[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control...

Coordinated Attitude Control of Spacecraft Formation

without Angular Velocity Feedback: A Decentralized

Approach ∗

A. R. Mehrabian†, S. Tafazoli ‡, and K. Khorasani §

In this paper, motivated by recent developments of velocity-free spacecraft (SC) atti-tude control techniques and behavioral-based SC formation control a decentralized controlalgorithm for attitude coordination of SC formation without angular velocity feedback ispresented. Asymptotic stability of the SC formation is guaranteed by using Lyapunov anal-ysis and LaSalle’s theorem. The advantage of our proposed algorithm is that it requireslimited information exchange among the SC in the formation (only attitude informationexchange is necessary). Additionally, the proposed algorithm can be extremely useful whenangular velocity information of the SC in the formation in not available due to sensor fail-ures or communication constraints. Unlike other popular methods in the robotics areawhich tend to assume simple dynamics such as linear systems and single or double inte-grator dynamic models, in this paper, the full nonlinear attitude dynamics of the SC isconsidered to track fast time-varying reference trajectories.

Nomenclature

e Euler vectorθ Euler angle, radq The unit-quaternionq Vector part of the quaternionq4 Scalar part of the quaternionR Rotation matrix[]× Cross product matrixQ(q) Quaternion product matrixω Spacecraft angular velocity vector, rad/secE(q) Matrix based on quaternionu External torque (control effort), N-mFB Body frameFI Inertial frameJ Spacecraft moments of inertial matrix, kg-m2

δq Quaternion errorδω Angular velocity error, rad/secqjn Quaternion error between jth and nth spacecraftωjn Angular velocity error between jth and nth spacecraft, rad/secA Constant matrixB Constant matrix

∗This research is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada(NSERC) Strategic Projects.

†PhD Candidate, Department of Electrical and Computer Engineering, Faculty of Engineering and Computer Science,Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8. [email protected]

‡Adjunct Assistant Professor, Department of Electrical and Computer Engineering, Faculty of Engineering and ComputerScience, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8. Also with the CanadianSpace Agency (CSA), Saint-Hubert, QC, Canada J3Y 8Y9. [email protected]

§Professor, Department of Electrical and Computer Engineering, Faculty of Engineering and Computer Science, ConcordiaUniversity, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8. [email protected]

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-6289

Copyright © 2009 by A. R. Mehrabian, S. Tafazoli, and K. Khorasani. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

P Constant matrixQ Constant matrix[] First time derivative, sec-1

[] Second time derivative, sec-2

λp Controller’s constant proportional gainλd Controller’s constant derivative gain1 Identity matrixm Number of spacecraft in the formation

I. INTRODUCTION

Distributed spacecraft (SC) formation flying has received considerable attention in the passed few years1-12. This new technology allows SC to autonomously react to each other’s orbit changes quickly and more

efficiently. It enables the collection of different types of scientific data unavailable from a single satellite, suchas stereo views or simultaneously collecting data of the same ground scene at different angles. In addition,rather than flying all the instruments on one costly satellite, by using this technology it allows scientists toobtain unique measurements by combining data from several satellites. SC formation maintenance requirescontrol of relative distances among two or more SC. In some applications, e.g. deep space interferometry,in addition to maintaining a specific distance, the SC are required to maintain a specified relative attitudeduring formation maneuvers.12

Four general approaches are available for SC formation control, namely multi-input multi-output (MIMO),leader-follower (LF), virtual structure (VS), and behavioral.1 Roughly speaking in the LF approach, thereis a leader agent and a set of follower(s). The problem is how to design a controller to ensure that thefollower(s) indeed track the leader. The following approaches are typical control methods that are employedwithin this structure, namely: (1) proportional/derivative (PD) control;2 (2) feedback linearization designapproach;3 (3) Sliding mode control;7 and (4) LQR and H∞ control.8 In the VS approach, the SC behaveas rigid bodies embedded in a larger, virtual rigid body. The overall motion of the virtual structure asspecified by positions and orientations of the SC within it are used to generate reference trajectories for eachindividual SC to track using its own controllers.

In the behavioral approach, the control laws that govern the motion of each spacecraft are derived byweighting the importance of several desired behaviors including formation keeping and goal seeking.10, 11

The advantages of behavioral approach are:12 (1) it is a distributed (not centralized) strategy; (2) it is veryrobust (fault-tolerant); (3) it requires a low information exchange; and (4) it also provides more flexibilityin control algorithm design.

