ramirez-riberos et al

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Distributed Control of Spacecraft Formations via Cyclic Pursuit: Theory and Experiments Jaime L. Ramirez-Riberos, Marco Pavone, Emilio Frazzoli, and David W. Miller §  Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 DOI: 10.2514/1.46511 In this paper, distributed control policies for spacecraft formations that draw inspiration from the simple idea of cyc licpursui t arestudie d. Fir st studie d arecycli c-purs uitcontro l laws forboth sin gle - and double -integrat or mod els in three dimensions. In particular, control laws are developed that only require relative measurements of position and vel oci ty wit h res pec t to the two leadi ng nei ghbors in the ring topolo gy of cyc lic pursuit and tha t all ow convergence to a variety of symmetric formations, including evenly spaced circular and elliptic formations and evenly spaced Archimedes spirals. Second, potential applications are discussed, including spacecraft formation for interf erome tric imagi ng.Finally,experiment al results obtai ned by imple mentin g the aforementionedcontrol lawson the Synchronized Position Hold Engage Reorient Experimental Satellite testbed onboard the International Space Station are presented and discussed. I. Int rod uct ion I N RECENTyears, the idea of distributing the functionalities of a complex agent among multiple, simple and cooperative agents is attracting increasing interest in several application domains. In fact, mul ti- agen t sys tems pres ent sev eral adv ant ages . For exa mpl e, con siderin g an aero spa ce app lication, a clu ste r of spa cecr aft yin g in formation for high-resolution, synthetic-aperture imaging can act as a sparse aperture with an effective dimension larger than the one that can be achieved by a single larger satellite [ 1]. In general, the intrinsic parallelism of a multi-agent system provides robustness to failures of single agents; moreover, it is possible to reduce the total implementation and operation cost, increase reactivity and system rel iab ili ty , and add exi bil ity and mod ula rit y to mon oli thi c approaches. In this context, the problem of formation of geometric patterns is of par tic ula r int eres t, wit h eng ine erin g app lic ati ons inc lud ing dis tri but ed sensing usi ng mob ile sensor net wor ks, and spa ce missions with multiple spacecraft ying in formation (on which we wil l foc us the paper). Wi thi n the rob otics community, man y dist ribute d contro l strate gies have been recentl y propo sed for conv er- gence to geometric patterns. Justh and Krishnaprasad [ 2] presented two str ate gie s to achi eve , res pec tiv ely , rect ili nea r and cir cul ar formation, using all-to-all communication among agents. Jadbabaie et al. [3] formally proved that the nearest neighbor algorithm by Vicsek et al. [4] causes all agents to eventually move in the same dir ecti on, des pit e theabsenc e of cen tralizedcoordina tio n and des pit e the fact that each agent s set of nearest neighbors changes with time as the system evolves. Olfati-Saber and Murray [ 5] used potential- function theory to prescribe ocking behavior in connected graphs. On a similar line, Sepulchre et al. [ 6] proposed a potential-function metho dolog y to stabilize parall el formations or circula r format ions of unicy cle agents with constant speed. The agents exchange relative information under a limited communication topology. The conver- gence results are, however, only local. Lin et al. [ 7] exploited cyclic pursuit (where each agent i pursues the next i 1, modulo n) to achi evealign ment amo ng agen ts,whileMarsha ll et al. [8,9] exten ded the classic cyclic pursuit to a system of wheeled vehicles, each sub jectto a sin gle non holono mic con str ain t, andstudie d the pos sible equ ili bri um formati onsand the ir sta bil ity . Pal ey et al. [10] introd uced control strategiesto stabi lize symmetric format ions on con vex closed curves. Ren [11] int roduced Cart esi an coo rdi nat e coupli ng to existing consensus algorithms and derived algorithms for different types of collective motions (a more detailed discussion about the relation between our work and the work of Ren is provided in Secs. III.A and IV.C. Theproblemofformationofgeometricpatternshasbeenthesubject of inten siveresearch effor ts alsowithin the aerosp ace community, see the workby ScharfandPl oen [12] andreferen ces the rein ; see als o the very recent work of Chung et al. [13], where the authors present a contractiontheoryapproachtoachievesynchronizedformations.This app roa ch, howev er , req uir es a for m of hig her le velagreementon a set of predetermined trajectories. In an authoritative survey paper on formation yi ng, Scharf and Pl oen [12] pro pos e a di vis ion of formation yin g arch itectures int o thr ee mai n clas ses: namely, 1) multi ple-in put/multip le-ou tput (MIMO), in whichthe formationis treated as a single multiple-input, multiple-output plant; 2) leader/ follower, in which individual spacecraft controllers are connected hierarchically; and 3) cyclic, in which individual spacecraft con- trollers are connected nonhierarchically. According to Scharf and Pl oen [12], by all owi ng non hie rarc hic al con nect ions bet ween individualspacecraftcontrollers,cyclicalgorithmscanperformbetter than leader/ follo wer algori thmsand can distri bute contro l effort more evenly. Moreover, cyclic algorithms are generally more robust than MIMO algorithms, for which a local failure can have a global effect [12]. Cyclic algorithms can also be completely decentralized in the sensethat thereis neithe r a coordi natingagent nor insta bilit y result ing fromsingl e-po int fail ures[ 12]. The two pri mar y dra wba cks of cyc lic alg orithms are tha t the sta bil ity of these alg orithms and the ir information requirements are poorly understood [ 12]; in particular, the stability analysis of cyclic algorithms is dif cult, since the cyclic structure introduces feedback paths. Motivated by the pre vio us dis cus sion, the objective of this pap er is to pres ent a cla ss of cyc licalgor ith ms for for mat ionigh t, forwhicha rigorous stability analysis is possible and for which the information requirements are minimal. The starting point is our previous work [14], in which we developed distributed control policies that draw inspiration from the simple idea of cyclic pursuit and that guarantee Received 28 July 2009; revision received 4 March 2010; accepted for publication 25 March 2010. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper maybe made forpersonal or int ernal use , on con dit ion t hat the copi er pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in correspondence with the CCC. Ph.D. Candi date, Departme nt of Aeron autic s and Astr onauti cs, 77 Massachusetts Avenue, Room 37-351; [email protected]. Student Member AIAA. Ph.D. Candidate, Department of Aeron autics and Astronautics, 77 Massachusetts Avenue, Room 32-D712. Student Member AIAA. Associate Professor, Department of Aeronautics and Astronautics, 77 Massachusetts Avenue, Room 33-332. Senior Member AIAA. § Professor and Director of the Space Systems Laboratory, Department of Aeron autic s and Astro nauti cs, 77 Massa chuse tts Av enue, Room 37-32 7. Senior Member AIAA. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 33, No. 5, September October 2010 1655

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Distributed Control of Spacecraft Formationsvia Cyclic Pursuit: Theory and Experiments

Jaime L. Ramirez-Riberos,∗ Marco Pavone,† Emilio Frazzoli,‡ and David W. Miller §

 Massachusetts Institute of Technology, Cambridge, Massachusetts 02139DOI: 10.2514/1.46511

In this paper, distributed control policies for spacecraft formations that draw inspiration from the simple idea of 

cyclicpursuit arestudied. First studied arecyclic-pursuitcontrol laws forboth single- and double-integrator models

in three dimensions. In particular, control laws are developed that only require relative measurements of position

and velocity with respect to the two leading neighbors in the ring topology of cyclic pursuit and that allow

convergence to a variety of symmetric formations, including evenly spaced circular and elliptic formations and

evenly spaced Archimedes spirals. Second, potential applications are discussed, including spacecraft formation for

interferometric imaging.Finally,experimental results obtained by implementing the aforementioned control lawson

the Synchronized Position Hold Engage Reorient Experimental Satellite testbed onboard the International Space

Station are presented and discussed.

I. Introduction

I N RECENTyears, the idea of distributing the functionalities of a complex agent among multiple, simple and cooperative agents is

attracting increasing interest in several application domains. In fact,multi-agent systems present several advantages. For example,considering an aerospace application, a cluster of spacecraft flying information for high-resolution, synthetic-aperture imaging can act asa sparse aperture with an effective dimension larger than the onethat can be achieved by a single larger satellite [1]. In general, theintrinsic parallelism of a multi-agent system provides robustness tofailures of single agents; moreover, it is possible to reduce the totalimplementation and operation cost, increase reactivity and system reliability, and add flexibility and modularity to monolithicapproaches.

