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Improving reliability and efficiency of RTK ambiguity resolution with reference antenna array: BDS + GPS analysis and test

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Abstract

Rapid and reliable ambiguity resolution is the key to high-precision global navigation satellite system-based applications. To improve the efficiency and reliability of ambiguity resolution, one effective method is equipping the reference station with an antenna array of known geometry instead of a single antenna. In this contribution, the benefits of reference antenna array-aided real-time kinematic (RTK) positioning are investigated. The mathematical relations between the number of reference antennas and the float baseline and ambiguity solutions are explored, and the closed-form formula of ambiguity dilution of precision (ADOP) is further presented. It is demonstrated that the maximum decrease in ADOP or the improvement in precision of float solutions is approximately 29.29% with the increase in the number of reference antennas. Then, we analyze the impact of errors (noises or biases) on the float baseline and ambiguity solutions. Finally, the performance of the array-aided RTK is evaluated with raw BDS and GPS observations in terms of the ambiguity resolution success rate, false alarm, wrong detection alarm, and the time-to-first-fix (TTFF). It is demonstrated that the array-aided RTK can deliver improved ambiguity success rates with respect to the standard one-reference-antenna RTK, especially when only single-frequency, single-system observations from fewer satellites are available. Moreover, the array-aided RTK can provide much more robust ambiguity resolution in the presence of biases. And the performance of suppressing false alarm and wrong detection alarm is improved as well. Additionally, the TTFF with single-frequency observations is also significantly shortened. One order of magnitude improvements in TTFF are achieved for most of the cases from the standard one-reference-antenna solutions to the three-reference-antenna solutions. The results confirm that the array-aided RTK ensures more reliable and efficient ambiguity resolution.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (Grant No. 61601506) and the Fundamental Research Funds for the Central University, China University of Geosciences (Wuhan) (No. G1323541876).

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Authors and Affiliations

  1. Information and Navigation College, Air Force Engineering University, Xi’an, China

    Shaoshi Wu, Xiubin Zhao, Liang Zhang & Chunlei Pang

  2. Faculty of Information Engineering, China University of Geosciences (Wuhan), Wuhan, China

    Mingkui Wu

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  1. Shaoshi Wu
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  2. Xiubin Zhao
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  3. Liang Zhang
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Correspondence to Liang Zhang.

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Appendix

Appendix

Proof of (12)

According to (10) we have

$$ \begin{aligned} \varvec{Q}_{{\hat{\varvec{z}}_{1} \hat{\varvec{z}}_{1} }} & = \left( {\bar{\varvec{\varLambda }}^{\rm T} \left( {\varvec{Q}_{\varvec{\varPhi}} + \bar{\varvec{H}}\varvec{Q}_{{\hat{\varvec{b}}_{1} \hat{\varvec{b}}_{1} }} \bar{\varvec{H}}^{\rm T} } \right)^{ - 1} \bar{\varvec{\varLambda }}} \right)^{ - 1} \\ & {\kern 1pt} = \left( {\bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }} - \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}\left( {\bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \bar{\varvec{H}} + \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}} \right)^{ - 1} \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }}} \right)^{ - 1} \\ \end{aligned} $$
(20)

With \( \bar{\varvec{\varLambda }} = \varvec{e}_{n} \otimes\varvec{\varLambda} \), \( \bar{\varvec{H}} = \varvec{e}_{n} \otimes \varvec{G} \) and (8), we get the following transactional terms

$$ \begin{aligned} & \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }} = e \cdot \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}\\ & \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}} = e \cdot \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \varvec{G} \\ & \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \bar{\varvec{H}} = e \cdot \varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} \\ & \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}} = e \cdot \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G} \\ & \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }} = e \cdot \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}\\ \end{aligned} $$
(21)

where \( e\varvec{ = e}_{n}^{\rm T} \left( {\frac{1}{2}\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} \varvec{e}_{n} = \frac{2n}{n + 1} \).Taking (21) into (20) gives

$$ \begin{aligned} \varvec{Q}_{{\hat{\varvec{z}}_{1} \hat{\varvec{z}}_{1} }} & = \frac{1}{e} \cdot\varvec{\varLambda}^{ - 1} \left( {\varvec{Q}_{\varvec{\phi}}^{ - 1} - \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} + \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}} \right)^{ - 1} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} } \right)^{ - 1}\varvec{\varLambda}^{ - 1} \\ & {\kern 1pt} = \frac{1}{e} \cdot\varvec{\varLambda}^{ - 1} \left( {\varvec{Q}_{\varvec{\phi}} + \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G}} \right)^{ - 1} \varvec{G}^{\rm T} } \right)\varvec{\varLambda}^{ - 1} \\ & = \frac{n + 1}{2n}\varvec{Q}_{{\varvec{\hat{z}\hat{z}}}} \\ \end{aligned} $$

