Ramirezriberos Et Al

Upload
celticpenguin 
Category
Documents

view
217 
download
0
Transcript of Ramirezriberos Et Al
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 1/15
Distributed Control of Spacecraft Formationsvia Cyclic Pursuit: Theory and Experiments
Jaime L. RamirezRiberos,∗ 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 arecyclicpursuitcontrol laws forboth single and doubleintegrator 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,multiagent systems present several advantages. For example,considering an aerospace application, a cluster of spacecraft flying information for highresolution, syntheticaperture 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 multiagent 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 convergence to geometric patterns. Justh and Krishnaprasad [2] presentedtwo strategies to achieve, respectively, rectilinear and circular formation, using alltoall 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. OlfatiSaber and Murray [5] used potentialfunction theory to prescribe flocking behavior in connected graphs.On a similar line, Sepulchre et al. [6] proposed a potentialfunction
methodology to stabilize parallel formations or circular formations of unicycle agents with constant speed. The agents exchange relativeinformation under a limited communication topology. The convergence 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) multipleinput/multipleoutput (MIMO), in whichthe formation istreated as a single multipleinput, multipleoutput plant; 2) leader/ follower, in which individual spacecraft controllers are connectedhierarchically; and 3) cyclic, in which individual spacecraft controllers 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 resultingfromsinglepoint 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 percopy fee to the Copyright Clearance Center, Inc., 222 RosewoodDrive, Danvers, MA 01923; include the code 07315090/10 and $10.00 incorrespondence with the CCC.
∗Ph.D. Candidate, Department of Aeronautics and Astronautics, 77Massachusetts Avenue, Room 37351; [email protected]. Student Member AIAA.
†Ph.D. Candidate, Department of Aeronautics and Astronautics, 77Massachusetts Avenue, Room 32D712. Student Member AIAA.
‡Associate Professor, Department of Aeronautics and Astronautics, 77
Massachusetts Avenue, Room 33332. Senior Member AIAA.§Professor and Director of the Space Systems Laboratory, Department of Aeronautics and Astronautics, 77 Massachusetts Avenue, Room 37327.Senior Member AIAA.
JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS
Vol. 33, No. 5, September –October 2010
1655
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 2/15
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 cyclicpursuit algorithms, we
rigorously study cyclic distributed control laws for formationfl
ying,for both single and doubleintegrator 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 previouscyclicpursuit 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 doubleintegrator 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 deepspace missions, in which globalmeasurements may not be available. In Sec. V we discuss potentialapplications, including spacecraft formation for interferometricimaging and convergence to zeroeffort 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 cyclicpursuit 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].
1656 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 3/15
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 eigenvectors, and the same matrix U diagonalizes all circulant matrices.
F. Cyclic Pursuit for Single Integrators
In this section we review the cyclicpursuit 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 lefthalf complex plane.
2)If jj =n, twononzero eigenvalueslie on theimaginaryaxis,while all other nonzero eigenvalues lie in the open lefthalf complexplane.
3) If =n < jj < 2=n, two nonzero eigenvalues lie in the openrighthalf complex plane, while all other nonzero eigenvalues lie inthe open lefthalf 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 cyclicpursuit 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 singleintegrator dynamics.
III. CyclicPursuit Control Lawsfor SingleIntegrator 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 doubleintegrator dynamics in Sec. IV.
A. Cyclic Pursuit in 3D 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 cyclicpursuit 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 cyclicpursuit 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
RAMIREZRIBEROS ET AL. 1657
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 4/15
where k e2jk=n. L R has exactly three zero eigenvalues,and
1) If 0 jj < =n, all nonzero eigenvalues lie in the open lefthalf complex plane.
2)If jj =n, twononzero eigenvalueslie on theimaginaryaxis,while all other nonzero eigenvalues lie in the open lefthalf 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 righthalf complex plane, while all other nonzero eigenvalues lie in theopen lefthalf complex plane. The two eigenvalues in the open righthalf 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 lefthand 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 lefthand 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 lefthand complex plane.2) If kc 2sin=n sin =n, two nonzero eigenvalues lie
on the imaginaryaxis, while all other eigenvalues lie in the open lefthand complex plane.
3)If kc < 2 sin=n sin =n, twononzero eigenvalues lieinthe open righthand complex plane, while all other eigenvalues lie inthe open lefthand 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 splaystate 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 3D.
