# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 6 Evaluation on a Possible Lunar Constellation  
The baseline for the performance evaluation is the satellite navigation system proposed in [13; 14], exploiting pseudorange measurements from satellites in ELFO orbits, with a constellation of four satellites. A detailed description of the constellation is out of the scope of this work and has already been performed by the authors in [13; 14; 15; 50; 51], who showed that the constellation parameters are as follows: # of satellites and planes = 4, Semi-major axis (a) and Eccentricity \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Max number of user & 10 \\ Min/Max range & 600 m/20 km \\ Minimum SNR & 10 db \\ Antenna nor/vert dimension & 0.5 m \\ Signal Band & Ka \\ Signal Bandwidth & 2 MHz \\ Modulation Type & BPSK \\ BER & \(10^{-3}\) \\ Probability of lack of service & 0.99 \\ Message length (number of bits) & 1 ms (ca. 1000 bits) \\ N & 1000 \\ \(\sigma_{LE-pos}\) & 0.1 m for each coordinate \\ \(\sigma_{R}\) & 0.83 (+0.83) meters \\ \(\sigma_{AOA}\) & 0.01 (+0.01) degree \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Local sensor parameters and errors assumption in the ISL configuration. (e) equal to 9750 km and 0.6-0.7 respectively, Inclination (i) and Orbital Period (T) equal to 48-57 deg and 24 h, respectively. The user scenario consists of a spacecraft following a linear descending trajectory, starting at \((-89.34^{\circ},90^{\circ},20\,\) km) and ending at \((-89.98^{\circ},90^{\circ},0\,\) m) and the local sensor located at the south pole of the Moon at 10 m of altitude, coordinates \((-90^{\circ},0^{\circ},10\,\) m). The local sensor is considered usable for independent measurement (pseudorange, range, and angles) if the user is closer than 20 km; therefore, only satellite measurements can be exploited. The descending phase takes 100 min (well below the ephemeris validity period), and to evaluate the performance in different constellation conditions, three landing phases starting at three different constellation epochs were evaluated: the first starts at epoch 1350, the second starts at epoch 1600, and the third starts at epoch 2100. The timing of the three approaches was chosen so that it is possible to evaluate the proposed methods in three different constellation conditions: landing #1 has 4 satellites in view at all times, landing #2 has fewer than 3 satellites in view (not allowing localization with only satellites and satellites plus pseudolite configuration), and landing #3 has three satellites in view at all times. The CRLB and r.m.s. error (r.m.s error is computed on each epoch of the scenario exploiting 200 Monte Carlo runs of a Weighted Least Square positioning algorithm) were evaluated for each landing trajectory and compared in the following figures, one for each landing trajectory. Each figure reports both the vertical and the horizontal error to provide a clearer understanding of the error trends. In each figure, the vertical gray line shows the moment at which the PSL and the ISL start to be in view from the user, and the error is reported in log-scale to better distinguish the difference among the various methods. Finally, the local sensor configurations are as follows: only satellites (No local sensor), satellites plus pseudolite (PSL), satellites plus independent source of localization (ISL), satellites plus differential correction (DRS), satellites plus differential correction plus pseudolite (DRS+PSL), and satellites plus differential correction plus independent source of localization (DRS+ISL). Figure 4 shows the horizontal and vertical error for the first landing trajectory where all the techniques are available. Figure 5 shows the horizontal and vertical error for the second landing trajectory where, due to the lack of satellites, only the ISL is available for positioning. Finally, Figure 6 shows the horizontal and vertical error for the third landing trajectory where, using a pseudolite, it is possible to use the three satellites in view for positioning. Figure 4: Landing #1.