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988 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
Coordination of Groups of Mobile Autonomous
Agents Using Nearest Neighbor Rules
Ali Jadbabaie, Jie Lin, and A. Stephen Morse, Fellow, IEEE
Abstract—In a recent Physical Review Letters article, Vicsek
et al. propose a simple but compelling discrete-time model of
autonomous agents (i.e., points or particles) all moving in the plane
with the same speed but with different headings. Each agent’s
heading is updated using a local rule based on the average of its
own heading plus the headings of its “neighbors.” In their paper,
Vicsek et al. provide simulation results which demonstrate that
the nearest neighbor rule they are studying can cause all agents
to eventually move in the same direction despite the absence of
centralized coordination and despite the fact that each agent’s
set of nearest neighbors change with time as the system evolves.
This paper provides a theoretical explanation for this observed
behavior. In addition, convergence results are derived for several
other similarly inspired models. The Vicsek model proves to be
a graphic example of a switched linear system which is stable,
but for which there does not exist a common quadratic Lyapunov
function.
Index Terms—Cooperative control, graph theory, infinite prod-
ucts, multiagent systems, switched systems.
I. INTRODUCTION
I
N [1], Vicsek et al. propose a simple but compelling
discrete-time model of
autonomous agents (i.e., points
or particles) all moving in the plane with the same speed but
with different headings. Each agent’s heading is updated using
a local rule based on the average of its own heading plus the
headings of its “neighbors.” Agent
’s neighbors at time , are
those agents which are either in or on a circle of pre-specified
radius
centered at agent ’s current position. The Vicsek
model turns out to be a special version of a model introduced
previously by Reynolds [2] for simulating visually satisfying
flocking and schooling behaviors for the animation industry. In
their paper, Vicsek et al. provide a variety of interesting simu-
lation results which demonstrate that the nearest neighbor rule
they are studying can cause all agents to eventually move in the
same direction despite the absence of centralized coordination
and despite the fact that each agent’s set of nearest neighbors
change with time as the system evolves. In this paper, we
provide a theoretical explanation for this observed behavior.
Manuscript received March 4, 2002; revised December 2, 2002. Recom-
mended by Associate Editor A. Bemporad. This work was supported by the
Defense Advanced Research Project Agency (DARPA) under its SEC program
and by the National Science Foundation under its KDI/LIS initiative.
A. Jadbabaie was with Yale University, New Haven, CT 06520 USA. He is
now with the Department of Electrical and Systems Engineering and GRASP
Laboratory, University of Pennsylvania, Philadelphia, PA 19104 USA (e-mail:
jadbabai@seas.upenn.edu).
J. Lin and A. S. Morse are with the Center for Computational Vision and
Control, Department of Electrical Engineering, Yale University, New Haven,
CT 06520 USA.
Digital Object Identifier 10.1109/TAC.2003.812781
There is a large and growing literature concerned with
the coordination of groups of mobile autonomous agents.
Included here is the work of Czirok et al. [3] who propose
one-dimensional models which exhibit the same type of
behavior as Vicsek’s. In [4] and [5], Toner and Tu construct
a continuous ”hydrodynamic" model of the group of agents,
while other authors such as Mikhailov and Zanette [6] consider
the behavior of populations of self propelled particles with
long range interactions. Schenk et al. determined interactions
between individual self-propelled spots from underlying reac-
tion-diffusion equation [7]. Meanwhile, in modeling biological
systems, Grünbaum and Okubo use statistical methods to
analyze group behavior in animal aggregations [8]. This paper
and, for example, the work reported in [9]–[12] are part of a
large literature in the biological sciences focusing on many
aspects of aggregation behavior in different species.
In addition to thesemodeling and simulation studies, research
papers focusing on the detailed mathematical analysis of emer-
gent behaviors are beginning to appear. For example, Lui et al.
