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On Attitude Maneuver of Spinning Tethered Formation Flying Based on Virtual Structure Method

Koji Nakaya* Japan Aerospace Exploration Agency, Kanagawa, Japan, 2298510

Saburo Matunaga† Tokyo Institute of Technology, Tokyo, Japan, 1528552

This paper discusses attitude maneuvers of spinning tethered formation flying system. A virtual structure method is applied to control the formation. Modeling, formulation and attitude maneuver control architecture are explained. In this paper, the spinning tethered formation flying is assumed to be operated around the Earth. Numerical simulations are conducted for the maneuvers of the system keeping circular formation. Results of numerical simulations are considered in terms of accuracy of the formation, thruster force, tether tension and fuel consumption. Validity of the proposed control is indicated via results of the simulations.

Nomenclature i = Basis vectors of an earth-centered inertial coordinate frame I

jb = Basis vectors of spacecraft j body-fixed coordinate frame jB

v = Basis vectors of virtual structure body-fixed coordinate frame V jvsb = Basis vectors of rigid body j fixed coordinate frame

jvsB (Virtual structure consists of rigid body j )

jm = Mass of spacecraft j

jvsm = Mass of rigid body j

vsm = Mass of virtual structure

jI = Inertia matrix of spacecraft j

jvsI = Inertia matrix of rigid body j

vsI = Inertia matrix of virtual structure

jq = Position of spacecraft j measured in the coordinate frame I

cmq = Position of a center of mass measured in the coordinate frame I

vsq = Position of virtual structure measured in the coordinate frame I

jr = Position of rigid body j measured in the coordinate frame V

jε = Euler parameter of spacecraft j measured in the coordinate frame I : Tjjjjj ][ 4321 εεεεε =

jvsε = Euler parameter of rigid body j measured in the coordinate frame V : Tjvsjvsjvsjvsjvs ][ 4321 εεεεε =

vsε = Euler parameter of virtual structure measured in the coordinate frame I : Tvsvsvsvsvs ][ εεεεε =

YXω = Angular velocity of a coordinate frame X measured in a coordinate frame Y YXC = Direction cosine matrix from a coordinate frame Y to X ( yx YXC= )

Tether jk = Tether connecting spacecraft j and k

* Research Fellow, Institute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Member AIAA. † Associate Professor, Department of Mechanical and Aerospace Engineering, 2-12-1-I1-63 O-okayama, Meguro-ku, Member AIAA.

AIAA Guidance, Navigation, and Control Conference and Exhibit15 - 18 August 2005, San Francisco, California

AIAA 2005-6088

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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ljkS = Tether element l of Tether jk

ljkS

q = Position of tether element ljkS

ljkk = Spring constant between l

jkS and 1−ljkS

ljkc = Damping coefficient between l

jkS and 1−ljkS

L∆ = Natural length of tether element tE = Young’s modulus of tether

tA = Cross-section area of tether M = Mass of tether element

jjka = Position of tether tension controller of spacecraft j measured in the coordinate frame

jB kjka = Tether connection position of spacecraft k measured in the coordinate frame kB j

jkT = Controlled tether tension acted on jjka

kjkT = Tether tension acted on k

jka

jf = Thruster force of spacecraft j

jτ = Torque acted on spacecraft j jjkf = Force acted on spacecraft j from tether jk jjkτ = Torque acted on spacecraft j from tether jk

jkL = Length of tether jk

Ω = Orbital angular velocity µ = Gravitational parameter

I. Introduction N resent years, the concern with spinning tethered formation flying has been growing because the formation flying is able to reduce fuel consumption as well as precisely control relative position and attitude1-15. The

authors’ past studies have focused on the spinning tethered formation flying in orbit around the Earth, and considered formation deployment and elliptic formation change in terms of both numerical simulations and ground experiments1-3. These studies are characterized by using a virtual structure approach for formation control, which is one of various methods to control formation. However these studies limited the discussion to the situation that a spin axis direction of the formation was fixed in the inertia coordinate frame. It is necessary to change the spin axis direction for selecting observation targets when the formation flying is used for interferometry observation proposed by Mori and Matunaga4, 5, Quadrelli6, DeCou7, Maccone9, Quinn and Folta10, and so on. Therefore, this paper discusses attitude maneuvers of the spinning tethered formation flying in orbit around the Earth as shown in Fig. 1. The virtual structure approach is applied for the formation and maneuver control. Several studies have been made on maneuvers of tethered formation flying as shown in the following. Bombardelli, Lorenzini and Quadrelli14 studied on attitude maneuvers of tethered formation flying that consists of three spacecraft arranged in line. They made a thruster-time profile beforehand and applied it to the maneuvers. That is open-loop maneuver control strategy. DeCou8 also proposed open-loop maneuver control strategy for tethered formation flying which shape is triangle. However, these studies only discussed open-loop strategy. In this paper, we discuss feedback-controlled maneuvers using a virtual structure approach. This paper is organized as follows. In section II, an analytical model is introduced and equations of motion are derived. In section III, control architecture is explained. In section IV, the control is applied to the model, and the results of numerical simulations are discussed. In section V, conclusions are mentioned.

