Oscillator Models and Collective Motion
Spatial patterns in the dynamics of
engineered and biological networks
Derek A. Paley, Naomi E. Leonard, Rodolphe Sepulchre, and Daniel Grünbaum
contact: Dept. Mechanical and Aerospace Engineering
Princeton University, Princeton, NJ 08544
[email protected], fax (609) 258-6109
CSM-06-000 v0.03 — December 20, 2005
I NTRODUCTION
Coupled phase oscillator models are used to study the behavior of many natural and
engineered systems that represent aggregations of individuals. Examples include the heart’s
pacemaker cells, neurons in the brain, a group of fireflies, the central pattern generator for
a lamprey eel, and an array of superconducting Josephson junctions. In each case, the activity
of the individual is periodic; for example, each individual neuron, pacemaker cell or firefly fires
or flashes at regular intervals. To investigate how dynamics of interacting discrete individuals
can converge to a synchronized collective state, i.e., firing or flashing in unison, each individual
is modeled as an oscillator and the collective as a network of coupled oscillators [1], [2]. The
network model relies on interconnection among the individuals; if individual A is coupled to
individual B in the model, this means that individual A will adjust its oscillation in response to
what individual B is doing. For example, a firefly will modify the frequency of its flashing in
response to the flashing activity of its neighbors.
1
Synchronization here refers to the state in which all oscillators are in phase. However,
coupled oscillator models often exhibit a different and interesting family of equilibrium states
called incoherent states; these states involve anti-synchronization [3]. In an incoherent state, the
oscillator phases are randomly distributed around the unit circle such that their centroid is at
the origin. Since the phases “balance” themselves around the unit circle, we refer to these as
balanced states. In biological systems such as fish schools, synchronization and incoherence
have very different implications not only for primary group characteristics like movement, but
also for the capacity of groups to fulfill various biological roles such as predator avoidance.
We describe a framework for modeling, analysis and synthesis of collective motion that
extends coupled oscillator dynamics to include spatial dynamics. The phase of each oscillator
defines the direction of a particle moving in the plane at constant speed. To facilitate the analogy
with the earlier oscillator examples, note that in the case that each particle moves in a circle it
has a periodic activity that can be identified with its direction of travel since at regular intervals
it will instantaneously be heading north. Synchronization in our spatial framework implies that
all particles have the same heading and therefore at any given time they travel in a common
direction. On the other hand, the collective with phases in an incoherent or balanced state has
fixed center of mass. For example, when two particles head in opposite directions, their phases
are balanced and their center of mass is at rest.
The synthesis problem is to design the steering control for each particle so that collective
spatial patterns of interest emerge. We have derived control laws that stabilize, for instance, a
family of parallel motions and a family of circular motions with symmetric clustering of particles.
Our framework allows for limited communication among particles; for example, interconnection
2
can be fixed or even time-varying as is the case when individuals communicate only with
neighbors. For a class of patterns, we can therefore also solve the inverse problem, i.e., given a
pattern within the class, we can specify coupling that will produce such a pattern. Likewise, we
can analyze the network dynamics for bifurcations; changes in the nature of stability of steady
motions can be studied as a function of system and control parameters.
We are using these ideas and results both in application to engineered mobile sensor
networks and in study of the behavior of animal aggregations. For the former, the inverse problem
results allow us to systematically design control laws that yield sensor network patterns that can
be selected, for example, to optimize information content in the sampled data [4]. For the latter,
the inverse problem results and bifurcation analyses suggest the means to develop and study
simple models of interaction rules for animals in such a way that the group behavior resembles
observations in the field. At sea demonstrations of coordinated control strategies for formations
of underwater mobile sensors are described in [5]. A range of different animal models can be
found in [6], [7], [8], [9] and references therein. Some pertinent work on collective motion from
the engineering and physics literature includes [10], [11], [12], [13]. Our most recent work can
be found in [14], [15].
The engineering goal of the effort described in this paper is to develop a systematic and
provable design methodology for controlling collective motion of multi-agent systems, explicitly
taking advantage of possible strategies used by animals moving in groups to meet challenges
such as scalability and limited sensing. The biological goal of this work is to develop new means
to interpret data from biological aggregations in motion that exploits the provable dynamics of
simple models to clarify the relationship between individual choices and group behaviors.
3
We begin this paper with a presentation of the oscillator model with spatial dynamics.
We make the connection between synchronization of phases and linear momentum of the group
and describe how to design stabilizing steering control laws by taking the gradient of a phase
potential. The concepts are generalized to the case of limited communication by means of
interconnection graphs and the quadratic form induced by the corresponding Laplacian matrix.
Stabilization results are summarized and application to design of patterns for sensor coverage
by mobile sensor networks is discussed.
In the second half of this paper we use the coupled oscillator model with spatial dynamics
to study biological collectives. We show how the particle model resembles a model used to study
animal aggregations [16], [17]. We use this analogy to examine collective motion of the particle
model and show how we can vary parameters to produce a rich set of complex behaviors. We then
use the phase potential defined in our model to analyze experimental data on populations of fish,
specifically giant danios moving about in a one cubic meter tank [18], [17], [19]. As a preliminary
step, we present two-dimensional analyses of processes that are often three-dimensional in nature.
We conclude with a brief prospectus on this joint pursuit of engineering analysis of biological
groups and biological analysis of engineered groups.
O SCILLATOR M ODEL AND C OLLECTIVE M OTION
We use a common mathematical framework to describe a collection of either autonomous
robots or biological organisms. Each unit is a point mass or particle; we call the framework the
particle model. The particle model consists of N identical units which are defined by their
position and heading. We assume that the particles have unit mass, travel at unit speed, and can
4
maneuver only by steering and not by speeding up or slowing down.
We use complex notation for each particle’s position and velocity. Therefore, the position
of the kth particle is denoted by rk ∈ C ≈ R2 . Also, the velocity of the kth particle is eiθk =
cos θk + i sin θk , where θk ∈ S 1 . The 1-sphere, S 1 , is a circle and can be suitably parameterized
by angles in the interval [0, 2π) where 0 is identified with 2π. The steering control, uk , is a
feedback control law that is a function of particle positions and headings. To indicate a vector,
we drop the subscript and use bold: for example, r = (r1 , . . . , rN )T . Let 1 = (1, . . . , 1)T ∈ RN .
Later, we will use the real part of the Hermitian inner product defined by < x, y >= Re{x∗ y},
where x, y ∈ Cn , n ∈ N , and
∗
is the conjugate transpose. Using this notation, the particle
model can be expressed as the following continuous-time system,
ṙk = eiθk
(1)
θ̇k = uk (r, θ), k = 1, . . . , N.
Note that r ∈ CN and θ ∈ S 1 × · · · × S 1 = T N , the N -torus.
To provide intuition for the particle model (1), we give examples of motions that are
produced with simple control inputs. If the control is constant and zero, that is uk = 0 for all
k, then each particle moves in a straight line in the direction it was initially pointing, θk (0). If
the control is constant but not zero, uk = ω0 6= 0, then each particle travels around a circle with
radius |ω0 |−1 . The direction of rotation around the circle is determined by the sign of ω0 . For
use later, we define the center of the kth circle to be
ck = rk + iω0−1 eiθk .
(2)
Also, the center of mass of the particle group is
N
1 X
rj .
