## 3 Demonstration Experiment of MDPO
### MDPO Demonstration  
Unlike the experiment in the previous subsection, the MDPO algorithm assumes that only two visible satellites are available at each epoch. For the MDPO demonstration experiment, a set of GPS satellite pairs was selected so that \"phase angle\" and \"angular travel\" could be similar to those of the lunar MDPO simulation in [6]. The phase angle is the interior angle between the LOS vectors of the pair of satellites, and the angular travel is the angle at which one of the satellites moves during each observation period as shown in Figure 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**2 drms (m)** & **Mean HDOP** & **2 drms/HDOP (m)** \\ \hline
1.618 & 1.624 & 0.996 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The horizontal position errors and mean HDOP values. Figure 1: Relative positioning result using double-differenced pseudorange observations of all the visible GPS satellites: (**a**) the horizontal position errors; (**b**) the corresponding HDOP values. Figure 2: Phase angle between two satellites and angular travel during each observation period. Table 3 shows the satellite orbital parameters used in the previous lunar MDPO simulation study. In this case, a typical phase angle observed from the lander and the rover on the Moon was around 35 degrees, and each satellite moved about 3 degrees during the 30-s observation period. Figure 3 shows the sky plots of the GPS satellite pairs that provide similar phase angles to those of the lunar MDPO simulation. Each satellite of the pair has to be chosen from a different orbit plane since the separation angles of the GPS satellites in the same orbit plane are too wide to emulate the lunar MDPO constellation as shown in Table 3. The observation period was set to 450 s to achieve a similar angular travel value of 3 degrees. The observation period is much larger than that in the lunar MDPO simulation because the typical orbital period of GNSS satellites is about 24 h, whereas the simulated low-altitude lunar satellite completes one orbit every 2.4 h.  
Figure 3: Sky plots of the selected GPS satellite pairs: (**a**) PRN 01 and PRN 21; (**b**) PRN 12 and PRN 25; (**c**) PRN 16 and PRN 26; and (**d**) PRN 19 and PRN 17. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Orbital Elements** & **Satellite 1** & **Satellite 2** \\ \hline Altitude & 300 km & \\ \hline Inclination & \(0^{\circ}\) & \\ \hline Eccentricity & 0.0 & \\ \hline Right Ascension of the Ascending Node & \(0^{\circ}\) & \\ \hline Argument of Perilune & \(0^{\circ}\) & \\ \hline True Anomaly & \(0^{\circ}\) & \(-15^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Satellite orbital parameters used in the lunar MDPO simulation [6]. Figure 4 shows the horizontal position errors of the rover station and the corresponding HDOP values achieved by the MDPO algorithm using the real pseudorange observations. Each red circle in the horizontal position error plot has a radius of 2 drms from the true rover location. It should be noted that all the horizontal position estimates with a HDOP larger than 300 were excluded from the 2 drms calculation, as it indicates that the least-squares estimation tends to be singular. Fortunately, it is rather easy to avoid possible singularities even in real applications because the satellite orbits and the corresponding HDOP values can be predicted in advance. Figure 4: Relative positioning results using MDPO algorithm: the horizontal position errors and the corresponding HDOP values with (**a1,a2**) PRN 01 and PRN21; (**b1,b2**) PRN 12 and PRN25; (**c1,c2**) PRN 16 and PRN26, and; (**d1,d2**) PRN 19 and PRN17. Table 4 summarizes the results shown in Figure 4. In the MDPO demonstration experiment, all the 2 drms/HDOP values are consistent and about 1 m, which agrees with the 2\(\sigma\) double-differenced pseudorange error in Table 2 that was obtained with all the visible GPS satellites. This is a good indication that the MDPO algorithm worked properly with the real GPS observables and provided the expected performance.