Collective Circular Motion of Multi-Agent Systems in Synchronized and Balanced Formations With Second-Order Rotational Dynamics

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by  Anoop Jain

1
Collective Circular Motion of Multi-Agent Systems
in Synchronized and Balanced Formations With
Second-Order Rotational Dynamics
Anoop Jain and Debasish Ghose
Abstract—This paper considers the collective circular motion centroid move in the same direction. Note that, in this paper,
6 of multi-agent systems in which all the agents are required to “collectivemotion”and“formation”areusedinterchangeably.
1 traverse different circles or a common circle at a prescribed
Earlier work in [5] and [6] has focused on achieving syn-
0 angularvelocity.Itisrequiredtoachievethesecollectivemotions
chronizedandbalancedformationsinagroupofagentsunder
2 with the heading angles of the agents synchronized or balanced.
Insynchronization,theagentsandtheircentroidhaveacommon all-to-all and limited communication scenarios, respectively.
r
a velocity direction, while in balancing, the movement of agents In these papers, it is considered that the angular velocities of
M causes the location of the centroid to become stationary. The initial rotations of all the agents are the same and remains
agents considered are initially moving at unit speed around
constant at all times. Recently, the effect of heterogeneity
7 individualcirclesatdifferentangularvelocities.Itisassumedthat
in various aspects have been studied in the literature. For
1 the agents are subjected to limited communication constraints,
and exchange relative information according to a time-invariant example, [8] considers nonidentical linear velocities of the
] undirected graph. We present suitable feedback control laws for agents, and [9] considers heterogeneous control gains. In a
Y eachofthesemotioncoordinationtasksbyconsideringasecond- similar spirit, in this paper, we consider that the angular
order rotational dynamics of the agent. Simulations are given to
S velocities of the initial rotational motion of the agents are
illustrate the theoretical findings.
. nonidentical and are allowed to vary with time. This more
s
c Index Terms—Synchronization, balancing, multi-agent sys- generalscenarioisaddressedinthispaper.Inasimilarcontext,
[ tems,second-orderrotationaldynamics,desiredangularvelocity, the authors in [7], by assuming an all-to-all coupling among

limited communication.
2 agents, propose feedback controls to stabilize synchronized
v andbalancedcircularformationsatadesiredangularvelocity.
9 However, unlike [7], in this paper, we further assume that the
I. INTRODUCTION
7
communicationamongagentsisrestrictedandcanbemodeled
4
Therearevariousengineeringapplicationssuchastracking, as a time-invariant and undirected graph. Some related work,
3
0 surveillance,environmentalmonitoring,searching,sensingand but with all-to-all communication, has been presented in [10].
. datacollection,whereitisrequiredforthemulti-agentsystems There exists an ample literature related to the study of
1
to perform a particular collective motion [1] [4]. A multi- collective circular formation control. In [11], control laws
0 −
6 agent system might comprise ground vehicles, air vehicles, are proposed to stabilize collective circular motion of non-
1 underwater vehicles or a combination of these. In this article, holonomic vehicles around a virtual reference beacon, which
v: we focus on achieving collective circular motion that can be is either stationary or moving. In [12], authors propose a
i appliedinthescenariowherevehiclesarerequiredtoenclose, distributed circular formation control law for ring-networked
X
capture, secure or monitor a target or a search region. nonholonomic vehicles with local coordinate frames. In [13],
r Motivatedbytheseapplications,thecollectivemotionwhere Chen and Zhang propose a decentralized control algorithm
a
all the agents traverse i) different circles, or ii) a common to form a class of collective circular motion, in which the
circle at the prescribed angular velocity along with their vehicles are evenly distributed over the motion circle, and
heading angles in synchronized or in balanced states, are have the same rotational radius. The latter assumption is
consideredinthispaper.Synchronizationreferstothesituation relaxed in [14], where the agents move in circles around a
whenalltheagents,atalltimes,moveinacommondirection. common center, but with different radii. In [15] and, [16], the
A complementary notion of synchronization is balancing, in control algorithms to stabilize the collective motion around a
which all the agents move in such a way that their centroid, circularorbit,whichhaseitherafixedradiusandtime-varying
which is the average position of all the agents, remains fixed. center [15], or a fixed center and time-varying radius [16],
It is evident in synchronized formation that agents and their are proposed. An extension of these results is given in [17],
where a new framework based on affine transformations is
A.JainisagraduatestudentattheGuidance,ControlandDecisionSystem discussed to achieve more complex time-varying formations.
Laboratory (GCDSL) in the Department of Aerospace Engineering, Indian
In [18], the splay circular formation, characterized by equally
InstituteofScience,Bangalore,India(email:[email protected]).
D. Ghose is a Professor at the Guidance, Control and Decision System spaced arrangement of multiple robots, is stabilized by using
Laboratory (GCDSL) in the Department of Aerospace Engineering, Indian a modifiedKuramoto model [19].The stabilizationof circular
InstituteofScience,Bangalore,India(email:[email protected]).
motion under cyclic pursuit is given in the seminal paper
This work is partially supported by Asian Office of Aerospace Research
andDevelopment(AOARD). [20], and also discussed in [21] under dynamically adjustable
2
controlgains.Moreover,thecyclicpursuitproblemofvehicles 1 1
with heterogenous constant linear velocities is considered in 6 2 6 2
[22]. In [23], a Lyapunov guidance vector field approach is
used to guide a team of unmanned aircraft to fly a circular
5 3 5 3
orbit around a moving target with prescribed inter-vehicle
angular spacing. The circumnavigation problem for a team 4 4
(a) (b)
of unicycle-type agents, with the goal of achieving specific
circular formations and circling on different orbits centered at Fig.1. UndirectedcirculantgraphsforN=6.Both(a)and(b)arecirculant
a target of interest, is studied in [24]. graphsbutonly(b)isconnected.
It is to be noted that in the literature described above,
most of the attention is towards achieving a particular type of
is represented as
collective circular formation. However, in the present work,
the emphasis is given toward achieving the same along with r˙ =eiθk (1a)
k
a particular arrangement of the heading angles of the agents θ˙ =ω (1b)
k k
which could be a synchronized, balanced or a combination
ω˙ =u , k=1,...,N. (1c)
of both (usually called as symmetric phase pattern). These k k
formationsservesasthemotionprimitives,andcanbeutilized Here, complex notations are used to describe the position and
to get more general motion patterns [5]. velocityofeachagent.Fork=1,...,N,thepositionofthekth
The main contribution of this paper is to propose a limited agentisrk C,whilethevelocityofthekthagentisr˙k=eiθk=
communication based control strategy to stabilize aforemen- cosθ +isi∈nθ C, where, θ is the orientation of the (unit)
k k k
tioned collective circular motion of a group of agents with velocityvectoro∈fthekthagentfromtherealaxis,andi=√ 1
theirphasearrangementseitherinsynchronizedorinbalanced represents the standard complex number. The orientation−θ ,
k
formation, while allowing the angular velocities of individual, of the velocity vector, which is also referred to as the phase
initial circular motions, performed by the agents, to be dif- of the kth agent [19], represents a point on the unit circle S1.