The attitude control of a SC without angular velocity feedback has been studied by several researchers.Specifically, based on passivity arguments, Tsiotras13 extended the results reported in Ref. 14 for SC attitudestabilization using Modified-Rodrigues-Parameters (MRP). Later Akella15 extended the results that arereported in Ref. 13 and developed a tracking attitude controller for a SC. Recently, Tayebi16 proposed aquaternion-based dynamic output controller for the attitude tracking problem of a rigid SC without using theangular velocity information. Two advantages can be sought by using these methods, namely: (1) numericaldifferentiation difficulties due to the noise-corrupted attitude signals can be avoided and; (2) they are backedup with a rigorous stability analysis.

Design of decentralized attitude control for formation flying SC without requiring angular velocity willconsiderably reduce the communication requirements and eliminates the need for angular rate measurement.It also improves the robustness of the SC formation. It is shown in a recent study on SC on-orbit failures17

that mechanical gyroscopes, which are used for measuring the SC angular velocity, are the components withthe highest failure rates in the Attitude and Orbit Control subsystem (AOCS) of SC (see Fig. 9 in Ref. 17).Therefore, development of an efficient velocity-free control method for SC formation flying will considerablyincrease the chance of survival and completion of these important missions.

A decentralized attitude control law for SC formation without requiring angular velocity has appeared inRef. 12. In this paper, based on passivity analysis, the authors presented a decentralized behavioral controlarchitecture for SC in the ‘ring’ coordination architecture. However, the feedback control gains need to beadjusted for different initial SC attitudes. In addition, the final angular velocity of the SC formation is

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assumed to be zero in the reported analysis. More recently, the authors in Ref. 18 and Ref. 19 extended theresults reported in Ref. 16 to design a decentralized attitude controller for a rigid SC in formation based onLF and behavioral approaches, respectively.

In the present paper, inspired from Ref. 11 we propose a decentralized control law that is based on behav-ioral approach for coordinated attitude control of SC formation using passivity arguments. We extend theresults reported in Ref. 14 and Ref. 15 to design a controller without requiring angular velocity information.We assume that the final angular velocity of SC formation is non-zero. However, the control algorithmfor the case when the final angular velocity of SC is zero, which is desirable in deep space interferometrymissions, is shown to be easily derived from our developed analysis. The proposed control strategy is modelindependent in the sense that it does not require knowledge of the SC moments of inertia for the latter case.We conduct rigorous stability analysis for development of our proposed control algorithm. In contrast toexisting methods in the robotics research community,20, 21 which tends to assume rather simple dynamicssuch as linear systems and single or double integrator dynamic models, our proposed strategy in this pa-per primarily deals with complex dynamical systems consisting of nonlinear SC attitude dynamics that arecontrolled to track time-varying reference trajectories.

II. SPACECRAFT FORMATION STRUCTURE

Generally, there are three types of SC formation, namely:22 (1) trailing formations (e.g. Earth Observing-1 satellite program); (2) cluster formations (e.g. European Space Agency’s Cluster Mission); and (3) con-stellation formations (e.g. GPS satellites). Usually, these formations are made up of several smallsats whichweight less than 200 kg. In this paper, we consider a formation formed with m SC. All the SC in the forma-tion are 3-axis stabilized and fully actuated. This enables the SC to change their attitude from any initialattitude to any desired attitude rather quickly. We only consider the attitude dynamics of the SC, so thatthe developed algorithm can be applied to any SC formation where attitude synchronization is required.

Our main objective is to develop a decentralized control algorithm that enables the SC to first aligntheir attitudes within the formation, followed by the SC formation tracking a desired attitude specified forall the SC. This implies that the desired attitude assigned by the supervisor (ground station operator), isavailable to all the SC in the formation (there is no leader in the formation). In the development of thecontrol strategy, it will be assumed that the SC have no information about their own and other SC angularvelocity and only attitude information is available for exchange and feedback. This could occur due to eitherrate sensor failure17 or communication constraints among the SC.

A control algorithm is proposed below that relies only on the availability of attitude information forattitude alignment of the SC in the formation. Our analysis shows that stability of the formation can beguaranteed for different information flow architectures among the SC in the formation. In order to show thestability of the SC formation, first we consider a fully connected formation (aka ‘tight’ formation topology) asshown in Fig. 1. According to our derived analytical results, the stability of the formation can be guaranteedby using minimal information exchange among the SC as long as the flow is bi-directional and the SC graphis connected.23 It should be noted that in the tight formation topology, although the information flow iscentralized (attitude information of all the SC are available to all SC in the formation), the controller isdecentralized as each controller does not require the decision from the other controllers.

