 Open Access
 Total Downloads : 272
 Authors : Wantong Chen
 Paper ID : IJERTV5IS020002
 Volume & Issue : Volume 05, Issue 02 (February 2016)
 Published (First Online): 01022016
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
The SelfVerification of GNSS Multimode Single Epoch Attitude Determination:Method and Test
Wantong Chen
School of Electronics and Information Engineering Civil Aviation University of China
Tianjin, China
Abstract In recent years, there is a growing interest in GNSS (Global Navigation Satellite System) compass type attitude determination system. Multimode single epoch scheme is the point for the current research, since it is insensitive to cycle slips and it has a higher success rate. With this scheme, the coverage, integrity and availability can also be improved for the practical application. As a new type of compass, being different from the magnetic compass and gyrocompass, the correctness of resolved heading and elevation should be verified by internal fitting method or external fitting method. In this contribution, as one internal fitting method, the selfverification of GNSS multimode single epoch compass system is studied, based on the double collinear baselines with different lengths. Actual dynamic experiments based on L1/L2/B1 observations have been performed, the relative yaw and relative pitch are computed and the results verify the correctness of GNSS multimode single epoch compass system with the selfverification method.
KeywordsGNSS compass; selfverification; short baseline; integer ambiguity resolution; accuracy

INTRODUCTION
In the past decades, for attitude determination of vehicles, there are two widely used and fundamentally different types of compass: the magnetic compass and the gyrocompass. The magnetic compass contains a magnet that interacts with the earth's magnetic field and aligns itself to point to the magnetic poles. The gyrocompass contains a motorized gyroscope whose angular momentum interacts with the force produced by the earth's rotation to maintain a northsouth orientation of the gyroscopic spin axis, thereby providing a stable directional reference [1]. However, the accuracy of the magnetic compass is affected by the magnetic field intensity nearby the equipment, and the gyroscopes suffer from the error drift.
In recent years, there is a growing interest in GNSS (Global Navigation Satellite System) compass type attitude determination system. Carrier phase measurements from two antennas and an integer ambiguity resolution method are used to obtain precise attitudes such as yaw and pitch in this system. Compared with the magnetic compass and the gyrocompass, the GNSS compass can point to any desired direction without the abovementioned shortcomings [2]. For this technique, one antenna is assumed to be a reference and another is assumed to be a rover. By finding the baseline vector defined by two antennas, the vehicle attitude can be determined, namely the heading (or yaw) and elevation (or pitch).
This work is supported by the Scientific Research Fund of Civil Aviation University of China (Grant No.2013QD27X).
In order to acquire highprecision heading and elevation, the GNSS carrier phase measurements are usually employed. However, the phase observations are in essence affected by integer ambiguities: only the fractional part of the phase of the incoming GNSS signal can be measured [3]. Integer ambiguity resolution (IAR) is the process of resolving the unknown cycle ambiguities of the carrier phase data as integer, and many studies have been carried out to investigate the IAR method. More recent IAR methods make use of the Constrained LAMBDA (CLAMBDA) method to estimate the integer ambiguity, which is proved to be a fast, reliable estimator [4]. With this estimator, the successful ambiguity resolution can be achieved by utilizing instantaneous measurements, namely the single epoch ambiguity resolution, thus making IAR a total independence from carrier phase slips and losses of lock [5]. On the other hand, in order to improve the accuracy, the coverage, integrity and availability in the practical applications, the observables from multiple GNSS constellations (GPS, GLONASS, Galileo and Compass) are often utilized, namely the multimode scheme. It is desired from the perspective of users to exploit the possibilities and opportunities of fusing signals from different constellations so as to enhance coverage, accuracy, integrity, and availability. Thus, for modern GNSS compass system, various GNSS multimode single epoch schemes are often utilized.
As a new type of compass, the correctness of resolved heading and elevation should be verified. Two approaches can be utilized for validating the correctness: internal fitting and external fitting. For external fitting, other types of compass should be utilized such as gyrocompass, inertial navigation system and heading reference system. The major drawbacks of this method are the higher cost and the error drift of devices. Compared with the external fitting, internal fitting is easier to be achieved and no extra equipment is required, namely the selfverification method.
In this contribution, the selfverification of GNSS multi mode single epoch compass system is studied, based on the double collinear baselines with different lengths. The assessment of accuracy is also achieved with this scheme. Actual dynamic experiments based on L1/L2/B1 observations have been performed to verify the correctness of GNSS multimode single epoch compass system.

