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STAR PATTERN IDENTIFICATION : APPLICATION TO THE PRECISE ATTITUDE

DETERMINATION OF THE AURORAL SPACECRAFT


J.Cl . Kosik


Division Mathematiques Spatiales, CNES, Toulouse


Introduction :
The Auroral Spacecraft is one of the four satellites launched for the
Interball project by the Russian Space Agency in cooperation with the
international scientific community which are dedicated to magnetospheric
research. This spacecraft is subject to the effects of energy exchange
between the flexible booms and the main body. This energy pumping over
induced a rise of the nutation angle and the exponential rise was damped by
gas jets each 12 hours. As a consequence the spin rate was periodically
slightly modified and the images of the on-board cameras were blurred.
Orbit determination accuracy was also limited by the availability of only
one tracking station, the orbit error inducing a shift in the reference
magnetic field. The measured magnetic field was also perturbed by Field
Aligned Currents, thus the attitude determination went inaccurate in the
region of scientific interest. All these problems which were left aside in
a first intensive attitude production were recently reassessed and the
present status of their solution is described in this work. Of particular
interest was the possibility to use star information given by the canadian
auroral cameras. Auroral cameras cover a large field of view (more than 400
square degrees) and stars are observed outside the limb of the Earth. This
star information can be used as soon as the stars seen by the auroral
cameras can be identified. Several identification algorithms were
previously developed for the Gamma project. These algorithms have been
adapted to the Sun workstation. In the first part of this paper we briefly
review the strengths and weaknesses of these algorithms. For the Auroral
cameras we have selected the Distance-Orientation algorithm. This algorithm
requires an approximate attitude determination. Our immediate goal was the
obtention of this approximate attitude and the tools developed are
described in the second part of this work. A quaternion approach using the
Triad method was interested but failed in the FAC's regions. In another
attempt the attitude was calculated at the first point using the Triad
method and then numerically propagated using the Euler-Poinsot equations
and gradient gravity torques. This method gave a good average attitude with
the prerequisite of the right spin rate. It also had the advantage to
« ignore » FAC's regions. Star identification was possible after an
adequate choice of the parameters. A series of results for Orbit 689 and
the two cameras are presented. These results are discussed in the last part
of the work and future steps of this work are sketched.

THE IDENTIFICATION OF STARS:
Identification of star patterns was first used in CNES for
the Gamma Project. Three algorithms have been developped and tested on star
pattern images obtained in Pamir with a star sensor from Poland. One of
these algorithms, the polygonal algorithm, has been described by Gottlieb
in (Ref.1) . Two new algorithms, the Pivot algorithm and the Distance-
Orientation algorithm have been developed. We will briefly review these
three algorithms.

The Polygon Match :

This algorithm is described by Gottlieb (Ref.1). From a
set of measured stars, we arbitrarily select two stars, 1 and 2, and
compute the corresponding angular separation [pic]. We then look for all
pair of stars (i,j) in a finite region of the catalog around the
approximate boresight of the star sensor, such that :

[pic]

where [pic] is the uncertainty in the distance measurements of the star
sensors and [pic]is the angular separation calculated for the pair of stars
in the catalogue.


















FIG.1




Usually the number of star pairs is not negligible. Even if we obtain one
pair of stars, there are two possible identifications. In both cases it is
necessary to select another measured star, 3, from which we compute two
more separations, [pic] and [pic]. We now look for a third star, k, in the
catalogue that can be combined with the previous pairs such that :




[pic]
or
[pic]













FIG.2

If more than one pair is found the process must be repeated until the
identification is unambiguous.

Pole Technique :

If the number of stars is important enough, more than 7 stars, it is
possible to use the pivot algorithm . We choose a given measured star
(labeleled 1 in Fig.3) and call it the pole star. To this pole star we
associate all the measured distances to the other remaining distances, 8
distances for a total number of 9 measured stars. For each measured
separation we look for the pairs (i,j) in the catalog which satisfy the
distance criterion as defined in the previous paragraph. We obtain a set
[pic] where m identifies the measured star different from the pivot star.
The pole star must be found in each of these sets [pic]. The pole star is
thus the intersection of all the subsets [pic] (Fig .3) .






