I

Initial Attitude Attitude Maneuver Final Attitude Figure 1. Attitude Maneuver of Spinning Tethered Formation Flying

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II. Modeling

A. Tether Model In this paper, tether is treated as a collection of lumped masses, which are connected by springs and dashpot as

shown in Fig.2. Tether jk consists of N discrete masses ljkS ( Nl ,,1L= ). p

jkS ( np ,,1L= ) are located out of the spacecraft j , and q

jkS ( Nnq ,,1L+= ) are located inside the spacecraft j . ljkk and l

jkc indicate a spring constant and a

damping coefficient, respectively, in the following.

00

,0,

≤>

⎩⎨⎧ ∆

= ljk

ljkttl

jk

LAEk

αα ,

00

,0,

≤>

⎩⎨⎧

= ljk

ljk

ljkl

jkc

cαα (1)

ljkα is defined as shown in Eq.(2). A sign of l

jkα shows whether the tether element is longer than the natural length or not. Eqs.(1)(2) can simply represent a tether slack phenomenon.

11 −−∆−= l

jkljk SS

ljk qqLα (2)

Using the lumped mass model, the motion of the system is discontinuous at the time when the tether mass is deployed / retrieval from the spacecraft j . Therefore it is necessary to calculate and revise the value of the discontinuity so that the system conserves momentum and angular momentum. The new velocity of spacecraft and lumped mass, and angular velocity of spacecraft at the time after tether mass deployment / retrieval can be calculated from the law of momentum and angular momentum conservation1.

B. System Model Figure 3 shows an analytical model of the spinning tethered formation flying, which consists of three spacecrafts

connected by three tethers. The motion of the system is considered in the earth-centered inertial coordinate system. 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: jkj

jjkjj

j

jjj q

mm fffqq +++= 3||

µ&& (3)

Attitude: jkj

jjkjjjj

j

IBj

IBIBj q

jjj τττqIqωIωωI +++×Ω

=×+ 2

2

||3

& (4)

jf ，jτ in Eqs.(3)(4) are represented as follows with coordinate frames.

j

Tj fif = (5)

j

Tjj τbτ = (6)

jjkf ， j

hjf ， jjkτ ， j

kjτ in Eqs.(3)(4) are described in the following.

Spacecraft jSpacecraft k

jb kb

Tether jk

1jkS 2

jkS 1−njkS

njkS

Njk

nNjk SS L−

jjkT

kjkT

11 , jkjk ck 22 , jkjk ck njk

njk ck ,

Figure 2. Lumped Mass Model of Tether

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)()( jjkjS

jjkjS

jjk

jjk n

jknjk

T aqqaqqf +−+−= (7)

)()( 11 111

kjkj Sjkjj

kS

jkjjkjkj

jkj ck qaqqaqf &&& −+−−+−= α (8)

jjk

jjk

jjk faτ ×= (9)

jkj

jkj

jkj faτ ×= (10)

Tether length jkL is represented in the following. n means the number of tether elements outside the spacecraft j .

)( jjkjSjk n

jkLnL aqq +−+∆= (11)

The equations of motion for tether j are formulated as follows. In the case of nl <≤1 :

3

111 )()(

)()(

11

11

ljk

ljk

ljk

ljk

ljk

ljk

ljk

ljk

ljk

ljk

ljk

SSSSljkSS

ljk

ljk

SSljkSS

ljk

ljkS

qMck

ckM

qqqqq

qqqqq

µα

α

+−+−+

−+−=

++

−−

+++ &&

&&&& (12)

In the case of nl = :

3

)()( 11 ljk

ljk

njk

njk

njk

njk

njk SS

jjkSS

ljkSS

njk

njkS qMckM qfqqqqq µα ++−+−= −− &&&& (13)

The position of the center of mass is expressed as follows.