In this context, the problem of formation of geometric patterns isof particular interest, with engineering applications includingdistributed sensing using mobile sensor networks, and spacemissions with multiple spacecraft flying in formation (on which wewill focus the paper). Within the robotics community, manydistributed control strategies havebeen recently proposed for conver-gence to geometric patterns. Justh and Krishnaprasad [2] presentedtwo strategies to achieve, respectively, rectilinear and circular formation, using all-to-all communication among agents. Jadbabaieet al. [3] formally proved that the nearest neighbor algorithm byVicsek et al. [4] causes all agents to eventually move in the samedirection, despite theabsence of centralizedcoordination and despitethe fact that each agent ’s set of nearest neighbors changes with timeas the system evolves. Olfati-Saber and Murray [5] used potential-function theory to prescribe flocking behavior in connected graphs.On a similar line, Sepulchre et al. [6] proposed a potential-function

methodology to stabilize parallel formations or circular formations of unicycle agents with constant speed. The agents exchange relativeinformation under a limited communication topology. The conver-gence results are, however, only local. Lin et al. [7] exploited cyclicpursuit (where each agent  i pursues the next  i 1, modulo n) toachievealignment among agents,whileMarshall et al. [8,9] extendedthe classic cyclic pursuit to a system of wheeled vehicles, eachsubjectto a single nonholonomic constraint, and studied the possibleequilibrium formationsand their stability. Paley et al. [10] introducedcontrol strategies to stabilize symmetric formations on convex closedcurves. Ren [11] introduced Cartesian coordinate coupling toexisting consensus algorithms and derived algorithms for different types of collective motions (a more detailed discussion about therelation between our work and the work of Ren is provided in

Secs. III.A and IV.C.Theproblemofformationofgeometricpatternshasbeenthesubject of intensiveresearch efforts alsowithin the aerospace community, seethe workby Scharfand Ploen [12] andreferences therein; see also thevery recent work of Chung et al. [13], where the authors present a contractiontheoryapproachtoachievesynchronizedformations.Thisapproach, however, requires a form of higher levelagreementon a set of predetermined trajectories. In an authoritative survey paper onformation flying, Scharf and Ploen [12] propose a division of formation flying architectures into three main classes: namely,1) multiple-input/multiple-output (MIMO), in whichthe formation istreated as a single multiple-input, multiple-output plant; 2) leader/ follower, in which individual spacecraft controllers are connectedhierarchically; and 3) cyclic, in which individual spacecraft con-trollers are connected nonhierarchically. According to Scharf and

Ploen [12], by allowing nonhierarchical connections betweenindividualspacecraftcontrollers,cyclicalgorithmscanperformbetter than leader/follower algorithmsand can distributecontrol effort moreevenly. Moreover, cyclic algorithms are generally more robust thanMIMO algorithms, for which a local failure can have a global effect [12]. Cyclic algorithms can also be completely decentralized in thesensethat thereis neither a coordinatingagent nor instability resultingfromsingle-point failures[12]. The two primary drawbacks of cyclicalgorithms are that the stability of these algorithms and their information requirements are poorly understood [12]; in particular,the stability analysis of cyclic algorithms is dif ficult, since the cyclicstructure introduces feedback paths.

Motivated by the previous discussion, the objective of this paper isto present a class of cyclicalgorithms for formationflight, forwhicha rigorous stability analysis is possible and for which the informationrequirements are minimal. The starting point is our previous work [14], in which we developed distributed control policies that drawinspiration from the simple idea of cyclic pursuit and that guarantee

Received 28 July 2009; revision received 4 March 2010; accepted for publication 25 March 2010. Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper maybe made forpersonal or internal use, on condition that the copier pay the$10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 incorrespondence with the CCC.

∗Ph.D. Candidate, Department of Aeronautics and Astronautics, 77Massachusetts Avenue, Room 37-351; [email protected]. Student Member AIAA.

†Ph.D. Candidate, Department of Aeronautics and Astronautics, 77Massachusetts Avenue, Room 32-D712. Student Member AIAA.

‡Associate Professor, Department of Aeronautics and Astronautics, 77

Massachusetts Avenue, Room 33-332. Senior Member AIAA.§Professor and Director of the Space Systems Laboratory, Department of Aeronautics and Astronautics, 77 Massachusetts Avenue, Room 37-327.Senior Member AIAA.

JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS

Vol. 33, No. 5, September –October 2010

1655

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convergence to symmetric formations. The key features of thecontrol laws we proposed in [14] are global stability and thecapability to achieve a variety of formations: namely, rendezvousto a single point (circles and logarithmicspirals); moreover, thosecontrollaws are distributed and require the minimum number of communication links (n links for  n agents) that a cyclic structurecan have.

Specifically, the main contributions of this paper are threefold.First, building upon previous work on cyclic-pursuit algorithms, we

rigorously study cyclic distributed control laws for formationfl

ying,for both single- and double-integrator models in three dimensions.We also extend our control laws to deal with the (linearized) relativedynamics of spacecraft: e.g., in the Earth’s gravitational field. A keyfeature of our control laws is that, unlike those of Chung et al. [13],they do not require any agreement on a set of predeterminedtrajectories. Second, we discuss potential applications, includingspacecraft formation for interferometric imaging. Finally, we present and discuss experimental results ¶  obtained by implementing theaforementioned control laws on the Synchronized Position HoldEngage Reorient Experimental Satellite (SPHERES) testbed∗∗

onboard the International Space Station.The organizationof this paper is as follows.In Sec.II we introduce

basic concepts in matrix theory and we briefly review our previouscyclic-pursuit control laws [14], which were devised for single-

integrator models in two dimensions. In Sec. III we extend our previous results along two directions:

1)InSec. III.Aweaddress the case inwhichagentsmove inR3 andit is desired to control the center of the formation.

2) In Sec. III.B we study robust convergence to evenly spacedcircular formations with a prescribed radius.

In Sec. IV we extend our control laws to double-integrator modelsin three dimensions. In particular, we develop control laws that onlyrequire relative measurements of position and velocity with respect to the two leading neighbors in the ring topology of cyclic pursuit,and allow the agents to converge from any initial condition (except for a set of measure zero) to a single point, an evenly spaced circular formation, an evenly spaced logarithmic spiral formation, or anevenly spaced Archimedes spiral formation (an Archimedes spiral isa spiral with the property that successive turnings have a constant separation distance), depending on some tunable control parameters.Control laws that only rely on relative measurements are indeed of critical importance in deep-space missions, in which globalmeasurements may not be available. In Sec. V we discuss potentialapplications, including spacecraft formation for interferometricimaging and convergence to zero-effort orbits, and we argue that Archimedes spiral formations are very useful symmetric formationsfor applications. In Sec. VI we present and discuss experimentalresults obtained by implementing the proposed control laws on threenanospacecraft onboard the International Space Station (ISS), and inSec. VII we draw our conclusions.

II. Background

In this section, we provide some definitions and results from matrix theory. Moreover, we briefly review cyclic-pursuit controllaws for single integrators.

A. Notation

We let R>0 and R0 denote the positive and nonnegative realnumbers, respectively. We let I n denote the identity matrix of size n;we let  AT  and A denote, respectively, the transpose and theconjugate transpose of a matrix A. For an n n matrix A, we let eig A denote the set of eigenvalues of  A, and we refer to its ktheigenvalue as  A;k, k 2 f1; . . . ; ng (or simply as k when there is no

possibility of confusion). Finally, let j :1

p .

B. Kronecker Product 

Let A and B be m n and p q matrices, respectively. Then theKronecker product A B of A and B is the mp nq matrix:

 A B a11B . . . a1nB

..

. ...

am1B . . . amnB

264

375

If  A

is an eigenvalue of  A with associated eigenvector  A

and B

isaneigenvector of  B with associated eigenvector  B, then  AB is aneigenvalue of  A B with associated eigenvector  A B. Moreover,the following property holds:  A BC D  AC BD, where A, B, C, and D are matrices with appropriate dimensions.

C. Determinant of Block Matrices

If A, B, C and D are matrices of size n n, and AC CA, then

det

A B

C D

det AD CB (1)

D. Rotation Matrices

A rotation matrix is a real square matrix whose transpose is equalto its inverse and whose determinant is 1. The eigenvalues of a rotation matrix in two dimensions aree j, where is the magnitudeof the rotation. The eigenvalues of a rotation matrix in threedimensions are 1 and e j, where is the magnitude of the rotationabout the rotation axis; for a rotation about the axis 0; 0; 1T , thecorresponding eigenvectors are 0; 0; 1T , 1;  j; 0T 1;  j; 0T .

E. Circulant Matrices

A circulant matrix C is an n n matrix having the form 

C

c0 c1 c2 . . . cn1

cn

1 c0 c1 . . . ..

.

... c1

c1 c2 . . . . . . c0

266664

377775 (2)

The elements of each row of C are identical to those of the previousrow, but are shifted one position to the right and wrapped around. Adetailed treatise on circulant matrices can be found in the work byGray[15]. Thefollowing theoremsummarizes some of thepropertiesof circulant matrices and will be essential in the development of thepaper.

Theorem 1 (adapted from Theorem 7 in [15]): Every n ncirculant matrix C has eigenvectors

k

1 np 

1; e2jk=n; . . . ; e2jkn1=n

T ; k

2 f0; 1; . . . ; n

1

g(3)

and corresponding eigenvalues

k Xn1

p0

cpe2jkp=n (4)

and can be expressed in the form C U U , where U  is a unitarymatrix whosekth columnis the eigenvector k,and is the diagonalmatrix whose diagonal elements are the corresponding eigenvalues.Moreover, let C and B be n n circulant matrices with eigenvaluesfB;kgn

k1 and fC;kgnk1, respectively; then

1) C and B commute; that is, CB

BC, and CB is also a circulant 

matrix with eigenvalues

eig CB fC;kB;kgnk1

 ¶ 

Data available online at  http://ssl.mit.edu/spheres/video/CyclicPursuit [retrieved 17 May 2010].∗∗Data available online at  http://ssl.mit.edu/spheres/  [retrieved

17 May 2010].

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2) C B is a circulant matrix with eigenvalues

eig C B fC;k B;kgnk1

From Theorem 1 all circulant matrices share the same eigen-vectors, and the same matrix U diagonalizes all circulant matrices.