FIG.3







Distance-Orientation algorithm :

It is possible to improve the polygon technique if we have
a crude estimate of attitude. The crude estimate of the attitude enables
the projection of the star catalogue region onto the star sensor field of
view. The same stars should be found, projected onto a tilted
« theoretical » star sensor . As a consequence the adequate pairs should
verify the distance criterion as above, but should also have approximately
the same orientation : each pair can be considered as a vector and the
vectors both of measured stars and projected catalogue stars should have
approximately the same orientation (Fig.4). The distance-orientation
criterion can be expressed as :

[pic]


Where the last inequality expresses that the angle between the two vectors
should be small enough .














and







FIGURE 4



In this algorithm a set of catalogue pairs is obtained from the distance
criterion. The application of the orientation criterion leads to a subset.
Each pair which can satisfy the orientation criterion for angles around 0њ
and also for angles around 180њ. The obtention of an angle around 180њ
simply means that the order of the catalogue stars is opposite to the right
order, i .e., if a catalogue pair (1c,2c) compared to measured pair (1m,2m)
differs from about 180њ , then the catalogue star 1c corresponds to star 2m
and the catalogue star 2c corresponds to star 1m : the orientation-distance
algorithm is already more powerful than the distance criterion algoritm as
the aforementionned ambiguity is suppressed. If more than one pair of stars
is obtained it is necessary to select another measured star and continue
the process.

Algorithm efficiency :

The distance-orientation algorithm which requires a crude
attitude attitude is the most powerful algorithm. Results can be obtained
with 2 or 3 measured stars. The polygon match algorithm requires 4 to 5
stars. The least efficient algorithm is the pivot algorithm which requires
more then 7 stars. In any case the corresponding software for these
algorithms is quite complex and increases with the number of measured stars
and the number of stars of the catalogue in the boresight region. As a
consequence the identification of stars is the most difficult in the Milky
Way region. Some statistical studies have been performed by Kosik (Ref.2)

PRECISE ATTITUDE DETERMINATION OF STARS USING THE AURORAL CAMERAS:
A precise attitude
determination of the Interball Auroral spacecraft can be obtained using the
star patterns observed by the auroral cameras. The identification of
stars is obtained with the distance-orientation algorithm which requires
the knowledge of the crude attitude of the spacecraft. In order to avoid
the problems linked to Field Aligned Currents an Euler Poinsot scheme has
been developed.

Calculation of the approximate attitude :
The attitude of the spacecraft at the initial
point can be obtained using the sun and magnetic field information at this
point and the TRIAD method described by Lerner (Ref.3). This initial
attitude is propagated using the Euler-Poinsot equations and the gravity-
gradient torques. The Euler-Poinsot equations are solved by numerical
integration and the gravity gradient torques are calculated using the
position of the spacecraft given by the orbit file. The integration process
is stopped for the time of the aurora and star snapshot. Figure 5,6,7 show
the reference magnetic field components together with the measured magnetic
field components in the Solar Magnetospheric frame of reference. An
oscillation of the measured magnetic field components can be observed.

[pic]

FIGURE 5


[pic]
FIG.6



[pic]

Fig .7









Problems encountered in the calculation of the approximate attitude :
In the standard case the
tracing of the magnetic field measured and calculated gives an oscillating
curve for the measured magnetic field and a non-oscillating curve for the
theoretical magnetic field as in figures 5, 6, 7. However, the theoretical
magnetic field curve can deviate from the measured magnetic field as in
figure 8,9,10.


[pic]
Fig .8
[pic]
Fig.9


[pic]
Fig.10

This phenomenon is due to the non adequate spin rate in the Euler-Poinsot
integration. The adequate spin rate can be found through the minimization
of the difference between the modulus of the reference magnetic field and
the modulus of the measured magnetic field, difference calculated along the
orbit arc. Figures 8, 9, 10 correspond to a spin rate of 0.05236 radians/s
and figures 11, 12 and 13 correspond to a spin rate of 0.05246 radians/s.

[pic]
Fig.11
[pic]
Fig.12
[pic]
Fig.13



The adequate spin was found to be 0.05256 rad/s as shown in figures 14,15
and 16





[pic]

Fig .14

[pic]


Figure 15





[pic]
Fig.16


Another problem can be encountered when the determination of the attitude
of the initial point is performed in a perturbed magnetic field region such
as a FAC's region. Figures 17, 18 and 19 show the subsequent consequence on
the Euler-Poinsot integration process.

[pic]
Fig. 17
[pic]
Fig.18
[pic]
Fig.19



Figures 20,21 and 22 show a typical case of orbitography offset : the
position of the spacecraft at some time t is different from the one by the
orbitgraphy . The calculated reference field does not coincide with the
measured magnetic field due to this orbit shift.
[pic]
Fig.20
[pic]

Fig.21
[pic]
Fig.22

Calculation of the precise attitude :
With the knowledge of the star camera orientation in
the spacecraft frame of reference is is thus possible to calculate the star
camera boresight in the inertial frame. We set a magnitude limit, 5 in the
case of the auroral camera, and define an enlarged field of view of the
star camera and thus take into account the attitude uncertainty effects.
From the catalogue of stars we can extract a subset of stars which belong
to the auroral camera field of view. We then project these stars onto
the star camera field of view. Part of the aurora camera field of view is
hidden by the Earth and only stars outside the limb of the Earth will be
available for the identification as shown in the figure below.
[pic]




