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Formation Deployment Control for Spinning Tethered Formation Flying -Simulations and Ground Experiments-

Koji Nakaya*, Masafumi Iai, Kuniyuki Omagari, Hideyuki Yabe and Saburo Matunaga† Tokyo Institute of Technology, Tokyo, Japan, 1528552

This paper discusses formation deployment for a spinning tethered formation flying system. The proposed control method is based on a virtual structure approach, that is one of various strategies and approaches for conventional multi-spacecraft formation control. We consider two types of formation deployment; spin angular velocity is in advance calculated 1) using an angular momentum profile determined, and 2) using a tether tension profile determined. We discuss results of numerical simulations in terms of maximum thrust and tether tension. The simulation model consists of three spacecrafts and tethers represented using a lumped mass model. We also introduce an experimental set-up using two-dimensional micro-gravity simulators to evaluate the proposed control methods, and refer to the current status of the experiment.

I. Introduction N resent years, methods for constructing synthetic aperture radar (SAR) and interferometry observation system by formation flying in orbit have been proposed1. Spacecrafts of formation flying system generally need so much

fuel to precisely keep the required relative position and attitude, and their lifetimes become to be short. For improving the problem, a spinning tethered formation flying system has been proposed as one solution and has been researched theoretically and experimentally2-6. In addition, the spinning tethered formation flying has been applied to deploy and maintain large membrane structures such as a solar sail spacecraft7.

Figure 1 shows an instance for realizing spinning tethered formation flying. An initial system which consists of spacecrafts launched simultaneously (step (a)), or which is constructed by docking of spacecrafts launched individually (step (a’)(a’’)) reaches operational state (step (c)) through a step of formation deployment (step (b)). Mori treated spinning tethered formation flying as a kind of a tethered service satellite system, and proposed feed-forward tether tension control for the formation control2. In his research mass and flexibility of tethers are not taken into consideration. Quadrelli studied spinning tethered formation flying to achieve a space interferometer for deep space3. He used a lumped mass model for tethers and discussed spacecraft system. However discussion about a thruster profile and tether tension is insufficient. Kim developed control law for NASA’s SPECS4 mission5. In this research each spacecraft was assumed to be a particle so that attitude motion was neglected. Decou studied formation control architecture for spinning tethered formation flying in a geostationary orbit6. He dealt with formation as a rigid body, and derived control input of each spacecraft from the rigid body motion. However, this was an open-loop control method, and each spacecraft was assumed to be a particle so that attitude motion was neglected.

* Graduate Student, Department of Mechanical and Aerospace Engineering, 2-12-1-I1-63 O-okayama, Meguro-ku, Student Member AIAA. † Associate Professor, Department of Mechanical and Aerospace Engineering, 2-12-1-I1-63 O-okayama, Meguro-ku, Senior Member AIAA.

I

(b) Formation Deployment

(c) OperationalState

(a) SimultaneousLaunch

(a’) IndividualLaunch

(a’’) Dockingin Orbit

Figure 1. Spinning Tethered Formation Flying

AIAA Guidance, Navigation, and Control Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-4896

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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This paper discusses formation deployment control for the spinning tethered formation flying in a circular orbit around the Earth. This paper is organized as follows. In section II, a system model is introduced and equations of motion are derived. The system model consists of three rigid bodies to consider spacecraft attitude motion, and tethers modeled using lumped mass. In section III, control architecture is explained. Over the past years, several studies have been made on formation control in several research fields: aircrafts, spacecrafts and mobile robots. The formation control is mainly classified into three categories, namely, leader-following, behavioral and virtual structures8. In this paper we propose a control method based on the virtual structure approach8-11, which takes advantage of easy prescription of coordinated behavior of the entire formation. In section IV, methods generating two types of a deployment command are mentioned. One is deployment in the case that spin angular velocity is calculated using an angular momentum profile determined in advance, and the other is deployment in the case that spin angular velocity is calculated using a tether tension profile determined in advance. In section V, numerical simulation is conducted to confirm the control method of section III and IV. Results of the simulation also are discussed. In section VI, ground experiment set-up is introduced and the present status of the experiment is referred. In section VII, conclusions are mentioned.

II. Modeling and Formulation

A. System model Figure 2 shows an analytical model that consists of three spacecrafts connected by tethers. i represents the

earth-centered inertial coordinate system. jb is the body-fixed coordinate system of spacecraft j . Each spacecraft and tether are assumed as follows.

Spacecraft j ( 3,2,1=j ): jM is mass. jI is the moment of inertia. IBjω is angular velocity. jq is a position of the c.m. of the body and jε represents a quaternion of the body.

