1

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.