F. Cyclic Pursuit for Single Integrators

In this section we review the cyclic-pursuit control laws wedevised in [14]. Let there be n ordered mobile agents in the plane,with their positions at time t 0 denoted by

x it  xi;1t; xi;2tT  2 R2

(i 2 f1; 2; . . . ; ng), where agent i pursues the next i 1 modulo n.The dynamics of each agent are described by a simple (vector)integrator [14]:

_x i ui; ui Rxi1 xi (5)

where the dot represents differentiation with respect to time, andR, 2 ; , is the rotation matrix:

R cos sin sin cos

Note that this control law is naturally amenable to distributed implementation. Let  x xT 

1 ;xT 2 ; . . . ;xT 

n T ; the dynamics of theoverall system can be written in compact form as

_x L Rxwhere L is the circulant matrix:

L 1 1 0 0

0 1 1 0

..

. ...

1 0 0 1

2664

3775 (6)

In our previous work [14] we have proven the following:

Theorem 2: L R has exactly two zero eigenvalues, and1) If 0 jj < =n, all nonzero eigenvalues lie in the open left-half complex plane.

2)If jj =n, twononzero eigenvalueslie on theimaginaryaxis,while all other nonzero eigenvalues lie in the open left-half complexplane.

3) If =n < jj < 2=n, two nonzero eigenvalues lie in the openright-half complex plane, while all other nonzero eigenvalues lie inthe open left-half complex plane.

The following theorem, proven in thework of Pavone andFrazzoli[14], characterizes the formations that can be achieved with controllaw (5).

Theorem 3: A team of agents starting at any initial condition(except for a set of measure zero) inR2n and evolving under Eq. (5)exponentially converge:

1) If 0 jj < =n, to a single limit point.2) If jj =n, to an evenly spaced circular formation.3) If  =n < jj < 2=n, to an evenly spaced logarithmic spiral

formation.The center of the formation is the initial center of mass of the

agents.The key features of the cyclic-pursuit control law (5) are global

stability and the capability to achieve in a  distributed  fashion a variety of formations: namely, rendezvous to a single point (circlesand logarithmic spirals). As satisfying as these results are, they stillhave several limitations, as follows:

1) They are derivedfor agents operating inR2 and the center of theformation is determined by the initial positions of the agents.

2) Circular formations occur when two eigenvalues are on theimaginary axis (which is nonrobust from a practical point of view)and their radius is also determined by the initial positions of theagents.

3) The agents have a single-integrator dynamics.

III. Cyclic-Pursuit Control Lawsfor Single-Integrator Models

In this section, we extend the results of Pavone and Frazzoli [14]along two directions:

1)InSec. III.Aweaddress the case inwhich agentsmove inR3 andit is desired to control the center of the formation.

2) In Sec. III.B we study robust convergence to evenly spacedcircular formations with a prescribed radius.

We will consider double-integrator dynamics in Sec. IV.

A. Cyclic Pursuit in 3-D with Control on the Center of the Formation

Let therebe n orderedmobile agents in the space, their positions at time t 0 denoted by

x it  xi;1t; xi;2t; xi;3tT  2 R3; i 2 f1; 2; . . . ; ng

and let  x xT 1 ;xT 

2 ; . . . ;xT n T . The dynamics of each agent are

described by a simple vector integrator:

_x i kgui; kg 2 R>0 (7)

Henceforth, without loss of generality, we assume kg 1. Consider the following generalized cyclic-pursuit control law:

u i Rxi1 xi kcxi; kc 2 R0 (8)

where the indices are evaluated modulo n (henceforth, agent indicesshould always be evaluated modulo n) and whereR, 2 ; ,is the rotation matrix (with rotation axis 0; 0; 1T , assuming anagreement on the direction of the z axis among the agents):

R cos sin 0

sin cos 0

0 0 1

0@

1A (9)

Note that control law (9), as before, is naturally amenable todistributed implementation. The overall system can be written incompact form as

_x L R kcI 3nx (10)

where L isdefinedinEq.(6). When kc 0, the cyclic-pursuit controllaw in Eq. (8) reduces, given a ring topology, to the control lawproposed by Ren [16] (which, in turn, is a generalization of our control law in [14]). The following theorem (adapted from Theorem 3.2 in [16]) characterizes the spectrum of  L R, and alsodescribes the formations that can be achieved under the control law(8) when kc 0. Note that when kc 0 the center of the formation isthe initial center of mass of the agents (as it can be easily verified).After the presentation of this theorem, we will consider the casekc > 0, and we will show how in this case the center of the formationcan be controlled to the origin. In the following, by evenly spacedcircular formation we mean that all agents follow circular trajectorieson a circle with identical radius and center and which lies on a common plane; moreover, all agents are evenly spaced on the circle.Evenly spaced spiral formations are defined similarly.

Theorem 4 (adapted from [16]): L R is diagonalizable for all 2 ; , and its eigenvalues and corresponding eigenvectors aregiven by

k k 1; k 0; 0; 1; 0; 0; k; . . . ; 0; 0; n1k T 

k k 1e j

k 1; j; 0; k; jk; 0; . . . ; n1

k ; jn1k ; 0T 

k

k

1

e j

k 1;  j; 0; k;  jk; 0; . . . ; n1

k ;  jn1k ; 0T 

k 2 f1; . . . ; ng

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where k e2jk=n. L R has exactly three zero eigenvalues,and

1) If 0 jj < =n, all nonzero eigenvalues lie in the open left-half complex plane.

2)If jj =n, twononzero eigenvalueslie on theimaginaryaxis,while all other nonzero eigenvalues lie in the open left-half complexplane. The two eigenvalues on the imaginary axis are  j2sin=n, with corresponding eigenvector 

n1, and  j2sin=n, with corresponding eigenvector 

1 .

3) If =n < jj < 2=n, two eigenvalues lie in the open right-half complex plane, while all other nonzero eigenvalues lie in theopen left-half complex plane. The two eigenvalues in the open right-half complex plane have real parts both equal to 2 sin=nsin =n, and have corresponding eigenvectors

n1 and 1 .

The agents starting at any initial condition (except for a set of measure zero) in R3n and evolving under Eq. (10) with kc 0exponentially converge:

1) If 0 jj < =n, to a single limit point: namely, their initialcenter of mass.

2) If jj =n, to an evenly spaced circular formation, whoseradius is determined by the initial positions of the agents.

3) If  =n < jj < 2=n, to an evenly spaced logarithmic spiralformation.

The center of the formation is the initial center of mass of the

agents. Remark 1: When kc 0, the control law in Eq. (8) only requires

themeasurement of therelative positionxi1 xi; however, it usesa rotation matrix that is common to all agents. Hence,control law (9)requires that all agents agree upon a common orientation, but it doesnot require a consensus on a common origin.

We now turn back our attention to the case kc > 0. Inthis case, thecenter of the formation is no longer determined by the initialpositions of the spacecraft, instead it always converges,exponentially fast, to the origin. In fact, when kc > 0 the eigenvaluesof  L R are shifted toward the left-hand complex plane by anamount precisely equal to kc, while the eigenvectors are left unchanged . Specifically, the following theorem is a simpleconsequence of Theorem 3.

Theorem 4: Assume kc > 0; then if  0 jj =n, all of theeigenvalues are in the left-hand complex plane. If, instead, =n <

jj < 2=n we have the following:1) If kc > 2sin=n sin =n, all of the eigenvalues are in

the open left-hand complex plane.2) If kc 2sin=n sin =n, two nonzero eigenvalues lie

on the imaginaryaxis, while all other eigenvalues lie in the open left-hand complex plane.

3)If kc < 2 sin=n sin =n, twononzero eigenvalues lieinthe open right-hand complex plane, while all other eigenvalues lie inthe open left-hand complex plane.

Accordingly, by appropriately selecting and kc (recall that theeigenvectors are left unchanged), the agents, starting at any initialcondition(except fora setof measure zero) inR3n and evolvingunder Eq.(10), exponentially converge to theorigin(case 1 in Theorem4 or 

when 0 jj =n), or to an evenly spaced circular formationcentered at the origin (case 2 in Theorem 4), or to an evenly spacedlogarithmic spiral formation centered at the origin (case 3 inTheorem 4). Simulation results are presented in Fig. 1, in whichseven agents reach a circular formation centered at the origin.

 Remark 2: When kc > 0, the control law inEq.(8) requires that theagents agree on a common reference frame (i.e., both a commonorigin and a common orientation); in particular, each agent needs tomeasure its relative position (xi1 xi) and know its absolute

position xi. Remark 3: Notethatthe center ofthe formation can bechosento beany point inR3. Assume, in fact, that we desire a formation centeredat xc 2 R3. Then if we modify the control law (8) according to

u i Rxi1 xi kcxi xc; kc 2 R>0

it is immediate to see that the center of the formation will convergeexponentially to xc.

B. Convergence to Circular Formations with a Prescribed Radius

Circular trajectoriesfor the systemin Eq.(5) occur onlyfor a valueof , such that twononzero eigenvalues are on theimaginaryaxis andall other nonzero eigenvalues have negative real part, which makesthis behavior not robust from a practical point of view. In this section

we address theproblem of robust convergence toa circular motionona circle of prescribed radius around the (fixed) center of mass of thegroup, with all agents being evenly spaced on the circle. Here, byrobust  we mean that the circular formation is now a locally stableequilibrium of a nonlinear system. The key idea is to make therotation angle a function of the state of the system.

