# Modeling Phase Transition in Self-organized Mobile Robot Flocks.

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**ABSTRACT:**In this paper, we study how a self-organized mobile robot flock can be steered toward a desired direction through externally guiding some of its members. Specifically, we propose a behavior by extending a previously developed flocking behavior to steer self-organized flocks in both physical and simulated mobile robots. We quantitatively measure the performance of the proposed behavior under different parameter settings using three metrics, namely, (1) the mutual information metric, adopted from Information Theory, to measure the information shared between the individuals during steering, (2) the accuracy metric from directional statistics to measure the angular deviation of the direction of the flock from the desired direction, and (3) the ratio of the largest aggregate to the whole flock and the ratio of informed individuals remaining with the largest aggregate, as a metric of flock cohesion. We conducted a systematic set of experiments using both physical and simulated robots, analyzed the transient and steady-state characteristics of steered flocking, and evaluate the parameter conditions under which a swarm can be successfully steered. We show that the experimental results are qualitatively in accordance with the ones that were predicted in Couzin etal. model (Nature, 433:513–516, 2005) and relate the quantitative differences to the differences between the models. KeywordsSwarm robotics-Self-organization-FlockingNeural Computing and Applications 09/2010; 19(6):849-865. · 1.76 Impact Factor - SourceAvailable from: Hande Çelikkanat[Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we study how and to what extent a self-organized mobile robot flock can be guided to move in a desired direction by informing some of the individuals within the flock. Specifically, we extend a flocking behavior that was shown to maneuver a swarm of mobile robots as a cohesive group in free space, avoiding obstacles. In its original form, the flock does not have a preferred direction and would wander aimlessly. In this study, we extend this behavior by "informing" some of the individuals about the desired direction that we wish the swarm to move. The informed robots do not signal that they are "informed" (a.k.a. unacknowledged leadership) and instead guide the swarm by their tendency to move in the desired direction. Through experimental results with physical and simulated robots we show that the self-organized flocking of a robot swarm can be effectively guided by an informed minority. We use two metrics to measure the accuracy of the flock direction, and the ability to stay cohesive. We show that the accuracy of guidance increases with 1)the flock size, 2)the "stubbornness" of the informed individuals to align with the desired direction, and 3)the ratio of the informed individuals.

Page 1

Modeling Phase Transition

in Self-organized Mobile Robot Flocks?

Ali Emre Turgut, Cristi´ an Huepe, Hande C ¸elikkanat,

Fatih G¨ ok¸ ce, and Erol S ¸ahin

KOVAN Res. Lab., Dept. of Computer Engineering

Middle East Technical University, Turkey

{aturgut,hande,fgokce,erol}@ceng.metu.edu.tr,

cristian@northwestern.edu

Abstract. We implement a self-organized flocking behavior in a group

of mobile robots and analyze its transition from an aligned state to an

unaligned state. We briefly describe the robot and the simulator plat-

form together with the observed flocking dynamics. By experimenting

with robotic and numerical systems, we find that an aligned-to-unaligned

phase transition can be observed in both physical and simulated robots

as the noise level is increased, and that this transition depends on the

characteristics of the heading sensors. We extend the Vectorial Network

Model to approximate the robot dynamics and show that it displays an

equivalent phase transition. By computing analytically the critical noise

value and numerically the steady state solutions of this model, we show

that the model matches well the results obtained using detailed physics-

based simulations.

1 Introduction

Flocks of birds, herds of quadrupeds and schools of fish stand as fascinating

examples of self-organized coordination, where groups of individuals coherently

move and maneuver in space as a collective unit [1,2]. Although it has long

been studied in biology, it was Reynolds [3] who first demonstrated flocking

in artificial systems, showing that realistic flocking behavior can be obtained

in computer animation using a number of simple behaviors. Reynolds’ seminal

work generated interest in many different fields.

In robotics, Matari´ c [4] made one of the earliest attempts to obtain flocking

in a group of robots by combining safe-wandering, aggregation, dispersion, and

homing behaviors. She was able to demonstrate that a group of robots can “flock”

towards a common homing direction while maintaining a cohesive grouping.