At this stage we have two subsets : the subset of catalogue stars and the
subset of measured stars. The identification of the measured stars can take
place using the orientation-angle algorithm described above. In this
algorithm we have the possibility, depending on the number of the measured
stars, to choose 2, 3 or 4 measured stars. After adjustment of the
parameters, the identification will be successfull if only one duo, trio or
quartet of catalogue stars is found. Now we have the positions of the
identified stars both in the star camera frame of reference and in the
inertial frame of reference. It is thus possible to use the TRIAD method or
a least squares method to obtain the transformation matrix between the
inertial frame of reference and the star camera frame of reference. Hence
we can deduce from the attitude of the star camera the attitude of the
spacecraft. The Auroral spacecraft has two cameras which deliver two star
patterns at the same time. It is thus possible to estimate the accuracy of
the attitude. The attitude of the Auroral spacecraft has been calculated
for orbit 689 and the two on-board cameras. The results are presented in
Table I. In the best cases a precision of 0.01 degree can be achieved.


TABLE I

|Date |Hour |Camera |Number of |Number of |Euler |
| | | |stars in |measured |rotatio|
| | | |the field |stars |n |
| | | |of view | | |
|97/2/11|2h 18mn |0 |40 |3 |4.847 |
| |41s | | | | |
|same |same |1 |29 |3 |5.161 |
| | | | | | |
|97/2/11|2h 20mn |0 |38 |3 |4.983 |
| |41s | | | | |
|same |same |1 |27 |3 |4.875 |
| | | | | | |
|97/2/11|2h 22mn |0 |41 |3 |4.954 |
| |40s | | | | |
|same |same |1 |27 |2 |5.089 |
| | | | | | |
|97/2/11|2h 24mn 39|0 |44 |3 |5.896 |
|same |same |1 |29 |2 |5.878 |
| | | | | | |
|97/2/11|2h 26mn 39|0 |40 |3 |5.113 |
|same |same |1 |26 |2 |5.222 |
| | | | | | |
|97/2/11|2h 28mn 39|0 |37 |2 |5.048 |
|same |same |1 |22 |2 |5.074 |
| | | | | | |
|97 /2/1|2h 30mn 38|0 |40 |3 |5.839 |
|1 | | | | | |
|same |same |1 |25 |2 |4.248 |





It is possible to evaluate the discrepancy between the approximate attitude
obtained with the triad method plus the Euler-Poinsot extrapolation and the
precise attitude obtained with the star cameras : One can calculate the
generalized Euler rotation between the precise spacecraft attitude and the
approximate attitude as mentionned by Markley (Ref.4). It is simply defined
as :

[pic]

where [pic] is the trace of the transformation matrix [pic] between the
approximate spacecraft attitude frame and the precise spacecraft attitude
frame.
From table I , one can notice that this angle [pic] is around 6 degrees.

Conclusion :
The auroral cameras on-board the Auroral spacecraft offer a unique
possibility to test star identification algorithms. Presently only one
algorithm has been tested and intensively used, the distance-orientation
algorithm. Other algorithms, like the polygonal match and the pivot
algorithm will be adapted to the Sun workstation. Frames containing up to
10 measured stars offer the possility to test the pivot algorithm also
developed for the Gamma project. This algorithm did not work properly for
Gamma because of the insufficient number of measured stars. Results
obtained with the distance-orientation algorithm indicate a discrpancy of
about 6 њ between the attitude calculated with the magnetic field and the
Sun vectors and the attitude calculated with the stars for orbit 689. Early
results for orbit 713 seems to corroborate this figure. Further work will
continue with the star camera data base and the other algorithms.
Consequences for the classical attitude will be drawn by a reevaluation of
the magnetic measurements (biases,offsets). The attitude motion of the
spacecraft will require more fundamental work on the possible influence of
flexible booms or other energy exchange effects.


REFERENCES

Ref .1 : Gottlieb D.M. : Star Identification Techniques, in Wertz, p 259.

Ref.2 : Kosik J.Cl. : Star Pattern Identification Aboard an Inertially
Stabilized Spacecraft, Jal. of Guidance, Control, and Dynamics, Vol.14, 2,
p.230, 1991.

Ref. 3 :Lerner G. : Three-Axis Attitude Determination, in Wertz, p.420.

Ref.4 : Markley F.L. : Parametrization of the Attitude, in Wertz, p.410.

Ref.5 :Wertz J.R. : Spacecraft Attitude Determination and Control ,
D.Reidel Publishing Company, 1974, Dordrecht, Holland.








-----------------------
*

1m



*

2m

* 3m

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First step : We look for stars i,j
Such that

[d(i,j) - d12 ] < ?

We usually obtain many pairs

* 5m

* 1m

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d13

d14

d15

*

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3m

4m

5m

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E3

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Is equivalent to



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