Specifically, let there be n orderedmobile agents in the plane, their positions at time t 0 denoted by xit  xi;1t; xi;2tT  2 R2

(i 2 f1; 2; . . . ; ng), where agent i pursues the next  i 1 modulo n.The kinematics of each agent is described by

_x i kgui; ui Rixi1 xi (11)

where the rotation angle i is now a function of the state of thesystem:

i

n kr kxi1 xik; k; r 2 R>0 (12)

Without loss of generality, we assume kg 1. In Eq. (12) theconstant  k is a gain, while r is the desired interagent distance.Intuitively, if the agents are close to each other with respect tor, theywill spiral out, since i > =n; conversely, if they are far from eachother with respectto r, theywillspiralin, since i < =n.Itiseasytosee that a splay-state formation (this term has been introduced in thework of Paley et al. [17]), whereby all agents move on a circle of radius r=2sin=n around the (fixed) center of mass of the group,with all agents being evenly spaced on the circle, is a  relativeequilibrium for the system. The next theorem shows that suchequilibrium is locally stable.

0 2 4 6 8 10−20

−10

0

10

20

30

40

t

  x

−10

0

10

20

30

−10

0

10

−5

0

5

Vehicle trajectories

Fig. 1 Convergence to circular trajectoriescenteredat theorigin; left:firstcoordinate as a function of time foreach agent andright: trajectoriesin 3-D.

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Theorem 5: A splay-state formation is a locally stable relativeequilibrium for system (11) and (12).

Proof: Wefirst consider a sequence of coordinate transformationssuch that a splay-state formation is indeed an equilibrium point (andnot a  relative equilibrium). Consider the change of coordinatespi : xi1 xi (i 2 f1; 2; . . . ; ng). In the new coordinates, thesystem becomes (the index i is, as usual, modulo n)

_p i Ri1pi1 Ripi (13)

where

i

n kr kpik

By introducing polar coordinates, i.e., by letting the first coordinatepi;1 %i cos #i and the second coordinate pi;2 %i sin #i, with %i 2R0 and i 2 R, the system becomes, after some algebraicmanipulations (see Appendix A for the details),

_% i %i1 cos#i1 #i i1%i1 %i cosi%i (14)

_# i %i1

%i

sin#i1 #i i1%i1 sini%i (15)

i%i

n kr %i (16)

where we have made explicit the dependence of  i on %i. By letting’i #i1 #i, we obtain

_%i %i1 cos’i i1%i1 %i cosi%i_’i %i2

%i1

sin’i1 i2%i2 sini1%i1

%i1

%i

sin’i i1%i1 sini%i

i%i

n kr %i

Define % : %1; . . . ; %nT  and ’ : ’1; . . . ; ’nT ; in the newsystem of coordinates %-’, a splay-state formation corresponds to an

equilibrium point % r; . . . ; rT  and

’ 2

n; . . . ;

2

n; 2n 1

nT 

In compact form, we write

_%_’

 f %; ’ (17)

The linearization of system (17) around the equilibrium point %; ’ is

_%i cos=n%i1 %i kr sin=n%i1 %i r sin=n’i

_’i

k cos=n 1

rsin=n

%i2 2%i1 %i

cos=n’i1 ’i

Without loss of generality we set r 1; the linearized system can bewritten in compact form as

_%_’

anL 2ksnI n snI n

bnL2 cnL

%’

: P

%’

where sn : sin=n, cn : cos=n, an : cn ksn, bn :kcn sn, and L is defined in Eq. (6). The spectrum of  P ischaracterized by the following Lemma.

 Lemma 1: The matrix P has 2n 3 eigenvalues with negative realpart, and three eigenvalues with zero real part. The eigenvalues withzero real part are 1 0, and 2;3 2 jsn; the correspondingeigenvectors v1, v2, and v3 are

v1 1n; 2k1nT ; v2 1; 2bnej=n1T ; v3 v2

where 1n 1; 1; . . . ; 1T  2 Rn, 1 is the eigenvector for k 1 inEq. (3) and v indicates the complex conjugate of v.

Proof: The proof of this lemma is presented in Appendix B. □

System(17) isconstrained toevolve ona subsetof R2n.Toseewhythis is the case, recall that from the definition of pi we haveXn

i1

pi 0

or, equivalently, Xn

i1

R#1pi 0

In polar coordinates, these constraints become

Xn

i1

%i cos#i #1 0

and Xn

i1

%i sin#i #1 0

Thus, in the system of coordinates %-’, the following two constraintsmust hold at all times:

g1%; ’ Xn

i1

%i cos

Xi1

k1

’k

0; g2%; ’

Xn

i1

%i sin

Xi1

k1

’k

0

Moreover, by definition of ’, thefollowing constraint must hold at alltimes:

g3%; ’ Xn

i1

’i 0

Let g%; ’ g1%; ’; g2%; ’; g3%; ’T  and define

M : f%; ’ 2 R2n: g%; ’ 0g R2n

Note that  %; ’ 2M. The Jacobian of  g%; ’ evaluated at theequilibrium point is

G

1 cos2=n . . . cos2n 1=n P

ni2 r sin2i 1=n . . . r sin2n 1=n 0

0 sin2=n

. . . sin

2

n

1=n

Pn

i2

r cos

2i

1=n

. . . r cos

2

n

1=n

0

0 0 . . . 0 1 . . . 1 10@ 1A

Let  B%; ’ be the open ball of radius > 0 centered at point %; ’ in R2n. The rank of G is clearly 3; then there exists > 0,

such that  ~M : M\ B%; ’ R2n is a submanifold of R2n.The

tangent space of  ~M at  %; ’, which we call T %;’ ~M, is an

invariant subspace of P [since ~M, by construction, is invariant under 

Eq. , i.e., f %; ’ 2 T %;’ ~M for all %; ’ 2 ~M] and has dimension

2n 3. Picka basis fw1; . . . ; w2n3g of T %;’ ~M and complete it toa basis W  of R2n. Then with respect to this basis, P takes the upper-triangular form:

P1;1 P1;2

032n3 P2;2

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where 032n3 is the zero matrix with three rows and 2n 3

columns. Since oursystem is constrained to evolve, at %; ’, along

the tangent space T %;’ ~M, the local stability of the equilibrium point is solely determined by the eigenvalues of P1;1.

We next show that the three eigenvalues of  P2;2 are exactly thethreeeigenvalues of P thathave realpartequalto zero.It ispossible toshow (see Appendix C) that 

G vi ≠ 0 for each i 2 f1; 2; 3g

where vi, i 2 f1; 2; 3g, are the three eigenvectors associated to thethree eigenvalues with zero real part. Therefore, we have

viT %;’ ~M, i 2 f1; 2; 3g. Let  yi be the components of  vi withrespect to the basis W ; define yi;1 as the vector of components withrespect to fw1; . . . ; w2n3g, and yi;2 as the vector of components

with respect to the remaining basis vectors in W . Since viT %;’ ~M,vector  yi;2 is nonzero. Since vi is an eigenvector of  P with eigen-value i, we can write

P1;1 P1;2

032n3 P2;2

yi;1

yi;2

i

yi;1

yi;2

and therefore P2;2yi;2

iyi;2; i.e., i is an eigenvalue of  P2;2,

i 2 f1; 2; 3g. Since we have eigP1;1 eigP n eigP2;2, weconclude, by using Lemma 1 that all eigenvalues of  P1;1 havenegative real part. Therefore, the equilibrium point  %; ’ islocally stable. □

 Remark 4: The results in this section easily generalize to the three-dimensional case. Specifically, let the kinematics of each agent bedescribed by _xi Rixi1 xi, where

x it  xi;1t; xi;2t; xi;3tT  2 R3

is now a three-dimensional vector, Ri is the three-dimensionalrotation matrix in Eq. (9), and

i

n k

r

  xi1;1

 xi;1

2

 xi1;2

 xi;2

2q

Note that the coordinates xi;3t evolve independently of the other coordinates and converge to

1

n

Xn

i1

 xi;30

(this statement relies on elementary results about the eigenvalues of circulant matrices). Moreover, the coordinates xi;1t and xi;2tevolve exactly as in the two-dimensional case. By applyingTheorem 6 we conclude that a splay-state formationis a locally stablerelative equilibrium also in this three-dimensional case.

IV. Cyclic-Pursuit Control Laws for Double-Integrator Models

In this section, we study cyclic-pursuit control laws for doubleintegrators. We first present, in Sec. IV.A, a control law that requireseach agent to be able to measure its absolute position and velocity;then in Sec. IV.B, we design a control law that only requires relativemeasurements of position and velocity. As before, let 

x it  xi;1t; xi;2t; xi;3tT  2 R3

be the position at time t 0 of the ith agent, i 2 f1; 2; . . . ; ng,andlet x xT 

1 ;xT 2 ; . . . ;xT 

n T . Moreover, let R be the rotation matrix inthree dimensions with rotation angle 2 ; and rotation axis

0; 0; 1

T  [see Eq. (9)]. The dynamics of each agent are now

described by a double-integrator model:

x i ui (18)

A. Dynamic Cyclic Pursuit with Reference Coordinate Frame

Consider the following feedback control law

ui kdRxi1 xi R _xi1 _xi kckdxi

kc kd _xi; kd 2 R>0; kc 2 R (19)

Note that each agent needs to measure both its absolute position (if kc ≠ 0) and its absolute velocity (if  kc ≠ kd). The overalldynamics of the n agents are described by

_x

x

0 I 3n

kd A A kdI 3n

x : Cx (20)

where A : L R kcI 3n and L is the matrix defined inEq. (6). The following theorem characterizes eigenvalues andeigenvectors of C.

Theorem 6 : Assume that kd is not an eigenvalue of  A. Theeigenvaluesof thestate matrixC inEq.(20) are the union of1) the3n eigenvalues of A, and 2) kd, with multiplicity 3n.