[13] use Lyapunov methods and Leonard et al. [14] and Ol-
fati and Murray [15] use potential function theory to understand
flocking behavior, and Ögren et al. [16] uses control Lyapunov
function-based ideas to analyze formation stability, while Fax
and Murray [17] and Desai et al. [18] employ graph theoretic
techniques for the same purpose.
The one feature which sharply distinguishes previous ana-
lyzes from that undertaken here is that this paper explicitly takes
into account possible changes in nearest neighbors over time.
Changing nearest neighbor sets is an inherent property of the
Vicsek model and in the other models we consider. To ana-
lyze such models, it proves useful to appeal to well-known re-
sults [19], [20] characterizing the convergence of infinite prod-
ucts of certain types of nonnegative matrices. The study of in-
finite matrix products is ongoing [21]–[26], and is undoubtedly
producing results which will find application in the theoretical
study of emergent behaviors.
Vicsek’s model is set up in Section II as a system of
simultaneous, one-dimensional recursion equations, one for
each agent. A family of simple graphs on
vertices is then
introduced to characterize all possible neighbor relationships.
Doing this makes it possible to represent the Vicsek model
as an
-dimensional switched linear system whose switching
signal takes values in the set of indices which parameterize
the family of graphs. The matrices which are switched within
the system turn out to be nonnegative with special structural
properties. By exploiting these properties and making use of a
classical convergence result due to Wolfowitz [19], we prove
that all
agents’ headings converge to a common steady state
0018-9286/03$17.00 © 2003 IEEE
JADBABAIE et al.: COORDINATION OF GROUPS OF MOBILE AUTONOMOUS AGENTS 989
heading provided the agents are all “linked together” via
their neighbors with sufficient frequency as the system evolves.
The model under consideration turns out to provide a graphic
example of a switched linear system which is stable, but for
which there does not exist a common quadratic Lyapunov
function.
In Section II-B, we define the notion of an average heading
vector in terms of graph Laplacians [27] and we show how
this idea leads naturally to the Vicsek model as well as to
other decentralized control models which might be used for the
same purposes. We propose one such model which assumes
each agent knows an upper bound on the number of agents in
the group, and we explain why this model has convergence
properties similar to Vicsek’s.
In SectionIII, we consider a modified version of Vicsek’s dis-
crete-time system consisting of the same group of
agents, plus
one additional agent, labeled 0, which acts as the group’s leader.
Agent 0 moves at the same constant speed as its
followers but
with a fixed heading
. The th follower updates its heading just
as in the Vicsek model, using theaverageof its own heading plus
the headings of its neighbors. For this system, each follower’s
set of neighbors can also include the leader and does so when-
ever the leader is within the follower’s neighborhood defining
circle of radius
. We prove that the headings of all agents
must converge to the leader’s provided all
agents are “ linked
to their leader” together via their neighbors frequently enough
as the system evolves. Finally, we develop a continuous-time
analog of this system and prove under condition milder than im-
posed in the discrete-time case, that the headings of all
agents
again converge to the heading of the group’s leader.
II. L
EADERLESS COORDINATION
The system studied by Vicsek et al. [1] consists of au-
tonomous agents (e.g., points or particles), labeled 1 through
,
all moving in the plane with the same speed but with different
headings.
1
Each agent’s heading is updated using a simple local
rule based on the average of its own heading plusthe headings of
its “neighbors.” Agent
’s neighbors at time , are those agents
which are either in or on a circle of pre-specified radius
cen-
tered at agent
’s current position. In the sequel denotes
the set of labels of those agents which are neighbors of agent
at time . Agent ’s heading, written , evolves in discrete-time
in accordance with a model of the form
(1)
where
is a discrete-time index taking values in the nonnegative
integers
, and is the average of the head-
ings of agent
and agent ’s neighbors at time ; that is
(2)
where
is the number of neighbors of agent at time .
Observe that the preceding heading update rule maps headings
with values
into a heading with a value also in .
1
The Vicsek system also includes noise input signals, which we ignore in this
paper.
