Attitude Control Dynamics of Spinning Solar Sail “IKAROS” Considering Thruster Plume

Osamu Mori, Yoji Shirasawa, Hirotaka Sawada, Ryu Funase, Yuichi Tsuda, Takanao Saiki and Takayuki Yamamoto (JAXA), Morizumi Motooka and Ryo Jifuku (Univ. of Tokyo)

Abstract In this paper, the attitude dynamics of IKAROS, which is spinning solar sail, is presented. First Mode Model of out-of-plane deformation (FMM) and Multi Particle Model (MPM) are introduced to analyze the out-of-plane oscillation mode of spinning solar sail. The out-of-plane oscillation of IKAROS is governed by three modes derived from FMM. FMM is very simple and valid for the design of attitude controller. Considering the thruster configuration of IKAROS, the force on main body and sail by thruster plume as well as reaction force by thruster are integrated into MPM. The attitude motion after sail deployment or reorientation using thrusters can be analyzed by MPM numerical simulations precisely.

スラスタプルームを考慮したスピン型ソーラーセイル「IKAROS」の姿勢制御運動

森 治，白澤 洋次，澤田 弘崇，船瀬 龍，津田 雄一，佐伯 孝尚，山本 高行（JAXA）， 元岡 範純，地福 亮（東大）

摘要 本論文ではスピン型ソーラーセイル IKAROS の姿勢運動について示す．スピン型ソーラーセイル

を解析するために一次面外変形モデルおよび多粒子モデルを導入する．まず，一次面外変形モデ

ルから導出される 3 つのモードが IKAROS の面外運動を支配していることを示す．このモデルは

非常に簡易であり，姿勢制御設計に有効であると言える．一方，多粒子モデルに対しては，

IKAROS のスラスタ配置を考慮して，スラスタの反力だけでなくスラスタプルームが本体や膜面

に及ぼす力を組み込み，セイル展開後およびスラスタによるマヌーバ後の姿勢運動を数値シミュ

レーションにより詳細に解析する． 1. Introduction

A solar sail 1) is a space yacht that gathers energy for propulsion from sunlight pressure by means of a membrane. The Japan Aerospace Exploration Agency (JAXA) successfully achieved the world’s first solar sail technology by IKAROS (Interplanetary Kite-craft Accelerated by Radiation Of the Sun) mission 2) in 2010. JAXA is also studying the extended solar sail mission toward Jupiter and Trojan asteroids exploration via hybrid electric photon propulsion 3) as shown in Fig. 1. There are two types of solar sail as shown in Fig. 2. First is the mast type 4, 5), which uses some rigid support structure to deploy and maintain the sail. The other is the spin type which uses spinning centrifugal force. The deployment motion and attitude control of mast type are simpler than those of spin type. A lot of solar sail missions of mast type are studied as shown in Table 1. On the other hand, spin type can be accomplished with lighter-weight mechanisms than mast type because it does not require rigid structural elements. Thus spin type should be selected in case of a large solar sail. IKAROS demonstrates a spin type solar sail whose area is 200m2 for extended solar sail whose area is 2000m2. In this paper, the attitude dynamics of IKAROS, which is spinning solar sail, is presented. The out-of-plane oscillation modes of IKAROS are derived from First Mode Model of out-of-plane deformation (FMM) 6) analytically. Multi Particle Model (MPM) 7) is used

for the numerical model. Considering the thruster configuration of IKAROS, the force on main body and sail by thruster plume as well as reaction force by thruster are integrated into MPM. The Fast Fourier Transform (FFT) results from IKAROS flight data are compared with those from simulation data.

