Steering self-organized robot flocks through externally guided individuals

SWARM ROBOTICS

Steering self-organized robot flocks through externallyguided individuals

Hande Celikkanat • Erol Sahin

Received: 15 August 2009 / Accepted: 26 February 2010

� Springer-Verlag London Limited 2010

Abstract In this paper, we study how a self-organized

mobile robot flock can be steered toward a desired direc-

tion through externally guiding some of its members.

Specifically, we propose a behavior by extending a previ-

ously 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, adop-

ted 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 flock-

ing, 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 et al. model (Nat-

ure, 433:513–516, 2005) and relate the quantitative dif-

ferences to the differences between the models.

Keywords Swarm robotics � Self-organization �Flocking

1 Introduction

The coordination and control of large numbers of robots is

a challenge that defies most classical approaches to single-

robot control. Swarm robotics [7, 11], which takes its

inspiration from natural swarms such as ant colonies, aims

to tackle this challenge by analyzing the coordination

mechanisms that underly the self-organized operation of

natural swarms [4] and using them to ‘‘engineer’’ self-

organization in large groups of robots.

Self-organized systems are touted for their robustness

and scalability through their reliance on local interactions

[2]. However, the local interactions that lead to the emer-

gence of self-organization also make it difficult for an

external user to impose full control over the system. For

instance, the problem of making a flock of sheep follow a

desired trail is completely different, and we claim more

difficult, than the problem of making a single sheep do the

same. In the first case, the non-linear local interactions

among the sheep would give the flock a ‘‘mind of its own’’

and this limits the amount and extent of control an external

user can exert.

Most of the ongoing studies in swarm robotics (see [7–9,

11] for a collection of papers) have focused on developing

and analyzing coordination strategies that lead to self-

organization in groups of robots. In these studies, the

limitations of controllability due to the use of the self-

organization approach have been neglected so far, leaving

the question of how useful the approach can be in real-

world use, unanswered.

In this study, we are interested in how and to what extent

we can control the behavior of a swarm robotic system.

Specifically, we extend the flocking behavior proposed in

[53] by informing some of the robots about the preferred

direction in which we wish the swarm to move. The

H. Celikkanat � E. Sahin (&)

Kovan Research Lab., Department of Computer Eng,

Middle East Technical University, Ankara, Turkey

e-mail: [email protected]; [email protected]

H. Celikkanat

e-mail: [email protected]

123

Neural Comput & Applic

DOI 10.1007/s00521-010-0355-y

informed robots do not signal that they are ‘‘informed’’ and

instead guide the rest of the swarm by their tendency to

move in the preferred direction. We present the results of

experiments on both physical and simulated robots, and we

analyze the conditions that a self-organized flock of robots

can be effectively guided by a minority of informed robots

within the flock.

1.1 External control of swarm systems

A number of previous studies have investigated how the

behavior of swarms can be externally guided or influenced.

In an interesting study, Vaughan et al. [55] used a robotic

sheepdog to herd a duck flock to a predefined goal point.

Using an overhead camera system to track the position and

the orientation of the robot, as well as the size and center of

mass of the duck flock, they were able to compute the

instantaneous direction of motion for the robot to guide the

ducks to a desired goal point. In [24], Lien et al. studied

how a flock can be steered toward a desired path by an

external shepherd, and they proposed a set of strategies to

develop and improve behaviors such as herding, covering,

and patrolling, using simulated agents. In a later study [25],

the approach was extended to the case of multiple shep-

herds which have no means of explicit communication. In

[21], Kazadi and Chang studied the conditions under which

a swarm can be vulnerable to perturbations generated by

adversarial agents which aim to control the overall

behavior of the swarm.

1.2 Control of swarm systems via informed individuals

A different approach toward the control of swarms is to

embed externally controlled individuals. In this approach,

these embedded individuals affect the decision or the

steering of the swarm toward a desired direction via only

their local interactions and without explicitly signaling

their status. In [32], Correll et al. explored how a cow herd

can be controlled by changing the behaviors of some

individuals. Specifically, the authors devised and imple-

mented a mountable device that induces stress on cows,

which in return increases their tendency to aggregate, by

moving toward the herd center. It was shown that, by

stressing a fraction of the cows, the herd can be guided

toward a final goal position. The authors used a stress

model that propagates within a radius of detection, and

decays in time which was then used to analyze the per-

formance of the method with respect to parameters such as

the number of stressed cows, the radius of detection. Ward

et al. [57] proposed that decision making in swarms may

depend on quorum responses and used replica conspecifics

to manipulate a binary (left/right direction) preference task

in fish. The results indicated a quorum response, which was

shown to decrease the likelihood of amplifying a wrong

choice made by a small number of individuals through

indiscriminative mimicry of others. Furthermore, as the

group size increased, the non-linear quorum response

became more accurate than independent decision making

or weak linear response. Halloy et al. [15] manipulated the

collective shelter selection process of cockroaches with

robots that were socially integrated into the group. When

the robots were programmed to prefer darker shelters like

the cockroaches, robots and cockroaches would select a

common shelter, suggesting a collective decision making in

the mixed society. In contrast, when the robots were pro-

grammed to prefer lighter shelters, in most cases, they were

able to lead the whole group to prefer the lighter shelter

despite being in a minority.

Reebs [40] studied the foraging behavior of fish schools

and showed that a minority of informed individuals can

guide the whole school. In this study, 12 golden shiners

were trained in a tank to find food in a brightly lit corner. It

was shown that the flock was still guided toward the correct

corner, despite the replacement of informed individuals

(specifically 7, 9, and 11 individuals were removed) with

naive (a.k.a untrained) ones. The probability of choosing

the correct corner was shown to be proportional to the

number of informed individuals in the group. Similarly,

Seeley et al. [47] investigated the flight of honey bee

swarms in order to move to a new nest site when the hive

becomes too crowded. They have shown that only 5% of

the swarm members visit the new nest site prior to swarm

lift-off and are therefore aware of the goal direction. Such a

small ratio of informed individuals therefore seems capable

of guiding the rest of the swarm.

In Romey [42], the authors investigated on a model the

dependence of a flock’s trajectory on varying behavioral

rules of individuals, depending on interagent distance,

agents’ speeds, biases (indicating leadership), and ran-

domness. The model showed that, more ‘‘repulsive’’ indi-

viduals tended to localize at central regions, whereas

increasing the number of such agents resulted in a higher

group turning rate and a slower total velocity. Increased

variety in the individual speeds, although did not disturb

group cohesion, diminished the total forward velocity.

When individuals with occasional random behaviors were

introduced, the first few affected the group trajectory more

significantly than the rest. At reasonable amounts of noise,

the overall effect on the trajectory was small. Finally, few

agents (1 or 2 in a flock of 8) with minor biases toward a

preferred direction had a strong influence on the trajectory.

Increasing the number of such individuals decreased the

turning rate, while at the same time, increasing the degree

of compliance with the preferred direction.

Couzin et al. [6] proposed a model to explain how a

minority of informed individuals affect the decision-making


123

process in animal flocks. In their model, the naive indi-

viduals arranged their positions and alignments according

to their neighbors, whereas the informed ones used also the

desired direction information. The simulations of the model

showed that the accuracy of the flock increased with the size

of flock and with the importance given to the desired

direction information. This study is one of the main inspi-

rations behind the work reported in this paper, and its

relation to our work is further discussed in Sect. 6.

