8/22/2019 2008_Lei_Li (1)

1/6

AbstractThe problem of multiple robots system formation

using a distributed control method is studied in this paper. The

main idea of this paper is that uses swarm flocking controlalgorithm to implement the biods model of Reynolds among

multi-robots. With the help of graph theory, we propose a

provably-stable flocking control law, which ensures that the

internal group formation is stabilized to a desired shape, while

all the robots velocities and directions converge to the same.

Player/Stage simulation results show that the proposed method

can be efficiently applied to multiple robots formation control.

With the characteristic of Player/Stage, the algorithm in this

paper can be implemented on the real robots with so few or no

changes.

I. INTRODUCTION

HERE has been a significant interest in the control of

multiple robots moving in a formation or performing a

coordinated task for the Multi-Robot Systems (MRS)researchers over resent years [1-5]. This is because there are

many potential benefits of such systems performing tasks

which are difficult for a single robot, including overall system

robustness, intelligence, enhanced performance, and

flexibility. Solutions for formation control problem of MRS

are currently widely applied in search and rescue operations,

landmine removal, remote space exploration and mapping,

and also the control of satellites.

Fundamentally, there are two groups of approaches in

controlling multiple mobile robots formation, namely:

centralized control and decentralized control. The first group

use a centralized unit that supervises the whole group and

commands the individual robots based on the global

information [2, 3]. And the second group use distributed

methods for achieving the coordination on the basis of localinformation and decisions [4, 5]. Those systems that require

global information or broadcast communication may have a

lack of scalability or high costs of the physical setup but allow

more accuracy in forming a large range of robotic formations.

On the contrary, systems using only local communication and

sensor data, while limited in variety and precision of

formations, tend to be more scalable, more robust, and easier

Manuscript received December 10, 2007. This work was supported in part

by the NationalNatural Science Foundation of China under Grant 60475031.

Bin Lei is with the School of Logistics Engineering, Wuhan University of

Technology, Wuhan, 430063, P.R.China ( phone:86-027-86535227; e-mai:

lbhaha@whut.edu.cn).

Wenfeng Li is with the School of Logistics Engineering, Wuhan

University of Technology, Wuhan, 430063, P.R.China (e-mail:liwf@whut.edu.cn).

Fan Zhang is with the School of Logistics Engineering, Wuhan University

of Technology, Wuhan, 430063, P.R.China (e-mail: ailexy@126.com).

to build [1].

In nature, flocking is a kind of ubiquitous self-organized

phenomenon, such as flock of birds, school of fish, andcrowds of people. And it is a form of collective behaviour of

large number of interacting agents with a common group

objective. The study of flocks/schools and swarms has

attracted a lot of researchers in different fields such as biology,

physics, robotics and control engineering [6-14].

Understanding the mechanisms and operational principles in

them can provide useful ideas for developing formation

control, distributed cooperative control and coordination of

multiple mobile autonomous agents/robots.

In 1987, Reynolds introduced a computer model called

boids that simulates the motion of bird flocks or fish schools.

Each bird has a local control strategy, but the desirable group

behaviour is achieved. This model suggests that flocking is the

combined result of three simple steering rules, which each

agent independently follows [7]:

Separation: steer to avoid crowding local flockmates.

Alignment: steer towards the average heading of local

flockmates.

Cohesion: steer to move toward the average position of

local flockmates.

A similar model of Reynolds was proposed in 1995 by

Vicsek et al. [8]. In this model, headings of each agent are

updated as the average of the headings of agent itself with its

nearest neighbours plus some additive noise. Numerical

simulations indicate the spontaneous emergence of coherent

collective motion, resulting in the headings of all agents to

converge to a common value. A rigorous proof of

convergence for Vicseks model was given by Jadbabaie et al.

[9] in 2003. They proved that the alignment strategy leads tothe result that all the agents headings converge to a common

heading. The important contribution of the work in [9] is that

connectivity in all times is not needed and connectivity of the

network on average is sufficient for alignment of the agents.

