1

Stability Analysis of Social Foraging SwarmsVeysel Gazi and Kevin M. Passino

Abstract— In this article we specify an M-member “individual-based” continuous time swarm model with individuals that movein an n-dimensional space according to an attractant/repellent ora nutrient profile. The motion of each individual is determined bythree factors: (i) attraction to the other individuals on long dis-tances, (ii) repulsion from the other individuals on short distances,(iii) attraction to the more favorable regions (or repulsion fromthe unfavorable regions) of the attractant/repellent profile. Theemergent behavior of the swarm motion is the result of a balancebetween inter-individual interactions and the simultaneous inter-actions of the swarm members with their environment. We studythe stability properties of the collective behavior of the swarm fordifferent profiles and provide conditions for collective convergenceto more favorable regions of the profile.

I. INTRODUCTION

Swarming, or aggregations of organisms in groups, can befound in nature in many organisms ranging from simple bacte-ria to mammals. Such behavior can result from several differentmechanisms. For example, individuals may respond directly tolocal physical cues such as concentration of nutrients or distri-bution of some chemicals (which may be laid by other individu-als). This process is called chemotaxis and is used by organismssuch as bacteria or social insects (e.g., by ants in trail followingor by honey bees in cluster formation). As another example, in-dividuals may respond directly to other individuals (rather thanthe cues they leave about their activities) as seen in some higherorganisms such as fish, birds, and herds of mammals.

Evolution of swarming behavior is driven by the advantagesof such collective and coordinated behavior for avoiding preda-tors and increasing the chance of finding food. For exam-ple, in [1], [2] Grunbaum explains how social foragers as agroup more successfully perform chemotaxis over noisy gradi-ents than individually. In other words, individuals do much bet-ter collectively compared to the case when they forage on theirown. Operational principles from such biological systems canbe used in engineering for developing distributed cooperativecontrol, coordination, and learning strategies for autonomousmulti-agent systems such as autonomous multi-robot applica-tions, unmanned undersea, land, or air vehicles. The develop-ment of such highly automated systems is likely to benefit frombiological principles including modeling of biological swarms,coordination strategy specification, and analysis to show that

This work was supported by the DARPA MICA program via the AFRL un-der contract F33615-01-C-3151. Veysel Gazi was also partially supported byTUBITAK (the Scientific and Technical Research Council of Turkey). VeyselGazi is currently with Atilim University, Department of Electrical and Electron-ics Engineering, Kizilcasar Koyu, Incek, Golbasi, 06836 Ankara, TURKEY(veysel gazi@atilim.edu.tr). He was originally with The Ohio State Univer-sity, Department of Electrical Engineering, 2015 Neil Ave., Columbus, OH43210, USA. Kevin Passino is with The Ohio State University, Departmentof Electrical Engineering, 2015 Neil Avenue, Columbus, Ohio 43210, USA(passino.1@osu.edu).

group dynamics achieve group goals. In this article we de-velop a simple model of swarming in the presence of an at-tractant/repellent or a nutrient profile and analyze its stabilityproperties for different profiles. We show collective conver-gence to more favorable regions of the profile. The model thatwe develop here can be viewed as a representation of cohesivesocial foraging of swarms.

Biologists have been working on understanding and model-ing of swarming behavior for a long time [3], [4], [5], [6], [7],[8], [9]. There are two fundamentally different approaches thatthey have been considering for analysis of swarm dynamics.These are spatial and nonspatial approaches. In the spatial ap-proach the space (environment) is either explicitly or implicitlypresent in the model and the analysis. It can be divided intotwo distinct frameworks which are individual-based (or La-grangian) framework and continuum (or Eulerian) framework[6]. In the individual-based models the basic description is themotion equation of each (separate) individual and therefore itis a natural approach for modeling and analysis of complex so-cial interactions and aggregations. The general understandingwithin this framework now is that the swarming behavior is aresult of an interplay between a long range attraction and a shortrange repulsion between the individuals. The work by Brederin [3], where he suggested a simple model composed of a con-stant attraction term and a repulsion term which is inverselyproportional to the square of the distance between two mem-bers is one of the early works within this framework. Similarly,in [4] Warburton and Lazarus also considered an individual-based swarm model and studied the effect on cohesion of afamily of attraction/repulsion functions.