?The works of A.E. Turgut, F. G¨ ok¸ ce and E. S ¸ahin are supported by T¨UB˙ITAK un-

der grant no: 104E066. The work of C. Huepe is supported by the National Science

Foundation under Grant No. DMS-0507745. H.C ¸elikkanat acknowledges the partial

support of the T¨UB˙ITAK graduate student research grant. F. G¨ ok¸ ce is currently en-

rolled in Faculty Development Program (¨OYP) in Middle East Technical University

on behalf of S¨ uleyman Demirel University.

M. Dorigo et al. (Eds.): ANTS 2008, LNCS 5217, pp. 108–119, 2008.

c ? Springer-Verlag Berlin Heidelberg 2008

Page 2

Modeling Phase Transition in Self-organized Mobile Robot Flocks109

In [5], Kelly and Keating used robots that can sense the obstacles as well as

the relative range and bearing of their neighbors through a custom-made active

infrared system. The robots used a radio-frequency system to elect one of them

as the leader which would then wander in the environment while being followed

by the others. In a recent study, Hayes et al. [6] proposed a “leaderless distributed

flocking algorithm” consisting of two simpler behaviors: collision avoidance and

velocity matching, using local center-of-mass calculations based on emulated

range and bearing information. Additional studies in robotic flocking have been

carried out in works such as [7,8].

Flocking has also attracted interest in physics, where various models have

been proposed to study the emergence of order in such systems. The emergence

of order corresponds to the collective self-alignment of the group to a common

heading direction as a result of the interactions among its individuals. In a

pioneering study, Vicsek et al. [9] proposed the Self-Driven Particles (SDP) model

to explain the emergence of order in biological swarms. The SDP model uses

massless and volumeless particles that move at a constant speed in a square arena

with periodic boundary conditions. The heading of each particle is updated to

the average direction of motion of its local neighbors and a noise term which is

added to account for uncertainties in its inputs and control. Simulations of this

model revealed that particles align if the system is above a critical mean density

or if the magnitude of the noise is below a critical value. In a follow-up study,

Gregoire et al. [10] extended the SDP model to add attraction and repulsion

among the particles in their local neighborhood, thus achieving self-organized

motion in an open domain.

Aldana et al. [11] proposed the Vectorial Network Model (VNM) to study

the emergence of long-range order in systems with mostly local interactions. In

the VNM, particles are placed in a two-dimensional lattice and their positions

remain fixed. Each heading thus only determines here a pointing direction for

a given lattice site. As in the SDP, headings are updated to the average point-

ing directions of its inputs, but here only some of these inputs are taken from

neighboring lattice sites and the rest from randomly chosen (possibly distant)

sites. The VNM simulations show that long-range order does not emerge at any

nonzero noise value if the neighbors are chosen only locally. However, long-range

order does emerge for sufficiently small noise values if a small fraction of the in-

puts are chosen from random particle sites. An interesting aspect of the VNM is

that an analytic expression was obtained in [11] to describe its order-to-disorder

phase transition for the case with no local interactions, where all inputs are taken

from random particle sites.

Despite many efforts to control and model flocks in robotics and statistical

physics, these two lines of research have remained relatively disconnected from

each other. An exception can be found in the recent study in

link was established between the behavior of multi-robot systems and phase

transitions. There have been two main reasons behind the failure to integrate

both perspectives. First, until recently, true self-organized flocking behavior like

the one observed in nature had not been achieved in robotic systems. Indeed,

[12], where a

Page 3

110A.E. Turgut et al.

(a)

0

1

2

3

4

57

6

(b)(c)(d)

Fig.1. (a) Photo of a Kobot. (b) Top-view of a Kobot sketch showing the body (circle),

the IR sensors (small numbered rectangles), and the two wheels (grey rectangles). (c-

d) Starting from a disordered state, 7 Kobots negotiate a common heading and advance.

White arrows on the robots indicate the forward direction.

the previous experimental studies in robotics used either a virtual or an explicit

leader [5] to guide the group or assumed that a target heading (or destination)

was sensed by the whole group [4,6,13]. Moreover, in some of these studies [6],

the authors used “emulated sensors”. Second, the assumptions required by the

models developed in physics were often considered to be too unrealistic to be

linked with studies conducted in robotics. The SDP model, for example, uses

massless and volumeless mobile particles.