1.3 Flocking

In a seminal study [41], Reynolds showed that flocking

behavior can emerge from local interactions, and he was

able to obtain realistic looking flocking behavior in com-

puter animation. In this study, Reynolds assumed that the

individuals could sense the bearing, range and orientation

of their neighbors. The simulated boids (short for bird-oid)

were able to flock by (1) avoiding collisions, (2) aligning

their headings with nearby flockmates, and (3) moving to

the center of their nearby flockmates. Kwong and Jacob

[23] further developed this behavior to include an addi-

tional goal vector and a random noise vector. They used

evolutionary techniques to adapt the weights of these five

behaviors, the maximum acceleration and the optimal

distance between the boids, resulting in a variety of

behaviors such as ring and figure-eight formations. In [27],

the effect of the dynamically changing neighborhood

relations on these formations was studied, and the swarms

were trained to imitate given average neighbor count pro-

files to produce complex behaviors. In [28], a game plat-

form was designed on this model, where a user tries to herd

the swarm by controlling one of the agents. In [19], an

extension to the L-systems were proposed, in which the

L-system is not realized by a single agent, but by a swarm

of agents. The agents’ behavior is determined by the given

model, whose parameters are determined by the specific

L-system grammar.

In [17], the authors investigated varying density distri-

butions of fish schools on a realistic individual-based

model, where individuals are allowed to have varying

lengths, representing small and large fish. When the indi-

viduals prefer being in close proximity with their ‘‘kins’’,

agent groups form in arbitrary locations. When the agents

prefer being in the proximity of similar sized agents,

concentric size groups form, with smaller fish occupying

the center. When the small agents avoid large agents, the

tendency reverses, with large agents being in the center and

small agents occupying the periphery. Finally, under all

conditions, the density is high at the front of the flock, due

to high turning rates of the leading individuals.

In multi-robotic systems, Mataric [30] claimed to

achieve flocking in a group of robots, in the form of

collective homing. She showed that by combining safe-

wandering, aggregation, dispersion, and homing behaviors,

a group of robots (which can localize themselves by sta-

tionary beacons and broadcast this information) can

‘‘flock’’ toward a homing direction. Later, Kelly and

Keating [22] used robots that were able to sense the relative

range and bearing of neighbors by a custom-made infrared

(IR) system. The group of robots first elected a leader

through a negotiation phase that took place through wire-

less communication. Then, the leader wandered in the

environment and the rest of the group followed. In a more

recent study, Hayes and Tabatabaei [16] proposed a lead-

erless flocking algorithm, for a group of robots that were

assumed to sense the range and bearing of their neighbors

to compute the center-of-mass (CoM) of the group, and its

heading toward a goal. The proposed behavior consisted of

collision avoidance and velocity-matching flock centering

behaviors. The CoM was used for the cohesion of the

group, and the change in CoM was used to align the robots.

Although the algorithm was implemented on the Webots

simulator, the authors had to emulate the sensors using an

overhead camera system on physical robots. Nembrini

et al. [34] developed a set of behaviors to achieve aggre-

gation, collective obstacle avoidance, and collective taxis

toward a beacon. The behaviors were developed on a

swarm of seven physical robots that were equipped with a

set of IR sensors for obstacle detection, an omni-directional

IR system for robot detection and a wireless communica-

tion system. Although the simulations were successful, the

experiments with physical robots were reported to suffer

from the shortcomings of the hardware.

Recently, Turgut et al. [52, 53] presented a self-orga-

nized flocking behavior for a robot swarm, without using

emulated sensors [16], a priori knowledge of the goal

direction [30, 35], or a leader [22] to guide the flock. It was

shown that the flock can self-organize to move toward a

common direction in a completely distributed way using

only local interactions among the individuals. The pro-

posed behavior was able to make a group of robots (both

physical and simulated) move straight in open environment

and avoid obstacles on the path of the flock.

1.4 Modeling studies in statistical physics

and control theory

In addition to Couzin et al.’s study [6], there has also been

studies toward modeling flocks within statistical physics

and control theory. In a seminal work, Vicsek et al. [56]

proposed the self-driven particles (SDP) model to study the

emergence of the self-aligned motion of massless particles

under local interactions to explain phenomenon observed

in biological systems [3]. The SDP model uses massless

particles which move at constant speed in a square region


123

with periodic boundary conditions. The heading of each

particle is updated to the average heading of its neighbors

within its local interaction range perturbed by noise. The

results of simulations revealed that particles undergo a

phase transition from an unaligned state (all particles

moving in different directions) to an aligned state (all

particles moving in the same direction) above a certain

density or below a certain noise value [10]. The emergence

of global alignment from only local interactions of the

particles seemed to contradict the Mermin–Wagner theo-

rem [31], which stated that ordered phase (aligned state in

the SDP) cannot emerge in one- or two-dimensional sys-

tems having local interactions at non-zero temperatures

(corresponding to the existence of noise in the SDP) unless

long-range interactions exist.

In a subsequent study [50], Toner and Tu extended the

work of Vicsek et al. to propose a ‘‘quantitative theory of

flocking’’. In this study, the authors claimed that flocks are

‘‘a non-equilibrium dynamical system’’ and hence are not

constrained by the predictions of the Mermin–Wagner

theorem. They provided a theoretical analysis of how

flocks can break symmetry by moving toward an arbitrary

direction as a whole. Recently, Nagry et al. [33] argued

that the diffusion and relative displacement of the particles

among each other would provide the long-range interac-

tions required for the global alignment. In a recent follow-up

to this discussion, Aldana et al. [1] proposed the vectorial

network model (VNM), in which stationary particles

updated their headings to the average of their neighbors

perturbed by noise. However, unlike the SDP model, the

neighbors of a particle are not only picked up randomly

from the local neighborhood but also randomly from the

entire group. It was shown that the system would undergo a

phase transition from an unaligned to an aligned state when

there is at least one random neighbor in the neighboring set

of particles and the noise is also below the critical value. In

the case of totally local neighbors, the system stays in an

unaligned state unless noise is set to zero in accordance

with the predictions of the Mermin–Wagner theorem [31].

Gregoire et al. [14] extended the SDP model by adding an

attraction/repulsion term based on the local bearing and

range measurement of neighboring particles to generate

cohesive motion in open space. In this model, the heading

of each particle is updated to the weighted sum of the

heading adjustment (with the inclusion of a noise term) and

attraction/repulsion terms. In simulations, coherently

moving clusters in open space were achieved by the proper

selection of the coefficients of these terms. In a recent

study, Huepe et al. [18] investigated the underlying

dynamics of the original [56] and extended [14] SDP

models. They found that compared to the extended model,

the original SDP model creates high and unrealistic local

density values during the unaligned-to-aligned state phase

transition and is not suitable for modeling natural or robotic

swarms.

Within control theory, interest in the control and

analysis of flocking has also been on the rise recently.