THE BASIC MODEL OF GNSS COMPASS
A. Attitude Estimation
For GNSSbased attitude determination, two antennas
are
In this case, the conditional leastsquares solution for b and its variance matrix are both required for the estimator. The solution to the minimization problem follows therefore as
a arg min a a 2 min b a b 2
often attached to a vehicle, and then a baseline vector defined as a vector from reference antenna to another antenna can be determined using GPS relative positioning
aZ
b b a
n Qa
bR3 , b l
Qba
(8)
technique. The yaw and pitch of the vehicle can thus be computed from the resolved baseline vector b. If the baseline vector from reference antenna to another antenna is parameterized with respect to the local EastNorthUp frame, the heading and the elevation can be computed from the baseline components (coordinates) bE, bN and bU as
This can be solved by the Constrained (C) LAMBDA method with high efficiency and high success rate [8].
C. Error Propagation of GNSS Compass
The baseline vector in the local EastNorthUp frame can be expressed as follows:
E N
E N
tan1 b b
(1)
bE l sin cos
(9)
tan1 b b2 b2
(2)
b bN l cos cos
U N E
bU
l sin
B. GNSS Compass Model
where is the nonlinear operator and
l T .
With a prior baseline length, the GNSS compass model reads as [6]:
Linearization of these nonlinear observation equations can be given as [9]
y
y
E y = Aa Bb, D y Q , a Zn , b R3 , b l
(3)
bE l0 cos 0 cos0
b l sin cos
l0 sin 0 sin 0
l cos sin
sin 0 cos0 (10)
cos cos
N
0 0 0 0 0 0 0 0
where y is the given GNSS data vector, and a and b are the
bU 0
l0 cos0
sin 0
l
ambiguity vector and the baseline vector of order n and 3
where the given Taylor point of expansion is
respectively. E(Â·) and D(Â·) denote the expectation and
l T
and terms of second order and higher
dispersion operators, respectively, and A and B are the given design matrices that link the data vector to the unknown parameters. The variance matrix of y is given by
0 0 0 0
order have been neglected. The inverse of equation (10) becomes
the positive definite matrix Qy, which fully characterizes
cos 0
l cos
sin 0 0
l cos
the statistical properties of the given GNSS data vector.
si0 s 0
0 0

c s
cos bE (11)
Since the baseline length is often known in practical
n 0 in 0
os 0 in 0
0 b
l l
l N
applications, this priori given baseline information can be
l 0
0 0 b
treated as a useful constraint as well. In Equation (3), l
denotes the known baseline length, which is assumed to be
sin 0 cos0
cos 0 cos0
sin0 U
constant. Note that the GNSS compass model (3) involves two types of constraints: the integer constraints on the ambiguities and the length constraint on the baseline vector. For this model, once a is resolved, the leastsquares solution for b, namely the conditional leastsquares solution,
With the law of variance and covariance (vc) propagation, the vc matrix of can be calculated by
Q = J Q J
Q = J Q J
T (12)
b(a)
where J is the matrix that link the baseline error vector to
can be written as
b a BT Q1B1 BT Q1 y Aa
(4)
the attitude error parameters in (11). Note that
is
Q
Q
b(a)
y y determined by the design matrix B and the positive definite
The corresponding variance matrix is given as
matrix Qy, see also (5). With 2 , 2 and 2
being the
Q BT Q1B1
(5)
E N U
ba y
diagonal elements of
Q , we have the following
To solve the GNSS model (3), one usually applies the
leastsquares principle and this amounts to solving the following minimization problem:
expressions [10]:
cos
b(a)
2 2 sin
2 2
min y Aa Bb 2
2 0 E 0 N
(13)
aZn ,bR3 , b l Qy
(6)
l 2 cos 2
e 2 min a a 2
min
b a b 2
2 sin
0 0
sin 2 2 cos sin 2 2 (cos )2 2
Qy aZ
n Qa
bR3 , b l
Qba
0 0 E
0 0 N
0
0
l 2
0 U (14)
where 2 T Q1 and e is the least squares
Qy y
2 sin
sin 2 2 cos
cos 2 2 (sin )2 2 (15)
l 0 0 E
0 0 N 0 U
residuals. Moreover, the following cost function can be formulated [7]:
Equation (13) and (14) indicates that the accuracies of the estimated attitude angles found by using carrier phase are
min a a 2 min b a b 2
(7)
inverse proportional to the length of the baseline used. In
aZn Qa bR3 , b l Q
ba
other words, the accuracies of heading and elevation can be further improved for the longer baseline. Thus, if the
baseline is long enough, it can be treated as the reference system with high accuracy.
For the purpose baseline, with L being the baseline length, the accuracies of yaw and pitch are given by