III. SPACECRAFT ATTITUDE DYNAMICS AND KINEMATICS

We review the spacecraft (SC) attitude dynamics and kinematics in the section. Unit quaternion is usedto present the attitude of the SC. The unit quaternion, for the jth SC is defined as:

qj =

[ej sin( θj

2 )cos( θj

2 )

]=

[qj

qj,4

](1)

where ej = (ej,1, ej,2, ej,3) is the Euler axis, θj is the Euler angle, qj is the vector part, and qj,4 is the scalarpart of the quaternion, subject to the constraint:

q2j,4 + qT

j qj = 1

.

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SC1 SC2

SC3 SC4

SCn

SCm

Figure 1. The schematic of a ‘tight’ coordination architecture.

A rotation matrix is a 3 × 3 matrix that represents the attitude of one reference frame with respect toanother. For example, the rotation matrix denoted by R(qjn) represents the rotation from frame Fn to F j .The rotation matrix is related to the quaternion through:24

R(qj) = (q2j,4 − qT

j qj)1 + 2qjqTj − 2qj,4q×

j (2)

where ω× represents the cross product matrix, i.e.:

ω× =

⎡⎢⎣ 0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

⎤⎥⎦ (3)

Single subscript qj represents the attitude of the corresponding reference frame F j with respect to theinertial frame FI . Similarly, double subscripts qjn represents the attitude of the corresponding referenceframe Fn with respect to frame F j .

In general, +qjn and −qjn both represent the same rotation matrix, and this sign ambiguity can beresolved by using the kinematic equation below:24

qjn =12

[−ω×

jn ωjn

−ωTjn 0

]qjn (4)

≡ 12E(qjn)ωjn (5)

where ωjn = [ωjn,1, ωjn,2, ωjn,3]T is the angular velocity vector of the jth SC with respect to the nth SC,which is defined below:

ωjn = ωj − R(qjn)ωn (6)

where ωj is the angular velocity vector of the body frame of the jth SC, FBj , with respect to the inertial

frame FI . In addition, the matrix E(q) is given as:

E(q) =

⎡⎢⎢⎢⎣

q4 −q3 q2

q3 q4 −q1

−q2 q1 q4

−q1 −q2 −q3

⎤⎥⎥⎥⎦ (7)

The inverse of the quaternion is defined as:

q−1 = [−q1 − q2 − q3 q4]T (8)

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The following equations corresponding to the relative states of the jth and the nth SC will be used subse-quently:11

R(qjn) = RT (qnj) (9)qnj = −qjn = −R(qnj)qjn (10)

The individual SC in the formation is modeled as a rigid body. The equations of motion for a rigid bodyin absence of any external disturbance torque is governed by:

Jj ωj = uj − ω×j Jj ωj (11)

where Jj is the moment of inertia matrix of the jth SC expressed in the body frame, FBj , and uj is the

external torque expressed in FBj . The formulation given above can be extended to the SC with momentum

exchange actuators, e.g. control moment gyroscopes (CMGs), similar to the method that was presented inRef. 24 or the variable speed CMGs (VSCMGs), similar to the method that was presented in Ref. 5.

IV. SPACECRAFT ATTITUDE ERROR DYNAMICS

For a SC in a formation we define two error measures. These measures are the station-keeping andformation-keeping attitude state errors. The station-keeping error is defined as the attitude state error of anindividual SC with respect to its absolute desired attitude state. The station-keeping error, δqj , is definedas:

δqj = Q(q� −1j )qj (12)

where q�j is the desired attitude of the SC formation and the matrix Q(q) is defined as:

Q(q) =

⎡⎢⎢⎢⎣

q4 −q3 q2 q1

q3 q4 −q1 q2

−q2 q1 q4 q3

−q1 −q2 −q3 q4

⎤⎥⎥⎥⎦

The station-keeping angular velocity error, δωj, is defined as:

δωj = ωj − R(δqj)ω�j (13)

where ω�j is the absolute desired angular velocity vector expressed in the absolute desired reference frame.

The first derivative of the station-keeping angular velocity error is obtained as:15

δωj = ωj − R(δqj)ω�j + ω×

j R(δqj)ω�j (14)

We can now state the governing equations for the attitude error δqj and the angular velocity error δωj

as follows:

δqj =12E(δqj)δωj (15)

Jj δωj = uj − ω×j Jj ωj + Jj

(−R(δqj)ω�j + ω×

j R(δqj)ω�j

)(16)

If we consider that the desired angular velocity of the SC in the formation is zero, i.e. ω�j = ω�

j = 0,(16) simplifies to:

Jj δωj = uj − ω×j Jj ωj (17)

which implies that in this case we have:δωj ≡ ωj

Formation-keeping error, for the jth SC is the attitude state error of the jth SC with respect to the otherSC in the formation. The relative attitude error between the jth and the nth SC is defined as:

qjn = Q(q−1n )qj

= Q(δq−1n )δqj

(18)

