﻿ BDS中长基线三频RTK算法研究
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 大地测量与地球动力学  2022, Vol. 42 Issue (1): 5-8, 20  DOI: 10.14075/j.jgg.2022.01.002

### 引用本文

GAO Meng, WANG Shunshun, WANG Cao, et al. Research on the Algorithm of Triple-Frequency BDS RTK between Middle-Range Baseline[J]. Journal of Geodesy and Geodynamics, 2022, 42(1): 5-8, 20.

### Foundation support

National Natural Science Foundation of China, No.41904037, 42074012, 42030109; Key Research and Development Program of Liaoning Province, No, 2020JH2/10100044; Scientific Research Project of the Education Department of Liaoning Province, No.LJ2019QL008.

### 第一作者简介

GAO Meng, PhD, lecturer, majors in GNSS high precision positioning, E-mail: gaomeng512@163.com.

### 文章历史

BDS中长基线三频RTK算法研究

1. 辽宁工程技术大学测绘与地理科学学院, 辽宁省阜新市玉龙路88号，123000

1 BDS中长基线三频整周模糊度确定 1.1 双差超宽巷与宽巷整周模糊度解算

 $\begin{array}{l} \Delta \nabla N_{uv, (0, - 1, 1)}^{pq} = \left( {\frac{{\Delta \nabla \mathit{\Phi }_{uv, 3}^{pq}}}{{{\lambda _3}}} - \frac{{\left. {\Delta \nabla \mathit{\Phi }_{uv, 2}^{pq}} \right)}}{{{\lambda _2}}}} \right) - \\ \;\;\;\;\;\frac{{{f_3} - {f_2}}}{{{f_3} + {f_2}}} \cdot \left( {\frac{{\Delta \nabla P_{uv, 3}^{pq}}}{{{\lambda _3}}} + \frac{{\Delta \nabla P_{uv, 2}^{pq}}}{{{\lambda _2}}}} \right) \end{array}$ (1)

 $\Delta \nabla \bar N_{uv, (0, - 1, 1)}^{pq} = \frac{{\sum\limits_{{\rm{epoch = 1}}}^n {\Delta \nabla N_{uv, (0, - 1, 1)}^{pq, e}} }}{n}$ (2)

BDS的B2-B3超宽巷组合和B1-B3宽巷组合双差载波相位观测方程为：

 $\begin{array}{l} \Delta \nabla \mathit{\Phi }_{uv, (0, - 1, 1)}^{pq} = \Delta \nabla \rho _{uv}^{pq} + \Delta \nabla T_{uv}^{pq} - {\mu _{(0, - 1, 1)}} \cdot \\ \frac{{\Delta \nabla \kappa }}{{f_1^2}} - {\lambda _{(0, - 1, 1)}} \cdot \Delta \nabla \mathit{N}_{uv, (0, - 1, 1)}^{pq} + \Delta \nabla {\varepsilon _{\mathit{\Phi , }(0, - 1, 1)}} \end{array}$ (3)
 $\begin{array}{l} \Delta \nabla \mathit{\Phi }_{uv, (0, - 1, 1)}^{pq} = \Delta \nabla \rho _{uv}^{pq} + \Delta \nabla T_{uv}^{pq} - {\mu _{(1, 0, - 1)}} \cdot \\ \frac{{\Delta \nabla \kappa }}{{f_1^2}} - {\lambda _{(1, 0, - 1)}} \cdot \Delta \nabla \mathit{N}_{uv, (1, 0, - 1)}^{pq} + \Delta \nabla {\varepsilon _{\mathit{\Phi , }(1, 0, - 1)}} \end{array}$ (4)

 $\begin{array}{l} \Delta \nabla \hat N_{uv, (1, 0, - 1)}^{pq} = \frac{{\Delta \nabla \mathit{\Phi }_{uv, (0, - 1, 1)}^{pq} - \Delta \nabla \mathit{\Phi }_{uw, (1, 0, - 1)}^{pq}}}{{{\lambda _{(1, 0, - 1)}}}} + \\ \;\;\frac{{{\lambda _{(0, - 1, 1)}}}}{{{\lambda _{(1, 0, - 1)}}}} \cdot \Delta \nabla N_{uv, (0, - 1, 1)}^{pq} + \frac{{{\mu _{(0, - 1, 1)}} - {\mu _{(1, 0, - 1)}}}}{{{\lambda _{(1, 0, - 1)}}}} \cdot \\ \;\;\;\;\;\;\;\;\frac{{\Delta \nabla \kappa }}{{f_1^2}} - \frac{{\Delta \nabla {\varepsilon _{\mathit{\Phi }, (0, - 1, 1)}} - \Delta \nabla {\varepsilon _{\mathit{\Phi }, (1, 0, - 1)}}}}{{{\lambda _{(1, 0, - 1)}}}} \approx \\ \;\;\;\;\;\;\;\frac{{\Delta \nabla \mathit{\Phi }_{uv, (0, - 1, 1)}^{pq} - \Delta \nabla \mathit{\Phi }_{uv, (1, 0, - 1)}^{pq}}}{{{\lambda _{(1, 0, - 1)}}}} + \frac{{{\lambda _{(0, - 1, 1)}}}}{{{\lambda _{(1, 0, - 1)}}}} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\Delta \nabla N_{wv, (0, - 1, 1)}^{pq} \end{array}$ (5)