R=
N j=1
5
(3)
Under certain types of feedback controls, the particle model has important symmetric
properties. Let rkj = rk − rj and θkj = θk − θj be the relative position and heading of particles
j and k. Suppose we restrict the steering control to be a function only of the relative positions
and headings, that is
uk = uk (rk1 , . . . , rkN , θk1 , . . . , θkN ), k = 1, . . . , N.
(4)
Controls of this form preserve the continuous symmetries of the system (1) that make it invariant
to rigid rotation and translation of the particle group. The closed-loop dynamics evolve on a
reduced space whose components describe only the shape of the particle group and not its
position or orientation. Equilibria of the reduced dynamics are called relative equilibria and can
be only of two types [10]: parallel motion in a constant direction, which is characterized by
θkj = 0 and θ̇k = 0 for all j, k ∈ {1, . . . , N }; and circular motion about the same circle, which
is characterized by θ̇kj = 0 and ck = c0 ∈ C for all k, where ck is given by (2). A preliminary
goal of our work is to provide feedback control laws that stabilize these relative equilibria.
Phase Model
A powerful way to interpret the particle model is to independently consider the subsystem
of particle headings or phases. In order to do this, we split the control into three terms: a constant
term, the spacing control and the heading control. The heading control depends only on the
particle headings, so that
uk = ω0 + uspac
(r, θ) + uhead
(θ), ω0 ∈ R, k = 1, . . . , N.
k
k
(5)
The phase model is given by
θ̇k = ω0 + uhead
(θ), ω0 ∈ R, k = 1, . . . , N.
k
6
(6)
The system (6) is a system of coupled phase oscillators with identical natural frequency, ω0 .
In the case that uhead
depends only on the relative phases, the phase model is invariant to rigid
k
rotation of all the phases.
Next, we introduce terminology used to describe systems of coupled phase oscillators and
outline how this terminology relates to solutions of the particle model. The particle headings j and
k are phase-locked if θ̇kj = 0. Synchronized phase arrangements are phase-locked arrangements
for which θkj = 0 for all pairs j and k and correspond to parallel motion in the particle model.
The order parameter, p(θ), is a measure of synchrony of phase oscillators, given by [20],
p(θ) =
N
1 X iθk
e .
N j=1
(7)
The magnitude of the order parameter, 0 ≤ |p(θ)| ≤ 1, is proportional to the level of synchrony of
the phases and |p(θ)| = 1 for synchronized phases. Balanced or incoherent phase arrangements
satisfy |p(θ)| = 0. Taking the derivative with respect to time of (3) using (1) and comparing to
(7), we observe that the order parameter is equal to the velocity of the center of mass of the
particles or, equivalently, the average linear momentum of the group. The magnitude of the order
parameter is the speed of the center of mass. Therefore, balanced phase arrangements correspond
to particle motion with a fixed center of mass.
Controlling Linear Momentum
The phase model interpretation demonstrates that controlling the linear momentum of the
particle group is equivalent to controlling the coherence of the particle phases. For this purpose,
7
we introduce the rotationally symmetric phase potential
U1 (θ) =
N
|p(θ)|2 ,
2
(8)
which is maximum for synchronized phase arrangements and minimum for balanced phase
arrangements. The gradient of U1 (θ) is given by
∂U1
= < ieiθk , p(θ) >, k = 1, . . . , N.
∂θk
The rotation symmetry of U1 (θ) implies invariance of the potential to rigid rotation of the particle
phases and, consequently, its gradient is orthogonal to 1:
N
X
∂U1 T
1=
< ieiθk , p(θ) > = N < ip(θ), p(θ) > = 0.
∂θ
k=1
(9)
We choose the heading control in the phase model (6) to be the signed gradient of U1 (θ),
uhead
= −K1 < ieiθk , p(θ) >, K1 6= 0,
k
(10)
for k = 1, . . . , N . The control (10) guarantees that the potential U1 (θ) evolves monotonically
along solutions of (6). In fact, using (9), we obtain
U̇1 (θ) =
N
X
∂U1 T
θ̇ = −K1
< ieiθk , p(θ) >2 .
∂θ
k=1
The phase model (6) with the gradient control (10) is equivalent to
N
K1 X
θ̇k = ω0 +
sin θkj .
N j=1
(11)
The system (11) with K1 < 0 is a simplified Kuramoto model of coupled phase oscillators
with identical natural frequencies [20]. The Kuramoto model is a commonly studied model
of oscillator synchronization. We also consider K1 > 0 which stabilizes the balanced phase
arrangements. In fact, Lyapunov analysis provides the following global stability result.
8
The phase model (6) with the gradient control (10) forces convergence of all solutions
to the critical set of U1 (θ). If K1 < 0, then only the set of synchronized phase arrangements is
asymptotically stable and every other equilibrium is unstable. If K1 > 0, then only the balanced
set where |p(θ)| = 0 is asymptotically stable and every other equilibrium is unstable [14]. The
critical points of U1 (θ) other than the synchronized and balanced phase arrangements satisfy
sin θkj = 0 for all j, k ∈ {1, . . . , N } and are all saddle points. These critical points correspond
to two synchronized clusters of particle phasors separated by π radians with an unequal number
of phasors in each cluster, which we call the unbalanced (2, N )-patterns.
The particle model (1) with the gradient control (10) gives rise to the four distinct types
of motion shown in Figure 1, depending on whether or not ω0 = 0 and the sign of K1 . In
particular, for ω0 = 0, the particles move along straight trajectories. For ω0 6= 0, the particles
move around circles. In either case, if K1 < 0, then the headings are synchronized; if K1 > 0,
then the center of mass of the particles is fixed. The case with ω0 = 0 and K1 < 0 is an example
of the parallel formation and is the only example in Figure (1) of a relative equilibria. Later,
we will present spacing controls which stabilize the relative equilibria with circular motion but
first we extend the phase model to systems with limited communication.
L APLACIAN Q UADRATIC F ORMS
Implicit in the gradient control (10) is the assumption that each particle has access to the
relative heading of every other particle in computing its control. In this section, we extend the
heading control to scenarios in which this is not necessarily true. The heading interconnection
graph or heading topology describes which relative headings are available to each particle for
9
feedback. In fact, we will also consider spacing controls which are constrained by the spacing
interconnection graph. We describe a methodology for defining gradient controls for the phase
and particle models with limited communication.
Graph Laplacians
Let G denote a communication or sensing graph. In this setting, the vertices or nodes of G
are particles and the edges are unweighted communication or sensing links. The set of neighbors
of vertex k, denoted by Nk , refers to the indices of the particles whose relative states are available
for calculating the kth control. The edges leaving vertex k point towards the members of Nk so
that the out-degree of the vertex, dk , is the cardinality of Nk . A graph is (strongly) connected
if there is a possibly directed path spanning one or more edges between every pair of vertices.
A time-varying graph is weakly connected if it is not connected but the graph formed by taking
the union of edges over a finite time interval is connected.
We use the following matrix definitions [21]: D = diag(d) ≥ 0 is the degree matrix;
A ∈ RN ×N , where Akj = 1 if j ∈ Nk and 0 otherwise, is the adjacency matrix; and B ∈ RN ×e ,
where Bkf = −Bjf = −1 if edge f ∈ {1, . . . , e} connects vertex k to vertex j and 0 otherwise,
is the incidence matrix; the positive semi-definite matrix, L = D − A = BB T , is the Laplacian
matrix or graph Laplacian. The Laplacian is symmetric for undirected graphs, in which every
edge is bidirectional and the in-degree equals the out-degree.