Tjjj

Tj

Tj qqqq ][ 321iiq == (1)

Tjj

Tj

Tjjjjj z )]2(cos)2sin([][ 4321 φφεεεεε == (2)

where, jz represents a unit vector in the direction of the axis of rotation, and jφ is the angle of the rotation. A coordinate transformation from i to jb using a direction cosine matrix is shown as follows.

ib IBj

jC= (3)

jka is a position of the tether connection point. cmq means the position of the center of mass.

jkT

jjk aba = (4)

∑∑==

=3

1

3

1 jj

jjjcm MM qq (5)

Tether j ( 3,2,1=j ): it indicates the tether released from spacecraft j . The tether model is explained in the next section.

B. Tether model Many studies have been made on tether modeling, and precise models were proposed in the past studies12,13.

However, the precise models require complex formulation and a large amount of CPU time. Because the main objective of this paper is discussion of the system dynamics, the lumped mass model as shown in Fig.3 is sufficient to represent mass and appropriate flexibility of tethers14,15. The lumped mass model has the advantage of easy expression for them. In this model tether is assumed to be a collection of particles, which are connected by springs and dashpot.

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Tether j consists of N discrete mass j

iS ( Nj ,,1= ). jlS ( nl ,,1= ) is located out of the spacecraft j , and

jmS ( Nnm ,,1+= ) is located inside the spacecraft j . j

lk and jlc indicate a spring constant and a damping coefficient,

respectively, in the following.

00

,0,

≤>

∆

= jl

jlttj

lLAE

kαα

00

,0,

≤>

= jl

jltj

lC

cαα (6)

where L∆ means the natural length of the tether element. tE is Young’s modulus and tA is cross-section area of the tether. j

lα is defined as follows.

ji

ji SS

jl qqL

11

−−∆−=α (7)

A sign of jlα shows whether the tether element is longer than the natural length or not. Eqs.(6)(7) can simply

represent tether slack phenomenon. kf is controlled tether tension, which is worked on the nearest mass jnS to the

spacecraft j .

C. Discontinuity of Tether Motion Using the tether lumped model, the motion of the system is discontinuous at the time when the tether mass is

deployed / retrieval. Therefore it is necessary to calculate and revise a value of the discontinuity so that the system conserves momentum and angular momentum. Figure.4 shows tether mass deployment condition. e is defined as a unit vector from j

nS 1+ to jnS . Velocity at the time t when j

nS 1+ is released from spacecraft j is represented as follows using geometry restriction between j

nS and jnS 1+ .

1b

2b

3b

Spacecraft 1

1m 1I

Spacecraft 2

2m 2I

Spacecraft 3

3m 3I

11S 1

1−nS 1nS

111 Nn SS +

12f21f

331 Nn SS +

31S3

2S3

1−nS3nS

13f

31f

i

1q

221 Nn SS +

12a

13a

Tether 1

Tether 2

Tether 3

12S

33S

32Sq

Figure 2. Analytical Model

Spacecraft jSpacecraft k

jS1j

nS 1−

jnS

11,ck

jS2

22 ,cknn ck ,

jN

jn SS 1+

jb kb

Tether j

jf

Figure 3. Tether Lumped Mass Model

Spacecraft j

jnS 1+j

nS

e( )−tj

nSq

( )( )eeq ⋅−tjnS ( ) ( )( )eeqq ⋅−=+

+tt j

nj

n SS 1

Figure 4. Tether Mass Deployment

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( ) ( )( )eeqq ⋅−=++

tt jn

jn SS 1

(8)

where −t ， +t show the time immediately after / before the time t , respectively. The law of momentum conservation is represented as follows.

( ) ( )+=− tPtP (9)

where ( )−tP and ( )+tP represent momentum of the time −t and +t , respectively. From this equation, the new velocity of spacecraft j at the time +t can be calculated. The same applies to the angular velocity. From the law of angular momentum conservation, the new angular velocity of spacecraft j at the time +t can be calculated. The tether mass deployment occurs when the following relationship is satisfied.

Ljn

jn SS ∆≥−

+1qq (10)

In the case of the tether retrieval, the new velocity and angular velocity of spacecraft j at the time +t can be calculated from the law of momentum and angular momentum conservation of the collision phenomenon between the tether mass and the spacecraft.

D. Equation of Motion The center of the system is assumed to rotate around the Earth with angular velocity Ω . The equations of motion

for the spacecraft j are as follows.

Translation: jjjcntjj

jjj

qMM fffqq ′+++= 3||

µ (11)

Attitude: jjjcntjjjj

IBj

IBIBj

qjjj τττqIqωIωωI ′+++×

Ω=×+ 2

2

||3 (12)

where jcntf , jcntτ represent control thrust and torque vectors, respectively, worked on the spacecraft j with thrusters and inner torquers. jf , jτ mean force and torque vectors generated by tether j , as follows.