THE SELFVERIFICATION OF GNSS COMPASS
cos 2 2 sin 2 2
(16)
MA
MA
2
MA E MA N
With the discussion on the error propagation and the
MA,
L2 cos 2
accuracy assessment of GNSS compass, a new method is
sin sin 2 2 cos sin 2 2 (cos )2 2 (17)
proposed for the selfverification of GNSS multimode single epoch compass system. That is, without any other
2
MA,
MA MA E MA MA N MA U
s2 L2
kind of compass, the correctness of GNSS compass attitude determination are achieved by the internal fitting.
For the reference baseline, with sL being the baseline
length, the accuracies of yaw and pitch are given by

The Basic Principle
cos 2 2 sin 2 2
2
MB E MB N
(18)
In order to achieve the selfverification of GNSS compass, the proposed method utilizes double collinear baselines but with distinctly different lengths. One is the
MB,
sin
L2 cos 2
MB
MB
sin 2 2 cos sin 2 2 (cos
)2 2 (19)
purpose baseline equipped for the vehicle, and the other is treated as the reference system, which is much longer than
2
MB,
MB MB E MB MB N MB U
s2 L2
the purpose baseline.
Firstly, both baselines should be setup in the collinear way. In general, at least three antennas are employed and set up in the same straight line, which is shown in Fig.1.
Note that the baseline placement can also affect the
accuracy of GNSS compass, since the errors of heading and elevation are related to the direction of baseline vector. By disregarding the installation error of both baselines, we have
sL
,
(20)
Antenna B
Antenna A
L
Antenna
M
MA MB MA MB
Hence, the accuracies of reference baseline have the following relationships with those of the purpose baseline:
2
MA,
2 2
s
s
MB,
2
,
,
MA,
2 2
s
s
MB,
(21)
Fig.1 Double collinear baselines with three antennas
C. The Multifrequency Singleconstellation Model
If either baseline vector from reference antenna to another antenna is parameterized with respect to the local EastNorthUp frame, the heading and the elevation of both baselines can be computed as follows:
The purpose baseline is setup with Antenna M and
b
b
(22)
Antenna A in Fig.1, namely MA, and the baseline length is
1 MA,E 1
MA
MA,U
L. The reference baseline is setup with Antenna M and
bMA,N
b2 b2
MA tan , tan
MA tan , tan
Antenna B, denoted as MB, and the baseline length is s
MA,N MA,E
times longer than MA. To make sure that the accuracy of
1 bMB,E
1 bMB,U
(23)
reference baseline is high enough, the length of baseline
MB tan
b , MB tan 2 2
MB should be long and the times s is large.
MB,N
bMB,N bMB,E
Second, the phase centers of all the antennas are required to be stable enough, thus making the drift error of
Then the relative heading and relative elevation can be computed as follows:
phase center minimized. If the drift error is very large, the
,
(24)
attitude angles of long baseline may not be close enough to true attitude angles, thus making the reference baseline inaccurate even if all the geometric centers of the three antennas are in the same straight line. Hence, the surveying antennas with very stable phase center are required for the
MAMB MA MB MAMB MA MB
Hence, the accuracies of relative heading and relative elevation have the following relationship with those of reference baseline:
selfverification procedure.
2 s2 1 2
, 2
s2 1 2
(25)
B. The Accuracy Assessment
MAMB,
MB,
MAMB,
MB,
Note that Equation (5) is nothing to do with the baseline
For the standard deviation, we have
MA MB,
p>MA MB,
MB,
MB,
MA MB,
MA MB,
MB,
MB,
placement and the design matrix B is determined by the
s2 1
,
s2 1
(26)
satellite geometry and the positive definite matrix Qy is determined by the noise levels of observables. Thus, with the same type GNSS receivers and the same observing period, both baseline MA and baseline MB have the same
variance matrices Q , indicating that 2 , 2 and 2 of
Without the true attitude angles, the selfverification procedure can thus be achieved with
E MAMB 0, E MAMB 0
MAMB, MA, , MAMB, MA,
(27)
(28)
ba E N U
the both baseline vector are the same.
s2 1 s
MB,
s2 1
s MB,


