1. Introduction
Solar sail is one of challenging technologies for
interplanetary cruising and has been investigated
by many researchers in the world. It is attractive
because of its unique thrust system which uses
solar pressure, not propellants. Thus, solar sail is
effective for missions that have long duration such
as deep space exploration.
In Japan, several types of spinning solar sail are
investigated, and the Laboratory for Space
Systems (LSS) have proposed a tether-controlled
spinning solar sail is a type of tethered formation
flight. In one style of the system, three satellites
are located on vertices of a large thin triangular
membrane, as shown in Fig. 1.
Figure 1 Tether-Controlled Spinning Solar Sail
The system spins up and the centrifugal forces
are applied on the satellites by their mass. The
satellites and the vertices of the membrane are
connected with tethers to control relative distances
between them. The proposed type of the solar sail
has the following characteristics:
1) Structural Stability: Maintained during
membrane deployment using controlled tethers
2) Membrane tension can be supported and
controlled using boundary tethers
3) Spinning stability depends on dynamics of
membrane and tether control
The authors have studied the membrane
dynamics during its deployment using tether
length control and membrane forwarding
mechanisms by numerical simulations as well as
by hardware experiments[1-6], and its usefulness
and feasibility are shown.
In addition to membrane deployment, attitude
control is also an important issue for solar sail
technology. The authors propose C.P.-C.M. offset
torque generation method for attitude maneuver,
where C.P. means the centre of pressure due to
solar radiation and C.M. means the centre of mass
in the system. The offset between C.P. and C.M.
can be generated by controlling its tether length as
shown in Fig. 2, and the offset induced torque is
generated for attitude maneuver. This torque
generation method is one of unique methods used
for tethered spinning solar sail, so that in this
paper this method is considered.
Figure 2 C.P.-C.M. Offset Generation
The paper deals with numerical simulations for
dynamics of the membrane and the sub-satellites
during the tether length change. In the numerical
simulations, analytical model of multiple massspring-
damper is applied to the membrane, the
sub-satellites and the tethers. The dynamical
behaviour and C.P.-C.M. offset are observed in the
simulations using a 70m-scale solar sail model.
Also, estimation of attitude angle control rate and
attitude maneuver simulations are conducted and
the results are shown.
2. C.P.-C.M. Offset Torque Generation
2.1 Attitude Control Methods
The following control methods can be used for
the attitude maneuver of spinning solar sail:
1) Thruster
Thruster can generate large torque, but
propellant is needed, and frequent use of thruster
loses the solar sail�fs major advantage. Thus,
thruster should be an assistant actuator for long
duration mission such as deep space exploration.
2) Solar Pressure
This type is very small torque generator, but no
propellant is necessary. Solar pressure type
actuator can be classified as follows:
a) Direction-changeable petals on the subsatellites
b) Light reflectivity-changeable device, for
example, liquid crystal material on membrane
c) C.P.-C.M. offset torque generated with
changing ballast position or tether length etc.
Type a) and b) are promising method for attitude
control. On the other hand, type c) is one of unique
methods for the tether-controlled spinning solar
sail which have proposed by the authors, so that
this paper focuses on type c), and the possibility of
this torque generation method is investigated.
2.2 C.P.-C.M. Offset Generation for Tether-
Controlled Spinning Solar Sail
Tether-controlled spinning solar sail, as shown in
Fig. 1, is able to change its formation with tether
length control. Relative position of a sub-satellite
to membrane can be changed by changing the
length of tethers between other sub-satellites and
membrane, shown in Fig. 2. Tether length change
makers an offset between the C.P. and C.M., and
the in-plane offset torque will be generated due to
its displacement. This C.P.-C.M. offset torque is
used for attitude maneuver in this paper. Note that
C.P. of the system is nearly equal to C.P. of the
membrane.
3. Numerical Simulation
The authors conducted numerical simulations to
know dynamic behaviour of the sailcraft during
attitude maneuver with controlling its tether length.
In this section, an analytical model and simulation
results are shown.
3.1 Analytical Model
For an analytical model, a typical configuration
of tether-controlled spinning solar sail is applied
[7], and the assumptions are as follows:
1) Sub-Satellites
In this paper, sub-satellite�fs attitude is
neglected for simplicity. Thus, it is approximated
as point mass (nominal 10kg).
