Nomenclature
v The star velocity on CCD arrays (pixel=s)
p Velocity coefficient on CCD arrays (pixel=(rad=s))
T Observation time on the CCD arrays (s)
x X-axis on CCD coordinates (pixel)
! Angular velocity in body frame (rad=s)
f(x) Line Spread Function(LSF) forone second observation
1 Introduction
These days, nano-satellites attract interest, as
a way to launch observation and demonstration
equipment frequently with practical cost. Nano-
JASMINE (Nano-Japan Astrometry Satellite Mission
for INfrared Exploration )is currently underdevelopment
at Intelligent Space System Laboratory
(ISSL) Univ. of Tokyo in cooperation with National
Astronomical Observatory of Japan (NAOJ)
The main purpose of the project is to prove and
demonstrate the key technologies required for JASMINE
(Japanese Astrometry Satellite Mission for
Infrared Exploration) in a real space environment.
The mission is to measure the 3D positions of
stars to an accuracy of 1.8 mas(miri-arcsecond).
In order to get accurate star position data, Nano-
JASMINE should be stabilized to less than 740 mas
/ 8.8s(equivalent to 4e-7rad/s) accuracy during observation.
Currently there are not sensors available
for a nano-satellite to get accurate attitude information
as is required by the mission.
2 Nano-JASMINE and ADCS
overview
Table 1 and Table 2 show the outline of
Nano-JASMINE and ADCS hardware respectively.
1
Figure 1: Nano-JASMINE
Table 1: Specifications of Nano-JASMINE mission
Size: 40 40 40 cm
Mass: 14kg
Attitude Control : Three Axis Stabilization
Orbit: Sun-synchronous Orbit
Mission Life: 2 years
Nano-JASMINE should be stabilized attitude to a
high accuracy during observation. Currently there
are not sensors and methods to determine attitude
such a high accuracy. To solve this problem, Nano-
JASMINE will determine and control attitude using
star blurred image data from mission telescope.
Specifically, this method assesses the quality of the
star image based on how blurred it appears. To
get star blurred star image, satellite should stabilized
with sensors such as FOG (Fiber Optical
Gyro).Nano-JASMINE stabilize its attitude step
by step before observation. Fig.2 shows outline of
control strategy of Nano-JASMINE mission. This
paper focuses on observation phase.
3 Control attitude strategy to
using star images
Nano-JASMINE will determine and control attitude
using star blurred image data. In the simulation,
Nano-JASMINE is fed synthesized star image
or PSF (Point Spread Function) data as depicted
in Fig.5. As shown in Fig.4, each star image contains
attitude information about two axes. If the
satellite is unstable in X-or Y-axis directions (coordinate
definition is shown in Fig.3), the PSF will be
Table 2: ADCS hardware overview
Sensors
STT ~ 2
FOG ~ 3
SunSensor ~ 6
GAS ~ 3
GPS ~ 1
MissionTelescope ~ 1
Actuators
RW ~ 4
MTQ ~ 3
Figure 2: The outline of Control Phase in Nano-
JASMINE mission
extracted in the Y-axis direction of the CCD pixel
coordinate. If unstable in the Z-axis, the image will
be extracted in the X-axis. The satellite will obtain
star image from two directions which are both captured
on a single CCD imager (X-axis and Y-axis
in the satellite body-frame). This means the observation
direction decides which axis information is
contained within a star image: for y-axis observations
the XZ-plane is the focus, in the x-axis information
about the YZ-plane is obtained. The angular
velocity magnitude can be gained from the LSF
(Linear spread function) which may be calculated
from the PSF compressed information as shown in
Fig.4. To get attitude and be useful information
from LSF, several problems should be solved.
Firstly, the star image comes from two directions
2
Figure 3: Nano-JASMINE coordinates definition
Figure 4: PSF(Point Spread Function)and
LSF(Line Spread Function)
Figure 5: Star image from mission telescope in the
simulation
onto a single field of view (X-axis and Y-axis in
satellite body-frame) based on mission demands.
This requires a further step in the process whereby
the satellite must estimate which direction a particular
star image comes from (X-axis and Y-axis)
before determining attitude. Secondly, it is simple
to obtain magnitude of angular velocity from
the star image LSF; however measuring a precise
vector is more difficult.