ferent. With this purpose, in this paper, we consider identical In(1),ω Ristheangularvelocityofthekth agent,which
k
agents moving in a planar space at constant unit linear speed isdetermined∈bythefeedbackcontrolu R.Ifthecontrollaw
k
with second-order rotational dynamics. Thus, the dynamics of u is constant and equal to ω =0, then∈the kth agent travels
k k
eachagentisrepresentedbyastatevector,whichincludesthe at constant unit linear speed on(cid:54)a circle of radius ρ = ω 1.
k k −
position, heading angle and angular velocity of each agent as The direction of rotation around the circle is determi|ned|by
itselements.Thesecond-orderrotationalmodelisparticularly the sign of ω . If ω >0, then the kth agent rotates in the
k k
relevant in the context of planar rigid-body motion, where a anticlockwise direction, whereas, if ω <0, then the kth agent
k
dynamic vehicle model must account not only for motion of rotates in the clockwise direction.
theagent’scenterofmass,butalsoforrotationalmotionabout Lettheinitialmotionofalltheagentswithdynamics(1)be
thecenterofmass[25].Weusesecond-orderrotationalmodel governedbytheopen-loopcontrolu =0, k.Inthissituation,
k
to derive feedback controls that are adequate to regulate the the kth agent moves in a circular orbit of∀radius ω 1 with
k −
orientations as well as the angular velocities of the agents. angular velocity ω . Our aim is to seek a feedb|ack| control
k
The outline of the paper is as follows. In Section II, we de- u , k such that the collective motion of agents, subjected
k
∀
scribethesystemmodelandformulatetheproblem.InSection to limited communication constraints represented by a time-
III, control laws are proposed to stabilize collective motion invariant undirected graph, converge to a circular motion at
of agents on different circles at desired angular velocity with desired angular velocity (and hence desired radius of the
their phase arrangement either in synchronized or in balanced circlingorbitsinceradiusofrotation= angular velocity 1for
−
| |
states. The control laws to stabilize collective motion around an agent circling at unit linear speed) with their phase angles
a common circle of desired radius as well as center with their either in synchronized or in balanced states. We assume that
phasearrangement,againeitherinsynchronizedorinbalanced agents can exchange information only about their orientations
states, is proposed in Section IV. The control strategy to according to the underlying interaction network, and they are
stabilizesymmetricbalancedpatternsisproposedinSectionV. globally provided the information about the desired angular
InSectionVI,wecombinetheresultsoftheprevioussections velocity Ω . In addition, when it is required for the agents
d
andproposecontrolalgorithmstostabilizesymmetriccircular to move around a common circle, the information about the
formations suitable for mobile sensor network applications. desired center c (of the common circle) is also globally
d
Finally, Section VII concludes the paper. providedtothem.Notethatissueofcollisionavoidanceamong
agents is not considered in this work.
II. SYSTEMDESCRIPTIONANDPROBLEMSTATEMENT
B. Notations
A. Agent Model
We introduce a few additional notations that are used in
Similar to [7] and [25], the collective second-order rota- this paper. We use bold face letters rrr =(r ,...,r )T CN,
1 N
tional dynamics of N identical agents, moving in a planar θθθ =(θ ,...,θ )T TN, where, TN is the N-torus, wh∈ich is
1 N
space, each assumed to have unit mass and unit linear speed, equaltoS1 ... ∈S1 (N-times),andωωω=(ω ,...,ω )T RN,
1 N
× × ∈
3
to represent the vectors of length N for the agents’ positions, for k=1,...,N. Then, the vectors
headings and angular velocities, respectively. Next, we define
theinnerproduct z ,z oftwocomplexnumbersz ,z Cas fff(l)=ei(l−1)φ, l=1,...,N, (2)
1 2 1 2
(cid:104) (cid:105) ∈
z1,z2 =Re(z¯1z2),wherez¯1 representsthecomplexconjugate defineabasisofN orthogonaleigenvectorsofL.Theunitary
(cid:104) (cid:105)
of z1. For vectors, we use the analogous boldface notation matrix F whose columns are the N (normalized) eigenvectors
(cid:104)juwwwg,azzzt(cid:105)e=trRanes(pwwwo∗∗∗szzze) ofofrwwwwww.,zzTzh∈eCnNo,rmwhoefrezzzwww∗∗∗CdNenoistesdetfihneedconas- d(1ia/g√N0,)λfff(,l)..d.,iaλgonaliz0eiLs t,hteha(treiasl,)Ldia=goFnΛalFm∗,awtrhixereofΛth=e
zzz = zzz,zzz 1/2. The vectors 000 and 111 are u∈sed to represent by eigen{valu2es of LN}. ≥
00(cid:107)0=(cid:107) (0(cid:104),0,.(cid:105)..,0)T RN, and 111=(1,1,...,1)T RN, respec-
∈ ∈
tively.
III. CONTROLDESIGN
The design of control laws is described in this section. At
C. Representation of Limited Communication Topology
first,aphasepotentialW (θθθ)isdescribed,theminimizationof
1
In the framework of multiagent systems, communication which corresponds to synchronized formation, and its maxi-
among agents is described by means of a graph. A graph is a mizationcorrespondstobalancedformation.Then,apotential
pairG =(V,E),whereV = v1,...,vN isasetofN nodesor function G(ωωω) whose minimization results in the collective
verticesandE V V isase{tofedgeso}rlinks.ElementsofE motion of all the agents at a desired angular velocity, is
⊆ ×
are denoted as (v ,v ) which is termed an edge or a link from proposed.Finally,thecontrollawu isobtainedbyminimizing
j k k
vj tovk.Anundirectedlinkbetweennodesvj andvk indicates a composite potential function consisting ofW1(θθθ) and G(ωωω)
that the information can be shared from node v to node v as described below.
j k
and vice versa. A graph G is called an undirected graph if
it consists of only undirected links. The node v is called a
j A. Achieving Synchronized and Balanced Formations
neighbor of node v if the link (v ,v ) exists in the graph G.
k j k
Theaveragelinearmomentumofagroupofagentsplaysan
Inthisarticle,thesetofneighborsofnodev isrepresentedby
j
N . A complete graph is an undirected graph in which every important role in stabilizing their synchronized and balanced
j
pair of nodes is connected, that is, (v ,v ) E, j,k N. formations. It is maximized in synchronized formation and
j k
The Laplacian of a graph G, denoted by L ∈=[l ]∀ RN∈ N, minimizedinbalancedformation.From(1),theaveragelinear
jk ×
∈ momentum, p , of a group of N-agents, is given by,
is defined as [26], θ
ljk=|N1,j|, iiff kj=Nk j pθ = N1 k∑=N1eiθk =|pθ|eiΨ, (3)
− ∈
0 otherwise which is also referred to as the phase order parameter [19].
