1 Introduction
A space solar power system (SSPS) is investigated as
a new energy source that is expected as an alternative
to fossil fuel.[1][2] We generate electric power by the
SSPS, e.g. generate electricity using solar power satellites
(SPS�fs), convert it into microwave or laser, transmit
it to antennas on the earth, re-convert it into electricity
using the on-ground power receiving plants, and supply
the electric power to cities. We need to discuss system
feasibility about architecture and dynamics because
a typical SPS is several kilometers long.
By using the principle of minimum potential energy
and a numerical simulator, this study evaluates stabilities
of equilibrium points of the original and some derived
static balance SPS�fs proposed by USEF (Institute for Unmanned
Space Experiment Free Flyer) in 2006.
In the principle of minimum potential energy, we consider
a sum of the potential energy of gravity, the potential
energy of centrifugal forces, and the strain energies in
panels and tethers. The equilibrium point is calculated
by the Newton-Raphson method. The stability of the
equilibrium point is evaluated through the eigen-values
of second derivatives of total potential energy.
In the numerical simulator, we model a structurally
flexible satellite as the multi-body system that is composed
of rigid bodies connected with others by springs.
Dynamics of each rigid body is formulated by Newton-
Euler equations. [4][5]
In addition, we have developed the numerical simulator
that models the SPS�fs as flexible multi-body systems and
analyses. We additionally analyze the SPS�fs as flexible
multi-body systems, whereas the above-mentioned analyses
are based on the rigid multi-body system models.
The analysis using flexible multi-body systems yields a
result different from that using rigid multi-body system,
when a 9 unit SPS similar to the static balance system is
analyzed�D
2 �f06 USEF SPS Models
2.1 Definition of coordinate frames
Coordinate frames are set as follows. The coordinate
frames are illustrated in Fig. 1. The ��I is the inertial
frame of reference, ��O is the orbital frame of satellite,
and ��j is the Body j fixed frame. The ��I is the Earth-
Centered-Inertial (ECI) reference frame with its origin
OI at the geocentric. The origin OO of ��O is placed at
the center of mass of the entire system. The ��O has a set
of orthonomal bases vector [iO jO kO]. The vector iO,
jO, and kO point to the directions of the orbital motion,
the normal line of the orbital plane, and the geocentric,
respectively. The origin Oj of ��j is placed at the center
of mass and ��j consists of the inertial principal axes of
j
j
j
i
i
i
k
k
k
O
O
O
O
O O
I
I
I I
I
j j
j
j
j
j
j
O
O
O
O
S
S S ,
r
r
r
orbit
Fig. 1: Satellite modeled by rigid multi-body system, and
coordinate frames
z
y
x
2500 [m]
500 [m]
475 [m] 2375 [m]
5000 [m]
Line 1
Line 2
Line 3
Line 4
}Line 5
}
}
}
} Unit5 Unit4 Unit3 Unit2 Unit1
Unit6
Unit11
Unit16
Unit21
Tethers
Buses
Panels
Fig. 2: �f06 USEF SPS model
Body j. The rj is the position vector of Oj from OI , rO
is the position vector of OO from OI , and rj;O is the position
vector of Oj from OO . The vector !j expresses the
angular velocity of ��j from ��I , the vector !O expresses
the angular velocity of ��O from ��I , and the vector !j;O
expresses the angular velocity of ��j from ��O.
2.2 Modeling and outline of SPS�fs
2.2.1 Modeling and outline
The feasibility of a �f06 USEF SPS (Fig. 2) is examined.
The feature of the system dynamics of the SPS is
as follows. The whole system consists of the power generation
and transmission part located near the earth in
parallel to the surface, the distributed bus part located
on reverse-side, and the tethers connecting them. The
power generation and transmission part is composed of
the power generation and transmission panels, which generates
the electric power by the solar cells and transmits
the microwave by the phased array antenna facing on the
earth. The whole system consists of 25 homogeneous basic
units being connected each other. Each basic unit
is composed of the power generation and transmission
panel, the bus, and the four tethers. Each basic unit can
work as a SPS. The outline of the system is listed below.