1658 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 5/15
Theorem 5: A splaystate formation is a locally stable relativeequilibrium for system (11) and (12).
Proof: Wefirst consider a sequence of coordinate transformationssuch that a splaystate 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 splaystate 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 uppertriangular form:
P1;1 P1;2
032n3 P2;2
RAMIREZRIBEROS ET AL. 1659
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 6/15
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 eigenvalue 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 threedimensional 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 threedimensional vector, Ri is the threedimensionalrotation 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 twodimensional case. By applyingTheorem 6 we conclude that a splaystate formationis a locally stablerelative equilibrium also in this threedimensional case.
IV. CyclicPursuit Control Laws for DoubleIntegrator Models
In this section, we study cyclicpursuit 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 doubleintegrator 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. □
1660 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 7/15
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 velocity with respect to the velocity of agent i 1. The control law inEq. (22) can be considered a generalized cyclicpursuit 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. Moreover, 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 pathplanningproblem, in which the objective is to ensure that at least one agent
RAMIREZRIBEROS ET AL. 1661
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 8/15
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 lefthalf 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 eigenfunctions 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
lefthalf 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 thedoubleintegrator 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 CyclicPursuit 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 interferometric 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 cyclicpursuit 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
1662 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 9/15
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 uv 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 trajectories 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 loweffort trajectories. Consider the dynamicsystem
x i f xi; _xi ui f xi; _xi f xi; _xi unomi (28)
which has a zeroeffort (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 nearnatural trajectories (i.e. theoretically they require no control effort);in the following section, cyclicpursuit 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 righthandedorthogonal 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 righthanded orthogonal frame.
Consider a formation of spacecraft that use the cyclicpursuit 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 nonKeplerian reference orbit and including theJ 2 term are well approximated 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.
RAMIREZRIBEROS ET AL. 1663
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 10/15
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 righthanded 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 righthanded 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 timevarying functions of the differencein orbit inclination (see [20] for the details). Zeroeffort trajectories(i.e., trajectories with ui 0) for the above dynamic model areshown to be
xt x0 cos!Rt
1 sp
1 sp
2
1 sp y0 sin!Rt
1 s
p
xcct
yt x0
2
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
2
1 sp
1 sp cos!Rt
1 s
p ; 0
T
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 decentralizedcyclicpursuit controller in Eq. (30) with a transformation matrix
T
1sp
2
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 loweffort 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 areGPSlike. 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 3D view,
trajectories with respect to reference point x cc; center: 3Dview,trajectories with respect to nonKeplerian circular orbit;and right:control effortversustime. Note that as t ! 1, u ! 0.
1664 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 11/15
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 velocitytracking 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).
RAMIREZRIBEROS ET AL. 1665
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 12/15
testbed onboard the ISS: the objective is to demonstrate the closedloop 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 xz 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 threespacecraft 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 cyclicpursuit 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 xz 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 cyclicpursuit 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 closedloop 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
T
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.
1666 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 13/15
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
RAMIREZRIBEROS ET AL. 1667
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 14/15
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 SanzOtero 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.
References
[1] Carpenter, J., Leitner, J., Folta, D., and Burns, R., “Benchmark Problems for Spacecraft Formation Flying Missions,”AIAAGuidance,Navigation, and Control Conference and Exhibit, Austin, TX, AIAAPaper 20035364, 2003.
[2] Justh, E., and Krishnaprasad, P. S., “Steering Laws and Continuum Models for Planar Formations,” Proceedings of the 2003 IEEE
Conference on Decision and Control , IEEE Press, Piscataway, NJ,2003, pp. 3609–3614.
[3] Jadbabaie, A., Lin, J., and Morse, S., “Coordination of Groups of Mobile Autonomous Agents Using Nearest Neighbor Rules,”Proceedings of the 41st IEEE Conference on Decision and Control ,IEEE Press, Piscataway, NJ, 2002, pp. 2953–2958.
[4] Vicsek, T., Czirok, A., BenJacob, E., Cohen, I., and Shochet, O.,“Novel Type of Phase Transition in a System of SelfDriven Particles,”Physical Review Letters, Vol. 75, Nos. 6–7, 1995, pp. 1226–1229.doi:10.1103/PhysRevLett.75.1226
[5] OlfatiSaber, R., and Murray, R. M., “Flocking with ObstacleAvoidance: CoOperation with Limited Communication in MobileNetworks,” Proceedings of the IEEE Conference on Decision and
Control , IEEE Press, Piscataway, NJ, Dec. 2003, pp. 2022–2028.