We give examples of the graph Laplacian for two undirected graphs. Let KN denote the
complete graph of N vertices, each with degree N −1. Let CN denote an undirected cyclic graph
of N vertices, each with degree 2. Both types of graphs are circulant, which means that their
10
graph Laplacians are circulant matrices. Circulant matrices with N ≥ 2 columns are completely
defined by their first row since each other row starts with the last component of the previous row
followed by the first N − 1 elements of the previous row. Let Lk be the kth row of the graph
(KN )
Laplacian. Then, for the complete graph, we have L1
(CN )
cyclic graph, we have, L1
= (N − 1, −1, . . . , −1) ∈ RN . For a
= (2, −1, 0, . . . , 0, −1) ∈ RN . Circulant graphs are also d0 -regular,
meaning dk = d0 for all k ∈ {1, . . . , N }. We have d0 = N − 1 for KN and d0 = 2 for CN .
We list relevant properties of the eigenstructure of the graph Laplacian here. Let λ =
(λ1 , . . . , λN )T , be the vector of eigenvalues of L in ascending order of their real component.
By the Gers̆gorin disc theorem [22], we observe that 0 ≤ Re{λk } ≤ 2d∞ , where d∞ is the
maximum vertex out-degree and k ∈ {1, . . . , N }. We also have that L1 = 0 and the multiplicity
of the zero eigenvalue is the number of connected components of the graph. As a consequence,
the Laplacian matrix of a connected graph has one zero eigenvalue and N − 1 strictly positive
eigenvalues. Furthermore, the normalized eigenvectors of an undirected Laplacian, for which
L = LT , can be chosen to form a unitary basis.
Let L be a connected, undirected graph Laplacian. Associated with L is the Laplacian
quadratic form given by
QL (x) =
1
< x, Lx >, x ∈ CN ,
2N
(12)
which is zero only for x ∈ span{1} and positive otherwise [12]. Using L = BB T , equation
(12) can be interpreted as the scaled sum of the squares of the lengths of the edges of the graph
plotted in the complex plane.
11
Quadratic Phase Potential
With this formalism in hand, we return to the phase potential U1 (θ) and extend it to
connected, undirected graphs. Let z = eiθ be the complex vector whose components zk = eiθk
are on the complex unit circle. The Laplacian phase potential is [23]
W1 (θ) = QL (z) =
1
< z, Lz > .
2N
(13)
In the case that L = L(KN ) , one can compute the relation
W1 (θ) =
N
− U1 (θ).
2
(14)
The minimum value of the Laplacian phase potential occurs for synchronized phase arrangements
and the maximum value occurs for the balanced phase arrangements that maximize (13).
One can also show that W1 (θ) is rotationally symmetric. Choosing the heading control
in the phase model (6) to be the signed gradient of W1 (θ),
uhead
=
k
K1
< ieiθk , Lk eiθ >, K1 6= 0,
N
(15)
for k = 1, . . . , N , guarantees that the potential W1 (θ) evolves monotonically for a connected,
undirected Laplacian L. In fact, using the rotational symmetry, we obtain
N
∂W1 T
K1 X
θ̇ = 2
< ieiθk , Lk eiθ >2 .
Ẇ1 (θ) =
∂θ
N k=1
(16)
The phase model (6) with the gradient control (15) is equivalent to
θ̇k = ω0 +
K1 X
sin θkj , K1 6= 0.
N j∈N
(17)
k
The system (17) with K1 < 0 is a simplified Kuramoto model of identical coupled phase
oscillators with limited communication.
12
Using Lasalle’s invariance principle, one can show that the phase model under the control
(15) converges to the largest invariant set for which Ẇ1 (θ) = 0, which corresponds to
< ieiθk , Lk eiθ >= 0, k = 1, . . . , N.
(18)
Let θ̄ ∈ T N be a phase arrangement which satisfies (18), then θ̄ is a critical point of the phase
potential W1 (θ) [15]. For example, eigenvectors of the graph Laplacian are critical points of the
phase potential. Synchronized critical points correspond to the eigenvector 1, that is θ̄ = θ0 1,
where θ0 ∈ S 1 . The set of synchronized critical points exists for any graph and is a global
minimum of the phase potential. Balanced critical points satisfy 1T eiθ̄ = 0 and live in the union
of the spans of the eigenvectors of all the eigenvalues of L other than zero. The only other
critical points that we have identified are the unbalanced (2, N )-patterns for which sin θjk = 0
for all connected j and k.
A sufficient condition for balanced critical points to exist is that the graph is circulant,
that is, the Laplacian is a circulant matrix. The components of the eigenvectors of any circulant
matrix form symmetric phase patterns on the complex unit circle. The fact that these symmetric
patterns remain invariant under linear dynamics with circulant connectivity was observed in
[24]. Let V be the matrix whose mth column is an eigenvector of a circulant matrix given by
2π
Vkm = ei N (m−1)(k−1) , k, m = 1, . . . , N . We observe that if eiθ̄ is an eigenvector of a circulant
graph Laplacian other than 1, then the phasors eiθ̄k are evenly-distributed around the unit circle.
In the case of the cyclic graph CN , the heading graph for each balanced critical point forms
a generalized regular polygon. The eigenvectors and corresponding eigenvalues of L(CN ) are
shown in Figure 2 and the stability of the balanced critical points is characterized in [25].
13
S TABILIZING S PATIAL PATTERNS
We now summarize our approach to stabilizing spatial patterns in solutions of the
particle model with possibly limited communcation. The circular formation is a circular relative
equilibrium in which the particles travel around the same circle. Stabilizing formations on a
more general class of closed curves is not discussed here. Symmetric patterns of the particles
in the circular formation are isolated using a composite potential as described below.
Circular Formation
Recall that the circular formation is characterized by the algebraic condition c = c0 1,
c0 ∈ C, where ck is defined in (2). We use the Laplacian quadratic form to define a spacing
potential which is minimum in the circular formation. This potential is given by
QL (c) =
1
< c, Lc >,
2N
(19)
where L is the Laplacian of a connected, undirected graph. Observe that the velocity of the
center of each circle is given by differentiating (2) with respect to time along solutions of (1),
which we rearrange to obtain
ċk = eiθk (1 − ω0−1 uk ), k = 1, . . . , N.
(20)
Using (20), the potential (19) evolves along the solutions of the particle model (1) according to
Q̇L (c) =
N
1 X
< eiθk , Lk c > (1 − ω0−1 uk ).
N k=1
(21)
We assume controls of the form (5) with, for the moment, uhead
= 0. Choosing
k
uspac
= ω0
k
K0
< eiθk , Lk c >, K0 > 0, k = 1, . . . , N,
N
14
(22)
guarantees that QL (c) is nonincreasing. Using (5), (21), and (22), we obtain
N
K0 X
Q̇L (c) = − 2
< eiθk , Lk c >2 ≤ 0.