Orbit: geostationary orbit (GEO)
Power generation scale: 1 GW
Power transmission method: microwave
Satellite configuration:
(a) passive gravity gradient stabilization
Table 1: Spring ratio of hinge between units
Spring ratios of hinge between units along x-axis
kx 3.78��106 [N/m] k��x 1.60��103 [Nm/rad]
ky 1.47��107 [N/m] k��y 2.27��103 [Nm/rad]
kz 3.78��106 [N/m] k��z 3.07��1011 [Nm/rad]
Spring ratios of hinge between units along y-axis
kx 1.33��107 [N/m] k��x 2.05��103 [Nm/rad]
ky 3.41��106 [N/m] k��y 1.45��103 [Nm/rad]
kz 3.41��106 [N/m] k��z 2.50��1011 [Nm/rad]
(b) 25 basic units are connected
Size of satellite:
(a) total system: 2.5 ��2.375��5 [km]
(b) basic unit: 0.5��0.475��5 [km]
Satellite mass:
(a) total mass: 2.64��106 [kg]
(b) basic unit mass: 1.056��105 [kg]
Each bus is a rigid body cylinder with 5:6��104 [kg] in
mass, 10 [m] in diameter, and 15 [m] in length. The tether
is made by a Kevlar wire of 4:0��10!3 [m] in diameter, its
Young�fs modulus is 7.0��10!3 [Pa]�Cand 45 [kg] in mass
for 5000 [m] long�DIt is so light that the tether�fs mass is
neglected in analysis.. The tether must generate only the
tensile force that is proportional to the extensional deformation.
The power generation and transmission panel
part is considered as a single homogeneous plate whose
size is 2:5 [km]�� 2:375 [km] �� 0:02 [m] and flexural rigidity
par unit width is 1.52��103 [Nm2/m]�DIn the rigid
multi-body models, the panel of each unit is considered
as a rigid body plate and the flexibility is modeled by
hinges between unit panels. Each hinge is connected by
a spring with rotation and translation deformations that
realize a deformation equivalent to the flexible panel part.
The suffix of spring rate indicates the direction of deformation,
where ��x�C��y, and ��z represent the rotations
about x�Cy, and z axes, respectively.
2.2.2 Derived �f06 USEF SPS Systems
The above system is called the original system in this
study. The left figure of Fig. 3 (a) is the system seen from
!x direction. The original system has the problem that
the panel part is bent as shown in the right of Fig. 3 (a)
because moments of forces are applied to hinges between
lines. This study discusses the reason why the moment
of force is generated that causes the deformation of the
panel part. Both of the gravity and the centrifugal force
are statically applied to each part as illustrated in Fig. 4.
The gravity and the centrifugal force do not balance in
the unit being outside the orbital plane because the gravity
is pointing the geocentric but the centrifugal force is
parallel to the orbital plane. Therefore, the external force
fBj and fPj act on each part. The moments of forces
about hinges k and j are:
O?Hk = OrHkr;P k �� OfPk + O�}kr �� OfBk (1)
O?Hj =OrHjr;P j �� OfPj + O�}jr �� OfBj
+ !OrHjl;P j ! OrHjr;P j���� OfHk
(2)
It is inconvenient that the panel part is bent by the moments
of forces. The following static balance systems (1-
z
y
(a) Original system
5000 [m]
5.6 10 [kg]
4
1.0 10 [kg]
6
z
y
0.031[rad0.053[rad] ]
154.5[m]
13.63[m]
265.5[m]
z
y
5000 [m]
1.0 10 [kg]
6
(b) Static balance system (1-1)
z
y
1.0 10 [kg]
6
5.6 10 [kg]
4
(c) Static balance system (1-2)
z
y
1.0 10 [kg]
6
4.00
10 [m]
3
(d) Static balance system (2)
2.03
10 [m]
2
5.00
10 [m]
3
7.00
10 [m]
3
4.94
10 [m]
5.6 10 [kg]
4
2.14
10 [m]
3
6.43
10 [m]
3
7.85
10 [m]
3
2.35
10 [kg]
4
7.13
10 [kg]
4
9.09
10 [kg]
4
Fig. 3: Equilibrium points of the original and some derived static balance systems
1), (1-2), and (2) are designed, where the mass distribution
and/or the position arrangement of the buses are
modified to avoid this inconvenience. These systems are
summarized as follows.