In the Eulerian framework the swarm dynamics are describedusing a continuum model of the flux, namely concentration orpopulation density (i.e., a model in which each member of theswarm is not considered as individual entity, but the swarm is acontinuum described by its density in one, two, or three dimen-sional space) described by partial differential equations of theswarm density. The basic equation of the Euclidian models isthe advection-diffusion-reaction equation, where advection anddiffusion are the joint outcome of individual behavior and envi-ronmental influences, and the reaction term is due to the popula-tion dynamics. See, for example, [7] where the authors presenta swarm model which is based on non-local interactions of theswarm members. Their model consists of integro-differentialadvection-diffusion equations with convolution terms that de-scribe attraction and repulsion.

In the nonspatial approaches the population level swarmingdynamics are described in a non-spatial way in terms of fre-quency distributions of groups of various size. They assumethat groups of various sizes split or merge into other groupsbased on the inherent group dynamics, environmental condi-

tions, and encounters of other groups. See, for example, [9]where the authors present a general continuous model for an-imal group size distribution (a nonspatial patch model). Theyconsider a population with fixed size that is divided into groupsof various dynamic sizes. The drawback of the nonspatial ap-proaches is that they need several “artificial” assumptions aboutfusion and fission of groups of various sizes in order to describeand analyze the population dynamics.

Each of the above approaches has its advantages and dis-advantages. A comparative study is presented by Durrett andLevin in [8], where they compare four different approaches tomodeling the dynamics of spatially distributed systems by us-ing three different examples, each with different realistic bio-logical assumptions. They show that the solutions of all themodels do not always agree, and argue in favor of the discrete(individual based) models that treat the space explicitly. In a re-cent study in [10] Parrish and her colleagues survey similaritiesand differences between different models of swarm aggrega-tions and present preliminary results of efforts to unify all themodels within single framework. A good background and areview of the swarm modeling concepts and literature such asspatial and nonspatial models, individual-based versus contin-uum models and so on can be found in [5] and [6]. See also [11]and [12] and references therein for other related work. Othergeneral references are the books by Edelshtein-Keshet [13] andMurray [14].

In parallel to the mathematical biologists there are a numberof physicists who have done important work on swarming be-havior [15], [16], [17], [18], [19], [20]. The general approachthe physicists take is to model each individual as a particle andstudy the collective behavior due to their interaction. Many ofthem assume that particles are moving with constant absolutevelocity and at each time step each one travels in the averagedirection of motion of the particles in its neighborhood withsome random perturbation. They try to study the affect of thenoise on the collective behavior and to validate their modelsthrough extensive simulations.

For many organisms, swarming often occurs during “socialforaging” and with the focus in this paper of studying the in-teractions between inter-individual cohesion mechanisms cou-pled with effects from the environment, particularly the attrac-tant/repellent or nutrient profiles, there are other areas of rele-vant study. First, note that foraging theory is described in [21].The recently popular “ant colony optimization” is an optimiza-tion method based on foraging in ant colonies and is discussedin [22]. There, the focus is on biomimicry for the solution ofcombinatorial optimization algorithms (e.g., shortest path al-gorithms) and swarming as we study it here is not considered.In [23] the author shows that chemotactic behavior of E. colicoupled with evolutionary and “elimination/dispersal events”provides for a non-gradient distributed and parallel optimiza-tion procedure that can be used for adaptive control and coop-erative control problems. Also, the author there used a simi-lar characterization of an “attractant-repellent profile” to ours,and also studied swarm behavior as a distributed optimizationmethod. Member-member swarming mechanisms are differentfrom here, and are only considered from an optimization per-spective. Stability analysis was not considered in [22], [23].

Another optimization method based on swarming behavior isthe particle swarm optimization method [24], [25]. Althoughstill there is no rigorous stability proof for its operation, it seemsthat it is a very effective method in function minimization.

