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ISSN 10642307, 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 pseudoranges, Doppler

pseudospeeds, 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 singleantenna 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 UniversitetskiiTat’yana2 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 flatnoiselike 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 skewsymmetric 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 skewsymmetric 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 noiselike 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 pseudoranges

and Doppler pseudospeeds 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

Vol . 54

No. 4

2015

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 floatestimations of integer

uncertainties and their covariations by means of the estimation algorithm, and, then, the follow

ing optimization problem is solved by the known LAMBDAmethod [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 UniversitetskiiTat’yana2 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

654

JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL

Vol . 54

No. 4

2015

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 meansquare 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

°

rightangled triangle with the

length of the side face of 1m and all the visible satellites, the meansquare 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 pseudoranges 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

≈°

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

sitetskiiTat’yana2 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.

REFERENCES

1. A. A. Golovan and N. A. Parusnikov,

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 HotineMarussi Symposium on Mathematical Geodesy, Rome,

2009

.

5. V. V. Aleksandrov, A. D. Belen’kii, D. I. Bugrov, A. V. Lebedev, S. S. Lemak, and W. F. Guerrero Sanchez,

“TelemetryBased Estimate of Orientation Accuracy for the Tat’yana2 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