)()(

jkjS

jkjSjj

jn

jnf

aqqaqqf

+−+−

= (13)

jjkj faτ ×= (14)

jf ′ , jτ′ show force and torque vectors worked on the spacecraft j from tethers excluding tether j . In the case that the spacecraft j received force from tether k , jf ′ , jτ′ are represented in the following.

( ) ( )kk Sjkjk

Sjkjkk

j ck11 111 qaqqaqf −+−−+−=′ α (15)

jjkj faτ ′×=′ (16)

)( jkjSj jnLnL aqq +−+∆= (17)

where jL is the length of tether j . The equations of motion for tether j are formulated as follows. In the case of 1=i :

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( ) ( ) ( ) ( ) 312221111 111212111

jjjjjjjjj SSj

SSj

SSjj

Skjkj

Skjkjj

Sj qmckckm qqqqqqaqqaqq µαα +−+−+−++−+= (18)

In the case of ni <<1 :

( ) ( ) ( ) ( ) 3111 1111

ji

ji

ji

ji

ji

ji

ji

ji

ji

ji

ji SS

jiSS

jiSS

ji

jiSS

jiSS

ji

jiS

ji qmckckm qqqqqqqqqq µαα +−+−+−+−=

++−− +++ (19)

In the case of ni = :

( ) ( ) 3

11j

nj

nj

nj

nj

nj

nj

n SSj

njSSj

nSSj

nj

nSj

n qmckm qfqqqqq µα ++−+−=−−

(20)

III. Control Method

A. Virtual Structure Architecture A Virtual structure approach is one of approaches for conventional multi-spacecraft formation control. In the

approach, the entire formation is treated as a single structure. In this approach, the control is derived in three steps. First, the desired dynamics of the virtual structure is defined. Second, the motion of the virtual structure is translated into the desired motion for each spacecraft, and finally, tracking control for each spacecraft is derived9-11. Figure 5 shows the control architecture applying the virtual structure approach. The system jS represents the spacecraft j and the system jK represents the local controller. The system F is the formation control and the system G is a supervisor.

ju represents control forces and torques which are input vectors from jK to jS , and output vector jy represents the position and attitude vectors. The coordination variable ξ is broadcast from F to all spacecrafts. jz indicates a performance of keeping formation. Gy is input from G toF , and Fz indicates formation condition.

B. Application to Spinning Tethered Formation Flying In this section, application of the virtual structure architecture to spinning tethered formation flying is explained.

1. Virtual Structure Model Figure 6 is a virtual structure model for the system shown in Fig.2. The virtual structure consists of three rigid

bodies that are placed on a concentric circle at 120 degrees intervals. The center of mass of the virtual structure and that of each rigid body are assumed to be connected by a massless rod. v is the body-fixed coordinate system attached to the c.m. of the virtual structure. 3v represents a spin axis. jvsb is the body-fixed coordinate system of rigid body j ( 3,2,1=j ). jvsm is mass. jvsI is the moment of inertia of rigid body j . jkvsa means a virtual tether connection position in coordinate v . jr is the position of the c.m. of rigid body j measured from v as follows.

Tjjj

Tj

Tj rrrr ][ 321vvr == (21)

A scaling variableλ is introduced to represent expansion of the virtual structure.

T][ λλλ=λ (22)

Then jr is rewritten in the following.

G

F

Supervisor Fz

1K

1S

Local Control

Spacecraft #1

1y

1u

1z

Local Control

Spacecraft #3

Gy

3u

3y

Formation Control

3zCoordination Variable ξ

3K

3S2S

2yLocal Control

Spacecraft #2

2u

2K2z

Figure 5. Control Architecture using Virtual Structure Approach

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0jj rr Ξ= (23)

where, )( λλλdiag=Ξ and 0jr represents initial value of jr . The position of the origin vsq and angular velocity IV

vs/ω of the coordinate v measured from coordinate i are

Tvsvsvs

Tvs qqq ][ 321iq = (24)

Tvs

TIVvs ]00[/ ωvω = (25)

vsI means the moment of inertia of the virtual structure.

vvI vsT

vs I= (26)

∑=

+=3

1)~~(

jj

Tjjvsjvsvs rrmII (27)

where, jr~ is a skew-symmetric matrix of jr . A quaternion of coordinate jvsb measured from v is shown as follows.

Tjvsjvs

Tjvsjvs

TTjvsjvsjvsjvsjvs z ][])2cos()2sin([][ 4321 εεφφεεεεε ′′′=== (28)

where TTz ]100[v= ， 3)1(2 −= jjvs πφ . A quaternion of coordinate jvsb measured from v is also shown as follows.

Tvsvs

Tvsvs

Tvs

Tvsvsvsvsvs z ][])2cos()2sin([][ 4321 εεφφεεεεε ′′′=== (29)

where vsz is a unit vector in the direction of the axis of rotation, and vsφ is the angle of the rotation. A coordinate transformation from i to v using a direction cosine matrix is shown as follows.

iv IVC= (30)

The equations of motion for the virtual structure are as follows.

