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GNSS receiver autonomous integrity monitoring(RAIM) algorithm based on robust estimation

Yuanxi Yanga,b,c, Junyi Xuc,*

a State Key Laboratory of Geo-information Engineering, Xi'an 710054, Chinab Xian Research Institute of Surveying and Mapping, Xi'an 710054, Chinac Beijing Satellite Navigation Center, Beijing 100094, China

a r t i c l e i n f o

Article history:

Received 23 December 2015

Accepted 22 February 2016

Available online 29 April 2016

Keywords:

GNSS

Integrity

Receiver autonomous integrity

monitoring (RAIM)

Robust estimation

Fault detection

* Corresponding author.E-mail address: xujunyi-025@163.com (J.

Peer review under responsibility of Instit

Production and Hosting by Elsev

http://dx.doi.org/10.1016/j.geog.2016.04.004

1674-9847/© 2016, Institute of Seismology, Ch

Communications Co., Ltd. This is an open acce

a b s t r a c t

Integrity is significant for safety-of-life applications. Receiver autonomous integrity

monitoring (RAIM) has been developed to provide integrity service for civil aviation. At first,

the conventional RAIM algorithm is only suitable for single fault detection, single GNSS

constellation. However, multiple satellite failure should be considered when more than

one satellite navigation system are adopted. To detect and exclude multi-fault, most cur-

rent algorithms perform an iteration procedure considering all possible fault model which

lead to heavy computation burden. An alternative RAIM is presented in this paper based on

multiple satellite constellations (for example, GPS and BeiDou (BDS) etc.) and robust esti-

mation for multi-fault detection and exclusion, which can not only detect multi-failures,

but also control the influences of near failure observation. Besides, the RAIM algorithm

based on robust estimation is more efficient than the current RAIM algorithm for multiple

constellation and multiple faults. Finally, the algorithm is tested by GPS/BeiDou data.

© 2016, Institute of Seismology, China Earthquake Administration, etc. Production and

hosting by Elsevier B.V. on behalf of KeAi Communications Co., Ltd. This is an open access

article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

How to cite this article: Yang Y, Xu J, GNSS receiver autonomous integrity monitoring (RAIM) algorithm based on robustestimation, Geodesy and Geodynamics (2016), 7, 117e123, http://dx.doi.org/10.1016/j.geog.2016.04.004.

1. Introduction

It is very important for GNSS-based navigation or posi-

tioning in real time to know the quality and safety of the

system which is usually described by the “integrity”. The

Xu).

ute of Seismology, China

ier on behalf of KeAi

ina Earthquake Administra

ss article under the CC BY

integrity includes the ability of navigation system to provide

timely warnings to users when the system or some of its

components are not trusted for navigation. Therefore, the

integrity relates to reliability of the information supplied by

the navigation system. Especiallywhen it is required to deliver

Earthquake Administration.

tion, etc. Production and hosting by Elsevier B.V. on behalf of KeAi

-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

g e o d e s y and g e o d yn am i c s 2 0 1 6 , v o l 7 n o 2 , 1 1 7e1 2 3118

an alarm of any malfunction to users when failure detected

within a given period of time and with a given probability [1].

Receiver autonomous integrity monitoring (RAIM) has

been developed before the full operation of GPS in late 1990s

[2e4] for the consideration of providing integrity service for

civil aviation. Many works have been done to enhance the

performance of RAIM. However, the application of RAIM was

limited as a supplemental navigation means for en route

through non-precision approach (NPA) phases of flight due to

its low availability.

In the beginning, most RAIM algorithms have been devel-

oped considering only single satellite constellation [5] and

least squares estimation. The residual of the measurements

are the basic variable to identify the integrity, thus the

redundant measurements are necessary. It is not so difficult

to realize the RAIM performance if the residuals reflect the

measurement error or the satellite status reliably. In the

situation of single satellite constellation, the average visible

satellite is about 7e8 which cannot provide enough

redundancy.