Following the works above, Tanner et al. [10, 11]

considered a group of mobile agents moving in the plane with

double-integrator dynamics. They introduced a set of control

laws that enable the group to generate stable flocking motion,

but these control laws cannot regulate the final speed and

heading of the group. R. Olfati-Saber [12, 13] presented a

theoretical framework for design and analysis of distributed

flocking algorithms. They introduced three kinds of agents:

-agents, -agents and -agents, and gave a universal

definition of flocking for particle systems with similarities to

Lyapunov stability analysis.The models of agent in most of present flocking researches

are really simple: an agent is a point in the complex plane with

Stable Flocking Algorithm for Multi-Robot Systems Formation

Control

Bin Lei, Wenfeng Li, Fan Zhang

T

1544

978-1-4244-1823-7/08/$25.00 c2008 IEEE

8/22/2019 2008_Lei_Li (1)

2/6

no kinematics constraints of motion. But in our work, the

agents are simulated wheeled mobile robots. In this paper, we

use virtual forces to propose a distributed controller for MRS

formation control. It is shown that in a connected graph (time

invariant or not), asymptotically, all velocities are the same,

collisions are avoided, all potentials are minimized. One

advantage of this controller is that it needs only the smallest

local information to achieve the steady state for the whole

group. This is similar with the behaviours of agents found in

nature which may not be capable of performing thecalculations demanded by the general rule in real time, and

retains a straightforward implementation.

II. MODEL OF FLOCK MEMBER

A. Model of Mobile Robot.

In this paper, we built a model of Pioneer-2dx like mobile

robot which is shown in Fig.1. The reference point with

coordinates (x, y) is the midpoint of the rear wheels. We

assume that the distance between both rear and front axles is

l=1, and denote w as the speed of the front wheel of the robot

and as the angle between the front wheel and the main

direction of the robot. Simple computation shows that the

model of the robot is:

uuw

w

w

w

y

x

21

1

0

0

0

0

0

1

0

0

0

0

0

sin

sincos

coscos

+

+

=

(1)

Let us do not care about the direction of the front wheels.

We may still simplify the model and get a kinematic function

which is expressed as the following control system:

+

=

1

0

0

0

sin

cos

vy

x

(2)

where the control values of each robot are its translational

velocity v and angular velocity . The simplified dynamics

equations of the robot are given by,

mFuFmuvmt

t

t

t /=== (3a)

IFluFlIuI nn

n

n /=== (3b)

where m is the mass of the robot, Iis its moment of inertia, Fnand Ft are the normal component and tangential component of

resultant force acting on the robot respectively and l denotes

the moment arm of the forces.

And from (3) we can get the following equations

v(t+t) = v(t) + utyt (4a)

(t+t) =(t) + unyt, (4b)

these two values will be as the control input values for each

robot.

Fig.1. The robot model.

B. Problem Description.

Consider a group of Nmobile robots moving on the plane,

with dynamics expressed by double integrators:

ii vr=

n

i

t

iii uuuv +== , i = 1, . . . ,N (5)

where ri = (xi, yi)T

is the position of robot i, vi = (xi ,yi)T

its

velocity, and ui = ( ut, u

n)

Tits acceleration inputs, see Fig.2.

Let the relative position vector between robot i and j be

denoted rij = ri rj. Robot i is steered via its acceleration input

ui. Let R be the communication radius of robot i. Robots

beyond this range are assumed not to affect robot i. In addition,

we give the following assumptions:

y Robots are of the same model and satisfy non-slipping

and pure-rolling constraints.

y The workspace is flat and without obstacles.

y Each robot can extract necessary information via its

communication equipment.

y A collision is assumed to have occurred when the

coordinates of two robots coincide.

The problem can be described as: giving initial positions

and orientations of a group of robots, the objective is to design

the control input (5) so that if connectivity is maintained in the

group,

y formation is established,

y an overall motion that satisfies the limitation of

communication range R,

y no collision among each robot,

y velocity of each robot is synchronized,

y and pair-wise distances between robots are stabilized to

steady state values within a given range.

This is being understood technically as a convergenceproperty on the robot velocity vectors and their relative

distances. We will present a realization of the control law (5)

that achieves the control objective in the following section.

v

y

x

l

FnFt

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1545

8/22/2019 2008_Lei_Li (1)

3/6

Fig.2. Control forces acting on robot i and the neighbours of it.

III. FLOCKING ALGORITHM

At the first time, we briefly present the main graph theoretic

[14] terminology used in this paper.

An (undirected) graph G consists of a vertex set, V, and an

edge set E, where an edge is an unordered pair of distinct

vertices in G. If x, y V, and (x, y) E, then x and y are said

to be adjacent, with adjacency matrixA

= [a

ij]. The degreematrix ofG is a diagonal matrix (A) with diagonal elements

which are row-sums ofA. The symmetric matrix defined as

L = (A) A

is called the Laplacian matrix of G. It is known that the

Laplacian matrix captures many topological properties of the

graph. Among those, is the fact that L is always positive

semidefinite, it has zero as a single eigenvalue whenever the

graph is connected and the associated eigenvector is the

n-dimensional vector of ones, 1n. The second largest

eigenvalue, 2 is known to convey a lot of information about

the structure of the graph and its connectivity.