EXPERIMENTS
This section presents the evaluation of the propose self verification procedure based on actual dynamic tests. The accuracies of yaw and pitch are also compared with different baseline lengths.

Platform and Test Environment
In order to achieve the propose selfverification procedure for GNSS single epoch compass, the actual GNSS measurements are collected with three NovAtels OEM628 boards, which are designed with 120 channel and can tracks all current and upcoming GNSS constellations and satellite signals including GPS, GLONASS, Galileo and Compass. Configurable channels optimize satellite availability in any condition, no matter how challenging. For this experiment, the GPS L1/L2 and Compass (or BDS) B1 are exploited for constructing the GNSS multimode single epoch compass model. In order to minimize the multipath interference, three TrimbleÂ® Zephyr Model 2 antennas are utilized for this experiment, and this type of antenna has outstanding low elevation satellite tracking performance and extremely precise phase center accuracy and it also supports the GPS L1/L2 and Compass (or BDS) B1 bands.
0
30
30
330
330
30
300
300
20
60
60
45
25
32 60
31 75
90
90
270
270
90
240
240
14 29
120
120
150
150
210
210
22
180
180
Fig.3 The constellation of GPS satellites
30
30
330
330
0
30
60
60
300
300
14 45
60
10 75
270
270
7
90
90
90
4
120
120
240
240
3 1
210
210
9
150
150
180
180
Fig.4 The constellation of Compass satellites
10
Fig.2 The experiment system and environment
The proposed method has been tested processing actual data collected during a dynamic experiment, in which a car was equipped with three antennas and the shorter baseline length is 0.325m and the longer baseline length is 1.625m, as is shown in Fig.2. The car is moving along a narrow rectangle block about 4 laps and both ends of the rectangle block are arcshaped. During about 410 seconds observation, the number of available satellites equals eight for GPS and five for Compass most of the time. The constellation of GPS satellites in this experiment is shown in Fig.3 and the constellation of Compass satellites is shown in Fig.4, and each satellite is discernible by its PRN number. Note that the star symbol denotes the geostationary satellites of Compass. The numbers of visible satellites are given in Fig.5.
9
8
Number of Visible Satellite
Number of Visible Satellite
7
6
5
4
3
GPS
Compass
GPS
Compass
2
1
0
0 50 100 150 200 250 300 350 400
Time (second)
Fig.5 The number of visible satellites

Comparison of Attitude Determination
4
The heading/yaw and elevation/pitch are resolved based
on the model (8) with Constrained (C) LAMBDA method. 3
Relative Elevation (degree)
Relative Elevation (degree)
The yaw and pitch results are demonstrated in Fig.6 and 2
Fig.7, respectively.
1
350
300
250
Yaw (degree)
Yaw (degree)
200
150
0.325m baseline 1.625m baseline
0
1
2
3
4
0 50 100 150 200 250 300 350 400
Time (second)
100 Fig.9 The relative pitch of the purpose baseline and the reference baseline
50
0
0 50 100 150 200 250 300 350 400
Time (second)
Fig.6 The yaw comparison for 0.325m baseline and 1.625m baseline
4
3
2

