The Use of a Spinning Dissipator for Attitude Stabilization of Earth- Orbiting Satellites
Rapid Control of Attitude Angles for Spinning Solar …...1 Rapid Control of Attitude Angles for...
Transcript of Rapid Control of Attitude Angles for Spinning Solar …...1 Rapid Control of Attitude Angles for...
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Rapid Control of Attitude Angles for Spinning Solar Sail
Utilizing Spin Rate Change with Reflectivity Control Devices
By Takumi KUDO, Kenshiro OGURI, and Ryu FUNASE1)
1)Department of Aeronautics and Astronautics, The University of Tokyo, Tokyo, Japan
This paper suggests a new way of controlling a spinning solar sail rapidly with a reflectivity control device. The basic idea
of the control method of this study is that the smaller the sail’s nominal spin rate is, the faster the attitude maneuver
completes. In order to reduce the spin rate, this study uses RCD input to control not only the sail’s attitude angles but also
the sail’s nominal spin rate. 1296 cases of initial and target attitude were simulated, and it was found that in approximately
28.7% of all cases, attitude trajectory time was considerably smaller than a post method. This study also found a new
characteristic of attitude trajectory when the spin rate is controlled by RCD input.
Key Words: Reflectivity control device, Spinning solar sail, Spin rate control, Attitude control
Nomenclature
�̃� : Azimuth angle relative to the solar
direction: �̃� ∈ ℝ
𝛿 : Elevation angle relative to the solar
direction: 𝛿 ∈ ℝ
𝛺 : The sail’s nominal spin rate: 𝛺 ∈ ℝ
𝜙 : Phase angle where RCD switches from
off to on: 𝜙 ∈ ℝ
𝛼𝑠 : The azimuth angle of the solar direction:
𝛼𝑠 ∈ ℝ
𝛿s : Elevation angle of the solar direction:
𝛿𝑠 ∈ ℝ
Subscripts
n : The suggested method
p : The past method
1. Introduction
Attitude control of spinning solar sail using a reflectivity
control device (RCD) has been attracting much attention in
recent years because the device doesn’t need any fuel for
attitude control. RCD is a liquid crystal device which can
change its reflectivity characteristic by electrically switching on
and off; turning RCD on increases the ratio of specular
reflection, and turning RCD off increases the ratio of diffuse
reflection. Switching on and off varies the influence of SRP on
dynamics of spacecraft, which realizes attitude control of a
spinning solar sail. In fact, possibility of attitude control by
RCD input has been verified by IKAROS mission 1).
Attitude control methods by RCD input have been proposed
in recent years. For example, Oguri et al analytically derived a
time-optimal attitude control law from Extended-GSSM3), 4),
which is a dynamics model of a spinning solar sail controlled
by RCDs.
Although the time-optimal attitude control law puts an
assumption that the sail’s nominal spin rate is maintained
constant, a control law which takes advantage of the spin rate
change should be proposed. In fact, on the condition that the
spin rate is not maintained constant, faster solutions with regard
to the attitude trajectory time appear when a numerical
calculation is conducted.
Therefore the present paper proposes a faster strategy of
attitude maneuver regarding time. The method proposed in this
paper takes advantage of the spin rate control. The
consideration of spin rate control realizes a faster attitude
control strategy of a spinning solar sail with RCD control input.
Fig. 1 Concept of attitude control system utilizing reflectivity control
device. The RCD pictures of the figures are cited from 1)
2. Attitude Dynamics Model of Spinning Solar Sail with
RCD Input
In order to derive faster attitude control law of spinning solar
sail with RCD input than the law suggested in 3), this section
defines the sail’s attitude, the RCD input and the differential
equation of the attitude to be used in this paper.
2.1. Definition of Sail’s Attitude Angles, Spin Rate and
RCD input
This paper uses the sail’s state variable (�̃�, 𝛿, 𝛺) defined in
5); attitude angle α̃ and 𝛿 are the azimuth and elevation
angles measured from the sun direction vector s, respectively,
and 𝛺 is the sail’s nominal spin rate (see Fig. 2). 𝜙 is a phase
angle measured from the intersection line between the ecliptic
2
and the plane vertical to the sail’s nominal spin rate direction.