The design is modified as Fig. 3 (b) so that the system
is statically balanced (at an equilibrium point) when it is
at the design point. The moment of force about the hinge
becomes 0 when the line of action of the resultant force
f on the unit runs through the hinge. Accordingly, the
modified system would not result in the bending panel
part. The buses in line 1, line 2, and line 3 are modified
as mB1 = 2:33 �� 104 [kg], mB2 = 7:13 �� 104 [kg], and
mB3 = 9:09��104 [kg] in mass, respectively. Other buses
in line 4 and line 5 are symmetrical to line 3. This system
is called the static balance system (1-1).
The moment of force about the hinge becomes 0 when
the tether lengths are modified as Fig. 3 (c). The tethers
in line 1, line 2, and line 3, respectively, are l1 =
2:14��103 [m]�Cl2 = 6:43��104 [m]�Cand l3 = 7:85��104 [m]
long, and the other tethers in line 4 and line 5 are symmetrical
to line 3. This system is called the static balance
system (1-2).
However, the numerical simulation has shown that the
design point is not stable by considering slack of tethers.
Then, the positions of the buses are modified again
to make the design point be the true equilibrium point
where the tethers do not slack. The buses in line 1,
line 2, and line 3, respectively, are at [!747:0;!3735],
[!425:5;!4735], and [0:000;!6734] [m] in y and z directions,
and the other buses in line 4 and line 5 are symmetrical
to line 3. This system shown in Fig. 3 (d) is
called the static balance system (2).
The numerical simulations have shown that the design
points in the static balance systems (1-1) and (1-2)are
not stable by considering slack of tethers [6], though the
j
k
k
rO
S
SO
I
O
O
Line k Line j
rBj,O
rP j,O
q j
q k
rHj r,P j
rHjl,Pj
djr d j l
Hinge j
Hinge k
fBj
fP j
fHk
Fig. 4: Position of Line
simulation results are omitted. The design point in the
static balance system (2) is stabilized without slack of
tethers as follows.
3 Stability Analysis of Equilibrium Point
Using Potential Energy
The potential energy, which is caused by gravitational
force and centrifugal force, is considered as well as strain
energy of panels and tethers. The total potential energy
is defined as the sum of these potential energies. The
equilibrium point is calculated by applying the Newton-
Raphson method to the first derivative of total potential
energy. To evaluate the stability of the equilibrium point,
eigen-values of second derivatives of total potential energy
are calculated. If all signs of the eigen-values are
positive, the equilibrium point is stable. In the analysis,
following assumptions are introduced:
1. The size of satellite is so small compared to orbital
radius that the higher orders of these terms can be
neglected.
2. The displacement along x axis is assumed to be uniform.
Therefore, the deformation in y ! z plane is
considered.
3. The orbit is assumed to be circular.
3.1 Definition of the symbols and models
In this section, we treat five panels shown in Fig. 2
along the same line as one panel part. The rotation angle
of each panel part j(= 1; : : : ; 5) around iO axis relative
to ��O is described as ��j . In the same way, we treat five
buses along the same line as one bus part. The subscript
corresponds to the line number shown in Fig. 2. The configuration
of the panel and bus parts is shown in Fig. 4.