Accuracy Assessment of Relative Attitude Angles
As shown in Table I, the average and standard deviation of relative attitude angle measurements of dynamic this experiment are given.
Table Head 
Mean Value (degree) 
Standard deviation (degree) 
Relative Yaw 
0.0088 
0.7782Â° 
Relative Pitch 
0.0013 
1.2539Â° 
Table Head 
Mean Value (degree) 
Standard deviation (degree) 
Relative Yaw 
0.0088 
0.7782Â° 
Relative Pitch 
0.0013 
1.2539Â° 
TABLE I. RELATIVE ACCURACY ASSESSMENT
Elevation (degree)
Elevation (degree)
1
0
1
2
3 0.325m baseline
1.625m baseline
4
0 50 100 150 200 250 300 350 400
Time (second)
Fig.7 The pitch comparison for 0.325m baseline and 1.625m baseline
As is shown, the accuracy of 1.625m baseline is much higher than that of the 0.325m baseline. However, the yaw and pitch angles of both baselines are consistent. The resolved relative yaw and pitch are also given in Fig.8 and Fig.9, respectively.
[2] 
Review, vol. 13, pp.140149, March 1998. C.H. Tu, K.Y Tu, F.R. Chang and L.S. Wang, GPS compass: a 

novel navigation equipment, IEEE Trans Aerosp Electron Sys, vol. 33, pp. 10631068, July 1996. 

[3] 
A. Leick, GPS satellite surveying, 3rd ed., Wiley, New York, 2004, pp.324337. 

[4] 
P.J.G Teunissen, Integer least squares theory for the GNSS Compass, Journal of Geodesy, vol. 83, pp. 115, January 2010. 

[5] 
P.J. Buist, The Baseline Constrained LAMBDA Method for Single Epoch, Single Frequency Attitude Determination Applications, Proceedings of IONGPS, Forth Worth, Texas, USA, 2007. 

[6] 
P.J.G Teunissen, The LAMBDA method for the GNSS compass, Artif. Satellites, vol. 41, pp. 89105, July 2007. 

0 50 
100 
150 
200 
250 
300 
350 
400 
[7] 
C. Park, P.J.G Teunissen, Integer least squares with quadratic 
[2] 
Review, vol. 13, pp.140149, March 1998. C.H. Tu, K.Y Tu, F.R. Chang and L.S. Wang, GPS compass: a 

novel navigation equipment, IEEE Trans Aerosp Electron Sys, vol. 33, pp. 10631068, July 1996. 

[3] 
A. Leick, GPS satellite surveying, 3rd ed., Wiley, New York, 2004, pp.324337. 

[4] 
P.J.G Teunissen, Integer least squares theory for the GNSS Compass, Journal of Geodesy, vol. 83, pp. 115, January 2010.  
[5] 
P.J. Buist, The Baseline Constrained LAMBDA Method for Single Epoch, Single Frequency Attitude Determination Applications, Proceedings of IONGPS, Forth Worth, Texas, USA, 2007. 

[6] 
P.J.G Teunissen, The LAMBDA method for the GNSS compass, Artif. Satellites, vol. 41, pp. 89105, July 2007. 

0 50 
100 
150 
200 
250 
300 
350 
400 
[7] 
C. Park, P.J.G Teunissen, Integer least squares with quadratic 
2.5
2
1.5
Since the mean values of both relative yaw and relative pitch are both close to zero, the consistency of both baselines is thus be verified, see Equation (27). Note that we do not know the true attitude angles of both baselines and no other extra device is utilized for the fitting. It is not difficult to find that the reference baseline is five times longer than the purpose baseline. Thus, with Equation (28), it can also be inferred that the yaw accuracy of reference baseline is 0.156Â° and the pitch accuracy of reference baseline is 0.251Â°.
With the actual experimental results above, the correctness of GNSS multimode single epoch attitude determination can be proved based on the selfverification scheme.
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