Fig. 1 IKAROS mission and extended solar sail mission

©NASA

Mast type

JAXA

Spin type

Fig. 2 Mast type and spin type

Table 1 Solar sail missions

Mission Who Sail size [m2] Launch Type

IKAROS JAXA 200 Launched on May 2010 SpinNanoSail-D2 NASA 20 Launched on Nov. 2010 MastLightsail-1 TPS 32 Planning in 2011 MastCube Sail EADS - Surrey 25 Planning in 2011 MastCube Sail CU Aerospace 20 Planning in 2012 MastLightsail-2 TPS 100 Planning in 2013 MastUltra Sail CU Aerospace 100 Planning in 2015 Mast

Extended solar sail JAXA 2000 Planning in 2019 Spin

2. Dynamics Model of Spinning Solar Sail

This section presents IKAROS sail design and introduces two dynamics model of spinning solar sail. The first one is a simplified analytical model and the other one is a precise numerical model incorporating the flexibility of the sail. 2.1. IKAROS Sail Design Fig. 3 shows IKAROS sail design. The shape of the sail is a square whose diagonal distance is 20m. The sail membrane is made of polyimide resign whose thickness is 7.5 m. It is connected the main body by tethers. The shape of the main body is a cylinder whose diameter is 1.6m. A tip mass whose weight is 0.5kg is attached to each tip of the membrane in order to support the spinning deployment. Thin film solar cell and steering device are attached on the membrane. Therefore IKAROS sail is not uniform.

Steering device

Thin film solar cell

Tether

20m Tip mass

Main body

Fig. 3 IKAROS sail design

2.2. First Mode Model of out-of-plane deformation (FMM) An analytical dynamics model of the sail is derived considering first mode of deformation of the sail. Here we consider circular spinning solar sail configuration for analytical dynamics model shown in Fig. 4. The spinning solar sail consists of a main body and a large sail connected with the main body, which rotate around the Z-axis at a rate of . In this research, we adopted the dynamic model considering the first mode of sail deformation 6) proposed by Nakano, et al. The conservation laws of angular momentum and the equations of motion of sail are

012

12 xyIII (1)

012

12 yxIII (2)

02

1 22 IIJI yx (3)

02

1 22 IIJI xy (4)

where I1: moment of inertia (MOI) of sail Ix: MOI of main body around X, Y-axis Iz: MOI of main body around Z-axis I: MOI of overall spacecraft around X, Y-axis (=Ix+I1/2) J: MOI of overall spacecraft around Z-axis (=Iz+I1)

x, y, z: angular velocity around three axes I1 and I2 are defined as

b

a

r

rdrrrhI 3

1 2 (5)

b

a

r

r a drrrrrhI 22 2 (6)

where (r): density of sail [kg/m3] h: thickness of sail [m] ra: inner radius of sail [m] rb: outer radius of sail [m] and constitutes the first-order mode of the out-of-plane sail deformation w as follows.

cossin,, trrtrrtrw aa (7)

where corresponds to the phase of the spin motion of the main body. Here we analyze the attitude dynamics of the solar sail. State equations of the system can be described as the following equation.

xx

Adt

d (8)

Where

Tyx x (9)

01000

00100

000010

000001

01000

10000

2

1

2

2

1

2

2

12

2

12

sc

sc

sc

sc

NI

I

NI

I

NI

I

NI

I

A

(10)

x

x

I

III

I

I 21

2

1 2

1

(11)

1x

zsc I

IN (12)

It can be said that the dynamical property of the system is determined by the following three parameters which can be calculated by the moment of inertia of the main body and the sail.

scNI

I,,

1

2

(13)

The characteristic equation of the system can be derived as the following equation.

012det

2

2

2

1222222

I

INsNssAsI scsc

(14) It is found that the following three modes of oscillation constitute the nutation of the solar sail.

)(,, 210210 BBBBBB

(15)

Table 2 shows them. In the IKAROS configuration, I2/I1=0.78. In the cases of ra=0 (sail is as large as main body) and ra=rb (sail is much larger than main body), I2/I1=1 and I2/I1=0, respectively. In all cases, one of

three modes of oscillation is nearly equal to the spin rate . It is caused by nutation motion. The other two modes are caused by sail motion.