In [14], we reported the first true self-organized flocking on a group of mobile

robots and showed that the robots can maneuver in an environment as a cohesive

body while avoiding obstacles on their path. In this paper, we first demonstrate

that the emergence of order in a robot flock depends on the amount of noise in

the heading alignment behavior. We then extend the VNM to model the order-

to-disorder phase transition that occurs in these robotic systems as the noise

level is increased. Finally, we compare the predictions of the proposed model to

the results obtained from robots.

2 Experimental Framework

We used a custom-built mobile robot platform, called Kobot, and its physics-

based simulator, that we refer to as CoSS, in our experiments. Kobot is a CD-

sized (with a 12 cm diameter), light-weight, differentially driven robotic plat-

form (Figure 1(a)). It possesses an active Infrared Short-Range Sensing (IRSS)

system designed for short-range proximity measurements. This system utilizes

modulated infrared signals to minimize environmental interference and crosstalk

among the robots. It consists of eight sensors placed evenly at 45◦intervals (see

Figure 1(b)), each of which is capable of sensing kin-robots and obstacles within

a 21 cm range in seven discrete levels at 18 Hz.

The Virtual Heading Sensor (VHS) consists of a digital compass and a wireless

communication module to receive the relative headings of neighboring robots.

The VHS module measures its own heading with respect to the sensed North

at each control step and broadcasts it to other robots through wireless

communication. Each robot receives the broadcasted heading values within its

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Modeling Phase Transition in Self-organized Mobile Robot Flocks111

communication range. We define the set of robots that are “heard” by a given

robot as its VHS neighbors. The angular difference between the broadcasted val-

ues and its own heading allows a robot to compute its relative heading with

respect to its VHS neighbors and adjust it as needed. This operation of the VHS

module assumes that the sensed North remains approximately the same among

the robots within communication range.

Flocking Behavior. The flocking behavior [14] consists of a heading alignment

and a proximal control combined in the weighted vector sum:

a =1

8h + p,

where h is the heading alignment vector, p is the proximal control vector, and a

is the desired acceleration vector. The heading alignment vector h, which is used

to align the robot with the average heading of its neighbors, is calculated as:

?

where N denotes the set of VHS neighbors, θjis the heading of the jthneighbor

and ? · ? calculates the Euclidean norm.

The proximal control behavior uses readings obtained from the IRSS to avoid

collisions and to maintain cohesion between the robots. For each IR sensor, a

virtual force proportional to the square of the difference between the measured

distance and the desired distance is assumed. The desired distance (ddes) is

defined as a finite value for other robots and ∞ for obstacles, in order to keep

a fixed distance to its peers while moving away from obstacles. The normalized

proximal control vector p is therefore given by:

p =1

8

k

where k ∈ {0,1,···,7} denotes the sensor positioned at angle φk =

respect to the x-axis (see Figure 1(b)) and fkis calculated as:

⎧

⎩

ties of the robot. The forward velocity u is modulated depending on the deviation

of the desired acceleration vector from the current direction, as given by:

?

0

where ˆ acis the current direction of the robot parallel to the y-axis of the body-

fixed reference frame. The angular velocity ω is controlled by a proportional

controller:

ω =1

2(? ˆ ac−? a).

h =

j∈Neiθj

??

j∈Neiθj?

?

fkeiφk

π

4k with

fk=

⎨

−(dk−ddes)2

(dk−ddes)2

1

8

1

8

if dk≥ ddes

otherwise.

The desired acceleration vector is mapped to the forward and angular veloci-

u =

0.7

?

a

?a?.ˆ ac

?

if

otherwise

a

?a?.ˆ ac≥ 0

Page 5

112A.E. Turgut et al.

The Order and Average Order Metrics. We evaluate the flocking perfor-

mance by using the well-studied measure of order ψ [9], which corresponds to

the average alignment of the group and is computed as:

?????

where M is the number of robots and θk is the heading of the kthrobot. The

order can take any value between 0 and 1. When individuals are aligned and

the system is ordered we have ψ ≈ 1, and when individuals are not aligned and

the system is disordered we have ψ ≈ 0. When the steady-state performance of

the flocking behavior is considered, we will use the time average of the order,

denoted by¯ψ.

