1. Introduction
Japan Aerospace Exploration Agency (JAXA) is
currently planning the next generation Earthmagnetosphere
observation mission called gSCOPEh
(cross-Scale Coupling in Plasma universE, Fig.1)1. Its
predecessor gGEOTAILh, launched in 1992, played a
big role for understanding the ion behavior in the
Earthfs magnetotail and triggered interests toward
further microscopic scale (electrons scale) phenomena.
Following this successful work of GEOTAIL, SCOPE
aims at observing the Earthfs magnetotail where the
ions and electrons interact with each other, with 5
satellites flying in formation. To fully resolve the
time-domain behavior and spatial distribution of the
magnetospheric phenomena, a simultaneous
observation by spatially distributed electro-magnetic
instruments is essential.
From the formation design point of view, there are
several requirements to be satisfied, which include not
only the optimality of observation, but also constraints
stemming from the operationality, the physical layout
of the onboard components and so on.
In the SCOPE mission, the fundamental requirements
for formation design come from (1) observation
optimality, (2) relative orbit determination accuracy,
Fig.1: SCOPE Mission Image
2
and (3) inter-satellite communication link capability
constrained by antennae layout of each satellite. The
(2) requirement is needed because in the SCOPE
mission, the inter-satellite radiometric ranging and
clock synchronization via inter-satellite
communications are employed. (3) is needed to
comply with the actual antennae directivity limitations
As the first step of designing the SCOPE formation,
one wants to obtain all the possible answers which
satisfy (1), (2) and (3) at the same time. Here what is
needed is surveying a whole solution space, not
focusing on the highly accurate local minimum, but
rather on approximated answers which represent all
the possible types of solutions.
This paper proposes a formation design strategy which
surveys a whole solution space. This method employs
the simulated annealing (SA) algorithm to obtain the
globally optimum solution. In this paper, the method
is validated through analyzing the well-known
optimum formation in low-earth orbit, and then
applied to SCOPE formation design. Finally, based on
the solution types obtained by SA, a heuristics to
design the optimum SCOPE formation is introduced.
2. Formation Optimality Index
Before constructing the design method, this section
introduces several indices used in the following
formation design.
2.1. Observation Optimality (FDOP-O)
Consider the multi-point simultaneous observations by
spatially distributed satellites fleet. The observation
objective is to obtain the spatial differential coefficient
of the observation field. The observation of i-th
satellite, which is located at the relative position
vector ( , , )T x y z = r can be written as;
( ) Y f = r (1)
Suppose the field to be observed is linear, i.e.;
( ) , ,
T f f f f
x y z
? ? =? ? ? ?
r (2)
then from the observation of i-th and j-th satellites, the
difference of observations ij j i Z Y Y = ? can be
expressed as;
( )T T
ij j i ij Z f f = ? = r r r (3)
Eq.(3) implies that the quantity f can be calculated
from the observation differences. To obtain
3-dimensional f value, at least 3 different ij Z are
required, which means at least 4 satellites are needed
to obtain f .
Practically f is determined by the least-square
method as follows;
( )( ) 1 T
ij ij ij ij f Z ? = r r r (4)
where rij is the relative position vector from i-th to j-th
satellite. If the field is supposed to be isotropic, the
error covariance for equation (4) can be calculated
from the following equation;
( )1 2 T
ij ij
? = P rr (5)
where 2 is the error variance of the observation.
Therefore the estimation quality can be estimated
from the following quantity;
( )1 Tr T
ij ij
? ? ?
? ? ? ? r r (6)
which represents the dilution of precision for the
observation quantity. (6) implies that the well-known
tetrahedron formation is optimum, and in addition, the
larger formation is better. By eliminating the size
effect from (6), we finally obtain the formation
dilution of precision for observation as follows;
( )1 2
ij max(r )Tr T
ij ij FDOPO ? ? ? = ? ? ? ? r r (7)
Here the index is normalized by the maximum relative
distance among all the two satellites combinations.
2.2. Formation Geometry Determination
Optimality (FDOP-G)
Consider the quality of formation geometry
determination from the relative ranging information.