1.2 双差窄巷整周模糊度解算

BDS的B1-B2和B1-B3无电离层组合双差伪距和载波相位观测方程为：

 $\begin{array}{l} \Delta \nabla P_{uv, {\rm{I}}{{\rm{F}}_{12}}}^{pq} = \Delta \nabla \rho _{uv}^{pq} + \Delta \nabla T_{{\rm{dry }}, uv}^{pq} + {\rm{MF}}_u^{pq} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{\rm{RZT}}{{\rm{D}}_{{\rm{wet }}, uv}} + \Delta \nabla {\varepsilon _{P, {\rm{I}}{{\rm{F}}_{12}}}} \end{array}$ (6)
 $\begin{array}{l} \;\;\;\;\;\;\;\;\Delta \nabla \mathit{\Phi }_{uv,{\rm{I}}{{\rm{F}}_{12}}}^{pq} = \Delta \nabla \rho _{uv}^{pq} + \Delta \nabla T_{{\rm{dry}},uv}^{pq} + {\rm{MF}}_u^{pq} \cdot\\ {\rm{RZT}}{{\rm{D}}_{{\rm{wet }}, uv}} - {\lambda _{(2.487\;2, - 1.487\;2, 0)}} \cdot \Delta \nabla N_{uv, 1}^{pq} - \frac{{{f_2}}}{{{f_1} + {f_2}}} \cdot \\ \;\;\;\;\;\;\;\;{\lambda _{(1, - 1, 0)}} \cdot \Delta \nabla N_{uv, (1, - 1, 0)}^{pq} + \Delta \nabla {\varepsilon _{\mathit{\Phi }, {\rm{I}}{{\rm{F}}_{12}}}} \end{array}$ (7)
 $\begin{array}{l} \;\;\;\;\;\;\Delta \nabla \mathit{\Phi }_{uv,{\rm{I}}{{\rm{F}}_{13}}}^{pq} = \Delta \nabla \rho _{uv}^{pq} + \Delta \nabla T_{{\rm{dry}},uv}^{pq} + {\rm{MF}}_u^{pq} \cdot\\ \left. {{\rm{RZT}}{{\rm{D}}_{{\rm{wet }}, uv}} - {\lambda _{(2, 943\;7, 0, - 1, 943\;7}}} \right) \cdot \Delta \nabla N_{uv, 1}^{pq} - \frac{{{f_3}}}{{{f_1} + {f_3}}} \cdot \\ \;\;\;\;\;\;\;\;\;\;\;\;{\lambda _{(1, 0, - 1)}} \cdot \Delta \nabla N_{uv, (1, 0, - 1)}^{pq} + \Delta \nabla {\varepsilon _{\mathit{\Phi }, I{{\rm{F}}_{13}}}} \end{array}$ (8)

 $\mathit{\boldsymbol{L}}\left( i \right) = \mathit{\boldsymbol{A}}\left( i \right)\mathit{\boldsymbol{B}}\left( i \right)$ (9)