The phase arrangement θθθ is synchronized if the modulus of
walhloewres,t|hNejr|episretsheentcaatirodninoalfittyheofsetvheerasletprNopje.rtTiehsisodfeafingirtaiopnh thephaseorderparameter(3)equalsone,thatis,|pθ|=1.The
phase arrangementθθθ is balanced if the phase order parameter
in the form of matrix properties of its Laplacian L. It is well
(3) equals zero, that is, p =0 [19].
known that the Laplacian L of an undirected and connected θ
Thus,thestabilizationofsynchronizedandbalancedforma-
graphG is(P1)symmetricandpositivesemi-definite,and(P2)
tions is accomplished by considering the potential
has an eigenvalue of zero associated with the eigenvector 111,
that is, Lxxx=0 iff xxx=111x0. U(θθθ)=(N/2) pθ 2, (4)
| |
In this article, we will also use the notion of a circulant
graph. A graph G is circulant if and only if its Laplacian L which reaches its unique minimum when pθ =0 (balanced)
is a circulant matrix, that is, L is completely defined by its and its unique maximum when all phases are identical (syn-
chronized). Based on this potential function, the design of
first row [27]. Each subsequent row of a circulant matrix is
control law for the all-to-all communication among agents
thepreviousrowshiftedonepositiontotherightwiththefirst
may be accomplished [10]. However, in order to account for
entry of the row equal to the last entry of the previous row.
limitedcommunicationamongagents,wemodifythepotential
An example of an undirected circulant graph, consisting of
function (4) in the following manner [6].
6 nodes, is shown in Fig. 1. Note that the Laplacian for the
graphs in Figs. 1(a) and 1(b) are, respectively, given by L = Let P = IN (1/N)111111T, where, IN is an N N-identity
circ(1,0,0, 1,0,0) and L =circ(2, 1,0,0,0, 1), wheare, matrix, be a pr−ojection matrix which satisfies P2×=P. Let the
circ(z)repre−sentsthecirculabntmatrixw−ithzbeing−itsfirstrow. vectoreiθθθ berepresentedbyeiθθθ =(eiθ1,...,eiθN)T CN.Then,
∈
As both La and Lb are circulant matrices, both the graphs Peiθθθ = I 1111111T eiθθθ =eiθθθ p 111. (5)
shown in Fig. 1 are circulant, but only the graph shown in N−N − θ
Fig. 1(b) is connected. (cid:18) (cid:19)
Using (5), one can obtain
Now, we state the following lemma from [28], which
describes various properties of an undirected circulant graph Peiθθθ 2= eiθθθ,Peiθθθ =N(1 p 2), (6)
θ
that will be useful in proving the results in this paper. (cid:107) (cid:107) −| |
Lemma 1: Let L be the Laplacian of an undirected circu- whichiszero(minimum(cid:68))when (cid:69)p =1(synchronizedforma-
θ
lant graph G =(V,E) with N vertices. Set φ =(k 1)2π/N tion), and equates to N (maxim|um)| when p =0 (balanced
k θ
− | |
4
formation). Since P is (1/N) times the Laplacian of the algebraic equations
completegraph,theidentity(6)suggeststhattheoptimization
∂W
ofU(θθθ) in (4) may be replaced by the optimization of m = ieimθk,Lkeimθθθ =0, 1 k N. (13)
∂θk ≤ ≤
1 (cid:68) (cid:69)
W1(θθθ)=QL(eiθθθ)= eiθθθ,Leiθθθ , (7) Leteimθθθ beaneigenvectorofL witheigenvalueλ R.Then
2 Leimθθθ =λeimθθθ, and ∈
which is a Laplacian quadratic form(cid:68)associate(cid:69)d with L, and
∂W
is positive semi-definite. Note that, for a connected graph, the m = ieimθk,L eimθθθ =λ ieimθk,eimθk =0, k, (14)
k
quadraticform(7)vanishesonlywheneiθθθ =eiθc111,whereθc ∂θk ∀
S1 is a constant (see property P2), that is, the potentialW1(θθθ∈) which imp(cid:68)lies that mθθθ i(cid:69)s a cr(cid:68)itical point o(cid:69)f Wm(θθθ). Since
is minimized in the synchronized formation. graph G is undirected, the Laplacian L is symmetric, and
hence its eigenvectors associated with distinct eigenvalues are
The time derivative ofW (θθθ), along the dynamics (1), is
1 mutually orthogonal [29]. Since G is also connected,111 spans
N ∂W N ∂W the kernel of L. Therefore, the eigenvector associated with
W˙1(θθθ)=k∑=1(cid:18)∂θk1(cid:19)θ˙k=k∑=1(cid:18)∂θk1(cid:19)ωk. (8) λsyn=ch0roinsizeeimdθθθ. =Alesoiθc111eifmoθθθr,Lanyeimθθθθc ∈=S01,iwfhaincdhoimnlpyliiefsemimθθθθθθ =is
Note that eiθc111foranyθc S1.Thisprovesthefirstpartofthetheorem.
∂W 1 N ∂ Next,weassu∈met(cid:10)hatG isacir(cid:11)culantgraph,thenLemma1
1 = ∑ eiθj,L eiθθθ
j provides the following equivalent expression of the phase
∂θ 2 ∂θ
k j=1 k
(cid:68) (cid:69) potential (12).
= 21 j∑=N1(cid:32)(cid:28)∂∂eθiθkj,Ljeiθθθ(cid:29)+(cid:42)eiθj,∂(L∂θjekiθθθ)(cid:43)(cid:33) Wm(θθθ)= 12 F∗eimθθθ,ΛF∗eimθθθ . (15)
(cid:68) (cid:69)
1 Let www=F eimθθθ, above equation can be written as
= ieiθk,L eiθθθ ∑ eiθj,ieiθk ∗
k
2(cid:32)(cid:68) (cid:69)−j∈Nk(cid:68) (cid:69)(cid:33) W (θθθ)= 1 www,Λwww = 1 ∑N w 2λ . (16)
= ieiθk,Lkeiθθθ =− ∑ sin(θj−θk), (9) m 2(cid:104) (cid:105) 2k=2| k| k
(cid:68) (cid:69) j∈Nk Since matrix F is unitary, www = eimθθθ =√N. Thus,
where, L is the kth row of the Laplacian L. Substituting (9) (cid:107) (cid:107) (cid:107) (cid:107)
k 1 N
in (8), we get W (θθθ)= www,Λwww λ , (17)
m max
2(cid:104) (cid:105)≤ 2
N
W˙ (θθθ)= ∑ ieiθk,L eiθθθ ω . (10) where,λmax isthemaximumeigenvalueofL.Themaximum
1 k k
of W (θθθ) is attained by selecting eimθθθ as the eigenvector
k=1 m
(cid:68) (cid:69) of L associated with the maximum eigenvalue. Since eimθθθ
For the reasons which will be addressed in Section V, let is orthogonal to 111, that is, it satisfies 111Teimθθθ =0, and thus
us define the mth phase order parameter pmθ and the phase corresponds to the phase balancing of mθθθ (see (11)). This
potentialW (θθθ), respectively, as completes the proof.
m
Remark 1: Form=1,thephasepotentialW (θθθ)minimizes
1 N 1
p = ∑eimθk, (11) whenallthephasessynchronize.Thisstatecorrespondstothe
mθ mN
k=1 situation when all the agents, at all times, move in a common
1 direction. On the other hand, the potential W1(θθθ) maximizes
Wm(θθθ)= eimθθθ,Leimθθθ , (12) when all the phases balance. This state corresponds to the
2
where, m N(cid:44) 1,2,3,... ,(cid:68)and eimθθθ =((cid:69)eimθ1,...,eimθN)T. Tmhoitsiofinxoefdalplotihnet aisgeanctstuaablloyutthaefipxoedsitpioonintRsuocfhtthheatcpeθnt=roi0d.