N k=1
Lyapunov analysis gives the following global stability result.
is given by
Consider the model (1) with the control (5) where uhead
= 0 and uspac
k
k
(22). Then for ω0 < 0 (resp. ω0 > 0), the set of clockwise (resp. counter-clockwise) circular
formations of radius |ω0 |−1 is globally asymptotically stable and locally exponentially stable
[15]. By Lasalle’s invariance principle, solutions converge to the largest invariant set for which
Q̇L (c) = 0. In this set, θ̇ = ω0 1 and c = c0 1. The spacing control (22) preserves the rotation
and translation symmetries of the particle model, so the steady-state position of the center of the
circular formation, c0 ∈ C, is determined by the initial positions and headings of the particles.
Also, the center of mass of the particles is not necessarily fixed. We show next how to control
the speed of the center of mass simultaneously to stabilizing the circular formation.
Assume for now that the heading and spacing interconnection graphs are identical.
Consider the composite potential formed by taking a linear combination of the spacing potential
QL (c) and the phase potential W1 (θ), given by
V (r, θ) = K0 QL (c) + K1 ω0−1 W1 (θ), K0 > 0, K1 6= 0, ω0 6= 0.
(23)
We simultaneously stabilize the circular formation and the speed of the center of mass by
choosing the control (4) with uhead
given by (15) and uspac
given by (22):
k
k
uk = ω0 +
K1 − K0
K0
< ieiθk , Lk eiθ > +ω0
< eiθk , Lk r > .
N
N
(24)
Taking the time derivative of (23) along the solutions of (1) using (13) and (19), we obtain
N
1 X
V̇ (r, θ) = − 2
(K0 < eiθk , Lk c > +K1 ω0−1 < ieiθk , Lk eiθ >)2 ≤ 0.
N k=1
15
By Lasalle’s invariance principle, solutions converge to the largest invariant set for which
V̇ (r, θ) = 0. In this set, θ̇ = ω0 1 which implies that both c and W1 (θ) are constant. From
this, one can show that the synchronized circular formation, characterized by c = c0 1 and
|p(θ)| = 1 is asymptotically stable for K1 < 0. If, in addition to being connected, the graph G
is also circulant, then one can show that balanced circular formations characterized by c = c0 1
and |p(θ)| = 0 are asymptotically stable for K1 > 0 [15]. These spatial patterns are shown in
Figure 3 for a cyclic graph. Increasing the gain K1 > 0 in Figure 3(c) produces the balanced
(2, N )-pattern since that configuration maximizes the potential W1 (θ).
Isolating Symmetric Patterns
Next, we generalize the phase potential in order to isolate symmetric patterns of particles
in the circular formation. Let 1 ≤ M ≤ N be a divisor of N . An (M, N )-pattern is a symmetric
arrangement of N phases consisting of M clusters uniformly spaced around the unit circle,
each with
N
M
synchronized phases. For any N , there exist at least two symmetric patterns: the
(1, N )-pattern, which is the synchronized state, and the (N, N )-pattern, which is the splay state,
characterized by N phases uniformly spaced around the circle. All symmetric patterns other
than the synchronized state are balanced. To characterize (M, N )-patterns, we define the mth
moment of the particle phasor distribution as
p(mθ) =
N
1 X imθk
e
,
mN k=1
(25)
where m is a positive, rational number. Symmetric (M, N )-patterns satisfy |p(mθ)| = 0 for
m = 1, . . . , M − 1 and M |p(M θ)| = 1 [14].
To isolate symmetric patterns, we design phase potentials which are minimum in the
16
desired pattern. The phase potential (13) can be generalized for this purpose by taking the
Laplacian quadratic form (12) of the mth power of the particle phasors,
Wm (θ) = QL (
1 m
1
z )=
< z m , Lz m >,
m
2N m2
where L is the Laplacian of a connected, undirected graph. The procedure for stabilizing
symmetric patterns of particles with limited communication is not discussed here. Instead, we
summarize the results for the complete graph. Analogous to Wm (θ), a natural generalization of
the potential U1 (θ) are the potentials [14]
Um (θ) =
N
|p(mθ)|2 ,
2
where p(mθ) is given by (25). Note that the minimum and maximum of Um (θ) correspond
to balancing and synchronization of the phasors raised to the mth power. This leads to the
following prescription for phase potentials which are minimum for symmetric phase patterns.
Let 1 ≤ M ≤ N be a divisor of N . Then θ ∈ T N is an (M, N )-pattern if and only if it is a
global minimum of the potential
U
M,N
(θ) =
M
X
Km Um (θ)
(26)
m=1
with Km > 0 for m = 1, . . . , M − 1 and KM < 0 [14].
As before, we combine the circular formation spacing potential with a phase potential
to drive the particles to the circular formation in a particular phase arrangement. The composite
potential which isolates an (M, N )-pattern circular formation is given by
V M,N (z) = K0 QL(KN ) (c) + U M,N (θ).
The following result can be proved with Lyapunov analysis. Each (M, N )-pattern circular
formation of radius |ω0 |−1 | is an isolated relative equilibrium of the particle model (1) and
17
is exponentially stabilized by the control law (4) with uhead
given by the negative gradient of
k
(26) and uspac
given by (22) with L = L(KN ) [14]. The six symmetric patterns for N = 12 are
k
shown in Figure 4.
Symmetry Breaking
Up to this point, we have chosen controls of the form (5) which preserve the continuous
rotation and translation symmetries of the particle model. However, it is useful for certain
applications to break these and other symmetries, such as the permutation symmetry of the
particles implied by a circulant interconnection graph. We summarize the results for breaking
the continuous symmetries first. Afterward, we briefly mention the role that communication
topologies, which may differ for the spacing and heading controls, can play in engineered
collectives.
We separately break the rotation and translation symmetries by adding a reference heading
and reference beacon, respectively. Suppose that θ0 ∈ S 1 represents a reference heading with
dynamics θ̇0 = ω0 . One way to break the rotational symmetry is by coupling a single particle
to the reference heading as well as to the other particles. Without loss of generality, we couple
the N th particle to the reference heading. The potential W1 (θ) + 1 − cos(θ0 − θN ) is minimum
for θk = θ0 for all k ∈ {1, . . . , N }. In the case that ω0 6= 0, this corresponds to each particle
traveling around a circle with its heading synchronized with every other particle as well as with
the reference heading. In the case that ω0 = 0, this corresponds to parallel motion of the particle
group in the direction of the reference heading, θ0 . This symmetry breaking control can be used
to track piece-wise linear reference trajectories as shown in Figure 5 [14].
18
Similarly, suppose that the position of a static beacon in the complex plane is denoted by
c0 ∈ C. Then the positive, quadratic form
1
kc−c0 1k2
2N
is zero for ck = c0 for all k ∈ {1, . . . , N }
which corresponds to each particle traveling around a circle which is centered at the beacon c0 .
The gradient of this potential can be used instead of the spacing control (22) to fix the location of
the circular formation in the plane. An application of the translation symmetry breaking control
is illustrated in Figure 6 for the particle model with a constant drift vector field. If each particle
is assigned a different beacon, we obtain a form of collective motion that is well suited to a
particular class of optimal sampling problems with mobile sensor networks [4].