2) Membrane
Modelled as multiple mass points and connect
them with spring and damper. In this paper, for
simplicity, four mass points are placed on vertices
and centre of the triangular membrane.
Membrane is made of Al-evaporated polyimide
which light reflectivity is 0.85, and Yang�fs module
is GPa] [ 30�@= m E . The thickness is 0.75 [ƒÊm], and
its mass area density is ] m kg [ 10 0 . 1 2 2�@? �~ = t r .
The shape of membrane is regular triangle of 70
[m], so that the total mass is approximately 20 [kg].
3) Tether
Massless spring which generates only pulling
force. Yang�fs module is GPa] [ 123�@= t E ,sectional
area is ] [m 10 0 . 1 2 4�@? �~ = t A and mass line density
is ] m kg [ 10 7 . 9 3�@? �~ = t
r so that the mass of
tether is negligible.
Tension force acts on tether is modeled as
follows:
If d ij ij L L _ 3 ,
( )
( )
ij
i j
d ij ij ij ij L
L L k
r r
T
?
? = _ (1)
If else,
0 = ij T (2)
where
ij L : relative distance between i th and j th mass
point ( ) i j ij L r r ? =
d ij L _ : commanded length between j i ?
ij T : tension force acts on i th mass point by
j i ? tether
ij k : spring constant of j i ? tether
i r : position vector to i th mass point from the
origin
The analytical model is shown in Fig. 3.
Element numbers of the mass points are defined
as shown in Fig.4:
1-3 : Sub-satellites
4-7 : Membrane (4-6 are on vertices, 7 is on
center)
All sub-satellites are connected each other with
tether, for example, 1st and 2nd mass point. Also,
sub-satellite and the closest vertex of the
membrane are connected with tether, for example,
1st and 4th mass point.
Figure 3 Numerical Simulation Model
(Mass-Spring-Damper)
Figure 4 Mass Point Number Allocation
3.2 Tether Length Control Method
In this paper, simple method of tether length
change under no solar pressure. One of subsatellites
controls tether length sinusoidally, which
can be described as follows:
l l l sat t D + = (3)
where
t l : tether length between the sub-satellites
sat l : nominal distance between the subsatellites
l D : sinusoidal tether length variation
( ) t A l t t w sin = D
Definitions of parameters are shown in Fig.
5. An example of tether length change is
shown in Fig. 6. The sailcraft rotates at an initial
angular velocity c s w .
Figure 5 Tether Length Variation
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30
Time , sec
Controlled tether length , m
Satellite-Satellite
Satellite-Membrane
Figure 6 History of Tether Length
( [m] 30 = t A [s] 20 2 = = t t T w p )
3.3 Simulation Result
One successful example of numerical simulations
is shown in Fig. 7, where one sub-satellites
controls length of three tethers it has, to make
position offset between the C.P. and C.M. Fig.
7(a) shows shape variation of the sailcraft in 3-D
model, and Fig. 7(b) shows history of angular
momentum during the simulation. From the figure,
conservation of total angular momentum is
confirmed, while angular momentum exchange
between membrane and sub-satellites occurs.
Although tether length change is finished, residual
vibration is remained between them.
(a) 3-D View from Top
(b) History of Angular Momentum
Figure 7 One of Numerical Simulation Result
( [ ] sec 20�@= t T , [ ] rad/sec 1 . 0 / = c s w , [ ] m 90 = sat l , [ ] m 5 At = )
The exchange of angular momentum during
tether length change is cause of this residual
vibration, as shown in Fig. 8.
When this swing gets large, membrane breaks
bounds of tethers, as shown in Fig. 9. This
configuration is unacceptable because it may cause
a tangle of membrane and tether.
Figure 8 Excitation Vibration
( [ ] sec 20�@= t T , [ ] rad/sec 2 . 0 / = c s w , [ ] m 100 = sat l , [ ] m 15 At = )
Figure 9 Unacceptable Configuration
The vibration behavior dependency on the
following parameters is investigated:
sat l : initial tether length between sub-satellites
sat m : mass of sub-satellites
c s w : initial angular velocity of the sailcraft
t w :
tether length change rate
Maximum tether length amplitude max A which
dosen�ft induce the unacceptable configuration
during sinusoidal tether length change is obtained
by changing initial angular velocity of the sailcraft
and tether length change rate.