Jadbabaie et al. [20] investigated the stability conditions of

the aligned motion of particles in the SDP model, neglecting

the effect of noise on the heading calculation. They showed

that stability is ensured when the neighboring graph remains

connected within a finite time interval, with time divided

into infinitely many irregular intervals. A more relaxed

condition is also proposed, indicating stable motion even if

none of the neighboring graphs is connected but the union

remains connected within a finite time interval. Tanner

et al. [48, 49] proposed a stable control law for flocking in

free space based on range, bearing, and velocity informa-

tion of neighbors of a robot in close proximity. They

considered two cases. In the fixed-topology case, neighbors

are assumed to be fixed, whereas in the dynamic-topology

case, the neighbors are assumed to vary in time (it is

assumed that the flock remains connected despite changes

in the connectivity). The proposed control law included an

attraction/repulsion term (similar to the one used in [14])

depending on local distance measurement, and an align-

ment term depending on local velocity measurement in

both cases. The authors proved the existence of stable

flocking for both cases using Graph Theory and Lyapu-

nov’s stability theorem.

In [36], Olfati-Saber proposed a ‘‘theoretical framework

for the design and analysis of distributed flocking algo-

rithms’’ for particles that are assumed to sense the positions

of their local neighbors in a noise-free way. He approached

the problem as a distributed consensus problem [37]

where the particles aim to agree on a common velocity

vector. The author proposed three algorithms for flocking:

two for open-space flocking and one for flocking in envi-

ronments containing obstacles to be avoided. The first

algorithm consisted of a gradient-based attraction/repulsion

term and a velocity-matching term and was shown to be

equivalent to Reynolds’ behaviors [41]. The authors have

shown that this algorithm was ‘‘insufficient for creation of

flocking behavior’’ in large groups leading to regular

fragmentation. In the second algorithm, the authors

extended the first algorithm by including a third term which

acted as a moving rendezvous point to provide the flock

with a common objective. The third algorithm extended the

second algorithm with obstacle avoidance by modeling

obstacles as virtual agents that move on their periphery. It

was proven that the second and third algorithms can gen-

erate stable flocking. It should be noted that this result is

consistent with the predictions of the Mermin–Wagner

theorem.

The modeling studies reviewed earlier have used models

of individuals that are largely unrealistic and at least an


123

order of magnitude simpler than the models of physical

robots. First, these models have exclusively used massless-

[1, 6, 10, 14, 31, 33, 50, 56] or mass-particle [20, 36, 48,

49] models neglecting the effect of physical size in flock-

ing. Second, individuals are assumed to sense the heading

and range, or position (and velocity) of their neighbors.

This assumption, dating back to Reynolds, has proven to be

too problematic to implement on robots and has been the

main reason behind the lag in ‘‘implementing’’ flocking on

robots (see [52] for a complete discussion). Third, sensing

and actuation noise is neglected in some of the studies. It’s

especially interesting to note that studies in control theory

[20, 36, 48, 49] have almost exclusively neglected heading

noise, despite the existence of studies that relate this noise

to the emergence of flocking in statistical physics [1, 10,

14, 31, 33, 50, 56]. For instance, in studies within control

theory, individuals within the flock are assumed to sense,

communicate, and act synchronously and without any

delays, whereas in real world, robots operate asynchro-

nously and there exist both deterministic and stochastic

delays in their sensing, actuation, and communication. We

argue that the modeling studies stand on too many

assumptions and that the assumptions that they make differ

from each other, and heir applicability to robotic systems

remains a difficult challenge.

The study reported in this paper builds on our prior work

reported in [52, 53] and aims to understand how and to

what extent a mobile robot flock can be steered through

externally guided individuals within the flock. Specifically,

we extend the flocking behavior such that the flock, instead

of wandering aimlessly, tries to follow a desired direction

of motion that is externally provided by a user. Our

approach is mainly inspired by the works of [6, 40] and

[43] on the decision-making mechanisms. A preliminary

version of this work was published in [5].

2 Experimental platforms

2.1 Kobot robotic platform

Kobot is a CD-sized (12 cm diameter) mobile robot plat-

form that is developed specifically for swarm robotic

studies [52]. It has two differentially driven motors and

infrared (IR) sensors around its base. An IEEE 802.15.4/

ZigBee compliant wireless communication module with a

range of *20 m indoors is used for communication.

2.1.1 Infrared short-range sensing system

The infrared short-range sensing system (IRSS) is com-

posed of 8 IR sensors placed at 45� intervals around the

base (Fig. 1a). The sensors can sense artifacts within a

range of *20 cm at seven discrete levels at 18 Hz and also

distinguish kin-robots from other obstacles. Specifically,

the output of the kth sensor is a 2-tuple (ok, rk). ok 2f0; 1; . . .; 7g denotes the detection level to the object being

sensed (ok = 1 and ok = 7 indicate, respectively, a far and

nearby object. ok = 0 indicates no object is detected by the

sensor), and rk [ {0, 1} indicates whether the sensed

object is another kin-robot (rk = 1) or an ordinary obstacle

(rk = 0).

2.1.2 Virtual heading sensor

The virtual heading sensor (VHS), composed of a digital

compass and a wireless communication module, allows

the robots to sense the approximate relative orientations

of nearby robots. Specifically, the heading of the robot

with respect to the sensed North is measured in a

clockwise direction through the digital compass (Fig. 1b).

The robot broadcasts its heading information to other

robots within its communication range using the wireless

communication module. At the same time, the robot also

receives the heading information broadcasted by other

robots. The headings of the neighboring robots are con-

verted to the local reference frame of the robot by vec-

torial subtraction of the robot’s own heading. In effect,

the VHS virtually senses the relative headings of the

neighboring robots. This information can then be used for

aligning the robot with its neighbors. We would like to

note that the VHS does not assume the sensing of the

absolute North direction.

(a) (b)

Fig. 1 a The scaled sketch of Kobot. The rectangles indicate the IR

sensors located around the base. The discretized gray blob emanating

from the front IR sensor shows the sensing range of the sensor. b The

body-fixed reference frame. The x-axis coincides with the rotation

axis of the wheels. The forward velocity (u) is the velocity of the

robot along its y-axis. x is the angular velocity. vR and vL are the

velocities of the right and left motors. The heading of the robot, as

measured by the virtual heading sensor (VHS), is h, which is the angle

of the y-axis with the sensed North direction (ns). l is the distance

between the wheels


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2.2 The simulation environment

A physics-based simulator, called the Controllable-Swarm

Simulator (CoSS) [52], shown in Fig. 2b, was used for

conducting experiments with more robots than physically

available and for longer durations than possible in our

physical experimental setups. CoSS is developed using the

ODE (Open Dynamics Engine, URL: http://www.ode.org)

physics engine. In a previous study [52], we have verified

that the experimental results obtained within CoSS are in

agreement with the ones obtained from Kobots.

The IRSS is modeled in the simulator according to

systematic experiments [52]. Meanwhile, the VHS is

modeled using three parameters, namely, the range of the

communication, the number of robots whose broadcasted

heading values can be received at one control step (referred

to as the VHS neighbors), and the noise in the VHS. The

range and the number of VHS neighbors are set to 20 m

and 20, respectively. The noise on the VHS with the vec-

torial noise model [14] by adding a random noise vector to

the heading measurements of each robot:

h0 ¼ \feih þ geinog

where h is the actual heading, g is the magnitude of the

noise vector, and no is its direction chosen from a Gaussian

distribution described by Nðl ¼ h; r ¼ �p2Þ: \ð:Þ calcu-

lates the argument of the resulting vector.

