﻿ 基于加权水平精度因子的伪卫星基站选择方法
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 大地测量与地球动力学  2019, Vol. 39 Issue (10): 1070-1075  DOI: 10.14075/j.jgg.2019.10.016

### 引用本文

XUE Xiaofeng, Wang Ling. Pseudolite's Station Selection Method Based on Weighted Horizontal Dilution of Precision[J]. Journal of Geodesy and Geodynamics, 2019, 39(10): 1070-1075.

### 文章历史

1. 湖南大学电气与信息工程学院，长沙市麓山南路2号，410082

1 加权精度因子简介 1.1 精度因子

 ${\mathop{\rm cov}} ({\rm{d}}x) = {\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{H}}} \right)^{ - 1}}\mathit{\sigma }_{{\rm{UERE}}}^2 = \mathit{\boldsymbol{D}}\mathit{\sigma }_{{\rm{UERE}}}^2$ (1)
 $\mathit{\boldsymbol{H}} = \left[ {\begin{array}{*{20}{c}} {a_x^1}&{a_y^1}&{a_z^1}&1\\ {a_x^2}&{a_y^2}&{a_z^1}&1\\ \vdots & \vdots & \vdots & \vdots \\ {a_x^n}&{a_y^n}&{a_z^n}&1 \end{array}} \right]$ (2)

 ${\rm{GDOP}} = \sqrt {{D_{11}} + {D_{22}} + {D_{33}} + {D_{44}}}$ (3)
 ${\rm{PDOP}} = \sqrt {{D_{11}} + {D_{22}} + {D_{33}}}$ (4)
 ${\rm{HDOP}} = \sqrt {{D_{11}} + {D_{22}}}$ (5)

1.2 加权精度因子

 ${\mathop{\rm cov}} ({\rm{d}}\rho ) = {\mathit{\boldsymbol{W}}_n}\mathit{\sigma }_{{\rm{UERE}}}^2$ (6)

 $\begin{array}{l} \;\;{\mathop{\rm cov}} ({\rm{d}}x) = {\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{W}}_n^{ - 1}\mathit{\boldsymbol{H}}} \right)^{ - 1}}{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{W}}_n^{ - 1}{\mathop{\rm cov}} ({\rm{d}}\mathit{\rho }) \cdot \\ \mathit{\boldsymbol{W}}_n^{ - 1}\mathit{\boldsymbol{H}}{\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{W}}_n^{ - 1}\mathit{\boldsymbol{H}}} \right)^{ - 1}} = {\left( {{\mathit{\boldsymbol{H}}^{\rm{T}}}\mathit{\boldsymbol{W}}_n^{ - 1}\mathit{\boldsymbol{H}}} \right)^{ - 1}}\mathit{\sigma }_{{\rm{UERE}}}^2 = \mathit{\boldsymbol{Q}}\mathit{\sigma }_{{\rm{UERE}}}^2 \end{array}$ (7)

 ${{\rm{WGDOP}} = \sqrt {{Q_{11}} + {Q_{22}} + {Q_{33}} + {Q_{41}}} }$ (8)
 ${{\rm{WPDOP }} = \sqrt {{Q_{11}} + {Q_{22}} + {Q_{33}}} }$ (9)
 ${{\rm{ WHDOP }} = \sqrt {{Q_{11}} + {Q_{22}}} }$ (10)

2 伪卫星系统加权矩阵

2.1 基站测距误差

 ${\mathit{\sigma }_{th}} = \sqrt {\frac{{{B_n}D}}{{2C/{N_0}}}}$ (11)

 $\frac{{\mathit{\sigma }_i^2}}{{\mathit{\sigma }_{1{\rm{m}}}^2}} = \frac{{\frac{{{B_n}D}}{{2{C_i}/{N_0}}}}}{{\frac{{{B_n}D}}{{2{C_{1{\rm{m}}}}/{N_0}}}}} = \frac{{{C_{1{\rm{m}}}}}}{{{C_i}}}$ (12)

 ${C_R}(d) = \frac{{{C_T}{G_T}{G_R}{\lambda ^2}}}{{{{(4{\rm{ \mathsf{ π} }})}^2}{d^2}L}}$ (13)

 $\sigma _i^2 = d_i^2\sigma _{1{\rm{m}}}^2$ (14)

2.2 加权矩阵

 ${\mathop{\rm cov}} ({\rm{d}}\mathit{\rho }) = {\mathop{\rm diag}\nolimits} \left[ {\begin{array}{*{20}{c}} {\mathit{\sigma }_1^2}&{\mathit{\sigma }_2^2}& \cdots &{\mathit{\sigma }_n^2} \end{array}} \right]$ (15)