ψ =

1

M

M

?

k=1

eiθk

?????,

3Modeling the Virtual Heading Sensor

The properties of the flocking behavior depend on two specific characteristics

of the virtual heading sensor, namely: (1) the number of VHS neighbors that

can be simultaneously detected, and (2) the nature and amount of noise in the

digital compass measurements. Hence, we will study here the dependence of

the system on these characteristics by modeling them with the CoSS simulator.

The number of VHS neighbors N depends on the range of communication, the

number of robots within this range, and the frequency and duration of this

communication. Systematic analysis has shown that in the physical robots, the

number of VHS neighbors can be as large as 20 in large groups.

The noise in VHS solely depends on the noise characteristics of the digital

compass. In indoor environments, the presence of ferrous materials induce large

amounts of noise on the digital compass. We model this noise by adding a vector

of fixed magnitude in a random direction to each heading measurement [10,12].

The resulting noisy heading of the jthneighbor received as input by a given

−180 −120−60 060 120180

0

0.01

0.02

0.03

0.04

? ?h [degree]

(a)

p(x)

η=0.25

η=1

η=120

051015

time [s]

(b)

20 2530

0

0.2

0.4

0.6

0.8

1

ψ

η=0

η=1

η=15

0 50100

(c)

150 200250

0

0.2

0.4

0.6

0.8

1

time [s]

ψ

η=0

η=5

η=15

Fig.2. (a) Probability density function of the simulated noisy measurement inputs

of one VHS neighbor for different η values. (b) Evolution of ψ in time for 7 Kobots.

(c) Evolution of ψ in time for 7 robots in the CoSS simulator. In this figure and

subsequent figures the standard deviations are indicated using error bars.

Page 6

Modeling Phase Transition in Self-organized Mobile Robot Flocks113

robot is:

θj=? (eiθa+ ηeiξ)

where θa is the actual heading of the jthneighbor, η is the magnitude of the

noise vector, and? (·) is function that calculates the argument of a vector. The

noise-vector direction ξ is a delta-correlated random variable with a uniform

distribution in [−π,π].

The probability density functions of the simulated noisy measurements is

plotted in Figure 2(a) for various values of η. It is apparent that η determines

the standard deviation of the resultant noisy headings, as expected.

4Analysis of the Flocking Behavior

We investigate the effect of the sensing noise on the transient and on the steady-

state to characterize the flocking phase transition in our system. The transient

characteristics were investigated by conducting experiments with 7 Kobots and

simulating 7 robots in CoSS for 1 VHS neighbor. The steady-state characteristics

were investigated by simulating 100 robots in CoSS. The robots were initially

placed in a regular hexagonal formation with a center-to-center distance of 25 cm

having random orientations. Each experiment was repeated 10 times.

Transient Analysis. We varied the sensing noise on the VHS and measure the

evolution of ψ in time for 7 Kobots in an actual experiment and for 7 robots

in the CoSS simulator. In the Kobot experiments, the environmental noise is

assumed to be negligible, so we set it to correspond to the η = 0 case. Higher η

values are attained by adding noise artificially to each heading measurement. The

evolution in time of ψ for the experimental and simulation cases are plotted on

Figures 2(b) and 2(c). The results indicate that that ψ increases from a random

initial condition, approaching approximately 1 when η is low, while settling to

smaller ψ values when η is increased. Although the trends in the Kobot and CoSS

cases are the same, Kobots perform worse than the robots in the CoSS simulation

for the same amount of noise. This is probably due to the environmental noise

in the Kobot experiments, that adds an unpredictable base noise level to the

system.

Steady-State Analysis. We will now analyze the steady-state flocking behav-

ior by finding the value of ψ at which the system settles for various noise levels.

We consider the results of two sets of CoSS simulations: (1) runs with a small

and a large number of robots (7 and 100, respectively), but with only 1 VHS

neighbor, (2) runs with 100 robots and different numbers of VHS neighbors. Each

simulation was conducted for 1000 s and its steady-state ψ value was calculated

by averaging over the last 500 s. This guarantees that the steady-state condition

has been achieved since it takes less than 250 s for ψ to converge. The results

are plotted on Figures 3(a) and 3(b).

Figure 3(a) shows that, regardless of the flock size, an increase in η decreases

the order. However, the order shows a much smaller decrease for the small group

Page 7

114A.E. Turgut et al.