 $\mathit{\boldsymbol{A}}(i) = \left[ {\begin{array}{*{20}{c}} {\mathit{\boldsymbol{C}}(i)}&{{\bf{MF}}(i)}&{{{\bf{0}}_s}}&{}\\ {\mathit{\boldsymbol{C}}(i)}&{{\bf{MF}}(i)}&{{\lambda _{(2.487\;2, - 1.487\;2, 0)}} \cdot {\mathit{\boldsymbol{I}}_s}}&{}\\ {\mathit{\boldsymbol{C}}(i)}&{{\bf{MF}}(i)}&{{\lambda _{(2, 943\;7, 0, - 1, 943\;7)}} \cdot {\mathit{\boldsymbol{I}}_s}}&{} \end{array}} \right]$
 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{B}}(i) = }\\ {{{\left( {\begin{array}{*{20}{l}} \mathit{\boldsymbol{X}}&{{\rm{RZT}}{{\rm{D}}_{{\rm{wet }}, uw}}}&{\Delta \nabla N_{uv, 1}^1}&{\Delta \nabla N_{uv, 1}^2}& \cdots &{\Delta \nabla N_{uv, 1}^s} \end{array}} \right)}^{\rm{T}}}} \end{array}$
 $\begin{array}{*{20}{c}} {\mathit{\boldsymbol{X}} = {{\left( {\begin{array}{*{20}{l}} x&y&z \end{array}} \right)}^{\rm{T}}}}\\ {\mathit{\boldsymbol{L}}(i) = {{\left( {\begin{array}{*{20}{l}} {l_1^1}& \cdots &{l_1^s}&{l_2^1}& \cdots &{l_2^s}&{l_3^1}& \cdots &{l_3^s} \end{array}} \right)}^{\rm{T}}}} \end{array}$
 ${\bf{MF}}(i) = {\left( {\begin{array}{*{20}{l}} {{\rm{MF}}_u^1}&{{\rm{MF}}_u^2}& \cdots &{{\rm{MF}}_u^s} \end{array}} \right)^{\rm{T}}}$
 $\begin{array}{l} \;\;l_1^s = \Delta \nabla P_{uw, {\rm{I}}{{\rm{F}}_{12}}}^s - \Delta \nabla \rho _{uv}^s - \Delta \nabla T_{{\rm{dry}}, uv}^{pq}\\ l_2^s = \Delta \nabla \mathit{\Phi }_{uv, {\rm{I}}{{\rm{F}}_{12}}}^s - \Delta \nabla \rho _{uv}^s - \Delta \nabla T_{{\rm{dry}}, uv}^s + \\ \;\;\;\;\;\frac{{{f_2}}}{{{f_1} + {f_2}}} \cdot {\lambda _{(1, - 1, 0)}} \cdot \Delta \nabla N_{uv, (1, - 1, 0)}^s\\ l_3^s = \Delta \nabla \mathit{\Phi }_{uv, {\rm{I}}{{\rm{F}}_{13}}}^s - \Delta \nabla \rho _{uv}^s - \Delta \nabla T_{{\rm{dry}}, uv}^s + \\ \;\;\;\;\;\;\frac{{{f_3}}}{{{f_1} + {f_3}}} \cdot {\lambda _{(1, 0, - 1)}} \cdot \Delta \nabla N_{uv, (1, 0, - 1)}^s \end{array}$

2 算例与分析

 图 1 B2-B3双差超宽巷整周模糊度小数部分 Fig. 1 Fractional part of double differenced extra-wide-lane integer ambiguity for B2-B3

B2-B3双差超宽巷整周模糊度准确固定后，利用B2-B3超宽巷组合观测值和B1-B3宽巷组合观测值所受电离层延迟误差较接近的特点解算B1-B3宽巷整周模糊度。图 2为观测时段内卫星对应的B1-B3双差宽巷整周模糊度小数部分，可以看出，基线1和基线2的B1-B3双差宽巷整周模糊度小数部分绝大多数在0.5周以内。取平均值可以削弱残余电离层延迟误差对整周模糊度的影响，从而较为准确地固定B1-B3双差宽巷整周模糊度。

 图 2 B1-B3双差宽巷整周模糊度小数部分 Fig. 2 Fractional part of double differenced wide-lane integer ambiguity for B1-B3

 图 3 双差窄巷整周模糊度小数部分 Fig. 3 Fractional part of double differenced narrow-lane integer ambiguity

 图 4 RTK定位误差 Fig. 4 RTK positioning error

3 结语

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Research on the Algorithm of Triple-Frequency BDS RTK between Middle-Range Baseline
GAO Meng1     WANG Shunshun1     WANG Cao1     ZHU Huizhong1     XU Aigong1
1. Faculty of Geomatics, Liaoning Technical University, 88 Yulong Road, Fuxin 123000, China
Abstract: As integer ambiguity is affected by atmospheric delay error, it is difficult to correctly fix ambiguity between BDS middle-range baseline. We propose the algorithm of triple-frequency ambiguity resolution between BDS middle-range baseline. We use the combination of MW to determine B2-B3 extra-wide-lane integer ambiguity, and then solve the B1-B3 wide-lane integer ambiguity using the closer characteristic of ionospheric delay error between B2-B3 extra-wide-lane and B1-B3 wide-lane combination observation. We use the ionosphere-free combination observation to estimate narrow-lane integer ambiguity and relative zenith tropospheric delay error, and finally the real-time dynamic positioning is realized. We test this algorithm by the measured BDS middle-range baseline. The results indicate that the method can fix the triple-frequency carrier phase ambiguity between middle-range baseline correctly and effectively, and positioning accuracy to the centimetre level can be achieved.
Key words: Beidou navigation satellite system; triple-frequency; real-time kinematic(RTK); integer ambiguity; ionospheric delay error