Superscript O( ) indicates that the variables are described
in the coordinate frame ��O�D
3.2 Gravitational and centrifugal potential
energies
Under the assumptions 1 and 3�CThe general gravitational
and centrifugal potential energies of a rigid body
are approximated as follows.
Gravitational potential energy:
U(g)
j = !
1
rO mj 1 !
1
2r2
O ??Or2
j;Ox + Or2
j;Oy ! 2Or2
j;Oz

+
Orj;Oz
rO +
1
2r2
O
tr ??OIj

!
3
2r2
O ??OR2
j31Ij11 + OR2
j32Ij22 + OR2
j33Ij33

(3)
Centrifugal potential energy:
U(c)
j = !
1
2
!2
O mj r2
O + Or2
j;Ox + Or2
j;Oz ! 2rO
Orj;Oz	 +OR2
j21Ij11 + OR2
j22Ij22 + OR2
j23Ij33
(4)
By omitting the constant term, the sum of gravitational
and centrifugal potential energies for No. j panel part is
described as:
U(g+c)
Pj = ! !2
O �E1
2mPj !!Or2
P j;Oy + 3Or2
P j;Oz��
!2 (IPj22 ! IPj33) sin2 ��j?
(5)
In the same way, the sum of gravitational and centrifugal
potential energies for No. j bus part is described as:
U(g+c)
Bj = ! !2
O �E1
2mBj !!Or2
Bj;Oy + 3Or2
Bj;Oz���C (6)
3.3 Strain energy
At first, the strain energy of tethers between the panels
and buses in line j are derived. The �}jl and �}jr indicates
the position vectors from left and right ends of panel and
bus as shown in Fig. 4. These vectors are described as
�}jl =rBj;O ! (rPj;O + rHjl;P j)
! �� ��1
2Lx !1
2Ly cos ��j !1
2Ly sin ��j ?T
�}jr =rBj;O ! (rPj;O + rHjr;P j)
! �� ��1
2Lx
1
2Ly cos ��j
1
2Ly sin ��j ?T
(7)
where Lx;Ly;Lz are the length of each panel along x; y; z
axes, respectively. Therefore, strain energy of tethers is
described as
U(th)
jl = ? 0 (�}jl < ln jl)
10 �� 1
2kth jl (�}jl ! ln jl)2 (�}jl �C ln jl)
U(th)
jr = ? 0 (�}jr < ln jr)
10 �� 1
2kth jr (�}jr ! ln jr)2 (�}jr �C ln jr)
(8)
where kth jl�Ckth jr are spring constants of left and right
tethers, and ln jl�Cln jr are natural length of left and right
tethers, respectively.
Then, the strain energy of panel parts is derived. The
flexibility of panels is modeled as translational and rotational
springs attached between panels. The relative angle
between the panel part j and panel part k is ��k ! ��j .
Under the assumption 2, there are the same five rotational
springs between the panel parts. Therefore, the
strain energy caused by the rotation between the panel
part j and panel part k is described as:
U(rot)
i = 5 ��
1
2k��x (��k ! ��j)2 (9)
The relative displacement between the edges of panel part
j and panel part k is defined as �}g i. In the relative displacement,
the displacements along tensile direction and
shear direction are described as �}t i and �}s i, respectively.
Under the assumption 2, there are the same five translational
springs between the panel parts. Therefore, the
strain energy caused by the translation between the panel
part j and panel part k is described as:
U(trans)
i = 5 ��
1
2 !ky�}2
t i + kz�}2
s i��2 (10)
3.4 Total potential energy
The total potential energy is described in terms of the
above potential energies as:
Utotal =
5 Xj=1
(U(g+c)
Pj + U(g+c)
Bj ) +
5 Xj=1
(U(th)
jl + U(th)
jr )
+
4 Xi=1
(U(rot)
i + U(trans)
i )
(11)
3.5 Procedure of analysis
Under the assumption 2 and constraint condition,
which the center of mass of the satellite is fixed to the
origin of orbital coordinate frame, following 23 variables
are independent.
x = ��1 �� �� �� ��5
OrP2;Oy �� �� �� OrP5;Oy
OrB1;Oy �� �� �� OrB5;Oy
OrP2;Oz �� �� �� OrP5;Oz
OrB1;Oz �� �� �� OrB5;Oz cccT (12)
We call x state variable. From the above constraint condition,
the position vector of panel part 1 is described
as:
OrP1;O = !