05

η

15

0

0.2

0.4

0.6

0.8

1

ψ

7 robots

100 robots

(a)

0 1224364860728496108120

0

0.2

0.4

0.6

0.8

1

η

ψ

N=1

N=3

N=7

N=10

N=15

N=18

N=20

(b)

Fig.3. (a) Plot of¯ψ measured in CoSS simulations containing 7 and 100 robots. The

horizontal lines in the boxes, the box-ends, and the additional error bars correspond

to the median and to its first and third quartiles, respectively. (b) Plot of¯ψ for 100

robots in CoSS for various N.

when η = 15. This is due to the finite size effects observed that are known

to blur the difference between order and disordered collective states in small-

sized statistical systems. In the rest, we will therefore focus on CoSS simulations

with 100 robots, which enables reasonable run times while minimizing finite-size

effects. It is apparent on Figure 3(b) that for all N values, ψ approaches 1 and

the system organizes in a coherent flock as the noise level η is lowered to zero.

If η is increased, ψ decreases and eventually approaches to 0. We also observe

that increasing N increases the order of the flock for a constant η. The order-to-

disorder transition that occurs by increasing the noise level in this robotic system

is equivalent to the second-order phase-transition observed in various statistical

physics systems. As in the physics context, we will refer to the η value at which

the system loses all order as the critical noise level ηc.

5Modeling the Phase Transition in Flocking

We are interested in modeling analytically the order-to-disorder phase transi-

tion observed in the flocking behavior of Kobots as a function of noise, and to

determine ηc for a given number of VHS neighbors. We will consider a simple

network-based model that includes the noise, interactions with randomly chosen

agents, and an inertia-like term that captures qualitatively various local interac-

tions that affect the Kobot dynamics. Our model does not, however, deal with all

the complexities of the behavior. Indeed, in order to allow analytical solutions,

it avoids any spatial description by representing agents as nodes interacting

through random switching connections in a network. This model is an extension

of the Vectorial Network Model (VNM) introduced in [11] as a network version

of the SDP model in [9].

The original VNM can be solved analytically and is known to display an

order-to-disorder transition similar to the one observed in Kobots, but it is not

well adapted to describe some of the details of our robotic system. For example,

the noise is introduced differently and the agents can turn in any direction at

Page 8

Modeling Phase Transition in Self-organized Mobile Robot Flocks115

every time-step, with no restrictions on the turning rate. We therefore introduce

a new network system to model the Kobots, the Stiff-Vectorial Network Model

(S-VNM), which we define as follows. At every time-step, each node updates its

heading hj(t) based on N inputs chosen randomly from any node in the system,

according to

hj(t + 1) = κeiθj(t)+ λ

N

?

k=1

eiθk(t)+ η

N

?

k=1

eiξk(t).

(1)

Here, the heading hj is a vector of arbitrary magnitude with? [hj] = θj, where

the heading of Kobot j is given (in this network representation) by the angle

θj associated to node j. The model parameters κ, λ, and η determine the rel-

ative importance of the persistence, interaction, and noise terms, respectively.

The latter two correspond directly to the terms implemented in the VHS control

part of the Kobot dynamics. Here, the noise is introduced as in [10], with a

fixed magnitude η in a random direction given by ξk, a delta-correlated random

variable uniformly distributed in [−π,π]. The persistence term models qualita-

tively an effective inertia that appears mainly due to the proximity interactions

between Kobots. These make it harder for a given robot to turn in response to

its VHS inputs or the noise since, if it is surrounded by other robots heading in

the same direction, these will block it from shifting its heading.

Analytical Treatment of the S-VNM. One of the main appeals of the S-

VNM lies in our ability to treat it analytically. We derive here a solution that

allows us to compute the critical noise value ηc(at which the order-to-disorder

transition occurs) in terms of κ, λ, and N. We will approach this problem as

follows. First, we compute the probability density function (PDF) of each term

in Equation (1). Then we impose that the PDF of θj(which is equal for all j) is

the same at time t and at time t+1, which, together with Equation (1), provides

us with a closed expression for the statistical steady state of this PDF. Finally,

we find the value of η at which a constant distribution for θjbecomes unstable.