Main body

Sail

w(r, , t)

XB

YB

ZB

ra rb

Fig. 4 First mode model of out-of-plane deformation

Table 2 Three modes of oscillation analyzed by FMM

I2/I1 B0 B1 B2 0.01 0.929 28.9 29.4 0.5 0.919 2.76 3.29 0.78 0.884 1.50 1.99 0.99 0.442 1.01 1.06 2.3. Multi Particle Model (MPM) When a numerical modeling method which can analyze the dynamics of spinning solar sail is required, the useful model includes Finite Element Method (FEM) 8). However, when the FEM is applied for the analysis of the dynamics of solar sail, it is thought that it takes huge time to achieve the valuable information about the attitude motion of solar sail if a lot of parameters are varied. So, Multi Particle Model (MPM) is used for the numerical model in this study. The characteristics of MPM are that it takes less time for dynamics simulation and can perform more stable analysis than FEM, because MPM is a model which substitutes the elements of sail for particles connected by springs and dampers. Fig. 5 illustrates the MPM of IKAROS. The sail is modeled by mass-spring network and the main body is modeled by rigid body. Mass of each particle is determined based on a designed value and actual measured value of membrane, tether, tip mass, thin film solar cell and steering device, and ununiform mass distribution is considered. The inter-particle tension T can be described as following form:

00

00

LLLKLLK

LLLKLLKT

(16)

where K, L0, L, α and denote spring constant, natural length of spring, distance between two particles, coefficient of compression stiffness and damping ratio, respectively. Assuming that the sail resists a compression slightly, nonlinear spring model using coefficients of compression stiffness are employed. The spring constant K is determined by applying the principle of virtual work on an element so as to satisfy the relations of strain energy. This model assumes that the stress in the direction along each spring depends only on the strain in the same direction, so that the elasticity matrix is approximated to be diagonal. This

model can also take into account the effect of bending stiffness of each element and crease stiffness of folding line by implementing rotational spring, however these characteristics have little effect on the global behavior of the sail, and are not considered in this study. For the scheme of numerical time integration, the explicit Runge-Kutta-Gill method is employed.

KC

Lx

z

y

Fig. 5 Multi particle model of IKAROS

3. Attitude Dynamics of IKAROS

This section shows the attitude dynamics of IKAROS. The out-of-plane oscillation modes of following two motions are analyzed. - motion after sail deployment - motion after reorientation using thrusters 3.1. Motion after Sail Deployment The deployment method of IKAROS consists of two stages as shown in Fig. 6. In the first stage deployment, each quarter of the sail is extracted like a Yo-Yo despinner and the sail forms a cross shape. If the deployment is performed dynamically, each quarter is twisted again around the main body just after the deployment. Therefore the first stage deployment is performed quasi statically by activating the guides that hold the sail through the relative rotation mechanism. The second stage deployment is performed dynamically by activating the guides and releasing the hold of the sail. The sail expands quickly to form a square shape. Fig. 7 shows the angular velocities of main body at second stage deployment. The oscillations of angular velocities of x and y axes are occurred and damped gradually after second stage deployment start, because the second stage deployment is performed dynamically. The spin rate of z-axis is 0.0417Hz (=). Fig. 8 shows the FFT results of x-axis angular velocities of IKAROS flight data and MPM numerical simulation data after sail deployment. The first peak frequency of flight data, 0.0371Hz (=0.890) is equal to that of numerical simulation data, 0.0366Hz (=0.878) as shown in Table 3. They are also equal to B0 (=0.884) which is one of three modes analyzed by FMM as shown in Table 2. Because it is nearly equal to the first peak is caused by nutation motion. The other peaks of flight data and MPM data are equal to B2 (= 1.99) and B1 (= 1.50), which are the other

two modes analyzed by FMM. Thus the motion after sail deployment is governed by three modes derived from FMM.