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The time derivative of qjn is defined in (5). The relative angular velocity vector of the jth SC withrespect to the nth SC, ωjn, that is defined in (6) can be re-written in terms of δωj and δωk, namely:

ωjn = δωj − R(qjn)δωn (19)

Our main goal in this paper is to design a decentralized controller for each SC which commands theactuators in order guarantee coordinated SC attitude and angular velocity alignment, i.e. qj → qn (orequivalently, qjn → 0) and ωjn → 0. This objective is designated as the ‘formation-keeping’ behavior.Furthermore, we have to ensure that the designed controllers guarantee that each SC attitude converges tothe commanded attitude, i.e. δqj → 0 and δωj → 0. This objective is designated as the ‘station-keeping’behavior. The constraint that we impose here is that there are no information exchanges regarding angularvelocity of the SC in the formation.

V. SINGLE SPACECRAFT ATTITUDE CONTROL WITHOUTVELOCITY FEEDBACK

In this section, we first introduce an attitude control law for a single SC. The synchronized attitudecontrol of SC formation will be considered in the next section. The following theorem is presented first toguarantee stability of a single SC without angular velocity feedback.

Theorem 1: Consider the jth SC error kinematics (15) and the dynamics (16) and let the control input,uj , be computed by using the following dynamic controller:{

uj = −δqj − 2ET (δqj)BT P(Azj + Bδqj) + Jj R(δqj) ω�j + Υ×

j JjΥj

zj = Azj + Bδqj

(20)

where A ∈ �4×4 is Hurwitz and B is given by:

B =

[13×3 03×1

01×3 0

](21)

and where 1 denotes an identity matrix, and matrix P is a symmetric, positive-definite solution of thefollowing Lyapunov equation:

AT P + PA = −Q (22)

for any 4 × 4 symmetric positive-definite matrix Q. Furthermore, we define Υj = R(δqj) δωj. Then theclosed-loop system (15), (16) and (20) is globally asymptotically stable.

Proof : Consider the following positive-definite radially unbounded Lyapunov function candidate:

Vj =12δωT

j Jjδωj +12δqT

j δqj +12(δqj,4 − 1)2 + 2(Azj + Bδqj)T P(Azj + Bδqj) (23)

The time derivative of this function along the trajectories of the closed-loop system (15), (16), and (20) canbe computed as follows:

d

dtVj =δωT

j Jjδωj + δqTj δ ˙qj + (δqj,4 − 1)δqj,4 + 2(zT

j Pzj + zTj Pzj)

=δωTj

[uj − ω×

j Jj ωj + Jj


j R(δqj)ω�j

)]+ δωT

j δqj

+ 2[(Azj + Bδqj)T Pzj + zT

j P(Azj + Bδqj)]

=δωTj [−δqj − 2ET (δqj)BT Pzj + Jj R(δqj) ω�

j + Υ×j JjΥj − ω×

j Jj ωj

+ Jj


j R(δqj)ω�j

)] + δωT

j δqj

+ 2[(zT

j AT + δqTj BT )Pzj + zT

j P(Azj + Bδqj)]

=δωTj

(Υ×

j JjΥj − ω×j Jj ωj + Jjω

×j R(δqj)ω�

j

)− 2δωT

j ET (δqj)BT Pzj + 2[zT

j AT Pzj + δqTj BT Pzj + zT

j PAzj + zTj PBδqj

]

(24)

In view of (13) and the fact that ω×ω = 0, it can be shown that:15

δωTj

(Υ×

j JjΥj − ω×j Jj ωj + Jjω

×j R(δqj)ω�

j

)= 0

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Thus, we have:

Vj = −2δωTj ET (δqj)BT Pzj + 2zT

j (AT P + PA)zj + 4δqTj BT Pzj

= −2zTj Qzj − 2δωT

j ET (δqj)BT Pzj + 4δqTj BT Pzj

(25)

Equation (15) can be re-written as δωTj ET (δqj) = 2δqT

j . Therefore, from (25) we obtain:

Vj = −2zTj Qzj ≤ 0 (26)

which is a negative semi-definite function. Note that since Vj is radially unbounded all the solutions remainbounded.