In recent years, engineering applications such as formationcontrol of multi-robot teams and autonomous air vehicles haveemerged and this has increased the interest of engineers inswarms. Some examples include [26], [27], and [28], where theauthors describe formation control strategies for autonomousair vehicles and multiple autonomous land vehicle teams, re-spectively. Similar work are the study of asynchronous dis-tributed control and geometric pattern formation of multipleanonymous (or identical) robots [29] and the study on coop-erative control and coordination of a group of holonomic mo-bile robots to capture/enclose a target by making group forma-tions [30]. Another related work is the social potential fieldsmethod for distributed control of groups of robots consideredby Reif and Wang in [31]. It is based on artificial force laws be-tween individual robots and robot groups, where the force lawsare inverse-power or spring force laws incorporating both at-traction and repulsion. It is an interesting and important work.However, it does not contain stability proof of the approach.

Other work on formation control and coordination of multi-agent (multi-robot) teams can be found in [32], [33], [34], [35],[36]. In [32] a feedback linearization technique using only lo-cal information for controller design to exponentially stabilizethe relative distances of the robots in the formation is proposed.Similarly, in [33], [34], the concept of control Lyapunov func-tions together with formation constrains is used to develop aformation control strategy and prove stability of the formation(i.e., formation maintenance). The results in [35], on the otherhand, are based on using virtual leaders and artificial potentialsfor robot interactions in a group of agents for maintenance ofthe group geometry. By using the system kinetic energy and theartificial potential energy as a Lyapunov function closed loopstability is proved. Moreover, a dissipative term is employed inorder to achieve asymptotic stability of the formation. In [36],the results in [35] are extended to the case in which the groupis moving in a sampled gradient field.

Important work on swarm stability is given by Beni andcoworkers in [37] and [38]. In [37] they consider a synchronousdistributed control method for discrete one and two dimensionalswarm structures and prove stability in the presence of distur-bances using Lyapunov methods. On the other hand, [38] is,to best of our knowledge, one of the first stability results forasynchronous methods (with no time delays). There they con-sider a linear swarm model and provide sufficient conditions forthe asynchronous convergence of the swarm to a synchronouslyachievable configuration.

Swarm stability under total asynchronism (i.e., asynchronismwith time delays) was first considered in [39], [40]. In [39]a one dimensional discrete time totally asynchronous swammodel is proposed and stability (swarm cohesion) is proved.The authors prove asymptotic convergence under total asyn-chronism conditions and finite time convergence under par-tial asynchronism conditions (i.e., total asynchronism with abound on the maximum possible time delay). In [40], on theother hand, the authors consider a mobile swarm model and

2

prove that cohesion will be preserved during motion under cer-tain conditions, expressed as bounds on the maximum possibletime delay. In [41] we obtained similar results to those in [39]for a swarm with a different mathematical model for the inter-member interactions and motions using some earlier results de-veloped for parallel and distributed computation in computernetworks in [42]. All of these stability investigations have beenlimited to either one or two dimensional space. Note that in onedimension, the problem of swarming is very similar to the prob-lem of platooning of vehicles in automated highway systems, anarea that has been studied extensively (see, for example, [43],[44], [45] and references therein).

Recently some results on the multidimensional case havebeen also obtained. For example, the work in [46], [47] isfocusing on extending the work in [39], [40] to the multidi-mensional case by imposing special constraints on the “leader”movements and by using a specific communication topology.

In [48], [49] we developed an “individual-based” contin-uous time synchronous model for swarm aggregations in n-dimensional space. We showed that for the given model theindividuals will form a cohesive swarm in a finite time, and weobtained an explicit bound on the swarm size. In [50] we ex-tended our results in [48], [49] to a more general class of attrac-tion/repulsion functions and allowed for unbounded repulsionfor collision avoidance. Note that in [48], [49], and [50] the mo-tion of the swarm members was based only on inter-individualinteractions and was not affected by the environment. In thisarticle we build on our earlier results in [48], [49], and [50] byconsidering a swarm which moves in an attractant/repellent pro-file (i.e., a profile of nutrients or toxic substances) and show col-lective convergence to (divergence from) more favorable (un-favorable) regions of the profile. The inter-individual interac-tions and the interactions with the environment in our modelare based on artificial potential functions, a concept that hasbeen used extensively for robot navigation and control [51],[52]. Therefore, our model here can be viewed as a type of asocial potential fields model similar to the one in [31] (where norigorous stability was considered). Therefore, the results hereare an initial step for a rigorous stability analysis of a more gen-eral social potential fields model. Another similar work to oursis the work in [30]. However, there the work is limited to twodimensions and the stability analysis is very limited. Anotherrelated article using potential functions similar to ours is [35]and [36]. However, in [35] the model does not incorporate en-vironmental effects and in [36] mostly quadratic gradient fieldsare considered. The results in this article were initially pub-lished in [53] and [54].