The state vector to be estimated is the formation shape,
3
which is written as;
[ ]T
1 2 2 3 3 3 1 1 1 n n n x x y x y z x y z ? ? ? ? = x ??
(8)
where the 0th satellite is supposed to be at the origin,
and the 1st satellite is supposed to be located on the
x-axis, the 2nd satellite is on the xy-plane. Hence the
dimension of estimation state vector becomes 3(n-2)
for n-satellite system.
From the relation between ranging output and relative
position vector ij i j r ? = ? r r , the following
observation equation is obtained;
ij ij
ij i j
i j
r r
r
? ?
? = +
r r
r r
(9)
Eq.(9) for all the satellites combinations are combined
to form the following matrix relation;
_ _ = y H x (10)
where
01 02 0 1 12
T
1 1 3 2 2 1
n
n n n n n
r r r r
r r r
?
? ? ? ? ?
? = ? ? ? ? ?
? ? ? ? ?
y ??
?? ?? (11)
is the observation vector. The formation geometry
estimation is now translated to minimize
( )( ) T
_ _ _ _
1
2
J= ? ? y H x y H x
hence the dilution of precision for geometry
determination is derived as follows;
( )1
_ _ Tr /( 1) T FDOPG n
? ? ? = ? ? ? ? ?
H H (12)
In (12), division by n-1 is introduced to eliminate the
satellite number dependency.
2.3. Formation Dynamics Determination
Optimality (FDOP-D)
Consider the quality of relative motion (or formation
dynamics) determination from the relative ranging
history information. The state vector to be estimated is
the relative position vector (with reference to the
inertial space), which can be written as;
[ ]T
1 1 1 2 2 2 1 1 1 n n n x y z x y z x y z ? ? ? = x ??
(13)
Different from (8), x includes the information not only
about the shape of the formation, but also the
orientation of the formation with reference to the
inertial space. Therefore the dimension of estimation
state vector becomes 3(n-1) for n-satellite system.
Denoting the transition matrix of x as ( ) 2 1 , r t t , the
observation equation is written as r = y H x . Then
the formation dynamics estimation is translated to
minimize
( )
( )
1 ( ) ( ) ( ,0) (0)
2
( ) ( ) ( ,0) (0)
T
r
r
J t t t
t t t dt
= ?
?
y H x
y H x i
??
hence the dilution of precision for dynamics
determination is derived as follows;
Fig.2: Samples Distribution in E-P Plane
Fig.3: Correlations Between FDOPO, FDOPG and E,P
4
1
T 1 Tr ( ,0) ( ) ( ) ( ,0) /( 1) T FDOPD t t t t dt n
T
? ? ? ? ? = ? ? ? ? ?
? ? ? ? ? ?
H H ??
(14)
where T is the orbital period. Again division by n-1 in
(14) is introduced to eliminate the satellite number
dependency.
2.4. Basic Characteristics of the Indices
The proposed indices FDOPO, FDOPG are compared
to the indices used in ESAfs CLUSTER mission2. In
CLUSTER, a generic evaluation of formation shape is
done by abbreviating the formation shape to the
pseudo-ellipsoid, and extracting the characteristic
values gelongationh E and gplanarityh P. These values
are derived in summary as follows.
First, define the tensor R;
T
ij ij = R rr (15)
The principle axes of the pseudo-ellipsoid are given
by the eigenvectors R(n) of R. If we suppose
(1) (2) (3) R R R ? ? (16)
then, their square roots represent respectively the
major, middle and minor semiaxis of the
pseudo-ellipsoid;
(1) (2) (3) , , a R b R c R = = = (17)
The elongation E and planarity P are defined using
(17) as follows;
1 ( / )
1 ( / )
E b a
P c b
= ?
= ?
(18)
E and P take the value between 0 and 1. The sphere
corresponds to (E,P)=(0,0). For 4-satellite formation,
(E,P)=(0,0) corresponds to the tetrahedron formation.
To survey the relation between E,P and FDOPO,
FDOPG, 1000 4-satellite formation samples are
generated as is shown in Fig.2. Each satellite in each
sample is generated randomly.