In other words, eigC eig A [fkdg. Moreover, theeigenvector of  C corresponding to the kth eigenvaluek 2 eig A, k 2 f1; . . . ; 3ng, is

k : k;1

k;2

k

kk ; k 2 f1; . . . ; 3ng

where k is the eigenvector of  A corresponding to k. The 3n(independent) eigenvectors corresponding to the eigenvalue kd

(that has multiplicity 3n) are

k : k;1

k;2

k1

d ek3n

ek3n

; k 2 f3n 1; . . . ; 6ng

where e j is the jth vector of the canonical basis inR3n.Proof: First, we compute theeigenvalues of C. The eigenvalues

of C are, by definition, solutions to the characteristic equation:

0

det

I 3n I 3n

kd A I 3n  A kdI 3n By using the result in Eq. (1), we obtain

0 det2I 3n  A kdI 3n kd A det kdI 3n detI 3n  A

Thus, the eigenvalues of C must satisfy 0 det kdI 3n and0 detI 3n  A; hence, the first part of the claim is proved.

By definition, the eigenvector  T k;1T 

k;2T  corresponding to theeigenvalue k, k 1; . . . ; 6n, satisfies the eigenvalue equation:

k

k;1

k;2

" #

0 I 3n

kd A A kdI 3n

" #k;1

k;2

" #

k;2

kd Ak;1  Ak;2 kdk;2

" #

Thus, we obtain

kk;1 k;2; kk;2 kd Ak;1  Ak;2 kdk;2

and therefore

kkd kk;1 kd k Ak;1 (21)

If k kd, then we have 3n eigenvectors given by k1d e j; e jT ,

 j f1; . . . ; 3ng. If, instead, k 2 eig A [note that, byassumption, kdeig A], we obtain from Eq. (21)

kk;1  Ak;1

and we obtain the claim. □

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We are now in a position to study the formations that can beachieved with control law (19).

Theorem 7 : Assume that kd is not an eigenvalue of  A. Thenagents’ positions starting at any initial condition (except for a set of measure zero) in R3n and evolving under Eq. (20) exponentiallyconverge:

1) If kc 0, to formations centered at the initial center of mass:a) If 0 jj < =n, to a single limit point.b) If jj =n, to an evenly spaced circle formation.

c) If =n < jj < 2=n, to an evenly spaced logarithmic spiralformation.2) If kc > 0, to formations centered at the origin,

a) If 0 jj =n, to a single limit point.b) If =n < jj < 2=n,

i) If kc > 2sin=n sin =n, to a single limit point.ii) If  kc 2sin=n sin =n, to an evenly spaced

circle formation.iii) If  kc < 2sin=n sin =n, to an evenly spaced

logarithmic spiral formation.Proof: As a consequence of Theorem 7, the eigenvectors of C

are linearly independent. Indeed, the eigenvectors k for  k 2f1; . . . ; 3ng are linearly independent, since thevectors k are(in fact,by Theorem 4, L R is diagonalizable for all 2 ; );moreover, the eigenvectors k for  k

2 f3n

1; . . . ; 6n

gare clearly

linearly independent. Since, by assumption, kdeig A, theindependence of the eigenvectors of C follows.

Then the proof is a simple consequence of Theorem 4, Theorem7,and the arguments in Sec. 3.5 of the work of Pavone andFrazzoli [14]. □

B. Control Law with Relative Information Only

Consider the following feedback control law:

u i k1R2xi2 xi1 xi1 xi k2R _xi1 _xi(22)

where k1 and k2 are two real constants (not necessarily positive). Inthis case, each agent only needs to measure its relative position withrespect to the positions of agents i 1 and i 2 [note that xi2 xi1 xi2 xi xi1 xi] and its relative veloc-ity with respect to the velocity of agent  i 1. The control law inEq. (22) can be considered a generalized cyclic-pursuit control law,whereby agent  i requires information from agent i 1 and agent i 2 (modulo n). Note also that the control law (22) uses a rotationmatrix that is common to all agents; hence, it requires that all agentsagree upon a common rotation axis, but it does not  require a consensus on a common origin. Indeed, in the case of spacecraft,agreement on the orientation can be easily achieved by using star trackers.

It is possible to verify that 

L2 1 2 1 0 . . . 0

0 1 2 1 . . . 0... ..

.

2 1 0 . . . . . . 1

2664 3775Then the overall dynamics of the n agents can be written in compact form as

_x

x

0 I 3n

k1L2 R2 k2L R

x : F x (23)

Let A : L R, and define

: k2

2

  

k2

22

k1

s(24)

The following theorem characterizes eigenvalues and eigenvectors of F .

Theorem 9: Assume that  ≠ 0. The eigenvalues of the statematrix F  in Eq. (23) are the union of 1) the 3n eigenvalues of  A, each one multiplied by , and 2) the 3n eigenvalues of  A,each one multiplied by .

In other words,eigF  eig A [ eig A. More-over, the eigenvector of  F  corresponding to the kth eigenvaluek 2 eig A, k 2 f1; . . . ; 3ng, is

k

: k;1

k;2 k

kk ; k

2 f1; . . . ; 3n

gwhere k is the eigenvector of  A corresponding to the eigenvaluek=. Similarly, the eigenvector corresponding to the ktheigenvalue 3nk 2 eig A, k 2 f1; . . . ; 3ng, is

3nk : 3nk;1

3nk;2

k

kk

; k 2 f1; . . . ; 3ng

where k is the eigenvector of  A corresponding to the eigenvaluek=.

Proof: First, we compute the eigenvaluesof F . Notethat bytheproperties of the Kronecker product, L2 R2 L R2 A2. The eigenvalues of  F  are, by definition, solutions to thecharacteristic equation:

0 det

I 3n I 3n

k1 A2 I 3n k2 A

Using the result in Eq. (1) we have that 

0 det2I 3n k2A k1 A2

detI 3n  AI 3n  A

Then the first part of the claim is proven.By definition, the eigenvector  k;1k;2T  corresponding to the

eigenvalue k, k 2 f1; . . . ; 6ng, satisfies the eigenvalue equation:

k

k;1

k;2" #

0 I 3n

k1 A2

k2 A" #k;1

k;2" #

k;2

k1 A2k;1 k2 Ak;2

" #

Thus, we obtain

kk;1 k;2; kk;2 k1 A2k;1 k2 Ak;2

and therefore,

2kk;1 k1 A

2k;1 k2 Akk;1

which can be rewritten as

kI 3n

 A

kI 3n

 A

k;1

0 (25)

Therefore, if  k 2 eig A (analogous arguments hold if k 2 eig A), the above equation is satisfied by letting k;1 beequal to k; in fact, in this case [note that k is the eigenvector of  A corresponding to the eigenvalue k= and that  ≠ 0],

kk;1 k

k  Ak  Ak;1

and the claim easily follows. □

By appropriately choosing k1, k2 and , it is possible to obtain a variety of formations. Here, we focus only on circular formationsandArchimedes spiral formations (an Archimedes spiral is a spiral withthe property that successive turnings have a constant separationdistance), which are arguably among the most important symmetricformations for applications. In particular, Archimedes spiralformations are useful for the solution of the coverage path-planningproblem, in which the objective is to ensure that at least one agent 

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eventually moves to within a given distance from any point in thetarget environment. More applications will be discussed in Sec. V.We start with circular formations.

1. Circular Formations with Only Relative Information

We start with the following lemma. Lemma 2: The vector 

wk 03n

1

k

where 03n1 is the zero matrix with 3n rows and 1 column, is a generalized eigenvector for the zero eigenvalues k, where kn; 2n; 3n.

Proof: The claim can be easily obtained by direct verification intothe equation F  kI 6nwk k. □

Theorem 10: Let   k2 2cos=2n and k1 k2=22sin2=2n. Moreover, assume that  =2n; then the system converges to an evenly spaced circular formation whose geometriccenter has constant velocity.

Proof: With theabovechoices for k1 and k2, it is straightforward toverify that  e j=2n. From Theorem 9, F  has exactly twoeigenvalues on the imaginary axis, a zero eigenvalue with algebraic

multiplicity 6 and geometric multiplicity 3, and all other eigenvaluesk in the open left-half complex plane with linearly independent eigenvectors. Then by using Theorem 9 and Lemma 2, it is possibleto show that as t ! 1, the time evolution of the system satisfies

xt_xt

xG

_ xG

t

_ xG

03n1

c1

w1dom

!w2dom

c2

w2dom

!w1dom

where xG and _ xG are the initial position and velocity of the center of the formation, c1 and c2 are constants that depend on the initialconditions, ! is a constant equal to 2 sin=n, and the eigen-functions wp

dom, p 2 f1; 2g, are given by

w1dom cos!t 1; sin!t 1; 0; . . . ; cos!t n; sin!t

n; 0; w2dom sin!t 1; cos!t

1; 0; . . . ; sin!t n; cos!t n; 0 (26)

where i 2i 1=n, i 2 f1; . . . ; ng. (Note that   _w1dom

!w2dom, _w2

dom !w1dom.) □

Next we show how to choose k1, k2 and to achieve Archimedesspiral formations; note that an Archimedes spiralis describedin polar coordinates by the equation %’ a’, with a 2 R>0.

2. Archimedes Spiral Formations with Only Relative Information

We start with the following lemma. Lemma 3: Let k1 k2=22 and assume =n. Then

wk 03n1

k

is a generalized eigenvector for the eigenvalue k=.