B IOLOGICAL C OLLECTIVES : M ODEL , S IMULATIONS , AND O BSERVATIONS
In this subsection, we describe a model for animal aggregations in which the animals
move at constant speed and are subject to a combination of steering behaviors; see, for example,
[16] and references therein. Without loss of generality we use the term “fish” to refer to a unit
in the collective. We assume that each fish’s response to every other sensed fish is determined
by its relative position and heading projected onto a horizontal plane. The perceptual range, or
sensory range, is defined by the union of three concentric zones. The response to a sensed fish is
either a repulsion, attraction, or orientation behavior vector, depending on which zone contains
the other fish. The directed sensing capability of each fish generates a blind spot directly behind
it. These zones, and the corresponding type of behavior, are illustrated in Figure 7.
Let rk ∈ C and θk ∈ S 1 be the position and heading of the kth fish. Let ρrep , ρori and
ρatt define the zone boundaries in Figure 7. For example, the orientation zone of the kth fish
is contained in the annulus {r | ρrep ≤ |r − rk | ≤ ρori }. The angular width of the blind spot
19
of the kth fish is denoted by αk . In this setting, we use graph theory to describe the sensory
interconnections generated by each zone. Note that the graphs may be directed due to the blind
spot in the perceptual zone and may be unconnected due to the limited size of the perceptual
range. In the directed case, the edges in the graph are oriented towards the sensed fish. We
(l)
denote by Nk the set of fish sensed by the kth fish in zone l ∈ {rep, ori, att}. We also assume
the fish have unit mass, travel at unit speed, and can maneuver only by steering.
Let q = 0, 1, 2, . . . denote the qth time step of a discrete model. The fish model is
rk (q + 1) = rk (q) + eiθk (q)
(27)
θk (q + 1) = θk (q) + T uk (q), k = 1, . . . , N,
where T 1 is the fish response latency and uk (q) is the steering behavior. Observe that the
fish model (27) is the discrete-time Euler method approximation to the continuous-time particle
model (1). Given concentric zones of repulsion, orientation and attraction, the corresponding
desired behavior vectors in the fish model are for k = 1, . . . , N [16]
X
vkrep (q) = −
r̂kj (q)
j∈Nkrep (q)
X
vkori (q) = eiθk (q) +
eiθj (q)
j∈Nkori (q)
vkatt (q) =
X
r̂kj (q)
j∈Nkatt (q)
where r̂kj (q) =
rkj (q)
|rkj (q)|
(l)
and Nk (q) are the neighbors of k in zone l at time q. Independent fish
motion can be introduced in the fish model by an additional, possibly random, behavior term
ωk (q). The steering behavior, uk (q), is
uk (q) = ωk (q)+ < ieiθk (q), Krep vkrep (q) + Kori vkori (q) + Katt vkatt (q) >,
20
(28)
where the gains satisfy Krep Kori > 0 and Krep Katt > 0. Next, we show that these
behaviors can be expressed using the graph Laplacian and are directly related to the steering
controls of the particle model.
The steering behavior (28) can be expressed in terms of the degree, D(l) , adjacency, A(l) ,
and Laplacian, L(l) , matrices of the sensory graph in zone l. Dropping the (q) notation, if the
repulsion zone of the kth fish is empty, we can rewrite (28) as
att
iθ
iθk
uk = ωk − Kori < ieiθk , Lori
k e > +Katt < ie , Lk r > .
(29)
Note the strong resemblance between the steering behavior (29) and the circular control (24).
In place of the deterministic natural frequency, ω0 , the fish behavior is a random turning rate
ωk . The alignment terms are identical for K1 − K0 = −Kori N < 0, which is consistent with a
behavior that seeks to align the fish velocities. The spacing terms differ only by a factor of i
which is a consequence of their differing motivations as described next.
(l)
−1
Let r̃k = d−1
k Lk r = rk − dk
P
(l)
j∈Nk
rj be the vector from the center of mass of
the neighbors of particle k in zone l to the position of particle k. The circular control of the
particle model stabilizes motion of the kth particle perpendicular to r̃k . In contrast, both the
attraction and repulsion behaviors in the fish model stabilize motion parallel to r̃k as appropriate
for aggregation with collision avoidance.
Although we focus on circular formations in the next subsection, we are not proposing
that the circular motion control is an accurate model of fish behavior. We justify the use of this
control for our analysis below for the following reason. One can interpret stabilizing the circular
formation as a example of activity consensus, i.e., individuals are “moving around” together.
Stabilizing parallel motion is another form of activity consensus in which individuals “move
21
off” together. We conjecture that we can study changes in collective motion that arise from
changing parameters related to the orientation behavior independently of the choice of activity;
the choice of circular motion is convenient because it keeps the center of mass of the particle
group rotating around a fixed point with radius less than or equal to |ω0 |−1 .
Another conjecture explored below is that we can use the heading control (15) with
K1 > 0 as a surrogate for the repulsion control of the fish model. This choice is justified by
considering the close encounter of two individual fish. At least locally, the maximum absolute
range rate between two fish in a near collision is obtained by anti-synchronizing headings.
Bifurcations of Collective Motion
Simulation studies suggest that fish may be able to influence group behavior merely by
selecting the number of influential neighbors and not by changing the behavior rule itself [19].
In this subsection, we study qualitative changes in collective motion exhibited by solutions of the
particle model that result not from changing the control law but from varying key parameters such
as the number of neighbors. We call these qualitative changes bifurcations and the corresponding
parameters bifurcation parameters, although rigorous bifurcation analysis is not discussed here.
Bifurcations in models of animal groups is a growing area of research. In [16], bistability
of parallel and circular motion was demonstrated in a three-dimensional fish schooling simulation.
This bistability generated hysteresis under slow variation of the parameter which determines the
outer radius of the orientation zone. An analytical result for N = 2 particles demonstrating
bistability of parallel and circular motion in the particle model appeared in [26], in which the
bifurcation parameter was the heading control coupling gain K1 . If some individuals in a group
22
have differing preferred or informed directions, then the group itself can bifurcate or split.
Simulations suggest that the percentage of informed animals in a group necessary to achieve
consensus decreases as the group size increases [27]. Analytical results classifying bifurcations
in the phase model when two individuals have different, preferred directions appears in [28]. In
this subsection, we demonstrate via simulation a related bifurcation that occurs in the particle
model in which the choice of parameters is motivated by results for biological aggregations.
In this subsection, we use the particle model (1) with the control (24). We assume that
the communication graphs, generated by perceptual zones as in the previous subsection, may
differ for the spacing and heading controls. Consequently, these graphs may be directed and
switching. Analytical results for stabilizing collective motion with directed and switching graphs
is the subject of ongoing work. Preliminary results inspired by [13] suggest that the results for
undirected, fixed graphs hold under small modifications to the control [15].
Motivated by [16] and [19], our example of a bifurcation occurs as a result of changing
the outer radius of the orientation zone. For simplicity, we assume there is no repulsion zone and
that the inner radius of both the orientation and attraction zones is zero, that is, the orientation
and attraction zones overlap. We choose ρatt |ω0 |−1 . Let ρ = ρori |ω0 | be the dimensionless
bifurcation parameter that corresponds to the ratio of the outer radius of the orientation zone
divided by the radius of the circular motion. The angular width of the blind spot is αk = 60◦
for all k ∈ {1, . . . , N }.
As a result of the spacing control, all particles converge to the circular formation as
shown in Figure 8. For K1 < 0, particles in overlapping orientation zones become synchronized.
