1. Introduction
Solar sail offers advantage over conventional propulsions
systems that missions arenft constrained by V ? available
from stored reaction mass. This advantage enables many
novel interesting high-energy space mission concepts.
Higher angular resolution in astronomical images
requires increasing apertures of telescope or increasing
baselines of interferometers. The mass of the support
structure of the telescopes increases accordingly, and
propellant for lunch and navigation of long baseline
telescopes is about to exceed technical and financial
boundaries. Formation flying is a revolutionary idea
overcome the mass constraint which combine satellites in
autonomous formation flight to behave just like a rigid
body[1].
The advantages of a mission near L2, as Beichman [2]
point out are, the simplicity and the inexpensive inserting
into Lissajous orbit from Earth. Moreover L2 is a place with
a constant cold temperature with half of the entire celestial
sphere available at all times, which suitable for missions
with heat sensitive instruments.
More on transfer from Earth to L2, it gives benefits such
as, less energy to achieve required which gives more mass
to be delivered, its distance relatively close to Earth
compare others Lagrange Points, availability of multiple
options due to its nature of flexibility and forgiveness.
Typical missions like XEUS (X-ray evolving-Universe
Spectroscopy) mission [3], which consists of two satellites
is main motivation of this research. The future of The
Gamma Ray Astronomy [4] on the development of
significantly higher sensitivity instrumentation, will be
2
solved potentially by formation flying.
Here is proposed, two satellites in a formation flying
placed around L2 point of Sun-Earth system as Mirror
Spacecraft and Detector Spacecraft. This formation flying
powered by solar sail and analyzed in a Restricted Three
Body Problem domain.
2. Solar Sails
Solar consist of large area gossamer structures with a
reflective coating which intercepts the solar photon flux. A
satellites receives impart momentum on the sail and the
reflection results in a reaction force, hence providing
double the force which would be imparted to absorbing
surface.
Fig. 1 Solar sail attitude angles
The sail accelerations can be expressed as follows [5],
2 2
2 cos cos s
s R
ภส
ล ย ี = ศ n (1)
Where ลthe sail reflectivity, Rs is the distance from the Sun,
ยis the pitch angle of the sail normal vector n to the
Sun-line and ตis th yaw angle of the sail of the sail normal
vector to the Sun-line and ภis the solar sail lightness
parameter defined as the ratio of solar radiation pressure to
gravitational attraction:
2
s
s
L
c
ภ
ฮ ส ะ
= (2)
where solar luminosity Ls=3.86x1026W, c is the speed of
light, สs is the solar gravitational parameter and ะis the
ratio of solar sail mass to surface area known as the loading
parameter.
3. Circular Restricted Three Body Problem
Fig.2 Cartesian coordinate system
Congruently with Farquhar work on translunar libration point [6],
the nonlinear equations of motion around L2 in non dimensional
form are expressed as below,
( ) ( ) 3 3
1 2
3 3
1 2
3 3
1 2
1 2 ( 1 ) 1
1 2
1
L L L x
y
z
x y x x x a
r r
y x y y y a
r r
z z za
r r
ส ส
ส ม ม ม
ส ส
ส ส
?
? ? + ? + = ? + + ? + +
?
+ ? =? ? +
?
= ? ? +
???? ??
???? ??
????
(3)
The linearized form of the above system of equations [7]
are
( )
( )
( ) L 3 3
2 2 1
2 1
Where,
1 B
1
L x
L y
L z
L L
x y B x a
y x B y a
z B z a
ส ส
ม ม
? ? + =
+ + ? =
+ =
? ? ? ? ? = + ? ?
+ ? ? ? ?
???? ??
???? ??
???? (4)
4. Control Strategy Fixed Formation
In the formation flying, the Mirror satellite is put the leader and
the Detector Satellite as the follower. The relative motion of the
follower respected to the leader are shown below
l f ? = r r ฯ ???? ???? ???? (5)
Hence,
( )
( )
2 2 1
2 1
x y L x x
y y L y y
y L z z
B a
B a
B a
ฯ ฯ ฯ
ฯ ฯ ฯ
ฯ ฯ
? ? + =
+ + ? =
+ =
???? ??
???? ??
????
(6)
For the formation frozen in rotating frame will require that
0
0
0
x x
y y
z
ฯ ฯ
ฯ ฯ
ฯ
= =
= =
=
???? ??
???? ??
????
(7)
Consequently the accelerations needed are
3
( )
( )
2 1
1
x L x
y L y
z L z
a B
a B
a B
ฯ
ฯ
ฯ
= ? +
= ?
=
(8)
analytically, for the formation to be fixed in inertial frame
the accelerations are
( )
( )
cos sin 0 2 1
sin cos 0 1
0 0 1
x L x
y L y
z Lz I
a t t B
a t t B
a B
ฯ
ฯ
ฯ
? ? + ? ? ? ? ? ?
? ? ? ? ? ? = ? ? ? ? ? ? ?
? ? ? ? ? ? ? ? ? ?? ?