Proof: The claim can be easily obtained by direct verification intothe equation F  kI 6nwk k. □

Theorem 11:Let k1 k2=22, andassume k2 > 0 and =n.Then the system converges to an Archimedes spiral formation whosegeometric center has constant velocity.

Proof: In this case we have 2 R>0, and thus k k3n

for allk 2 f1; . . . ; 3ng; as a consequence, the eigenvaluesof F  areeig A. Hence, F  has exactly two eigenvalues on theimaginary axis, each one with algebraic multiplicity 2 and geometricmultiplicity 1, a zero eigenvalue with algebraic multiplicity 6 andgeometric multiplicity 3, and all other eigenvalues

k in the open

left-half complex plane. Then byusing Theorem9 and Lemma 3,it ispossible to show that as t ! 1, the time evolution of the system satisfies

xt_xt

" #

 xG

_ xG

" # t

_ xG

03n1

" # d1

03n1

w1dom

" # d2

03n1

w2dom

" #

c1 d1tw1

dom

!w2dom

" # c2 d2t

w2dom

!w1dom

" #

where xG and _ xG are the initial position and velocity of the center of the formation; c1, c2, d1, and d2 are constants that depend on the

initial conditions; ! is a constant equal to 2sin=n; and theeigenfunctions wpdom, p 2 f1; 2g,aredefinedinEq.(26). Then agents

will perform spiraling trajectories; the radial growth rateis a constant equal to

 d1 d2

p , and the center of the formation moves with

constant velocity _xG defined by the initial conditions. □

  Remark 5: Under both control law (19) and control law (22),circular formations and Archimedes spiral formations only occur for specific values of  and of the gains, such that some eigenvalues areon the imaginary axis, which makes these behaviors not robust.Similar to in Sec. III.B, in order to robustify these formations, we arecurrently studying strategies to make the rotation angle a function of the state of the system.

C. Comparison with Existing Results in the Literature

It is of interest to discuss the relation between the control policiespresented in this section and a set of related results recently appeared[11].In his work, Ren [11] proposes the following control lawfor thedouble-integrator model in Eq. (18):

u i Xn

k1

qikRxi xk  _xi; i 2 f1; . . . ; ng

  2 R>0

where qik is the i; kth entry of a weighted adjacency matrix Qassociated to a weighted directed graphG (note that thering topologyis a special case). In Ren’s work [11], it is shown that if G  has a directed spanning tree, theagentswill eventually converge to a singlepoint, a circle or a logarithmic spiral pattern, depending on the valueof  ; the center of the formation is determined by the initialconditions and thevelocity needs to be measured in a common frameof reference. Our results in this section differ from the results in [11]along three fundamental dimensions:

1) We consider control on the center of the formation.2) We study control laws that only require relative measurements

of position and velocity.3) We consider a  novel  type of symmetric formation: namely,

 Archimedes spirals, which are especially useful for interferometricimaging.

We next describe possible applications for the control laws weproposed, and we present experimental results.

V. Applications of Cyclic-Pursuit Algorithms

In the past few years, cyclic pursuit has received considerableattention in the control community (see Sec. I); however to date, tothe best of our knowledge, no application has been proposed for which cyclic pursuit is a particularly effective control strategy. In thissection, we discuss application domains in which cyclic pursuit isindeed an ideal candidate control law.

A. Interferometric Imaging in Deep Space

Interferometric imaging, i.e., image reconstruction from inter-ferometric patterns, is an application of formationflight that has beendevised and studied for space missions such as NASA’s TerrestrialPlanet Finder (TPF) and ESA’s Darwin [18]. The general problem of interferometric imaging consists of performing measurements in a way that enough information about the frequency content of theimage is obtained. Interferometric imaging is indeed an idealapplication for the cyclic-pursuit algorithms we have presented inSec. IV.B. In fact, the solution to this problem is independent of theglobal positions of the spacecraft [19]. Moreover, missions like TPF

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and Darwin consider locations far out of the reach of GlobalPositioning System (GPS) signals and are expected to only rely onrelative measurements to perform reconfigurations and observationmaneuvers. We have recently proposed an heuristic, yet effectivecontrol strategy forthis coverage problem (referred to in the literatureas the u-v plane coverage problem), whereby the spacecraft perform  Archimedes spiral  trajectories. (Specifically, Archimedean spiralrelative trajectories are designed such that their expansion rate andvelocity are optimized to obtain suf ficient interferometric inform-

ation without leaving coverage gaps as they expand and cover anincreasing area of aperture [19].) Figure 2 shows simulated trajec-tories resulting from the application of control law (22); the initialpositions are random inside a volume of 10 km3. In the first casespacecraft converge to circular trajectories, while in the second casespacecraft converge to Archimedes spirals. The inertial frame for theplots is the geometric center of the configuration.

B. Reaching Natural Trajectories

In this section we modify the previous control laws to achieveconvergence to elliptical trajectories. Consider the application of a similarity transformation to the rotation matrix R, in other words,we now replace the rotation matrixR in the previous control lawswith TRT 1, where T  is a nonsingular matrix. Such similarity

transformation does not change the eigenvalues of L R (hence,Theorem 4 still holds), butchanges theeigenvectors of R to T.It is straightforward to see that the trajectories arising with theprevious control laws are then transformed according to

~x it T xit; i 2 f1; . . . ; ng (27)

in particular, circular trajectories can be transformed into elliptical trajectories.

Indeed, the above approach is useful to allow the system toglobally converge to low-effort trajectories. Consider the dynamicsystem 

x i  f xi; _xi ui  f xi; _xi  f xi; _xi unomi (28)

which has a zero-effort (ui 0) invariant set  xi , for which f xi ; _x

i unomi. If we use a controller for which the state reaches

xi t as t ! 1, thenthe control effort will tendtoui 0 as t ! 1.

In the case of the dynamics of relative orbits slightly perturbedfrom a circular orbit, elliptical relative trajectories are closed near-natural trajectories (i.e. theoretically they require no control effort);in the following section, cyclic-pursuit controllers are proposed aspromising algorithms for formation acquisition, maintenance, andreconfiguration.

1. Clohessy – Wiltshire Model 

The Clohessy–Wiltshire model approximates the motion of a spacecraft with respect to a frame that follows a circular orbit with

angular velocity !R and radius Rref  =!2R1=3, where is the

gravitational constant of Earth. The equations of motion xi  f xi;ui are

 x i 2!R _yi 3!2R xi uix; yi 2!R _ xi uiy

zi !2Rzi uiz

(29)

where the x, y and z coordinates are expressed in a right-handedorthogonal reference frame, such that the x axis is aligned with theradial vector of the reference orbit, the z axis is aligned with theangular momentum vector of the reference orbit, and the y axiscompletes the right-handed orthogonal frame.

Consider a formation of spacecraft that use the cyclic-pursuit controller 

ui  f xi kgkdTRT 1xi1 xi TRT 1 _xi1 _xi kckdxi kc kd=kg _xi (30)

with kg !R=2sin=n sin =n and

T 12

0 0

0 1 0

z0 cosz z0 sinz 1

24

35

where z0 and z are tunable parameters (their roles will become clear later). Then from the results in Sec. IV, we obtain, as t ! 1,

x it xi t T 

r sin!Rt ir cos!Rt i

0

" #; i 2 f1; . . . ; ng

where i 2i 1=n, and r is a constant that depends on theinitial conditions. Thus, we obtain

 xi t 1

2r sin!Rt i; y

i t r cos!Rt iz

i t zor sin!Rt i z

hence, the formation will converge to an evenly spaced ellipticalformation with an x:y ratio equal to 1:2, a y:z ratio equal to 1:z0,anda phasing between the x and z motion equal to z. By replacing theseequations into Eq. (29), it is easily shown that as xt ! xt, wehave u ! 0.

2. Including J 2 Perturbation

A more accurate model for themotion of a spacecraft considers theeffects of the J 2 gravitational term. Schweighart and Sedwick [20]showed that the equations of motion relative to a circular non-Keplerian reference orbit and including theJ 2 term are well approxi-mated by the linear system 

24

68

46

810

2

4

6

8

x, km

a)

y, km

−10

0

10

20

−10

0

10

20

2468

b)

Fig. 2 Convergence from random initial conditions to symmetric formations; left: circular trajectories and right: Archimedes spiral trajectories.