∈ { }
Now, we state the following lemmas from [28], which of the agents since R˙ =(1/N)∑kk=1r˙k = pθ. The use of the
describes various properties of the phase potentialW (θθθ) that phasepotentialWm(θθθ)form>1willbeelaboratedinSection
m
V, where we address the stabilization of symmetric phase
will be useful in proving the results in this paper.
arrangements.
Lemma 2: (Critical points of the Laplacian phase poten-
tial Wm(θθθ)) Let L be the Laplacian of an undirected and B. Achieving Desired Angular Velocity
connected graph G =(V,E) with N vertices. Consider the
The agents, initially rotating at different angular velocities,
Laplacian phase potential W (θθθ) defined in (12). If eimθθθ for
m are required to stabilize their collective motion at desired
all m N is an eigenvector of W (θθθ), then mθθθ is a critical
∈ m angular frequency Ωd (and hence achieve the desired radius
point of Wm(θθθ), and mθθθ is either synchronized or balanced. ρ = Ω 1). For this, we choose a potential function
d d −
The potential W (θθθ) reaches its global minimum if and only | |
m
if mθθθ is synchronized. If G is circulant, then W (θθθ) reaches 1 N
m G(ωωω)= ∑(ω Ω )2, (18)
its global maximum in a balanced phase arrangement. 2 k− d
k=1
Proof: The critical points of W (θθθ) are given by the N which is minimized when ω =Ω , k.
m k d
∀
5
ThetimederivativeofG(ωωω),alongthedynamics(1),yields largest invariant set since
N N d ∂W d
G˙(ωωω)=k∑=1(ωk−Ωd)ω˙k=k∑=1(ωk−Ωd)uk (19) dt(cid:18)∂θk1(cid:19)= dt(cid:68)ieiθk,Lkeiθθθ(cid:69)
= Ω eiθk,L eiθθθ + ieiθk, iΩ L eiθθθ =0, k,
d k d k
− − ∀
(cid:68) (cid:69) (cid:68) (cid:69) (27)
C. Composite Potential Function and Control Law on this set. Now, we analyze the critical points ofW (θθθ).
1
Analysis of the critical points ofW (θθθ): The critical points
In this subsection, the control law u , for the kth agent, 1
k ofW (θθθ) are given by the N algebraic equations
1
is proposed by constructing a composite potential function,
which ensures that all the agents travel around individual ∂W1 = ieiθk,L eiθθθ =0, 1 k N. (28)
k
circlesatadesiredangularvelocityΩd withtheirphaseseither ∂θk ≤ ≤
(cid:68) (cid:69)
in balanced or in synchronized states. For m=1, since (28) is the same as (13), it follows from
Theorem 1: Let L be the Laplacian of an undirected and Lemma 2 thatW (θθθ) minimizes in the synchronized state and
1
connected circulant graph G =(V,E) with N vertices. Con- maximizes in balanced state. Moreover, since maximization
sider system dynamics (1) with control law of the potential W (θθθ) corresponds to the global minimum
1
ofV (θθθ,ωωω), balanced formation is asymptotically stable. This
∂W 1
uk= K (ωk Ωd) 1 , k. (20) completes the proof.
− (cid:18) − − ∂θk (cid:19) ∀ Theorem 2: Let L be the Laplacian of an undirected and
For K >0, all the agents converge to a collective motion in connected circulant graph G =(V,E) with N vertices. Con-
which they travel around individual circles of the same radius
sider system dynamics (1) with control law
ρ = Ω 1 with their phase angles in balanced state.
d d −
| | ∂W
Proof: Consider a composite potential function u =K (ω Ω )+ 1 , k. (29)
k k d
N (cid:18) − ∂θk (cid:19) ∀
V (θθθ,ωωω)=K λ W (θθθ) +G(ωωω); K>0. (21) For K <0, all the agents converge to a collective motion in
1 max 1
2 −
(cid:18) (cid:19) which they travel around individual circles of the same radius
Note that 0 W1(θ) (N/2)λmax (Lemma 2 for m = 1) ρd = Ωd −1 with their phase angles in synchronized state.
≤ ≤ | |
which ensures that V1(θθθ,ωωω) 0. Using (8) and (19), the Proof: Consider a composite potential function
≥
time derivative of the potential function V (θθθ,ωωω) along the
1
V (θθθ,ωωω)= KW (θθθ)+G(ωωω); K<0 (30)
dynamics (1), is 2 1
−
N ∂W N Using (10) and (19), the time derivative of the potential
V˙1(θθθ,ωωω)=−Kk∑=1(cid:18)∂θk1(cid:19)ωk+k∑=1(ωk−Ωd)uk. (22) functionV2 along the dynamics (1), is
N ∂W N
With the control law (20), the time derivative of V1(θθθ,ωωω) V˙2(θθθ,ωωω)= K∑ 1 ωk+∑(ωk Ωd)uk (31)
results in − k=1(cid:18)∂θk (cid:19) k=1 −
V˙ (θθθ,ωωω)= K∑N (ω Ω )2 KΩ ∑N ∂W1 (23) With the control law (29), the time derivative ofV2 results in
1 k d d
− k=1 − − k=1 ∂θk V˙ (θθθ,ωωω)=K∑N (ω Ω )2 KΩ ∑N ∂W1. (32)
2 k d d
From (10), we note that k=1 − − k=1 ∂θk
N ∂W N Using (24), (32) becomes
∑ 1 = ∑ ∑ sin(θ θ )=0. (24)
j k
k=1 ∂θk −k=1j∈Nk − V˙2(θθθ,ωωω)=K∑N (ωk Ωd)2 0. (33)
− ≤
Using (24), (23) becomes k=1
N Since (θθθ,ωωω) TN RN is compact, it follows from the
V˙1(θθθ,ωωω)= K∑(ωk Ωd)2 0. (25) LaSalle’s inva∈riance×theorem [30] that, for K < 0, all the
− − ≤
k=1 solutions of (1) with the control law (29) converge to the
Since (θθθ,ωωω) TN RN is compact, it follows from the largest invariant set contained in V˙ (θθθ,ωωω)=0 , that is, the
2
∈ × { }
LaSalle’s invariance theorem [30] that, for K > 0, all the set
solutions of (1) with the control law (20) converge to the ∆ = (θθθ,ωωω) ω =Ω , k . (34)
s k d
largest invariant set contained in V˙ (θθθ,ωωω)=0 , that is, the { | ∀ }
set { 1 } In∆s,ωk=Ωd, ∀k,whichimpliesthateachagentrotateswith
angular speed Ω . Moreover, since u =ω˙ =0, k in the set
∆b={(θθθ,ωωω) | ωk=Ωd, ∀k}. (26) ∆s,itimpliesfromd (20)that(∂W1/∂θkk)=0k, k,w∀hichdefines
∀
In∆ ,ω =Ω , k,whichimpliesthateachagentrotateswith the critical points of W (θθθ). The set ∆ is itself the largest
b k d 1 s
∀
angular speed Ω . Moreover, since u =ω˙ =0, k in the invariant set since (27) holds. Following Lemma 2, since
d k k
∀
set ∆ , it implies from (20) that (∂W /∂θ )=0, k, which minimization of thepotentialW (θθθ)corresponds tothe global
b 1 k 1
∀
defines the critical points of W (θθθ). The set ∆ is itself the minimum of V (θθθ,ωωω), balanced formation is asymptotically
1 b 2
6
30 1 0.8 0.8
Limited communication Limited communication All to all communication
Y (m)−1210000 |p|θ0000....2468 All to all communication Angular frequencies (rad/sec)−−−000000......0246642 Angular frequencies (rad/sec)−−−000000......0246642
−2−020 −10 0 10 20 30 00 5 10 15 20 −0.80 10 20 30 −0.80 10 20 30
X (m) time (sec) time (sec) time (sec)
(a) (b) (c) (d)
F(aig).T2r.ajecBtoarlaienscionfgthoef Nag=ent6saugnednetrs,thceoncnoencttreodl lbaywas (g2r0ap)hwaitshsKho=wn1.in(bF)igA.v1e(bra),geonlininedarivmidoumalecnitrucmlesa,peparcohachhaevsinzgertohewditehsitriemder.a(dci)usCρodns=en|sΩusd|o−f1a=ng5ulmar.
