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Instantaneous BeiDou+GPS RTK positioning with high cut-off elevation angles

Abstract

As the Chinese BeiDou Navigation Satellite System (BDS) has become operational in the Asia-Pacific region, it is of importance to better understand as well as demonstrate the capabilities that a combination of BeiDou with GPS brings to positioning. In this contribution, a formal and empirical analysis is given of the single-epoch RTK positioning capabilities of such a combined system. This will be done for the single- and dual-frequency case, and in comparison with the BDS- and GPS-only performances. It will be shown that with the combined system, when more satellites are available, much larger than the customary cut-off elevations can be used. This is important, as such measurement set-up will significantly increase the GNSS applicability in constrained environments, such as e.g. in urban canyons or when low-elevation multipath is present.

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Acknowledgments

This work has been executed in the framework of the Positioning Program of the Cooperative Research Centre for Spatial Information (CRC–SI). The first author is the recipient of an Australian Research Council (ARC) Federation Fellowship (Project number FF0883188). All this support is gratefully acknowledged.

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Correspondence to P. J. G. Teunissen.

Appendix

Appendix

Proof of Theorem (Combined-system ADOP) The ambiguity variance matrix of the combined system can be expressed in the single-system variance matrices as

$$\begin{aligned} \begin{array}{l} Q_{\hat{a}\hat{a}}= \left[ \begin{array}{ll} Q_{\hat{a}_{G}\hat{a}_{G}} &{} 0\\ 0 &{} Q_{\hat{a}_{B}\hat{a}_{B}} \end{array} \right] - \left[ \begin{array}{c} Q_{\hat{a}_{G}\hat{b}_{G}} \\ Q_{\hat{a}_{B}\hat{b}_{B}} \end{array} \right] [Q_{\hat{b}_{G}\hat{b}_{G}}\\ \qquad \qquad +\,Q_{\hat{b}_{B}\hat{b}_{B}} ]^{-1} \left[ \begin{array}{c} Q_{\hat{a}_{G}\hat{b}_{G}} \\ Q_{\hat{a}_{B}\hat{b}_{B}} \end{array} \right] ^{T} \end{array} \end{aligned}$$

Upon taking the determinant, we get

$$\begin{aligned} \begin{array}{l} |Q_{\hat{a}\hat{a}}|=|Q_{\hat{a}_{G}\hat{a}_{G}}||Q_{\hat{a}_{B}\hat{a}_{B}}| |I-[Q_{\hat{b}_{G}\hat{a}_{G}}Q_{\hat{a}_{G}\hat{a}_{G}}^{-1}Q_{\hat{a}_{G}\hat{b}_{G}}\\ \qquad \qquad \,+\,Q_{\hat{b}_{B}\hat{a}_{B}}Q_{\hat{a}_{B}\hat{a}_{B}}^{-1}Q_{\hat{a}_{B}\hat{b}_{B}}][Q_{\hat{b}_{G}\hat{b}_{G}}+Q_{\hat{b}_{B}\hat{b}_{B}} ]^{-1}| \end{array} \end{aligned}$$

where we made use of the determinant property \(|I_{m}-BC|=|I_{n}-CB|\) for \(m \times n\) matrices \(B\) and \(C^{T}\). With the use of

$$\begin{aligned} Q_{\check{b}_{*}\check{b}_{*}}=Q_{\hat{b}_{*}\hat{b}_{*}}-Q_{\hat{b}_{*}\hat{a}_{*}}Q_{\hat{a}_{*}\hat{a}_{*}}^{-1}Q_{\hat{a}_{*}\hat{b}_{*}} \end{aligned}$$

we can then finally write

$$\begin{aligned} |Q_{\hat{a}\hat{a}}|=|Q_{\hat{a}_{G}\hat{a}_{G}}||Q_{\hat{a}_{B}\hat{a}_{B}}| |\frac{Q_{\check{b}_{G}\check{b}_{G}}+Q_{\check{b}_{B}\check{b}_{B}}}{Q_{\hat{b}_{G}\hat{b}_{G}}+Q_{\hat{b}_{B}\hat{b}_{B}}}| \end{aligned}$$
(18)

from which the expression (12) for the combined ADOP follows. \(\square \)

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Teunissen, P.J.G., Odolinski, R. & Odijk, D. Instantaneous BeiDou+GPS RTK positioning with high cut-off elevation angles. J Geod 88, 335–350 (2014). https://doi.org/10.1007/s00190-013-0686-4

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Keywords

  • BeiDou (BDS)
  • GPS
  • Multi-GNSS
  • Integer ambiguity resolution
  • Real time kinematic (RTK) Positioning
  • Cut-off elevation