 ${\mathop{\rm cov}} ({\rm{d}}x) = {\mathop{\rm diag}\nolimits} \left[ {d_1^2\quad d_2^2\quad \cdots \quad d_n^2} \right]\mathit{\sigma }_{1{\rm{m}}}^2$ (16)

 ${\mathit{\boldsymbol{W}}_n} = {\mathop{\rm diag}\nolimits} \left[ {\begin{array}{*{20}{l}} {d_1^2}&{d_2^2}& \cdots &{d_n^2} \end{array}} \right]$ (17)
3 基于基站对WHDOP贡献选站法 3.1 算法简介

 $\mathit{\boldsymbol{H}}_n^{\rm{T}}{\mathit{\boldsymbol{W}}_n}{\mathit{\boldsymbol{H}}_n} = \mathit{\boldsymbol{H}}_{n - 1}^{\rm{T}}{\mathit{\boldsymbol{W}}_{n - 1}}{\mathit{\boldsymbol{H}}_{n - 1}} + {\mathit{\boldsymbol{h}}^{\rm{T}}}\mathit{\boldsymbol{wh}}$ (18)

 $\begin{array}{l} {\mathit{\boldsymbol{A}}_{n - 1}} = {\left( {\mathit{\boldsymbol{H}}_{n - 1}^{\rm{T}}{\mathit{\boldsymbol{W}}_{n - 1}}{\mathit{\boldsymbol{H}}_{n - 1}}} \right)^{ - 1}} = \left( {\mathit{\boldsymbol{H}}_n^{\rm{T}}{\mathit{\boldsymbol{W}}_n}{\mathit{\boldsymbol{H}}_n} - } \right.\\ \;\;\;{\left. {{\mathit{\boldsymbol{h}}^{\rm{T}}}\mathit{\boldsymbol{wh}}} \right)^{ - 1}} = {\mathit{\boldsymbol{A}}_n} + \mathit{\boldsymbol{A}}_n^{\rm{T}}h\mathit{S}\mathit{\boldsymbol{h}}{\mathit{\boldsymbol{A}}_n} = {\mathit{\boldsymbol{A}}_n} + \mathit{\boldsymbol{E}} \end{array}$ (19)

 $\begin{array}{l} {\rm{WHDOP}}_{n - 1}^2 = {\mathit{\boldsymbol{A}}_{{n_{11}}}} + {\mathit{\boldsymbol{A}}_{{n_{22}}}} + {\mathit{\boldsymbol{E}}_{11}} + {\mathit{\boldsymbol{E}}_{22}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{WHDOP}}_n^2 + {\mathit{\boldsymbol{E}}_{11}} + {\mathit{\boldsymbol{E}}_{22}} \end{array}$ (20)

 $\Delta {\mathit{\boldsymbol{A}}_n} = {\mathit{\boldsymbol{E}}_{11}} + {\mathit{\boldsymbol{E}}_{22}}$ (21)

1) 根据接收机通道数量和用户对定位精度的要求确定选择基站的数目；

2) 计算每个基站对WHDOP的贡献值ΔAn

3) 确定ΔAn的最小值，并去掉其对应的基站；

4) 对剩余基站执行步骤2)~3)，直到剩余基站数目等于m时停止。

3.2 算法分析

4 仿真与分析 4.1 WHDOP选站性能仿真

 图 1 运动轨迹、定位误差频数分布 Fig. 1 The graphic of movement path, frequency distribution histogram of position error

4.2 基站对WHDOP贡献选站法仿真

 图 2 算法2与算法1的WHDOP值与差值曲线 Fig. 2 The WHDOP values and difference curves of algoeithm 2 and algorithm 1

 图 3 算法1与算法2基站选择耗时曲线 Fig. 3 Selection time curves of algoeithm 2 and algorithm 1
5 结语

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Pseudolite's Station Selection Method Based on Weighted Horizontal Dilution of Precision
XUE Xiaofeng1     Wang Ling1
1. College of Electrical and Information Engineering, Hunan University, 2 South-Lushan Road, Changsha 410082, China
Abstract: We propose a station selection method for pseudolite system based on weighted horizontal dilution of precision. This method gives different weights to each station according to the relative position of station and receiver, reducing the possibility that stationsthathave large measurement error areselected to participate in positioning. We analyze the mathematical models of pseudo-rangemeasurementerror, and the WHDOP expression is analyzed in detail. Station selection based on the contribution of WHDOP is designed. The simulations reveal that the method can select stations efficiently and accurately, thereby reducing the calculation amount of the receiver and improving the positioning accuracy of the system.
Key words: pseudolite; plans of PL-only; region location; precision dilution; station selection