The correlations between FDOPO, FDOPG and E,P
are shown in Fig.3. Seen from these figures, the
optimality of FDOPO and FDOPG corresponds to
(E,P)=(0,0). It can also be said that the optimality of
FDOPO also guarantees the FDOPG optimality.
3. Global Optimization Using Simulated Annealing
Algorithm
3.1. Simulated Annealing Optimization
To survey the whole solution space to find the
optimum formation shapes, the simulated annealing
(SA) optimization is employed. Some characteristics
of SA are described below;
? SA updates the solution by perturbing the previous
solution randomly.
? At first SA searches around the whole solution space,
rather than insatiably pursuing the optimality. As the
optimization step goes by, updates which increase
the optimality becomes preferred more by analogy
of a cooling process of steel gannealingh.
? SA is suitable for searching the discrete space, but it
is also useful for obtaining the approximated
optimum of the multimodal continuous space.
? The results of SA optimization are remained to be
refined, by investigating the proximity of the SA
solution in detail (by local minimum search).
The overall algorithm of SA is illustrated in Fig.4. In
the formation optimization, the SA process becomes
as follows;
(1) The initial solution, which is a set of relative state
vectors xi=(xi, yi, zi, vxi, vyi, vzi)T at a certain
orbital phase (i.e. at apogee), is generated
randomly.
(2) Chose 1 satellite (i-th satellite), and regenerate
the state vector xfi randomly.
(3) If the objective function J(xf) is smaller than J(x),
apply x:=xf, otherwise stochastically apply x:=xf
according to the process temperature T.
Fig.4: Simulated Annealing Algorithm
5
(4) Terminate the algorithm when J(x) is sufficiently
converged, otherwise update T by T:=RT and then
go to (2).
3.2. Validation: Cart-Wheel Motion
To validate the SA algorithm works properly, a
cart-wheel motion in LEO is tested. The fleet of
satellites in a circular orbit, having relative motion
plane 60 degree inclined from the orbital plane,
satisfying a certain relation between relative position
and velocity, would hold their relative geometry, while
the whole formation is rotating one cycle per orbital
period. It is called a gcart-wheel motionh, and it can
be solved via Hillfs equations easily.
If we chose the objective function so as just to keep
the geometry of the formation, and as a result of that,
if a cart-wheel motion is obtained by SA, it can be
said that the SA works properly for formation
optimization.
The objective function is designed so as to keep the
relative distances between all the satellites are kept to
L for whole through the orbit;
( )2
2
1 ( ) ij
i j n
J t L dt
C T
= ? r ?? (19)
To calculate the relative orbital motion over one
orbital period, the solution of the following relative
orbital motion, which can be found in many papers3
and described briefly in appendix, is used;
( )
T
3 2
3 2 i i i i i R R
? ?
+ ~ + ~ + ~ ~ =? ? ? ?
? ?
RR r ? r ? r ? ? r 1 r ?? ???? ??
(20)
where R is the radius, ? the orbital angular velocity,
the gravity constant, respectively.
Because the relative motion must be periodic in the
practical formation flight, the following additional
condition must hold;
3 1/ 2 3 / 2
(0) (2 )
(0) (1 ) (1 )
y e
x a e e
+ = ?
+ ?
??
or (21)
3 3/ 2 1/ 2
( ) (2 )
( ) (1 ) (1 )
y e
x a e e
? = ?
+ ?
??
where the depedent variable for x,y is the true
anomaly . The result of SA optimization is shown
in Fig.5, where the conditions are chosen as follows;
? 3 satellites in 200km circular LEO, L=1.0
? search space: (x,y,z)=[-1:1], (vx,vy,vz)=[-1e3:+1e3].
? SA setting: T0=100, R=0.9
? Number of trials: 100
In Fig.5, x is the radial direction, z the outer-plane
direction, and y completes the right-hand system. The
solid lines indicate the analytical solution of the
cart-wheel motion. As the numerical solution
distribution well aligns the analytical one, it can be
said that the SA algorithm works properly to generate
the optimum formation.
4. High Elliptic Formation Optimizations
4.1. FDOP-O Minimization Problem
The primary objective of SCOPE mission is to
perform multi-point simultaneous observations around
apogee, therefore consider the following objective
function;
210
150
( ) J FDOPO d = ??