∑∑==

=3

1

3

1 jj

jjjcm mm qq (14)

1b

2b

3b

Spacecraft 1

1m 1I

Spacecraft 2

2m 2I

Spacecraft 3

3m 3I

i

1q

Tether 12

Tether 23

Tether 31

112S 2

12S 112−nS nS12

Nn SS 121

12 L+

112a

131a

112T2

12T1

31T

331T

Nn SS 231

23 L+

Nn SS 311

31 L+

nS31

131−nS

331S

231S

131S

231Sq

Figure 3. Analytical Model for Spinning Tethered Formation Flying

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III. Control Method for Attitude Maneuver

A. Virtual Structure A Virtual structure approach is one of approaches for conventional multi-spacecraft formation control16-19. In the

approach, the entire formation is treated as a single structure. The control is derived in three steps in this approach. 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 derived. Figure 4 is suggested to be a virtual structure model for the system as shown in Fig.3. 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 massless rods. The position of the C.M. of rigid body j measured in the coordinate frame V as follows.

Tjjj

Tj

Tj rrrr ][ 321vvr == (15)

A scaling variablejλ is introduced to represent expansion/reduction of the virtual structure. This variable

jλ makes it possible to describe transformation of the virtual structure. Then jr is rewritten in the following.

Tj ][ 321 λλλ=λ (16)

0jjj rr Λ= (17)

where, )diag( 321 λλλ=Λ j and 0jr represents an initial value of jr . vsI means the inertia matrix of the virtual

structure.

vvI vsT

vs I= (18)

∑=

+=3

1)~~(

jj

Tjjvsjvsvs rrmII (19)

where, jr~ is a skew-symmetric matrix of jr . jvsε is described concretely as follows.

Tjvsjvsjvsjvsjvs ][ 4321 εεεεε = T

jvsjvsT

jvsjvsTz ][])2cos()2sin([ εεφφ ′′′== (20)

where, TTz ]100[v= represents a rotation axis， and 3)1(2 −= jjvs πφ represents a rotation angle in the coordinate frame V .

120°120°

v

1 vsb2 vsb

3 vsb

vsI

1vsm 1vsI

2vsm 2vsI

3vsm 3vsI

Virtual Tether 12

Virtual Tether 23

Virtual Tether 31

12vsa13vsa1v

2v 3v

Figure 4. Virtual Structure Model

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B. Description of Attitude Maneuver In this section, a method for describing attitude maneuvers is explained. Figure 5 shows relationship between

coordinate frames in this paper. j are basis vectors of a coordinate frame J , that means an inertia coordinate frame or an orbital frame depending on a situation. The both cases are explained in the following. When J is regarded as an inertia coordinate frame, an origin of the J frame is coincided with vsq , and each axis of the J frame is always equal to the initial V frame represented by 0V . On the other hand, when J is regarded as an orbital coordinate frame, an origin of the J frame is coincided with vsq , and each axis of the J frame is set as shown in Fig 6. 1j , 2j and 3j axis mean a zenith direction, a tangential forward direction and a perpendicular direction to an orbital plane, respectively. In the both cases, coordinate frame K , which basis vectors are k , indicates a spin coordinate frame to represent a spin axis direction. The relationship between the coordinate frame J and K is described in the following using a direction cosine matrix.

jk JKC= (21)

where,

)()( 21 ψθ CCC JK = (22)

In the Eq.(22), 1C and 2C mean a single axis rotation matrix around 1-axis and 2-axis, respectively. In this paper, a formation maneuver is expressed using an angle θ around 1-axis and an angle ψ around 2-axis. endθ and endψ indicate desired angles of θ and ψ , and they are described in the following formulas.

))(2)(sin( sesend tttt −−= πθθ (23)

))(2)(sin( sesend tttt −−= πψψ (24)

where, st and et represent maneuver start time and maneuver end time, respectively. These formulas are selected because derivations of them are continuous functions. The relationship between the coordinate frame K and V is represented as follows using a direction cosine matrix.

kv KVC= (25)

where,

)(3 ϕCC KV = (26)

In the Eq.(26), 3C means a single axis rotation matrix around 3-axis. ϕ means an angle around the spin axis, and is expressed in the following formula using spin angular velocity 3vs

Kω of the virtual structure measured in the coordinate frame K .

∫=t

vsK dt

0 3ωϕ (27)

Inertia Coordinate

Inertia Coordinateor

Orbital Coordinate

Spin Coordinate

v

Virtual Structure Coordinate

i j

j′

k

axis)(around 2jψ

axis)(around 1j′θ

axis)(around 3kϕ

IJC

JKC

KVC

Figure 5. Relationship between Coordinate Frames

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The absolute angular velocity IVω of the virtual structure is represented in the following.

KVJKIJIV ωωωω ++= (28)

Each term of Eq.(28) is concretely shown in the following.

