# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 3 Differential Reference Station  
A local sensor based on the Moon can produce and broadcast differential corrections as in GNSS [24, 36]. The local sensor can be installed on the lunar surface at the south pole region, i.e., the target area for future lunar exploration missions [1]. The station generates and transmits the pseudorange corrections based on the pseudorange measurements of all visible satellites. The station should be composed of a satellite navigation receiver and an antenna, used to compute the pseudorange and its position, and a communication system to send corrections to the user. Corrections can be sent directly to the user or via satellites, if the navigation system is combined with a satellite communication system, as often proposed [37, 38], and exploited by the user to reduce its error component. This configuration is the simplest and least expensive of the three proposed here, requiring only a data link receiver on the user side. The DRS generates the pseudorange corrections based on the pseudorange measurements from visible satellites:  
\[\delta\rho_{i}=\rho_{i}-R_{i,r}=ct_{r}+n_{i,r}\;\;\;i=1,...,N \tag{10}\]  
where \(R_{i,r}\) is the distance between the satellite and the reference stations (known), and \(n_{i,r}\) is the additive measurement error. The correction is sent to the user and exploited to reduce the error component:  
\[\rho_{i}-\delta\rho_{i}=R_{i,u}+ct_{u}-ct_{r}+n_{i,u}-n_{i,u}\;\;\;i=1,...,N \tag{11}\]  
On the Moon, the errors that can be reduced by differentiating the measurements are the satellite position error and the satellite clock error. Remembering that no important propagation errors (apart from multipath) are present and recalling that the satellite clock error is exactly the same for the reference and user, the final error contribution after the differential correction is as follows:  
\[n_{i,u}-n_{i,r}=(n_{i,u}^{S-p}-n_{i,r}^{S-p})+(n_{i,u}^{RX}-n_{i,r}^{RX})+(n_ {i,u}^{M}-n_{i,r}^{M}) \tag{12}\]  
where the terms in the last two couples (receiver and multipath errors, respectively) are uncorrelated but small and the first couple represents the residual error due to the satellite position error. Referring to Figure 2, the difference between satellite position errors (\(n_{i,u}-n_{i,r}\)) depends on the geometry and can be derived as in [23]: a satellite ephemeris error, \(\Delta\mathbf{R}\), causes a scalar ranging error \(E\), that is, the projection of the satellite vector error on the direction given by the unit vector pointing the satellite from the user, \(\mathbf{u}_{S}\):  
\[E=\Delta\mathbf{R}^{T}\cdot\mathbf{u}_{S}=\Delta\mathbf{R}^{T}\cdot\frac{ \mathbf{R}}{R} \tag{13}\]From Figure 2, \(\mathbf{R}=\mathbf{R}_{S}-\mathbf{r}\) and \(\mathbf{u}_{S}=\frac{\mathbf{R}_{S}-\mathbf{r}}{\left\|\mathbf{R}_{S}-\mathbf{r }\right\|}=\frac{\mathbf{R}}{R}\), and the variation in error due to the geographical displacement between the user and the reference, \(\Delta E\), is given by:  
\[\Delta E\cong\frac{\partial E}{\partial\mathbf{r}}\cdot\Delta\mathbf{r}=\frac {-\Delta\mathbf{R}^{T}\cdot\mathbf{d}}{R} \tag{14}\]  
where \(\mathbf{d}=\left[\Delta\mathbf{r}-\left(\mathbf{u}_{S}^{T}\cdot\Delta\mathbf{ r}\right)\mathbf{u}_{S}\right]\). Finally, noting that \(\mathbf{d}<\Delta\mathbf{r}\), the residual error absolute value is limited by:  
\[\left|\Delta E\right|\leq\frac{\left|\Delta\mathbf{r}\right|\cdot\left|\Delta \mathbf{R}\right|}{R} \tag{15}\]  
where \(\left|\Delta\mathbf{r}\right|\) is the user-local sensor distance and \(R\) is the satellite distance.