3 The steered flocking behavior

The steered flocking behavior is expressed as a weighted

vector sum of three terms:

a ¼ hþ b pþ c d

khþ b pþ c dk ð1Þ

where h is the heading alignment vector, p is the proximal

control vector, and d is the preferred direction vector. a is

the resultant desired heading vector, according to which a

robot calculates its own direction of motion. The relative

importance of the terms is controlled by b [ [0, ?), the

weight of the proximal control vector, and c [ [0, ?), the

weight of the direction preference vector.

3.1 Heading alignment

The heading alignment term, h, tries to align the robot with

the average heading of its neighbors and is calculated as:

h ¼P

j2N eihj

kP

j2N eihjk

where N denotes the set of VHS neighbors, hj is the

heading of the jth neighbor converted to the body-fixed

reference frame, and k � k calculates the Euclidean norm.

The heading values of the neighbors are obtained by the

VHS which collects the broadcasted heading values. These

values are then converted to the robot’s body-fixed refer-

ence frame by hj ¼ p2� ðhglobal

j � hÞ, where h is the robot’s

own heading measurement.

3.2 Proximal control

The proximal control term uses the IRSS readings to

maintain cohesion within the flock, as well as to prevent

collisions between the robots and with obstacles. To this

end, the kth IR sensor is assumed to generate a virtual

force, fk, which is defined as:

fk ¼�ðok�odesÞ2

C if ok � odes

ðok�odesÞ2C otherwise

(

ð2Þ

where C is a scaling constant (10 for rk = 1, 35 for rk = 0).

The proximal control vector, p, is computed as the

vector sum of the forces acting through the eight IR

sensors:

p ¼ 1

8

X

k

fkei/k ð3Þ

where k 2 f0; 1; . . .; 7g is the index of the sensor which is

located at /k ¼ p4k with the x-axis of the body-fixed ref-

erence frame.

Fig. 2 a A photo of 7 Kobots.

The digital compass modules

are placed on top of the plastic

masts of the robots in order to

minimize electromagnetic

interference from the body.

b A snapshot from CoSS


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http://www.ode.org

3.3 Direction preference

The direction preference vector, d, acts as a bias to

incorporate external guidance and is calculated as:

d ¼ dp � ac

where ac is the current heading vector of the robot coin-

cident with the y-axis of the body-fixed reference frame

and dp is the desired heading direction.

The weight of the direction preference term (c) is set to

0 in naive individuals and to a non-zero positive value in

informed ones.

3.4 Motion control

At each control step, a robot updates its forward (u), and

angular (x) velocities using the instantaneous desired

heading vector, a. The forward speed is calculated as:

u ¼ ða � acÞumax if a � ac� 0

0 otherwise

�

ð4Þ

where a and ac denote the desired and current heading

vectors, modulating the umax the maximum forward

velocity of the robot.

x ¼ ð\ac � \aÞKp ð5Þ

where Kp is the proportionality constant of the controller.

The rotational speeds of the right and left motors

(Fig. 1b) are calculated as:

NR ¼ u� x2

l� � 60

2pr; NL ¼ uþ x

2l

� � 60

2pr

where NR and NL are the rotational speeds (rotations per

minute) of the right and left motors, respectively, l is the

distance between the wheels of the robot (meters), u is the

forward velocity (meters per second) and x is the angular

velocity (radians per second).

4 Metrics

We propose three metrics to quantify the performance

steered flocking: namely, (1) the mutual information met-

ric, used for analyzing the time evolution of information

sharing between the informed and the naive robots, (2) the

accuracy of the flock in following 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.

4.1 Mutual information

The use of the mutual information concept in multi-agent

systems, was first suggested by Parunak et al. [39] as a

measure of correlation, through which the concepts of

coherent, collaborative, cooperative, competitive, and

coordinated can be defined. Recently, Sperati et al. [44]

used the mutual information between robots as a fitness

function to evolve coordinated behavior in a robot swarm.

They showed that maximizing mutual information in a

task-independent manner as the fitness function would

result in the emergence of coordination among the robots.

Sporns and Lungarella [45] used information theoretic

metrics, including mutual information, complexity, and

integration (the latter two metrics due to [51]) for evolving

coordinated behavior in a simulated creature. In all these

works, mutual information was shown to be a very effec-

tive, task-independent metric of shared information.

4.1.1 Formal definition

The mutual information gives ‘‘the reduction in uncertainty

of one variable due to knowledge of another. If knowledge

of Y reduces our uncertainty of X, then we say Y carries

information about X’’ [12]. Thus, it can be utilized as a

measure of the information transferred from an informed

robot to a naive one during flocking.

Mutual information is defined in terms of information

entropy. Adopting the notation of Feldman [12] and indi-

cating a discrete random variable with the capital letter X,

which can take values x 2 X , the mutual information is

defined as:

MI½X; Y� ¼ H½X� þ H½Y � � H½XY�¼ H½X� � H½XjY �¼ H½Y � � H½YjX�

where H[X] denotes the marginal information entropy of X,

H[XY] denotes the joint entropy of X and Y, and H[X|Y]

denotes the conditional entropy of X given Y. The complete

derivation is included in the ‘‘Appendix’’.

The mutual information metric has a number of favor-

able properties. First, MI[X, Y] is zero when there is no

statistical dependence between the two variables, in which

case H[XY] = H[X] ? H[Y], indicating no shared infor-

mation. Moreover, it is also zero when the marginal

entropies of the two variables H[X] and H[Y] are zero,

when there is already no uncertainty about either of the

variables. It is non-negative and bounded by some finite

maximum value. It is symmetric, that is MI[X, Y] =

MI[Y, X], agreeing with our intuition that information

sharing must be two-way. It has the capability of capturing

non-linear statistical dependencies, unlike other widely

utilized metrics such as Euclidean distance, the Pearson

coefficient, or covariance [44, 46]. For instance, quoting

from Steuer et al. [46], ‘‘a vanishing mutual information

does imply that two variables are independent, while for


123

the Pearson correlation this does not hold’’. Finally, it can

be calculated as a function of time; therefore, it can be used

to analyze the time evolution of shared information in a

dynamical system.

4.1.2 Methodology

In this study, we measure the mutual information between

a (randomly chosen) informed and a (randomly chosen)

naive robot. We calculate the shared ‘‘information’’ in

terms of these robots’ ability to move in the same direction.

Specifically, we denote the heading values of the informed

robot and the naive robot by the random variable X and Y,

respectively. The state space of X and Y is infinite and must

be discretized. Parunak and Brueckner [38] pointed to the

importance of not dividing an infinite state space into too

many states, in which case the possibility of observing two

random variables in the same state would be ‘‘vanishingly

small’’. Another issue is the concern of statistical signifi-

cance when trying to obtain the probability distribution of a

random variable from a finite number of observations. In

[26], it was suggested that three times more samples than

the possible states of a variable must be observed for a

faithful estimation of the probability distribution of the

variable. In light of these concerns, we divide the unit

circle into 8 discrete intervals of p/4 radians, and therefore

the number of states that an informed-naive robot pair can

be in becomes 8 9 8. We conduct 200 [ 3 9 8 9 8

experiments for estimating the probability distributions,

which provides us with a safe range for estimating the joint

probability distribution. In order to capture the dynamic

aspects of the shared information, we calculate the proba-

bility distributions p(X), p(Y), and p(X, Y) separately at

each time step. Our expectation is that X and Y will be

independent of each other at the initial phases of the

experiments and will become correlated as time advances,

provided that the necessary conditions are supplied.