Consider now the set H = {(δωj , δqj , zj) : V = 0}. On H we have zj = zj = 0, which from (20) impliesthat δ ˙qj = 0. This, from (15) implies that δωj = 0 since E(δqj) is nonsingular for all δqj . These results canbe used along with (13), (16) and (20) to demonstrate that δqj = 0. Since A is Hurwitz, (20) also impliesthat zj = 0. Thus, the largest invariant set in H is the set H = {(δωj , δqj , zj) ∈ H : δωj = δqj = zj = 0}.By invoking LaSalle’s invariance theorem25 and noting that V is radially unbounded, the closed-loop systemis globally asymptotically stable. This completes the proof of the theorem. �

We are now in a position to state the following result.Corollary 1: Consider the case when the desired angular velocity is zero, i.e. ω�

j = ω�j = 0. Then the

following control law can be used to stabilize the SC dynamics (17):

uj = −δqj − 2ET (δqj)BT P(Azj + Bδqj) (27)

where zj is defined in (20).Proof : This follows directly from Theorem 1, and can be shown by replacing δωj with ωj in the Lyapunov

function (23), and its time derivative (24). �

VI. COORDINATED SPACECRAFT ATTITUDE CONTROL WITHOUTVELOCITY FEEDBACK

The design of synchronized SC formation control reorientation is presented in this section by assumingthat no angular velocity measurement is available for each SC and for information exchange among the SC.The developed controller is based on the sum of the control action for the station-keeping behavior and thecontrol action for the formation-keeping behavior.

A. Station-Keeping Controller

Inspired from (20), the station-keeping control law for the jth SC is defined according to:{us

j = −λpjδqj − 2λd

jET (δqj)BT P(Azj + Bδqj) + Jj R(δqj) ω�j + Υ×

j JjΥj

zj = Azj + Bδqj

(28)

where superscript ‘s’ stands for the station-keeping behavior, λpj > 0, and λd

j > 0 are the controller constantparameters. The matrices A and B are defined in the previous section.

B. Formation-Keeping Controller

The control law for the formation-keeping behavior for the jth SC, ufj , is defined as:{

ufj = −∑m

n=1 λpjnqjn − ∑m

n=1 λdjn

(ET (qjn)BT P(Azjn + Bqjn) − R(qjn)ET (qnj)BT P(Aznj + Bqnj))

zjn = Azjn + Bqjn n = 1, 2, . . . , m, j �= n

(29)where m is the number of the SC in formation. In addition we have:

λpjn = λp

nj ≥ 0, λdjn = λd

nj ≥ 0. (30)

Through the constraints (30) we assume that the communication flows among the SC are bi-directional.Different coordination architectures can be built by proper selection of the formation flow gains λp

jn and λdjn.

For example, selecting λp12 = λp

21 = λd12 = λd

21 = 0 eliminates the connection between the SC1 and the SC2in the formation. We are now in the position to present our main results.

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VII. MAIN RESULTS

The main result of this paper is presented in the following theorem.Theorem 2: Consider m SC in a formation where the dynamic equations of the jth SC is given by (16).

The decentralized control law for the jth SC obtained by adding the station-keeping behavior (28) and theformation-keeping behavior (29) given as:

uj =usj + uf

j

= − λpj δqj − 2λd

jET (δqj)BT P(Azj + Bδqj) + Jj R(δqj) ω�j + Υ×

j JjΥj −m∑

n=1

λpjnqjn

−m∑

n=1

λdjn

(ET (qjn)BT P(Azjn + Bqjn) − R(qjn)ET (qnj)BT P(Aznj + Bqnj))

(31)

where zj , znj , j, n = 1, 2, . . . , m, j �= n are defined in (28) and (29) will guarantee that the closed-loopsystem signals remain all bounded and the SC formation is globally asymptotically stable.

Proof : Consider the following positive-definite radially unbounded Lyapunov function candidate:

Vj =12δωT

j Jjδωj +12λp

j δqTj δqj +

12λp

j (δqj,4 − 1)2 + 2λdj (Azj + Bδqj)T P(Azj + Bδqj)

+m∑

n=1

[λp

jn

(qT

jnqjn + (1 − qjn,4)2)

+ λdjn(Azjn + Bδqjn)T P(Azjn + Bδqjn)

] (32)

The time derivative of Vj along the trajectories of the closed-loop system is governed by:

Vj =δωTj Jjδωj + λp

j δqTj δ ˙qj + λp

j (δqj,4 − 1) δqj,4 + 2λdj (z

Tj Pzj + zT

j Pzj)

+12

m∑n=1

λpjnqT

jnωjn +m∑

n=1

λdjn

(zT

jnPzjn + zTjnPzjn

) (33)

From (25), the above equation simplifies to:

Vj = − 2λdj z

Tj Qzj − δωT

j

[m∑

n=1

λpjnqjn +

m∑n=1

λdjn

(ET (qjn)BT Pzjn − R(qjn)ET (qnj)BT Pznj

)]

