The Self-Verification of GNSS Multi-mode Single Epoch Attitude Determination:Method and Test

The Self-Verification of GNSS Multi-mode 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. Multi-mode 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 self-verification of GNSS multi-mode 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 multi-mode single epoch compass system with the self-verification method.

KeywordsGNSS compass; self-verification; short baseline; integer ambiguity resolution; accuracy

1. 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 north-south 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 above-mentioned 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 high-precision 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 multi-mode 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 multi-mode 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 self-verification method.

   In this contribution, the self-verification 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 multi-mode single epoch compass system.

2. THE BASIC MODEL OF GNSS COMPASS

   A. Attitude Estimation

   For GNSS-based attitude determination, two antennas

   are

   In this case, the conditional least-squares 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 East-North-Up 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 East-North-Up 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(·) and D(·) 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 least-squares solution for b, namely the conditional least-squares solution,

   With the law of variance and covariance (v-c) propagation, the v-c 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

   least-squares 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

3. THE SELF-VERIFICATION 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 self-verification of GNSS multi-mode 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

   1. The Basic Principle

   cos 2 2 sin 2 2

   2

   MB E MB N

   (18)

   In order to achieve the self-verification 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 Multi-frequency Single-constellation Model

   If either baseline vector from reference antenna to another antenna is parameterized with respect to the local East-North-Up 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:

   self-verification 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 self-verification 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,

4. 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.

1. Platform and Test Environment

   In order to achieve the propose self-verification 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 multi-mode single epoch compass model. In order to minimize the multipath interference, three Trimble® 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 arc-shaped. 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

2. 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

3. 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 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.

2.5

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 0.156° and the pitch accuracy of reference baseline is 0.251°.

With the actual experimental results above, the correctness of GNSS multi-mode single epoch attitude determination can be proved based on the self-verification scheme.

REFERENCES

Relative Yaw (degree)

Relative Yaw (degree)

1

0.5

0

-0.5

-1

-1.5

-2

-2.5

Time (second)

equality constraints and its application to GNSS attitude

Fig.8 The relative yaw of the purpose baseline and the reference baseline

determination systems, Int J Control Autom Syst, vol.7, pp. 566- 576, December 2009.

1. P.J.G Teunissen, G. Giorgi, P.J Buist, Testing of a new single- frequency GNSS carrier-phase compass method: land, ship and aircraft experiments, GPS Solutions, vol. 15, pp. 15-28, January 2011.

2. Park Chansik, Cho Deuk Jae, Cha Eun Jong, et al. Error Analysis of 3-Dimensional GPS Attitude Determination System. International Journal of Control, Automation, and Systems, vol. 4, pp. 480-485, April 2006.

3. W. Chen, H. Qin, Y. Zhang, T. Jin, Accuracy assessment of single and double difference models for the single epoch GPS compass, Advances in Space Research, vol. 49, pp. 725-738, Decemeber 2012.