End of proof. □

Proof of (15) and (16)

With the errors in the observations, the float solutions in (3) are expressed as

$$ \begin{aligned} & \hat{\varvec{b}} = \varvec{Q}_{{\varvec{\hat{b}\hat{b}}}} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \left( {\varvec{p} +\varvec{\varepsilon}_{{\varvec{p},m}} -\varvec{\varepsilon}_{{\varvec{p},r}} } \right) \\ & \hat{\varvec{z}} =\varvec{\varLambda}^{ - 1} \left( {\varvec{\phi}- \varvec{G\hat{b}} +\varvec{\varepsilon}_{{\varvec{\phi},m}} -\varvec{\varepsilon}_{{\varvec{\phi},r}} } \right) \\ \end{aligned} $$

Then the impacts of the errors on the float solutions as shown in (15) can be immediately obtained.For the array-aided case, the errors in float solutions caused by the errors in observations are

$$ \begin{aligned} & \Delta \hat{\varvec{b}}_{1} = \varvec{Q}_{{\hat{\varvec{b}}_{1} \hat{\varvec{b}}_{1} }} \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \varvec{E}_{\varvec{P}} \\ & \Delta \hat{\varvec{z}}_{1} = \varvec{Q}_{{\hat{\varvec{z}}_{1} \hat{\varvec{z}}_{1} }} \bar{\varvec{\varLambda }}^{\rm T} \left( {\varvec{Q}_{\varvec{\varPhi}} + \bar{\varvec{H}}\varvec{Q}_{{\hat{\varvec{b}}_{1} \hat{\varvec{b}}_{1} }} \bar{\varvec{H}}^{\rm T} } \right)^{ - 1} \left( {\varvec{E}_{\varvec{\varPhi}} - \bar{\varvec{H}}\Delta \hat{\varvec{b}}_{1} } \right) \\ \end{aligned} $$
(22)

Taking \( \varvec{Q}_{{\hat{\varvec{b}}_{1} \hat{\varvec{b}}_{1} }} = \left( {\bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \bar{\varvec{H}}} \right)^{ - 1} \), \( \bar{\varvec{H}} = \varvec{e}_{n} \otimes \varvec{G} \), and (8) into (22), the first equation of (22) is simplified as \( \Delta \hat{\varvec{b}}_{1} = \left( {\frac{{\varvec{e}_{n}^{\rm T} \left( {\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} }}{{\varvec{e}_{n}^{\rm T} \left( {\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} \varvec{e}_{n} }} \otimes \left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G}} \right)^{ - 1} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} } \right)\varvec{E}_{\varvec{P}} \). With \( \varvec{e}_{n}^{\rm T} \left( {\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} = \frac{1}{n + 1}\varvec{e}_{n}^{\rm T} \), \( \varvec{e}_{n}^{\rm T} \left( {\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} \varvec{e}_{n} = \frac{n}{n + 1} \), \( \varvec{Q}_{{\varvec{\hat{b}\hat{b}}}} = \left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G}} \right)^{ - 1} \), and \( \varvec{E}_{\varvec{P}} = \left[ {\begin{array}{*{20}c} {\varvec{\varepsilon}_{{\varvec{p},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{p},r_{ 1} }}^{\rm T} } & {\varvec{\varepsilon}_{{\varvec{p},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{p},r_{2} }}^{\rm T} } & \cdots & {\varvec{\varepsilon}_{{\varvec{p},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{p},r_{n} }}^{\rm T} } \\ \end{array} } \right]^{\rm T} \), we get

$$ \begin{aligned} \Delta \hat{\varvec{b}}_{1} & = \frac{1}{n}\left( {\varvec{e}_{n}^{\rm T} \otimes \left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G}} \right)\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} } \right)\varvec{E}_{\varvec{P}} \\ & = \varvec{Q}_{{\varvec{\hat{b}\hat{b}}}} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \left( {\varvec{\varepsilon}_{{\varvec{p},m}} - \frac{1}{n}\sum\limits_{i = 1}^{n} {\varvec{\varepsilon}_{{\varvec{p},r_{i} }} } } \right) \\ \end{aligned} $$