?? (22)
where the reference orbit is chosen to be a high
elliptic orbit for SCOPE((RE+3000km)~30RE). As is
shown in section 2.4, the minimization of FDOPO
automatically optimizes FDOPG as well.
The result of SA optimization is shown in Fig.6,
where the conditions are chosen as follows;
? 4 satellites in (RE+3000km)~30RE, sat0 at origin.
Fig.5: Cart-Wheel Motion Generated by SA
6
? Search space: (x,y,z)=[-1:1], (vx,vy,vz)= [-1e-5:1e-5].
Periodicity condition (21) must hold.
? SA setting: T0=100, R=0.9
? Number of trials: 100
The coordinate system for Fig.6 is the same as that of
Fig.5. The linear approximated analysis based on
Eq.(20) and Eq.(A1) imply that there are 4
geometrically equivalent formations for a given
FDOPO and FDOPG history, which are
(x,y,z)=(+,+,+), (+,+,-), (-,-,+) and (-,-,-). This sign
selectivity has been removed and unified to (+,+,+) to
clarify the pure formation shape dependency on
FDOPO. Fig.6 indicates that, of course it shapes a
tetrahedron formation, but in addition to that, there is
a plane symmetry about yz-plane, and viewed from
x-axis, 3 satellites (sat1,2,3) forms a cocentric circle.
In summary, the FDOPO optimum formation is
illustrated as Fig.7.
4.2. SCOPE Formation Optimization
In SCOPE mission, FDOPO is not an only thing to be
optimized. Because of the communication
performance limitation of the member satellite, there
are following two constraints in real operation;
(1) The relative distance between all two satellites
must be within 500km to assure the inter-satellite
communications.
(2) Each satellite is equipped with the limited
directivity antennae for inter-satellite
communication. All the satellites have
orbit-normal spin axes, and have the antenna link
coverage of }60deg. Therefore the relative
location of each satellite is limited to be within
}60deg elevation with reference to the orbital
plane.
Taking above (1) and (2) into account, the objective
function for SCOPE formation optimization is written
as follow;
( )
( )
210
150
1 2 2 ,
2
( )
asin
max ( )
min ( )
ij
i j
ij ij
ij
ij
J FDOPO d
z
k d
x y
r
k
r
=
? ? ? ? ? + ?? ? ? ? +? ? ?
?
+
?
??
??
??
(23)
where the two constraints are implemented as penalty
functions with weighting parameters k1 and k2. The
second term restricts the maximum relative elevation
between all the satellites combinations, and the third
term tries to minimize the variation of the relative
distance over one orbital period.
Fig.6: FDOPO Minimum Formation Generated by SA (Left:position, Right:velocity distributions)
reference orbit
(around apogee)
orbital plane
top view side view
Fig.7: Illustration of FDOPO Minimum Formation
7
Fig.8 is the result of SA optimization under the
following conditions;
? 3 satellites in 200km circular LEO, L=1.0
? k1=1.0, k2=0.01, ?=40deg
? search space: (x,y,z)=[-1:1], (vx,vy,vz)=[-1e3:+1e3].
Periodicity condition (21) must hold.
? SA setting: T0=100, R=0.9
? Number of trials: 100
Fig.8 indicates that the optimum formation for
SCOPE is not so much different in shape as that of
Fig.6, but the velocity distribution is narrowed. It is
understood as the effect of the two constraints,
restricting the relative motion of the satellites in a
whole orbital period.
5. Heuristics for Optimum SCOPE Formation
Based on the results obtained in section 4, the
heuristics of designing the optimum SCOPE
formation can be constructed as follow;
(1) Periodicity condition
All the satellites must have the same orbital
period.
(2) Symmetry condition
Sat1 and Sat2 are on the yz-plane symmetric
positions at apogee. Sat3 on the yz-plane at
apogee.
(3) Cocentric condition
The yz-plane projection of Sat1,Sat2 and Sat3
positions are cocentric at apogee.
