## 5 Conclusions  
In this study, we demonstrated the MDPO algorithm for the dual-satellite lunar navigation system with real GNSS pseudorange measurement obtained from GNSS stations on the ground. A pair of stations separated by about 5 km were selected to emulate a lander and a rover on the Moon.  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Values** & \\ \hline Propagation Distance, \(d\) & 300 km & 1000 km \\ \hline Transmitted Power, \(P_{T}\) & 0.2 W (\(-7\) dBW) & \\ \hline Frequency, \(f\) & \multicolumn{2}{c}{2500 MHz} \\ \hline \(C/N_{0}\) & 47.1 dB-Hz & 36.6 dB-Hz \\ \hline \hline \end{tabular}
\end{table}
Table 5: Link budget analysis of the lunar MDPO signals.  
Prior to the MDPO demonstration, a double-differenced pseudorange error assessment was performed using all the visible GPS satellites. Major pseudorange errors, such as satellite and receiver clock offsets and ionospheric delay, can be effectively removed from the double-differenced observations and the remaining error would be basically the sum of pseudorange noises. Thus, the distribution of the relative position error depends entirely on the variance of the range error and the satellite geometry, and the 2\(\sigma\) (95%) error of the double-differenced pseudorange observations can be obtained from the 2 drms on the horizontal position errors and the corresponding HDOP values. The result showed that the expected 2\(\sigma\) error of the double-differenced pseudorange observations was about 1 m.  
Next, we applied the MDPO algorithm using the same data set used in the double-differenced pseudorange error assessment. In this case, it was assumed that only two visible satellites were available at each epoch and four sets of satellite pairs were selected to emulate the satellite geometry resembling that of the previous lunar MDPO study. It was natural that the resulting horizontal positioning accuracies varied by satellite geometry, but the double-differenced pseudorange errors estimated from the 2 drms of horizontal positioning errors and corresponding HDOP values were consistent and about 1 m. This agrees with the result of the previous double-differenced pseudorange error assessment, and it can be concluded that the MDPO algorithm worked properly with the real GPS observables and provided the expected performance.  
Although the positioning accuracy of the MDPO algorithm tends to be lower because of its poor satellite geometry, it is still good enough for many lunar exploration activities. The MDPO algorithm is also suitable for the initial deployment stage of the lunar navigation satellite system because it only requires two satellites in view.  
We are currently working on a design study of a lunar navigation satellite system using a pair of 6U-size CubeSats to demonstrate the MDPO algorithm on the Moon. More details of the satellite design will appear in our future article.  
Conceptualization, T.E. and T.T.; methodology, T.E.; software, T.E. and T.T.; validation, T.E.; formal analysis, T.E. and T.T.; investigation, T.E.; resources, T.T.; data curation, T.E.; writing--original draft preparation, T.E.; writing--review and editing, T.E.; visualization, T.E.; supervision, T.E.; project administration, T.E.; funding acquisition, T.E. All authors have read and agreed to the published version of the manuscript. This work was supported by MEXT Coordination Funds for Promoting AeroSpace Utilization, Grant Number JPJ000959.  
Publicly available datasets were analyzed in this study. GNSS data in RINEX format can be found here: CDDIS [[[https://cddis.nasa.gov/](https://cddis.nasa.gov/)]([https://cddis.nasa.gov/](https://cddis.nasa.gov/)), accessed on 28 September 2022].  
The authors declare no conflict of interest.