Finally, we would like to note that the maximum value of

the mutual information metric, determined by the quanti-

zation of the heading state, is 3 (see ‘‘Appendix’’).

4.1.3 Finite size effects

A final issue about the calculation of the mutual informa-

tion is the bias introduced by the observation of a finite

number of samples. It was pointed out that when the

entropy of a random variable is estimated from the obser-

vation of a finite number of samples, the estimation is

‘‘systematically biased downwards’’ [13] and that this bias

can be removed from the estimated entropies [46] as:

H½X� ¼ ~H½X� þ a� 1

2 � b

where ~H½X� is the estimated entropy, a is the number of

discretized states of the random variable X, b is the number

of observed samples, and H[X] is the true entropy.

4.2 Accuracy

The mutual information metric can only measure the

degree of alignment between the informed and naive robots

and has no notion of the desired direction. Therefore, it

cannot distinguish whether a commonly converged upon

direction is also aligned with the desired direction. Toward

this end, we adopt the accuracy metric of Couzin et al. [6]

to measure the flock’s degree of alignment with the desired

direction.


The accuracy metric depends on the angular deviation of

the direction of the flock from the desired direction. The

angular deviation is analogous to the standard deviation

from linear statistics for inherently directional data, and the

accuracy metric can be defined as:

Accuracy ¼ 1� S00=2

where S0

0denotes the angular deviation of a group of

vectors from a desired direction. The complete derivation

is included in the ‘‘Appendix’’. The accuracy metric

becomes 1 when the angular deviation is minimum and 0

when the angular deviation is maximum. We expect the

accuracy metric to be as high as possible in a desired

scenario.

4.2.2 Methodology

The accuracy is calculated via two different methods. In

the steady- state analyses, the direction of motion is cal-

culated for each experiment. The direction of motion is

defined as the direction vector between the position of the

center of mass at times t and tf, where time t marks a

moment in the steady-state phase and tf is the end of the

experiment. Therefore, we do not consider the transient

dynamics of the system, and only consider the converged

direction of motion.

Once the direction of motion is calculated for all

experiments conducted with a certain parameter set, we

compute the angular deviation, where each sample vector

corresponds to the direction of motion in one experiment.

We then calculate a single steady-state accuracy value

which measures the overall performance in all the

experiments.

In the transient analyses, we vectorially sum the heading

values of individual robots and obtain the average heading

of the flock at each time step in every experiment. In order


123

to calculate the angular deviation at each time step, we take

the average heading vector of the flock at each experiment

as a sample vector. We thus plot the time evolution of the

accuracy associated with these experiments. In Kobot

experiments, the tracking of the center of mass of the flock

of a moving flock was not possible and hence we used

the scheme used in the transient analyses, by collecting the

heading values of individual robots and extracting the

accuracy from the average heading of the flock. Since

the accuracy metric is already a measure of variance, error

bars are not shown in the plots.

4.3 Largest aggregate

The cohesiveness of the flock can be captured by analyzing

connected aggregates. Specifically we used the ratio of the

largest aggregate to the whole flock and the ratio of

informed individuals remaining with the largest aggregate

(denoted by /), as a combined metric of flock cohesion.


Assuming that the swarm is represented by a graph

G ¼ ðV; EÞ, the set of vertices V ¼ 1; 2; . . .; n denote the

robots. The set of edges E � fði; jÞ : i; j 2 V; i 6¼ jg con-

tains the pair (i, j) if robots i and j are within IR sensing

range of each other. The partitioning of the graph into

connected aggregates is then conducted as in Algorithm 1.

Algorithm 1 The clustering algorithm

1: initialize n aggregates with Ai ¼ fvig2: while 9Ai;Aj such that i 6¼ j; k 2 Ai; l 2 Ajðk; lÞ 2 E do

3: merge Ai and Aj

4: end while

5 Experimental results

We implemented the steered flocking behavior using the

following behavioral parameters on both simulated and

physical robots: namely, b = 4, umax = 7cm/s, Kp = 0.5,

odes set to 3 for kin-robots and 0 for obstacles. The sensi-

tivity analysis for these parameters had previously been

conducted in [52], and the optimal parameter set is utilized

here. In the simulations, the g parameter, which sets the

amount of heading noise in VHS, is set to 1. We denote the

ratio of informed robots in the flock as q. We will now

present three experiments conducted with physical and

simulated robots in order to provide a preliminary dem-

onstration of the behavior before moving on to quantitative

evaluations. In these experiments, we set the weight of the

direction preference vector c = 1.

Figure 3 shows snapshots from a sample scenario with

Kobots. Initially, all the robots in the group are naive (hence

no preferred direction), and the flock self-organizes to move

as a whole toward the upper right part of the view as shown

in the leftmost snapshot. Then, at the time of the second

snapshot, 4 of the 7 Kobots are ‘‘informed’’ by an external

user that the preferred direction of motion is 90� to the right

of their individual current headings. Since the Kobots have

already aligned their headings previously, this creates the

effect of their all turning to the same direction simulta-

neously. Through the local interactions of these informed

robots, the flock is steered to move toward the lower right

part of the view as can be seen in the last snapshot.

In order to demonstrate the transient performance of the

behavior in complying with the extreme changes in the

preferred directions, we conducted three experiments using

Kobots with the informed robot ratio q set to 2/7, 4/7, and

7/7. In these experiments, the preferred directions of the

informed Kobots are reversed every 30 s. The headings of

all the robots during each of these experiments are shown

in circular plots in Fig. 4. As expected, with each reversal

of preferred direction, the flock goes through a transition,

which can be seen as tails within the circular bands. As q,

the ratio of informed robots, increases the tails become

shorter, indicating that the flock becomes aligned with the

preferred direction faster.

Figure 5 shows the recorded robot trajectories from the

steering of 100-robot flocks in CoSS with the informed

robot ratio q set as (a) 10/100, (b) 50/100, (c) 100/100. The

direction preference vectors for informed robots are set to

90� (indicating upwards) 0� (indicating left), 270� (indi-

cating downwards), and 180� (indicating right) at times 0,

120, 240, and 360 s, respectively. The small, black circles

show the robots, whereas the (red) traces indicate their

trajectories. It can be seen that in all three cases, the flock

can be steered along the desired direction, despite the ratio

of informed robots being as low as q = 10/100 in the first

case. As expected, the response to changes in the preferred

direction is more rapid in the q = 50/100 and q = 100/100

cases.

5.1 Transient characteristics

We analyze the transient characteristics of steered flocking

using 100 simulated robots in four sets of experiments,

namely, set 1: [q = 10/100, c = 0.5], set 2: [q = 10/100,

c = 1.0], set 3: [q = 1/100, c = 1.0], and set 4: [q =

1/100, c = 10.0]. The robots were initialized in a hexag-

onal formation with 25 cm inter-robot distances at random

orientations and the desired heading for the informed

robots was set as 90�. The informed robots were picked at


123

random. The robots were simulated for 1,000 s (corre-

sponding to 10,000 simulation steps), and 200 experiments

were conducted.