If we approximate the condition (3) to the coplanar
condition, in which Sat0, Sat1 and Sat3 are on the
same orbital plane, the conditions for each satellite at
apogee can be listed as follows;
? Sat0: all 0 (on the reference orbit)
? Sat1: / , 0, / tan( /2) z y x p z v y x l = = = = ??
? Sat2: / , 0, / tan( /2) z y x p z v y x l = = = =? ??
? Sat3: 0, / tan( ) x x y z z y l = = = = = ?? ?? ??
where
3 3/ 2 1/ 2
(2 )
(1 ) (1 )
e p
a e e
? = ?
+ ?
and l is the size of the formation, the inplane angle
(Sat2-0-1), the outer-plane angle (elevation of
Sat3 with reference to the orbital plane), a the
semimajor axis. Substituting above conditions to
Eq.(A9), we get the following formation state vector
for each satellite at apogee;
[Sat1][Sat2]
( ) ( )
( ) ( ) ( )
( )
cos / 2 , sin / 2 , 0
( )sin /2 ,
1 1
2 ( )cos /2 , 0
1
x
y z
x l y l z
e v l
e e
e v l v
e
= =} =
? =
? ? + +
? = =
?
??
?? ?
(24)
[Sat3]
0, cos , sin
0, 0, 0 x y z
x y l z l
v v v
= = =
= = =
(25)
Fig.8: SCOPE Optimum Formation Generated by SA (Left:position, Right:velocity distributions)
8
Originally there are 6(n-1)=18 degree-of-freedom
design parameters for 4-satellite formation, but now
from (24) and (25), the number of design parameters
is reduced to only 4, that are ,, and l.
Because we know, from section 2, that the tetrahedron
is the optimum shape, the unit size formation can be
assumed as =60deg, =53.5deg, l=1.0. Therefore
only remaining parameter now is .
The simulation examples are shown in Fig.9 and
Fig.10 with different . FDOPD is 5.0 for Fig.9
and 8.0 for Fig.10, respectively. Seen from these
figure, the outer-plane relative elevations are kept
always below 60deg, and the FDOPO and FDOPG
become small enough around apogee. It is also
important to note that, approximately corresponds
to the perigee/apogee relative distance ratio. This
perigee/apogee relative distance ratio can be
approached to 1.0 at the cost of increasing FDOPO,
FDOPG and FDOPD.
6. Conclusions
The new formation design strategy using simulated
annealing (SA) optimization was discussed. It was
shown that SA is useful to survey a whole solution
space of optimum formation, taking into account
realistic constraints. It was also revealed that this
method was not only applicable for circular orbit, but
also for high-elliptic orbit formation flying.
Fig.9: Heuristic Design of SCOPE Formation (=2.0)
Fig.10 Heuristic Design of SCOPE Formation (=5.0)
9
The developed algorithm was applied to SCOPE
formation design. A distinctive and useful heuristics
was found by investigating SA results, showing the
effectiveness of the proposed design process.
Reference
1. Fujimoto, M., Saito,Y., Tsuda,Y., Shinohara,I.,
Kojima,H., gThe scientific targets of the SCOPE
missionh, COSPAR04-A-01348, D2.3/E3.3/
PSW2-0016-04, COSPAR 2004
2. Paschmann, G.., Daly, P. W., Analysis Methods for
Multi-Spacecraft Data., SR-001, ISSI Scientific
Reports, 1998
3. How, J. P., Inalhan, G., gRelative Dynamics &
Control of Spacecraft Formations in Eccentric Orbitsh,
AIAA Paper, AIAA-2000-4443, 2000
10
Appendix
The analytical solution for Eq.(20) is derived as
follows3;
( )
( )
2
1 2 3 2
1 2
3 4
5 6
cos ( ) sin 2 sin ( ) cos
1 cos
( ) 1 cos 2 1 cos ( )
1 1 1 sin
1 cos 1 cos
sin cos ( )
1 cos 1 cos
( ) c
e x e d e H d d
e
y e d e e H d
d d
e e
z d d
e e
x e
? ? ? ? = + ? + ? ? ? ? ? ? ? ? ? ? ? + ? ?
? ? = + + + ? ? ? ? ? ?
? ? ? ? ? ? + + + ? ? ? ? ? ? + + ? ? ? ? ? ?
