# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## Validation and Analysis
### Kinematic Experiment
#### 3.2.3 Positioning Results of Different Reference Receivers  
To evaluate the performance of MBRTK for different number of reference receivers, the redundancy for the scheme-one, scheme-two, scheme-five and scheme-six are shown in Figure 10. The average redundancy for scheme-one and scheme-two is 9.4 and 9.3, which are almost the same because of the same processing strategy with the traditional SBRTK model, while the average redundancy for scheme-five and scheme-six with MBRTK is 26.7 and 40.0. The more the reference receivers are, the stronger the reliability of MBRTK model is. Meanwhile, the ADOP values [51,52] quantify the priori precision and geometry of ambiguities, which can be computed and illustrated in Figure 11 for the  
Figure 8: Number of observed satellites (**left**) and redundancy (**right**) for three different schemes. Figure 9: Positioning errors for scheme-three, scheme-four, and scheme-five in N, E, and U components. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & \multirow{2}{*}{**Systems**} & \multicolumn{3}{c}{**RMS/cm**} & \multicolumn{3}{c}{**STD/cm**} & \multicolumn{1}{c}{**Number of**} & \multicolumn{1}{c}{**Number of**} & \multicolumn{1}{c}{**Supcess Rate**} \\ \cline{3-10}  & & **N** & **E** & **U** & **N** & **E** & **U** & **Epochs** & **Success** & **Success** \\ \hline Three & GPS & 5.5 & 2.3 & 6.3 & 3.7 & 2.3 & 5.5 & 4621 & 2840 & 61.5\% \\ Four & BDS & 3.4 & 1.6 & 5.7 & 2.5 & 1.5 & 4.7 & 4621 & 3905 & 84.5\% \\ Five & GPS/BDS & 3.0 & 1.5 & 4.6 & 1.5 & 1.4 & 3.9 & 4621 & 4363 & 94.4\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Statistics of three schemes with different satellite systems for kinematic experiment where STD=standard deviation. four schemes. We can see that, the ADOP values of all the four schemes can remain below 0.30 and 0.15 cycles for 99% and 90% of the observation segment, respectively. The ADOP values of scheme-five and scheme-six is the smaller than those of scheme-one and scheme-two, and scheme-six with three reference receivers has the optimal ADOP values. This is, thus, a promising indication that faster successful AR is possible for MBRTK model using multiple reference receivers. Figure 12 shows the positioning errors of TBR9 after the first convergence for scheme-one, scheme-two, and scheme-six in N, E, and U components, respectively. Considering the positioning errors of scheme-five discussed in the previous section, the corresponding statistics of four schemes with different reference receivers are listed in Table 6. The RMS values of four different schemes are (3.3, 1.5, and 4.9) cm, (3.3, 1.8, and 5.4) cm, (3.0, 1.5, and 4.6) cm, and (3.0, 1.5, and 4.5) cm in N, E, and U components, respectively, while the corresponding STD values for four schemes are (1.9, 1.4, and 4.3) cm, (2.1, 1.5, and 4.2) cm, (1.5, 1.4, and 3.9) cm, and (1.5, 1.4, and 4.0). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Schemes**} & \multirow{2}{*}{**Model**} & \multicolumn{3}{c}{**RMS/cm**} & \multicolumn{3}{c}{**STD/cm**} & \multicolumn{3}{c}{**Number of Epochs**} & \multicolumn{1}{c}{**Number of Success**} & \multirow{2}{*}{**Success Rate**} \\ \cline{3-3} \cline{5-9}  & & **N** & **E** & **U** & **N** & **E** & **U** & \\ \hline One & SBRTK & 3.3 & 1.5 & 4.9 & 1.9 & 1.4 & 4.3 & 4621 & 3906 & 84.5\% \\ Two & SBRTK & 3.3 & 1.8 & 5.4 & 2.1 & 1.5 & 4.2 & 4621 & 3972 & 86.0\% \\ Five & MBRTK & 3.0 & 1.5 & 4.6 & 1.5 & 1.4 & 3.9 & 4621 & 4363 & 94.4\% \\ Six & MBRTK & 3.0 & 1.5 & 4.5 & 1.5 & 1.4 & 3.9 & 4621 & 4471 & 96.8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Statistics of four schemes with different reference stations for kinematic experiment. Figure 11: Ambiguity dilution of precision (ADOP) values of four schemes with different reference receivers. Figure 10: Redundancy for four schemes with different reference receivers.