This corresponds to the critical noise level ηcabove which there is a stationary

distribution with hj pointing in any direction with the same probability (the

disordered state), and below which such solution is unstable, thus drifting the

system to a distribution with a preferred direction for hj(the ordered state). In

what follows we will use the fact that the magnitude of h does not intervene in

the dynamics of the S-VNM to rescale Equation (1) by dividing it by λ. We thus

define ˜ κ = κ/λ and ˜ η = η/λ and use these rescaled variables below, dropping

the tildes until the end of this calculation to simplify the notation.

We first consider the noise term, which can be viewed as the total displacement

that results after taking N steps of length 1 in a two-dimensional random walk.

Using this analogy, we can apply well-known random walk results and write the

PDF of the position on the x y plane after N steps as

Pη(x,y) =

1

Nη2πe−x2+y2

Nη2.

(2)

Page 9

116A.E. Turgut et al.

We will need below the two-dimensional Fourier transform of Eq. 2, that can

be readily computed to obtain

ˆPη(λx,λy) = e−1

4Nη2(λ2

x+λ2

y).

(3)

We now carry out the calculation of the PDF of the interaction term, which

can be quite involved. Fortunately, this result was already obtained in [11]. As-

suming that all the θk have the same PDF and are statistically independent

(which is true if the N inputs are all picked at random from any node in a large

system), the Central Limit Theorem is used in [11] for large enough N values

to find an approximate Gaussian expression for this PDF. Computing again its

Fourier transform we obtain:

ˆPNθ(λx,λy) = eiN(Δ1,0λx+Δ0,1λy)−N

2(σ2

cλ2

x+σ2

sλ2

y+2σ2

csλxλy)

(4)

where σ2

Δ1,1(t)−Δ1,0Δ0,1. Here, Δm,n(t) denotes the instantaneous cosine-sine moment

of the angle distribution, given by

?π

in which Pθ(α;t) is the PDF of the angle that describes the heading of each

particle. Note that Eq. (4) converges very rapidly to the exact PDF of the inter-

action term as N is increased, providing a very good approximation for N > 5.

The derivations above furnish expressions for the PDF of every element of

equation (1) in terms of the PDF of the direction of a single particle Pθ(α;t).

However, as they stand these expressions are far too complicated to find the

functional form of Pθ(α;t) as a stationary solution of Eq. (1). To continue the

calculations, we will thus concentrate in solutions close to ηc. If η > ηc, the

system is in the disordered regime and we know that all hj must point in any

direction with the same probability. This corresponds to having a uniform dis-

tribution Pθ(α) =

stationary solution of the dynamics since the system becomes organized and the

symmetry in the pointing directions of all hjmust be broken. Therefore, a small

perturbation about the Pθ(α) solution must grow in this regime. Without loss

of generality, we can write a generic small perturbation of Pθ(α) as:

1

2π+ δ cos(α),

where δ ? 1. Using this form for Pθ(α;t), Eq. (4) becomes:

c(t) = Δ2,0(t) − [Δ1,0(t)]2, σ2

s(t) = Δ0,2(t) − [Δ0,1(t)]2, and σ2

cs(t) =

Δm,n(t) =

−π

Pθ(α;t)cosm(α)sinn(α)dα,

1

2π. If η < ηc we know that this PDF cannot be a stable

Pθ(α) =

(5)

ˆPNθ(λx,λy) = e

iπNδλx−1

2N

?

(1

2−π2δ2)λ2

x+

λ2

2

y

?

.

(6)

We now can find the combined PDF of the interaction and noise terms by

computing the inverse Fourier transform of the product ofˆPη(λx,λy) times

ˆPNθ(λx,λy) to obtain

Page 10

Modeling Phase Transition in Self-organized Mobile Robot Flocks117

PNθη(x,y) =

1

πN?(1 + η2)(1 + η2− 2π2δ2)e

−1

N

?

(x−Nπδ)2

1+η2−2π2δ2+

y2

1+η2

?

.

(7)

¿From this result we find the PDF of the Right Hand Side (RHS) of equation (1)

by calculating the convolution of PNθηwith Pθ(α) (the PDF of the persistence

term). The resulting expression is:

?π

Expanding Eq. 8 to first order in the small δ perturbation, then expressing the

resulting equation in polar coordinates (R,Φ) and integrating over R we finally

obtain the PDF of the RHS

1

2π+ δΓ cos(Φ),

where

PRHS(x,y) =

−π

PNθη(x − κcosθ,y − κsinθ)Pθ(θ)dθ.

