﻿ 低秩条件下双星TDOA和FDOA无源定位算法<sup>*</sup>
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A passive location algorithm based on TDOA and FDOA of dual-satellite in the condition of unfiled rank
ZHOU Longjian, LUO Jingqing
Electronic Engineering Institute of PLA, Hefei 230037, China
Received: 2016-10-18; Accepted: 2017-01-07; Published online: 2017-01-19 15:11
Foundation item: National Natural Science Foundation of China (60801044); Natural Science Fundamental Research Project of Shanxi Province of China (2013JQ8020)
Corresponding author. LUO Jingqing, E-mail:luojingqing001@126.com
Abstract: To solve the problem that the position vector of the first satellite, the position difference vector between dual satellites and the speed difference vector between dual satellites are coplanar when the fixed emitter with known altitude is located by dual satellites using time difference of arrival (TDOA) and frequency difference of arrival (FDOA), an analytic solution in three-dimensional spaces was proposed. The condition for no solution is only that the two position vectors and two speed vectors of dual satellites are collinear. The paper analyzes the two conditions of both coplanar but non-collinear and collinear. The analytic solutions of different conditions are given. The problem of location can be simplified into solving quadratic equation with one unknown when the position vectors of dual satellites are collinear, which alleviates the complexity of the problem solving and reduces the computational complexity. Besides, the positioning accuracy is enhanced in nadir point of dual satellites when the three vectors are coplanar. The algorithm was proved useful by simulation experiments.
Key words: time difference of arrival (TDOA)     frequency difference of arrival (FDOA)     passive location     analytic solution     dual-satellite

1 双星时频差定位原理

 图 1 双星时频差无源定位示意图 Fig. 1 Schematic of dual-satellite passive location using TDOA and FDOA
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2 定位解析解

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2.1 共面但不共线

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1) 若q3=0，即两卫星的位置矢量X1, X2共线，则可以直接利用式(16)，通过求解一元二次方程得到r1的值。

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2) 若q3≠0，联立式(8) 和式(16)，可得

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2.2 共线

1) 若矢量X1X2V1V2在同一直线上，此时无法观测。

2) 矢量X1X2V1V2在同一直线上，而X1X2V1V2不在同一直线上。

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3 性能分析 3.1 误差理论分析

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3.2 仿真分析

 图 2 两卫星位置矢量共线时定位误差分布 Fig. 2 Location error distribution when position vectors of dual satellites are collinear

 图 3 三矢量共面但不共线时定位误差分布 Fig. 3 Location error distribution when three vectors are co-planar but non-collinear

 图 4 三矢量不共面时定位误差分布 Fig. 4 Location error distribution when three vectors are not co-planar

4 结论

1) 给出了卫星1位置矢量、两卫星位置矢量差以及两卫星速度矢量差三者共面情况下的解析解，解决了三元高次非线性方程组在线性转化过程中由于信息丢失造成无法观测的问题。

2) 指出了双星TDOA和FDOA无源定位过程中无法观测情况，即当且仅当两卫星位置矢量和速度矢量四者共线时无法观测。

3) 当两卫星位置矢量共线时，可以将复杂的一元六次方程求解问题简化为一元二次方程求解，减小虚根数量，降低求解复杂度。

4) 双星无源定位时当卫星1位置矢量、两卫星位置矢量差以及两卫星速度矢量差三者共面时可以提高星下点某些特定区域定位精度。

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文章信息

ZHOU Longjian, LUO Jingqing

A passive location algorithm based on TDOA and FDOA of dual-satellite in the condition of unfiled rank

Journal of Beijing University of Aeronautics and Astronsutics, 2017, 43(10): 2040-2046
http://dx.doi.org/10.13700/j.bh.1001-5965.2016.0811