[ ] frame)coordinateorbitalanasregardedis(when

frame)coordinateinertiaanasregardedis(when

00 J

JTT

IJ

⎩⎨⎧

Ω=

j0

ω (29)

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

00

0

0)( 1

θψθ

&

&CTJK kω (30)

[ ]TvsKTKV

300 ωvω = (31)

i1i 2i

3iEarthj

1j

3j

2j

vsq

Tangential Forward Direction

Perpendicular Direction to Orbital Plane

Zenith Direction

Figure 6. Definition of Orbital Coordinate Frame J

C. Equation of Motion for Virtual Structure The equations of motion for the virtual structure are described as follows.

Translation: 3vsvsvsvsvs qmm qq µ=&& (32)

Attitude: vs

IVvs

IVvs

IVIVvs tωIωIωωI =+×+ //// && (33)

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

vsT

vs tvt = (34)

)( // dIVIVvsvsevsevs ωωKKt −−−= ωε (35)

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

)()( djjjjj KdK λλλλλ λλ&&&&

& −−−−= (36)

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where, λK and λ&K are controller gains. The desired position, velocity, Euler parameter and angular velocity for each spacecraft are described in the following. U means an unit matrix.

0jj

TIVvs

dj rCqq Λ+= (37)

00

~jj

IVTIVjj

TIVvs

dj rCrCqq Λ+Λ+= ω&&& (38)

⎥⎦

⎤⎢⎣

⎡′′′

⎥⎦

⎤⎢⎣

⎡′′′−

′′′−′′=

vs

vs

jvsT

jvs

jvsjvsjvsdj

Uεε

εεεεε

ε (39)

IVdIB j ωω = (40)

D. Control of Thruster, Tether and Wheel In the general virtual structure approach, normal

PD control is conducted using current states and desired states17-19. However, for the spinning tethered formation flying, we can derive a characteristic control method. Because the system rotates around the center of mass of the whole system under the condition that spacecraft 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, spin angular velocity and position of the center of mass 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. 1. Thruster for Controlling Spin Angular Velocity and Out-of-Plane Motion

Thrusters for controlling spin angular velocity and out-of-plane motion are considered in the coordinate frame V as shown in Fig.8. Let cmr represent a position of the center of the mass of the real formation as follows.

cmT

cm rvr = (41)

∑∑==

=3

1

3

1 jj

jjjcm mrmr (42)

Let cmr ′ and jr′ represent projections of cmr and

jr onto the 21vv plane, and jr is defined as follows.

cmjj rrr ′−′=ˆ (43)

θ indicates the angle between djr and

jr , and 3jr represents the out-of-plane displacement of spacecraft j .

jr∆ is defined as shown in the following.

j

djj rrr ˆ−=∆ (44)

Thruster Control Tether Control

Spin Angular Velocity

Out of Plane Motion (Attitude Maneuver)

Spin plane

Spacecraft

Spin Radius

CM Position Figure 7. Control Target Motions using Thruster and Tether

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Here, jr and

jr∆ are the vectors on the 21vv plan. Therefore we can set [ ]TTj yx 0ˆ vr = . u′ is defined as

follows, that is a normal vector to jr on the 21vv plan.

TT xy ]0[ −=′ vu (45)

The unit vector u describing a direction of thruster for spin angular velocity control is represented in the following.

TTj uu ]0[||/)sign( 21vuuuru =′′′⋅∆= (46)

Thruster control for spin angular velocity and out-of-plane motion is derived as follows. The control makes θ and

3jr to be zero using PD gains; θK ，

θ&K ， rK ， rK &

[ ]TjrjrT

j rKrKKKuKKu 3321 )()( &&&&&& −−++= θθθθ θθθθvf (47)

2. Thruster for Controlling Position of the Center of Mass

For controlling position of CM, thrusters of each spacecraft are used. When this control is applied to the formation, it is important to avoid tether slack condition as shown in Fig.9 and keep the equilateral triangle formation. Thrusters for controlling the position of the center of mass are considered in the coordinate frame V as shown in Fig.10. Let

je )3,2,1( =j represent a thruster direction for the control. When the formation keeps the desired equilateral triangle, two tethers connecting spacecraft j are on the xy plane of the coordinate frame

jB . je

is limited to the first quadrant and the fourth quadrant of the xy plane of the coordinate frame jB . In this case, the

angle between je and the tether direction becomes an obtuse angle. Therefore, thruster force orients to the tether

extending direction, and the tether slack condition is avoidable. Let tr represent the vector that indicate the position of the origin of the coordinate frame V as the desired CM position. The force tf needed for controlling the CM position is defined as follows using PD gains; tK ,

tK &.

ttttt KK rrf &&−−= (48)

The force tf of Eq.(48) is assumed to be generated using two adjacent je .Here, the formation is assumed to be an

equilateral triangle. Therefore, the angle between two adjacent je is 120 degrees. The selection of

je , whether je

1v2v

3v

v

djr

cm′r

cmrjr

j′r

θ3jr

ˆ jrˆ jr

djr θ

Thruster Direction Unit Vector

u

jr∆

Figure 8. Definition of Thruster Force for Controlling Spin Angular Velocity and Out-of-Plane Motion

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and 1+je are selected or

1−je and je are selected, is decided using signature of the angle ϕ between tf and

je as shown in Fig. 11. Element

tjf of tf along je direction is derived in the following.