speeds(frequencies)atdesiredvalueΩd=0.2rad/sec(limitedcommunication).(d)ConsensusofangularspeedsatdesiredvalueΩd=0.2rad/sec(all-to-all
communication)
20 350 Limited communication 350 All to all communication 0.8 Limited communication 0.8 All to all communication
Y (m)−−111055050 Heading Angles (deg)11223050505000000 Heading Angles (deg)11223050505000000 Angular frequencies (rad/sec)−−−000000......0246642 Angular frequencies (rad/sec)−−−000000......0246642
−15 0 0 −0.8 −0.8
−20 −10 0 10 20 0 5 10 15 20 25 0 5 10 15 20 25 0 10 20 30 0 10 20 30
X (m) time (sec) time (sec) time (sec) time (sec)
(a) (b) (c) (d) (e)
(Faig).T3r.ajeScytonrciehsroonfiztahteioangeonftNsu=nd6eargtehnetsc,ocnotrnonlelcatwedsb(2y9a)gwriatphhKas=sho1w.n(bin)CFiogn.1se(bn)s,uosnininhdeivadidinugalacnigrcleless,(leimacihtehdacvoinmgmthuenidceastiiorend).r(acd)iuCsoρndse=ns|Ωusd|i−n1h=ea5dimng.
angles (all-to-all communication). (d) Consensus of angular speeds−at desired value Ωd=0.2 rad/sec (limited communication). (e) Consensus of angular
speedsatdesiredvalueΩd=0.2rad/sec(all-to-allcommunication).Sincetheagentscontinuetorotatearoundtheirrespectivecirclesinsynchronizedfashion,
theirheadingangleskeepincreasingwithtime.
stable. This completes the proof. value Ω is faster under all-to-all communication scenario as
d
Remark 2: In the previous analysis, the invariant sets ∆ expected.
b
and ∆ , defined by (26) and (34), respectively, look similar,
s
however, they are different since ∆ contains phases θθθ corre-
b
sponding to the balanced formation, while ∆ contains phases IV. MOTIONONACOMMONCIRCLE
s
θθθ corresponding to the synchronized formation.
In this section, we achieve synchronization and balancing
Example 1: In this example, Theorems 1 and
of a group of agents around a common circle of desired
2 are demonstrated through simulation of N = 6
radius ρ = Ω 1 as well as center c , which is fixed. This
agents connected via a graph as shown in Fig. 1(b). situationdis s|ho−dwn| in Fig. 4, where, tdhe kth agent rotates in
Let the initial positions, initial heading angles and
the anticlockwise direction on a circle of radius ρ = Ω 1
nonidentical initial angular velocities of the agents be d | −d |
and center c . Without loss of generality, we assume Ω >0.
rrr(0) = ((1, 1),(10,3),( 1, 5),( 5,1),(12,5),( 4,10))T, d d
θθθ(0) = (3−0 ,45 ,120 ,−75 −,90 ,6−0 )T, and −ωωω(0) = Therefore, at equilibrium, all the agents, which may initially
◦ ◦ ◦ ◦ ◦ ◦ be rotating in clockwise and anticlockwise directions, move
(0.2, 0.3,0.4, 0.5,0.6, 0.8)T, respectively. Although
− − − in the anticlockwise direction on a common circle of desired
the initial locations of the agents are given for representing
radius as well as center. The position r of the k-th agent in
the trajectories of the agents in the simulation, the locations k
Fig. 4, is given by
themselves are not important so far as the objective of
synchronization is concerned. Even with different locations, r =c iρ eiθk. (35)
k d d
−
the convergence properties will be the same, although the
In order to stabilize collective motion of all the agents around
trajectories will be different.
a common circle of radius ρ = Ω 1 > 0 and center c ,
Fig. 2 shows balancing of the agents around individual d −d d
which is fixed, we introduce an error variable e =r (c
chiernccleesdeastirdeedsirraeddiusanρgdul=arΩvdel−oc1i=ty5ωmd =of0c.i2rcurlaadr/soercb,itsanads iρdeiθk),∀ k, and choose a potential function as,k k− d−
| |
expected. On the other hand, Fig. 3 shows synchronization 1 N 1 N
of agents on individual circles at the same desired angular S(rrr,θθθ) = ∑ ek 2= ∑ rk cd+iρdeiθk 2
2 | | 2 | − |
frequencyvelocity.Inallfiguresinthispaper,thetrajectoryof k=1 k=1
the centroid is shown in black. Note that, in these figures, the = 1 ∑N r c +iρ eiθk,r c +iρ eiθk (,36)
k d d k d d
convergenceoftheangularspeeds(frequencies)tothedesired 2 − −
k=1
(cid:68) (cid:69)
7
wcdhichiρdiseiθnk,onk-.neIgtatmiveeanasndthabtectohme emsinziemroizawtihoenneovferS(rrrrk,θθθ=) Y axis eiθk k−-tiheiθakgent
− ∀
corresponds to the situation when all the agents move on a ρd
common circle of radius ρd =Ω−d1>0 centered at the fixed rk cdΩd
point c .
d
The time derivative of the potential function (36) along the
dynamics (1), yields Xaxis
S˙(rrr,θθθ)= ∑N rk cd iρdeiθk,eiθk(1 ρdωk) (37) aFnigd.c4e.ntOerricedn.tationofthekthagentonacircleofdesiredradiusρd=Ω−d1>0
− − −
k=1
(cid:68) (cid:69)
Using linearity of inner product [32] in (37), we get
all the solutions of (1) with the control law (40) converge to
N
S˙(rrr,θθθ) = ∑ rk cd,eiθk (1 ρdωk) thelargestinvariantsetcontainedin U˙1(rrr,θθθ,ωωω)=0 ,thatis,
− − { }
k=1 the set
(cid:68) (cid:69)
N Γ= (rrr,θθθ,ωωω) ω =Ω , k . (45)
∑ρ ieiθk,eiθk (1 ρ ω ) (38) { | k d ∀ }
d d k
− −
k=1 (cid:68) (cid:69) In Γ, ωk=Ωd, ∀k, which implies that each agent rotates with
Since ieiθk,eiθk =0, (38) is simplified to angularspeedΩd.Moreover,sinceuk=ω˙k=0, k,intheset
∀
Γ, it implies from (40) that
(cid:10) (cid:11) N
S˙(rrr,θθθ)= ∑ rk−cd,eiθk (1−ρdωk). (39) κ rk cd,eiθk +K∂W1 =0, (46)
k=1(cid:68) (cid:69) − ∂θk
Theorem 3: Let L be the Laplacian of an undirected and for all k. Let Γ b(cid:68)e the larges(cid:69)t invariant set in Γ. Taking the
b
connected circulant graph G =(V,E) with N vertices. Con- time-derivative of (46) in the set Γ yields
sider system dynamics (1) with control law
d d ∂W
uk=−κρd(ωk−Ωd)+Ω2d(cid:18)κ(cid:68)rk−cd,eiθk(cid:69)+K∂∂Wθk1(cid:19)(4.0) Using (27),κ(d4t7)(cid:68)rbke−cocmde,seiθk(cid:69)+Kdt(cid:18)∂θk1(cid:19)=0. (47)
For K >0 and κ >0, all the agents converge to a circular d
r c ,eiθk = r c ,iΩ eiθk + eiθk,eiθk =0.
