# LCNS Positioning of a Lunar Surface Rover Using a DEM-Based Altitude Constraint
## 5 Results  
### Baseline  
As indicated in Section 4.3, the positioning algorithms used for both the baseline solution (LCNS only) and the solution considering the DEM are similar (i.e., the same measurement model and process noise are used). The only difference occurs when the number of satellites is less than 4, in which case the range-rate observations are no longer used (as these require an accurate estimation of the clock drift). This section shows the results of the covariance analysis using the simulation parameters defined in Section 4.4. Figure 10a shows the number of visible satellites during the full scenario of approximately 3.1 days, while Figure 10b shows the horizontal dilution of precision (HDOP) of the solution with (red line) and without (blue line) the utilisation of DEM together with the LCNS measurements. It can be clearly seen that the LCNS constellation introduced in Table 1 is optimized to ensure good coverage in the South pole, as 52% of the time, at least 4 satellites are visible. When comparing the HDOP of the solution with and without DEM, the added value of the DEM measurement is evident. While the HDOP of the satellite-only solution (without DEM) has peaks that reach values up to \(10^{4}\), the HDOP when using an additional DEM measurement is always below 10. Furthermore, the availability can be extended to the periods in which only 3 satellites are available, as the 3 satellite measurements and the additional DEM measurement allow the receiver to solve its three-dimensional position and clock bias. When less than 3 satellites are available, no solution is computed, so no HDOP is provided. It is important to note that, within this contribution, it is assumed that the rover will continue moving even when no LCNS-based solution is available. This is possible using just IMU-based propagation or a visual-based odometer. When more than 3 satellites are again visible, the LCNS-based solution is re-started, considering the initial uncertainties in Table 6.  
Figure 11a shows the horizontal (2D) covariance with (red line) and without (blue) the utilisation of the DEM. The behaviour of the horizontal covariance closely follows the behaviour of the HDOP, as reported in Figure 10b. The peaks at the beginning of each tracking period starting at 300 m (note that Figure 11a reports 3-sigma) indicate the  
Figure 10: Geometry of the scenario. (**a**) shows the number of visible satellites in time, while (**b**) shows the HDOP considering both the LCNS-only and LCNS + DEM scenarios. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Threshold to enable DEM measurement & 150 m \\ Multiplier \(n\) for \(\sigma_{DEM}\) & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: DEM simulation parameters. convergence time needed as the filter is initialized at a position uncertainty of 100 m (as reported in Table 6). During the periods of good HDOP (satellite only), the performances between the satellite only and satellite and DEM solution are very similar; however, in case the geometry of the solution with the satellites degrades, having the DEM in the solution avoids a degradation of the solution accuracy. Furthermore, when only 3 satellites are visible, the LCNS-only solution cannot be computed, while with the additional DEM measurement, the rover is able to determine its position with reasonable performance (around 75 m 3-sigma). Figure 11b shows the uncertainty (1-sigma) of the DEM measurement. The peaks (around 10 m) are caused by the initial rover uncertainty, which is set to 100 m (refer to Table 6) at the beginning of each tracking period. Once the receiver covariance has converged, it can be seen that the DEM uncertainty is primarily governed by the data component (refer to Section 4.5.2 for the definition of the data component), which is distributed between 0 and 1 m (refer to Figure 8b).