RCDs are turned on between the angular range of [𝜙, 𝜙 + π].
Fig. 2 Definition of the sail’s state variables (�̃�, 𝛿, 𝛺) and input 𝜙. The
azimuth angle, the elevation angle, the spin rate of the spin axis, and the
phase angle in this figure are �̃�, 𝛿, 𝛺, and 𝜙, respectively
2.2. Introduction of the Differential Equation of the State
variables
The differential equations of the state variables are written as
follows: 𝑑
𝑑𝑡{�̃�𝛿
} = 𝑭(𝛺, 𝑢) {�̃�𝛿
} − �̇�𝑠 + 𝑢𝒉(𝛺, 𝜙) (1)
𝑑𝛺
𝑑𝑡= 𝑢𝑔(�̃�, 𝛿, 𝜙) +
𝐶
𝐼𝑠 (2)
where u (0 ≤ 𝑢 ≤ 1) is the magnitude of RCD input between
the phase angular range of [𝜙, 𝜙 + π]; �̇�𝑠 is the time rate of
change of the sun direction vector; 𝐼𝑠 is the sail’s moment of
inertia around the nominal spin rate direction. C is the shape
and optical parameter of the sail’s membrane. Although it is
found that C changes as the spin rate 𝛺 slightly changes2), this
paper regards that change is negligible, and thinks of C as a
constant value.
This paper defines u as 1.0 because when u equals 1.0, the
difference of SRP force between RCD-on range and RCD-off
range becomes the largest, which offers the largest SRP torque.
In this paper, the largest SRP torque is assumed to realize faster
maneuver than all the other values of u.
The matrix 𝑭(𝛺, 𝑢), the vectors 𝐚�̇�, h(𝛺, 𝜙), and the scalar
𝑔(�̃�, 𝛿, 𝜙) are written as follows:
𝑭(𝛺, 𝑢) =1
𝐼𝑠𝛺[𝐴 + 𝑢𝛥𝐴 −(𝐵 + 𝑢𝛥𝐵)𝐵 + 𝑢𝛥𝐵 𝐴 + 𝑢𝛥𝐴
] (3)
𝐚�̇� = {𝛼�̇�
𝛿�̇�} (4)
𝒉(𝛺, 𝜙) =1
𝐼𝑠𝛺[
sin𝜙 cos𝜙−cos𝜙 sin𝜙
] {𝐻1
𝐻2} (5)
𝑔(�̃�, 𝛿, 𝜙) =1
𝐼𝑠({𝛥𝐷 𝛥𝐸} [
−sin𝜙 cos𝜙−cos𝜙 −sin𝜙
] {�̃�𝛿
}
+ 𝛥𝐶)
(6)
where Δ𝐴, Δ𝐵, Δ𝐶, Δ𝐷, Δ𝐸, 𝐻1, 𝐻2 are the RCD’s location and
optical parameters defined in 4), 5). This paper and 3) use those
differential equations (1), (2).
3. Existing Time-Optimal Attitude Control Law3)
3.1. Problem of the Assumption
Although the existing time-optimal attitude control law puts
an assumption that the sail’s spin rate is constant3), the spin rate
was found to considerably change during the attitude maneuver.
Because of that, it has been found that the sail cannot arrive at
the target attitude when the existing control law is adopted (see
Fig. 3).
In Fig. 3, the assumption of the purple curve is that the spin
rate 𝛺 is constant, and therefore only Eq.(1) is integrated.
However, if the spin rate is not controlled by any devices except
for the RCD, Eq.(2) also has to be integrated. The blue curve in
Fig. 3 is the integration result of both of Eq.(1) and Eq.(2). As
in Fig. 3 (left), the spin rate of the blue curve changes. In Fig. 3
(right), although the purple curve strictly arrives at the target
attitude (20.0, 20.0) deg, the blue curve approximately arrives
at (-15.0, -10.0) deg and cannot arrive at the target attitude. The
error of the attitude angle is so large (approximately 10 deg)
that it cannot be neglected.
