Mathematical Problems in Engineering

Volume 2018, Article ID 2563202, 6 pages

https://doi.org/10.1155/2018/2563202

## Research on GNSS Receiver Autonomous Integrity Monitoring Method Based on M-Estimation

Correspondence should be addressed to Pengfei Zhang; nc.ude.cun@fpgnahz

Received 29 November 2017; Accepted 22 February 2018; Published 28 March 2018

Academic Editor: Andrés Sáez

Copyright © 2018 Pengfei Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Receiver Autonomous Integrity Monitoring (RAIM) method is an effective means to provide integrity monitoring for users in time. In order to solve the misjudgment caused by the interference of gross error to the least squares algorithm, this paper proposes a RAIM method based on M-estimation for multiconstellation GNSS. Based on five programs, BDS, GPS/BDS, and GPS/BDS/GLONASS at the current stage, the future Beidou Global Navigation Satellite System, and the future GPS/BDS/GLONASS/Galileo system, the new RAIM method is compared with the traditional least squares method by simulation. The simulation results show that, with the increase of constellations, RAIM availability, fault detection probability, and fault identification probability will be improved. Under the same simulation conditions, the fault detection and identification probabilities based on M-estimation are higher than those based on least squares estimation, and M-estimation is more sensitive to minor deviation than least squares estimation.

#### 1. Introduction

Receiver Autonomous Integrity Monitoring (RAIM) is an effective method of integrity monitoring [1]. With the rapid development of Chinese Beidou navigation system and European Galileo system and recovering of Russia’s GLONASS constellation, interoperability among the four global navigation satellite systems (GNSS) has become an inevitable trend. In addition, research on multiconstellation RAIM algorithm has also been promoted [2–5]. RAIM can respond quickly and completely to the satellite fault and aerial abnormality, without any external intervention; it can also provide alarm information for users timely and effectively. Moreover, the user-level integrity monitoring can be realized simply and the input cost is low [6]. Therefore, research on RAIM algorithm is very necessary, and reliable RAIM algorithm can guarantee the integrity of navigation and positioning results. In 1987, Kalafus first introduced the concept of RAIM, and the consistency check for the current time redundancy variable was carried out, and at least five visible satellites are required for fault detection; at least six visible satellites are required for fault identification and exclusion. Afterward, many domestic and foreign experts proposed different snapshot algorithm and filter algorithm [6–9], but most of the above studies are based on GPS, GLONASS, and Galileo systems. The RAIM method research on the Beidou satellite navigation system and its combination with other GNSS is still in its infancy and development stage. This paper mainly applies the snapshot algorithm to multiconstellation fault detection and identification and proposes a RAIM method based on M-estimation for multiconstellation. Five programs, BDS, GPS/BDS, GPS/BDS/GLONASS at the current stage, the future Beidou Global Navigation Satellite System, and the future GPS/BDS/GLONASS/Galileo system, are compared with the traditional RAIM method based on least squares in simulation analysis.

#### 2. RAIM Method Based on Least Squares for Multiconstellation

Suppose that the receiver pseudorange observation equation iswhere is a vector, representing the pseudorange measurement error for each satellite.

And the least square positioning solution isMakeand then

Pseudorange residual vector:Make is called residual sensitivity matrix; then the residual sum of squares can be expressed as

is symmetric, idempotent, and its sum of squares of each row and column is equal to the corresponding diagonal element, and the sum of each row or column is equal to 0.

Make , and thenwhere is the number of visible satellites and is the number of navigation satellites involved in the calculation.

The error in the a posteriori unit weight of the pseudorange residual vector is

Therefore, the unit weight error of the pseudorange residual vector is calculated by the sum of squares of the pseudorange residuals. Under the normal circumstance of the system, the residual of the pseudorange is small, and the a posteriori unit weight error is also small. When the deviation of the measurement pseudorange is large, will become larger and it needs to be detected. Assuming that there is no fault, each component of the distance residual vector is independent of the normal distribution random error with the mean zero and variance . Because the residual sensitivity matrix is a real symmetric matrix whose rank is equal to , according to the statistical distribution theory, obeys chi-square distribution with freedom degree ; if there is a fault and the mean value of the distance residual vector is not zero, then obeys the noncentral chi-square distribution with freedom degree [9]. Therefore, can be used as a test statistic. Let the test statistic be

The detection threshold can be calculated by the allowed maximum false alarm probability . The false alarm is an indication that the user is notified of a positioning fault when no positioning fault occurs. The detection threshold is obtained by detecting the probability density function from the detection threshold to the infinite integral. This problem belongs to hypothesis test problem in mathematical statistics. Suppose the following: : no fault occurred. : a fault occurred.