? ? ? ? = + ? ? ? ? + + ? ? ? ?
=
( ) ( )
( )
2
2 2
1 2 2 3
3
2 2
1 2
2
3 2
sin 2 sin cos os 2 cos ( ) 2 sin ( )
1 cos 1 cos
sin
( ) sin 2 ( ) 2 sin ( ) 2 cos ( )
cos sin sin cos
1 cos 1 cos 1 cos
e e d e H e H d
e e
d
y e d eH e H e H d
e e d
e e e
? ? ? ? + + + ? ? ? ? ? ? ? + + ? ?
+ ? ? ? ?
? ? = ? + ? + ? ? ? ? ? ?
? ? ? ? + + + +
+ ? ? + + ? ? ( )
( ) ( )
4 2
2
5 6 2 2
cos sin sin sin cos ( )
1 cos 1 cos 1 cos 1 cos
d
e e z d d
e e e e
? ?
? ?
? ? ? ?
? ? ? ? ? ? ? ? = + + ? +
+ + ? ? ? ? + + ? ? ? ?
(A1)
where
( ) ( ) 5 / 2 2 2 3 ( ) 1 1 sin sin cos
2 2
eE e H e e E E E ? ? ? =? + ? + + ? ? ? ?
(A2)
(E)f is -differentiation of (E), e the eccentricity and E
the eccentric anomaly. -differentiation and
t-differentiation is related by the following equation;
2
3 2 3 / 2
(1 cos )
(1 )
e
a e
+ =
?
?? (A3)
(A1) indicates that the inplane motion and outer-plane
motion are independent of each other, and the relative
distance history is equivalent for the sign
combinations of (x,y,z)=(+,+,+), (+,+,-), (-,-,+),
(-,-,-).
The periodicity condition (21) is equivalent to d2=0.
When this periodicity holds, the state vector at perigee
can be expressed using d1,c,d6 as follows;
3
1 4
6
1
3
5
( )
1 ( ) 1
1
1 ( )
1
( )
1 ( ) 1
1
1 ( )
1
x d
y ed d
e
z d
e
x ed
y d
e
z d
e
=?
? ? = + + ? ? ? ? ? ? + ? ?
? ? =? ? + ? ?
=? ? ? ?
? ? = + ? ? + ? ?
? ? =? ? + ? ?
(A4)
In the same way, the state vector at apogee can be
expressed as follows;
3
1 4
6
1
3
5
( )
1 ( ) 1
1
1 ( )
1
( )
1 ( ) 1
1
1 ( )
1
x d
y e d d
e
z d
e
x ed
y d
e
z d
e
=
? ? = ? + ? ? ? ? ? ? ? ? ?
? ? =?? ? ? ? ?
= ? ? ? ? ?
? ? = ? + ? ? ? ? ?
? ? = ?? ? ? ? ?
(A5)
If d1 is selected so that the inplane relative distance is
preserved between apogee and perigee, it is derived
from (A4) and (A5) that;
[ ] [ ] * *
1 4 1 4
*
1 4 2
1 1 1 1
1 1
1
1
e d d e d d
e e
d d
e
? ? ? ? + + = ? + ? ? ? ? + ? ? ? ? ?
? =
?
(A6)
Let denote the offset ratio of d1 from d1
*.;
*
1 1 d d = (A7)
Substituting (A6),(A7) into (A5), we get;
3
*
1
*
1
3
(0)
1 (0)
1 1
(0)
1 (0) 1
1
x d
y d
e e
x ed
y d
e
= ?
? ? = ? + ? ? ? + ? ?
= ?
? ? = + ? ? + ? ?
(A8)
and
3
*
1
*
1
3
( )
1 ( )
1 1
( )
1 ( ) 1
1
x d
y d
e e
x ed
y d
e
=
? ? = ? + ? ? + ? ? ?
= ?
? ? = ? + ? ? ? ? ?
(A9)
If <1, the formation size at apogee becomes larger
than the size at perigee, while if >1, the size at
apogee becomes smaller than the size at apogee.
When =1, the inplane distance at apogee is equal to
that of perigee.