# Exploiting Lunar Navigation Constellation for GNC Enhancement in Landing Missions
## 4 Results
### Nominal Scenario Performance  
The current analysis entails and compares two different controllers in the presented scenario. In particular, these two controllers have been obtained through two separated tuning procedures: a minimisation of the final trajectory control error (case A) and a minimisation of the propellant expenditure (case B), whilst maintaining a final error below the threshold of \(500\,\mathrm{m}\). Such a threshold is considered enough for the main braking phase, which is targeted by this study, before handling the final descend where a precision below \(90\,\mathrm{m}\) is targeted. The results from a GNC point of view are displayed for the two cases in Figures 7 and 8, where both the estimation and control errors are plotted over the simulated time on a logarithmic scale. The total number of LNS satellites in view by the lander is plotted in Figure 9. Note that the control error plot starts slightly before \(t=1\,\mathrm{h}\), i.e., when the propulsive leg begins. Figure 4: Representation of the constellation orbits in the LME_2000 inertial frame. Figure 5: Representation of the trajectory in the IAU_MOON reference frame. The trajectory followed by the spacecraft is plotted in blue, overlapped on the guidance for the thrusted arc in red. Figure 6: Zoomed view of the natural and thrusted arcs, on the plane of the trajectory. Figure 7: Navigation and control error of the landing trajectory for case A, plotted at logarithmic scale. The navigation errors depend on the number of satellites visible (see Figure 9), for both cases, evolving practically identically for the non-propelled phase. Indeed, for the first part of the trajectory the error remains quite high, until a new servicer enters the visibility of the lander, just before \(t=0.4\) h. At that instant, the navigation error drops to the order of \(\sim\)100 m and then reduces even more as soon as the number of satellites in view increases up to a peak of five servicers. The reached estimation accuracy stabilises in the order of a few metres in this part. When the control action starts, the initial oscillatory behaviour of the thrust acceleration is reflected on both the navigation and control errors. The former recovers the accuracy of a few metres after stabilisation. It is useful to remark that the filter is in a stable and converged state just before the start of the control phase, which is reflected by the low values in the covariance matrix, \(\mathbf{P}\). This effect gives a high stiffness to the filter, which prevents its convergence to the new correct status at the beginning of the thrusting phase. In order to overcome this issue, the covariance matrix is re-initialised to 10% of its initial condition, \(\mathbf{P}_{0}\), at the beginning of the thrusted arc. In such a way, the filter is made more flexible and driven to convergence to the new state. The control error instead behaves differently in the two cases, as seen in both Figures 7, 8, and 10, where such errors are plotted in a Local Vertical Local Horizontal (LVLH) frame relative to the set-point trajectory alongside the propellant mass expenditure. Case A, tuned for best control error performance, after the initial oscillating behaviour, is able to reach a final error of 55.6 m, which is kept stable during the whole phase, at the expense of 13.3 kg of propellant mass. On the other hand, case B reaches a coarser value of 380.4 m, using only less than half of the propellant mass, with a value of 6.6 kg. The performance of the two controllers is summarised in Table 1, where the values of the PID coefficients are also displayed. Figure 8: Navigation and control error of the landing trajectory for case B, plotted at logarithmic scale. Figure 9: Number of LNS satellites in visibility. It is possible to see, in particular by the vertical component of the error in Figure 10, how case A presents higher frequency oscillations, reflecting the more aggressive behaviour of this tuning condition.