Because of this, it makes sense to represent headings at any
finite time
, as real numbers in . Of course it is entirely
possible that in the limit as
, a heading might approach
the value
; any such limiting value is interpreted as a heading
of 0. Analogous statement apply to all other models considered
in the sequel. Accordingly, throughout the paper headings at any
finite time
, are represented as real numbers in .
The explicit form of the update equations determined by
(1) and (2) depends on the relationships between neighbors
which exist at time
. These relationships can be conveniently
described by a simple, undirected graph
2
with vertex set
which is defined so that is one of the
graph’s edges just in case agents
and are neighbors. Since
the relationships between neighbors can change over time, so
can the graph which describes them. To account for this we
will need to consider all possible such graphs. In the sequel we
use the symbol
to denote a suitably defined set, indexing the
class of all simple graphs
defined on vertices.
The set of agent heading update rules defined by (1) and (2),
can be written in state form. Toward this end, for each
,
define
(3)
where
is the adjacency matrix of graph and the diag-
onal matrix whose
th diagonal element is the valence of vertex
within the graph. Then
(4)
where
is the heading vector and
is a switching signal whose value at time
, is the index of the graph representing the agents’ neighbor
relationships at time
. A complete description of this system
would have to include a model which explains how
changes
over time as a function of the positions of the
agents in the
plane. While such a model is easy to derive and is essential for
simulation purposes, it would be difficult to take into account in
a convergence analysis. To avoid this difficulty, we shall adopt
a more conservative approach which ignores how
depends on
the agent positions in the plane and assumes instead that
might
be any switching signal in some suitably defined set of interest.
Our goal is to show for a large class of switching signals
and for any initial set of agent headings that the headings
of all
agents will converge to the same steady state value
. Convergence of the to is equivalent to the state
vector
converging to a vector of the form where
. Naturally, there are situations where
convergence to a common heading cannot occur. The most
obvious of these is when one agent—say the
th—starts so
far away from the rest that it never acquires any neighbors.
Mathematically, this would mean not only that
is never
2
By an undirected graph on vertex set
V
=
f
1
;
2
;
...
n
g
is meant
V
together with a set of unordered pairs
E
=
f
(
i; j
):
i; j
2Vg
which are called
’s edges. Such a graph is simple if it has no self-loops [i.e.,
(
i; j
)
2E
only if
i
6
=
j
] or repeated edges (i.e.,
E
contains only distinct elements). By thevalence
of a vertex
v
of is meant the number of edges of which are “incident” on
v
where by an indicant edge on
v
is meant an edge
(
i; j
)
of for which either
i
=
v
or
j
=
v
. The adjacency matrix of is an
n
2
n
matrix of whose
ij
th
entry is 1 if
(
i; j
)
is one of ’s edges and 0 if it is not.
990 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 6, JUNE 2003
connected
3
at any time , but also that vertex remains an
isolated vertex of
for all . This situation is likely to be
encountered if
is very small. At the other extreme, which
is likely if
is very large, all agents might remain neighbors
of all others for all time. In this case,
would remain fixed
along such a trajectory at that value in
for which
is a complete graph. Convergence of to can easily be
established in this special case because with
so fixed, (4)
is a linear, time-invariant, discrete-time system. The situation
of perhaps the greatest interest is between these two extremes
when
is not necessarily complete or even connected for
any
, but when no strictly proper subset of ’s vertices
is isolated from the rest for all time. Establishing convergence
in this case is challenging because
changes with time and (4)
is not time-invariant. It is this case which we intend to study.
Toward this end, we denote by
the subset of consisting
of the indices of the connected graphs in
. Our
first result establishes the convergence of
for the case when
takes values only in .
Theorem 1: Let
be fixed and let
be a switching signal satisfying , . Then
(5)
where
is a number depending only on and .
It is possible to establish convergence to a common heading
under conditions which are significantly less stringent that those
assumed in Theorem 1. To do this we need to introduce sev-
eral concepts. By the union of a collection of simple graphs,
, each with vertex set , is meant the
simple graph
with vertexset and edgeset equalingthe union
of the edge sets of all of the graphs in the collection. We say that
such a collection is jointly connected if the union of its members
is a connected graph. Note that if such a collection contains at
least one graph which is connected, then the collection must be
jointly connected. On the other hand, a collection can be jointly
connected even if none of its members are connected.
