A Problem of Satellite Orientation Determination by Spaced Satellite Antennas and Angular Rate Sensors

Abstract

The estimation models are given in terms of a scheme with the deep integration of the SNS–ARS data. Several assessments of the accuracy of the integrated navigation solutions are discussed using the model trajectory and angular motion of the UniversitetskiiTat’yana2 satellite.

Full text

ISSN 10642307, Journal of Computer and Systems Sciences International, 2015, Vol. 54, No. 4, pp. 651–655. © Pleiades Publishing, Ltd., 2015.
Original Russian Text © A. Dzhepe, A.V. Kozlov, A.A. Nikulin, 2015, published in Izvestiya Akademii Nauk. Teoriya i Sistemy Upravleniya, 2015, No. 4, pp. 155–159.
651
INTRODUCTION
The problem of integrating the angular rate sensors (ARS) with several antennas spaced on the object
case is topical for defining the angular position of a satellite in orbit. Let us discuss the informational
aspects of this task.
The system of spaced satellite antennas is a source of information of two types: the position and rate
data on the point motion that is the phase center of the base antenna and the data on the position of the
object case, which were expressed by the assessments of the coordinates of the base vector joining the
phase centers of the antennas. The sensors of the angular rate allow one to estimate a change in space ori
entation.
Two types of integration of satellite and inertial navigation systems are known: the weakly coupled and
close (deep) integrated systems. In the first variant, the secondary SNS data, including the coordinates
and rates, as well as the yaw (true course), roll, and pitch (at three and more spaced antennas with the
geometry providing the conditionality of the problem) angles, for the multielement SNS, are used for
integration solutions. Only the raw and pitch angles can be determined in the presence of two spaced
antennas installed along the longitudinal axis of the object.
In the second variant, the primary satellite measurements, which are the code pseudoranges, Doppler
pseudospeeds, and phase measurements received from the base antenna and other spaced ones, are used.
The advantage of the second method is that the complex processing of the satellite information for a small
number of visible satellites, when the secondary SNS data are not formed, is possible. The integration
models for inertial navigation systems and singleantenna SNS are well known and described in [1]. The
specific nature of the problem of the ARS–SNS deep integration with spaced antennas is in using the pri
mary phase satellite measurements from several antennas. Let us describe the corresponding models.
1. THE GENERAL FORMULATION OF THE PROBLEM
The problems of deep integration in one form or another are technically reduced to the solution of the
linear stochastic estimation problem in the general form:
(1.1)
Here,
x
is the state vector with the following components:
Errors of determination of the satellite orientation;
Parameters of instrumental errors of inertial sensors ARS; and
Integer uncertainties of phase measurements;
is the matrix of a linear dynamic system; is the measurement vector; is the measurement matrix;
is the vector of random error of a dynamic system; and is the vector of random measurement errors.
The models of matrix parameters
A
and
H
are based on the models of orientation error equations and
linearized models of primary satellite data. For the sake of certainty, let us select the standard Kalman for
() () () (), () () () ().
dx t Atxt qt zt Htxt rt
dt =+ = +
AzH
qr
A Problem of Satellite Orientation Determination by Spaced
Satellite Antennas and Angular Rate Sensors
A. Dzhepe, A. V. Kozlov, and A. A. Nikulin
Moscow State University, Moscow, Russia
Received November 17, 2014; in final form, January 15, 2015
Abstract
—The estimation models are given in terms of a scheme with the deep integration of the
SNS–ARS data. Several assessments of the accuracy of the integrated navigation solutions are dis
cussed using the model trajectory and angular motion of the UniversitetskiiTat’yana2 satellite.
DOI:
10.1134/S1064230715030053
CONTROL SYSTEMS
OF MOVING OBJECTS