Increasing the size of the orientation zone (decreasing M in Figure 8) stabilizes fewer, larger
23
synchronized clusters of particles. This result resembles the bifurcation result of [16] in the
following way: increasing the size of the orientation zone switches the distribution of particle
headings from incoherent or swarm-like to synchronized or parallel. The results differ because, in
our simulations, the particles remain in the circular formation while in [16] they switch between
swarm-like, circular, and parallel collective motion.
For K1 > 0, the particles cluster into the symmetric patterns appropriate for the size
of the orientation zone. Because positive gain on the alignment control corresponds to antisynchronization, clusters form with non-overlapping orientation zones. We illustrate results for
π
ρori = ρ|ω0 |−1 where ρ = 2 sin M
, M = 2, 8, and 12, which correspond to the chord length that
separates M evenly spaced clusters on a circle of unit radius. Note that, although the clusters
are possibly not equal in size, this method generates symmetric patterns for which the number
of clusters is not necessarily a divisor of M .
Analysis of Fish Data
We analyzed actual fish schooling data for evidence of synchronization, characterized by
small values of the phase potential W1 (θ) given by (13) with L = Lori . We examined trajectories
of individual fish within 4− and 8−fish populations of giant danios Danio aequipinnatus which
were obtained using stereo videography and a computerized tracking algorithm; see [17] for
details on the data collection. Each experiment contains ten minutes of data collected in a one
meter cubic tank. Continuous fish trajectories were generated from the raw data using smoothing
splines to remove frame-rate noise. We calculated smoothed unit velocity estimates by averaging
over a five second moving window. Only the horizontal components of position and velocity
24
were used in the analysis. This was reasonable for an initial analysis because many of the most
significant spatial patterns evident in these data were primarily two-dimensional [18].
As a first step, we calculated the value of the phase potential as a function of time using
the switching sensory graphs generated by the realistic orientation zones defined in [17]. Let
b = 5.3 cm denote the body length of a fish. The orientation zone of the kth fish is contained in
the annulus defined by {r | 1.4b ≤ |r − rk | ≤ 2.4b} and with the trailing blind sector αk = 60◦ .
A snapshot of each data set is shown in Figure 9 with orientation zones shown in grey. The
4-fish data set exhibits periods of highly synchronized collective motion. The two 8-fish data
sets, panels (b) and (c) in Figure 9, exhibit markedly different schooling behavior which we
refer to as tight milling and diffuse milling, respectively. A limitation of this choice of sensory
graphs is that low values of the phase potential can also be generated if the orientation zones
are empty and the graph is only weakly connected, as in the case of diffuse milling.
In order to overcome this limitation, we recalculated the phase potential using a sensory
graph generated by the nearest n neighbors of each fish. In the left column of Figure 10, we
plot the scaled potential
2
W1 (θ)
n+1
so that the scaled magnitude is always in the interval [0, 1]
for any n. In each graph, we plot the results for n = 1 and n = N − 1 for a representative
thirty second period starting from t = 30 seconds. In the right column of Figure 10, we plot the
histograms of the scaled potentials for all ten minutes of each data set for n = 1 and n = N − 1
as well as the histogram of the difference between the two values of the phase potential. For
n = 1, if the nearest neighbor is in the zone of repulsion {r | |r − rk | ≤ 1.4b}, we use the
second nearest neighbor. Note that n = N − 1 corresponds to the complete graph, in which case
2
W1 (θ)
n+1
= 1 − |p(θ)|2 by (14) and (8). This metric is inversely proportional to the mobility
25
of the center of mass of the group, which may be constrained by the relative small tank.
Using the histograms in Figure 10, we make the following observations. The 4-fish data
has a bimodal distribution of the scaled phase potential histogram for n = N − 1. The peak
near zero corresponds to synchronized collective motion. The peak at 0.75 corresponds to three
synchronized fish and one anti-synchronized fish, which is the only unbalanced (2, N )-pattern
for N = 4. For the n = 1 histogram, the single peak near zero provides the strongest evidence
for local synchronization behavior. For both 8-fish data sets, the n = N − 1 scaled potential
histograms have a single peak at unity which is consist with the milling behavior and indicates
that the group center of mass was predominantly fixed. Although each n = 1 histogram has a
single peak as well, the mean is not at zero. Nonetheless, evidence for local synchronization
is provided by the histogram of the difference between the values of the scaled potential for
n = N − 1 and n = 1 which is nearly always positive. The result that both the 8-fish data sets
contain evidence for local synchronization despite differences in the group behavior, suggests
that the heading component of the individual fish behaviors may be similar in both data sets.
P ROSPECTUS : C OLLABORATIVE E NGINEERING
AND
B IOLOGICAL A NALYSIS
Investigation of collective motion represents an overlap of interests between engineers
and biologists who are motivated by very different sets of basic scientific questions. To engineers,
biological aggregations is one of many areas in which organisms confront “design problems”
that have close engineering analogs. These organisms may have found, through natural selection,
highly effective solutions that provide performance benchmarks and inspiration, if not actual
blueprints, for engineers tasked with coordinating large systems of interacting agents. To
26
biologists, collective motion or, more generally, social response to neighbors, is ubiquitous,
ranging from quorum sensing in bacteria to schooling in fish. Yet, it has proven impossible so
far to establish mechanistic relationships between realistic individual behaviors and their group
characteristics, or to distinguish those characteristics of groups that are the primary benefits
of natural selection from those that are incidental or even disadvantageous. The results in this
paper illustrate a profitable dialogue between biologists, whose observations help motivate novel
analytical directions, and engineers, whose mathematical formalisms suggest new and insightful
ways of interpreting biological data as measured against idealized but understandable models.
An example of how this conversation may lead to fundamental insights is in the
area of scalability. Engineers designing controls for coordinated N -member groups face the
impracticality of information exchange between members as N gets large. Very large social
groups abound in nature; these organisms appear to have solved the problem. However, in nearly
all biologically-inspired simulations of collective motion, large groups lack robustness and tend
to fragment easily into smaller groups unless they also contain explicit mechanisms, such as a
finite spatial domain or common directional preferences, that suppress the tendency.
We suggest that our analyses and related work look forward in at least two useful
directions. First, they provide a starting point for “reverse-engineering” algorithms from observed
trajectories. Biologists’ abilities to infer underlying behavior from movement observations
has been far more limited, and far more limiting, than their abilities to simulate movement
from hypothetical behavioral algorithms. As suggested by our analysis of fish trajectories, any
analytical infrastructure that permits bidirectional inferences, that is movement from algorithm
and algorithm from movement, is a big step forward, even if the underlying models are simplified
27
compared to real biological behaviors.
Second, our analyses of connected communication graphs have established an explicit
link that has largely been lacking between control theory and the distance-mediated attraction,
repulsion, and alignment neighborhoods that are the basis of most social grouping simulations.
One interpretation of our results is a biological hypothesis that dynamics within a fish school
continually tend towards locally stable ordered states but are continually perturbed by surrounding
neighborhoods of individuals with distinct and possibly incompatible ordered states. If so, our
analysis suggests explicit and general predictions of what those states’ characteristics must
be, and how the balance between stability and perturbation would be statistically reflected in
movement data. Analytical results that tie general provable results to reasonable biological models
promise to move the coordinated group discussion incrementally towards general necessary and
sufficient conditions for attaining specific group properties.
