# LCNS Positioning of a Lunar Surface Rover Using a DEM-Based Altitude Constraint
## 4 Covariance Analysis
### Measurement Covariance Modelling
#### 4.5.2 DEM Covariance Model
However, depending on the receiver horizontal covariance (denoted by \(\sigma_{2D}\)), the receiver may actually be located in another pixel with a different height. For example, assuming a horizontal receiver uncertainty of \(\sigma_{2D}\) = 9 m and a DEM resolution of 5 m per pixel,the rover can be in any of the pixels indicated by 1-9. Subsequently, the standard deviation of all these pixels is used to evaluate the rover's contribution to the uncertainty of the DEM measurement. Note that only the pixels for which the distance between the center of that pixel and the center of the pixel denoted with \(Rx\) is less than \(\sigma_{2D}\) are used to evaluate \(\sigma_{rx}\). It is assumed that this approach will give a first-order estimate on the accuracy of the DEM measurement. A more complicated approach (e.g., considering the position history to select more probable pixels) can be considered for future work. However, in case the rover uncertainty is below 5 m (equal to the resolution of the DEM), at least the surrounding pixels (see Figure 7, pixels indicated 1-9) are used to compute the standard deviation, as the rover may be located in any of these pixels. Figure 8a,b shows the standard deviation of the terrain surrounding each pixel considering a rover uncertainty (1\(\sigma\)) of, respectively, 10 m and 100 m. Figure 8a shows that a rover uncertainty of 10 m leads to a surrounding terrain standard deviation not larger than 8 m and actually much lower along the actual traverse. A 100 m receiver uncertainty leads to a surrounding terrain standard deviation up to 25-30 m (Figure 8b). The actual traverse shows lower values (below 15 m), confirming the good selection of the traverse (e.g., far from steep cliffs). Figure 9 shows the variation of the terrain along the rover traverse introduced in Section 2 in terms of the standard deviation of all pixels within the circle covered by the horizontal uncertainty of the rover as well as the absolute variation between the minimum and maximum height experienced within the selected pixels. Note that both the standard deviation and absolute deviation considering a receiver uncertainty of 10 m are fairly low (below 5 m in most of the epochs). However, increasing the rover standard deviation to 100 m, the absolute variation attains significant values (e.g., above 50 m); nevertheless, the standard deviation remains low. Figure 7: Conceptual drawing showing how the uncertainty related to the DEM measurement is evaluated based on the rover’s horizontal uncertainty. The overall DEM uncertainty is modelled by Equation (17), which contains the terms \(\sigma_{Data}\) and \(\sigma_{Rover}\), which, respectively, represent the uncertainty related to the data component (i.e., interpolation and orbital errors) as well as the DEM uncertainty induced by the rover horizontal uncertainty (which is defined as the standard deviation of all the pixels that are captured within a circle surrounding the propagated rover position consistent with Figure 7). \(\sigma_{Data}\) and \(\sigma_{rover}\) are assumed to be uncorrelated. \[\sigma_{DEM}=\sqrt{\sigma_{Data}^{2}+\sigma_{rover}^{2}} \tag{17}\]  
Finally, Table 8 shows the the parameters used in the DEM scenario. A threshold is used to disable the DEM aiding in case the current position uncertainty is too large. In such a case, the uncertainty of the receiver would lead to meaningless results. Furthermore, the \(n\) denotes the multiple of \(\sigma_{DEM}\) (i.e., 3-sigma for \(n=3\)) used in the measurement covariance matrix \(\Sigma_{k}\) to account for the geological error component. Figure 8: Standard deviation (\(\sigma_{rx}\)) of the lunar terrain surrounding the rover traverse (in red). (**a**) shows the standard deviation of the terrain height considering all pixels which have their centre within 10 m of the query point, while (**b**) shows the standard deviation for all pixels which have their centre within 100 m of the query point. The projection is polar stereographic, where (0, 0) denotes the South Pole.