# GPS/BDS RTK Positioning Based on Equivalence Principle Using Multiple Reference Stations
## 5 Conclusions  
In this paper, we present a new algorithm of MBRTK based on the equivalence principle for precise kinematic positioning. First, equivalent DD observation equations using SD observations are obtained through the equivalent transform to eliminate the receiver clock errors. Second, considering the redundancy of state information, a modified Kalman filter with parameter constraints is proposed to estimate the position-velocity states and equivalent DD ambiguities. The most outstanding feature of the MBRTK is that it has a rigorous model to solve the correlative characteristics of differenced observations for multiple reference receivers, as well as high positioning accuracy and availability could be achieved.  
From the experiment and analysis, the following conclusions can be obtained:  
1. For the static experiment, the MBRTK has better positioning accuracy than SBRTK. The MBRTK results with two and three reference receivers have the improvement of positioning accuracy by approximately (14%, 13%, and 14%), and (21%, 17%, and 19%) in N, E, and U components, respectively, validating the feasibility and reliability of the proposed MBRTK.
2. For the kinematic experiment, combined GPS/BDS positioning for MBRTK increases the number of available satellites and redundancy of the positioning model. Compared with single GPS and single BDS positioning, the combined GPS/BDS positioning has the accuracy improvement by approximately (45%, 35%, and 27%) and (12%, 6%, and 19%) in N, E, and U components and the availability improvement by approximately 33% and 10%, respectively.
3. For the analysis of reference receivers, the MBRTK model with multiple reference receivers can significantly increase the redundancy and provide the smaller ADOP values in favor to improve the performance of AR in comparison to the SBRTK model. The positioning results with two and three reference receivers can provide the similar accuracy of 3.0 cm, 1.5 cm, and 4.5 cm level in N, E, and U components, which have the accuracy improvement by approximately (9%, 0%, and 6%) and (9%, 16%, and 16%) compared with two SBRTK results, respectively. Meanwhile, the MBRTK results with two and three reference receivers have the availability improvement by approximately 10% in the success rate.
4. Due to the different models adopted by IE software 8.60 and MBRTK, there exists systematic errors between them. Compared with the positioning results of IE software 8.60 with one reference receiver, the MBRTK can achieve the stable accuracy with the cm-level, and have a rigorous and reliable model to keep a high availability of kinematic positioning.  
Our future work will extend this algorithm to multi-frequency and multi-GNSS positioning considering the computational efficiency, and focus the distribution of multiple reference receivers to improve the precision, availability, and reliability of positioning results.  
[PERSON], [PERSON] and [PERSON] conceived and designed the experiments; [PERSON] performed the experiments, analyzed the data, and wrote the paper; [PERSON], [PERSON] and [PERSON] helped in the discussion and revision. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This study is financially supported by the National Natural Science Foundation of China (Grant No. 41874032, 41731069 and 41931076) and the National Key Research & Development Program of China (2016 YFB0501701).  
**Acknowledgments:** We are grateful to the anonymous reviewers and editors for their helpful and constructive suggestions, which significantly improved the paper quality. The authors would like to thank the Curtin GNSS Research Centre of Curtin University and the Center for Satellite Navigation and Remote Sensing of Shandong University, Weihai, for publicly sharing their GNSS data. Figure 7 is edited by using the Google Earth, which is also acknowledged.  
**Conflicts of Interest:** The authors declare no conflict of interest.