1
mP10@
5 Xj=2
mPj;O
OrPj;O +
5 Xj=1
mBj;O
OrBj;O1A
(13)
The following first and second derivatives of total potential
energy are derived analytically.
f(x) =h @Utotal
@x1 �� �� �� @Utotal
@x23 iT (14)
J(x) =h @f
@x1 �� �� �� @f
@x23 i (15)
The equilibrium point x? is searched by applying
Newton-Raphson method to the first derivative of total
potential energy Utotal. Following calculation is iterated
until jjf(xk)jj decreases within an allowable error.
xk+1 = xk ! J!1(xk)f(xk) (16)
The eigen-values of J(x?) are used to judge if the total
potential energy has the minimum value at the obtained
equilibrium point. If all eigen-values are positive,
the total potential energy has the minimum value at the
equilibrium point. Therefore, the equilibrium point is
stable. If the above iterative calculation could not converge,
equilibrium point might not exist. In such a case,
following additional calculation is carried out to confirm
the non-existence of equilibrium point.
At first, a tentative equilibrium point is searched under
the assumption that tethers can exert resorting force
against compression. Then, the tension of tether is
checked whether the tether is compressed or not. If some
tethers are compressed, spring constants of compressed
tethers are set to zero, and f�CJ are recalculated. At
last, the equilibrium point is recalculated with the initial
condition of tentative equilibrium point under above
assumption.
3.6 Results of analysis
Through the analysis, the equilibrium point is obtained.
At the equilibrium point, rotation angle of each
panel part and displacement of panel parts and bus parts
from the no-strain state are shown in the following.
Result of �f06 USEF original SPS system
The right figure of Fig. 3 (a) is the equilibrium point.
The rotation angles of panel parts are:
�� ��1 ��2 ��3 ��4 ��5 ? = �� 0:053 0:031 0:000 !0:031 !0:053 ? [rad]
(17)
The displacements of panel parts along the y�Cz axes from
the no-strain state, uPj , are symmetric with respect to
line 3.
�� OuP1
OuP2
OuP3 ? = �E 0:56 0:11 0:00
!13:62 6:30 13:64 �C [m]
(18)
The displacements of bus parts along the y�Cz axes from
the no-strain state, uBj , are also symmetric with respect
to line 3.
�� OuB1
OuB2
OuB3 ? = �E 265:5 154:5 0:0
!6:60 8:68 13:63 �C [m]
(19)
The elongation of each tether is as follows.
" u(th)
1l u(th)
2l u(th)
3l u(th)
4l u(th)
5l
u(th)
1r u(th)
2r u(th)
3r u(th)
4r u(th)
5r #
= �E 5:4 4:9 6:0 7:0 6:6
6:6 7:0 6:0 4:9 5:4 �C�� 10!3 [m]
(20)
All tethers are not slacked at the obtained equilibrium
point. This equilibrium point is stable because all eigenvalues
of J are positive.
Result of static balance system (1-1)
In this system, the calculation for search of equilibrium
point cannot converge. Under the assumption that tethers
can exert resorting force against compression, a tentative
equilibrium point is as follows.
Fig. 3 (b) is the tentative equilibrium point. The rotation
angles of panel parts are:
�� ��1 ��2 ��3 ��4 ��5 ? = �� 0:000 0:000 0:000 0:000 0:000 ? [rad]
(21)
The displacements of panel parts along the y�Cz axes from
the no-strain state, uPj , are symmetric with respect to
line 3.