⎩⎨⎧

==≠+=

)0()0(sin/)tan/(tantantan

αααβαβα

ttj

ttj

ff

ff (49)

Thruster force for controlling the position of the center of mass is described as follows.

)0( >= tjjtjj ff ef (50)

where, in the case of 0≤ϕ , ||ϕα = , ϕπβ −= 32 , and in the case of 0>ϕ , ϕα = , |32| πϕβ −= .

Tether

Spacecraft

Thruster

Slack

Thrust Force

Figure 9. Tether Slack Condition by Thruster Force

Tether

Tether

jb xy

z

Spacecraft j

Plane

1v2v

v

Plane

trtf

2e

1e

3e

C.M.

Projection of C.M.

Spacecraft 1

Spacecraft 2

Spacecraft 3

Tether

Projection of Tether

Envelope for Controlling C.M.

je

Figure 10. Definition of Thruster for CM Position Control

tf

je

1+jetjf

1+tjf

ϕ

ϕπ −32tf

ϕje

1−je

tjf

1−tjf32πϕ −

)0( <ϕ )0( >ϕ

α

β α

β

Figure 11. Thruster Selection for CM Position Control

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3. Tether Tension Control Tether tension control is derived as follows. LK and

LK & are PD gains1.

)()( djkjkL

djkjkL

djk

jjk LLKLLKTT &&

& −+−+= (51)

djkT means the equilibrium tension for keeping the present spin radius.

( ))3sin(

)6(sin~~

π

πωω dj

dj

dj

dj

dIVdIVjd

jk

rrrrmT

⋅−=

&& (52)

4. Wheel Control Wheel control is derived to make

jε and IBj

jω become djε and dIB

jjω , respectively. eK and ωK are PD gains.

)(dIB

jIB

jeejjjKK ωωετ ω −−−= (53)

IV. Numerical Simulation In this section, numerical simulations are conducted to consider attitude maneuvers of the spinning tethered

formation flying.

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

system. (1) Shape of spacecraft:

kg50=jm , 2kgm)083.2083.2083.2(diagI j =

m]025.025.0[ Tjjka −=

(in the case of )1,3(),3,2(),2,1(),( =kj ) m]025.025.0[ Tk

jka −−= (in the case of )2,3(),1,2(),3,1(),( =kj ) (2)Tether:

mkg /1075.1 3−×=ρ (linear density) 2/9800 mmNEt = ， 233.1 mmAt = ， 01.0=j

jkc ,

Total tether length: m200 ，Tether elements: 39 (3) Orbital motion: Orbital altitude: km800=H , Radius of the Earth:

km137.378,6=eR , 2314 sm10986.3 ×=µ 3)( eRH +=Ω µ , Orbital period: 4.6052=T s

(4) Virtual Structure: Initial scaling variable: [ ]Ts 111=λ , m]0sin0.100cos0.100[0

Tjvsjvsjr φφ= , where, 3)1(2 −= jjvs πφ

Tevs RHq ]00[ += , T

evs RHq ])(00[ Ω+=& , Initial spin angular velocity: rad/s02.03 =vsKω

In the initial condition, the spin plane of the virtual structure is assumed to coincide with the orbital plane as shown in Fig. 12. The maneuver angles described in Eq.(23) and Eq.(24) are defined as deg0=endθ , deg45=endψ . (5) Spacecraft:

The initial condition of each spacecraft is assumed to be correspondent with the condition acquired from the initial virtual structure.

i1i 2i

3iEarth

j v= (t=0)

1j

2j

3j

Circular Orbit ( plane)2i 3i

Initial Circular Trajectory of

Formation

jk

1j3j

2j1k

2k

3k

Figure 12. Coordinate Frames in Numerical Simulations.

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B. Results of CM Position Control Before we turn to attitude maneuver simulations, it is necessary to evaluate CM position control. For evaluating

the control, let an initial position of cmq set T

vsvsvscm qqqq ]0.1000.100[ 321 ++= , and we assume that attitude maneuver is not executed during CM position control. Figure 13(a) shows a CM position error, and it indicates that the error converges to zero with the control. Figure 13(b) indicates thruster force of spacecraft 1 that is generated along 1e direction as shown in Fig.10. It shows that thruster force is generated periodically. The reason of the periodical thruster force is that Eq.(50) is satisfied periodically because of the formation spin motion. When spacecraft 1 generates no thrust force, spacecraft 2 and 3 generate thruster force for CM position control. Figure 13(c) shows tether tension. An important point to emphasize is the fact that all spacecraft keep almost equilibrium tether tension. This figure means there is no slack in the tethers. Figure 13(d) shows a relative position error between spacecraft 1 and 2. The error 12E is described by the following equation.

