Artificial Formation Forces for Stable Aggregationof Multi-Agent SystemSamitha W Ekanayake and Pubudu N Pathirana

School of Engineering and TechnologyDeakin University

Geelong, Victoria 3217Australia

[swek, pubudu] @deakin.edu.au

Abstract- This paper introduces, a robust and stablealgorithm based on artificial formation forces, for Multi-AgentSystem (MAS) aggregation in 2D space. The MAS model withartificial forces; consists of inter-member collision avoidanceelement, formation generation element and a velocity baseddamping element; is analysed for stability and convergence.Computer simulations are used to illustrate stability andconvergence, and to demonstrate effectiveness of the algorithm.

Keywords-Aggregation, Formation, Multi-agent System,Swarm, Stability analysis

I. INTRODUCTION

Corporative robotics, where multiple robots perform col-laborative tasks in decentralized manner, has been a researchinterest for robotics researchers across the world. Distinctadvantages of MAS, such as robustness to failures of itsmembers, flexibility in motion, adaptive behavior and lowcost of individual members, compared with single robot basedsystems, make them suitable for vast range of applications(i.e. Nano-robotics for medical applications, Defense , Search& Rescue etc,). Natural swarms or social foraging insects;such as bees, ant colonies etc, sets the conceptual frameworkfor MAS. Researches and studies on group behavior of suchanimals/ insects [1], [2], [3], [4], mathematical modeling ofgroup behavior [5], [6] etc, forms the foundations for MASresearch.Formation /aggregation of agents for coordinated task is amongfundamental behaviors of MAS, as well as in biologicalswarms [1], [3]. Many researchers have performed variousmethods for aggregation of MAS. Gazi and Passino introducedbiologically inspired individual attraction/ repulsion basedswarm aggregation model in [7], [8], [9], with stability analysisand defining a time and physical bounds for convergence.Behavior based, which is more nature oriented, swarm for-mation and obstacle avoiding method [10] was practicallyimplemented and tested in real-world, while Soyal and Sahin[11] introduces a aggregation method using simple reactivebehaviors. Study by Nuruse etal [12] is different approach inbehavior based aggregation, where emotion and affection likestatus determines the aggregation behavior of agents.In this paper we present a swarm model for multi-agentsociety that will converge around a predefined location in 2D

space. The model assumes that the agents as point objectswith no physical dimensions and having same properties ineach agent (such as mass, mobility, etc,.). The formationalgorithm uses inter-individual repulsion forces in order toavoid collisions between agents, global formation forces whichconverges the swarm around a predetermined location anda friction-like damping force that brings swarm to a halt.Also in this model, disturbances, sensor errors, limitationsin control model of the mobile agent and other practicaldifficulties were not considered, while assuming that eachagent is having its own localization capabilities. With recentadvances in communication, computing and networking, theassumption of knowledge ofpositions without error become apractical statement.An outline of the paper is as follows: Formation algorithmtogether with the swarm model is introduced in section II.Stability and behavior analysis of the model were carried outin sections III and IV. In section V we present computersimulations, to demonstrate the effectiveness of the aggregationalgorithm and to illustrate results obtained in section III.

II. DYNAMIC FORMATION ALGORITHM

Consider a multi-agent system or swarm consisting of Nmembers in two dimensional euclidean space. The state of themember i is described by

xiz ZiXt =l

\ i )where zi, represents the position vector of jth member in 2Dspace. (We assume that each agent knows positions of all themembers in the swarm with no time delay.)Before stating the aggregation algorithm we definea = ( 1 0 ), Q = ( 0 1 ). Also, we use R forreal number and R+ to represent positive real number.The state of the whole swarm,x ( Xl X2 X3 .XN )T is determined incontinuous time dynamic model described by,

= A*x+B*u (1)

1-4244-0555-6/06/$20.00 ( 2006 IEEE Page 1 29 ICIA 2006

Where,

A

A00

0

B = [ B

0 0 ...A 0 ...0 A ...

0 0 ...

B B ...

000

A2 NxN

B T

and

A= 0 i (1) Bc of,The control function u in (1) consists of,

'U = ( Ul zU2 'U3 ..UN )T1where

ui (F tt + Friobot friction)kmass

Where, kfriction C R+ is a constant which determines thepower of the damping action. An analysis of kfriction, withrespect to stability, is presented in section III.