+12

m∑n=1

λpjnqT

jnωjn +m∑

n=1

λdjn

(zT

jnPzjn + zTjnPzjn

) (34)

In order to show the stability of the SC formation, we consider the sum of the component Lyapunovfunctions. The composite Lyapunov function is given by:

V = −m∑

j=1

2λdj z

Tj Qzj −

m∑j=1

m∑n=1

δωTj

(λp

jnqjn + λdjn


))

+12

m∑j=1

m∑n=1

λpjnqT

jnωjn +m∑

j=1

m∑n=1

λdjn

(zT

jnPzjn + zTjnPzjn

) (35)

Using (6), (10) and (30) we can show that:

12

m∑j=1

m∑n=1

λpjnqT

jnωjn =m∑

j=1

m∑n=1

λpjnδωT

j qjn (36)

Furthermore, according to (30) and noting the fact that qT R(q) = qT we obtain:m∑

j=1

m∑n=1

λdjn

(zT

jnPzjn + zTjnPzjn

)= −

m∑j=1

m∑n=1

λdjnzT

jnQzjn

+m∑

j=1

m∑n=1

δωTj

(λd

jn


)) (37)

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Consequently, from (35)-(37) we obtain:

V = −m∑

j=1

2λdj z

Tj Qzj −

m∑j=1

m∑n=1

λdjnzT

jnQzjn ≤ 0 (38)

Consider now the set H = {(δωj, δqj , zj , ωjn, qjn, zjn) : V = 0}. On H we have zj = 0, which from (28)implies that δ ˙qj = 0. This from (15) implies that δωj = 0. It can be further shown that by using (13), (16)and (20) one gets δqj = δqn = 0; thus from (18) we have qjn = 0. Since A is Hurwitz, (20) also implies thatzj = 0. Similarly, zjn = 0 results in having qjn = 0 from (29) and from (5) we have ωjn = 0.

Therefore, the largest invariant set in H is the set H = {(δωj , δqj , zj , ωjn, qjn, zjn) ∈ H : δωj = δqj =zj = ωjn = qjn = zjn = 0}. By invoking LaSalle’s invariance theorem25 and since V is radially unbounded,it follows that the system is globally asymptotically stable. This completes the proof of the theorem. �

To demonstrate stability preservation of the SC formation with minimal communication exchange achievedby using the above control laws, we consider the following typical example. Consider a SC formation thatis formed from three SC (m = 3) with only two bi-directional connections (see Fig. 2). For this system withminimal connections we have:

λpj , λd

j > 0, j = 1, 2, 3

λp23 = λp

32 = λd23 = λd

32 = 0,

λp21 = λp

12, λp31 = λp

13, λd21 = λd

12, λd31 = λd

13

(39)

Based on the parameters that are given above the Lyapunov function (32) is a positive-definite radi-ally unbounded function. According to (38), the time derivative of the Lyapunov function along with thetrajectories of the system is given by:

V = −3∑

j=1

2λdj z

Tj Qzj − λd

21(zT21Qz21 + zT

12Qz12) − λd31(z

T31Qz31 + zT

13Qz13) ≤ 0 (40)

First note that all the terms in the equation above are quadratic so it is a negative semi-definite function.Having the first term in (40) equal to zero implies δωj = δqj = zj = 0, which according to (18) yieldsqjn = 0. Semi-negativeness of the second term in (40) and knowing the fact that A is Hurwitz impliesz12 = z21 = ω12 = ω21 = 0. Similarly, noting semi-negativeness of the third term in the above equationwe can conclude z13 = z31 = ω13 = ω31 = 0. From (6) and knowing the fact that the rotation matrix forthe unit quaternion attitude representation (2) is nonsingular, we conclude that ω32 = ω23 = 0. Obviously,z32, z23 are not defined for this system. Therefore, using LaSalle’s invariance theorem25 and given that V isradially unbounded, it follows that the system is globally asymptotically stable.

The global asymptotic stability of the SC formation having a larger number of SC in the formation whennot all SC are directly connect to each other although the SC formation graph is connected, can be shownin a similar manner. We now have the following Corollary.

SC1 SC2 SC3

Figure 2. Spacecraft formation having three members with minimal connection (information exchange).