formation in which they travel around a common circle of dt k− d k− d d
radius ρ = Ω 1 > 0 and center c in the anticlockwise (cid:68) (cid:69) (cid:68) (cid:69) (cid:68) (cid:69) (48)
directiondwith th−deir phase angles in badlanced state. Since eiθk,eiθk =1, (48) can be written as
Proof: Consider a composite potential function
(cid:10) (cid:11) r c ,iΩ eiθk = 1. (49)
k d d
N − −
U (rrr,θθθ,ωωω)=κS(rrr,θθθ)+ρ K λ W (θθθ) +ρ3G(ωωω); K>0,κ>0. (cid:68) (cid:69)
1 d 2 max− 1 d It is easy to check that (49) is satisfied only if
(cid:18) (cid:19) (41)
r =c iρ eiθk, k, (50)
Using (10), (19) and (39), the time derivative of the potential k d− d ∀
functionU1(rrr,θθθ,ωωω) along the trajectories of (1), is whichisthepositionofthekthagentrotatingaroundacircleof
radius ρ and center at c (see Eq. (35)). This implies that, in
N d d
U˙1(rrr,θθθ,ωωω)=κ ∑ rk cd,eiθk (1 ρdωk) thesetΓb,alltheagentsconvergetoacommoncircleofradius
k=1(cid:68) − (cid:69) − ρd =Ω−d1 >0 and center cd. Moreover, by substituting (50)
+K∑N ∂W1 ( ρ ω )+ρ3∑N (ω Ω )u (42) in (46), we get (∂W1/∂θk)=0, ∀k, which defines the critical
k=1(cid:18)∂θk (cid:19) − d k dk=1 k− d k points of W1(θθθ) belonging to the set Γb. Since maximization
of the potential W (θθθ) corresponds to the global minimum
1
Using (24), (42) can be rewritten as of U (rrr,θθθ,ωωω), balanced formation is asymptotically stable
1
N N ∂W (Lemma 2). This completes the proof.
U˙1(rrr,θθθ,ωωω)= ∑ κ rk cd,eiθk +K∑ 1 Theorem 4: Let L be the Laplacian of an undirected and
k=1(cid:32) (cid:68) − (cid:69) k=1 ∂θk (cid:33) connected circulant graph G =(V,E) with N vertices. Con-
N
(1 ρ ω )+ρ3∑(ω Ω )u (43) sidersystemdynamics(1)withcontrollaw(40).ForK<0and
× − d k d k− d k κ>0,alltheagentsconvergetoacircularformationinwhich
k=1
they travel around a common circle of radius ρ =Ω 1>0
Under control law (40), the time derivative of U (rrr,θθθ,ωωω) d −d
1 and center c in the anticlockwise direction with their phase
d
results in
angles in synchronized state.
N
U˙ (rrr,θθθ,ωωω)= κρ4∑(ω Ω )2 0. (44) Proof: Consider a composite potential function
1 − d k− d ≤
k=1 U (rrr,θθθ,ωωω)=κS(rrr,θθθ) ρ KW (θθθ)+ρ3G(ωωω); K<0,κ>0.
Since(rrr,θθθ,ωωω) RN TN RN iscompact,itfollowsfromthe 2 − d L d (51)
∈ × ×
LaSalle’s invariance theorem [30] that, for K>0 and κ >0, Underthecontrol(40),thetimederivativeofU (rrr,θθθ,ωωω)along
2
8
20 20 0.8
Y (m)−−111055050 Y (m)−−111055050 Angular frequencies (rad/sec)−−−000000......0246642
−1−510 0 10 20 30 −1−510 0 10 20 30 −0.80 100 200 300
X (m) X (m) time (sec)
(a) (b) (c)
Fig. 5. Synchronization and balancing of N=6 agents, connected by a graph as shown in Fig.1(b), around the common circle of desired radius ρd=
t|Ωhed|c−o1nt=ro5llmawan(d40d)ewsiritehdKce=nter1cda=nd(κ20=,50)..1(.a)(cB)aClaonncseednsfuosrmoaftiaonnguulnadrevretlhoecictioenstraotldlaewsire(4d0v)awluitehΩKd==00.5.2arnadd/κse=cf0o.1r.b(abla)nScyendcfhorromniazteiodnf.ormationunder
−
the trajectories of (1), yields by the formations in which the agents are in phase balancing
along with a symmetrical arrangement of their phase angles
N
U˙ (rrr,θθθ,ωωω)=U˙ (rrr,θθθ,ωωω)= κρ4∑(ω Ω )2 0. (52) around the desired common circle [5].
2 1 − d k− d ≤
k=1 Let1 M N beadivisorofN.Asymmetricarrangement
≤ ≤
Since U˙ (rrr,θθθ,ωωω)=U˙ (rrr,θθθ,ωωω), the proof follows the same ofN phasesconsistingofM clustersuniformlyspacedaround
2 1
steps as used to prove Theorem 3. However, in this case, let the common circle, each with N/M synchronized phases,
Γ bethelargestinvariantsetinΓdefinedin(45).Thus,itcan is called an (M,N)-pattern. For instance, the (1,N)-pattern
s
be concluded that all the agents converge to a common circle corresponds to the synchronized state and the (N,N)-pattern
of radius ρ =Ω 1>0 and center c in the set Γ . Moreover, correspondstotheso-calledsplaystate,whichischaracterized
d −d d s
since minimization of the potential W (θθθ) corresponds to the by N phases uniformly spaced around the common circle.
1
global minimum of U2(rrr,θθθ,ωωω), synchronized formation is We now state the following lemma [28] that is useful in
asymptoticallystable(Lemma2)inthesetΓs.Thiscompletes proving the results of this section.
the proof.
