﻿﻿ ﻿ 改进M估计的抗多个粗差定位解算方法
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Robust Positioning Algorithm with Modified M-estimation for Multiple Outliers
TONG Haibo,ZHANG Guozhu
College of Electronic Science and Engineering,National University of Defense Technology,Changsha 410073,China
First author: TONG Haibo (1984—),male,PhD candidate,majors in GNSS receiver autonomous integrity monitoring (RAIM),robust positioning algorithms,weak signal acquisition and tracking technology.E-mail： hbo.tong@gmail.com
Abstract:The possibility of multiple outliers should not be neglected in measurements for the increment number of the navigation satellites,and the RAIM based on the single-outlier hypothesis cannot provide effective restraining against multiple outliers. The robust estimation has gained attention widely. A robust positioning algorithm is presented for resisting multiple outliers which make the traditional M-estimator ineffective. The robustness can be improved by the modified positioning algorithm. To handle the biased convergence problem,the robust estimate of the initial values is realized by modifying the S-estimation with available satellite number in real time. Positioning results of the actual GPS measurements show that the proposed method resists multiple outliers effectively.
Key words: global navigation satellite system     positioning     robust estimation     fault detection     M-estimation     S-estimation

1 引 言

2 传统M估计抗差定位的局限性

GPS定位解算时，初始值x0=[x0y0z0b0]T处线性化的观测方程后，得

M估计主要通过选择合适的函数ρ，自动为含粗差观测值分配小的加权值，从而达到抑制粗差的目的。根据观测噪声模型的不同，国内外学者对多种有效的ρ函数进行了深入的研究[1516]。从抑制幅值较大的粗差方面考虑，本文采用文献[16]中的双边加权函数

(1) 采用最小二乘计算结果作为初值。

(2) 根据式(4)计算伪距残差矢量并估计方差，再由式(9)得到等价权矩阵。

(3) 由M估计的迭代式(8)计算第k+1次估计结果，其中k=0,1,2,…。

(4) 返回步骤 (2)，直到max|xk+1xk|小于门限值。

3 改进后的M估计定位解算方法

GPS用户通常可见卫星数为7～12颗。当伪距观测值数量有限时，S估计的抗差性能会有所下降[18]，崩溃点(breakdown point,BP)描述了该估计所能容忍的含粗差样本比例上限[19]。样本数为n时，S估计的BP可通过式(12)计算

 n 8 9 10 11 12 13 14 K 2.937 3.189 2.561 2.756 2.349 2.51 2.212

4 实测GPS数据处理结果分析

GPS实测数据采用美国联邦航空局提供的参考站观测数据，文件为“Acv_EPak_1330_1616_06”，其观测值更新频率为1次/s，持续时间为1h，可见卫星数为10～12颗，图 2(c)中给出了可见卫星数随时间的变化情况。采用斯坦福大学开发的Matlab工具包SGMP[20]进行LS定位解算。目前，该工具包仅提供最小二乘的定位解算，将基于Huber函数的M估计[8]和本文改进后的抗差估计分别替换RAIM得到传统的M估计和本文改进后的M估计定位结果。为比较3种定位算法的抗差性，设计了4个场景：一是不含粗差，即粗差为0；二是指定PRN3卫星含粗差且粗差随时间增大，最大幅值为300m；三是指定PRN3和PRN6两颗卫星含粗差，两个粗差随机生成且互不相关，服从0到300的均匀分布；四是指定PRN3、PRN6和PRN9共3颗卫星同时含有粗差且3个粗差之间相互独立，服从场景三中的均匀分布。

 图 2 改进后的M估计 Fig. 2 The modified M-estimate

 粗差个数 RAIM 传统M估计 改进后的M估计 σx σy σz σx σy σz σx σy σz 0 0.964 0.650 0.979 0.867 0.660 1.266 0.827 0.623 1.393 1 2.713 3.032 3.975 0.895 0.669 1.529 0.926 0.656 1.566 2 9.381 12.107 10.862 13.609 8.032 61.048 0.882 0.627 1.299 3 30.532 28.460 46.037 26.612 13.742 59.683 3.433 1.253 9.359

 m 粗差数 0 1 2 3 改进前 1.520 25.599 80.051 142509.131 改进后 26.294 26.273 25.847 46.553

 图 1 传统M估计 Fig. 1 The traditional M-estimate

(1) 无粗差时，3种算法的定位精度相差不大，由于M估计是次优估计，所以传统的和改进后的M估计的位置误差都略大于带RAIM功能LS算法。两种M估计均以精度的略微下降来换取抗差性能的提高。

(2) 当有1颗卫星含粗差时，传统的和改进后的M估计的定位精度基本不受粗差影响。当粗差污染严重增至两颗卫星时，带RAIM功能的定位精度变差；传统M估计在少部分历元上抑制了粗差，但是产生100m以上偏差的概率明显增加，定位精度无法满足需要；改进后的M估计有效地抵制了两个粗差，最大三维定位偏差仍在10m以内，且精度与0和1个粗差时基本一致。

(3) 随着可见卫星数的增多，改进后的M估计抗差性能增强。在含3个粗差的图 2(c)中，当卫星数为12时，三维定位偏差在5m以下；当卫星数较少为11时，定位误差达20m左右，当卫星数较少到10颗时，定位误差将近300m。同时该结果也初步验证了式(13)的准确性：从式(13)可知当卫星数大于等于12颗时，改进后的M估计才能有效抑制3个粗差。

4 结 论

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http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0055

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#### 文章信息

TONG Haibo，ZHANG Guozhu.

Robust Positioning Algorithm with Modified M-estimation for Multiple Outliers

Acta Geodaetica et Cartographica Sinica,2014,43(4):366-371.
http://dx.doi.org/10.13485/j.cnki.11-2089.2014.0055