We picked one exemplary case from each set of

experiments and plotted the time evolution of the robot

headings in Fig. 6. Figure 6a depicts the initial configura-

tions of the robots. In the experiment plotted in Fig. 6b,

only 10 of the robots are informed and the weight of the

direction preference behavior is set to half of the heading

alignment behavior, that is c = 0.5. The headings of the

informed and naive robots are shown with white and black

points, respectively. It can be seen that the informed robots

first align themselves with the rest of the flock (at around

t = 15 s) and then slowly converge to the desired direction

together with the rest of the flock (at around t = 75 s). The

flock is able to converge to the desired heading in a safe

manner and is stable in this direction thereafter.

In the second plot, shown in Fig. 6c, 10 robots are

informed as in the previous case, but the weight of the

direction preference behavior is doubled to equal the

weight of the heading alignment behavior, that is c = 1.0.

The time evolution of robot headings shows clearly that

the naive individuals were able to align toward a common

heading around - 45� as early as 10 s. At this point,

however, the disagreement between the naive and

informed robots became visible. It can be seen that in

comparison with the first case, the informed individuals

try to turn toward the preferred direction more hastily,

with little concern about whether the rest of the flock are

aligned with them or not. However, this disagreement was

resolved during the next 50 s, and the whole flock was

able to move in the preferred direction. At this point, we

should note that, the placement of the informed individ-

uals within the flock can have a significant effect on the

performance. Informed individuals that are placed close to

the periphery of the flock have a high risk of getting

disconnected from the flock during the initial negotiation

phase.

Fig. 3 Snapshots of steered flocking with 7 Kobots. At the time of the second snapshot, 4 of the robots are commanded to turn 90� to the right of

their current direction. White lines indicate the heading directions of the Kobots. See text for details

(a) (b) (c)

Fig. 4 Sample runs with 7 Kobots where the informed robot ratio q is

set as a 2/7, b 4/7, c 7/7. The plots depict the headings of the Kobots

with respect to time. The time axis is shown with radially growing

circles. Each dot on a circle denotes the heading of one Kobot at the

specific time step associated with that circle. A dot at the of a circleindicates a heading toward 90� direction, whereas a dot at the bottomof a circle indicates a heading toward 270� direction. The headings of

the Kobots at time t = 0 is plotted on the innermost circle, and the

headings of the Kobots at time t = 120 is plotted in the outermostcircle. The informed robots are commanded to go in a 90� direction

in t = 0–30 s, a 270� direction in t = 31–60 s, a 90� direction in

t = 61–90 s, and a 270� direction in t = 91–120 s. The time steps in

which the preferred direction is changed are indicated with boldcontinuous lines

Fig. 5 The trajectories from the

steering of 100-robot flocks in

CoSS. In the experiments, the

informed robot ratio q is set to

a 10/100 and b 50/100,

c 100/100


123

In experiments plotted in Fig. 6d and e, only 1 informed

robot was used to steer a 100-robot flock. In the experiment

plotted in Fig. 6d, the weight of the direction preference

vector c is set to 1.0. It can be seen that initially the

informed robot aligns with the rest of the flock for short

time. However, the informed robot soon breaks off from the

flock, yet remains within the VHS communication range of

the rest of the flock. Hence, while the informed robot aligns

itself closer to its preferred direction, its alignment remains

influenced by the rest of the flock that is heading toward a

different direction. At 350 s, the informed robot gets out of

the VHS range of the rest of the flock and aligns itself

immediately with the preferred direction.

In the last experiment, plotted in Fig. 6e, the weight of

the preferred direction is ten times the weight of the

heading alignment behavior, that is c = 10.0. As a conse-

quence of this, the informed robot tries quickly to turn to its

own preferred direction. Note that the heading of the

informed robot varies greatly as it collides with other

robots that get in its way and it tries to get rid of them.

However, as soon as it gets out of the flock, it turns to the

desired direction and continues in this direction until the

end of the experiment. Since the weight of the preferred

direction is significantly more than the weight of the

heading alignment behavior, its direction is not much

affected by the common alignment of the flock, even while

it stays within the VHS range of the rest of the flock, unlike

the previous case.

5.1.1 Information sharing

Figure 7a plots the initial 500 s of mutual information

averaged over 200 experiments for each set. It can be seen

that in set 1, the mutual information increases rapidly due

to the quickly established common alignment between the

informed and the naive robots. In set 2, however, since the

informed robots are more persistent in following their

desired direction, the mutual information increases slowly.

In both of these cases, the mutual information stabilizes

around 2 and does not reach its maximum value of 3. This

stems from the fact that alignment of the robots is often

noisy, due to the proximal control term which tries to keep

the flock cohesive.

In set 3, the mutual information can be seen to be on the

rise during the initial but short period in which the

informed robot tries to comply with the common align-

ment. However, in time, the informed robot turns to its own

direction, and the mutual information decreases again. In

set 4, the mutual information never increases, indicating an

early split of the informed and naive robots on the direction

of motion.

5.1.2 Accuracy of direction of motion

Figure 7b plots the evolution of accuracy in time calculated

over 200 experiments. It can be seen that, in set 1 and set 2

(which contained 10 informed robots), the accuracy

increases with time, meaning that an informed robot ratio qof 10/100 is sufficient for steering the flock. On the other

hand, as can be seen in set 3 and set 4, q = 1/100 is

insufficient. Finally, as expected, the accuracy increases

quicker when the weight of the direction preference vector

c is higher.

5.2 Steady-state characteristics

In this section, we analyze the steady-state characteristics

of the steered flocking behavior. In experiments with

Fig. 6 The time evolution of

headings in four CoSS

experiments picked out from

a set 1, b set 2, c set 3, and

d set 4. Note that only the initial

200 s of the first two

experiments is plotted in order

to highlight the transient

characteristics


123

100-robot flocks (conducted solely in simulation), the

accuracy is calculated between t = 875 s and tf = 1,

000 s, whereas in experiments with 7-robot flocks (con-

ducted with physical and simulated robots), the accuracy is

calculated between t = 20 s and tf = 60 s. The experiments

using physical and simulated robots are repeated 5 and 200

times, respectively.

5.2.1 Effect of the weight of direction preference behavior

The weight of direction preference behavior (c) is an

important parameter that affects the performance of the

steered flocking behavior. In order to analyze this effect,

we conducted experiments using two different flock sizes

(namely, 100-robot and 7-robot flocks), varying both cand q.

Figure 8a shows the accuracy values obtained from the

experiments in a split plot. We can make couple of

observations. First, when the informed robot ratio q is high

(that is q = 50/100 or q = 100/100), the accuracy remains

close to maximum for all c values. The accuracy values

obtained for q = 1/100 remain low, indicating that it is

impossible to steer the direction of a 100-robot flock using

only a single informed robot. For q = 10/100, it can be

seen that the accuracy values start from a low value and

approach to 0.8 upto c = 1.0. It is interesting to note that

the accuracy drops to slightly lower values for high cvalues, as shown on the right portion of the split plot. This

is due to the fact that the more ‘‘stubborn’’ the informed

robots are to move in their preferred direction, the more

probable that they will be separated from the flock. If they

head toward the desired direction much sooner than the rest

of the robots, they can get out of the VHS range of the flock

and can no longer affect the alignment of the flock.