Fig. 3 Comparison of the spin rate change (left) and attitude trajectory
(right) between the different assumptions of the spin rate. The initial spin
rate, the initial attitude angles, and the target attitude angles are 2.0 rpm,
(20.0, 20.0) deg, (-20.0, -20.0) deg, respectively. The assumption of the
purple curve is that the spin rate is constant during the attitude trajectory,
and that of blue is that the spin rate changes during the attitude maneuver.
The shape and optical parameters of the simulation are those of Table. 1.
3.2. One way to Apply the Existing Time-Optimal Control
Law to the Plant where the Spin Rate Changes
If the sail needs to arrive at the target attitude utilizing that
control law, it needs to be recalculated in a short period of time.
At each calculation, the spin rate equals to the spin rate at the
end of the last calculation. By conducting that recalculation, the
sail is able to arrive at the target attitude. It is explained by the
fact that, at each recalculation, the spin rate change is so small
that the assumption of the existing time-optimal control law that
the spin rate is constant almost holds true. As in Fig. 4 the sail
almost strictly arrives at the target in spite of the assumption
that the spin rate is not maintained to be constant.
Fig. 4 Spin rate change and the attitude trajectory when the recalculation
method of this subsection is applied. The initial spin rate, the initial attitude
angles, and the target attitude angles are 2.0 rpm, (20.0, 20.0) deg, (-20.0, -
20.0) deg, respectively. The parameters of the simulation are those of Table.
2 and Table. 3.
Although this recalculation method cannot be concluded to
be the proper way to apply the existing time-optimal control
3
method to the plant where the spin rate changes, this research
regards that recalculation method has been the fastest control
law with regard to the attitude trajectory time except for the
control law suggested in Section 4.
3.3. Existence of a Faster Solution than the Recalculation
Method
In many cases, maneuver time of that recalculated time-
optimal control law has been similar to the numerical
calculation result. However, maneuver time of the numerical
calculation sometimes has been found to be strikingly smaller
than that of the recalculated time-optimal control law (see Fig.
5). In the case of Fig. 5, trajectory time of the numerical
calculation result is 0.5 days smaller than that of the
recalculation method of Subsection 3.2. This paper considers
the cause of the problem should be that the existing time-
optimal control law doesn’t control the spin rate. Fig. 5 indicates
that when the spin rate is controlled to decrease, a faster attitude
control law would be obtained. This supposition is consistent
with the fact that a low angular momentum probe can change
its attitude rapidly.
Fig. 5 Comparison of the spin rate change (left) and the attitude trajectory
(right) between the recalculated existing time-optimal attitude control law
(blue) and the numerical time-optimal solution (green)
4. Proposed Method: Rapid Attitude Control Strategy
Utilizing the Spin Rate Change
The current investigation derives a faster attitude control law
than the existing time-optimal attitude control law referenced
in Section 3 of this paper. The suggested control method utilizes
the spin rate control. The basic idea of the control law is that
the smaller the spin rate is, the faster the time to finish the
attitude maneuver is. The small spin rate means small angular
momentum, and the slight angular momentum realizes quick
attitude maneuver.
4.1. Rapid Attitude Control Strategy
The suggested method utilizes the characteristics described
in Subsection 4.1 and 4.2 and consists of three phases. This
method utilizes RCD to decrease the spin rate to reduce the
angular momentum. As described before, low angular
momentum should be related to a faster attitude trajectory with
respect to the trajectory time. Each of the three phases is
explained below (see Fig. 6).
In Phase 1, the spin rate is decreased by putting the RCD
input of Eq. (10). In Phase 2, the spin rate is kept to be constant
by putting one of the RCD inputs of Eq. (11). In Phase 3, the
existing time-optimal control law is conducted. As stated in
Section 3, due to the spin rate change during the attitude
trajectory, the time-optimal control law is continuously solved
in a short period of time.
There are three degrees of freedom in this suggested method,
and numerical calculations are needed to obtain the most rapid
attitude maneuver with regard to the trajectory time. Two of the
degrees of freedom are time to change from Phase 1 to Phase 2,
and that from Phase 2 to Phase 3(𝑡1 and 𝑡2, respectively). The
other one is the two inputs of Phase 2 (𝜙2).