Lemma 1: Let G be a graph on n vertices with Laplacian L.

Then for any vector x,

i) L satisfies the following sum-of-squares (SOS)

property: ,)(2

1 2

),(

i

Eji

jij

T xxaLxx

=

ii) Let G be a connected graph, then

,0||||

min)(22>=

x

LxxL

T

as x ranges over all nonzero vectors orthogonal to 1n.

Proof: All these results are well-known in the field of

algebraic graph theory and can be found in [14].

Definition 1 (Neighbouring graph): [10] The

neighbouring graph, G = {V, E}, is an undirected graphconsisting of a set of vertices (nodes), V = {r1, . . . ,rN },

indexed by the agents in the group, and a set of edges, E= {(ri,

rj) V V | ||ri -rj||

+ R, to capture the

fact that beyond this distance there is no robot interaction.

Define the following function as

)/||(||)||(||||)(|| RrkDrhrf ijijij = (7a)

abe

eabba

abe

abebash

cs

cs

cs

cs

+

=

+

+

+=

)1(

2)(

2)( , (7b)

where h(s) is an uneven sigmoid function with parameters thatsatisfy 0 < a < b ,c>1 and guarantee h(0) = 0.

The collective potential function V(r) of a group of robots is

a nonnegative attractive/repulsive pair-wise potential

function

with a global minimum at ||rij|| = D and a finite cut-off at R.

Then, define the following potential energy function as

=

=

i Nij

ijij

i Nij

ij

i

i

RrkDrH

rFrV

,

,

)/||(||)||(||2

1

||)(||2

1)(

(8a)

.)1ln()(

)()( bsea

b

c

badsshsH cs +

+== (8b)

The functions H(s) and h(s) are depicted in Fig.3.

i

j

x

y R

i

riFi

Qi

vi

i

1546 2008 IEEE Congress on Evolutionary Computation (CEC 2008)

8/22/2019 2008_Lei_Li (1)

4/6

Fig.3. The attractive/repulsive function H(s) and h(s) with parameters a=1,

b=5, c=3 and D=2.

From function (7), the virtual forces acting on the agent i

(see Fig.2) are defined as

.))(/||(||

||||/)||)((||

=

=

i

i

Nj

ijiji

ij

Nj

ijiji

vvRrkQ

rrrrrfF(9)

Then, the control law in (5) can be written as

./))sin((

/)cos(

IlFQu

mFu

iiii

n

i

iii

t

i

+=

= (10)

Thus, (4) becomes

./))sin(()()(

/)cos()()(

ItlFQttt

mtFtvttv

iiiiii

iiiii

++=+

+=+

(11)

IV. STABILITY ANALYSIS

Stability analysis of small swarm or flock agents were

considered in [12-18]. In [16], Beni et al. consider a

synchronous distributed control method for discrete one and

two dimensional swarm structures and prove stability in the

presence of disturbances using Lyapunov methods. Swarm

stability under total asynchronism (i.e., asynchronism with

time delays) was first considered in [17]. A one dimensional

discrete time totally asynchronous swarm model is proposed

and stability (swarm cohesion) is proved. The authors prove

asymptotic convergence under total asynchronism conditions

and finite time convergence under partial asynchronism

conditions. In this section, the stability analysis method is

mainly according to reference [13].

A group of robots using the control law (11) obtain the

following structural dynamics:

-

=

=

vrLrVvm

vr

)()(

(12)

where L is the m-dimensional graph Laplacian matrix,

defined by

mILL =

where denotes the Kronecker product. This

m-dimensional Laplacian satisfies the following SOS

property:

.||||2

1 2

),(

i

Eji

jij

T xxaxLx

= (13)

Before presenting the stability analysis of flocking behavior

under the control law (11), we need to define the following

structural Hamilton function:

H(r, v) = V(r) + T(v) (14)

where = 2)2/1()( iivmvT is the kinetic energy of themulti-robot systems in the moving frame.

Theorem 1: Consider a group of robots applying the

control law (11) with structural dynamics (12). Let c = {(r,v) :

H(r,v) c} be a level-set of the Hamilton function, such that

for any solution starting inc the robots form a flock t 0 .

Then, the following statements hold

i) all robots asymptotically move with the same velocity,

ii) given c < c*

= F (0), no robot internal collisions occur

for all t 0.

Proof: Taking the derivative of H(r, v), we have

0||||2

1

)(

)()(),(

2

),(

)13(

)12(

==

+=

+=+=

i

Eji

jiji

T

i

ii

T

ii

T

i

ii

T

ii

T

i

vvavLv

vLVvVv

vmvVvvTrVvrH (15)

which means the structural energy H(r, v) is monotonically

decreasing for all t 0 . In addition, H(r, v) c for all t 0 that

implies c is an invariant set.