In the figure, we also plotted the accuracy values

obtained from 7-robot flock experiments (with the

informed robot ratio q set to 1/7) conducted with both

physical and simulated robots. These experiments lasted

for 60 s, due to the limitations of the physical testing arena,

and hence the flocks cannot be considered to fully enter the

steady-state phase. That said, based on this data, we can

nevertheless make two conclusions. First, the results

obtained from physical and simulated robots are mostly in

agreement, with Kobot flocks starting to perform better

even at lower c values. Second, the accuracy values

obtained with q = 1/7 are closer to the ones obtained with

q = 10/100, showing that the dynamics captured in the

CoSS are in agreement with those obtained from the

physical robots.

Figure 8b and c evaluates the degree of cohesion within

the flock with respect to the informed robot ratio q for

different weights of the direction preference vector c.

Ideally, both the ratio of the largest aggregate and / should

be 100%, indicating that there exist a single aggregate that

contains all the individuals of the flock and that none of the

informed robots left the largest aggregate. A number of

observations can be made from these plots. First at

q = 0.01, where there is a single informed robot, although

the flock seem to remain as a whole for all values of c, a

careful look into / shows that it varies between 100% and

0% indicating that the only informed individual may have

left the flock. As a consequence of this, the largest aggre-

gate does not contain any informed individuals and it

moves toward an arbitrary direction. The plots show that

there are only three sets of q and c values for which both

the ratio of the largest aggregate and / remain close to

100%; namely (q = 0.5, c = 0.2), (q = 0.8, c = 0.2),

(q = 0.5, c = 1). The plot suggests that for lower values of

the informed robot ratio q, the weight of the direction

preference vector c that would prevent fragmentations

would be less than 0.2.

5.2.2 Effect of flock size

In order to study the effect of flock size, we set the weight

of the direction preference vector c to 0.2, based on the

experiments reported in the previous subsection, and

plotted the accuracy for different the informed robot ratios

(a) (b)Fig. 7 The time evolution of amutual information and

b accuracies averaged over

200 experiments for sets 1–4conducted in CoSS


123

q in Fig. 9a. The results show that when the duration of the

simulation is long enough, we achieve a high level of

accuracy for all values of q. On this figure, we have also

plotted the corresponding results obtained with 7 Kobots

moving for 60 s along with the results of the same exper-

iments conducted in CoSS. The results show that the results

of CoSS simulations are in accordance with the ones

obtained from Kobots and that accuracy tends to remain

low for lower values of q since the experiment duration is

short.

Figure 9b and c plot ratio of the largest aggregate and /with respect to flock size under different the informed robot

ratios q. The results clearly show that for a given q value,

the / value tends to improve with flock size, indicating that

larger flocks can be controlled using relatively small pro-

portions of informed individuals. For instance, the distri-

bution of the / values obtained for q = 0.1 can be seen to

shrink, with its median value approaching 100% as the

flock size gets larger. We claim that this result is due to the

fact that informed individuals are more likely to get

‘‘trapped’’ within the rest of flock in large flocks.

6 Discussion

In this section, we relate the predictions of Couzin et al.’s

model [6] to our results. Toward this end, we will now state

the assumptions that were made by Couzin et al.’s model.

First and foremost, the individuals are assumed to be

massless particles as opposed to circular differential-drive

robots which have a non-zero size and mass. Second, the

individuals are assumed to sense the position and heading

of its neighbors within its local interaction range on an

individual basis (that is a single position and heading

sensing per neighbor), whereas our robots can only make

anonymous range readings using line-of-sight (that is more

than one range reading can be obtained for a close-by

neighboring robot and that occluded neighboring robots

may not be sensed at all). Third, the range of heading

alignment and attraction interactions is assumed to be 6

times the range of repulsion interactions. In our study,

however, the attraction range is only twice the repulsion

range and that the range alignment interaction is *100

times the attraction range (though the number of inter-

actions is limited by the maximum number of VHS

neighbors which is set to 20).

Despite these differences in the models, the predictions

of Couzin et al.’s model can still be related with the

experimental results reported in this paper. First, both

studies show that accuracy increases asymptotically to

maximum with the ratio of informed individuals (Fig. 9a in

this paper, and Fig. 1a in [6]). Our results are in accordance

with the predictions of Couzin et al.’s model that the flock

can achieve a high level of accuracy only when the ratio of

informed individuals is more than a critical ratio. However,

Couzin et al.’s model predicts that this critical ratio

depends on the size of the flock and that as flock size

increases, the proportion of informed individuals needed to

(a) (b)

(c)

Fig. 8 a The average accuracies measured at tf is plotted against the

weight of the direction preference vector c for different informed

robot ratios. The experiments with Kobots and CoSS are repeated 5

and 200 times, respectively. Note that the data obtained from c = 5

and 10 are plotted on the right portion of the split plot. b, c The ratio

of the largest aggregate to the flock size and / at tf is plotted against

the informed robot ratio q for different c values. The data are obtained

from 200 simulations of 100-robot flocks in CoSS. The ends of the

boxes and the horizontal line in between correspond to the first and

third quartiles and the median values, respectively. The top and

bottom whiskers indicate the largest and smallest non-outlier data,

respectively. The data in between the first and third quartiles lie

within the 50% confidence interval, while the data in between the

whiskers lie within the 99.3% confidence interval


123

achieve a desired accuracy value decreases. Although this

relation is not immediately visible in results plotted in

Fig. 8a where the accuracy values remain high for even

small flock, results plotted in Fig. 9b and c expose the

existence of a relation at low informed robot ratios (q)

more clearly. We argue that the relative smallness of this

effect can be attributed to the large interaction range of the

VHS system which can have upto 20 VHS neighbors

within a 20 m radius. As a consequence of this, in small

flocks, the heading alignment works globally. In a specu-

lative experiment to test our hypothesis, we varied the

number of VHS neighbors and plotted the accuracy values

for different informed robot ratios in Fig. 10. The plot

shows that the accuracy for a given ratio of informed

individuals, accuracy increases with the number of VHS

neighbors, and explains why such a relation was not visible

in our experiments.

Second, although both studies agree qualitatively that

an increase in the weight of direction preference increa-

ses the tendency of fragmentation in the flock, the frag-

mentation tendency we observe at large informed robot

ratios (such as informed robot ratio (q) = 0.5) is more

than the one predicted by Couzin et al.’s model (Fig. 2 in

[6]). We argue that the reason behind this discrepancy

lies behind the difference between the ratio of proximal

attraction and repulsion ranges of individuals. As sum-

marized elsewhere, in Couzin et al.’s model, the attrac-

tion range is chosen to be 6 times the repulsion range,

whereas in our robots, the attraction range is merely

twice the range of repulsion. Hence, in our flocks, the

individuals that lie at the boundary of the flock are more

prone to splitting from the rest, especially during the

initial alignment period.

7 Conclusions

The problem of how and to what extent the behavior of a

swarm can be controlled is an important challenge for the

deployment of swarm robotic systems in the real world. In

this paper, we addressed this problem by studying the

steering of a self-organized robot flock toward a desired

direction through the external guidance of a minority.

Specifically, we extended a behavior that was shown to

generate self-organized flocking in a group of mobile

robots by adding it a direction preference component. The

individuals in the flock were split into two groups; namely,

naive and informed. All individuals were concerned with

flocking together with their neighbors, whereas the

informed ones were also given a preference to head in a

certain direction.