Corollary 2: Consider the case when the desired SC formation angular velocity is zero, i.e. ω�j = ω�

j = 0.Then the following control law can be used to stabilize the SC formation dynamics:

uj =usj + uf

j

= − λpj δqj − 2λd

jET (δqj)BT Pzj −m∑

n=1

λpjnqjn

−m∑

n=1

λdjn

(ET (qjn)BT P(Azjn + Bqnj) − R(qjn)ET (qnj)BT P(Aznj + Bqjn))

(41)

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Proof : It follows directly from Theorem 2, and can be shown by replacing δωj with ωj in the Lyapunovfunction (32) and its time derivative (33). �

Remark 1: Examples of formation architectures that can be constructed with four SC with guaranteedstability by using our proposed control laws are shown in Fig. 3. The ‘ring’ topology in shown in Fig. 3(a).In Fig. 3(b), a LF-like topology is shown where the attitude of all SC in the formation is available to SC1only while the desired attitude is available to all the SC in the formation. A ‘Minimal’ formation topologywhere the formation is constructed by requiring minimum information flows is shown in Fig. 3(c).

SC1 SC2

SC3 SC4

SC1 SC2

SC3 SC4

(a) (b)

SC1 SC2

SC3 SC4

(c)

Figure 3. Examples of different formation architectures that can be constructed by using our proposed controllaws with guaranteed asymptotic stability: (a) ‘ring’ topology; (b) LF-like topology; (c) minimal connectiontopology.

Remark 2: Other approaches for designing velocity-free controllers for the SC formation may be con-sidered. For example, one may try to use an observer to estimate the velocity similar to the one that isintroduced in Ref. 26. However, a separation principle-like property considered in this paper is conjecturedbut not shown formally. Thus, the closed-loop stability is not analyzed.14 In addition, the observer-basedcontroller introduced in Ref. 27 requires knowledge of the SC moment of inertia matrix and the reactionwheels (RW) angular velocities (if it is used in the AOCS subsystem), which makes the control law quitecomplex. The results obtained in Ref. 28 on the velocity-free quaternion-based tracking controller cannot beimplemented here either since only local stability of the closed-loop system was shown to be guaranteed.

VIII. SIMULATIONS

The performance of our proposed control law is demonstrated in simulations in this section. For thepurpose of simulations, we consider four SC in the formation with different initial attitudes and differentmoments of inertia. The attitude of the SC in the formation is available to all the SC in the formation.However, the SC have no information regarding their own and other SC angular velocities and momentsof inertia. The final formation angular velocity is assumed to be zero in the simulations. The controllerparameters are selected as:

λpj = 0.01, λd

j = 0.1, λpjn = 0.2, λd

jn = 0.6, for j, n = 1, 2, 3, 4 and j �= n (42)

which represents a ‘tight’ formation topology. We select higher gains for the formation-keeping behavior whencompared to the station-keeping behavior in order to emphasize on the requirement of formation-keeping.In addition, the matrices A and Q are chosen as:

A = −1 × 14×4, Q = 1.5 × 14×4

Based on the matrices selected above, the solution to the Lyapunov equation (22) is found as:

P = 0.75 × 14×4

The initial attitude error of the four SC in the formation are given by:

δq1(0) = [0, 0, 0.8, 0.6]T , δq2(0) = [0, 0, 0, 1]T ,

δq3(0) = [0, 0.2, 0, 0.9798]T , δq4(0) = [0.9, 0, 0, 0.4359]T

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The moments of inertia of the four SC are given below:

J1 = diag([10, 5, 15]),J2 = diag([8, 3, 20]),

J3 = diag([12, 7, 12]),J4 = diag([12, 10, 12]),

Note that the initial conditions for the linear dynamical systems zj , zjn, j, n = 1, 2, . . . , m, j �= n arechosen randomly between zero and one. Using the SC and the controller parameters as indicated above, thetime response of the SC in the formation are obtained. The error quaternions are shown in Fig. 4 duringthe first 400 seconds of the maneuver.

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 1

0 50 100 150 200 250 300 350 400−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

δ q 2

SC1SC2SC3SC4

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 3

time [sec]0 50 100 150 200 250 300 350 400

−0.5

0

0.5

1

δ q 4

time [sec]

Figure 4. Error quaternions for the ‘tight’ formation with four SC during the first 400 seconds.

As can be seen from this figure, the SC in the formation first align their attitudes. Subsequently, theytry to reduce the formation attitude error (station-keeping behavior is dominant in this period). This wasexpected according to the selected weights of the station-keeping and the formation-keeping behaviors in(42). It is interesting to note that due to the priority of the formation-keeping behavior, the second SC whichhas initial station-keeping error of zero moves away from its initial attitude and goes towards other SC inthe formation. This behavior makes the formation-keeping error zero, although increases the station-keepingerror. Approximately after 150 seconds, the formation-keeping maneuver ends (implying that the four SCare all in the same attitude and the formation attitude error is very close to zero). Consequently, the SC inthe formation try to reduce the formation attitude error as time evolves.