Translation: 3vsvsvsvsvs qmm qq µ= (31)

Attitude: vsIV

vsvsIV

vsvsIV

vsIV

vsvs tωIωIωωI =+×+ //// (32)

vst represents control torque working on the virtual structure as follows.

vsT

vs tvt = (33)

)( // dIVvs

IVvsvsvsevsevs ωωKKt −−−= ωε (34)

where vseε is the relative quaternion, and vseK and vsKω are controller gains. A superscript d means the desired value. When dλ and dλ are given, λ is defined as follows.

)()( dd KK λλλλλ λλ −−−−= (35)

where λK and λK are controller gains.

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2. Desired Motion of Each Spacecraft

The desired motion of each spacecraft is derived from the virtual structure motion mentioned in the previous section. The desired position, velocity, quaternion and angular velocity are described in the following.

IVvs

dIB

vs

vs

jvsT

jvs

jvsjvsjvsjvsvs

dj

jTIVIV

vsjTIV

vsdj

jTIV

vsdj

j

UrCrCqq

rCqq

ωω

εε

εεεεε

εεε

ω

=

′′′

′′′−

′′′−′′==

Ξ+Ξ+=

Ξ+=

00

0

~ (36)

where U means an unit matrix, and djL , d

jL represent the desired tether length and velocity.

3. Control of each spacecraft In the general virtual structure architecture, normal PD control is conducted using current states and desired

states. However, for the spinning tethered formation flying system, we can derive a characteristic control method. Because the system rotates around the center of mass of the whole system under the condition that spacecrafts are connected by tethers, the system can control the spin radius using only tether tension and centrifugal force while the system must use thrusters to control out-of-plane motion and spin angular velocity as shown in Fig.7. This is an especially important feature of the formation flying. Control of thrusters, wheels and tethers is explained in the following parts.

Thruster control: thrusters are used to control spin angular velocity and out-of-plane motion. Thruster control is considered in the coordinate system v as shown in Fig.8. Let jr′ represent projections of jr onto the 21vv plane. Thruster control is derived as follows.

−−

+′××′′××′

==

33

)()()(

jrjr

jdjj

jdjj

Tjcnt

Tjcnt

rKrK

KKrrrrrr

fθθ θθ

vif (37)

where θK ， θK ， rK ， rK are controller gains, θ means the angle between djr and jr′ , and 3jr represents the out-

of-plane displacement of the spacecraft j . Wheel control: the control is derived as follows.

)( dIBj

IBjeejcnt

jjKK ωωε ω −−−=t (38)

120°120°

v

1 vsb2 vsb

3 vsb

vsI

1vsm 1vsI

2vsm 2vsI

3vsm 3vsI

Virtual Tether 12

Virtual Tether 23

Virtual Tether 31

12vsa

13vsa1v2v 3v

Figure 6. Virtual Structure Model

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where eε is the relative quaternion , and eK and ωK are controller gains. Tether control: the control is derived as follows.

)()( djjL

djjL

djj LLKLLKff −+−+= (39)

where LK and LK are controller gains. djf means the equilibrium tension for keeping the present spin radius2.

3

)(3sin3sin

6sin)( 22 dddMdddMfdvsj

dvsjd

j−

=+−

=ω

πππω (40)

where d represents the desired spin radius as follows.

djrd = (41)

IV. Command for formation deployment The following section explains command generations for a spin radius and angular velocity during formation

deployment. This paper deals with two methods for generating deployment command as shown in Table 1. One is the method using an angular momentum profile. The other is the method using a tether tension profile.

A. Spin Radius Figure 9 shows a general command of a spin radius and a time differential of it. Let 1t and 2t represent start time

and end time of the deployment, respectively. a means a constant value of the time differential. 11 ttm − and 22 mtt − are defined as rise time and fall time, respectively. In this paper the following formulas describe a command of the time differential of the spin radius.

Out of Plane Motion

Spin Angular Velocity

Thruster Control Tether Control

Spin plane

Spacecraft

Spin Radius

Figure 7. Control Target of Thruster and Tether

1v2v

3vjr

jr′djr

θ

3jrv

Figure 8. Definitions for Thruster Control

Table 1. Deployment Type in This Paper Given informationbefore deployment

Generated from the giveninformation Calculated value

・Spin radius profile・Initial angular velocity・Final angular velocity・Spin radius profile・Initial angular velocity・Final angular velocity

Deployment 1

Deployment 2

・Angular acceleration

・Angular velocity

・Angular momentum profile

・Tether tension profile

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≤≤+−−=<<=

≤≤+−−−=><=

)(5.0))()(cos(5.0)(

)(5.0))()(cos(5.0)(0

22222

21

11111

21

tttattttadtttad

tttattttadttorttd

mmm

mm

mm

π

π (42)

The command of the spin radius is acquired using integral calculus of the formulas above. γ is defined as follows.