Fig. 6 Conceptual diagram of the suggested method
In this paper, numerical calculations were carried out to
achieve the most rapid attitude trajectory with regard to time.
Enough sets of 𝑡1 , 𝑡2 and 𝜙2 are chosen and simulations
were conducted per set in order to obtain the fastest set.
Specifically, in the simulation of this paper, 𝑡1 and 𝑡2 were
chosen from 0.75, 1.5, 2.25, 3.0 days, and 𝜙2 was chosen from
the Eq. (11). Although the attitude trajectory time of typical
space probes does not exceed as long as one day, it usually takes
about or more than a day for spinning solar sail with RCD input
to complete the attitude maneuver2), 3), 4), 5). Hence, the time
length order of candidates of 𝑡1 and 𝑡2 are not improper.
4.2. Minimum Spin Rate Differentiation Input
In this Subsection, the input of Phase 1 (see Fig. 6) is derived.
To obtain an RCD input which realizes the maximum spin rate
declination at each attitude angle, Eq. (2) is transformed as
follows: 𝑑𝛺
𝑑𝑡= 𝐴(�̃�, 𝛿) sin(𝜙 + 𝜓) + 𝐶 + Δ𝐶 (7)
where
𝐴(�̃�, 𝛿) = √(Δ𝐷�̃� + Δ𝐸𝛿)2
+ (ΔE�̃� − Δ𝐷𝛿)2 (8)
tan𝜓 =ΔEα̃ − ΔDδ̃
ΔDα̃ + ΔEδ̃ (9)
Because 𝐴(�̃�, 𝛿) ≥ 0 and the value 𝜙 + 𝜓 which minimizes
the trigonometric function sin(𝜙 + 𝜓) is 𝜙 + 𝜓 = 𝜋/2 , the
minimum value of the left-hand side of Eq. (7) at each attitude
angle (�̃�, 𝛿) is obtained by substituting 𝜙 + 𝜓 = 𝜋/2 into the
right-hand side of Eq. (7). Finally, the following equation is
obtained by transforming the equation 𝜙 + 𝜓 = 𝜋/2. Eq. (10) is
an RCD input which realizes the maximum spin rate declination
at each attitude angle.
𝜙�̇�𝑚𝑖𝑛𝑖𝑚𝑢𝑚=
𝜋
2− tan−1
Δ𝐸�̃� − Δ𝐷𝛿
ΔD�̃� + Δ𝐸𝛿 (10)
4.3. Constant Spin Rate Input
In this subsection, the input of Phase 2 (see Fig. 6) is derived.
RCD inputs which realize the constant spin rate are derived by
substituting 0 for 𝑑Ω/𝑑𝑡 in Eq. (2). Those inputs are written
as follows:
𝜙 = sin−1 𝜓1 − tanψ2, 𝜙 = π − sin−1 𝜓1 − tan𝜓2
(11)
where
4
𝜓1 =𝐶 + Δ𝐶
√(ΔDα̃ − ΔEδ̃)2
+ (Δ𝐸�̃� − Δ𝐷𝛿)2
(12)
𝜓2 =Δ𝐸�̃� − Δ𝐷𝛿
Δ𝐷�̃� + Δ𝐸𝛿 (13)
5. Spacecraft Specification
In this paper, the shape of the sail is assumed to be a
deformed circular membrane, and RCD is put on the outer edge
of the circle (Fig. 7). r is the radius of the sail; n is the average
normal vector of the sail membrane; ξ and η are average
outer-plane and twist deformation angles of the membrane
respectively2); h is the offset length from the spacecraft’s center
of gravity.
Fig. 7 Shape and RCD model of the sail in this research
This research used the optical and shape constants on Table.
1 to calculate the optical and shape parameters on Table. 2. The
relations of the optical and shape parameters of Eq. (1)-(6) with
the sail’s shape and its reflectivity are explained in 2) and 5).
In this research, the initial spin rate and the sun vector
differentiation are also identified on Table. 3. The initial spin
rate 2.0 is not improper because, in the case of IKAROS, the
sail’s spin rate is maintained between 1 – 2.5 rpm2) during
the nominal mission phase, and the spin rate 2.0 falls within
that range. The sun direction vector differentiation on Table.