To solve the first problem, the skew angle of
the star image is estimated and used to determine
pointing in the XY plane. Fig.5 shows a sample
field of view from the mission telescope. Stars
within the image may be skewed at inclines in a mutually
perpendicular frame. This is because satellite
gets image data from two directions simultaneously.
As a result, the star distortion contains
information of observation direction.
Figure 6: The star image from mission telescope
(both X-axis and Y-axis are positive)
Figure 7: The star image from mission telescope
(X-axis is negative and Y-axis are positive)
Fig.6 and Fig.7 show how to decide which direction
the star image comes from. In the case of
Fig.6(the angular velocity of both X-axis and Yaxis
is positive, in other words, the signs of X-and
Y-axis are agreed), PSF or star image inclined to
two sides. In this case, satellite can easily decide
which direction PSF comes from. In the case of
Fig.7(the angular velocity of X-axis is positive and
3
Y-axis is negative, In other words, the signs of Xand
Y-axis are different), PSF inclined to only one
side. In this case, satellite cannot decide direction.
If Nano-JASMINE keeps the sign of angular velocity
to agree during observation, satellite can decide
his attitude.
Also, the second problem will be solved with the
skew angle of star image. The satellite will try to
keep the sign of angular velocity during observation
(to agree X-and Y-axis sign). If satellite signature
change, the star images from mission telescope
inclined only one side, because the sign of
X-and Y-axis are different. That is to say,satellite
can know if the sign of angular verbosity is change
or not. If star image become to be incline to only
one side, satellite estimate the sign of angular velocity
change, then change the sign of angular velocity
to be X-and Y-axis are agreed . Satellite can get
attitude from star skew angle information.
4 Relationship between PSF
and attitude stability
In this section ,the relationship between LSF and
angular velocity will be stated.
The variance of LSF is defined as follow in continual
model.
V ar(x) = E(x ! 1)2 (1)
= E(x2) ! 12 (2)
Where 1 is the average of LSF and Var(x) is
the varianse of LSF. After several seconds observation,
LSF defined as follow.
Z
1
!1
Z T
0
f(x)dtdx = 1 (3)
Z
1
!1
f(x)dx =
1
T
(4)
Where T is observation time and f(x) is PSF
(point Spread Function) in the continual model
in the case of one second observation. Nano-
JASMINE will get star image whose apparent magnitude
is different. As Eq.3 , LSF is normalized to
be treated easily.
If satellite is unstable , LSF on the CCD arrays
moves to one side. In this case ,LSF is calculated
as follow,
Z T
0
f(x ! vt)dt (5)
Where v is velocity on the CCD coordinate
(pixel/s),and T is observation time. Then the variance
of LSF can be calculated as follow,
Z
1
!1
x2
Z T
0
f(x ! vt)dtdx ! (
Z
1
!1
x
Z T
0
f(x ! vt)dtdx)2 (6)
In Eq.6 second term represents the average of LSF.
The first term of Eq.6 is calculated with the commutativity
of convolution as follow,
Z
1
!1
x2
Z T
0
f(x ! vt)dtdx (7)
=
Z
1
!1
Z T
0
x2f(x ! vt)dtdx (8)
=
Z
1
!1
Z T
0
(x ! vt)2f(x)dtdx (9)
=
Z
1
!1
(x2T + v2T2=3 ! xvT2)f(x)dx (10)
In Eq.10, the first term represents the variance of
LSF in the case of v = 0, and is calculated as follow,
Z
1
!1
Z T
0
x2f(x)dtdx (11)
Eq.10 shows the LSF variance when the satellite
attitude is stable.
The third term in Eq.10 equal to be zero, because
average of LSF in one second observation is defined
as 0.
Z
1
!1
xf(x)dx = 0 (12)
The second term in Eq.6 also is calculated with
the commutativity of convolution.
4
(
Z
1
!1
x
Z T
0
f(x ! vt)dtdx)2 (13)
= (
Z
1
!1
(x ! vt)
Z T
0
f(x)dtdx)2 (14)
= (!vt=2)2 (15)
From Eq.10 and Eq.15 ,Eq.6 is represents as follow.