From LaSalles invariance principle, all the solutions of (12)

starting in c converge to the largest invariant set in E= {(r,v)

c : H = 0}. However, since the group of robots constitutes

a dynamic flock for all t 0, G is a connected graph for all t

0. Thus, based on (15), we conclude that the velocities of all

robots match to the same value (v1 = = vn), which proves

part i). Moreover, the configuration r asymptotically

converges to a fixed r*

= D.

Assume two distinct robots m, n collision at time t1, for all t

0, we have

.)0(||)()((||

||)(||2

1||)()((||))((

*

},{\ },,,{\

cFtrtrF

rFtrtrFtrV

nm

nmi Nmmij

ijnm

i

==

+=

Hence, V(r(t1)) c*. But from (14) we have

V(r(t)) = H(r(t), v (t)) - T(v (t)) H(r(t), v (t)) c < c*

for all t 0. This is in contradiction with the above inequality.

Therefore, no two robots collide at any time t 0, which

proves part ii).

Remark 1: The assumption in Theorem 1 rarely hold for a

universal set of initial states and thus fragmentation emerges

instead of flocking. We will demonstrate these phenomena in

the next sections.

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1547

8/22/2019 2008_Lei_Li (1)

5/6

V. SIMULATIONS

In order to verify the validity and the performance of the

algorithm presented in the above sections, we have

implemented it written in Java and simulated it in a robot

simulation software Player/Stage.

Player [19] provides a network interface to a variety of

robots and sensor hardware and supports multiple concurrent

client connections to devices, creating new possibilities for

distributed and collaborative sensing and control.

Stage [20] simulates a population of mobile robots moving

in and sensing a two-dimensional bitmapped environment.

Stage devices present a standard Player interface so few or no

changes are required to move between simulation and

hardware. Several controllers designed in Stage have been

demonstrated to work on real robots.

In this section, we present several simulation results of 2-D

flocking. The following parameters remain fixed throughout

all simulations:D = 2m, R = 4m, a = 1, b = 5 and c = 3 for h(s).

The max speed of each robot is 0.6m/s, and m = 1kg, l=1m, I

=1 for each robot.

With a random position within a ball R = 5m, initial

velocities were also selected randomly with arbitrary

directions, fragmentation will be created, see Fig. 4. This is

because of lacking of existence of a group objective for all ofthe robots. Fragmentation is a generic form of collective

behavior of robots applying control law(11). This behavior is

insensitive to the type of probability distribution of the initial

position of the robots. Apparently, for the case of a highly

dense initial connected graph with small initial velocity

mismatch, one might expect that the group of robots form a

flock. Fragmentation phenomenon can be viewed as lack of

cohesion in a group of robots [13].

A special initial state of 5 robots is demonstrated in Fig.

5(a). We use the nearest neighbor principle which means we

ignore all neighbours but the nearest one within the neighbour

set (Ni =1). In matrix terms, this modification is intended to

take the adjacency matrix of the connectivity graph in the

original problem and reduce it to a minimal connected form

(modulo row / column swaps). Thus, converge speed is slower

than the following case, but decreasing the computing demand

of each robot. See fig. 5(b) (c) (d) (e).

Through the simulation, we find that the control law (11)

only guarantees the creation of flocking in some special initial

states. It is not surprising that a large number of robots with a

random set of initial states, flocking behavior is not created.

To create flocking for a generic set of initial states, we should

give each robot a group objective. Virtual leader[21] [22] or

-agent[13] may be two good choices and this will be a future

work for us.

VI. CONCLUSIONS

In this paper, we presented a distributed control algorithm

based on virtual force and nearest neighbour principle for

multi-robots formation control and implemented it for the

double integrator model in MRS. Simulation results show that

the proposed method can be efficiently applied to multiple

robots formation control.

Fig. 4. Fragmentation for 5 robots applying control law (11).

(a) t= 0s (b) t= 25s

(c) t= 50s (d) t= 100s

(e) Steady state.

Fig. 5. Flocking for 5 robots (nearest neighbour Ni=1).

REFERENCES

[1] C. A. Nguyen, Q. P. Ha, S. Huang, and H. Trinh. Observer-based

decentralized approach to robotic formation control. In Proceedings ofthe 2004 Australian Conference of Robotics and Automation, pages

18, Canberra, Australia, December 2004.

[2] C. Belta and V. Kumar, "Trajectory design for formations of robots by

kinematic energy shaping", Proc. IEEE International Conference on

Robotics & Automa tion, Washington DC, pp.2593-2598, 2002.

