# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## 2 Methodology
### The Redundancy of MBRTK  
The redundancy is an important indicator to illustrate the reliability of the model, which is computed as the number of observations minus the number of unknown parameters for single epoch [49]. In Table 1, we give the number of observations, the number of unknown parameters and the redundancy of MBRTK compared with the SBRTK. Because the Doppler observations are not adopted to estimate the velocity parameters in Equation (7), the velocity components of system states are only predicted in the time update step and are not adjusted in the measurement update step of Kalman filter. Thus, the velocity state parameters are not considered for the calculation of redundancy in Table 1. \(g\) and \(b\) are the satellite number of single GPS and single BDS. \(r\) is the number of reference stations. Because of the equivalence between SBRTK and MBRTK, the condition of MBRTK is the same as the conventional SBRTK, as shown in the fifth column of Table 1. The redundancy is given based on the following assumptions, as (1) the ambiguities are estimated as time-continuous values, (2) the equivalent DD observation equations can be established in intra-system for both single GPS and single BDS, (3) multiple reference receivers can track the same satellites simultaneously.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Number of** & **Number of** & **Number of** & **Number of** & **Condition** \\  & **Observations** & **Unknows** & **Redundancies** & **Constraints** & \\ \hline SBRTK:GPS & \(2(g-1)\) & \(3+(g-1)\) & \(g-4\) & \(0\) & \(g\geq 4\) \\ SBRTK:BDS & \(2(b-1)\) & \(3+(b-1)\) & \(b-4\) & \(0\) & \(b\geq 4\) \\ SBRTK:GPS + BDS & \(2(g-1)+2(b-1)\) & \(3+(b-1)+(b-1)\) & \(g+b-5\) & \(0\) & \(g\geq 4\) \& \(b\geq 4\) \\ MBRTK:GPS & \(2g\) & \(3r+(g-1)\) & \(r(g-2)\) & \(3(r-1)\) & \(g\geq 4\) \& \(k\
eq r\geq 1\) \\ MBRTK:BDS & \(2 rb\) & \(3r+r(b-1)\) & \(r(b-2)\) & \(3(r-1)\) & \(b\geq 4\) \& \(r\geq 1\) \\ MBRTK:GPS + BDS & \(2(g+b)\) & \(3r+r(g-1)+r(b-1)\) & \(r(g+b-1)\) & \(3(r-1)\) & \(g\geq 4\) \& \(k\geq 4\) \& \(r\geq 1\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The number of observations, unknowns, and redundancy for single epoch where SBRTK = single baseline real-time kinematic, MBRTK = multiple baseline real-time kinematic, GPS = global positioning system, and BDS = Beidou navigation system.  
As an example, considering the GPS/BDS model for MBRTK, the number of SD observations for both carrier phase and code is \(2r(g+b)\). The number of unknown parameters consists of \(3r\) receiver positions, \(r(g-1)\) ambiguities for GPS, and \(r(b-1)\) ambiguities for BDS. The number of observation redundancy is \(r(g+b-1)\). If there are multiple reference receivers (\(r\geq 2\)) with more data accumulated, the redundancy of MBRTK can increase rapidly, as well as \(3(r-1)\) constraint equations can be obtained, which can improve the estimable accuracy of unknown state parameters and enhance the strength of MBRTK. Generally, the more the number of reference receivers is, the greater the redundancy improvement is.