(8)

PRHS(Φ) =

(9)

Γ =

√π e

2?N(1 + η2)

−κ2

2N(1+η2)

?

(N + κ)I0

?

κ2

2N(1 + η2)

?

+ κI1

?

κ2

2N(1 + η2)

??

. (10)

Here In(·) are the Modified Bessel Functions of the first kind, usually defined

mathematically as the solutions to the differential equation: z2y??+ zy?− (z2+

n2)y = 0.

In the stationary case, the LHS of Equation 1 is equal to Pθ(φ). Hence, the

LHS and the RHS have the same form and the condition for a statistically

stationary solution Pθ(Φ) = PRHS(Φ) becomes

1

2π+ δ cos(Φ) =

1

2π+ λΓ cos(Φ).

(11)

The condition for Pθ(α;t) = 1/(2π) to be a stable stationary solution of the

probability distribution associated to the dynamics described in Eq. (1) is there-

fore given by Γ < 1. Setting Γ to 1, we thus find the critical noise level ηcat

which the order-to-disorder transition occurs.

While Γ = 1 fully determines ηc implicitly in terms of the model parame-

ters, this condition cannot be easily inverted to obtain an expression for ηc. We

can find an explicit approximate form for ηc, however, by carrying out certain

approximations in the regime that we are considering. We set κ and λ to 1.5

and 22, respectively, to capture the dynamics of the flocking behavior. For these

coefficients,

expressions, the terms containing κ are small when compared to those containing

other coefficients and, thus, they can be neglected. By then substituting η = ˜ ηλ

we obtain the following simple approximate expression for ηc

?

κ2

2N(1+η2)? 1, which makes I0() ∼ 1 and I1() ∼ 0. In the resulting

ηc= λ

Nπ

4

.

(12)

Page 11

118A.E. Turgut et al.

0102030

η

4050 60

0

0.2

0.4

0.6

0.8

1

¯ψ

N=1(S−VNM)

N=1(CoSS)

N=7(S−VNM)

N=7(CoSS)

(a)

0510

N

1520

0

20

40

60

80

100

ηc

CoSS

S−VNM analytic

(b)

Fig.4. (a) Phase transition diagram obtained numerically using S-VNM and CoSS

simulations. (b) Critical noise values obtained using the analytical solution of S-VNM

and CoSS simulations.

Results Using the S-VNM. The S-VNM can be utilized in two ways to pre-

dict the phase-transition in flocking. On one side, the S-VNM can be easily and

efficiently implemented numerically to obtain the full phase-transition diagram

for the stationary flocking solutions resulting from a given set of parameters. On

the other side, we can use the analytical solution found above for the S-VNM to

predict ηcas a function of N.

The steady-state response was investigated by simulating the S-VNM numer-

ically. The simulations were performed with 100 particles for 10000 time-steps

using N = 1 or N = 7.¯ψ for the last 5000 steps is plotted in Figure 4(a) as a

function of η. On the same plot,¯ψ of analogous CoSS simulations is also dis-

played. It is apparent that predictions of the S-VNM are in close agreement with

the CoSS results for N = 1. A slight deviation is observed in the N = 7 case as

the system becomes organized.

Figure 4(b) displays the predicted critical noise value for the flocking behavior

obtained using Eq. 12 together with results from CoSS simulations as a function

of N. The two results are in close agreement both for small and large N values.

However, we should note that Eq. 12 is actually only meant to be valid for

relatively large values of N (typically at least N > 5) due to the use of the

Central Limit Theorem in the analytical treatment.

6Conclusion

In this paper we studied self-organized flocking in a group of mobile robots.

We consider this work as a first step towards linking the mathematical models

of flocking proposed in statistical physics with the results obtained in robotic

systems. In this particular study, we showed the existence of an order-to-disorder

phase transition in flocking and that the amount of noise as well as the number of

neighbors with which each robot interacts determines the characteristics of this

transition. We have extended the Vectorial Network Model to incorporate the

dynamics of our robots and showed that the steady-state order characteristics

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Modeling Phase Transition in Self-organized Mobile Robot Flocks119

predicted by the model matches the ones obtained for the robotic system. This

analysis shows that the proximal interactions among the robots can be crudely

approximated by the inclusion of a stiffness term in the model.

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