10

-7

12

-8

11

-6

-5

9876

Position X (m)

PositionY

(m)

-9

1548 2008 IEEE Congress on Evolutionary Computation (CEC 2008)

8/22/2019 2008_Lei_Li (1)

6/6

[3] M. Egerstedt and X. Hu, "Formation Constrained Multi-Agent Control",

Proc. IEEE Int. Conf. Robotics & Automation, Seoul, Korea,

pp.3961-3966, 2001.

[4] H. Yamaguchi, T. Arai, and G. Beni, "A distributed control scheme for

multiple robotic vehicles to make group formations", Robotics and

Autonomous Sy stems, vol. 36, no. 4, pp. 125-147, 2001.

[5] S. Carpin and L.E. Parker, "Cooperative Leader Following in a

Distributed Multi- Robot System", Proc. IEEE International Conference

on Robotics & Automation, pp. 2994-3001, 2002.

[6] A. Okubo, Dynamical aspects of animal grouping: swarms, schools,

flocks, and herds, Adv. Biophysics., vol. 22, 1986, pp. 194.

[7] Reynolds, Flocks, birds, and schools: a distributed behavioral model,Computer Graphics, vol. 21, 1987, pp. 25-34.

[8] T. Vicsek et al. Novel type of phase transitions in a system of

self-driven particles. Physical Review Letters, 75:12261229, 1995.

[9] A. Jadbabaie, J. Lin, and A.S. Morse, Coordination of groups of mobile

autonomous agents using nearest neighbor rules, IEEE Trans. Auto.

Control, vol. 48, 2003, pp. 988-1001.

[10] Tanner H G, Jadbabaie A, Pappas GJ. Stable flocking of mobile

agents, Part I: Fixed Topology. Proc. Of 42nd IEEE Conference on

Decision an d Control , 2003:2010 - 2015.

[11] Tanner H G, Jadbabaie A, Pappas GJ. Stable flocking of mobile

agents, Part II: dynamic topology. Proc. of 42nd IEEE Conference on

Decision an d Control, 2003:2016 - 2021.

[12] R. Olfati-Saber and R. M. Murray, Flocking with obstacle avoidance:

Cooperation with limited communication in mobile networks, in Proc.

42nd IEEE Conf. Decision and Control, vol. 2, Dec. 2003, pp:

20222028.

[13] R. Olfati-Saber, Flocking for multi-agent dynamic systems:

algorithms and theory, IEEE Trans. Automatic Control, vol. 51, pp.401420, 2006.

[14] C. Godsil and G. Royle, Algebraic Graph Theory, Vol. 207 of

Graduate Texts in Mathematics. New York: Springer-Verlag, 2001.

[15] Veysel Gazi and Kevin M. Passino, Stability Analysis of Swarms,

IEEE Trans. Auto. Control, vol. 48, 2003, pp. 692-697

[16] G. B en i and P. Liang, Pattern recon figu ration in

swarmsconvergence of a distributed asynchronous and bounded

iterative algorithm, IEEE Trans. Robot. Automat, vol. 12, pp. 485490,

June 1996.

[17] Y. Liu, K. M. Passino, and M. M. Polycarpou, Stability analysis of

M-dimensional asynchronous swarms with a fixed communication

topology, IEEE Trans. Auto. Control, vol. 48, no. 1, pp. 7695, Jan.

2003.

[18] E. W. Justh and P. S. Krishnaprasad, A simple control law for UAV

formation flying, Inst. Sys. Res., Univ. Maryland, College Park, MD,

Tech. Rep. 2002-38, 2002.

[19] Brian P. Gerkey, Richard T. Vaughan, Most Valuable Player: A

Robot Device Server for Distributed Control, Proc. IEEE/RSJ Intl.

Conf. on Intelligent Robots and System, 2001, pp. 1226.

[20] Richard T. Vaughan. Stage: a multiple robot simulator, Institute for

Robotics and Intelligent Systems Technical Report (IRIS-00-393),

University of Southern California, 2000.

[21] H. Shi, et al, Virtual Leader Approach to Coordinated Control of

Multiple Mobile Agents with Asymmetric Interactions, Proceedings of

the 44th IEEE Conf. on Decision and Control, and the European

Control Conference, 2005, pp.6250-6255.

[22] H. Shi, et al., Flocking of Multi-Agent Systems with a Virtual

Leader, Proc. Of the 2007 IEEE Symposium on Artificial Life, 2007,

pp.287-294.

2008 IEEE Congress on Evolutionary Computation (CEC 2008) 1549