(9)
5. Study Cases
Here three type of study cases are examined to derive the
control strategy mathematically. The basic assumptions on these
cases are that the formation is placed nearby L2 point and the
shadow area around L2 is not accounted
5.1. The leader has no control capability and the follower has
solar sail with two gimbal
The relative position in rotating frame of the follower
subsequently can be written as
1 0 0
2 1
sin sin
1 0 0 sin cos
1
cos
1 0 0
L
x
y
L
z
L
B A
A
B
A
B
ฯ ย ี
ฯ ย ี
ฯ ย
? ?
? ? ? ? ? ? ? ? ?
? ? ? ? ? ? =? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?
? ?
? ? ? ?
(10)
The absolute distance between the leader and the follower is
( ) 2 2 2 , x y z ฯ ย ี ฯ ฯ ฯ = + + (11)
Fig. 3 The contour for the Euclidian distance between
spacecraft
Fig.4 Accessible Space
From the contour graph we can found the accessible space
is the limited to
1
2
3
4
48 72
108 132 66 108
228 252
288 312
ี
ี ย
ี
ี
? ? ?
? ? ? ? ? ?
? ? ?
? = ?
?? ??
?? ?? ?? ??
?? ??
?? ??
(12)
In this case the acceleration of the solar sail approached
[8,9] by
( ) 2 2
0 sin sin
0
A a
with
ย ำ ี
ำ
? +
?
(13)
and with
2
6
2
2
2.0 10
10 and for
6
L
N P m
Kg
m
ะ
? ? ~
=
(14)
for the follower with mass i s 500 Kg, the total area needed
for maintaining distance with the leader for 10 m, is 300
m2.
5.2. Both satellites has solar sail with two gimbal
For this case, the solar sail control accelerations are as follows
( )
( )
( )
2 2 1 1
2 2 1 1
2 1
sin sin sin cos
sin cos sin cos
cos cos
x
y
z
a A
a A
a A
ย ี ย ี
ย ี ย ี
ย ย
? = ?
? = ?
? = ?
(15)
Hence we can form a cost function for minimizing the drifting
motions of the formation as
( )
( ) ( )
( )
( )
( )
2 2
2 2 1 1
2 2
2 2 1 1 2 1
1 2 2 1 1
2 2 2 1 1
3 2 1
[ sin sin sin sin
sin cos sin cos cos cos ]
sin sin sin cos
sin cos sin cos
cos cos
x
y
z
H A
a
b
ย ี ย ี
ย ี ย ี ย ย
ษ ย ี ย ี ฯ
ษ ย ี ย ี ฯ
ษ ย ย ฯ
฿ ?
+ ? + ?
+ ? ? + ? ? ? ?
? ? ? ? ? ?
+ ? ? ? ? ? ?
(16)
4
The simplest analytical solutions is
1 2 0 0 or ย ย = = (17)
which for ย2=0
( )
( )
1
1
1
1
1 1 1
2
1 1
cos 1
tan
sin sin
tan
sin cos
z
x
y
x
y
c
b
a
a b
a b
ฯ
ย
ฯ
ี
ฯ
ฯ ย ี
ี
ฯ ย ี
?
?
?
? ? = ? ? ?
? ?
? ?
= ? ? ? ?
? ?
+
=
+
(18)
where
2 1
1
L
L
L
A a
B
A b
B
A c
B
=
?
=
?
=
(19)
5.3. The leader has solar sail with one gimbal and the follower has
solar sail with two gimbal
In this third case, the formation is fully controllable, because
the number degree of freedom is the same with the number of
equations available.
6. Conclusions
Here in this paper has been derived the control strategy
for formation flying around L2 point. The visibility of this
application also have been shown, that for maintaining the
relative motion between the leader and the follower for 10
m approximately need 300 m2 are of sail.
Future detail study needs to be done with more accurate
mathematical modeling of dynamical motions at the
restricted three body problem, to include perturbations
components. The satellite attitude dynamics also need to be
considered for implementing a vigorous formation control.
References
[1] http://sci.esa.int/science-e/www/object/index.cfm?fobj
ectid=37936
[2] C. Beichman, et al., Searching for life with the
Terrestrial Planet Finder: Lagrange Point options for a
formation flying interferometer, Advances in Space
Research 34 (2004) 637-664.
[3] http://sci.esa.int/science-e/www/object/index.cfm?fobj
ectid=38535
[4] http://sci.esa.int/science-e/www/object/index.cfm?fobj
ectid=36959
[5] J. Bookles, C.McInnes, Control of Lagrange point
orbits using solar sail propulsion, Acta Astronautica
(2007).
[6] R. W. Farquhar, Quasi-periodic orbits about the
translunar libration point, Celestial Mechanics 7(1973)
458-473.
[7] D. L. Richardson, Analytic construction of periodic
orbits about the collinear points, Celestial Mechanics
22 (1980) 241-253.
[8] K. Tarao and J. Kawaguchi, A control configured small
circular Halo orbit around L2, Paper AAS 05-425,
presented at the AAS/AIAA Astrodynamics Specialist
Conference, Lake Tahoe, California, August 7-11, 2005
[9] J. Kawaguchi and M. Yoshimura, On a representative
solar sail deep space flight based at the quay around L2
point, Paper IAC-020A.7.03, presented at the 53rd
International Astronautical Congress, The Word Space
Congress-2, Houston, Texas, October 10-19, 2002.