# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 5 Independent Source of Localization  
A new type of local sensor is proposed, which supplies signals for three additional measurements: elevation, azimuth, and range, allowing, when the number of navigation satellites is too low, a complete autonomous 3D positioning or joint position estimation with the satellites in view. This configuration is the most expensive and complex, it comprises a satellite navigation receiver and an independent localization system receiver, for the user, and the local sensor is composed of four transmitters/receivers equipped with phased array sector antennas (e.g., with an azimuth coverage angle of 90 degrees and an elevation coverage angle of 60 degrees each) to cover all the possible azimuth angles. Each sector antenna must allow measurement of the horizontal and vertical angles of arrival of the incoming signals by exploiting digital beam-forming techniques. Additionally, the use of two-way communication enables the measurement of the distance between the spacecraft/rover and the local sensor. The working principle of the ISL can be outlined in the following steps: the user interrogates the local sensor using an omnidirectional antenna, the local sensor measures the angle of arrival of the interrogation and sends back a reply containing the measured angle with a fixed time of reply, the user computes the range exploiting the travel time and reads the angle of arrival on the reply, and the user computes its position exploiting the three measurements and the measurement from satellites, if available. As mentioned before, the user must be equipped with a specific transponder to communicate with the local sensor and the coverage area is limited due to visibility and link budget. The elevation and azimuth measurements in East North Up (ENU) coordinates centered on the local sensor are given by the following expression:  
\[\hat{E}l(\theta_{ENU})=El(\theta_{ENU})+n_{El}=\tan^{-1}\left(\frac{z^{ENU}l}{R _{H}}\right)+n_{El} \tag{18}\]  
\[\hat{Az}(\theta_{ENU})=Az(\theta_{ENU})+n_{Az}=\tan^{-1}\left(\frac{y^{ENU}l}{ x^{ENU}}\right)+n_{Az} \tag{19}\]  
where \(R_{H}=\sqrt{\left(x^{ENU}\right)^{2}+\left(y^{ENU}\right)^{2}}\) is the horizontal distance between the user and the local sensor. These measurements must be expressed in Lunar-Centered Lunar-Fixed (LCLF) coordinates to solve the general localization problem, exploiting the following transformation matrix:  
\[\mathbf{X}^{ENU}=\mathbf{G}(\mathbf{X}-\mathbf{X}_{0}) \tag{20}\] \[\mathbf{G}=\left[ \begin{array}{ccc}-\sin\gamma&\cos\gamma&0\\ -\cos\gamma\sin\varphi&-\sin\gamma\sin\varphi&\cos\varphi\\ \cos\gamma\cos\varphi&\sin\gamma\cos\varphi&\sin\varphi\end{array} \right]\]  
where \(\mathbf{X}^{ENU}=\left(x^{ENU},y^{ENU},z^{ENU}\right)^{T}\) is the 3D coordinates vector of the user in ENU coordinates, \(\mathbf{X}_{0}=\left(x_{LE},y_{LE},z_{LE}\right)\) is the local sensor position in LCLF coordinates, \(\mathbf{X}\) is the position vector of the user in LCLF, and \(\gamma\) and \(\varphi\) are the longitude and latitude, respectively, of the local sensor. Again, the \(\mathbf{H}\) matrix and the covariance matrix of the measurement can be used to compute the system performance. The \(\mathbf{H}\) matrix can be computed as the Jacobian of the composite function, obtaining:  
\[\mathbf{H}_{Az/El}=\mathbf{H}_{Az/El}^{ENU}\cdot\mathbf{G} \tag{21}\]  
where \(\mathbf{H}_{Az}^{ENU}=\left[\frac{-y^{ENU}}{R_{H}^{2}},\frac{x^{ENU}}{R_{H}^{ 2}},0\right]\) and \(\mathbf{H}_{El}^{ENU}=\left[\frac{-x^{ENU},z^{ENU}}{R_{H}^{2}\cdot R_{H}},- \frac{y^{ENU},z^{ENU}}{R_{D}^{2}\cdot R_{H}},\frac{R_{H}}{R_{3D}^{2}}\right]\).