�� OuP1
OuP2
OuP3 ? = �E 0:00 0:00 0:00
0:00 0:00 0:00 �C [m]
(22)
The displacements of bus parts along the y�Cz axes from
the no-strain state, uBj , are also symmetric with respect
to line 3.
�� OuB1
OuB2
OuB3 ? = �E 0:08 0:06 0:00
0:00 !0:01 !0:01 �C [m]
(23)
The elongation of each tether is as follows.
" u(th)
1l u(th)
2l u(th)
3l u(th)
4l u(th)
5l
u(th)
1r u(th)
2r u(th)
3r u(th)
4r u(th)
5r #
= �E 6:0 12:9 9:7 2:3 !1:0
!1:0 2:3 9:7 12:9 6:0 �C�� 10!3 [m]
(24)
The elongation of two tethers is negative. These tethers
should be slacked and spring constants of slacked tethers
should be set to zero. The equilibrium point is recalculated
with the initial condition of tentative equilibrium
point under above assumption that spring constants of
slacked tethers are zero. However, the valid equilibrium
point cannot be found. Therefore, the equilibrium point
could not exist for this system.
Result of static balance system (1-2)
In this system, the calculation for search of equilibrium
point cannot converge. Under the assumption that
tethers can exert resorting force against compression, a
tentative equilibrium point is as follows.
Fig. 3 (c) is the tentative equilibrium point. The rotation
angles of panel parts are:
�� ��1 ��2 ��3 ��4 ��5 ? = �� 0:000 0:000 0:000 0:000 0:000 ? [rad]
(25)
The displacements of panel parts along the y�Cz axes from
the no-strain state, uPj , are symmetric with respect to
line 3.
�� OuP1
OuP2
OuP3 ? = �E 0:00 0:00 0:00
0:00 0:00 !0:01 �C [m]
(26)
The displacements of bus parts along the y�Cz axes from
the no-strain state, uBj , are also symmetric with respect
to line 3.
�� OuB1
OuB2
OuB3 ? = �E 0:01 0:07 0:00
0:00 !0:01 !0:02 �C [m]
(27)
The elongation of each tether is as follows.
" u(th)
1l u(th)
2l u(th)
3l u(th)
4l u(th)
5l
u(th)
1r u(th)
2r u(th)
3r u(th)
4r u(th)
5r #
= �E 2:6 16:9 15:0 3:0 !0:5
!0:5 3:0 15:0 16:9 2:6 �C�� 10!3 [m]
(28)
The elongation of two tethers is negative. These tethers
should be slacked and spring constants of slacked tethers
should be set to zero. The equilibrium point is recalculated
with the initial condition of tentative equilibrium
point under above assumption that spring constants of
slacked tethers are zero. However, the valid equilibrium
point cannot be found. Therefore, the equilibrium point
could not exist for this system.
Result of static balance system (2)
Fig. 3 (c) is the equilibrium point. The rotation angles
of panel parts are:
�� ��1 ��2 ��3 ��4 ��5 ? = �� 0:000 0:000 0:000 0:000 0:000 ? [rad]
(29)
The displacements of panel parts along the y�Cz axes from
the no-strain state, uPj , are symmetric with respect to
line 3.
�� OuP1
OuP2
OuP3 ? = �E 0:00 0:00 0:00
0:00 0:00 0:00 �C [m]
(30)
The displacements of bus parts along the y�Cz axes from
the no-strain state, uBj , are also symmetric with respect
to line 3.
�� OuB1
OuB2
OuB3 ? = �E 0:10 0:00 0:00
0:00 0:00 0:00 �C [m]
(31)
Fig. 5: Dynamics of SPS of USEF in 2006: static balance
system (1-2)
The elongation of each tether is as follows.