R EFERENCES
[1] A. T. Winfree. Biological rhythms and the behavior of populations of coupled oscillators.
J. Theor. Biol., 16:15–42, 1967.
[2] Y. Kuramoto. Self-entrainment of a population of coupled non-linear oscillators. In Int.
Symp. on Math. Problems in Theor. Phys., pages 420–422. Kyoto Univ., January 1975.
[3] S. H. Strogatz. From Kuramoto to Crawford: exploring the onset of synchronization in
populations of coupled oscillators. Physica D, 143:1–20, 2000.
[4] N. E. Leonard, D. Paley, F. Lekien, R. Sepulchre, D.M. Fratantoni, and R.E. Davis.
28
Collective motion, sensor networks and ocean sampling. Proceedings of the IEEE, 2005.
Special issue on “The Emerging Technology of Networked Control Systems,” to appear.
[5] E. Fiorelli, N.E. Leonard, P. Bhatta, D.A. Paley, R. Bachmayer, and D.M. Fratantoni. MultiAUV control and adaptive sampling in Monterey Bay. Submitted to IEEE J. Ocean. Eng.
[6] A. Okubo. Diffusion and ecological problems: mathematical models. Springer, 1980.
[7] D. Grünbaum and A. Okubo. Modelling social animal aggregations. In S.A. Levin, editor,
Lecture Notes in Biomathematics 100, pages 296–325. Springer-Verlag, 1994.
[8] D. Grünbaum. Schooling as a strategy for taxis in a noisy environment. Evolutionary
Ecology, 12:503–522, 1998.
[9] G. Flierl, D. Grünbaum, S. Levin, and D. Olson. From individuals to aggregations: the
interplay between behavior and physics. J. Theor. Biol, 196:397–454, 1999.
[10] E.W. Justh and P.S. Krishnaprasad. Equilibria and steering laws for planar formations.
Systems and Control Letters, 52(1):25–38, 2004.
[11] A. Jadbabaie, J. Lin, and A.S. Morse. Coordination of groups of mobile autonomous agents
using nearest neighbor rules. IEEE Trans. Automatic Control, 48(6):988–1001, 2003.
[12] R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching
topology and time-delays. IEEE Trans. Automatic Control, 49(9):1520–1533, 2004.
[13] L. Moreau. Stability of multiagent systems with time-dependent communication links.
IEEE Trans. Automatic Control, 50(2):169–182, 2005.
[14] R. Sepulchre, D. Paley, and N. Leonard. Stabilization of planar collective motion, part I:
All-to-all communication. Submitted to IEEE Trans. Automatic Control.
[15] R. Sepulchre, D.A. Paley, and N.E. Leonard. Group coordination and cooperative control
of steered particles in the plane. Submitted to the Workshop on Group Coordination and
29
Cooperative Control, Tromso, Norway, May 2006.
[16] I.D. Couzin, J. Krause, R. James, G.D. Ruxton, and N.R. Franks. Collective memory and
spatial sorting in animal groups. J. Theor. Biol., 218(1):1–11, 2002.
[17] S.V. Viscido, J.K. Parrish, and D. Grünbaum. Individual behavior and emergent properties
of fish schools: a comparison of observation and theory. Marine Ecology Progress Series,
273:239–249, 2004.
[18] D. Grünbaum, S. Viscido, and J.K. Parrish. Extracting interactive control algorithms from
group dynamics of schooling fish. Number 309 in Lecture Notes in Control and Information
Sciences, pages 104–117. Springer-Verlag, 2005.
[19] S.V. Viscido, J.K. Parrish, and D. Grünbaum. The effect of population size and number
of influential neighbors on the emergent properties of fish schools. Ecological Modelling,
183:347–363, 2005.
[20] Y. Kuramoto. Chemical oscillations, waves, and turbulence. Springer-Verlag, 1984.
[21] C. Godsil and G. Royle. Algebraic Graph Theory. Number 207 in Graduate Texts in
Mathematics. Springer-Verlag, 2001.
[22] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1985.
[23] R. Sepulchre, D. Paley, and N.E. Leonard. Graph Laplacian and Lyapunov design of
collective planar motions. In Proc. Int. Symp. Nonlin. Theory Appl., October 2005. Bruges.
[24] J.A. Marshall and M.E. Broucke. On invariance of cyclic group symmetries in multiagent
formations. In Proc. 44th IEEE Conf. Decision Control, pages 746–751, Dec. 2005. Seville.
[25] J. Jeanne, N.E. Leonard, and D. Paley. Collective motion of ring-coupled planar particles.
In Proc. 44th IEEE Conf. Decision Control, pages 3929–3934, Dec. 2005. Seville.
[26] D. Paley, N.E. Leonard, and R. Sepulchre. Collective motion: Bistability and trajectory
30
tracking. In Proc. 43rd IEEE Conf. Decision Control, pages 1932–1937, 2004. Nassau.
[27] I.D. Couzin, J. Krause, N.R. Franks, and S.A. Levin. Information transfer, decision-making,
and leadership in animal groups. Nature, 433:513–516, 2004.
[28] B. Nabet, N.E. Leonard, I. Couzin, and S. Levin. Leadership in animal group motion: A
bifurcation analysis. Submitted to 2006 Symp. Math. Theory of Networks and Systems.
AUTHOR B IO ’ S
Derek A. Paley is a graduate student in the Department of Mechanical and Aerospace
Engineering at Princeton University. His research is partially supported by a NSF Graduate
Research Fellowship, a Princeton University Gordon Wu Graduate Fellowship, and the Pew
Charitable Trust grant 2000-002558.
Naomi E. Leonard is a Professor of Mechanical and Aerospace Engineering at Princeton
University. Her research is partially supported by ONR grants N00014–02–1–0826, N00014–
02–1–0861, and N00014–04–1–0534.
Rodolphe Sepulchre is a Professor of Electrical Engineering and Computer Science at the
Université de Liège. This paper presents research results of the Belgian Programme on
Interuniversity Attraction Poles, initiated by the Belgian Federal Science Policy Office. The
scientific responsibility rests with its authors.
Daniel Grünbaum is an Associate Professor in the School of Oceanography and Adjunct
Associate Professor in the Department of Biology at the University of Washington. The fish
tracking data presented here were generated in collaboration with Julia K. Parrish and Steve V.
Viscido with the support of N.S.F. grant CCR-9980058.
31
150
150
100
100
50
y 50
y
0
!50
0
!100
!150
!50
0
(a)
y
50
x
100
!100
20
10
10
0
y
!10
0
!10
!20
!20
!30
!30
!20
!10
(c)
Figure 1.
100
x
20
!30
0
(b)
0
x
10
20
30
!30
(d)
!20
!10
0
10
20
x
The four different types of collective motion obtained with the phase control (10)
for N = 12. The position of each particle, rk , and its velocity, eiθk , are illustrated by a red
circle with an arrow. The center of mass of the group, R, and its time-derivative, Ṙ = p(θ),
are illustrated by a crossed black circle with an arrow. (a) ω0 = 0 and K1 < 0; (b) ω0 = 0 and
K1 > 0; (c) ω0 6= 0 and K1 < 0; and (d) ω0 6= 0 and K1 > 0. Only (a) is a relative equilibria
of the particle model (1).
