## 5 Evaluation for Lunar NHRO Orbiters for NAV/COM Applications
Monte Carlo runs were carried out with an initial error on the reference orbit generated for each run with a Gaussian distribution, zero mean, and a standard deviation of 100 m for the position error and 0.1 mm/s for the velocity error. The error on the inter-satellite range measurements was considered Gaussian distributed with 1 m of standard deviation for each measurement session, and the error on the range rate is considered with a standard deviation of 0.06 mm/s. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Orbit type & 9:2 lunar synodic resonance \\ Orbit Period & 6.59 days \\ Perilune Radius & 3196–3557 km \\ Apolume Radius & 70,000 km \\ Orb.Period (non-dimensional units) & 1.5112 \\ \([x_{0}^{1},y_{0}^{1},z_{0}^{2},z_{0}^{1},z_{0}^{1},y_{0}^{1},z_{1}^{1}]\) & \([1.0221,0,-0.1821,-0,-0.1033,0]\) \\ \([x_{0}^{2},y_{0}^{2},z_{0}^{2},z_{0}^{2},y_{0}^{2},z_{0}^{2},z_{0}^{2}]\) & \([1.01282,-0.03468,-0.14507,-0.04860,-0.06772,0.2042]\) \\ \([x_{0}^{3},y_{0}^{2},z_{0}^{3},z_{0}^{3},y_{0}^{2},z_{0}^{3}]\) & \([1.01284,0.03442,-0.14582,0.04809,-0.06846,-0.20189]\) \\ \hline \end{tabular}
\end{table} 
Table 1: Reference Orbit and spacecraft initial condition for the spacecraft used in the simulation. The orbit period is referred to the normalized CRTBMP frame. For the Earth–Moon system, the time normalization constant is \(t^{*}=4.3644\) days. Figure 6: Elevation angle of the Lunar Gateway and the two anchors with respect to a user located at the Moon South Pole. To evaluate the performance of the Orbit Determination process, the Distance Root Mean Square (DRMS) of the estimated trajectories with respect to the nominal one was computed: the estimated, \(\mathbf{X}_{t}^{c}\), and the nominal, \(\mathbf{X}_{t}\), trajectories were propagated for three hours, and the norm of the displacement vector (distance) for each epoch is used to compute the DRMS for the given spacecraft:  
\[DRMS_{m}=\sqrt{\sum_{t_{span}}e_{i}^{2}} \tag{36}\] \[e_{l}=\|\mathbf{X}_{l}-\mathbf{X}_{t}^{c}\| \tag{37}\]  
where \(m\) indicates the \(m\)-th satellite, and \(l\) indicates the epoch of the propagation. Finally, the average of the DRMS error for all the spacecraft (Gateway and anchors) was considered:  
\[DRMS_{mean}=\sum_{j}DRMS_{j}/N_{spacecraft} \tag{38}\]  
For comparison, a classic orbit determination process exploiting an Earth-based measuring station performing the same measurements (range and range rate) was also simulated for the same time spans and measurement rate (measurements are taken only from the ground, without the use of any ISL). Examples of results for a time span of 9.89 days (1.5 T) and measurement time interval of 30 min are reported in Figures 7 and 8, where the histograms for the DRMS orbit error (in meters) for 500 Monte Carlo runs are reported for the two cases: with and without the range-rate measurements, respectively, and compared with the ground-based approach. It is possible to note that, with this configuration, the obtained performance is similar to the one obtained with the classical Earth-based OD method. Moreover, the results show that sub-meter positional DRMS error can be obtained by the use of range and range-rate measurements with a totally autonomous approach. In Table 3, the 90 th percentile for the DRMS errors computed for different simulations, varying the time span and measurement intervals, are reported. Apart from the first value of the time span, the proposed Autonomous OD gives similar or better results than the OD from the Earth and with a sub-meter DRMS error for a large part of the tested cases. Concerning the synchronization, the performances are calculated with the same approach: the clock was propagated with the nominal and estimated parameters, and the RMS clock bias was computed for different epochs in a time span of three hours.