)()( 1212 tttt mm −−=γ (43)

B. Command Using an Angular Momentum Profile (Deployment I) An increment/decrement of angular momentum needed to achieve formation deployment is calculated from an

initial and a final condition. The case that all the change of the angular momentum is given to the system before deployment is not practical since angular velocity of the system becomes too fast. Therefore, we must give the change during deployment and / or after deployment. Let 1H represent a change of angular momentum during deployment, and 2H represent a change of angular momentum after deployment. Also let allH represent total change of angular momentum needed to achieve formation deployment. The following equation is satisfied.

21 HHH all += (44)

Equation (44) is rewritten using a parameter X as follows.

allXHH =1 (45)

allHXH )1(2 −= (46)

If 1=X , it means all the change of angular momentum is given during deployment. Since we assume that the system rotates with maximum angular velocity at the initial condition in this paper, we do not consider the case that a change of angular momentum is given before deployment.

2-D motion of the virtual structure is considered to generate the profile of angular momentum. Angular momentum vsH of the virtual structure is represented as follows.

ωIH vs = (47)

Where 3,vsII = (z element of vsI ). ω means spin angular velocity of the virtual structure. The time differential of it is

NIIH vs =+= ωω (during deployment) (48)

NIH vs == ω (after deployment) (49)

1t 1mt 2mt 2t

a

t t

d d

sd

ed

1t 1mt 2mt 2t Figure 9. Spin Radius (left) and Time Differential (right)

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N represents torque worked on the virtual structure. Thrusters of spacecraft generate the torque. (Fig.10)

dFH vs ⋅= 1 (during deployment) (50)

evs dFH ⋅= 2 (after deployment) (51)

where 1F , 2F represent thruster force during / after deployment, and d and ed represent spin radius during / after deployment, respectively. Though 1F , 2F are possible to be arbitrary value, they are assumed to be constant in this paper. The profile of vsH is showed in Fig.11. sdt ， edt ， sat ， eat of Fig.11 are set to avoid sudden change of vsH .

0=vsH （ dstt _< ）

sdsdssvs dFttttdFH 1_1_1 5.0))/()(cos(5.0 +−−−= π （ 1_ ttt ds <≤ ） )(1 tdFHvs = （ 21 ttt <≤ ）

edeevs dFttttdFH 12_21 5.0))/()(cos(5.0 +−−= π （ dettt _2 <≤ ） 0=vsH （ asde ttt __ <≤ ） (52)

easasevs dFttttdFH 2_3_2 5.0))/()(cos(5.0 +−−−= π （ 3_ ttt as <≤ ） evs dFH 2= （ 43 ttt <≤ ）

eaeevs dFttttdFH 23_32 5.0))/()(cos(5.0 +−−= π （ aettt _4 <≤ ）

Where sd ， ed indicate the initial and final spin radius, respectively. We can get ω from Eqs.(48)(49) using vsH of Eq.(52). In the following part of this paper we call this type of the command generation “Deployment I”.

C. Command Using a Tether Tension Profile (Deployment II) A command for achieving desired tension during deployment is considered here. Equation (40) is rewritten as

follows.

dMfd jdj

dvs )3( +=ω (53)

d and d of the equation are acquired from (42). Therefore dvsω is determined by substituting d

jf into (53). In this paper d

jf is set as follows.

)(5.0))()(cos()(5.0 121sesed

j TTttttTTf ++−−−−= π (54)

where sT and eT represent tether tension of the initial and final spin condition, respectively. In the following part of this paper we call this type of the command generation “Deployment II”.

Nd

F

Virtual Structure Real Model Figure 10. Torque of Virtual Structure

vsH

t1t 2tsdt edt sat eat3t 4t

1H2H

)(td

Figure 11. vsH Profile

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V. Numerical Simulation In this section, numerical simulations are conducted to confirm the efficiency of the proposed control method

and to make clear the features of two types of the deployment. Results of the simulations also are discussed.

