# Exploiting Lunar Navigation Constellation for GNC Enhancement in Landing Missions
## 2 Simulation Environment
### User Navigation Equipment
#### 2.2.1 Radiofrequency-Based Measurements  
In this work, the user LNS receiver terminal retrieves from any \(i\)th visible element of the constellation the range \(\rho\), the range-rate, \(\dot{\rho}\), and the servicer ephemeris under the form of the state vector \(\mathbf{x}_{s,i}\). When dealing with such measurements, the observable data is usually defined as _pseudorange_\(\tilde{\rho}\) and _pseudorange-rate_\(\tilde{\rho}\), in order to indicate that the two measurements are affected by errors. In particular, looking at the range measurement, the geometric range obtained from the \(i\)th servicer can be defined as per Equation (5):  
\[\rho_{i}=c\Delta t_{i}=\|\mathbf{r}_{s,i}-\mathbf{r}\| \tag{5}\]  
where \(c\) is the speed of light and \(\Delta t_{i}\) is the time required for the signal to travel from the servicer position, \(\mathbf{r}_{s,i}\), to the user one, \(\mathbf{r}\). To simplify the analysis, the availability of the signal from the \(i\)-th servicer to the user is constrained only by point-to-point and Field-Of-View (FOV) considerations. In _one-way_ ranging, this time difference is obtained with standard Time-of-Arrival (ToA) techniques by subtracting the servicer clock-time at signal emission from the user clock-time at reception, i.e., \(\Delta t_{i}=t_{u}-t_{s,i}\). However, this measure is affected by errors since both the receiver and servicer clock present some offsets with respect to the LNS system time. As such, the measured time difference from the user terminal will provide the pseudorange, as per Equation (6):  
\[\tilde{p}_{i}=c\left(t_{u}+\delta t_{u}-t_{s,i}-\delta t_{s,i}\right)=\rho_{i}+c \delta t_{u}+\varepsilon_{\delta t_{u,i}} \tag{6}\]  
where \(t_{u}\) is the system time at which the signal would have reached the user in the absence of errors, \(\delta t_{u}\) is the receiver clock bias, \(t_{s,i}\) is the system time at which the signal left the \(i\)th service, and \(\delta t_{s,i}\) is the offset of the \(i\)th service clock from the system time. From this equation, it is clear that a bias in the measurements is present. Additionally, there are other possible sources of errors in the pseudorange readings, such as interference effects, receiver noise, instrumental delays, multipath losses, and relativistic effects. With this in mind, the final pseudorange formulation is reported in Equation (7), where all these error contributions (including \(\varepsilon_{\delta t_{u,i}}\)) are collapsed in a single component, \(\varepsilon_{\rho_{i}}\). \[\tilde{p}_{i}=\rho_{i}+c\delta t_{u}+\varepsilon_{\rho_{i}} \tag{7}\]  
A similar reasoning can be performed for the pseudorange-rate observable, which can be derived directly from the observed frequency Doppler shift, \(\Delta f_{D}\), but again is inherently affected by the same effects as in the previous case, summed up in a \(\varepsilon_{\rho_{i}}\) term in addition to the time bias derivative, \(\dot{\delta t}_{u}\), as shown by Equation (8). \[\tilde{p}_{i}=c\frac{\Delta f_{D}}{f_{s,i}}+c\dot{\delta t}_{u}+\varepsilon_{ \rho_{i}} \tag{8}\]  
In this work, to simulate these effects and generate the measurements fed to the filter, we implemented the model in Equations (9) and (10). \[\tilde{p}_{i} =\|\tilde{\mathbf{r}}_{s,i}-\mathbf{r}\|+b_{c}+\varepsilon_{\rho _{i}} \tag{9}\] \[\tilde{p}_{i} =\left(\tilde{\mathbf{v}}_{s,i}-\mathbf{v}\right)\cdot\left( \tilde{\mathbf{r}}_{s,i}-\mathbf{r}\right)/\rho+d_{c}+\varepsilon_{\dot{p}_{i}}\] (10) \[\tilde{\mathbf{x}}_{s,i} =\left[\tilde{\mathbf{r}}_{s,i}^{\top},\quad\tilde{\mathbf{v}}_ {s,i}^{\top}\right]^{\top}=\mathbf{x}_{s,i}+\left[\varepsilon_{r}^{1\times 3}, \quad\varepsilon_{v}^{1\times 3}\right]^{\top} \tag{11}\]  
For both the observables, all the error effects are directly collected in single additive Gaussian noise terms, except for the receiver clock bias, \(b_{c}\), and drift \(d_{c}\), contributions, which are treated as an additional parameter to be estimated.