In order to demonstrate convergence of the SC formation to the desired attitude, the time response of thesystem is shown for 1800 seconds in Fig. 5, which clearly shows the convergence of the SC states. The controlefforts for the SC in the formation are shown in Fig. 6 for the first 400 seconds in three different channels. Itis clear that the control efforts magnitude are reasonable and converge to zero as the SC align their attitudeand the formation attitude error decreases. These results confirm that our analytical conclusions obtainedearlier are substantiated. It is interesting to note that in the simulations convergence of the closed-loopsystem states appears to be exponential which is an advantage of our designed controller.

In order to demonstrate the performance and stability of our proposed control laws when links amongthe SC are disconnected two scenarios are considered. In the first scenario, it is assumed that the secondand the third SC are no longer connected to each other, while all the other links are present. In this casewe have, λp

23 = λp32 = λd

23 = λd32 = 0. In the second scenario, we assume that the first and the fourth SC as

well as the second and the third SC are not connected. This essentially realizes the ring topology as shown

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0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.5

0

0.5

1

δ q 1

0 200 400 600 800 1000 1200 1400 1600 1800−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

δ q 2

SC1SC2SC3SC4

0 200 400 600 800 1000 1200 1400 1600 1800−1

−0.5

0

0.5

1

δ q 3

time [sec]0 200 400 600 800 1000 1200 1400 1600 1800

−0.5

0

0.5

1

δ q 4

time [sec]

Figure 5. Error quaternions for the ‘tight’ formation with four SC during the first 1800 seconds.

0 50 100 150 200 250 300 350 400−1.5

−1

−0.5

0

0.5

1

u 1 [N.m

]

Control effort vs. time

SC1SC2SC3SC4

0 50 100 150 200 250 300 350 400−0.2

0

0.2

0.4

0.6

u 2 [N.m

]

0 50 100 150 200 250 300 350 400−1.5

−1

−0.5

0

0.5

1

u 3 [N.m

]

time [sec]

Figure 6. Control effort for the four SC in the ‘tight’ formation during the first 400 seconds.

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0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 1

0 50 100 150 200 250 300 350 400−0.4

−0.2

0

0.2

0.4

0.6

δ q 2

SC1SC2SC3SC4

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 3

time [sec]0 50 100 150 200 250 300 350 400

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

δ q 4

time [sec]

Figure 7. Error quaternions for the formation with four SC with no connection between the second and thethird SC during the first 400 seconds.

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 1

0 50 100 150 200 250 300 350 400−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

δ q 2

SC1SC2SC3SC4

0 50 100 150 200 250 300 350 400−1

−0.5

0

0.5

1

δ q 3

time [sec]0 50 100 150 200 250 300 350 400

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

δ q 4

time [sec]

Figure 8. Error quaternions for the ‘ring’ formation with four SC during the first 400 seconds.

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in Fig. 3(a). The time responses of the SC in the formation are shown in Fig. 7 and Fig. 8, respectively. Insimulations, it can be observed that the response of the SC in the formation does not change significantlyby loosing connections between SC and the formation-keeping error, in both cases, converge to zero almostat the same time as it converged in the ‘tight’ formation architecture (around 200 seconds). However, wehave noticed rapid responses and early misalignment among the SC that are not connected to each other inthe latter case (refer to the first 100 seconds in Fig. 8 and compare it with the first 100 seconds of Fig. 4).The significance is that transient delays are introduced in the attitude alignment for a few seconds at thebeginning of the maneuver which is not surprising.

IX. CONCLUSIONS

This paper presents a decentralized approach for coordinated attitude maneuver for SC formation withoutrequiring and feeding back angular velocity information. The proposed algorithm is specially useful when theinformation exchanges among the SC in a formation is limited, i.e. when only attitude position informationexchanges are possible among the SC. The proposed algorithm can also be used when one or more of the SCrate gyroscopes, which are commonly used for measuring SC angular velocity, has failed and the SC angularvelocity information is no longer available. The global asymptotic stability of the SC formation by using ourproposed control algorithm is guaranteed through Lyapunov analysis and LaSalle’s theorem. The stabilityanalysis is also shown to assist one in determining the minimum of information exchanges among the SCwhile guaranteeing the asymptotic stability of the formation. It is also important to emphasize that in ourproposed method, the desired (final) angular velocity of the SC formation does not need to be zero. Wehave shown that a simplified version of the proposed control algorithm can be employed when the desiredangular velocity of the SC formation is zero (as desired in deep space interferometry missions). In this case,the proposed control strategy is model independent in that it does not require knowledge of the SC momentsof inertia matrices.

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