# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 2 Satellites-Only and Performance Evaluation
In the first case, assuming an unbiased estimator for the problem given in Equation (1), \(\hat{\mathbf{\theta}}(\mathbf{\rho})\), the Cramer-Rao inequality is given by the following expression [25]:  
\[\mathbf{C}(\mathbf{\theta})=\mathrm{E}\left\{(\hat{\mathbf{\theta}}(\mathbf{\rho})-\mathbf{ \theta}(\mathbf{\rho}))(\hat{\mathbf{\theta}}(\mathbf{\rho})-\mathbf{\theta}(\mathbf{\rho}))^{T} \right\}\geq\left[\mathbf{J}(\mathbf{\theta})\right]^{-1} \tag{5}\]  
Here, \(\mathbf{C}(\mathbf{\theta})\) represents the error covariance matrix related to the estimated vector of unknowns, while \(\mathbf{J}(\mathbf{\theta})\) refers to the Fisher Information Matrix (FIM) and its elements have the following form (in case of Gaussian distributed zero mean additive error):  
\[\begin{split}& J(\mathbf{\theta})_{m,k}=\frac{\partial\mathbf{\rho}( \mathbf{\theta})}{\partial\mathbf{\theta}_{m}}\mathbf{N}(\mathbf{\theta})^{-1}\frac{ \partial\mathbf{\rho}(\mathbf{\theta})}{\partial\mathbf{\theta}_{k}}+\\ &+\frac{1}{2}tr\left(\mathbf{N}(\mathbf{\theta})^{-1}\frac{\partial \mathbf{N}(\mathbf{\theta})}{\partial\mathbf{\theta}_{m}}\mathbf{N}(\mathbf{\theta})^{-1} \frac{\partial\The satellite position error, which depends on the ephemeris error in the navigation message and it is assumed equal to 10 m (1\(\sigma\)) [27; 28; 29; 30; 31; 32]. In particular, the most recent lunar satellite navigation-related study [31] can be considered as the source for the reference satellite position performance for this work, assuming the ephemeris validity period (and their renewal time) is smaller than 4 h.
2. The satellite clock (\(n_{S-Ck}\)) error, which results from clock time deviations and is assumed to be as proposed in [14; 33] with Deep Space Network synchronization every 24 h, obtaining an error within 10 ns with respect to UTC [34].
3. The multipath error, where the open-sky Brenner's multipath model is used to estimate the multipath error [35] with the following assumptions [14]: Binary Phase Shift Keying (BPSK) signals, rectangular spreading symbols with a 1 \(\times\) 1.023 MHz spreading code rate, and a receiver front-end bandwidth of 8 MHz.
4. For the receiver error (\(n_{RX}\)), the same assumptions of [14] are used, obtaining a receiver error standard deviation of 0.95 m.  
In the CRLB computation, if the satellite position is not directly added to the pseudorange, typically, the matrix \(\mathbf{N}\) is considered diagonal, with the elements on the diagonal computed as follows, as in the case of computation of Dilution Of Precision (DOP) [24]:  
\[\sigma_{PR}^{2}=\sigma_{RX}^{2}+\sigma_{S-p}^{2}+\sigma_{M}^{2}+\sigma_{S-Ck}^ {2} \tag{7}\]  
where \(\sigma_{RX}^{2},\sigma_{S-p}^{2},\sigma_{M}^{2},\sigma_{S-Ck}^{2}\) are the variances of the receiver measurement error, satellite position error, multipath error, and satellite clock error, respectively. Having \(\mathbf{C}(\mathbf{\theta})\), it is possible to compute the standard deviation for the 3D horizontal and vertical position errors [24; 36]:  
\[\sigma_{3D}=\sqrt{\mathbf{C}(\mathbf{\theta})_{11}+\mathbf{C}(\mathbf{ \theta})_{22}+\mathbf{C}(\mathbf{\theta})_{33}} \tag{8}\] \[\sigma_{Hor.}=\sqrt{\mathbf{C}(\mathbf{\theta})_{11}+\mathbf{C}(\mathbf{ \theta})_{22}}\] \[\sigma_{Vert.}=\sqrt{\mathbf{C}(\mathbf{\theta})_{33}}\]  
Alternatively, as mentioned before, the performance can be evaluated by simulating the scenario and the user localization algorithm and evaluating the final error statistics using Monte Carlo runs. In detail, the simulation steps are:  
1. Trajectory Simulation: Nominal satellite and user trajectories are generated. 2. Satellite Position Error: Gaussian, uncorrelated noise is added to each coordinate of the satellite trajectory. 3.