652
JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
Vol . 54
No. 4
2015
DZHEPE et al.
mulation of the estimation problem (1.1). Let us consider that the random errors are described by a vector
random flatnoiselike process with the given intensity and zero expected value.
2. MODEL EQUATIONS OF THE OBJECT ORIENTATION
The orientation of the object will be characterized by the orientation matrix
C
of the coordinate system
Ms
(
is the origin of coordinates, is its name) related to the object case with respect to the Greenwich ref
erence triad
O
η
, which is traditionally used in the satellite navigation problems. The orientation angles are
uniquely determined by the elements of the matrix. To estimate
С
'
of the orientation matrix
C
in the nav
igation computer system, the Poisson kinematic equation is numerically integrated
(2.1)
by using the measurements of of the ARS and specified conditions
С
'(
t
0
)
. Hereinafter, (a prime) is
used to denotate the measurement results or calculation with the measured values,
is the skewsymmetric matrix formulated according to the measured vector of the absolute angular speed
of the object ; is the matrix constructed by vector
u
= [0 0
u
3
]
T
, the Earth’s angular rota
tion speed,
u
3
≈
15.041
grad/h.
Let us introduce the error of the ARS measurements (gyroscopic drift)
ν
and the small rotation vector
β
(
t
)
characterizing the error of the model orientation matrix
С
'(
t
)
[2]:
,
Here,
E
is the identity matrix 3
×
3 in size, is the skewsymmetric matrix put in correspondence with
vector
β
.
In linear approximation, the following equation is true [2]:
(2.2)
Let us use the simplest model for the gyroscopic drift
ν
, when this error is presented by the sum of the
unknown constant and flat noise:
(2.3)
Here, is the vector random flat noiselike process with the given intensity and zero expected value.
3. MODELS OF LINEARIZED MEASUREMENTS OF SNS
Being well known, the models of satellite measurements in the case of using the code pseudoranges
and Doppler pseudospeeds from the base antenna [1] are not described below. Let us consider the cor
rectional models of differential combinations of phase measurements from the spaced antennas showing
the special nature of the ARS—SNS integration problem.
Let us use the simplified model of phase measurement obtained on the
j
th antenna for the
i
th sat
ellite since the receivers are beyond the atmosphere of the Earth [3]:
Here, is the distance between the
j
th antenna and
i
th satellite; and
f
are the wave length and the fre
quency of a radio signal; are the errors of the receiver and satellite hours, is the integer uncer
tainty of phase measurement; and is the random error of phase measurement.
M
s
ˆ
''''CCCu=ω −


'ω'∗
32
31
21
''
0
ˆ''
'0
''
0
⎡⎤
ω−ω
⎢⎥
⎢⎥
ω= −ω ω
⎢⎥
⎢⎥
ω−ω
⎣⎦
T
'123
'''
[]ω=ωωω u

'ω=ω−ν ˆ
()'.CE C=+β
ˆ
β
ˆ'.β=ωβ+ν

0.qβ
ν=ν +
qβ
j
i
Zϕ
()
.
j
js
jj
i
iiii
ZfTN
ϕ
ρ
= + Δτ − Δ + + Δϕ
λ
j
i
ρλ
,i
TΔτ Δ
j
i
N
j
s
i
Δϕ

JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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A PROBLEM OF SATELLITE ORIENTATION DETERMINATION 653
The coordinates of all the base vectors connecting the phase centers of the antennas are known in
the axes of . Let us denote the model value of the vectors in the Greewick axes
О
η
:
by means of the model orientation matrix
С
'
of the coordinate system
Ms
'
with respect to the Greenwich
triad.
Then, we implement two transformations.
1. Let us form the second differences of phase measurements: , where
index
m
denotes the reference (master) antenna,
k
identifies any (slave) antenna, and 0 is the leading sat
ellite. It is assumed that the constant biases compensate each other in the given measurements.
The obtained differences are proportional to the project of the base vector on the difference of the direc
tion vectors:
(3.1)
Here, are the satellite and object coordinates in the Greenwich axes
О
η
,
, which are the
second differences of the integer of uncertainties and flat measurements.
2. Let us form the measurements in the small , where is the useful signal
of first difference , which was calculated by means of the satellite solutions:
.
Then, we obtain
(3.2)
Measurement (3.2) can be formed for any combination of two or more satellite antenna. The features
of the model (3.2) and, correspondingly, the estimation problem (1.1) are the unknown integers .
This approach is used to solve the integer uncertainties . At first, the floatestimations of integer
uncertainties and their covariations by means of the estimation algorithm, and, then, the follow
ing optimization problem is solved by the known LAMBDAmethod [4], which idea is an optimization of
the enumeration by a set of integer vectors through the transformation of the weighting matrix :
(3.3)
where is a set of integer vectors with dimension
n
. Further, the components obtained as a result
of solving problem (3.3) are subtracted from measurements (3.2) and, therefore, excluded from the state
vector of the estimation problem (1.1).
4. MODELING
To investigate the testability of the state vector of the system specified by Eqs. (2.2) and (2.3) by mea
surements (3.2), the numerical modeling was carried out based on the characteristic of the trajectory and
angular motion of the UniversitetskiiTat’yana2 satellite [5].
The orbit parameters: the semimajor axis is 7210220 m, the eccentricity is 0.00321, and the orbit incli
nation is 98.8 grad.
k
s
l
Ms
T
''
kk
s
lCl
η
=
00
()
kmk mk
iii
ZZZ ZZ
ϕϕϕ ϕϕ
Δ= − − −
*
,TΔτ Δ
TT
0
0
()()
1*.
i
sat sat
kkk
iii
mm
i
ZlN
ϕη
⎡⎤
η−η η−η
Δ= − +Δ+Δϕ
⎢⎥
λρρ
⎣⎦
,
i
sat
ηη *
,
k
ii
NΔΔϕ
comp
kkk
iii
zZZ
ϕϕϕ
Δ=Δ −Δ
comp
k
i
Zϕ
Δ
k
i
zϕ
Δ
TT
0
0
(')(')
1'
''
i
comp sat sat
kk
imm
i
Zl
ϕη
⎡⎤
η−η η−η
Δ= −
⎢⎥
λ⎢⎥
ρρ
⎣⎦
TT
0
0
(')(')
1ˆ*
'.
''
i
sat sat
kkk
iii
mm
i
zlN
ϕη
⎡⎤
η−η η−η
Δ= − β+Δ +Δϕ
⎢⎥
λ⎢⎥
ρρ
⎣⎦
k
NΔ
k
NΔ
k
NΔ