3 is determined for the spacecraft to rotate 360 degrees
around the sun per year on the ecliptic plane.
Finally, the parameters on Table. 2 and Table. 3 are employed
in the simulations of this research.
Table. 1 Sail’s shape and optical constants and the initial spin rate used
in this paper
Name of the Parameter Value Unit
Sail’s Radius: r 5.5 [m]
Sail’s Offset: h -0.20 [m] Density of the Membrane 1.270e+3 [kg/m2]
Thickness of the Membrane 4.800e-6 [m] Sail’s Outer-Plane
Deformation Angle: 𝜂
1.577 [deg]
Sai’s Twist Deformation
Angle: 𝜉
-5.830e-3 [deg]
Sail’s Specular Reflection
Constant: 𝐶spe
0.72 -
Sail’s Diffuse Reflection 0.16 -
Constant: 𝐶dif
RCD Area: 𝐴RCD 30.00 [m2] RCD Radius: 𝑟RCD 5.000 [m]
RCD Height: ℎRCD -0.20 [m]
Distance from the Sun 1.0 [AU]
Table. 2 Shape and optical parameters derived from Table. 2 Name of the Parameter Value Unit
Moment of Inertia around
the Spin Axis: 𝐼s 53.88 [kgm2]
Membrane Parameter: A -1.47e-7 [kgm2rad s⁄2
] Membrane Parameter: B 2.79e-5 [kgm2rad s⁄
2]
Membrane Parameter: C -2.49e-7 [kgm2rad s2⁄ ] RCD Parameter: Δ𝐴 0 [kgm2rad s⁄
2]
RCD Parameter: Δ𝐵 4.10e-6 [kgm2rad s⁄2
] RCD Parameter: Δ𝐶 0 [kgm2rad s⁄
2]
RCD Parameter: Δ𝐷 0 [kgm2 s⁄2
] RCD Parameter: Δ𝐸 -6.53e-5 [kgm2 s⁄
2]
RCD Parameter: 𝐻1 0 [kgm2rad2 s⁄2
] RCD Parameter: 𝐻2 -2.61e-5 [kgm2rad2 s⁄
2]
Table. 3 Initial spin rate and the sun direction vector differentiation used
in this paper Name of the Parameter Value Unit
Initial Spin Rate: 𝛺0 2.0 [rpm]
𝛼�̇� 2.02e-7 [rad s]⁄
𝛿�̇� 0.0 [rad s⁄ ]
6. Simulation 1: Simulation to Calculate the Ratio of the
Effective Case
In this section and the next section, two types of simulation
were conducted respectively using the parameters on Table. 2
and Table. 3. Both simulation results were compared with the
existing time-optimal control law explained in Section 3.2
to verify the effectiveness of the suggested control method.
In the first simulation, various sets of initial and target
attitude were chosen to conduct the control method
introduced in Section 4. On the other hand, in the second
simulation, the initial attitude was fixed and various target
attitudes were chosen. In both simulations, some features of
the suggested method were discovered.
6.1. Simulation Conditions
In this simulation, 1296 sets of the initial and target attitude
angles were simulated. Concretely, the initial and target
azimuth and elevation angles were chosen from (-30.0, -18.0, -
6.0, 6.0, 18.0, 30.0) [deg], and all of the cases of the initial and
target azimuth and elevation angles were simulated.
6.2. Simulation Results
Fig. 8 is the histogram of the simulation results whose
conditions were determined in Subsection 6.1. In Fig. 8, 𝑡𝑛 is
the time to complete an attitude maneuver when the suggested
method introduced in Subsection 4.1 was applied. 𝑡𝑝 is the
time to complete the attitude trajectory if the existing time-
optimal control method explained in Subsection 3.2 was used.
Therefore, if 𝑡𝑛/𝑡𝑝 smaller than 1.0, the suggested method in
Subsection 4.1 is faster than the existing method in Subsection
3.2, and vice versa.
