error at \(t_{S}\). In particular, the two plots, the upper for case A and the lower for case B, present histograms with differentiation between successful and failed samples with respect to the control error at the start of the landing phase. Such errors are the consequence of the imposed initial state dispersion to each Monte Carlo sample, which, as previously highlighted, build up to high values quite easily. The outcome of this visualisation is able to assess that, for the more aggressive controller, the basin of convergence is limited to an initial error between 50 km, while for the milder controller of case B, such a boundary is much larger, around 130 km. Both boundary values are very large, hinting that the proposed approach is in any case feasible to provide reliable navigation to landing missions, even though further optimisations in the control algorithms are still possible. The histogram presented in Figure 15 reports instead the propellant mass consumption of the successful samples for case A and case B, stressing again how the tuning parameters of case B provide less expensive manoeuvres, with an upper boundary of approximately \(\sim\)30 kg, in contrast to the \(\sim\)50 kg of case A. With such Monte Carlo analysis, we gained clear insights on the statistical behaviour of the landing GNC strategy for both cases, which provide reasonable alternatives in the trade-off between landing precision and robustness. In order to widen the robustness analysis of such a GNC strategy, the Monte Carlo samples could be generated to include other additional dispersed parameters. In particular, the presented analysis, with the goal of being representative of a safe landing operation, considers only scenarios with very good LNS visibility, with a total number of servicers reaching five before starting the propelled phase.