1668 RAMIREZRIBEROS ET AL.
8/3/2019 Ramirezriberos Et Al
http://slidepdf.com/reader/full/ramirezriberosetal 15/15
[6] Sepulchre, R., Paley, D., and Leonard, N. E., “Stabilization of Planar Collective Motion with Limited Communication,” IEEE Transactionson Automatic Control , Vol. 53, No. 3, April 2008, pp. 706–719.doi:10.1109/TAC.2008.919857
[7] Lin, Z., Broucke, M., and Francis, B., “Local Control Strategies for Groups of Mobile Autonomous Agents,” IEEE Transactions on
Automatic Control , Vol. 49, No. 4, April 2004, pp. 622–629.doi:10.1109/TAC.2004.825639
[8] Marshall, J.,Broucke, M. E.,and Francis, B.A., “Pursuit Formationsof Unicycles,” Automatica, Vol. 42, No. 1, Jan. 2006, pp. 3–12.
doi:10.1016/j.automatica.2005.08.001[9] Marshall, J., Broucke, M. E., and Francis, B. A., “Formations of Vehicles in Cyclic Pursuit,” IEEE Transactions on Automatic Control ,Vol. 49, No. 11, 2004, pp. 1963–1974.doi:10.1109/TAC.2004.837589
[10] Paley, D., Leonard, N. E., and Sepulchre, R., “Stabilization of Symmetric Formations to Motion Around Convex Loops,” Systems and
Control Letters, Vol. 57, No. 3, March 2008, pp. 209–215.doi:10.1016/j.sysconle.2007.08.005
[11] Ren, W., “Collective Motion from Consensus with CartesianCoordinate Coupling,” IEEE Transactions on Automatic Control ,Vol. 54, No. 6, June 2009, pp. 1330–1335.doi:10.1109/TAC.2009.2015544
[12] Scharf, D. P., and Ploen, F. Y. H. S. R., “A Survey of Spacecraft Formation Flying Guidance and Control (Part II): Control,” Proceedings of the 2004 American Control Conference, Boston, 2004, pp.2976–2985.
[13] Chung, S., Ahsun, U., and Slotine, J.J. E., “Application of Synchronization to Formation Flying Spacecraft:LagrangianApproach,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 2, March 2009,
pp. 512–526.doi:10.2514/1.37261
[14] Pavone, M., and Frazzoli, E., “Decentralized Policies for GeometricPattern Formation and Path Coverage,” Journal of Dynamic Systems,
Measurement, and Control , Vol. 129, No. 5, 2007, p. 633.doi:10.1115/1.2767658
[15] Gray, R., “Toeplitz and Circulant Matrices: A Review,” Foundationsand Trends in Communications and Information Theory, Vol. 2, No. 3,2005, pp. 155–239.doi:10.1561/0100000006
[16] Ren, W., “Collective Motion from Consensus with CartesianCoordinate Coupling—Part I: SingleIntegrator Kinematics,” Proceed
ings of the 2008 IEEE Conference on Decision and Control , IEEEPress, Piscataway, NJ, 2008, pp. 1006–1011.
[17] Paley, D., Leonard, N. E., and Sepulchre, R., “Stabilization of Symmetric Formationsto Motion Around Convex Loops,” Systems and Control Letters, Vol. 57, No. 3, March 2008, pp. 209–215.doi:10.1016/j.sysconle.2007.08.005
[18] Fridlund, C., “The Darwin Mission,” Advances in Space Research,Vol. 34, No. 3, 2004, pp. 613–617.doi:10.1016/j.asr.2003.05.031
[19] Chakravorty, S., and Ramirez, J., “Fuel Optimal Maneuvers for Multispacecraft Interferometric Imaging Systems,” Journal of Guidance,Control, and Dynamics, Vol. 30, No. 1, Jan. 2007, pp. 227–236.doi:10.2514/1.20178
[20] Schweigart, S., and Sedwick, R., “HighFidelity Linearized J 2 Modelfor Satellite Formation Flight,” Journal of Guidance, Control, and Dynamics, Vol. 25, No. 6, 2002, pp. 1073–1080.doi:10.2514/2.4986
RAMIREZRIBEROS ET AL. 1669