" u(th)
1l u(th)
2l u(th)
3l u(th)
4l u(th)
5l
u(th)
1r u(th)
2r u(th)
3r u(th)
4r u(th)
5r #
= �E 4:8 8:5 11:8 3:4 2:8
2:8 3:4 11:8 8:5 4:8 �C�� 10!3 [m]
(32)
All tethers are not slacked at the obtained equilibrium
point. This equilibrium point is stable because all eigenvalues
of J are positive.
3.7 Comparison with numerical simulations
The analyzed results using the principle of minimum
potential energy are compared with numerical simulations.
A numerical simulation of the original system with
some dumping obtains a stationary state in orbit as time
goes, which can be considered as a statically stable state.
The obtained state in Fig. 3 is as same as the result in
section 3.6.
The simulation result of the static balance system (1-2)
is illustrated every 7000 [s] in Fig. 5. It is not possible to
maintain the stability at the design point because tethers
slack, and the system collapses as time goes. The instability
is caused by bus position shifting since the slack
tethers cannot generate compression forces. The static
balance system (1-1) is also unstable and cannot maintain
the stability at the design point.
A numerical simulation shows that the static balance
system (2) converges to the stationary state at the design
point after perturbation torques are applied in hinges.
Hence, stability of the design point can be confirmed,
whereas its illustration is omitted.
3.8 Summary of the results
The numerical simulations and the obtained results using
the principle of minimum potential energy agree with
each other. As a result, it can be said that both analysis
methods are valid.
The original system has the stable equilibrium point
where the panel part is bent. At the equilibrium state,
the center of mass of panel part 3 (center) displaces about
26 [m] to the nadir with respect to that of panel part 1
(tip).
For static balance system (1-1)�C(1-2), tentative equilibrium
states are found under the assumption that tethers
can exert resorting force against compression. If
Sj
iO
rjO ,
SO
j
kO
O
orbit
Flexible body j
Virtual rigid body j
Sjnorig
Sjn
sjn
ujn
r jn,j
jn j
Fig. 6: Relation between flexible body and virtual rigid body
spring constants of slacked tethers are set to zero, however,
no valid equilibrium state are found near the tentative
equilibrium state for static balance system (1-1),
(1-2).
The static balance system (2) has the stable equilibrium
point at the design point where the panel part is
flat.
4 Flexible Body Analysis on �f06 USEF
SPS
4.1 Outline of method of flexible body
analysis
When the numerical simulator calculates the external
forces, it is necessary to know the position and orientation
of each part in flexible bodies. According to the definition
of mechanical analysis software ADAMS [7, 8, 9], the
position and orientation in flexible body is considered as
the sum of the rigid body motion and that caused by the
elastic deformation as illustrated in Fig. 6.
Nomenclature for flexibility is given below. The ��j is
the reference coordinate frame used for flexibility description
in ADAMS, ��jn is a node coordinate frame in each
node of a flexible body, and ��jnorig is a coordinate frame
in each node when the flexible body is not deformed. The
ujn is an elastic deformation vector of the flexible body
from the origin Ojnorig of ��jnorig to the origin Ojn of the
��jn.
The position of ��jn of a flexible body is described as
the position of ��jnorig added by the elastic displacement
of the node:
rjn;O = rj;O + sjn + ujn (33)
= rj;O + sjn + cjnq (34)
where the elastic displacement ujn is given by a mode
representation, cjn is a modal matrix for translation,
and q is a vector of the modal coordinates. The node
velocity of the flexible body is obtained by the following
equation.
