# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## 1 Introduction  
Global Navigation Satellite Systems (GNSSs) have been extensively used for scientific and commercial applications in geodesy, geodynamics, transportation, and other industries [1; 2; 3; 4; 5]. With the rapid development of GNSSs over the last several decades, the global users will be able to use the multi-constellation and multi-frequency observations to improve the reliability and availability [6; 7; 8; 9; 10]. In many emerging GNSS applications, such as precise positioning of drones and unmanned vehicles, the accuracy and availability of real-time kinematic (RTK) are crucially of paramount importance. However, the performance of single-baseline RTK (SBRTK) degrades greatly due to signal blockage and multipath errors. Many research efforts have been expanded to improve the RTK positioning accuracy and availability in urban areas. The mostly investigated method is the increasing use of observations including multiple satellite systems and reference or rover receivers [11; 12; 13]. Combing the global positioning system (GPS) and Beidou navigation system (BDS) positioning, the number of available satellites for users increases, which has a potential to improve the positioning accuracy and availability [14; 15; 16; 17]. As the observations were inevitably contaminated by cycle slips and gross errors, [PERSON] et al. [18] and [PERSON] et al. [19] applied a robust Kalman filter to detect the outliers and achieved good performances. To obtain the redundancy of the frequencies overlap between the systems, a priori corrections for differential inter-system biases (DISBs) was adopted to obtain the better ambiguity resolution (AR) and improve the positioning performance [20; 21; 22]. In addition, the multiple reference stations approach can provide better positioning accuracy [23; 24], which is recommended when precise kinematic positioning is required. The concept of the multiple rover constraints algorithm with multiple antennas on a common moving platform has been introduced to increase the reliability and accuracy for attitude determination [25] and RTK positioning [26; 27]. However, in the above-mentioned studies, a model of single-baseline solution (SBS) is generally adopted to process the observational data. In particular, when multiple reference receivers are used, the SBS model is not appropriate, because multiple reference receivers can cause the inconsistencies in coordinates of rover receiver, which would directly affect the positioning accuracy and availability. Since the introduction by [PERSON] and [PERSON] [28], the equivalence between the undifferenced and the differenced observations has attracted increasing attention from the satellite geodesy community. [PERSON] and [PERSON] [29] pointed out that the undifferenced and differenced observation equations would result in the same normal equation as long as the parameters in the undifferenced observation equations were independent. [PERSON] [30; 31] proposed the equivalent observation equation to unify the undifferenced and differenced methods. Afterward, [PERSON] and [PERSON] [32] developed the simplified equivalent equations to obtain the multi-baseline solution (MBS) using corresponding pseudo-observations. [PERSON] and [PERSON] [33] developed the simplified equivalent equations in the case of each receiver tracking different satellites with elevation-dependent weights, whose computation efficiency was significantly improved comparing to that of the SBS. [PERSON] and [PERSON] [34] proposed a new model of MBS based on the equivalence principle for static positioning. However, the carrier phase model for the multi-baseline RTK (MBRTK) has not been studied in the kinematic application. In MBRTK, the high-dimensional ambiguity needs to be estimated. It is often not necessary to resolve all the ambiguities to achieve the required accuracy for the final position [35]. [PERSON] et al. [36; 37] and [PERSON] et al.