Second stage deployment (dynamic)

First stage deployment (quasi static)

Guide Fig. 6 Deployment method of IKAROS

10000 2000010

20

30

-1

0

1

Time , [sec]

Z S

pin

Ra

te ,

[de

g/s

]

X,Y

Sp

in R

ate

, [d

eg

/s]

Second stage deployment start

Sp

in r

ate

(z-a

xis)

[d

eg/s

]

An

gula

r ve

loci

ties

(x,

y-ax

es)

[deg

/s]

ωz

ωy

ωx

Fig. 7 Angular velocities of main body at second stage

deployment

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-6

10-5

10-4

10-3

10-2

10-1

100 wx

frequency (Hz)

SPEC

TR

UM

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-5

10-4

10-3

10-2 wx

frequency (Hz)

SPEC

TR

UM

B1 B2

(a) IKAROS flight data

B0

(b) MPM numerical simulation data

Fig. 8 FFT results of x-axis angular velocities after sail deployment

Table 3 First peak of motion after sail deployment

First peak [Hz] Fight data 0.0371 (=0.890) MPM data 0.0366 (=0.878)

3.2. Motion after Reorientation using Thruster IKAROS has following three methods of reorientation. - reorientation using thrusters - reorientation using steering devices 9) - reorientation using solar pressure torque The oscillatory motion of the sail is occurred due to the impulsive torque by thrusters. We consider the attitude maneuver using thrusters. IKAROS has four thrusters on the main body to perform spin up, spin down, reorientation as shown in Fig. 9. - spin up: thrusters 1 and 3 - spin down: thrusters 2 and 4 - reorientation: thrusters 1 and 2, or thrusters 3 and 4 In this thruster configuration, the thruster plume impinges on main body and sail. Thus the force on main body and sail by thruster plume as well as reaction force by thruster should be considered as shown in Fig. 10. Figs. 11 and 12 shows plume flow model and plume impingement model. Plume flow and plume impingement are formulated by source flow method and free molecule flow, respectively. The force by thruster plume is integrated into multi particle model by following three steps as shown in Fig. 13. 1) Calculate the force on center of triangle element of sail using plume flow and impingement models. 2) The force is decomposed into normal and shearing forces. 3) These forces are distributed to three particles. When the spin rate of z-axis is 0.247Hz (=), the reorientation using thrusters 3 and 4 is performed. Fig. 14 shows the angular velocities of main body. The oscillations of angular velocities of x and y axes are occurred due to the impulsive torque by thruster and thruster plume. Fig. 15 shows the FFT results of x-axis angular velocities of IKAROS flight data and MPM numerical simulation data after reorientation using thrusters. The first peak frequency of flight data, 0.0488Hz (=0.198) is equal to the second peak frequency of numerical simulation data, 0.0469Hz (=0.190); the second peak frequency of flight data, 0.0391Hz (=0.158) is equal to the first peak frequency of numerical simulation data, 0.0371Hz (=0.150) as shown in Table 4. They are equal to B2 (=1.99) and B1 (=1.50) which are two of three modes analyzed by FMM as shown in Table 2. Because they are not equal to , the first and second peaks are caused by sail motion. The other peaks of flight data and MPM data are equal to B0 (= 0.884), which are the other one modes analyzed by FMM. Thus the motion after re-orientation using thrusters is governed by three modes derived from FMM.

Tank

Thruster 1

Thruster 4Thruster 3

Thruster 2

Main body

Fig. 9 Attitude maneuver using thrusters

Membrane

Main body

Reaction force by thruster

Thruster plume

Force on membrane by thruster plume

Force on main bodyby thruster plume

Thruster

Fig. 10 Reaction force by thruster and force on main body and sail by thruster plume

Plume axisNozzle

Continuum

Transition (freezing) region

Free molecule flow

Flow types in a plume expanding into vacuum

r

,r

Fig. 11 Plume flow model: source flow method

Specular reflectionDiffuse reflection

Incident plume

wT

wPiP

(2 ) i wp p p : Reflection coefficients(These parameters are independent)

i ,

i

Fig. 12 Plume impingement model: free molecule flow

Center of triangle elements

Thruster

Normal vector of triangle element

Plume axis

Sharing force

Normal force

Fig. 13 Integrating force by thruster plume into multi

particle model

ωy

ωx

Re-orientation

Fig. 14 Angular velocities of main body when

reorientation using thrusters is performed

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-6

10-5

10-4

10-3

10-2

10-1 wx

frequency (Hz)