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 x i 2!Rc _yi 5c2 2!2R xi u xi 3

4K J 2 cos2 kt

yi 2!Rc _ xi uyi 12K J 2 sin2 kt

zi q2zi 2lq cosqt uzi

where, again, the x, y, and z coordinates are expressed in a right-handed orthogonal reference frame, such that the x axis is alignedwith the radial vector of the reference orbit, the z axis is alignedwith the angular momentum vector of the reference orbit, and the

y axis completes the right-handed orthogonal frame; moreover,c  

1 sp 

,

s 3J 2R2e

8r2ref 

1 3cos2iref 

K J 2 3!2RJ 2R2

e

rref 

sin2iref 

Re is the nominal radius of the Earth,

k c 3J 2R2e

2r2ref 

cos2iref 

rref  and iref  are parameters of the reference orbit, q is approximatelyequal to c!R,and and l are time-varying functions of the differencein orbit inclination (see [20] for the details). Zero-effort trajectories(i.e., trajectories with ui 0) for the above dynamic model areshown to be

 xt  x0 cos!Rt 

1 sp 

 

1 sp 

1 sp  y0 sin!Rt

 1 s

 xcct

yt  x0

1 sp  1 s

p  sin!Rt 

1 sp 

y0 cos!Rt 

1 sp 

ycct

zt lt m sinqt (31)

with

xcct

cos2 kt cos!Rt 

1 sp 

; sin2 kt

1 sp  

1 sp  cos!Rt

 1 s

p ; 0

where , , and m are constants that depend on the reference orbit parameters and are not discussed here, for brevity. (For the details we

refer the reader to the work of Schweigart and Sedwick [20].) As inthe previous section, by defining the control coordinates in a reference frame centered in xcct, and using the decentralizedcyclic-pursuit controller in Eq. (30) with a transformation matrix

T  

1sp 

1sp  0 0

0 1 0

z0 cosz z0 sinz 1

24

35

and kg !R  1 sp  =2sin=n, it is straightforward to show that the formation will converge to elliptical trajectories centered at thepoint xcct. Then as t ! 1,

x t ! T r sin!Rt ir cos!Rt i

0

24

35

and thus, as t ! 1, the trajectories for  xt, yt are those describedin Eq. (31), and we have that 

u x ! 0; uy ! 0

uz ! !2R q2 2lq cosqt 2lq cosqt

The last term corresponds to the cohesive force required to maintainthe formation when the orbits are not coplanar (i.e. the spacecraft havedifferent inclination and thus different J 2 secular drift rates). For zo ! 0, then q !R, l ! 0 and the theoretical required thrust converges to zero.

Figure 3 shows simulation results for the control laws described inthis section. Thesystem is simulatedusingdynamicsincludingthe J 2terms. Theresults show convergence from randominitial positionstothe desired orbits: i.e., an evenly spaced elliptical formation in thedesired plane.It is also shown (Fig.3b) how thecontrol effortreducesas the spacecraft reach the desired low-effort trajectories. In this casethe orbits are coplanar and the differentialJ 2 effects in thez directionconverge to 0.

Although the achieved trajectories are not natural trajectories for a free orbiting body, the proposed decentralized control law allows

convergence to ellipticalformations that are nearly natural andwouldrequire low fuel consumption for their maintenance.

VI. Experimental Results ona Microgravity Environment 

The previous control laws have been tested on the SPHEREStestbed onboard the International Space Station. SPHERES is anexperimental testbed consisting of a group of small vehicles with thebasic functionalities of a satellite (see footnote ∗∗). Their propulsionsystems use compressed CO2 gas, and their metrology systems areGPS-like. Each vehicle has a local estimator that calculates a global

Fig. 3 Convergence to elliptical trajectories with dynamics including J 2 terms. The dots indicate the positions after three orbits. Left 3-D view,

trajectories with respect to reference point x cc; center: 3-Dview,trajectories with respect to non-Keplerian circular orbit;and right:control effortversustime. Note that as t ! 1, u ! 0.

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estimate of the state from measurements of the time of flight of ultrasound pulses generated by fixed beacons. The system uses a single TDMA based RF channel to communicate its state toneighboring spacecraft. Figure 4 shows a picture of three SPHERESspacecraft onboard the ISS.

The dynamics of each spacecraft are well approximated by a double integrator. For the tests presented in this section, we used a velocity-tracking control law to track the velocity profile in Eq. (11)(where the rotation angle is a function of the state of the system). In

fact, in preliminary ground experiments we noticed that control law(8), whichusesa constant rotation angle, was not able to stabilize theradius of circular formations. This was expected, since, as discussedin Sec. III.B, circular formations are notrobustunder control law (8).

We set k 1, while the gain kg (and, consequently, the frequencyof rotation)was setin eachmaneuverto make thecentripetal force for the circular motion equal to 0:11N  (half the saturation level of thethrusters). We next describe twotestsperformedusingthe SPHERESFig. 4 Picture of three SPHERES satellites performing a test onboard

the ISS. (Photograph credit: NASA—SPHERES).

0 100 200 300

−1

0

1

Sphere Logical ID 1

   P  o  s ,  m

 

Px

Py

Pz

Man

0 100 200 300−0.1

0

0.1

   V  e   l ,  m   /  s

 

Vx

Vy

Vz

Man

0 100 200 300

−1

0

1

Sphere logical ID 2

 

0 100 200 300−0.1

0

0.1

 

0 100 200 300

−1

0

1

Sphere logical ID 3

 

0 100 200 300−0.1

0

0.1

 

time, s time, s time, s

a) c)b)

Fig. 5 Experimental results from test 1: position and velocity vs time (from left to right: satellites 1, 2, and 3).

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6  

x, m

 

  z ,  m

Sat1

Sat2

Sat3

−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

x, m−0.5 0 −0.5

−0.4

−0.2

0

0.2

0.4

0.6

x, m−0.5 0 0.5

−0.4

−0.2

0

0.2

0.4

0.6

x, m

−0.5

0

0.50 0.2 0.4

−0.4

−0.2

0

0.2

0.4

0.6

y, m

x, m

d)b)a) c)

Fig. 6 Experimental results from test 1: x- z plane and sequence of maneuvers a–d.

0 50 100 150 200

−1

0

1

Sphere logical ID 1

   P  o  s ,  m

 

Px

Py

Pz

Man

0 50 100 150 200

−0.1

0

0.1

   V  e   l ,  m   /  s

 

Vx

Vy

Vz

Man

0 50 100 150 200

−1

0

1

Sphere logical ID 2

 

0 50 100 150 200−0.1

0

0.1

 

0 50 100 150 200

−1

0

1

Sphere logical ID 3

 

0 50 100 150 200−0.1

0

0.1

 

b) c)a)

time, s time, stime, s

Fig. 7 Experimental results from test 2: position and velocity vs time (from left to right: satellites 1, 2, and 3).

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testbed onboard the ISS: the objective is to demonstrate the closed-loop robustness of the approach.

A. Test 1

The first test comprised the following series of maneuvers, withthe initial conditionsbeing x1 0; 0:1; 0:2T , x2 0; 0:1; 0:2T , x3 0; 4; 0T , and zero velocity (with respect to the ISS):

In maneuver a, two spacecraft perform a rotation maneuver in the x-z plane with a radius r 0:3 m;

In maneuver b, a change in the desired radius is commanded andthe spacecraft spiral out to achieve a circular formation withr 0:4 m;

In maneuver c, a third spacecraft joins the formation and thesystem reconfigures into a three-spacecraft evenly spaced circular formation with r 0:35 m;

In maneuver d, a similarity transformation T  is applied to therotation matrix, and the spacecraft achieve an elliptical formationwith eccentricity equal to 0.8.

The objective was to test convergence to evenly spaced circular formations and robustness with respect to changes in the number of agents. Figure 5 shows global position and velocity of eachspacecraft (with respect to the ISS); Fig. 6 shows the trajectoriesperformed by the spacecraft during maneuvers a –d. Experimentalresults (see, in particular, Fig.6) demonstrate the effectiveness of theproposed cyclic-pursuit controller.

B. Test 2

The secondtest comprisedthe followingseriesof maneuvers, withthe initial conditionsbeing x1 0; 0:1; 0:2T , x2 0; 0:1; 0:2T , x3 0; 4; 0T , and zero velocity (with respect to the ISS):

In maneuver a, three spacecraft perform an evenly spaced rotationmaneuver in the x-z plane with radius r

0:35 m.

In maneuver b, a similarity transformation T  is applied to therotation matrix, and the spacecraft achieve an elliptical formationwith eccentricity equal to 0.8.

Figure 7 shows global position and velocity of each spacecraft (with respect to the ISS); Fig. 8 shows the trajectories performedby the spacecraft during maneuvers a and b. Also in this case,experimental results (see, in particular, Fig. 8) demonstrate theeffectiveness of the proposed cyclic-pursuit controller (seefootnote ¶ ).

VII. Conclusions

In this paper we studied distributed control policies for spacecraft formationthat draw inspiration from thesimple idea of cyclicpursuit.We discussed potential applications and we presented experimentalresults. This paper leaves numerous important extensions open for further research. First, all of the algorithms that we proposed aresynchronous: we plan to devise algorithms that are amenable to

asynchronous implementation. Second, we envision studying theproblem of convergence to symmetric formations in presence of actuator saturation. Finally, to further assess closed-loop robustness,we plan to perform additional tests onboard the ISS.

Appendix A: Representation of Nonlinear Approachin Polar Coordinates

Assume that kpik %i > 0. Note that pi %iR#ie1, where e1

is the first vector of the canonical basis: i.e., e1 1; 0T . Then wecan write pi1 as

pi1 %i1

%i

%iR#i1e1 %i1

%i

%iR#i1R#iR#ie1

%i1

%i

R#i #i1pi (A1)

Moreover, it also holds that 

p T i R pi kpik2 cos  %2

i cos  for any   2 R (A2)

First, wefind thedifferentialevolution forthe magnitudeof pi:i.e.,for %i. We have

_% i dkpikdt

pT i _pi

kpik 1

%i

pT i Ri1pi1 Ripi

By using Eqs. (A2) and (A2) we then obtain

_%i 1

%i

pT i

Ri1

%i1

%i

R#i #i1pi Ripi

%

icos

#

i1 #

i

i1 %

icos

iWe now find the differential evolution for the phase of pi: i.e., for 

#i. Taking time derivative in both hands of the identitypi;1 sin #i pi;2 cos #i 0, we easily obtain

_#i pi;1 _pi;2 pi;2 _pi;1

%2i

1

%2i

pT i R

2

_pi

1

%2i

pT i R

2

Ri1pi1 Ripi

Then by using Eqs. (A1) and (A2) we obtain

_

#i 1

%2i p

i R

2Ri1%i1

%i R#i #i1pi Ripi %i1

%i

sin#i1 #i i1 sini

−0.5 0 0.5

−0.5

0

0.5

x, m

  z ,  m

−0.5

00.5

−0.20

0.2

−0.5

0

0.5

y, mx, m

  z ,  m −0.5 0 0.5

−0.5

0

0.5

x, m

  z ,  m

a) b)

Fig. 8 Experimental results from test 2: x- z plane and sequence of maneuvers a and b.