121212 qqqq −−−= ddE (54)

The blue data in Fig.13(d) indicate the relative position error calculated without the tether model as shown in Fig.2, and the black data indicate the error calculated with the tether model. The error with the tether model is -0.015m, while the error without tether model is almost zero. In the case of the minus values of the error, Eq.(54) indicates that relative distance between spacecraft 1 and 2 of real system is longer than that of the virtual structure. This phenomenon is caused by effect of tether extension. In any case, the error is small, and the formation is surely kept during the CM position control. These figures indicate that the proposed CM position control described in Eqs.(48)-(50) works well. If position accuracy better than -0.015m is required, direct ranging system is needed to measure distance between spacecraft. In this paper, we only use tether length and tether velocity data.

0 1000 2000 30001

1.5

2

Time, sec

Ten

sion

, N SC1 SC2 SC3

Figure 13(c). Tether Tension during CM Position Control.

0 1000 2000-0.03

-0.015

0

0.015

0.03

Time, sec

Err

or, m

with Tether Model without Tether Model

Figure 13(d). Relative Position Error between SC1 & 2 during CM Position Control.

0 1000 2000 30000

0.1

0.2

Time, sec

Thr

ust,

N

Figure 13(b). Thruster Force of SC1 for CM Position Control.

0 1000 2000 30000

50

100

150

Time, sec

CM

Pos

ition

, m

Figure 13(a). Position Error of CM.

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C. Results of Attitude Maneuver Simulations In this section, the coordinate frame J is regarded as

the inertia coordinate frame, and attitude maneuvers of the circular formation are considered. Figure 14(a) shows the motion of the spin axis of the formation and the trajectory of spacecraft 1 during the maneuver. From Fig. 14(a), it is cleared that the formation achieved the planned 45 degrees attitude maneuver. Tether tension data during the maneuver is indicated in Fig. 14(b). Parameter t of the figure means time duration for the maneuver. t equals se tt − shown in Eq.(23) and Eq.(24). In the case of t is short, oscillation of tether tension is confirmed. The longer t is, the smaller the tether tension oscillation is. Figure 14(c) shows thruster force for controlling out-of-plane motion during the maneuver. The stronger thruster force is needed when time duration t is small. Figure 14(d) indicates a relative position error between spacecraft 1 and 2. It is cleared that the error is extremely small, and the formation is tightly kept during the maneuvers. The total impulse of thruster for the attitude maneuver is indicated in Fig. 14(e) as an indicator of fuel consumption during the maneuver. It is confirmed that fuel consumption is increased when time duration t is small. For example, if t is changed from 100 sec to 300 sec, the fuel consumption is fairly reduced. Therefore, selection of time duration for the maneuver is important to reduce fuel consumption. It is also cleared that the fuel consumption gradually becomes a constant value when t is longer. The proposed attitude maneuver control as mentioned in the section III didn’t consider tether dynamics. However, it is cleared that the proposed control works effectively from these figures.

Initial Trajectory

Final Trajectory

Initial Spin AxisFinal Spin Axis

45 deg.

Figure 14(a). Spin Axis Direction and Trajectory of Spacecraft.

0 500 1000 1500-0.03

-0.015

0

0.015

0.03

Time, sec

Err

or, m

t= 100sec t= 300sec t= 600sec t= 900sec t=1200sec

Figure 14(d). Relative Position Error between Spacecraft 1 & 2.

0 1000 2000 30000

50

100

150

200

t= 700sec t= 800sec t= 900sec t=1000sec t=1100sec t=1200sec

Time, sec

∫|f|

dt, N

s

t= 100sec t= 200sec t= 300sec t= 400sec t= 500sec t= 600sec

Figure 14(e). Total Impulse of Thruster for Attitude Maneuver

0 500 1000 15000.8

1

1.2

1.4

Time, sec

Ten

sion

, N

t= 100sec t= 300sec t= 600sec t= 900sec t=1200sec

Figure 14(b). Tether Tension for Each Maneuver Time t .

0 500 1000 1500-3

-1.5

0

1.5

3 t= 100sec t= 300sec t= 600sec t= 900sec t=1200sec

Time, sec

Thr

ust,

N

Figure 14(c). Thruster Force for Out-of-Plane Motion.