SPEC

TR

UM

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.510

-9

10-8

10-7

10-6

10-5

10-4

10-3 wx

frequency (Hz)

SPEC

TR

UM

B1 B2B0

(a) IKAROS flight data

Fig. 15 FFT results of x-axis angular velocities after

reorientation using thrusters

Table 4 First and second peaks of motion after reorientation using thruster

First peak [Hz] Second peak [Hz] Fight data 0.0488 (=0.198 0.0391 (=0.158) MPM data 0.0371 (=0.150) 0.0469 (=0.190) 4. Conclusions

In this paper, the attitude dynamics of IKAROS was presented. 1) First Mode Model of out-of-plane deformation (FMM) and Multi Particle Model (MPM) were introduced to analyze the out-of-plane oscillation mode of spinning solar sail. 2) Three modes of out-of-plane oscillation were derived from FMM analytically. One of them is caused by nutation motion, because it is nearly equal to the spin rate. The other two modes are caused by sail motion. 3) The out-of-plane oscillation was occurred after sail deployment. The first peak frequency of flight data was

equal to that of MPM numerical simulation data. It is equal to a mode which is caused by nutation motion. 4) The out-of-plane oscillation was also occurred after re-orientation using thrusters. The first and second peak frequencies of flight data were equal to those of MPM numerical simulation data. They are two modes which are caused by sail motion. 5) These motions are governed by three modes derived from FMM. 6) FMM is very simple and valid for the design of attitude controller. MPM is the model incorporating the flexibility of the sail and valid for precise numerical simulations.

References

[1] R. M. Colon, “Solar Sailing, Technology, Dynamics and Mission Applications,” Springer-Praxis, 1999. [2] O. Mori, Y. Tsuda, H. Sawada, R. Funase, T. Yamamoto, T. Saiki, K. Yonekura, H. Hoshino, Minamino, T. Endo and J. Kawaguchi, “World's First Mission of Solar Power Sail by IKAROS,” International Conference on Space, Aeronautical and Navigational Electronics, SANE2010-95, 2010. [3] J. Kawaguchi, “A Solar Power Sail Mission for A Jovian Orbiter and Trojan Asteroid Flybys,” COSPAR04-A-01655, 2004. [4] G. Greschik and M. M. Mikulas, “Design Study of a Square Solar Sail Architecture,” Journal of Spacecraft and Rockets, Vol.39, pp.653-661, 2002.

[5] J. D. Hinkle, P. Warren and L. D. Peterson, “Geometric Imperfection Effects in an Elastically Deployable Isogrid Column,” Journal of Spacecraft and Rockets, Vol.39, pp.662-668, 2002. [6] T. Nakano, O. Mori and J. Kawaguchi, “Stability of Spinning Solar Sail-craft Containing A Huge Membrane,” AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA-2005-6152, 2005. [7] Y. Shirasawa, O. Mori, Y. Miyazaki, H. Sakamoto, M. Hasome, N. Okuizumi, H. Sawada, S. Matunaga, H. Furuya and J. Kawaguchi, “Evaluation of Membrane Dynamics of IKAROS Based on Flight Result and Simulation Using Multi-Particle Model,” 28th International Symposium on Space Technology and Science, 2011-o-4-05v, 2011. [8] Y. Miyazaki, H. Sakamoto, Y. Shirasawa, O. Mori, H. Sawada, M. Yamazaki and IKAROS Demonstration Team, “Finite Element Analysis of Deployment of Sail Membrane of IKAROS,” 28th International Symposium on Space Technology and Science, 2011-o-4-06v, 2011. [9] R. Funase, Y. Shirasawa, Y. Mimasu, O. Mori, Y. Tsuda, T. Saiki and J. Kawaguchi, “Fuel-free and Oscillation-free Attitude Control of IKAROS Solar Sail Spacecraft Using Reflectivity Control Device,” 28th International Symposium on Space Technology and Science, 2011-o-4-09v, 2011.