Lemma 3: Let 1 M N be a divisor of N. An arrange-
Example 2: In this example, the simulation results are ≤ ≤
ment θθθ of N phases is an (M,N)-pattern if and only if, for
presented for the same 6 agents as considered in Example 1.
all m 1,...,M 1 , the phase arrangement mθθθ is balanced
Fig. 5 depicts the synchronization and balancing of the and th∈e{phase arr−ang}ement Mθθθ is synchronized.
agents around a common circle at desired angular speed
As mentioned in the above lemmas, the mth harmonic of
Ω =0.2 rad/sec (and hence desired radius ρ =5 m) and
d d the potential W (θθθ) defined in (12) thus plays an important
desired center c =(20,5). Balanced formation is shown in m
d role in stabilizing symmetric phase patterns. Moreover, for
Fig.5(a),andsynchronizedformationisshowninFig.5(b).In
various values of m, different symmetric balanced patterns
Fig. 5(c), the convergence of the angular speeds of the agents
arise as shown in Fig for N=6. See [5] for a more detailed
to a desired value Ω =0.2 rad/sec, is shown in balanced
d description.
formation only since the plot for synchronized formation is
Following Lemma 3, we choose an (M,N) phase potential
similar.
as
Note that these figures are obtained by setting different
values of gains K and κ. The selection of these gains is M 1K N K
WM,N(θθθ)= ∑− m λ W (θθθ) MW (θθθ), (53)
arbitrary and depends upon a particular problem of interest. m2 2 max− m −M2 M
m=1 (cid:18) (cid:19)
By selecting gains K and κ appropriately, we can make the
with K >0 for 1 m M 1, and K <0. The global min-
system of agents to stabilize in a desired formation, at faster m M
imumofWM,N(θθθ)≤achi≤eved−(only)whenW (θθθ)ismaximized
or slower convergence rates. m
for1 m M 1,andW (θθθ)isminimized.Thiscorresponds
Itisworthnotingthat,inFig. 5(a),thefinalpositionofthe M
≤ ≤ −
to the situation when θθθ,2θθθ,...,(M 1)θθθ are balanced and
centroid of the group coincides with the center c =(20,5) of
d −
Mθθθ is synchronized (Lemma 2), and hence the minimization
the common circle. This is due to the fact that, in balanced
of (53) gives rise to an (M,N)-pattern (Lemma 3).
formation, the linear momentum p =p =0, which causes
θ 1θ
(35) to reduce to cd =(1/N)∑Nk=1rk when summed over all Substituting forWm(θθθ) from (12) into (53), yields
k on both the sides, which is the average position of all the
1M 1K
agents, or the position of their centroid. WM,N(θθθ)= ∑− m Nλ eimθθθ,Leimθθθ
2 m2 max−
m=1
(cid:16) (cid:68) (cid:69)(cid:17)
K
V. SYMMETRICBALANCEDPATTERNSAROUNDTHE M eiMθθθ,LeiMθθθ , (54)
−2M2
COMMONCIRCLE (cid:68) (cid:69)
whose time derivative along the dynamics (1), is given by
In this section, we aim to achieve symmetric balanced
N M
pρat=terωnso1f>th0eaagsewntesllaraosucnedntaecrocm.mTohnesceircplaettoefrndsesairreeddreafidnieuds W˙M,N(θθθ)=−∑ ∑ Km ieimθk,Lkeimθθθ ωk, (55)
d d− d k=1m=1
(cid:68) (cid:69)
9
20 20 20
15 15 15
10 10 10
m) 5 m) 5 m) 5
Y ( 0 Y ( 0 Y ( 0
−5 −5 −5
−10 −10 −10
−1−510 0 10 20 30 −1−510 0 10 20 30 −1−510 0 10 20 30
X (m) X (m) X (m)
(a) (b) (c)
Fig. 6. Different symmetric balanced patterns of 6 agents, connected by a graph as shown in Fig.1(b), around the common circle of desired radius
ρ(bd)=(3Ω,6−d)1p=at5temrn.a(ncd)d(e6s,i6r)edpactetnertenr:csdpl=ay(2fo0r,m5)a,tiuonnd.erthecontrollaw(56)withK=1,κ=Km=0.1for1≤m≤M−1,andKM=−0.5.(a)(2,6)pattern.
Using (59), (58) can be rewritten as
Block1 Block2 Block3
N M
G1 G2 G3 V˙(rrr,θθθ,ωωω)= ∑ κ rk cd,eiθk +K ∑ Km ieimθk,Lkeimθθθ
G=G1 G2 G3 k=1(cid:32) (cid:68) − (cid:69) m=1 (cid:68) (cid:69)(cid:33)
(cid:2) (cid:2) N
(1 ρ ω )+ρ3∑(ω Ω )u . (60)
Fig. 7. Block interaction of 12 agents with 3 blocks, each containing 4 × − d k d k− d k
agents. k=1
Under the control law (56), the time derivative of V(rrr,θθθ,ωωω)
results in
N
V˙(rrr,θθθ,ωωω)=U˙ (rrr,θθθ,ωωω)= κρ4∑(ω Ω )2 0. (61)
1 − d k− d ≤
k=1
where, L is the kth row of the Laplacian L.
k Since V˙(rrr,θθθ,ωωω) =U˙ (rrr,θθθ,ωωω), the proof follows the same
1
steps as used to prove Theorem 3. However, in this case, let
Theorem 5: Let L be the Laplacian of an undirected and
Ξ be the largest invariant set in Γ defined in 45. Thus, it can
connected circulant graph G =(V,E) with N vertices. Con-
be concluded that all the agents converge to a common circle
sider system dynamics (1) with control law
of radius ρ =Ω >0 and center c in the set Ξ. Moreover,
d d d
uk= κρd(ωk Ωd) it follows from Lemma 2 that the maximization of Wm(θ)
− − for 1 m M 1, and minimization of W (θ) corresponds
M ∂W ≤ ≤ − M
+Ω2d(cid:32)κ rk−cd,eiθk +Km∑=1Km ∂θkm(cid:33) (56) tpohathseeagrlroabnaglemmeinnitmsθuθθm,2θθθof,.t.h.e,(pMoten1t)iθaθθlaVre(rrrb,θθθal,aωωωnc)e.dTahnuds,Mthθθθe
(cid:68) (cid:69) −
is synchronized in the set Ξ, and hence give rise to an (M,N)
with K >0 for 1 m M 1, and K <0. For K>0 and
m M
≤ ≤ − phase pattern (Lemma 3). This completes the proof.
κ>0,alltheagentsconvergetoacircularformationinwhich
Example 3: In this example, the simulation results are
they travel around a common circle of radius ρ =Ω 1>0
d −d presented for the 6 agents considered in Example 1. Fig. 6
and center c in the anticlockwise direction with their phase
d
shows the different symmetric balanced patterns of the agents
angles in the (M,N)-pattern.
around the common circle of desired radius ρ =Ω 1=5 m
d −d
and center c = (20,5). The arrangement in Fig. 6(c) is
Proof: Consider a composite potential function d
the splay state, in which the 6 agents are at equal angular
V(rrr,θθθ,ωωω)=κS(rrr,θθθ)+ρdKWM,N(θθθ)+ρd3G(ωωω) (57) separation of 60◦, as expected.
Remark 3: We may assume that agents move on different
with K >0 for 1 m M 1, and K <0. Using (19),
m M altitudes as in [8] when they are in synchronization around
≤ ≤ −
(39) and (55), the time derivative of the potential function
a common circular orbit. Approaches to collision avoidance
V(rrr,θθθ,ωωω) along the dynamics (1), is
among agents will be explored in future.