II. THE SWARM MODEL

We consider a swarm of M individuals (members) in an n-dimensional Euclidean space. We model the individuals aspoints and ignore their dimensions. The position of individ-ual i is described by xi ∈ R

n. We assume synchronous motionand no time delays, i.e., all the individuals move simultaneouslyand know the exact relative position of all the other individuals.Let σ : R

n → R represent the attractant/repellent profile or the“σ-profile” which can be a profile of nutrients or some attrac-tant or repellent substances (e.g., food/nutrients, pheromones

laid by other individual, or toxic chemicals). Assume that theareas that are minimum points are “favorable” to the individu-als in the swarm. For example, assume that σ(y) < 0 representsattractant or nutrient rich, σ(y) = 0 represents a neutral, andσ(y) > 0 represents a noxious environment at y. (Note that σ(·)can be a combination of several attractant or repellent profiles.)

We consider the equation of motion of each individual i de-scribed by

xi =−∇xiσ(xi)+M

∑j=1, j 6=i

g(xi− x j), i = 1, . . . ,M, (1)

where g(·) represents the function of mutual attraction and re-pulsion between the individuals and is an odd function of theform [50]

g(y) =−y[

ga(‖y‖)−gr(‖y‖)]

, (2)

where ga : R+ → R

+ represents (the magnitude of) the attrac-tion term and it has a long range, whereas gr : R

+ → R+ rep-

resents (the magnitude of) the repulsion term and it has a shortrange, and ‖ · ‖ is the Euclidean norm. As in [50] it is assumedthat yga(‖y‖) = ∇yJa(‖y‖) and ygr(‖y‖) = ∇yJr(‖y‖), whereJa(·) and Jr(·) are the artificial social potential functions of theattraction and repulsion between the individuals, respectively.The functions ga(·) and gr(·) are chosen such that on large dis-tances attraction dominates, on short distances repulsion dom-inates, and there is a unique constant distance δ where attrac-tion and repulsion balance. In other words, we assume thatthere exists δ such that ga(δ) = gr(δ), and for ‖y‖> δ we havega(‖y‖) > gr(‖y‖) and for ‖y‖< δ we have gr(‖y‖) > ga(‖y‖).This is consistent with biological observations [3], [4], wherethe inter-individual attraction/repulsion is based on an inter-play between attraction and repulsion forces with the attractiondominating on large distances, and the repulsion dominating onshort distances. The distance δ, on which the attraction andrepulsion between two individuals balance is called the equi-librium distance in the biological literature. However, we willnot use this term here in order to avoid any confusion with theterminology in stability theory.

The term −∇xiσ(xi) represents the motion of the individualstowards regions with higher nutrient concentration and awayfrom regions with high concentration of toxic substances. Notethat the implicit assumption that the individuals know the gra-dient of the profile at their position is not very restrictive sinceit is known that some organisms such as bacteria are able toconstruct local approximations to gradients [23].

We would like to emphasize that even though we get our in-spiration from biological swarms, our model constitutes alsoessentially a kinematic model for swarms of engineering multi-agent systems. In the context of multi-agent (i.e., multi-robot)systems the profile σ(·) constitutes an artificial potential fieldthat models the environment containing obstacles or threats tobe avoided (analogous to toxic substances) and targets to bemoved towards (analogous to food). In systems with real agents(with their specific dynamics) the trajectories generated by ourmodel can be used as reference trajectories for the agents tofollow or track.