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In the numerical simulations given above, the initial angular velocity of the formation is fixed to rad/s02.03 =vs

Kω . In the following, we fix maneuver time to sec300=t and change the initial spin angular velocity. Figure 15(a) shows tether tension for each angular velocity. For each angular velocity, there is tension drifting during the attitude maneuver. Figure 15(b) shows thruster force for controlling out-of-plane motion during the maneuver. The total impulse of thruster for the attitude maneuver appears in Fig.15(c). In this figure, we should notice that the total impulse of rad/s001.03 =vs

Kω is larger than that of rad/s002.03 =vsKω . In terms of angular

momentum of the virtual structure, that are assumed to be a rigid body, the total impulse for the maneuver in the case of rad/s001.03 =vs

Kω should be smaller than the case of rad/s002.03 =vsKω . From this result, it is inferred that

advantage of tethered formation flying, that are small fuel consumption to achieve precise relative position and attitude control, is lost when the initial angular velocity is small and the tether tension becomes weak. Figure 15(d) shows a relative position error between spacecraft 1 and 2. It is cleared that the error becomes small when the initial angular velocity becomes small except the case of rad/s001.03 =vs

Kω . In the case of rad/s001.03 =vsKω , the relative

position error is oscillate.

0 500 1000 1500-1

-0.5

0

0.5

1

w=0.02rad/s w=0.01rad/s w=0.005rad/s w=0.002rad/s w=0.001rad/s

Time, sec

Thr

ust,

N

Figure 15(b). Thruster Force for Out-of-Plane Motion.

0 1000 2000 30000

50

100 w=0.02rad/s w=0.01rad/s w=0.005rad/s w=0.002rad/s w=0.001rad/s

Time, sec

∫|f|

dt, N

s

Figure 15(c). Total Impulse of Thruster for Attitude Maneuver.

0 500 1000 1500-0.05

-0.025

0

0.025

0.05 w=0.02rad/s w=0.01rad/s w=0.005rad/s w=0.002rad/s w=0.001rad/s

Time, sec

Err

or, m

Figure 15(d). Relative Position Error between Spacecraft 1 & 2.

0 1000 2000 30000

0.5

1

1.5

w=0.02rad/s w=0.01rad/s w=0.005rad/s w=0.002rad/s w=0.001rad/s

Time, sec

Ten

sion

, N

Figure 15(a). Tether Tension for Each Initial Angular Velocity.

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In the following parts, tether motion during the maneuver is considered. We set the coordinate frame O which basis vectors are represented by o to observe the tether12 motion. The y axis of o is set along the direction from tether connecting point 1

12a to 212a . The z

axis of o is parallel to the z axis of 1b . The x axis of o is set to form right-handed system as shown in Fig.16

(a). Figure 16(b) shows the tether12 motion measured from the coordinate frame O with maneuver duration

sec100=t . From the figure, tether vibration during the attitude maneuver is observed. Maximum vibration amplitude for each maneuver time is plotted in Fig.16(c). It is cleared that the vibration amplitude becomes sharply large when the maneuver time becomes short. Figure 16(d) indicates maximum vibration amplitude for each initial angular velocity. The maneuver time is fixed to

sec300=t . It is cleared that the vibration amplitude becomes sharply large when the initial angular velocity becomes small. In the case of rad/s001.03 =vs

Kω or rad/s002.03 =vsKω , the amplitude is especially large. Large

amplitude of the tether vibration is supposed to lead increasing fuel consumption as shown in Fig.15(c).

1b

2b

3b

Spacecraft 1

Spacecraft 2

Spacecraft 3

Tether 12

Tether 23

Tether 31

112S 2

12S 112−nS nS12

nS31

131−nS

331S

231S

131S

x

y

z

o

Figure 16(a). Coordinate Frame for Observing Tether12 Motion

Figure 16(b). Tether Motion Measured From Coordinate Frame O

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V. Conclusion In this paper, attitude maneuvers of spinning tethered formation flying are considered. First, the feedback

maneuver control based on the virtual structure approach is proposed. Second, the control is applied to the analytical model, which consists of three rigid spacecraft and tethers modeled by lumped masses, and validity of the proposed control is indicated via numerical simulations. The following facts are also indicated,

(a) Even if the proposed control is applied, a constant error of relative position is remained. The error is caused by effect of tether extension. In this paper, tether length and tether velocity data are only used to measure distance between spacecraft. For more precise control, direct ranging system is required.

(b) When angular velocity is small and tether tension is small, amplitude of tether vibration during maneuvers becomes large. The large amplitude of the vibration leads increasing relative position error, and consequently increasing fuel consumption.