Fortunately, GLONASS from Russia is recovering to full

constellation, Chinese BeiDou navigation satellite system has

been established in 2012 [6], and Galileo from Europe is in

steady progress [7,8]. The multiple constellations provide us

abundant measurements for improving the positioning

accuracy, and provide us more possibilities for monitoring

the user's integrity. The benefits of the multi-constellation

were demonstrated [8,9]. However, the multiple satellite

faults affect the current RAIM reliability, because the

current methods used in fault detection is the Chi-square

test which is suitable for single failure identification. Many

efforts have been done to deal with double or multi-fault

satellite scenarios [10,11]. The ICAO Advanced RAIM

(ARAIM) working group has been promoting the

development of RAIM algorithm for multiple constellation

and multiple faults for aviation. The integrity monitoring

for two satellites faults was discussed [12]. A group

separation (GS) RAIM method was designed for dual

constellation, such as GPS and Galileo [13]. Three solutions

were calculated, one was based on all-in-view satellite

(including GPS and Galileo), and the other two solutions use

GPS or Galileo respectively. The separation between the all-

in-view solution and either of the independent system was

used as a test statistic. If any one of the two statistics

exceeds a detection threshold, the observation from that

system will be abandoned entirely. This GS method can

handle multiple faults within one of the system. It'sobvious that the GS method did not take full advantage of

the observations. There is a bargain between integrity and

precision. A modification of GS method called optimally

weighted average solution (OWAS) was proposed later [14],

the position solution of this method was the weighted

average of the two individual systems. Both methods

assume that multiple faults occur in one system, while the

other system is healthy. An detailed explanation of an

optimized multiple hypothesis solution separation (MHSS)

algorithm for RAIM was given, all the possible fault mode

were considered to detect and exclude the fault satellite

[15]. Though these fault detection and exclusion procedure

are used in GNSS situation by considering all the possible

fault modes, its efficiency is low because one has no

foreknowledge about the number of faults [16].

Considering the available dynamic model, they developed

a RAIM scheme based on Kalman filtering [17]. Thesemethods

are based on Baarda outlier detection and identification

procedure [18]. If a failure satellite creates a significant

outlier, the residuals of measurements based on the least

squares estimates may be seriously contaminated, the

corresponding variance scale may be very large, in this case

the RAIM detection may fail [19,20].

An alternative approach for RAIM is presented in this paper

based on multiple GNSS and robust estimation. GPS/BeiDou

real data is used as an example to verify the new method.

2. RAIM based on least squares estimation

A RAIM needs reference coordinates to calculate the mea-

surement errors. Usually the raw observations are scalar pseu-

doranges of satellites to receiver lines. If the ionospheric free or

ionospheric-corrected pseudorange measurements are avail-

able, and the locations of the static or kinematic receivers are

assumed to be approximately (xr, yr, zr), the receiver clocks are

supposed to have been synchronized to certain accuracy, then

the observation equation for the pseudorange is as follows [21]:

ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðxi � xrÞ2 þ

�yi � yr

�2 þ ðzi � zrÞ2q

þ cDtr þ Di (1)

where, ri is the geometric range between the user antenna and

the ith satellite, Dtr is the user receiver clock offset, c is the

speed of light, Di is the sum of measurement errors.

In practice, we usually use the residuals of the measure-

ments as variables to identify the problematic satellites or

measurement outliers. Assume that the raw measurement

and estimated measurement vectors are L and bL respectively,

the residual vector is then expressed as

V ¼ bL � L (2)

Collecting observations from n satellites, we obtain n

observation equations

V ¼ AdbX � 3 (3)

where d bX is the estimated correction vector of the approxi-

mate receiver coordinate (x0r ; y

0r ; z

0r ) and clock offset parame-

ters of different systems, 3 is a vector of measurements minus

the calculated ones based on the approximated receiver co-

ordinate, and A is the design matrix, considering the differ-

ences between different satellite navigation systems, it can be

expressed as [16,22]:

A ¼

26666664

agps;1x agps;1

y agps;1z 1 0

« « « « «agps;n1x agps;n1

y agps;n1z 1 0

agnss;n1þ1x agnss;n1þ1

y agnss;n1þ1z 0 1

« « « « «agnss;nx agnss;n

y agnss;nz 0 1

37777775 (4)

where, gnss denotes the other navigation system such as

BeiDou or Galileo. For this paper, it refers to BeiDou. a is

expressed as

g e o d e s y and g e o d yn am i c s 2 0 1 6 , v o l 7 n o 2 , 1 1 7e1 2 3 119

ax ¼ xi � xuhðxi � xuÞ2 þ

�yi � yu

�2 þ ðzi � zuÞ2i1=2 (5)

ay ¼ yi � yuhðxi � xuÞ2 þ

�yi � yu

�2 þ ðzi � zuÞ2i1=2 (6)

az ¼ zi � zuhðxi � xuÞ2 þ

�yi � yu

�2 þ ðzi � zuÞ2i1=2 (7)

If the least squares principle is applied, then the estimation

dbX is obtained as

dbX ¼ �ATPA��1

ATP3 ¼ N�1ATP3 (8)

with N ¼ ATPA�1, P denotes the weight matrix of

measurements

P ¼

26641�s21 0 / 0

0 1�s22 / 0

« « 1 «0 0 / 1

�s2n

3775: (9)

The residual vector follows

V ¼ AdbX � 3 ¼ �AN�1ATP� I�3 ¼ ðH� IÞ3 (10)

withH¼AN�1ATP.H is an idempotentmatrix, that isH2¼H [23].