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Appendix B: Proof of Lemma 1

Proof: The eigenvalues of  P are solutions to the characteristicequation

0 det

I n anL 2ksnI n snI n

bnL2 I n cnL

Note that both matrix (I n anL 2ksnI n) and matrix bnL2 arecirculant; then since circulant matrices form a commutative algebra 

(see Sec. II), we can apply the result in Eq. (1) and obtain

0 detI n anL 2ksnI nI n cnL snbnL2 det2I n

2ksnI n an cnL|{z}: B

2ksncnL ancn snbnL2|{z}: C

det2I n B C det2I n B B2=4 S

where S : B2=4 C. Note that B and C are circulant, therefore S isalso circulant. SinceS is circulant,it canbe diagonalizedaccordingtoS

UDSU ,where DS is a diagonal matrix with theeigenvaluesof S

on the diagonal; accordingly, we have S1=2 UD1=2S U . Note that B

and S1=2 commute. In fact, since B is circulant, it can be diagonalizedvia the same orthogonal matrix U : B UDBU , where DB is a diagonal matrix with the eigenvalues of B on the diagonal; hence,

S1=2B UD1=2S U UDBU  UD1=2

S DBU  UDBD1=2S U 

UDBU UD1=2S U  BS1=2

Therefore, we have

0 det2I nn B B2=4 S detI n B=2

 

Sp 

detI nn B=2  

Sp 

Hence, the eigenvalues of  P are the union of the eigenvalues of 

(B=2  

Sp 

)and(B=2  

Sp 

). Since B and S1=2 are diagonalizedby the same similarity transformation U , we have

B

2 S1=2 U 

DB

2U  UD

1=2S U  U 

DB

2 D

1=2S

U  (A3)

Let B;k be the kth eigenvalue of B, and S;k be the kth eigenvalue of S, k 2 f1; . . . ; ng. Then from Eq. (A3), we have that 

eig P fB;k=2 1=2S;k gn

k1

Hence, we are left with the task of computing the eigenvalues of  Band S. Such eigenvalues can be easily found by using Eq. (4):

eigB f2ksn an cn1 e2jk=ngnk1

eigSf 2B;k=4 C;kgn

k1 f2ksn an cn1

e2jk=n2=4 2ksncn ancn snbn 2ksncn

2ancn 2snbne2jk=n ancn bnsne4jk=ngnk1

We first consider the eigenvalues of  B. By using the followingidentities

1 ek j 2sink=2e jk

2

1 1

21 e jk cosk=2e jk=2 (A4)

and after some algebraic manipulations omitted for brevity, theeigenvalues of B can be written as

B;k 2ksn 22cn ksn sink=ne jk=n2

k 2 f1; . . . ; ng (A5)

Hence, we have for k 2 f1; . . . ; ng,

ReB;k 2ksn 22cn ksnsin2k=nImB;k 22cn ksn sink=n cosk=n (A6)

Next, we consider theeigenvaluesof S. By using,again, theidentitiesin Eq. (A4), and with simple algebraic manipulations, we can write,for k 2 f1; . . . ; ng,

S;k k2s2

ncosk=ne jk=n2

ksncn s2n2 sink=ne jk=n

22

k2s2

n 4sin2k=n 4ksncn k2

s2n 4c2

nsin2k=nei2k=n

Note that k, cn,and sn are positive realnumbers;then the term insidethe square brackets is a positive real number. Therefore, we have for k 2 f1; . . . ; ng,

Re  S;kp k

2

s

2

n 4sin

2

k=n 4ksncn k

2

s

2

n

4c2sin2k=n1=2 cosk=n

which can be rearranged as

Re 

S;k

p ksn 2cn ksnsin2k=n2

4sin2k=ns2n sin2k=n1=2 B;1=22

4sin2k=ns2n sin2k=n1=2 (A7)

From Eqs. (A6) and (A7) it is straightforward to show that 1) For k n, we have

P;n B;n=2  

S;np ksn ksn 0

2) For k 1, k n 1,

Re 

S;1

p ReB;1=2 ksnc2

n 2s2ncn

Im 

S;1

p ks2

ncn 2s3n

Re 

S;n1

p ReB;n1=2 ksnc2

n 2s2ncn

Im 

S;n1

p ks2

ncn 2s3n

Therefore, we obtain

ReP;1 ReB;1=2 ReB;1=2 0

ImP;1 2cn ksnsncn ks2ncn 2s3

n 2sn

ReP;n1 ReB;n1=2 ReB;n1=2 0ImP;n12cn ksnsncn ks2

ncn 2s3n 2sn

3) For 1 < k < n 1, since sink=n2 > sin=n2, we have

Re  

S;k

p ReB;k=2

and thus ReP;k < 0 for k =2 f0; 1; n 1g.Now we proceed to show that  v1; v2; v3 in Lemma 1 are the

eigenvectors corresponding to the eigenvalues 0, 2sn j and 2sn jrespectively. First, consider the zero eigenvalue. Since L 1n 0n

[where 0n 0; 0; . . . ; 0T  2 Rn], it is easy to verify that 

Pv1

anL 2ksnI n snI nb

nL2 c

nL

1n

2k

1n

02n

Now consider the imaginary eigenvalue 2 2sn j. By replacing v2

into the eigenvalue equation we obtain

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anL 2ksnI n snI nbnL2 cnL

1

 f n;2k1

2sn j

1

 f n;2k1

Note that  L and L2 (which are both circulant matrices) satisfy,respectively,

L1 e2j=n 11 2sn jcn  jsn1

and

L21 e2j=n 121 4s2ne2j=n1

Hence, v2 is an eigenvector for P if and only if 

ane2j=n 1 2ksn1 2bne j=nsn1 2sn j1 (A8)

4bns2ncn  jsn21 cn2sn jcn  jsn2bne j=n1

4bne j=nsn j1 (A9)

By using the identities in Eq. (A4), the first condition can be verifiedaccording to

2sncn ksn je j=n 2snk 2snbne j=n 2sn j

2cn ksn j k  jcn  jsnej=n 2bne j=n

2sn kcnej=n 2bne j=n

Similarly, the condition in Eq. (A9) can be verified according to

2bns2ncn  jsn2 2cn jsncn  jsnbne j=n

2bne j=nsn j

sncn  jsn cn jcn  jsn   j; js2n c2

n  j

Similar arguments hold for 3 and v3 (which are complex conjugatesof 2 and v2). This concludes the proof. □

Appendix C: Proof That vi =2 T %;’~ M

Proof: It is enough to prove that at least one of the components of G vi is nonzero. The proof that G v1 ≠ 0 is trivial. We proceed toshow that r g1 v2 ≠ 0: i.e.,

1cos

2

n

cos

2n 1

n

Xn

i2

r sin

2i 1

n

r sin

2n 1

n

0

1

2bne j=n1

" # r %g1 1

r ’g1 2bne j=n1 ≠ 0

Both terms inthe abovesum can beshownto bereal and positive. For the first term, we have that 

r %g1 1 Xn1

k0

cos2k=ncos2k=n  j sin2k=n

Xn1

k0

cos22k=n  jXn1

k0

2sink=n Xn1

k0

cos22k=n

2 R>0

For the second term we have

r ’g1 2bne j=n1 Xn1

k0

Xn

ik1

sin2k=n2bne j=ne2k=n

2bn Xn1

k0 ake2k

1

j=n

where

ak Xn

ik1

sin2i=n Xk

i1

sin2i=n

Now we show that  Xn

k0

ake2k1j=n > 0

First, consider the following facts,

ak 0 8 k

ak1 ak sin2k 1=n < ak for 0 k < n =2 1

abn1=2c < ak 8 k ≠ dn 1=2e

ank Xnk

i1

sin2i=n Xn

mk1

sin2m=n ak

Xn1

k0

ake2k1j=n Xbn1=2c

k0

akekj=n ankekj=n qan1=2

Xbn1=2c

k0

akekj=n ekj=n qan1=2

Xn1

k0

e2k1j=n Xbn1=2c

k0

ekj=n ekj=n q 0

where q 0 if n is even. ThenXn1

k0

ake2k1j=n >Xn1

k0

abn1=2ce2k1j=n 0

Since bn sn kcn > 0, we have that  r ’g1 2bne j=n2 R>0.

The proof for  G v3 ≠ 0 is analogous; in particular, it requiresshowing that jr %g2 1 2 R>0 and jr ’g2 2bne j=n1 2R>0.

Acknowledgments

This work was supported in part by National Science Foundationgrants 0705451 and 0705453. The authors would like to thank theSynchronized Position Hold Engage Reorient Experimental Satelliteteam at the Space Systems Laboratory at Massachusetts Institute of Technology, especially Alvar Sanz-Otero and Christopher Mandy,for facilitating the implementation onboard the International SpaceStation and for their support in obtaining the experimental results.Any opinions, findings, and conclusions or recommendationsexpressed in this publication are those of the authors and do not necessarily reflect the views of the supporting organizations.

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