N
PΔ
N
PΔ
arg min ,
N
kn
kkk
zP
NZ
NNN
Δ
Δ∈
Δ= Δ−Δ

n
Zk
z
NΔ

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JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
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DZHEPE et al.
The satellite motion around the center of mass was specified by three rotation angles about the orbital
coordinate system according to [5]:
where are the amplitudes of changes of angles and , respectively; are the frequencies of
the orientation angle changes’; and are the initial phases.
The first differences of the phase measurements were modeled according to (3.1) with constant uncer
tainties and random errors with a meansquare deviation of order 0.01 m. The Poisson equation (2.1) was
integrated by using the model measurements of ARS with errors of te initial orientation of order 1 grad.
The ARS indication error was modeled according to description (2.3), = 30 grad/h (100 Hz). The
variants of solving the problem in the presence of the data from two or three satellite antennas and various
sets of satellites were considered.
After using the data from three antennas located in the vertices of 45
°
rightangled triangle with the
length of the side face of 1m and all the visible satellites, the meansquare deviation (MSD) of the errors
of all the vector components of the small rotation is about 0.1 grad (Fig. 1).
When only the data from three satellites from the 1000th to the 1500th s are available (Fig. 1), the MSD
of the error varies depending on the geometry of the visible constellation within 0.5 grad. For the weakly
coupled systems, the correction is absent in this time period, and the error increases up to 2 grad.
Using the data from the two satellite antennas placed along the axis
Ms
1
at a distance of 1 m from each
other, the first component
β
1
of vector
β
is defined only at the angular motion of the satellite, the MSD of
the error of angle estimation is 1.5 grad one hour after the execution of the algorithm (Fig. 2). The low
estimation accuracy is caused by the fact that
β
1
is observed indirectly through
β
2
and
β
3
. The MSD of the
estimations of the rotation angles around the second and third axes is approximately 0.1 grad. The data of
only three satellites are available from the 1000th to the 1500th (Fig. 2).
The errors
β
2
and
β
3
increase to 0.5 grad, while the error
β
1
has an insignificant change. For weakly
bounded systems, the correction is absent in this time period, and the error over all the angles increases up
to 2 grad. After the secondary data of SNS appear, the estimation error
β
1
decreases but still remains
slightly more than in deep bounded systems.
To verify a possible increase in accuracy, the model measurements were processed by applying the
coarser data on the pseudoranges from the spaced antennas (MSD of 1 m). There is no material improve
ment of the resulting estimation of the orientation in this case.
305
() 20
970
ttψ= + 
() sin( )
() sin( )
tA t
tA t
ϕϕϕ
ϑϑϑ
ϕ= ω+α
ϑ= ω+α /s
,0.5
2
,deg,
600
AA
ϕϑ
ϕϑ
š
π
ωω≈
,AA
φϑ ϕϑ,
ϕϑ
ωω
,
ϕϑ
αα
1
[]qβ
σ
10
0
0 500 1000 1500 2000 2500 3000 3500
Tim e, s
β
1
β
2
β
3
10
−
1
MSD of the orientation error, deg
Fig. 1.
The MSD of the orientation error under using three antennas.

JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL
Vol . 54
No. 4
2015
A PROBLEM OF SATELLITE ORIENTATION DETERMINATION 655
CONCLUSIONS
The models of the integration algorithm of multielement SNS–ARS in the problem of satellite orien
tation in the context of the deep integration scheme are presented. Modeling the motion of the Univer
sitetskiiTat’yana2 satellite and the corresponding measurement data showed the workability of the algo
rithms for determining the orientation by ARS and phase measurements from the spaced satellite anten
nas in the deep integration mode. The efficiency of the deep integration scheme compared to the weak
integration scheme is shown.
Using three satellite antennas allows one to define the orientation angles with the accuracy of order
0.1 grad for more than six visible satellites or of order 0.5 grad for three satellites. The rotation angle
around the base vector when applying two antennas is defined only at the angle motion with an accuracy
of order 1.5 grad.
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Navigation System Mathematical Foundation
, Pt. 2:
Applications of Optimal
Estimation Methods to Navigation Problems
, 2nd ed. (MAKS Press, Moscow, 2012) [in Russian].
2. A. A. Golovan and N. A. Parusnikov,
Navigation System Mathematical Foundation
, Pt. 1:
Mathematical Models
of Inertial Navigation
, 3rd ed. (MAKS Press, Moscow, 2011) [in Russian].
3. N. B. Vavilova, A. A. Golovan, N. A. Parusnikov, and S. A. Trubnikov,
Mathematical Models and Algorithms for
Processing GPS Navigation System Measurements, Standard Mode
(Mosk. Gos. Univ., Moscow, 2009) [in Rus
sian].
4. G. Giorgi, P. J. G. Teunissen, S. Verhagen, and P. J. Buist, “Integer ambiguity resolution with nonlinear geo
metrical constraints,” in
Proceedings of the VII HotineMarussi Symposium on Mathematical Geodesy, Rome,
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5. V. V. Aleksandrov, A. D. Belen’kii, D. I. Bugrov, A. V. Lebedev, S. S. Lemak, and W. F. Guerrero Sanchez,
“TelemetryBased Estimate of Orientation Accuracy for the Tat’yana2 Satellite,” Mosc. Univ. Mech. Bull.
66
,
72 (2011).
Translated by A. Evseeva
10
0
0 500 1000 1500 2000 2500 3000 3500
Tim e, s
β
1
β
2
β
3
10
−
1
MSD of the orientation error, deg
Fig. 2.
The MSD of the orientation error under using three antennas.
SPELL: 1. te