Then the following test statistic for the least square based

RAIM [24] can be used

test ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiVTPV

p(11)

if we assume that V is a normally distributed random variable

with zero mean and a standard deviation of si for all N satel-

lites in view, then the test statistic is a chi-square distributed

variable with N-4 degrees of freedom [24]. The threshold T can

be calculated by the number of satellite N and the possibility

of false alarm Pfa (10�7 in this paper).

A fault is detected when the test statistic is larger than the

threshold. If there are enough satellites, the fault exclusion

procedure starts by excluding one satellite every time to

determine which measurement is abnormal.

Fig. 1 e Visible satellite nu

3. RAIM statistics based on robustestimation and other available information

To ensure the reliability of the user integrity detection and

warning, it is necessary to get qualified residuals and the

corresponding variance scale which is a basic variable for

setting up the threshold. The reasonable variance scale de-

pends on the reliable residuals, and the reliable residuals rely

on the qualified parameter estimates. A robust estimation is a

reasonable alternative to get the reliable parameter estimates

in most cases [19,20,25], especially in the situation with mul-

tiple constellations.

An M-estimator of equation (3) is

dbXk ¼�ATPA

��1ATP3 ¼ N�1ATP3 (12)

where P is a robust equivalent weight matrix with the ele-

ments Pii as [19,26]:

Pki ¼

8>>>>>><>>>>>>:

Pki

�� ~Vki

�� � c0ða0; rÞ

Pki

c0�� ~Vki

�� c1 �

�� ~Vki

��c1 � c0

!2

c0ða0; rÞ<�� ~Vki

�� � c1ða1; rÞ

0�� ~Vki

��> c1ða1; rÞ

(13)

where, ~Vki ¼VkisVki

, Vk ¼ AdbXk � 3, and sVkiis the normalized

variance of Vk, c0 and c1 are determined by the significance

level a0 and a1 (the value of which are 0.01 and 0.001 respec-

tively) and the freedom degree r. c1 is taken as the threshold, if~Vki is larger than c1, a fault is declared and excluded by auto-

matically setting the corresponding weight as zero.

4. Calculation and analysis

Real data are used to verify the new RAIM method. The

UB240 BD2/GPS receivers from Beijing Unicore Communica-

tions Incorporation are used to collect data in July 14e15, 2013.

mber (cut angle 10�).

g e o d e s y and g e o d yn am i c s 2 0 1 6 , v o l 7 n o 2 , 1 1 7e1 2 3120

It is shown in Fig. 1 that the average visible satellite number of

GPS and BeiDou is about 13.

4.1. Performance comparison between robust estimationbased RAIM for single fault

A real satellite failure occurred on July 14, 2013. Therefore,

the data between 02:00 am-06:00 am is used to verify the

performance of robust estimation based RAIM for single fault.

To be convenient for comparison and analysis, four

schemes are adopted:

Fig. 2 e Position err

Fig. 3 e Position err

Scheme 1: navigation solution based on BeiDou;

Scheme 2: robust navigation solution based on BeiDou;

Scheme 3: navigation solution based on BeiDou/GPS;

Scheme 4: robust navigation solution based on BeiDou/GPS;

The result can be seen in Figs. 2e5.

(1) It can be seen from Figs. 2 and 3 that there are obvious

fault around epoch 1500. The position error is large than

25 m, and most of the problem satellite is excluded by

robust estimation. The position accuracy in Table 1

shows that the RMS of Scheme 1 and Scheme 2 are

or of Scheme 1.

or of Scheme 2.

Table 1 e Comparison of position accuracy for differentschemes (Unit:m).

95 RMS

E N U E N U

Scheme 1 3.788 6.441 10.471 3.109 3.468 7.468

Scheme 2 3.653 5.984 10.037 2.966 3.093 5.702

Scheme 3 1.701 1.792 3.319 1.311 1.149 3.125

Scheme 4 1.578 1.645 3.075 1.133 0.928 1.886

g e o d e s y and g e o d yn am i c s 2 0 1 6 , v o l 7 n o 2 , 1 1 7e1 2 3 121

respectively 7.458m and 5.702m. Yet for one epoch, the

fault is not correctly excluded.