Note also that, even though our model is a type of a kinematicmodel, it can be viewed as an approximation of a model with

3

point mass swarm member dynamics for some organisms suchas bacteria. To see this consider the point mass model in whichthe individuals move based on the Newton’s law miai = F i.This gives rise to the system of motion equations

xi = vi

mivi = ui,

where ui = F i is the total force acting on individual i. Now,suppose there is a velocity damping term of the form −kvvi inui, where kv > 0. In other words, assume that we have

ui =−kvvi + ui.

Now, note that for organism such as bacteria we have mi verysmall (i.e., we have mi≈ 0) and the viscosity of the environmentfor them is high. Therefore, we can take mi = 0. Substitutingthis in the above system of equations we obtain

xi =1kv

ui,

which is exactly the same model that we consider in this articlewith kv = 1 and

ui =−∇xiσ(xi)−M

∑j=1, j 6=i

[

∇xi Ja(‖xi− x j‖)−∇xiJr(‖xi− x j‖)]

.

Note that the above controller ui is an energy minimization con-troller of the form

ui =−∇xiE,

where E is the total artificial potential energy in the system andis given by

E =M

∑i=1

σ(xi)+12

M

∑i=1

M

∑j=1, j 6=i

[

Ja(‖xi− x j‖)− Jr(‖xi− x j‖)]

.

Therefore, each of the individuals in the swarm moves such thatto minimize the total artificial potential energy in the system.

One drawback of the model here is that each individual needsto know the relative position of all the other individuals. In bi-ological swarms, often each individual can see (or sense) onlythe individuals in its neighborhood because the ranges of theirsenses are limited. Therefore, in nature the attraction or “de-sire to stick together” depends only on the individuals that itcan sense. Therefore, the final behavior of the swarms de-scribed here may not be in “perfect harmony” with real bio-logical swarms. For example, in real swarms we may observeformation of several separate clusters or swarms instead of asingle swarm (as would be the case here). Moreover, in realswarms if the swarm arrives in the vicinity of two close valleysthen swarm splitting may occur, whereas here it may not nec-essarily do so. Nevertheless, the analysis here is a first step to-wards developing a comprehensive and rigorous stability theoryfor social foraging of swarms. Moreover, in engineering appli-cations the sensing limitations of the agents can be overcomewith technologies such as Global Positioning System (GPS).Note also that with restrictions on the initial relative positions

of the swarm members it may be possible to obtain local sta-bility results for swarms in which the individuals have limitedsensing (and therefore attraction) range. However, this will notbe considered here. One can think of our swarm as a (prede-fined) team (or a set) of agents (robots) which know each othersrelative position and are required to gather around targets andavoid obstacles or threats.

The objective here is to analyze the qualitative properties ofthe collective behavior (motions in n-space) of the individuals.To this end we define the center of the swarm as x = 1

M ∑Mi=1 xi.

Then, the motion of the center is given by

˙x = − 1M

M

∑i=1

∇xi σ(xi)− 1M

M

∑i=1

M

∑j=1, j 6=i

g(xi− x j)

= − 1M

M

∑i=1

∇xi σ(xi), (3)

since we have

1M

M

∑i=1

M

∑j=1, j 6=i

g(xi− x j) = 0,

which follows from the fact that g(·) are odd functions of theform of Eq. (2) and g(xi− x j) =−g(x j− xi) for all pairs (i, j).The above equation implies that the center of the swarm movesalong the average of the gradient of the profile evaluated at thecurrent positions of the individuals. However, this does not nec-essarily mean that it will converge to a minimum. Moreover,this does not imply anything about the motions of the individ-uals. In fact, the convergence properties of the swarm to mini-mum (or critical) points of the profile depends on the propertiesof the profile.