N
V˙(rrr,θθθ,ωωω)=κ ∑ rk cd,eiθk (1 ρdωk) VI. ACHIEVINGCOORDINATEDSUBGROUPS
− −
k=1
(cid:68) (cid:69) Motivated by mobile sensor network applications, as dis-
M N N
+Kk∑=1m∑=1Km ieimθk,Lkeimθθθ (−ρdωk)+ρd3k∑=1(ωk−Ωd)uk. cuusesssedmuinltip[6le],girnapthhsisastecatitoinm,ewteo pyrioelpdosmeuclotin-ltervoelllaswenstihnagt
(cid:68) (cid:69)
(58) patterns in a group of agents. By using such a multi-level
interaction among agents, a formation in which the agents are
Note that
arranged in subgroups around different circular orbits with a
N N
∑ ieimθk,Lkeimθθθ = ∑ ∑ sin(m(θj θk))=0 (59) symmetric pattern of their phase angles, can be obtained. In
k=1(cid:68) (cid:69) −k=1j∈Nk − this context, the different graphs, by considering both intra-
10
and inter subgroup coordination, are used in the steering Proof: Consider a composite potential function
controllaw,sothatsubgroupscanbeformedandregulated.A
N
particular application of such formations can be found in the U1b(rrr,θθθ,ωωω)=κSb(rrr,θθθ)+ρdbK 2λmax−Wˆ1(θθθ)
optimal mobile sensor coverage problem where it is required (cid:18) (cid:19)
3
forthesensorstomovearoundclosedcurveswithcoordinated + ρdb Gb(ωωω); κ>0,K>0, (64)
phasing for maximizing information in data collected in a (cid:110) (cid:111)
where,
spatially and temporally varying field (for instance, from an
oceInano)rd[3e1r]t.o use multi-level sensing networks, we divide the Sb(rrr,θθθ)= 12 ∑N rk−cbd+iρ0beiθk,rk−cbd+iρdbeiθk ,
k=1
(cid:68) (cid:69)
group of N agents into B blocks (subgroups) and refer to
Wˆ1(θθθ)=(1/2) eiθθθ,Lˆeiθθθ ,
each block by its block index 1 b B. For the sake of
convenience, we assume that there≤is n≤o interaction between 1 N (cid:68) (cid:69)2
subgroups unless specifically mentioned. Let the graph G Gb(ωωω)= 2 ∑ ωk−Ωbd ,
b k=1
(cid:16) (cid:17)
describe the interaction between all the agents in block b.
and λ is the maximum eigenvalue of Lˆ. Since (64) has
Collectively, the set of all interaction is defined by the graph max
G(cid:44) B G .Inthissituation,theLaplacianLˆ corresponding a structure similar to (41), the proof is similar to the proof of
to grapbh=1Gbis a block-diagonal matrix, each block of which Theorem 3 and hence omitted.
repre(cid:83)sentstheLaplacianofthegraphG .Forinstance,consider SimilarlythecorollariestoTheorems4and5canbestated,
b
and are not described here to avoid repetition.
agroupof12agentsdividedinto3blockscontaining4agents
in each block. Let their interaction network be represented by
B. Case 2: ρb=ρ for all 1 b B
a graph G =G1 G2 G3 as shown in Fig. 7. The Laplacian d d ≤ ≤
In such a situation, by assuming that the agents within a
matrix corresponding to this block interaction is given by
(cid:83) (cid:83) block can also interact with the agents in other blocks in ad-
Lˆ= 00L44××G441 00L44××G442 00L44××G443  (62) dlpeihvtaieoslenspteoonttsehinnetgiirainlnsettecwroorbrrlekoscpiknoninadtiesnirgnagcttloieotnhc,oeintdtiirsfoflperoaeslngstiobcrlioethmtmomuussinenicmceautitlohtine-
wLhGere=, c0i4rc×(42,rep1r,e0s,en1ts) aand4×L4G z=erocirmc(a3t,rix1,,an1d, L1)G1a=re tfoupnoctlioogni.esFocrainnsbtaenccoe,mcboinnseiddetroaggertouapsoinfg1l2eapgheanstesapsostehnotwianl
the3Laplacian o−fthe s−ubgraphs G (2=G )and−G ,−resp−ectively. in Fig. 7, now with two types of sensing networks comprising
1 3 2
Since Lˆ is a block diagonal matrix, its eigenvalues are the an intra- and an inter subgroup coordination. Assume that the
union of the eigenvalues of all block diagonal matrices [32]. intra subgroup interaction is all-to-all and the inter subgroup
In this illustration, although we assume that each subgroup is interactionisasshownFig.7.LetL˜=NP(=NIN 111111T),and
of the same size, however, this is not required. Lˆ,givenby(62),betheLaplaciancorresponding−totheintra-
We further assume that each agent is assigned to one and andintersubgroupcoordination,respectively.Then,withsuch
only one block, so that ∑Bb=1Nb=N, where Nb is the number amulti-levelinteraction,thepotentialfunctionin(64)becomes
of agents in the bth block. Also, let Fb = f1b,...,fNbb Ub(rrr,θθθ,ωωω)=κSb(rrr,θθθ)+ρ K Nλ W (θθθ) +ρ3 G(ωωω),
be the set of agent indices, and ρdb = Ωbd −1(cid:110)>0 and cbd(cid:111), 1 d (cid:18)2 max− 1 (cid:19) d (65)
respectively, be the desired radius, and center of the common
(cid:8) (cid:9) where,
circular orbit associated with the bth block. 1
Based on these notations, we now consider the following W1(θθθ)= eiθθθ,(Lˆ+L˜)eiθθθ . (66)
2
twocasesofcoordinatedsubgroupswithrespecttothedesired (cid:68) (cid:69)
b
radius ρb for a reason which will be apparent little later. ThepotentialU1(rrr,θθθ,ωωω)isminimizedbychoosingthecontrol
d
u = κρ (ω Ω )+ Ω 2
A. Case 1: ρb is different for all 1 b B k − d k− d { d}
d ≤ ≤ κ r cb,eiθk +K ieiθk, Lˆ +L˜ eiθθθ , (67)
In this situation, we have the following corollaries to × k− d k k
Theorems 3, 4 and 5. which (cid:104)acc(cid:68)ounts for bo(cid:69)th intr(cid:16)a-(cid:68)and i(cid:0)nter subgr(cid:1)oups(cid:69)(cid:17)co(cid:105)ordina-
Corollary 1: Fork Fb,considersystemdynamics(1)with
control law ∈ tion. Note that, unlike previous case of the different radius
of the circular orbits, here we can combine all the phase
2
uk=−κρdb(ωk−Ωbd)+ Ωbd κ rk−cbd,eiθk +K ieiθk,Lˆkeiθθθ , stabilizing potentials independently to get a single potential
(cid:110) (cid:111) (cid:16) (cid:68) (cid:69) (cid:68) (6(cid:69)3(cid:17)) function since ρdb = ρd for all b = 1,...,B. Thus, under
where, Lˆk is the kth row of the Laplacian Lˆ of a graph G = the control (67), the phase arrangement in which the entire
Bb=1Gb with Gb being an undirected and connected circulant group, as well as each block, are in balanced formation, is
subgraph. For K >0 and κ >0, all the agents belonging to obtained. The same behavior of agent’s phase arrangement
(cid:83)
the bth block converge to a circular formation in which they in synchronized and splay formations can be depicted under
move on a circle of radius ρb = Ωb −1 >0 and center cb suitable control laws.
d d d
in the anticlockwise direction with their phase angles in the Remark 4: It is evident from the above discussion that the
(cid:8) (cid:9)
balanced state. secondorderrotationalmodelisadequatetocontrolthephase