drjn;O
dt �P�P�P�PO
= d(rj;O + sjn + cjnq)
dt �P�P�P�PO
(35)
The attitude of node coordinate frame ��jn from the
orbit coordinate frame ��O is represented by a direction
cosine matrices as:
ORjn = ORj
jRjnorig
jnorigRjn (36)
z
y
x
Unit1
Unit4
Unit7
1500 [m]
500 [m]
1425 [m]
475 [m]
5000 [m]
Unit2 }Line 1
}Line 2
}Line 3
12.4[m]
Fig. 7: Static balance system (2) with 3 �� 3 units
where the rotational displacement is also given by the following
mode representation, and ajn is a modal matrix
for rotation:
ORjn = ORj
jRjnorig [I + [ajnq��]] (37)
The node angular velocity is described by the following
equation as well as the node velocity, where !jn;O is the
angular velocity of ��jn relative to ��O:
!jn;O = !j;O + !jn;j (38)
The angular velocity is also given by the modal matrix for
rotation ajn and the time derivative of modal coordinate
vector ?q:
!jn;O = !j;O +ajn?q (39)
It becomes very complex to calculate the external forces
to the flexible body considering the effect of elastic deformation.
Because the elastic displacement of the flexible
body is very small in the behavior of whole system, the
external forces are approximated by the external forces
applied to the virtual rigid body defined below. The virtual
rigid body is a virtual body without deformation
that is at the average position and orientation of the flexible
body in each moment, and its mass center position,
the linear and angular momentum are equivalent to the
original flexible body. Therefore, the external forces and
torques calculated for the virtual rigid body are applied
to the flexible body.
4.2 Analysis on flexible body model
The analysis model (Fig. 7) with 3��3 units is designed
as well as the static balance system (2) from the original
�f06 USEF SPS that stabilizes the design point avoiding
the tether slack. Fig. 8 shows the simulation result of the
system behavior every 60000 [s] where the rigid multibody
system is used as well as the previous analysis. It is
confirmed that the panel part is not bent and the static
balance condition is kept.
The model with the flexible panels is used for an analysis.
Fig. 9 shows the simulation result of the system
behavior 0 [s] and 180000 [s]. The displayed contour
shows the elastic displacement of panel from the initial
flat plane before deformation. In each panel, the four
Fig. 8: Dynamic behavior of static balance system (2) with
3 �� 3 units; rigid multi-body analysis
Fig. 9: Dynamic behavior of static balance system (2) with
3 �� 3 units; flexible multi-body analysis
-12.4 [m]
5000 [m]
-1.55 [m]
z
y
z
y
-3.21 [m]
-3.24 [m]
1.80 [m]
-3.12 [m]
-2.96 [m]
Fig. 10: Equilibrium points of static balance system (2) with
3��3 units; rigid body analysis (left) and flexible body analysis
(right)
corners are pulled in the direction toward the bus, and
the center part is pulled in the direction toward the geocentric.
The static balance system is designed so that the
moment of force about the hinge between units becomes
0. However, moments of force are not 0 excluding hinges.
The moments of force deform the panel when each panel
is modeled as a flexible body. There is a 5 [m] difference
in z direction between the center and the corners of the
panel.
Fig. 10 shows the equilibrium point of the rigid body
analysis and that of the flexible body analysis. Because
each panel is bent, its mass center moves toward the geocentric
and the bus shifts in opposite direction. Both
ends of the panel part move in the direction toward the
geocentric. Therefore, the bus parts in lines 1 and 3 get
away from line 2.
These show the necessity of the flexible body analysis.
It is necessary to discuss system design using the flexible
multi-body system, e.g. to increase number of tethers to
avoid panel bending, to confirm the panel strength being
enough to the load, etc.
5 Conclusion
This study has discussed the equilibrium points and
their stability of the original and the derived static balance
SPS�fs by using the principle of minimum potential
energy and the numerical simulations. The following results
have been shown. The two analysis methods are
both valid. The original system has the stable equilibrium
point whereas the panel part is bent. The static
balance system (2) has the stable equilibrium point at
the design point where the panel part is flat. If the static
balance system is designed by a rigid body, the analysis
using a flexible body may have a result different from that
of rigid body when the elastic deformation in the panel
becomes remarkable. Therefore, it will be necessary to
design SPS�fs considering structural flexibility more accurately
in the future.
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