III. BEHAVIOR OF MULTI-AGENT SYSTEM

In this section the behavior of the multi-agent systemas a whole (whole swarm) is analysed and conditions for

(2) convergence obtained. Please note that we use term swarm,to represent MAS in the following text. In our analysis, wholeswarm is considered as one complete object, where motion is

(3) governed by the resultant force, FR,-FR

att + robot friction

Where, FR is the resultant attraction force on the wholeswarm and is given by;

(4)kmass C R+, represents the mass of the robot, and consideredto be a constant for all the members in the MAS.The term Fi is the artificial force acting on each individual,which makes them move toward the desired location (Zd) andsettles around there. Friobot term in the control function avoidsthe inter-agent collisions and this function looks similar to therepulsive term described in [8], [13] etc, but the problem of de-creasing repulsion behavior for infinitesimally small distancesis avoided, thus inter-agent collision avoidance is guaranteed.Term FJrictionis the artificial friction force exterted on eachagent, this force term makes the agent to a complete stop whenthe forces are balanced, or when stable state is reached.Before analysing the behavior of the MAS, we define theartificial forces as follows. (note that for notational simplicity,we use Fatt, Fjriction and FriObot terms instead of functionalsFatt (Zi, Zd), Ffrictionr(Zi) and Fiobot(x) respectively.)Fa in (4) is the attraction force on jth agent from the shape,defined as

Fatt =katt(Zd-aXi) (5)where Zd is the destination/target vector and katt C R+, is aconstant determining the power of attraction force.Term FiobOt in (4), refers to the force acting on 'th robot dueto the location of other members of the swarm, is defined as

Fr0bOt = krobot 4 CVX- (6)

krobot C R+ is a constant determining, the power of interindividual repulsion and the distance between each other atstable state. Bounds for krobot is discussed in section IV, withrespect to the stability of the formation.

Term Fjriction in (2) is given by,

N

FR = Ftt = Nkatt (Zd -Zcm)i=l

(For the simplicity, we use zi and Zi instead of aXi and /3Xirespectively.)F R is the resultant friction force on the whole swarm,frictiongiven by;

N

Fricti onF Nkfriction (im)friction ~~ricinT)i=l

The term F/RboL corresponds to the resultant individual forcesacting on each robots, and it is given by,

N N (ziFrobot = 1J 1 Zi

i=l j=lj7Xi- zj).zjll3 0

Therefore the final expression for the Resultant force actingon the swarm is reduced to,

FR = Nkatt (Zd -Zcm) -Nkfriction (zcm)which consists of two artificial force components, firstone being the active component which exerts a force onZcm towards Zd and the second one being the inactivecomponent depends on the velocity of Zcm and behave as theresistive force. In above expressions, Zcm is the center ofmass of the swarm and 4cm is velocity of the center of mass

N N

Z (zi)of swarm, and are defined as, Zcm ZNcm N

A. Stability and convergenceLet e to be the error between target position and the current

position of the swarm, such that;

Ffriction kfriction (13Xi) (7) 6 = (Zcm -Zd)

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motion towards Zd

Proof:Direction of motion of the swarmIf we select a Lyanonov function candidate as Vcm =1M kl2 1 2, then will be bounded by,2 2

Vcm < -C|e|2Thus as Vcm e 2 > 0 and Vcm < 0, one can say that the2motion of the swarm is in the direction of decreasing c.

Fig. 1. kfriction = 0.2 Conditions for stability at the destinationThe model described in (8) can be converted to standeredexpression for frequency domain analysis as,

1G(s) M

C Ks2+M M

(9)

Then using the general frequency domain second-order transferfunction,

+ +22 + 2(bwnS nw

Fig. 2. kfriction = 1.5

Fig. 4. kfriction = 6-5 0 5 10 15x coordinate

Fig. 3. kfriction = 2

then e = Zcm and e Zcm= Therefore the equation of motionof the swarm, with respect to error, is as follows;

M =-C e-Kc (8)where M = Nkmass, K = Nkatt and C = Nkfriction.

Theorem 1: The center of mass of the swarm, (Zcm) willconverge to the center of mass of the contour, (Zd) andbecome stable at Zd,if kmass, katt, kfriction > 0, and(a) if kfriction < 20kmass katt, the swarm demonstratesunder damped dynamics around Zd and,(b) if kfriction > 2V/kmass katt, it will demonstrate damped

30 r

E

10

-E

1D

02

3o 0a-10c

-M-20 _

Li-30 F

Simulation steps

Fig. 5. Behaviour of the error between current and disired locations of Zcmfor some kf rictionvalues

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200

150

100

50

20

15

C10.

-5

-10

-15-31

30-

25

20

15

15

we obtain the corresponding value for ( from (9) is as below,C

2v M-Kkfriction

2 /kmass katt(10)

which proves our assertion. aNote that in proving above, we consider the vector basedsystem (8) to be a scalar second order ODE. This is possibleas all the artificial forces are acting along the vector e and wecan reduce the vector equation to a scalar system assuming alinear motion in the direction of e.Above Theorem says that the swarm described by (1) with thecontrol function described by (4) will move to the target pointZd, but it does not say anything about the motion of individualmembers.

IV. BEHAVIOR OF A SINGLE AGENT

In this section we investigate the behavior of an individualmember of a specific swarm, where members have approxi-mately equal inter-member repulsion function magnitudes. Wecall such a swarm, "X Swarm".