A. Initial Conditions We set the following parameters for the initial system.

(1) Shape of spacecraft:

kgM j 50= ， 2

083.2000083.2000083.2

kgmI j

=

ma Tjk ]025.025.0[−= in the case of )1,3(),3,2(),2,1(),( =kj

ma Tjk ]025.025.0[ −−= in the case of )2,3(),1,2(),3,1(),( =kj

(2) Tether:

]/[1075.1 3 mkg−×=ρ (linear density) ]/[9800 2mmNEt = ， ][33.1 2mmAt = ， 001.0=tC , total tether length: m64 ，tether elements: 15

(3) Orbit motion: orbit altitude: mH 800000= ，radius of the Earth: mRe 6378137= , gravitational constant 231410986.3 sm×=µ , orbital angular velocity: sradRH e /)( +=Ω µ (4) Virtual Structure:

1=sλ , 20=eλ , st 151 = , st 6152 = , mr Tjvsjvsj ]0sin0.1cos0.1[0 φφ=

mRHq Te

svs ]00[ += , smRHq T

esvs /])(00[ Ω+= ,

sradsvs /2.0=ω , srade

vs /04.0=ω ， 8.0=γ , 1=X Each time of Figure 11 is defined as follows

stttttttt easaedsd ]925,875,775,725,715,615,15,10[],,,,,,,[ 4321 = Fig.12 shows the initial condition of the virtual structure. The spin axis 3v is represented as follows.

TT ]02121[3 −= iv This vector indicates that the angle between the spin axis and the normal vector of the orbit plane is 45 degrees. The virtual structure is controlled to keep the spin axis in the coordinate system i during formation deployment. (5) Spacecraft: The initial condition of each spacecraft is assumed to be correspondent with the condition acquired from the initial virtual structure.

B. Result and Discussion of Numerical Simulation 1. Validity of the proposed control method

Figure 13 shows the trajectory of spacecraft 1 in Deployment I and II. These data show that spacecrafts surely follow the motion of the virtual structure and complete the deployment. Figure 14 indicates values of θ of the spacecraft 1 explained in Fig.7. The values of Deployment I and II are very small. In the case of Deployment II, the value of θ increases rapidly at the beginning of the deployment while the value becomes almost constant in the case of Deployment I. Figure 15 indicates a position error from the 21vv plane. Control for reducing out-of-plane errors works effectively for both Deployment I and II. Figure 16 shows a position error of the spacecraft 1. In the both deployment the error values are kept very small during deployment. The errors during the deployment are bigger than them after the deployment. Figure 17 indicates an absolute value of an error between the actual quaternion and

1i 2i

3i

i

3i2i

1ii

Circular Orbit

Virtual Structure

45 Spin Axis

Figure 12. Initial Condition of Virtual Structure

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the desired quaternion to show attitude control performance. In the both case, the errors are very small. Figure 18 shows that the errors between vsq and cmq are kept small during formation deployment. This phenomenon indicates the proposed control is able to keep the orbit of cmq . These results conclude that the proposed control method works effectively for the spinning tethered formation flying.

-20 -10 0 10 20

-20

-10

0

10

20

[m]

[m]

Deployment I spacecraft 1

-20 -10 0 10 20

-20

-10

0

10

20

[m]

[m]

Deployment II spacecraft 1

Figure 13. Trajectory of Spacecraft 1

0 3000 6000 9000 12000-1.5

-1

-0.5

0

0.5

1

1.5[×10-4]

Time [s]

r 13 [

m]

Deployment IDeployment II

Figure 15. Out-of-Plane Error of Spacecraft 1

0 200 400 600 8000

0.005

0.01

0.015

Deployment IDeployment II

Time [s]

q 1d -q

1 [m

]

Figure 16. Position Error of Spacecraft 1

0 200 400 600 8000

1

2

3

4

[×10-4]

Time [s]

|εd 1-

ε1|

Deployment IDeployment II

Figure 17. Attitude Error of Spacecraft 1

0 3000 6000 9000 120000

0.2

0.4

0.6

0.8

1[×10-4]Deployment IDeployment II

Time [s]

|qcm

-qvs

|

Figure 18. Position Error between vsq and cmq

0 200 400 600 8000

0.5

1

1.5

[×10-3]

Time [s]θ

[rad

]

Deployment IDeployment II

Figure 14. Angle Error of Spacecraft 1

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13

2. Discussion about Deployment I Firstly, effect of a variable X of Eqs. (45)(46) about Deployment I is discussed.

Tether tension: As we mentioned before, in the case of 1=X , it means allH is given during deployment, and in the case of 0=X , it means allH is given after deployment. In the case of 1>X , a larger amount of angular momentum than allH during the deployment and it is reduced after deployment to achieve allH . Figure 19 shows tether tension for each X . Tether tension during deployment for 0=X is extremely small and almost zero. The larger value of X becomes, the bigger tension is achieved during the deployment.

Thruster: Figure 20 shows thruster profiles for each X . It indicates that the larger X is applied, the bigger thrust is needed for the deployment. In the case of 1>X , reverse thrust is required to reduce excess of the angular momentum.

Required time to complete final condition: Required time to complete final condition in the case of 1=X is the shortest since the time of finishing deployment is equal to the time of achieving the final condition. In the case of another X , it takes longer time than 1=X to adjustment angular momentum. However, it is possible to shorten the time using a large thruster. The thruster achieves that sasa tt − is small.