Then the false alarm probability

According to the above equation, the detection threshold can be obtained. If , it indicates that the fault is detected, and the alarm will be sent to the user.

Fault detection is based on the test of pseudorange residuals sum of squares, and fault identification is based on the test of pseudorange residual element. And the basic idea is based on Baarda’s data snooping method [10], constructing a statistic based on least squares residual vector. The statistic obeys a certain distribution, and given a significant level, the residuals can be judged by the statistical test whether a gross error exists. From the relationship between residual and observation error, we can make the fault identification test statisticwhere represents the row and column element of the pseudorange residual vector cofactor matrix .

From (12), we can see that the statistical distribution of is consistent with .

When there is no fault, ; when there is a fault, .

Where is the statistic offset parameter, if satellite has a pseudorange error , then

visible satellites can obtain test statistics. Given the total false alarm probability , then the false alarm probability of each test statistic is , and the detection threshold of fault identification can be calculated from the false alarm probability of each test statistic.where

From the above formula we can calculate the identification threshold corresponding to each test statistic . Compared to the test statistic with the identification threshold, if , then the th satellite is faulty and should be excluded.

#### 3. RAIM Method Based on M-Estimation for Multiconstellation

M-estimation used in this paper is an iterative weighted least squares estimator. Different weights are applied to different points according to the pseudorange residual vector; that is, the points with small residuals are given a larger weight, while those with larger residuals are given a smaller weight. And weighted least squares estimation is then established, repeatedly iterating to improve the weight coefficient.

Different from the least squares making pseudorange residual sum as the extreme function, the extreme function of M-estimation iswhere , being the probability density function, take the derivative of and let it equal zero, make , and then

Make (the weighting factor), and introduce equivalent weight element ; then the above formula is rewritten aswhere is the equivalent weight matrix (diagonal matrix), and the element is ; bring it into the error equation; then

Thus the M estimated value of the robustness of the parameter vector iswhere .

There are many methods to construct the equivalent weight matrix, but the robust estimates are much the same, and a “normalized” residual index is used in each method, which is defined aswhere is median, and is residual scale. The Huber method is chosen; that is,where generally takes 1.345 [11].

The following fault detection and identification methods are similar to the least squares RAIM method, and construct test statistics: No fault assumption : . A fault assumption : .

is the noncentralization parameter, and is the a priori variance of the pseudorange residual; calculate the corresponding detection threshold from the corresponding false alarm probability. Then the fault detection and fault identification are carried out.

#### 4. Simulation Analysis

This paper designs the following five programs, using the self-compiled software for simulation analysis.

*Program 1*. It is the currently operating Beidou regional navigation system BD2 (5GEO + 5IGSO + 4MEO), with a total of 14 satellites.

*Program 2*. It is the Beidou system, BDS, for the future global navigation satellite system (5GEO + 3IGSO + 27MEO), with a total of 35 satellites.

*Program 3*. It is the currently operating GPS (32 satellites) + BD2 (5GEO + 5IGSO + 4MEO), with a total of 46 satellites.

*Program 4*. It is the currently operating GPS (32 satellites) + BD2 (5GEO + 5IGSO + 4MEO) + GLONASS (24 satellites), with a total of 70 satellites.

*Program 5*. It is a complete combination of four major satellite navigation systems GPS (32 satellites) + BDS (5GEO + 3IGSO + 27MEO) + GLONASS (24 satellites) + Galileo system (27 satellites) in the future, with a total of 118 satellites.

The Beidou regional constellation BD2, GPS constellation, and GLONASS constellation all adopt the broadcast ephemeris of 2015-01-12. The Beidou system, BDS, for the future global navigation satellite system will be simulated with 35 satellites (5GEO, 3IGSO, and 27MEO); the Galileo system is simulated with 27 satellites; the specific parameters are shown in Table 1.