32
!1=0.00
!2=0.27
!3=1.00
!4=2.00
!5=3.00
!6=3.73
!7=4.00
!8=3.73
! =3.00
! =2.00
! =1.00
! =0.27
9
Figure 2.
10
11
12
Balanced critical points of the phase potential for a circulant graph Laplacian with
N = 12. The components of the eigenvectors (red circles) of a circulant graph Laplacian form
symmetric patterns on the unit circle. Each figure is labeled with the corresponding eigenvalue
for a cyclic graph Laplacian. The heading interconnections (blue lines) of a cyclic graph form
generalized regular polygons.
33
25
20
15
10
5
y
0
!5
!10
!15
!20
!25
!20
!10
0
x
10
20
!20
!10
0
x
10
20
!20
!10
0
x
10
20
(a)
25
20
15
10
5
y
0
!5
!10
!15
!20
!25
(b)
25
20
15
10
5
y
0
!5
!10
!15
!20
!25
(c)
Figure 3.
Stabilizing the circular formation with and without heading control. The spacing
and heading interconnection graphs are identical, cyclic graphs. Each figure has ω0 = 0.1 and
K0 = N ω0 . (a) Synchronized circular formation with K1 < 0; (b) circular formation with
K1 = 0; and (c) balanced circular formation with K1 > 0. Increasing the gain K1 > 0 in (c)
produces the balanced (2, N )-pattern which maximizes the potential W1 (θ).
34
20
20
20
10
10
10
y
0
0
0
!10
!10
!10
!20
!20
!20
0
20
!20
!20
0
20
!20
20
20
20
10
10
10
0
0
0
!10
!10
!10
0
20
y
!20
!20
!20
!20
0
20
x
!20
0
20
x
!20
0
20
x
Figure 4. Isolating the six symmetric patterns for N = 12 with the complete graph. The patterns
correspond to M = 1, 2, 3, 4, 6, and 12 evenly spaced clusters with ω0 = K0 = 0.1, Km > 0 for
m = 1, . . . , M − 1, and KM < 0. The top left is the synchronized state and the bottom right is
the splay state.
35
100
50
0
y !50
!100
!150
!200
0
100
200
300
x
400
500
Figure 5. Stabilizing the parallel formation with a reference heading. A sequence of rotational
symmetry breaking controls of the form (15) tracks a piecewise-linear reference trajectory with
ω0 = 0 and K1 < 0. The sequence of reference headings is π8 , − 3π
, π8 , and − 3π
.
8
8
20
20
10
10
y
y
0
!10
0
!10
!20
!!0
!40
!20
x
0
!20
20
!20
(a)
Figure 6.
(b)
!10
0
10
20
30
x
Stabilizing the circular formation with a fixed beacon in a drift vector field. Both
figures have ω0 = K0 = 0.1, K1 > 0, and drift velocity −0.1. (a) No symmetry breaking; (b)
symmetry breaking with c0 = 10 depicted by a black dot.
36
ZO
y
ZR
rk
Figure 7.
ZA
eiθk
θk
x
The concentric zones of attraction, ZA, orientation, ZO, and repulsion, ZR, in the
fish model (27), after [16]. The blind spot behind the fish is indicated by the blue lines and has
angular width αk .
37
25
30
20
15
20
10
y
5
y
10
0
0
!5
!10
!10
!15
!20
!20
!20
!10
0
10
20
!20
y
25
25
20
20
15
15
y
0
!5
!5
!10
!10
!15
!15
!20
!10
0
10
20
!20
!10
0
x
(b)
2$
25
20
20
1$
15
20
10
20
10
$
y
5
0
0
!$
!5
!10
!10
!1$
!15
!20
!20
10
x
10
y
20
5
0
!20
10
10
5
!20
Figure 8.
0
x
10
(c)
!10
x
(a)
!20
!10
0
10
20
!20
x
!10
0
x
A bifurcation in the circular formation that occurs when the outer radius of the
orientation zone (grey patches) is increased. The heading and spacing interconnection graphs
are generated by the attraction and orientation zones with ρatt |ω0 |−1 , ρori = ρ|ω0 |−1 , and
π
αk = 60◦ . The control parameters are ω0 = K0 = 0.1. The three rows correspond to ρ = 2 sin M
,
with (a) M = 2, (b) M = 8, and (c) M = 12. For K1 < 0 (left column), increasing the size of
the orientation zone (decreasing M ) stabilizes fewer, larger synchronized clusters of particles.
For K1 > 0 (right column), the particles cluster in symmetric patterns appropriate to the size of
the zone.
38
100
90
80
70
60
y (cm)
50
40
30
20
10
0
0
20
40
60
80
100
60
80
100
x (cm)
(a)
100
90
80
70
60
y (cm)
50
40
30
20
10
0
0
20
40
x (cm)
(b)
90
80
70
60
50
y (cm)
40
30
20
10
0
!10
0
(c)
Figure 9.
20
40
x (cm)
60
80
Actual fish school trajectories. The position (red circle), heading (black arrow),
trajectory (blue line), and orientation zone (grey patch) are plotted for each fish. (a) The 4-fish
experiment; (b) the tight milling 8-fish experiment; and (c) the diffuse milling 8-fish experiment.
39
n=N!1
n=1
0.8
Frequency
1
0.9
1000
500
0
0.6
0.5
0.4
Frequency
0.1
40
45
50
Time (seconds)
55
1
n=N!1
n=1
0.9
0.8
0
0.6
0.7
0.8
0.9
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
!0.2
0
0.2
0.4
0.6
0.8
1
Difference/012ue!re;5
1000
500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.7
0.8
0.9
1
n=N!1 (blue)
0.6
Frequency
2W1(!)/(n+1)
0.5
500
0
0.5
0.4
1000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
n=1 (red)
Frequency
0.3
0.2
0.1
40
45
50
Time (seconds)
55
1
n=N!1
n=1
0.9
0.8
1000
500
0
!0.4
60
!0.2
0
0.2
0.4
0.6
0.8
1
Difference (blue!red)
Frequency
35
(b)
1000
500
0
0.1
0.7
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n=N!1 (blue)
0.6
Frequency
2W1(!)/(n+1)
0.4
1000
0.7
0.5
0.4
1000
500
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n=1 (red)
Frequency
0.3
0.2
0.1
(c)
0.3
500
0
!0.4
60
Frequency
35
(a)
0
30
0.2
n=1/0re;5
0.2
0
30
0.1
1000
0
0.3
0
30
0
n=N!1/012ue5
Frequency
2W1(!)/(n+1)
0.7
35
40
45
50
Time (seconds)
55
60
1000
500
0
!0.4
!0.2
0
0.2
0.4
0.6
0.8
1
Difference (blue!red)
Figure 10. The value of the Laplacian phase potential for fish schooling data using the sensory
graph generated by the n nearest neighbors of each fish. The scaled potential
2
W1 (θ)
n+1
is plotted
for n = 1 and n = N − 1 in the left column for a representative thirty second period starting at
t = 30 seconds. The right column shows histograms of the scaled potential for n = N −1, n = 1,
and the difference between the two potentials for all ten minutes of (a) the 4-fish experiment;
(b) the tight milling 8-fish experiment; and (c) the diffuse milling 8-fish experiment.
40