In this research, if the ratio of the attitude trajectory time
𝑡𝑛/𝑡𝑝 of a set of initial and target attitude angles is equivalent
to or smaller than 0.98, the suggested method is called effective
in the attitude set. On the other hand, when 𝑡𝑛/𝑡𝑝 is equivalent
5
to or larger than 1.02, the proposed method is called
counterproductive in the attitude set. If 𝑡𝑛/𝑡𝑝 is between 0.98
and 1.02, the suggested method is called ineffective.
As in Fig. 8, in 28.7% of all cases, the suggested method was
effective, and in 3.5% of all cases, the proposed method was
counterproductive. At least in some cases, the suggested
method was considerably faster than the existing method in
Subsection 3.2. In this section, whether the number 28.7% is
large or not is not discussed.
Fig. 8 Histogram of the simulation results. The simulation condition is
explained in Subsection 6.1.
Fig. 9 is the simulation result whose 𝑡𝑛/𝑡𝑝 is the smallest.
The initial attitude is (-30.0, 30.0)[deg], and the target attitude
is (30.0, 30.0)[deg]. As in Fig. 9, the spin rate was reduced in
phase 1, and the spin rate is maintained constant in Phase 2. The
spin rate increased in Phase 3, but the cause of the inclination
is not understood.
The suggested method was approximately 9 days faster than
the existing method. It should be because the decreased spin
rate accelerated the attitude maneuver in the case of the
suggested control method, and the decreased spin rate
decelerated the attitude maneuver in case of the existing control
law. The spin rate and angular momentum correlated each other
and the smaller the angular momentum is, the faster the attitude
maneuver completes.
Fig. 9 Comparison of the spin rate change (left) and the attitude trajectory
(right) between the recalculated existing time-optimal attitude control law
of Subsection 3.2 (blue) and the suggested control method of Subsection
4.1 (red), when 𝑡𝑛/𝑡𝑝 is the smallest
Fig. 10 is the simulation result whose 𝑡𝑛/𝑡𝑝 is the largest.
The initial attitude is (6.0, 30.0)[deg], and the target attitude is
(30.0, 30.0)[deg]. As in Fig. 9, the spin rate was not decreased
in phase 1, and the spin rate is maintained constant in phase 2.
The reason why the suggested method was about 9 days
slower than the existing method should relate to the fact that the
spin rate was not satisfactorily decreased. However, the reason
why the time difference was as large as 9 days has not been
understood, and therefore the cause should be investigated in
the future.
Fig. 10 Comparison of the spin rate change (left) and the attitude
trajectory (right) between the recalculated existing time-optimal attitude
control law of Subsection 3.2 (blue) and the suggested control method of
Subsection 4.1 (red), when 𝑡𝑛/𝑡𝑝 is the largest
6.3. Some Features Concerning the Suggested Control
Method
There are certain characteristics about the attitude trajectory
of phase 1 and phase 2.
By substituting RCD inputs 𝜙 obtained from Eq. (11) into
Eq. (1) at each attitude angle (α̃, δ̃) , maps of the angular
velocity vector (α̇̃, δ̇̃) are calculated. Fig. 11 shows the
angular velocity maps of a sail whose optical and shape
parameters are written in Table. 1. The green vectors are the
angular velocity vectors, and the contours are the magnitude of
the vectors; the magnitude of the angular velocity vectors in the
red area is larger than that in the blue area.
It is apparent that the vectors in Fig. 11 describe circles
around the origin. That indicates that the attitude trajectory
should become a circle and the spin rate remains constant if
RCD input of Eq. (11) is put at each attitude angle. Several sets
of other shape and optical parameters are introduced to draw
maps like Fig. 11, and in each case, the angular velocity vectors
described a circle. Hence it can be stated that if the RCD input
calculated by Eq. (11) is put at each attitude angles, the attitude
trajectory draws a circle around the origin and the spin rate
remains constant.
Fig. 11 Attitude angle differentiation vector map when RCD input is put
to maintain the sail’s nominal spin rate constant
A map of the angular velocity vectors when the spin rate is
maintained constant by RCD input is obtained by substituting
the RCD input of Eq. (10) into Eq. (1) at each attitude angle
(�̃�, 𝛿). Fig. 12 shows the angular velocity maps of a sail whose
6
optical and shape parameters are written in Table. 1. The
meaning of the blue vectors and the contour is the same as the
blue vector and the contour in Fig. 11 respectively.