Secondly, effect of a variable γ about Deployment I is discussed. Figure 21 represents tether tension and Fig.22 shows thruster profiles for each γ in the case of 1=X . The result of Fig.21 clearly shows that tether tension grows up rapidly at the beginning of the deployment with small γ . This phenomenon is explained from Eq. (40) because small γ means small d . Figure 22 makes it clear that effect of γ is very small to thruster profiles.

0 200 400 600 800 1000-0.4

-0.2

0

0.2

0.4

Deployment I X=0.0Deployment I X=0.5Deployment I X=1.0Deployment I X=1.5Deployment I X=2.0

Time [s]

Thru

st [N

]

Figure 20. Thruster Profile for Each X

0 200 400 600 800 10000

1

2

3

4 Deployment I X=0.0Deployment I X=0.5Deployment I X=1.0Deployment I X=1.5Deployment I X=2.0

Time [s]

Teth

er T

ensi

on [N

]

Figure 19. Tether Tension for Each X

0 200 400 600 800 10000

1

2

3

Deployment I γ=0.0Deployment I γ=0.2Deployment I γ=0.4Deployment I γ=0.6Deployment I γ=0.8

Time [s]

Tet

her

Tens

ion

[N]

Figure 21. Tether Tension for Each γ

0 200 400 600 800 10000

0.05

0.1

0.15

Deployment I γ=0.0Deployment I γ=0.2Deployment I γ=0.4Deployment I γ=0.6Deployment I γ=0.8

Time [s]

Thru

st [N

]

Figure 22. Thruster Profile for Each γ

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14

3. Discussion about Deployment II In this section, effect of a variable γ about Deployment II is discussed. Figure 23 represents tether tension and

Fig.24 shows thruster profiles for each γ . The results of Fig.23 clearly show that tension for every γ achieves the command tension shown in Eq. (54). Figure 24 makes it clear that thruster profiles depend on γ , and maximum thrust depends on γ .

4. Comparison between Deployment I and II

Figure 25 and Fig.26 represent comparison of tether tension and thruster profiles between Deployment I and II. Figure 25 makes it clear that tether tension follows tension profile defined in advance in the case of Deployment II, while tether tension changes widely in the case of Deployment I. Consequently, reel system of Deployment I is required to have capability to control weak tension. This is not realistic since it is usually difficult to control the small tension. However, it is possible to keep tether tension in a controllable tension range by selection of proper X and γ . Figure 26 makes it clear that maximum thrust of Deployment I is smaller than one of Deployment II. Figure 26 also indicates that thruster profile changes during Deployment II, while it is almost constant during Deployment I.

VI. Ground Experiment The following section explains an experimental set-up using two-dimensional micro-gravity simulators

developed at Tokyo Institute of Technology, and present situation of our experiment.

A. Ground Experiment System Figure 27 shows schematic diagram of our ground experiment system. The system consists of 3m*5m flat floor,

three satellite dynamics simulators, a position determination subsystem using a CCD camera. Reel mechanism can be installed on the satellite dynamics simulator.

0 200 400 600 8000

0.5

1

1.5

Deployment II γ=0.0Deployment II γ=0.2Deployment II γ=0.4Deployment II γ=0.6Deployment II γ=0.8

Time [s]

Tet

her

Tens

ion

[N]

Figure 23. Tether Tension for Each γ

0 200 400 600 800

0

0.1

0.2

0.3

0.4Deployment II γ=0.0Deployment II γ=0.2Deployment II γ=0.4Deployment II γ=0.6Deployment II γ=0.8

Time [s]

Thr

ust [

N]

Figure 24. Thruster Profile for Each γ

0 200 400 600 8000

1

2

3

4Deploymant I γ=0.8 X=1.0Deploymant I γ=0.8 X=1.5Deploymant I γ=0.8 X=2.0Deploymant II γ=0.8

Time [s]

Teth

er T

ensi

on[N

]

Figure 25. Comparison of Tether Tension

0 200 400 600 800-0.4

-0.2

0

0.2

0.4

Deployment I γ=0.8 X=1.0Deployment I γ=0.8 X=1.5Deployment I γ=0.8 X=2.0Deployment II γ=0.8

Time [s]

Thr

ust [

N]

Figure 26. Comparison of Thruster Profile

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15

1. Satellite Dynamics Simulator Figure 28 represents the satellite dynamics simulator. This simulator is floated on a flat floor by air-bearings

using air pads to simulate 2-D micro-gravity motion. The simulator has eight thrusters to control position and attitude, where thrust force is 2.22N and torque is 0.444N per one thruster. Air tanks on the simulator supply air for thrusters and air-bearings. Attitude of the simulator is measured by an on-board gyro, and position is acquired from position determination subsystem via wireless LAN. Figure 30 represents system configuration of the simulator. Table 2 shows specifications of the simulator.