For the second equation of (22), since \( \varvec{Q}_{{\hat{\varvec{z}}_{1} \hat{\varvec{z}}_{1} }} = \left( {\bar{\varvec{\varLambda }}^{\rm T} \left( {\varvec{Q}_{\varvec{\varPhi}} + \bar{\varvec{H}}\varvec{Q}_{{\hat{\varvec{b}}_{1} \hat{\varvec{b}}_{1} }} \bar{\varvec{H}}^{\rm T} } \right)^{ - 1} \bar{\varvec{\varLambda }}} \right)^{ - 1} \), we have

$$ \begin{aligned} \Delta \hat{\varvec{z}}_{1} & = \left( {\bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }} - \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}\left( {\bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \bar{\varvec{H}} + \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}} \right)^{ - 1} \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{\varLambda }}} \right)^{ - 1} \\ & \quad \times \left( {\bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} - \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}\left( {\bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{P}}^{ - 1} \bar{\varvec{H}} + \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} \bar{\varvec{H}}} \right)^{ - 1} \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} } \right)\left( {\varvec{E}_{\varvec{\varPhi}} - \bar{\varvec{H}}\Delta \hat{\varvec{b}}_{1} } \right) \\ \end{aligned} $$
(23)

With (21) as well as \( \bar{\varvec{\varLambda }}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} = \varvec{e}_{n}^{\rm T} \left( {\frac{1}{2}\varvec{D}_{\text{A}} \varvec{D}_{\text{A}}^{\rm T} } \right)^{ - 1} \otimes \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} = \frac{2}{n + 1}\varvec{e}_{n}^{\rm T} \otimes \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \), \( \bar{\varvec{H}}^{\rm T} \varvec{Q}_{\varvec{\varPhi}}^{ - 1} = \frac{2}{n + 1}\varvec{e}_{n}^{\rm T} \otimes \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \), and \( \varvec{E}_{\varvec{\varPhi}} = \left[ {\begin{array}{*{20}c} {\varvec{\varepsilon}_{{\varvec{\phi},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{\phi},r_{ 1} }}^{\rm T} } & {\varvec{\varepsilon}_{{\varvec{\phi},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{\phi},r_{ 2} }}^{\rm T} } & \cdots & {\varvec{\varepsilon}_{{\varvec{\phi},m}}^{\rm T} -\varvec{\varepsilon}_{{\varvec{\phi},r_{n} }}^{\rm T} } \\ \end{array} } \right]^{\rm T} \), (23) can be simplified as

$$ \begin{aligned} \Delta \hat{\varvec{z}}_{1} & = \frac{1}{e}\left( {\varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}- \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} + \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}} \right)^{ - 1} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}} \right)^{ - 1} \\ & \quad \times \left( {\frac{2}{n + 1}\varvec{e}_{n}^{\rm T} \otimes \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} - \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} + \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}} \right)^{ - 1} \left( {\frac{2}{n + 1}\varvec{e}_{n}^{\rm T} \otimes \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} } \right)} \right)\left( {\varvec{E}_{\varvec{\varPhi}} - \left( {\varvec{e}_{n} \otimes \varvec{G}} \right)\Delta \hat{\varvec{b}}_{1} } \right) \\ & = \frac{1}{n}\left( {\varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}- \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} + \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}} \right)^{ - 1} \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1}\varvec{\varLambda}} \right)^{ - 1} \\ & \quad \times \left( {\varvec{e}_{n}^{\rm T} \otimes \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} - \varvec{\varLambda Q}_{\varvec{\phi}}^{ - 1} \varvec{G}\left( {\varvec{G}^{\rm T} \varvec{Q}_{\varvec{p}}^{ - 1} \varvec{G} + \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} \varvec{G}} \right)^{ - 1} \left( {\varvec{e}_{n}^{\rm T} \otimes \varvec{G}^{\rm T} \varvec{Q}_{\varvec{\phi}}^{ - 1} } \right)} \right)\varvec{E}_{\varvec{\varPhi}} -\varvec{\varLambda}^{ - 1} \varvec{G}\Delta \hat{\varvec{b}}_{1} \\ & = \frac{1}{n}\varvec{\varLambda}^{ - 1} \sum\limits_{i = 1}^{n} {\left( {\varvec{\varepsilon}_{{\varvec{\phi},m}} -\varvec{\varepsilon}_{{\varvec{\phi},r_{i} }} } \right)} -\varvec{\varLambda}^{ - 1} \varvec{G}\Delta \hat{\varvec{b}}_{1} \\ & =\varvec{\varLambda}^{ - 1} \left( {\varvec{\varepsilon}_{{\varvec{\phi},m}} - \frac{1}{n}\sum\limits_{i = 1}^{n} {\varvec{\varepsilon}_{{\varvec{\phi},r_{i} }} } - \varvec{G}\Delta \hat{\varvec{b}}_{1} } \right) \\ \end{aligned} $$

End of proof. □

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Wu, S., Zhao, X., Zhang, L. et al. Improving reliability and efficiency of RTK ambiguity resolution with reference antenna array: BDS + GPS analysis and test. J Geod 93, 1297–1311 (2019). https://doi.org/10.1007/s00190-019-01246-w

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