# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 5 Independent Source of Localization
Assuming also that the local sensor errors on \(x,y,z\) have the same standard deviation \(\sigma\), and recalling that the derivatives of the elevation and azimuth functions with respect to \((x,y,z)\) correspond to the elements of the Jacobian matrix \(\mathbf{H}_{Az/EL}^{ENU}\), the elevation and azimuth error variances are given by the following equations:  
\[\sigma_{El}^{2}= \sigma^{2}\Bigg{(}\frac{R_{H}^{2}\cdot R_{3D}^{2}}{R_{3D}^{4} \cdot R_{H}^{2}}\Bigg{)}=\frac{\sigma^{2}}{R_{3D}^{2}} \tag{28}\] \[\sigma_{Az}^{2}= \sigma^{2}\Bigg{(}\frac{(y-y_{LE})^{2}}{R_{H}^{4}}+\frac{(x-x_{LE} )^{2}}{R_{H}^{4}}\Bigg{)}=\frac{\sigma^{2}}{R_{H}^{2}} \tag{29}\]Finally, to compute the total angle error standard deviation, the two error contributions must be summed:  
\[\sigma_{EI/Az}=\sqrt{\sigma_{AOA,EI/Az}^{2}+\sigma_{EI/Az}^{2}} \tag{30}\]  
Note that the first contribution depends on the SNR, is proportional to (\(R_{3D}^{2}\)), and is predominant for large distances, while the second contribution is proportional to \(R_{3D}^{-2}\) and \(R_{H}^{-2}\) for elevation and azimuth measurements and is predominant for short distances (\(R_{H}\) can be also replaced by \(R_{3D}\), having a small overestimation of the error). Last but not least, recalling Equations (25) and (26), it is clear that the measurement performance also depends on other important parameters, such as the integration factor, \(N\), the instantaneous band of the signal, \(BW\), the antenna dimension, \(D\), and the carrier wavelength, \(\lambda\). For this reason, a preliminary design of the signals and protocols of the local sensor is proposed. Recalling Equation (26), higher frequencies guarantee higher performance with a small antenna. Moreover, recalling that the Interagency Operations Advisory Group (IOAG) has defined a series of standards and protocols for the lunar communication and navigation architecture [45], the Ka-band is here proposed to allow the use of a small antenna for simple integration on the ISL (\(D=0.5\) m is considered hereafter). The maximum coverage of the local sensor was fixed to 20 km, assuming a minimum SNR of 10 dB at 20 km. BPSK modulation was selected for robustness and reliability, obtaining a Bit Error Rate (BER) of \(10^{-3}\) at 20 km [42]. Finally, the \(BW\) was set to 2 MHz. A random access to the channel was chosen due to its simplicity, no need of channel management, the expected low number of simultaneous users, and throughput. Moreover, random access simplifies the time of flight measurement process, as in other navigation systems, such as the Distance Measurement Equipment (DME) [46]. The proposed realistic assumptions allow the selection of the parameters \(D\), \(SNR\), and \(\lambda\) to compute the needed standard deviation of the AOA. To obtain realistic values for \(N\), knowing the modulation scheme and the signal band, it is necessary to make additional assumptions about the duration of the message time and the number of bits. Three types of messages are proposed. A Local Element Initialization message (LEI) is broadcast periodically by the local sensor to let the user know that it is in coverage of the local sensor and to obtain the local sensor information, i.e., the sensor ID and its position. A User Interrogation message (UI) contains the user and the local sensor IDs, used to start the measurement process. It is periodically repeated during navigation. For a Local Element Reply message (LER), for each received UI, the local sensor computes the angles of arrival (elevation and azimuth), and the message replays after a fixed delay. The message contains the measured angles, the beacon station ID, and the user's ID. Upon receiving the LER, the user can compute the round-trip delay (i.e., the range) from the local sensor and decode the AOA information encoded on it. The working principle, data, and message flow of the proposed method are reported in Figure 3.  
The messages include a preamble, a data block, and a parity check field. The data block includes the message type that defines the data content.