2. Reel Mechanism The reel mechanisms are used for tether tension control. Functions of the reel mechanism include: reel in and out,

tension control, measurement of tether length. Figure 30 shows the reel mechanism. The mechanism has two DC motors (Motor A and B in Fig.30) to control inner and outer tension. Inner tension is controlled to be more than 0 N for avoidance of tether untied state. Tether length is measured by encoder of the Motor A. The mechanism also has a level winder to reel in and out tether equally on a rotating spool.

Ground Station

CCD Camera

Flat FloorThruster

SatelliteGyro

TetherReel

MechanismPC

Air Pad

Tank

Figure 27. Schematic Diagram of Ground ExperimentSystem

Figure.28 Satellite Dynamics Simulator

RS-232CWireless LAN

ADC

RS-485

DO

Gyro

Reel

Thruster Valves

Simulator on-board PCSatellite Simulator

- Attitude- Angular Velocity

- Tether Tension

- Command- Tether Length- Tether Velocity

- Position- Velocity

Figure 29. Satellite Dynamics Simulator System

Table 2. Specifications of Satellite DynamicsSimulator Size 0.6*0.6*0.69 [m]Wight 42 [kg]Moment of Inertia 2.3 [kgm2]Pressur in Air Tank 150 [kgf/cm2]Volume of Air Tnak 8.6 [l]Control Cycle 60 [ms]Communication Rate 10.0 [Mbps]

Roller

StrainGage A

OuterMechanism

InnerMechanism

Laser Displacement Sensor

LevelWinder

Reel

StrainGage C

MotorB

Motor AEncoder

StrainGage B

Figure 30. Reel Mechanism ( m2.0m4.0m14.0 ×× )

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3. Control System Figure 31 represents a control system block diagram of the satellite simulator and the reel mechanism for the

ground experiment. In this experiment, the virtual structure dynamics is computed offline, and distributed to each simulator. Since the simulator does not have wheels to control its attitude as we mentioned in the explanation on the satellite simulator, the simulator uses thrusters to control its attitude. Therefore, position control and attitude control are switched every control cycle.

B. Ground Experiment We conducted the ground experiment using the system as shown in the previous parts. In the current situation,

formation deployment motion is confirmed as shown in Fig.32. However, we must continue to turn up the system, for example, control gains, and to conduct the experiment. After valid experiment for data analysis, we compare the experiment data with the numerical simulations, and consider the proposed control method.

Virtual Structure Dynamics

θK

θK

jθ

jθ

PWM

eK

ωKPWM

eεGet

Thruster MappingSW

Simulator Dynamics

Get jj θθ ,dj

dj rr ,

jj rr ,jεjω

Get djf

dddvs ,,ω

djL

djL

LK

LKPID

ControllerTID

Motor Driver

Reel Dynamics

jf

jL

jL

-

-

+

+

+

++

+-

+

+

+

+

djε

djω

Tension Sensor

Gyro

Position Deternination

Subsystem

Figure 31. Control System Block Diagram of the Ground ExperimentSystem

Figure 32. Ground Experiment (Formation Deployment)

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17

VII. Conclusion In this paper, we firstly construct the analytical model for the spinning tethered formation flying system. The

model consists of three spacecrafts modeled by rigid bodies, and three tethers modeled by collection of lumped mass. Secondly, we proposed the control method for formation deployment control. The method is based on the virtual structure approach. We indicated features of the control for the spinning tethered formation flying, and referred thruster, wheel and tether tension control. Thirdly, we conducted numerical simulations for the two types of formation deployment; spin angular velocity is calculated using an angular momentum profile determined in advanced (Deployment I), and using a tether tension profile determined in advanced (Deployment II). From the numerical simulation we confirmed the proposed control method worked effectively. Moreover, we made it clear the features of Deployment I and II in terms of thruster and a tether tension profile, namely,

1) Maximum thrust of Deployment I is generally smaller than one of Deployment II. Furthermore, A thruster profile sharply changes during Deployment II, while it is almost constant during Deployment I.

2) Tether tension follows the tension profile defined in advance in the case of Deployment II, while tether tension changes widely in the case of Deployment I. However, it is possible to keep the tether tension in a controllable tension range by selection of proper X and γ in Deployment I.

We additionally explained our two-dimensional micro-gravity simulator, and mentioned the present status of the ground experiment as well as the control system of the satellite simulator and the reel mechanism. We continue the ground experiment to compare the experiment data and the numerical simulations, and to confirm the proposed control method.

Acknowledgments We are grateful to Mr. Shinji Masumoto of Tokyo Institute of Technology for giving us valuable advice. On

numerous occasions he helped operations for the ground experiments.

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