Fig. 10 and Fig. 11 show the summary of
numerical simulation result. In the analysis, the
following values of parameters are used:
[ ] kg 40 , 20 , 10 , 5 = sat m
[ ] m 80 , 90 , 100 = sat l
[ ] sec 5 , 10 , 20 , 40 = t T
[ ] sec rad 00 . 1 , 50 . 0 , 40 . 0 , 25 . 0 , 20 . 0 , 10 . 0 , 05 . 0 / = c s w
Fig. 10 shows linear dependency of maximum
tether length amplitude max A on angular velocity
rate t c s w w / with changing tether length between
the sub-satellites sat l . Fig. 11 shows the same
relationship with changing sub-satellite mass sat m .
It is found that the tether length controllability gets
larger when the initial tether length and subsatellite
mass are large.
0
10
20
30
40
50
60
70
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50


s/c / 

t
Max amplitude of tether extension , m
lsat = 100 [m]
lsat = 90 [m]
lsat = 80 [m]
Figure 10 Dependency of max A on t c s w w /
0
10
20
30
40
50
60
70
80
90
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50


s/c / ƒÖt
Max amplitude of tether extension , m
m_sat = 5 [kg]
m_sat = 10 [kg]
m_sat = 20 [kg]
m_sat = 40 [kg]
Figure 11 Dependency of max A on t c s w w /
4. Attitude Maneuver Rate Estimation
In this section, attitude maneuver rate obtained
by the proposed torque generation method is
estimated under the rigid body approximation.
Using Newton-Euler�fs equations,
h
dt
dh
M r �~ + = w (4)
where :
M : total torque acting on the system
h : angular momentum of the system
r w : angular velocity of despin body-fixed
frame to the inertial frame
Take an average of one cycle of the sailcraft�fs spin,
0 �â
dt
dh (5)
Thus, Eq.(4) can be rewrite as:
h
M
r �â w (6)
Fig. 12 shows obtainable torque by the C.P.-C.M.
offset method.
Estimated attitude maneuver rate r w at the
Earth, Mars and Jupiter are shown in Table 1.
Attitude maneuver rate of planned methods are
approximately 1-3 deg/day, thus the proposed
method is comparable to them for space
exploration missions.
Figure 12 Torque by the C.P.-C.M. Offset
Table 1 Estimation of Attitude Maneuver Rate
Tether Length Amplitude
1 [m] 3 [m] 5 [m]
Attitude
Control
Rate
[deg/day]
Earth 11 35 58
Mars 3.4 10 17
Jupiter 0.40 1.3 2.2
( [ ] sec rad 01 . 0 / = c s w , [ ] m 80 = sat l )
5. Attitude Dynamics under Solar
Pressure
In this section, attitude dynamics of the tethercontrolled
spinning solar sail is investigated by
numerical simulation under solar pressure with
changing its tether length. Analytical model is as
same as which was described in section 3.1. All
sub-satellites�f tethers were controlled sinusoidally.
Fig. 13 shows the trajectory of a head of normal
vector to the sub-satellite plane. Sub-satellite plane
can be inclined by the C.P.-C.M. offset torque, but
nutation and out of membrane vibration also
occurs. This result suggests some passive or active
nutation damping method is needed for attitude
maneuver of the sailcraft.
Figure 13 Time History of Normal Vector to
Sub-Satellite Plane
(Top: 3-D Graphics, Bottom: Component Graphs)
6. Conclusion
C.P.-C.M. offset torque generation method for
attitude maneuver of tether-controlled spinning
solar sail was proposed. Numerical simulations on
dynamic behavior during tether length control
were conducted with a simple mass-spring-damper
model. The parameter dependencies of the tether
length controllability were investigated for
excitation vibration between sub-satellites and
membrane. Rate of attitude maneuver using C.P.-
C.M. offset torque was estimated under solid body
assumption and suggested that comparable to the
other methods. Numerical simulation of attitude
maneuver by controlling tether length was
conducted under solar pressure. From the
simulation result, vibration of the membrane and
nutation motion of the sailcraft were found. Some
passive or active damping methods are under study
to suppress it.
References
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