References 1Nakaya, K., Iai, M., Omagari, K., Yabe, H. and Matunaga, S., “Formation Deployment Control for Spinning Tethered

Formation Flying -Simulations and Ground Experiments-,” AIAA Guidance, Navigation, and Control Conference and Exhibit, AIAA 2004-4896, Providence, Aug.16-19, 2004

2Nakaya, K., Iai, M., Mori, O. and Matunaga, S., “On Formation Deployment for Spinning Tethered Formation Flying and Experimental Demonstration,” 18th International Symposium on Space Flight Dynamics, Munich, Germany, Oct.11-15, 2004

3Nakaya, K. and Matunaga, S., “On Elliptic Formation Change and Deployment of Spinning Tethered Formation Flying,” Journal of the Japanese Society for Aeronautical and Space Sciences, (in Japnese) (to be published)

4Mori, O. and Matunaga, S.: Research and Development of Tethered Satellite Cluster Systems, Proceedings of 2000 IEEE/RSJ International Conference on Intelligent Robots and Systems, CD-ROM, Vol.3, F-AI-5-2, pp.1834-1840, Kagawa, Oct.30-Nov.5, 2000

5Mori, O. and Matunaga, S., “Coordinated Control of Tethered Satellite Cluster Systems,” AIAA Guidance, Navigation, and Control Conference and Exhibit CD-ROM, AIAA 2001-4392, Montreal, Aug.6-9, 2001.

6Quadrelli, M.: Modeling and Dynamics Analysis of Tethered Formations for Space Interferometry, AAS/AIAA Spaceflight Mechanics Meeting, Santa Barbara, CA, February 11-14, 2001

7DeCou, A. B.: Tether Static Shape for Rotating Multimass, Multitether, Spacecraft for ‘Triangle’ Michelson Interferometer, Journal of Gudance, Control, and Dynamics. 12 (1989), pp.273-275.

8DeCou, A. B.: Attitude and Tether Vibration Control in Spinning Tethered Triangles for Orbiting Interferometry, The Journal of the Astronautical Science, 41 (1993), pp.373-398

9Maccone, C.: Tethered System to Get Magnified Radio Pictures of the Galactic Center from a Distance of 550AU, Acta Astronautica, 45 (1999), pp.109-114.

10Quinn, D. A. and Folta, D. C.: A Tethered Formation Flying Concept for the SPECS Mission, Proceedings of the 23rd Rocky Mountain Guidance and Control Conference, Breckenridge, Colorado, Vol. 104 of Advances in the Astronautical Sciences, 2000, pp.183-196.

11Kim, M. and Hall, C. D.: Control of a Rotating Variable-Length Tethered System, 13th AAS/AIAA Space Flight Mechanics Meeting, Ponce, Puerto Rico, February 9-13, 2003.

0 0.005 0.01 0.015 0.020

0.5

1

Angular Velocity, rad/sec

Am

plitu

de, m

Figure 16(d). Maximum Vibration Amplitude for Each Angular Velocity.

0 300 600 900 12000

0.1

0.2

0.3

Maneuver Time, sec

Am

plitu

de, m

Figure 16(c). Maximum Vibration Amplitude for Each Maneuver Time.

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12Tragesser, S. G.: Formation Flying with Tethered Spacecraft, AIAA/AAS Astrodynamics Specialist Conference, AIAA 2000-4133, 2000

13Bombardelli C., Lorenzini E. C., and Quadrelli M. B.: Pointing Dynamics of Tether-Controlled Formation Flying for Space Interferometry, paper AAS 01-404, AAS/AIAA Astrodynamics Specialist Conference, Quebec City, July 30-August 2, 2001.

14Bombardelli, C., Lorenzini, E., C. and Quadrelli, M., B.: Retargeting Dynamics of a Linear Tethered Interferometer, Journal of Gudance, Control, and Dynamics. 27 (2004), pp.1061-1067.

15Pizarro-Chong, A. and Misra, A. K.: Dynamics of Multi-Tethered Satellite Formation, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, AIAA 2004-5308, Providence, Aug.16-19, 2004.

16Lewis, M. A. and Tan, K. H.: High Precision Formation Control of Mobile Robots Using Virtual Structures, Autonomous Robot, 4 (1997), pp.387-403.

17Beard, R. W. and Hadaegh, F. Y.: A Coordination Architecture for Spacecraft Formation Control, IEEE Transactions on control systems technology, 9 (2001), pp.777-790.

18Ren, W. and Beard, R. W.: Virtual Structure based Spacecraft Formation Control with Formation Feedback, AIAA Guidance and Control Conference, Monterey, CA, August 2002, AIAA Paper no. 2002-4963.

19Nakaya, K and Matunaga, S.: A study of attitude dynamics and control for a module-type large spacecraft, The 7th International Symposium on ArtificialIntelligence, Robotics and Automation in Space, Nara, Japan, May 2003