# Autonomous and Earth-Independent Orbit Determination for a Lunar Navigation Satellite System
## 4 Lunar Orbital Filter
### Kalman Filter Performances  
The proposed filter methodology is evaluated, considering various sensor combinations. The performances are presented in Table 3. Sample EKF results for a horizon-based and crater-based-only navigation filter are illustrated in Figures 11 and 12, respectively. Weights \(\gamma\) are used of 1.01 and 101 for the lunar horizon and craters respectively. An increase in noise is associated with distance to the lunar surface.  
Each performance is insufficient to satisfy the positioning objectives of an LCNS, with errors in the range of 100s of metres. Of note is the low accuracy of the lunar crater model, with a performance from 1 km to more than 20 km. This is due to the high variability of performance based on lunar surface proximity, as was illustrated in Section 3.4. This presents a challenge to the covariance model within the filter. The variability of the lunar horizon accuracy with altitude is much less, and so the EKF precision stabilises.  
\begin{table}
\begin{tabular}{|c c c c|} \hline \hline \multicolumn{1}{c}{**Sensor**} & \multicolumn{1}{c}{**Cross-Track [m]**} & \multicolumn{1}{c}{**Along-Track [m]**} & \multicolumn{1}{c}{**Radial [m]**} \\ \hline \hline \multicolumn{1}{c}{Lunar Horizon} & 234 \(\pm\) 492 & 1352 \(\pm\) 381 & 8 \(\pm\) 93 \\ \multicolumn{1}{c}{Lunar Craters} & 170 \(\pm\) 3531 & 1013 \(\pm\) 6620 & 170 \(\pm\) 20,754 \\ \multicolumn{1}{c}{Lunar Horizon and Craters} & 23 \(\pm\) 442 & 162 \(\pm\) 533 & 567 \(\pm\) 1376 \\ \hline \hline \end{tabular}
\end{table}
Table 3: EKF for position sensor combinations at 1\(\sigma\).  
Figure 10: Sensor and Kalman Filter model illustration.  
Figure 11: Lunar- horizon-only EKF position error for a spacecraft in ELFO. The error is in blue and the black illustrates the solution boundary at \(3\sigma\).  
Figure 12: Lunar-crater-only EKF position error for a spacecraft in ELFO. The error is in blue.  
Such a fusing may help in lunar ellipse or other poor limb-lighting conditions. In addition, if satellite pointing does not capture the limb perfectly but only the surface, the crater form could also be used to assist in stabilising the filter. Challenges to each sensor's performance are highlighted in Table 2.  
The ISL can support with performance, which is able to provide a relative measurement to the neighbouring satellite's range and range-rate. This technique may only be used in combination with other celestial sensors, which provide an absolute position reference to the Moon. The performances of the EKF methodology are presented in Table 4 as an average of all ten satellites, alongside a plot of performance over time of the position and velocity errors in Figures 13 and 14 respectively. A horizon-only approach is treated since it delivers the best performance in Table 3.  
However, the approach of using an ISL presents challenges to pointing, where it may not be possible for both a link to be maintained and a camera pointed towards the lunar limb. Times have been staggered in this model to cope with these challenges, with updates from 10-60 s from ISL and horizon sensors varying during operations.  
The performances indicated by Table 4 show that with the addition of the ISL, the performance and solution stability improve to around 100 m. This may be sufficient to support orbit determination of an LCNS but does not meet the requirement of 30 m.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**State** & **Cross-Track** & **Along-Track** & **Radial** \\ \hline Position [m] & \(1.3\pm 41.0\) & \(12.6\pm 28.5\) & \(19.9\pm 96.0\) \\ Velocity [mm/s] & \(4.4\pm 58.3\) & \(12.9\pm 50.4\) & \(4.3\pm 70.5\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: EKF for an ISL-assisted horizon-based optical navigation system at \(1\sigma\).  
Figure 13: Inter-satellite link with lunar horizon position error for 10 distributed spacecraft in ELFO.