# LCNS Positioning of a Lunar Surface Rover Using a DEM-Based Altitude Constraint
## 4 Covariance Analysis
### Simulation Settings  
Table 6 shows the main configuration settings for the covariance analysis grouped in three sets of parameters: LCNS ODTS uncertainty (which drives the SISE--signal in space error), the EKF process noise (which drives the definition of the process noise matrix \(Q\) defined in Equation (9)) and the initial state covariance of the EKF (which defines \(P_{0}\)).  
The ODTS uncertainty was selected to be in line with the parameters provided by previous work [29; 32], which are considered achievable at maximum age-of-data (AOD) (AOD was defined in line with the Galileo Service Definition Document (SDD) [44]). This parameter represents the system contribution to the user ranging error (also known as user equivalent range error, UERE). In this study, the ODTS uncertainty was considered to be the same in each direction (i.e., radial, tangential and along-track), and therefore, the uncertainties provided in Table 6 were directly mapped to the line-of-sight. This is clearly a simplification that aims to provide an upper bound of the performances. Two parallel studies to gain more insight into the ODTS uncertainty that can be achieved for a lunar navigation system are currently running at ESA. The outcomes of these studies will be taken into account in further studies to allow for a more realistic estimation of the SISE.  
Similarly to the ODTS uncertainty, the EKF process noise was selected to be in line with previous works [29; 32]. However, the magnitude of the clock process noise was adapted, and a higher process noise was associated with the clock drift (as this parameter drives the propagation of the clock bias). Furthermore, in this publication, the process noise related to the clock bias and drift was driven by the Allan deviation and frequency stability, respectively, assuming an OCXO (details on the Allan deviation and frequency stability of an OCXO are provided in Table 5). The process noise for the clock bias was set to 1 m/\(\sqrt{\text{s}}\), which is conservative considering the expected Allan deviation for an OCXO. The process noise for the clock drift was set to 10 m/s/\(\sqrt{\text{s}}\), which is within the spectrum of expected frequency stability seen for actual OXCOs (refer to Table 5).  
The process noise for the position and velocity was kept in line with previous works [32]. It has been shown that a velocity estimation with an accuracy of better than 1 cm/s is achievable using a tactical-grade IMU [45]. Therefore, the process noise provided in Table 6 is considered conservative. The process noise of 0.15 m/s/\(\sqrt{\text{s}}\) is in line with an LCNS-only solution considering a maximum rover velocity of 0.1 m/s. To understand how the positioning accuracy can be improved while using a tactical- or navigation-grade IMU, a sensitivity analysis is performed using different process noise values (related to the different IMU types reported in Table 4) for the position and velocity components, as further detailed in Section 5.3.  
Finally, the initial EKF position uncertainty was set to 100 m. This value seems realistic considering the work performed in previous publications [29; 32; 34] that demonstrated that the use of LCNS could lead to landing accuracies well within 100 m 3-sigma. In addition, the rover's initial position could be estimated over time using different techniques (LCNS, visual odometry, Earth tracking, etc.), allowing us to have an initial position uncertainty well within 100 m. The velocity initial uncertainty was set to 10 m/s, which is considered conservative, as the rover will not move faster than 0.1 m/s. The clock bias and drift initial uncertainty were selected in line with previous works [29; 32].