(2) It can be seen from Figs. 4 and 5 that the position ac-

curacy of BeiDou/GPS dual system is better than that of

BeiDou single system. Besides, by using observations of

Fig. 4 e Position err

Fig. 5 e Position err

BeiDou/GPS, the robust estimation based RAIM algo-

rithm, all fault observations are successfully excluded.

Therefore, the integrity of navigation will benefit from

the redundant observations when multiple satellite

navigation system are available.

4.2. Performance of the robust estimation based RAIMfor multi-fault detection and exclusion

Asmentioned before, when the satellite ranging signal error

is larger than 30 m or 4.42 times URA, the corresponding sat-

ellite service should be taken as a service failure [27]. In order to

verify the performance of the robust estimation based RAIM,

outliers with different sizes (30e100 m) are added to the data

or of Scheme 3.

or of Scheme 4.

Table 2 e Comparison of the successful rates.

Double fault False exclude Three fault False exclude

Sat. No Epochs Sat. No Epochs

C03,C05 e 0 G27,C05,C08 e 0

G31,C07 C10 274 G16,C02,C08 e 0

G31,C01 C04 35 G13,G27,C01 e 0

G13,C08 e 0 G32,C05,C10 e 0

G29,C02 C07 2 G32,G29,C10 e 0

G29,C03 C07 1 G32,C02,C03 G20,C07 5

G32,C10 e 0 G27,C03,C07 e 0

G06,C05 e 0 G31,G32,C01 C07 2

G06,C02 e 0 G31,C03,C07 C10 274

G27,C07 e 0 G06,C01,C04 e 0

G27,C10 e 0 G06,C03,C05 e 0

G16,C02 C07 1 G16,G29,C05 C07 1

G13,G27 e 0 G13,C01,C02 C05 14

C02,C07 C10 67 G16,G31,C04 C07 4

C01,C04 C07 21 G20,C05,C07 e 0

g e o d e s y and g e o d yn am i c s 2 0 1 6 , v o l 7 n o 2 , 1 1 7e1 2 3122

collected on July 15, 2013. Considering the possibility thatmore

than two faults are included in one navigation system or two

faults are included in both systems with simultaneously is

pretty small values, thus only the dual and three faults

scenario are discussed here. About 15 times of experiment

are performed for each fault mode.

The success exclusion of an outlier is defined by two ways:

(a) all the outliers are excluded and no normal measurements

are excluded; (b) all the outliers are excluded and somenormal

measurements may be miscancelled. The result is showed in

Table 2.

It can be seen from Table 2 that for all the tests, the

problem satellites are correctly excluded. This prove that the

robust estimation based RAIM can effectively exclude

multiple satellite fault. However, for both double fault and

triple fault mode, some of the normal satellite might be mis-

excluded. The longest time span of false exclude is about

274 epochs, satellite C07 has been false excluded for about 8

times. Usually it is the satellite of BeiDou that are false

excluded. This is probably because the measurement quality

of BeiDou is not as good as GPS. Though fault exclusion does

not affect the system integrity since there are enough

satellites, it should be carefully processed.

5. Conclusion

The global navigation satellite system is under fast devel-

opment now. However, the current based RAIM is theoreti-

cally based on the assumption that only one fault happens.

This is not reasonable for the GNSS scenario.

In this paper, an alternative approach for RAIM is pre-

sented based on robust estimation. Unlike the existing works

which use the simulated data, the real GPS/BeiDou data are

used to verify the newmethod. The integrity of navigationwill

benefit from the redundant observations when multiple sat-

ellite navigation systems are available. As more and more li-

ability critical applications except for aviation are concerning

integrity now, this may be a huge advantage. The robust

estimation based RAIM can effectively exclude multiple sat-

ellite fault.

However, some of the normal satellite may also be

excluded. Though fault exclusion does not affect the system

integrity since there are enough satellite, it should be carefully

processed. Besides, the new method is much more efficient

than the conventional LS based RAIM.

Acknowledgement

This project is supported by the National 863 project

(2013AA122501-1) and the National Natural Science Founda-

tion of China (41020144004, 41474015, 41374019, 41374003,

41274040).

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Yuanxi Yang, Ph D, researcher, Academi-cian of the Chinese Academy of Science.