Z
1
!1
(x2T)f(x)dx + v2T2=3 ! v2T2=4 (16)
Z
1
!1
(x2T)f(x)dx + v2T2=12 (17)
Velocity on CCD coordinate has relation to angular
velocity as follow,
v = p! (18)
Where p is coefficient defined as pixel velocity((
pixel/(rad/s))). Then relationship between angular
velocity and LSF variance is calculated as follow,
Z
1
!1
(x2T)f(x)dx + !2p2T2=12 (19)
Eq.19represents that relationship between angular
velocity and LSF variance is Quadratic function.
?2
LSF = av2 + c (20)
Where coefficient a and c are defined as follow,
c =
Z
1
!1
(x2T)f(x)dx (21)
a = p2T2=12 (22)
The coefficient a is determined only from observation
time. The coefficient c which is the variance
of LSF is determined from PSF shape. Fig.8 shows
the relation of LSF variance and angular velocity
in theory.
Figure 8: The relationship between LSF variance
and attitude stability in theory
5 Simulation results and conclusions
In Nano-JASMINE project, JASMINE simulator
is underdevelopment in order to support investigations
into error budgets, sampling control strategy,
scientific performances, etc.(Ref.6) In this research,
the feasibility of method to control and dynamic
model is analyzed with this simulator. The result
for simulation of the relationship between LSF variance
and stability is plotted in Fig.8.
Figure 9: PSF(Point Spread Function)and
LSF(Line Spread Function)
This relationship is shown as quadratic function
as theory. But the theory and simulation result
are not in 100 percent agreement, which means the
plots in the Fig.9 vary narrowly compare to theory
5
Figure 10: The relationship between LSF variance
and attitude stability in the simulation
because of noises. The dominant noise is shot noise.
Fig.10 shows how vary the LSF variance when the
satellite is stabilized ideally. In the case of bright
star, LSF variances vary relatively few. In the case
of dark star, LSF variances change widely, though
satellite attitude perfectly stabilize. From this simulation
result, satellite can decide accuracy of LSF
variance from the magnitude of stars. In this research,
attitude stability is decided more accurately
with Kalman Filter, that is to say, the brighter
star is gotten, the more accurate information satellite
treat as.(Ref.7) Fig.11 show simulation result of
this method. With estimation from the information
of stars brightness, satellite can get more accurate
angular velocity data. This method enable satellite
to determine angular velocity to 110!8rad=s and
be stabilized to less than 740 mas / 8.8s.
Based on simulation results, the feasibility of this
method to control attitude in Nano-JASMINE is
concluded .
References
[1] JASMINE team, hJASMINE project first design
reporth, 2003
[2] Naoteru Gouda, Other (2006). hJASMINE:
galactic structure surveyer.h SPIE
Astronomical Telescopes and Instrumentation
2006. 6265-143
[3] N. SAKO, Y. HATSUTORI, T. TANAKA,
S. NAKASUKA @(2006). hABOUT NANOFigure
11: Simulation result using the KalmanFilter
SATELLITE FOR INFRARED ASTROMETRY
(NANO-JASMINE) PROJECT.h International
Symposium on Space Technology and
Science, ISTS 2006-f-08
[4] Yoshiyuki Yamada, Other (2006). hJASMINE
Simulator.h SPIE Astronomical Telescopes
and Instrumentation 2006. 6265-142
[5] Nobutada Sako, Yoichi Hatsutori, Takashi
Tanaka, Takaya Inamori, Shinichi
Nakasuka,hNANO-SATELLITE ATTITUDE
STABILIZATION METHOD USING STAR
IMAGESh,IFAC 2007
@
[6] Yoshiyuki Yamada, Other (2006) @hNano-
JASMINE simulatorh @REPORT OF THE
NATIONAL ASTRONOMICAL OBSERVATORY
OF JAPAN @Vol.10, 1 | 22i2007j
[7] Takaya Inamori ,Shinichi Nakasuka (2007)
hMethod to stabilize a nano-satellite, using
star imageh The 51th The Space Science and
Technology Conference
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