In Fig. 12, the vectors are directed away from the origin,
which indicates that the attitude angle moves away from the
origin when the RCD input is put in order to maximize the spin
rate declination rate.
Fig. 12 Attitude angle differentiation vector map when RCD input is put
to minimize the sail’s nominal spin rate differentiation
7. Simulation 2: Simulation to Calculate the Ratio of the
Effective Case
In this section, the initial attitude was fixed and various
target attitudes were chosen. In both simulations, some
features of the suggested method were found.
7.1. Simulation Conditions
In this simulation, 36 sets of the initial and target attitude
angles were simulated. Concretely, the initial attitude was fixed
to (20.0, 20.0)[deg] and target azimuth and elevation angles
were chosen from (-30.0, -18.0, -6.0, 6.0, 18.0, 30.0) [deg]. All
of the cases of the initial and target azimuth and elevation
angles were simulated.
7.2. Simulation Results
Fig. 13 is a contour map of the various target attitudes. The
contour value is the calculation result of 𝑡𝑛 − 𝑡𝑝. 𝑡𝑛 and 𝑡𝑝
are defined in Subsection 6.2. In the blue area, the sign of 𝑡𝑛 −
𝑡𝑝 is the minus, and therefore when the target attitude lies in
the blue area, the suggested control method becomes faster than
the existing control method.
It can be said that the larger the distance from the initial
attitude to the target attitude is, the more effective the suggested
method should be.
Fig. 13 Contour map of the 36 target attitudes. The initial attitude is (20.0,
20.0)[deg] and the value of the contours is the calculation result of 𝑡𝑛 − 𝑡𝑝.
8. Conclusion
In this research, the attitude control method utilizing the spin
rate control is suggested, and the basic idea of the method is
that the smaller the spin rate is, the faster the attitude maneuver
completes because the spin rate and the angular momentum
correlated to each other.
Because of the simulation results in Section 7 and 8, it can
be said that the suggested method is sometimes faster than the
existing time-optimal control method in which the spin rate is
assumed to be constant. The ratio of the case where the
suggested method is considerably faster than the existing
method is not so large, and the way to define the three-phased
method is arbitrary. Therefore some method which is faster than
the existing method in many cases should exist.
There are some characteristics concerning the attitude
trajectory when RCD input is put to make the spin rate
differentiation smallest and the spin rate is maintained constant,
respectively. In the former case, the attitude angles should leave
from the origin. In the latter case, the attitude should rotate
around the origin. Although those discoveries do not directly
relate to the purpose of the research, they should provide useful
information when the spin rate is to be controlled in the future
study.
In Section 7, the effective area of the target attitudes was
found, and the knowledge proposes the way to use the
suggested method in actual spinning solar sail missions
utilizing RCD input. When the present attitude is obtained, a
map like Fig. 13 can be drawn. If the target attitude is in the blue
area, the suggested method should be made use of.
References
1) Funase, R., Shirasawa, Y., Mimasu, Y., Mori, O., Tsuda, Y., Saiki, T., & Kawaguchi, J. (2011). On-orbit verification of fuel-free
attitude control system for spinning solar sail utilizing solar radiation
pressure. Advances in Space Research, 48(11), 1740–1746.
2) T. Yuichi and T. Saiki, "Generalized Attitude Model for Spinning
Solar Sail Spacecraft," Journal of Guidance, Control, and
Dynamics, pp. 1-8, 2013.
3) Oguri, K., Kudo, T., & Funase, R. (2016). Attitude Maneuverability
Estimation for Preliminary Mission Design of Spinning Solar Sail
Driven by Reflectivity Control. AIAA/AAS Astrodynamics
Specialist Conference, (September), 1–19.
https://doi.org/10.2514/6.2016-5674
4) T. Furumoto. and R. Funase, "Attitude Control Model for Spinning
Solar Sail with Reflectivity Control Capacity," 30th International
Symposium on Space Technologies and Science (ISTS), pp. d-25,
July 2015.