(a)

(b)

(c)

Fig. 9 The effect of flock size for varying ratios of informed robots on a accuracy, b, c the ratio of the largest aggregate, and /. The experiments

with Kobots and CoSS are repeated 5 and 200 times, respectively. The plots in b, c used experimental data obtained from CoSS

Fig. 10 The effect of the number of VHS neighbors on the accuracy

for different


123

In this case, the tendency to flock together represented

the underlying self-organized dynamics of the swarm,

whereas the preference of some individuals to head in a

certain direction represented the dynamics induced by

external control. These two dynamics conflict with each

other, and the tension between them was analyzed to

determine the extent of external control that can be

applied. To this end, we proposed to use three metrics in

our analysis: (1) The ratio of the largest aggregate to the

whole flock and the ratio of informed individuals

remaining with the largest aggregate were used as a

combined metric of how well the underlying self-orga-

nized flocking dynamics was preserved. (2) The accuracy

metric measuring how the flock was aligned with the

preferred direction of informed individuals was used to

quantify the performance of the dynamics induced by

external control. (3) The mutual information metric to

quantify the extent of interaction between these two

dynamics. Based on these metrics, a flock can be said to

be steer-able if it can maintain its cohesion as a single

aggregate containing most of the informed individuals and

yet turn its heading toward the preferred direction, max-

imizing the mutual information between its informed and

naive individuals.

We conducted a series of experiments conducted with

physical robots and in physics-based simulation. Specifi-

cally we varied the weight of the direction preference

behavior, the ratio of informed individuals as well as the

size of the flock, and analyzed the extent of steer-ability of

flocks. From these experiments, we can draw a number of

conclusions. First, the steer-ability of a flock of a given size

is determined by the ratio of informed individuals as well

as the weight of the direction preference behavior. It is

shown to improve with the ratio of informed individuals in

the group as expected. When the weight of the direction

preference behavior is small, although the flock can

maintain its cohesion, the responsiveness of the flock to

turn into a desired direction decreases, along with the

accuracy of the flock at steady state. On the other hand,

when the weight becomes too large, then the flock can no

longer maintain its cohesion, since the informed individu-

als are likely to split away from the group. Therefore, there

exists an optimum value for the weight of the direction

preference behavior for a given ratio of informed individ-

uals. Second, for a given ratio of informed individuals, the

steer-ability of the flock increases with the size of the flock.

Although it may seem surprising at a first glance, this result

stems from the physical embodiment of the individuals and

that informed individuals are more likely to get ‘‘trapped?

in large flocks. Third, the steer-ability of a given flock

increases with the number of neighbors individuals per-

ceive for alignment (a.k.a number of VHS neighbors),

since it increases the mutual information between informed

and naive individuals.

This study can be extended on a number of fronts. First,

the flocking behavior can be modified to improve the

strength of cohesion by creating a virtual ‘‘surface tension’’

on the periphery of the flock. Second, the informed indi-

viduals can compute the mutual information between

themselves and the rest of the flock using VHS and use it to

adapt the weight of their direction preference behavior

during steering. Third, the ‘‘unacknowledged leadership’’

constraint, which was inspired from related studies in

biology, can be removed in swarm robotic systems in order

to improve the controllability of the flock. Fourth, the

dynamics of steering, in the presence of individuals with

different preferred directions needs to be investigated. In

addition to these, the dynamics of the flocking needs to be

analyzed from a theoretical perspective. Toward this end,

we have already extended the VNM model that was

developed in Statistical Physics (briefly reviewed in Sect.

1.1) to model the phase transition of the robots from an

unordered state (where all the robots have different head-

ings) to an ordered state (where all robots are aligned) in

the existence of noise [54]. We plan to extend this model to

the phase (which starts automatically after all the robots are

aligned), where the robots start moving together.

Acknowledgments This work is funded by TUB_ITAK (Turkish

Scientific and Technical Council) through the ‘‘KAR_IYER: Kontrol

Edilebilir Robot Ogulları’’ project with number 104E066. Addition-

ally, Hande Celikkanat acknowledges the partial support of the

TUB_ITAK graduate student fellowship. The simulations have been

performed on the High Performance Computing Center of the

Department of Computer Engineering, Middle East Technical

University.

Appendix

Mutual Information: Adopting the notation of Feldman

[12] and indicating a discrete random variable with the

capital letter X, which can take values x 2 X , the infor-

mation entropy is defined as:

H½X� ¼ �X

x2XpðxÞ � log2 pðxÞ

where p(x) is the probability that X will take the value of x.

H[X] is also called the marginal entropy of X, since it

depends on only the marginal probability of one random

variable. The marginal entropy of the random variable X is

zero if X always assumes the same value with p(X = x0) = 1

and maximum if X assumes all possible states with equal

probability. Having defined the marginal entropy of a

single random variable, this definition is easily extended to

the joint entropy of two random variables:


123

H½XY � ¼ �X

x2X

X

y2Ypðx; yÞ � log2 pðx; yÞ

as well as the conditional entropy of these two random

variables:

H½XjY � ¼ �X

x2X

X

y2Ypðx; yÞ � log2 pðxjyÞ

where p(x, y) is the joint probability that X will take the

value of x and Y will take the value of y, and p(x | y) is the

conditional probability that X will take the value of x given

that Y takes the value of y. Thus, the conditional entropy is

the entropy of X, given that Y is known.

Then, the mutual information MI[X, Y] is defined as:

MI½X; Y � ¼ �X

x2X

X

y2Ypðx; yÞ � log2

pðxÞ:pðyÞpðx; yÞ

or equivalently,

MI½X; Y� ¼ H½X� þ H½Y � � H½XY�¼ H½X� � H½XjY �¼ H½Y � � H½YjX�:

When p(x|y) becomes 1, the mutual information

MI[X, Y] is maximized to be H[X]. Note that, the value

of H[X] depends on the discretization of x. For instance, if

the value of random variable x is discretized into 8, then

p(x) becomes 18

leading to H½X� ¼ �8 � 18� log2

18¼ 3:

Angular Deviation: The angular deviation is calculated

as follows [29]. Let h1...hn denote a set of unit vectors

whose angular deviation is to be calculated. Then, their

(normalized) mean vector is the vector from (0, 0) to

ð �C; �SÞ, where

�C ¼ 1

n

Xn

i¼1

cos hi; �S ¼ 1

n

Xn

i¼1

sin hi:

Let �R ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�S2 þ �C2p

be the length of this normalized mean

vector and �x0 be its angle with the x-axis such that:

�C ¼ �R cos �x0; �S ¼ �R sin �x0:

Then, the angular deviation of these vectors around their

normalized mean vector is given by:

S0 ¼ 1� �R:

This intuitively means that the more aligned the vectors

are, i.e, the less the angular deviation is, the longer is the

mean vector. On the other hand, if they are scattered

around the unit circle in a random manner, then their vector

sum results in a shorter mean vector, denoting a greater

angular deviation from the mean.

The angular deviation around a specific direction can be

calculated as an extension of this formulation by letting adenote the angle of the desired direction with the x-axis. Then

�C0 ¼ �R cosð�x0 � aÞ; �S0 ¼ �R sinð�x0 � aÞ

give the components of the mean vector in the desired

direction, and

S00 ¼ 1� �C0

gives the angular deviation around this direction. In the

accuracy calculations, we utilize this extended formulation

which gives the angular deviation around the desired

direction of the flock.

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Steering self-organized robot flocks through externally guided individuals