It is natural to say that the
agents under consideration are
linked together across a time interval
if the collection
of graph
encountered along the
interval, is jointly connected. Theorem 1 says, in essence, that
convergence of all agents’ headings to a common heading is
for certain provided all
agents are linked together across each
successive interval of length one (i.e., all of the time). Of course
there is no guarantee that along a specific trajectory the
agents will be so linked. Perhaps a more likely situation, at least
when
is not too small, is when the agents are linked together
across contiguous intervals of arbitrary but finite length. If the
lengths of such intervals are uniformly bounded, then in this
case too convergence to a common heading proves to be for
certain.
3
A simple graph with vertex set
V
=
f
1
;
2
;
...
;n
g
and edge set
E
is
connected if has a “path” between each distinct pair of its vertices
i
and
j
where
by a path (of length
m
) between vertices
i
and
j
is meant a sequence of distinct
edges of
of the form
(
i; k
)
;
(
k ;k
)
;
...(
k ;j
)
. is complete if has a
path of length one (i.e., an edge) between each distinct pair of its vertices.
Theorem 2: Let be fixed and let
be a switching signal for which there exists an infinite sequence
of contiguous, nonempty, bounded, time-intervals
,
, starting at , with the property that across each
such interval, the
agents are linked together. Then
(6)
where
is a number depending only on and .
The hypotheses of Theorem 2 require each of the collec-
tions
, , to be jointly
connected. Although no constraints are placed on the intervals
, , other than that they be of finite length, the con-
straint on
is more restrictive than one might hope for. What
one would prefer instead is to show that (6) holds for every
switching signal
for which there is an infinite sequence of
bounded, nonoverlapping
but not necessarily contiguous in-
tervals across which the
agents are linked together. Whether
or not this is true remains to be seen.
A sufficient but not necessary condition for
to satisfy the
hypotheses of Theorem 2 is that on each successive interval
, take on at least one value in . Theorem 1 is thus
an obviously a consequence of Theorem 2 for the case when all
intervals are of length 1. For this reason we need only develop
a proof for Theorem 2. To do this we will make use of certain
structural properties of the
. As defined,each is squareand
nonnegative, where by a nonnegative matrix is meant a matrix
whose entries are all nonnegative. Each
also has the property
that its row sums all equal 1 (i.e.,
). Matrices with these
two properties are called stochastic [28]. The
have the addi-
tional property that their diagonal elements are all nonzero. For
the case when
(i.e., when is connected), it is known
that
becomes a matrix with all positive entries for
sufficiently large [28]. It is easy to see that if
has all positive entries, then so does . Such and
are examples of “primitive matrices” where by a primitive
matrix is meant any square, nonnegative matrix
for which
is a matrix with all positive entries for sufficiently large
[28]. It is known [28] that among the
eigenvalues of a prim-
itive matrix, there is exactly one with largest magnitude, that
this eigenvalue is the only one possessing an eigenvector with
all positive entries, and that the remaining
eigenvalues
are all strictly smaller in magnitude than the largest one. This
means that for
, 1 must be ’s largest eigenvalue and
all remaining eigenvalues must lie within the open unit circle.
As a consequence, each such
must have the property that
for some row vector . Any stochastic ma-
trices
for which is a matrix of rank 1 is called
ergodic [28]. Primitive stochastic matrices are thus ergodic ma-
trices. To summarize, each
is a stochastic matrix with pos-
itive diagonal elements and if
then is also primitive
and, as a result, ergodic. The crucial convergence result upon
which the proof of Theorem 2 depends is classical [19] and is
as follows.
Theorem 3 (Wolfowitz): Let
beafi-
nite set of ergodic matrices with the property that for each
sequence
of positive length, the matrix