One issue to note here is that as in [23] it is possible to viewthe foraging (and therefore social foraging) problem here asa distributed optimization problem (in which each individualis individually searching for the minimum) or optimal controlproblem, where the objective is to find the “optimal” controlpolicy or search strategy that will maximize, for instance, theenergy intake per time spent foraging. Here, we are not con-cerned with this problem. We specify the search strategy, whichis a distributed gradient search, and are concerned with stabilityor convergence properties of the strategy. Still, however, it is anoptimization or distributed function minimization problem andtherefore, our results here have some relevance to the optimiza-tion literature. Note that in nature there are many species with avariety of foraging or search strategies; some of these are mostcertainly not gradient-based and hence lie outside the scope ofthis work.

Note that the collective behavior in Eq. (3) has a kind of av-eraging (filtering or smoothing) effect. This may be importantif the σ-profile is a noisy function (or there is a measurementerror or noise in the system as discussed in [1], [2]). In otherwords, if the σ-profile were a “noisy function” and the indi-viduals were moving individually (without inter-individual at-traction/repulsion), then they could get stuck at a local minima,whereas if they swarm, since they are moving collectively, theother individuals will “pull” them out of such local minima.

4

This in turn will lead to the fact that the individuals will per-form better collectively (i.e., due to swarming) as seen in somebiological examples [1], [2].

In this article we will consider attraction/repulsion func-tions which are continuous and have linear attraction, i.e.,ga(‖xi−x j‖) = a for some a > 0 and all ‖xi−x j‖, and boundedrepulsion, i.e., gr(‖xi− x j‖)‖xi− x j‖ ≤ b for some b > 0 andall ‖xi− x j‖. The continuity assumption is needed in order toguarantee the existence and uniqueness of the solutions of thesystem. This assumption leads to the fact that g(·) vanishes atthe origin and brings a concern about collisions between the in-dividuals. However, by setting the magnitude of the repulsionhigh enough it is possible to avoid collisions at the expense ofgetting a larger swarm size. Another possibility is to choosegr(·) such that gr(‖xi− x j‖)‖xi− x j‖ → ∞ as ‖xi− x j‖ → 0 aswill be discussed later. One function that satisfies these condi-tions is [48]

g(y) =−y[

a−bexp(

−‖y‖2

c

)]

.

In the following sections we will first perform cohesion analy-sis for the swarm under conditions satisfied by several profilesfollowing which we will analyze the behavior of the swarm forseveral different profiles.

III. COHESION ANALYSIS

Before proceeding with analysis of the swarm behavior fordifferent profiles in this section we will analyze the cohesive-ness of the swarm under some general conditions satisfied byseveral profiles. To this end, we define the distance betweenthe position xi of individual i and the center x of the swarm asei = xi − x. The ultimate bound on the magnitude of ei willquantify the size of the swarm. Note that

ei = −∇xiσ(xi)−M

∑j=1, j 6=i

[

a−gr(‖xi− x j‖)]

(xi− x j)

+1M

M

∑j=1

∇x j σ(x j),

where gr(·) is such that the boundedness assumption is satis-fied. Defining a Lyapunov function as Vi = 1

2‖ei‖2 = 12 ei>ei,

and since ei = 1M ∑M

j=1(xi− x j) we obtain

Vi = −aM‖ei‖2 +M

∑j=1, j 6=i

gr(‖xi− x j‖)(xi− x j)>ei

−[

∇xi σ(xi)− 1M

M

∑j=1

∇x j σ(x j)

]>

ei. (4)

Now, if we can show that there is a constant ε such that forall ‖ei‖ > ε we have Vi < 0, then we will guarantee that inthat region ‖ei‖ is decreasing and eventually ‖ei‖ ≤ ε will beachieved. With this in mind we have two assumptions aboutthe profile. Note that these two assumptions do not have to besatisfied simultaneously.

Assumption 1: There exists a constant σ > 0 such that

‖∇yσ(y)‖ ≤ σ

for all y.Assumption 2: There exists a constant Aσ >−aM such that

[

∇xiσ(xi)− 1M

M

∑j=1

∇x j σ(x j)

]>

ei ≥ Aσ‖ei‖2

for all xi and x j.Note that Assumption 1 requires only that the gradient of

the profile to be bounded and is a very reasonable assumptionthat is satisfied with almost any realistic profile (e.g., plane andGaussian profiles). In contrast, Assumption 2 is a more restric-tive assumption. It requires the gradient of the profile at xi tohave a “large enough” component along ei so that the effect ofthe profile does not prevent swarm cohesion. Therefore, it maybe satisfied only by few profiles (e.g., a quadratic profile). Withthis in mind we state the following result.

