# Moon Sensor Station to Improve the Performance of Lunar Satellite Navigation Systems
## 5 Independent Source of Localization
The range measurement can be formulated as follows:  
\[R(\mathbf{\theta})=\sqrt{\left(x-x_{\text{LE}}\right)^{2}+\left(y-y_{\text{LE}} \right)^{2}+\left(z-z_{\text{LE}}\right)^{2}}+n_{R} \tag{22}\]  
obtaining, for the \(\mathbf{H}\) matrix:  
\[\mathbf{H}_{R}=\left[\frac{x-x_{LE}}{R_{3D}}\quad\frac{y-y_{LE}}{R_{3D}}\quad \frac{z-z_{LE}}{R_{3D}}\right] \tag{23}\]  
where \(R_{3D}=\sqrt{\left(x-x_{LE}\right)^{2}+\left(y-y_{LE}\right)^{2}+\left(z-z_{LE }\right)^{2}}\). The final \(\mathbf{H}\) matrix is composed of the three matrices relative to satellite measurements, sensor range, and angle measurements (zero padding is applied for compatibility):  
\[\mathbf{H}=\left[ \begin{array}{c}\mathbf{H}_{PR}\\ \mathbf{H}_{R},0\\ \mathbf{H}_{EL},0\\ \mathbf{H}_{AZ},0\end{array} \right];\ \mathbf{n}(\mathbf{\theta})=\left[ \begin{array}{c} \mathbf{n}_{PR}(\mathbf{\theta})\\ n_{R}(\mathbf{\theta})\\ n_{El}(\mathbf{\theta})\\ n_{Az}(\mathbf{\theta})\end{array} \right] \tag{24}\]As before, to evaluate the user accuracy, it is also necessary to derive the measurement error model for \(n_{R}\) and \(n_{AZ/EL}\). The range measurement error due to the receiver noise can be assumed as Gaussian distributed, zero mean, and standard deviation \(\sigma_{R}\) derived from the literature [41]:  
\[\sigma_{R}=\frac{c}{2\cdot\text{BW}\cdot\sqrt{N\cdot SNR}} \tag{25}\]  
where \(c\) is the speed of light, BW is the signal bandwidth, \(N\) is the integration factor (usually equal to the number of available measurements, for example, using more pulses or different snapshots of the incoming signal), and \(SNR\) is the signal-to-noise ratio. Angle measurement errors can be assumed composed of two different contributions: (1) a measurement error, with the assumption of Gaussian distribution, zero mean, and standard deviation \(\sigma_{AOA}\), taking into account the effect of the SNR on the measurement process in the receiver and (2) the error due to the station position uncertainty and/or the unknown user antenna-body displacement. The first contribution, \(\sigma_{AOA}\), can be computed recalling the antenna array parameters, with the following formula [41]:  
\[\sigma_{AOA}=\frac{\theta_{3}}{k_{m}\cdot\sqrt{2\cdot N\cdot SNR}} \tag{26}\]  
where \(\theta_{3}\) is the half-power beam width (for a generic aperture antenna, \(\theta_{3}=70*\frac{\lambda}{D}\)), \(k_{m}\) is a multiplicative factor that takes into account the antenna beam shape, \(N\) is the integration factor, and SNR is the signal-to-noise ratio. Finally, \(\lambda\) is the wavelength and \(D\) is the antenna dimension [42]. In the case of a super-resolution processor, Equation (26) becomes, assuming the use of the MUSIC algorithm [43]:  
\[\sigma_{AOA}=\sqrt{\frac{6}{m^{3}\cdot N\cdot SNR}} \tag{27}\]  
where \(m\) is the number of elements of the antenna (\(m=\frac{2D}{\lambda}\)) and \(N\) is the integration factor. In the following, \(N\) will be assumed equal to the number of bits of the preamble of the message discussed later on. The error due to the station position error can be derived from the propagation of uncertainty theory [44]. Recalling that the usual assumption is that the reference station position is previously calculated with a long-term observation or with independent localization systems such as the Laser Retroreflector Array (LRA) for Lunar Laser Ranging and/or X-band transponder for Range and Range Rate (RARR), and Delta Differential One-way Ranging (DDOR) exploiting Earth tracking stations, it is possible to assume that the station position error is very small, independent, and uncorrelated to the other sources.