Definition 1: A swarm is defined as "X swarm", if thereexists positive constants A, A, 6, , that satisfy the followingconditions simultaneously for all i, j and i t j.

1) dij > +A,2) For dij > 6, /3ij , (Constant)

lZi- zcmil A603 (60 + A\)3

4) llzi -zcij < A2hee cWhere

dij = llZ-zj 1,ij 1N

and Zi = iNi 1 (1 1)In the above definition, Zim is the center of mass of the swarmwithout the jth member. Also the condition l*z-Zi Tmj < Asets a limit for an agents' position relative to other agents.

Remark 1: The repulsion force between individual agentscan be described as follows, the term -i being the

~zi zj~directional component consists of a unit vector along the force;1

and the term 1l 1 2 being the magnitude of the forcewhich is inversely proportional to the distance between them.

Proof:

Frobot =krobot j (z3 z)J=l1,J7i ij

Using the definition of the "X Swarm", we haveN

Frobt =: krobotK E iFrobot krobot I{ j (ziJ=1,J7 i

*Zj)

krobot ; (N -1) (zi- zm)

Taking , + A)3, and using (z, Zi- m)

with Zcm is on Zd,N (z, zj) (N 1)

j=lj7i ~Zi Zj 3< 0 1e 1we can prove that Vi is bounded by,

Vi (N -1)krobot d

4 V >0.

Hence, the Lyapunov stability criteria is satisfied, which provesthe Theorem. a

V. SIMULATION AND RESULTS

In this section, a computer simulation was used to illustratethe results derived in previous sections and to demonstrate theeffectiveness and robustness of the swarm model for variousdisturbances. We consider robots/members having point massdynamics with unit mass (i.e. k,,as = 1), throughout thesimulations.

A. Behavior of the whole swarmFirst we illustrate the results derived in section III, i.e.

the behavior of whole swarm as a second order ODE. Todemonstrate the behavior with variation of kfriction (withZd = (-5, -3) and katt = 1), we present simulation re-sults for a range of kfriction values (6, 2,1.5, 0.2) in fig-ures 1 to 4 (representing different aggregation states; i.eover/critically/under/zerol damped) and in Fig. 5 we demon-strate the variation of error ei against time.

B. MAS behavior in sudden failure/death of membersSwarms in real-life applications, especially in hostile envi-

ronments, are subjected to face sudden erase of whole sectionof the swarm from disasters, or failure/malfunction of somemembers in random manner. In all those situations the existingmembers should rearrange the formation in order to continuethe mission. The series of simulation screen shots presentedin Fig. 6. shows the behavior of proposed swarm model insudden loss of agents.

C. MAS behavior in recruiting new members to the societyReplacements of dead/malfunctioning members, increasing

number of members to stay with increasing demands etc,results in adding new members to an existing MAS, whichmay already in stable state. Thus the MAS should rearrange theformation to recruit new members to the society. The series ofsimulation screen shots presented in Fig. 7. shows the behaviorof proposed model in recruiting new members in to an existingformation.

'In the zero damped situation, we include a small damping factor toeliminate infinitely large movements of agents

VI. CONCLUDING REMARKS

In this paper, we present an aggregation algorithm, basedon artificial formation forces, for a multi-agent system orswarm of robots with self localization and effective peer-to-peer communication capabilities. We showed that the motionof the whole swarm can be described as a second order ODEand analysed the behavior of it with respect to stability andconvergence. Further, the motion and behavior of a memberwas analysed with introducing a special swarm configurationcalled "X Swarm".Comparing with swarm aggregation models where membershaving minimum sensing and computational capabilities (suchas proximity sensors, simple logic based controllers etc), ourswarm model needs more advanced sensing, communicatingand computing power. With recent advances in technologies,such as wireless networking, GPS based self localization, etc.,this system can be effectively used in swarm aggregation ap-plications involving macro scale robots (e.g. land exploration,land-mine removal etc,).In our swarm model, the unbounded nature of the inter-member repulsion force (F0bodt), at infinitesimally small inter-member distances, may result implementation difficulties withreal-world actuators; but the same property, effectively avoidsinter-member collisions as it uses the maximum capacity ofactuators to repulse them selves away from each other.

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I_ * .* .

** 0

1 * .

20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0x coordinate

(a)T T r 1 r 1 T 1

-8

.- 10

-12

-14

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* 0

* 0

* 0-12

-16

-20 -18 -16 -14 -1 -10 -8 -6 -4 -2 0x coordinate

(C)

* .

2 B 14 12 10 8 6 -4 2x coordinate

(b)

20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0x coordinate

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0*0

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* . *

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-8 * * *

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-8

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-10I

-12

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* . 0)_. * *

.*0* * *

>~~~ _0

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(b)

i0_. .

*._*X 0.0 *

[_ ~* 0

20 -8 -16 -14 -12 -10 -8 B6 -4 -20x coordinate

(d)Fig. 7. Behavior of the MAS to adapt new members

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