Lemma 1: Consider the swarm described by the model inEq. (1) with g(·) as given in Eq. (2) with linear attraction (i.e.,ga(‖xi−x j‖) = a for some a > 0 and all ‖xi−x j‖) and boundedrepulsion, (i.e., gr(‖xi− x j‖)‖xi− x j‖ ≤ b for some b > 0 andall ‖xi− x j‖). Then, as t → ∞ we have xi(t)→ Bε(x(t)), where

Bε(x(t)) = y(t) : ‖y(t)− x(t)‖ ≤ ε

and• If Assumption 1 is satisfied, then

ε = ε1 =(M−1)

aM

[

b+2σM

]

,

• If Assumption 2 is satisfied, then

ε = ε2 =b(M−1)

aM +Aσ.

Proof: Case 1: From the Vi equation we obtain

Vi ≤ −aM‖ei‖2 +M

∑j=1, j 6=i

gr(‖xi− x j‖)‖xi− x j‖‖ei‖

+

∥

∥

∥

∥

∥

∇xi σ(xi)− 1M

M

∑j=1

∇x j σ(x j)

∥

∥

∥

∥

∥

‖ei‖

≤ −aM‖ei‖[

‖ei‖− b(M−1)

aM− 2σ(M−1)

aM2

]

,

which implies that as long as ‖ei‖> ε1 we have Vi < 0. Aboveto obtain the last inequality we used the fact that gr(‖xi −x j‖)‖xi− x j‖ ≤ b and the inequality

∥

∥

∥

∥

∥

∇xiσ(xi)− 1M

M

∑j=1

∇x j σ(x j)

∥

∥

∥

∥

∥

≤ 2σ(M−1)

M,

which follows from Assumption 1.Case 2: Similarly using Assumption 2 one can show that Vi

satisfies

Vi ≤−(aM +Aσ)‖ei‖[

‖ei‖− b(M−1)

aM +Aσ

]

.

5

Therefore, we conclude that as long as ‖ei‖> ε2 we have Vi < 0.

This result is important because it proves the cohesivenessof the swarm and provides a bound on the swarm size, definedas the radius of the hyperball centered at x(t) and containingall the individuals. Therefore, in order to analyze the collectivebehavior of the swarm we need to consider the motion of thecenter.

In species that engage in social foraging it has been observedthat the individuals in swarms desire to be close (but not tooclose) to other individuals. In the mean time, they want to findmore food. The balance between these desires determines thesize of the swarm (herd, flock or school). Our model capturesthis by having an inter-individual attraction/repulsion term andalso a term due to the environment (or the nutrient profile) af-fecting their motion. In the results above, the resulting swarmsizes depend on the inter-individual attraction/repulsion param-eters (a and b) and the parameters of the nutrient profile (σand Aσ). Moreover, the dependence on these parameters makesintuitive sense. Larger attraction (larger a) leads to a smallerswarm size, larger repulsion (larger b) leads to a larger swarmsize, larger σ (fast changing landscape) leads to a larger swarm.These concepts are present in foraging theory in biology andmodel the balance of the desire of the individuals to “stick to-gether” with the desire to “get more food” that was created byevolutionary forces. Note that for Assumption 2 to be satisfiedwe have the condition Aσ >−aM. The threshold Aσ =−aM isthe point at which the inter-individual attraction is not anymoreguaranteed to “hold the swarm together” since it is counterbal-anced by the repulsion from the profile. In other words, beyondthat threshold the repulsion from the center of the profile (i.e.,toxic substances) is so intense that the “desire to keep awayfrom the center of the profile” dominates (or is more plausiblethan) the “desire to stick together.” Therefore, if this conditionis not satisfied we cannot anymore guarantee cohesiveness ofthe swarm, i.e., it can happen that the swarm members movearbitrary far from each other. This helps to quantify the in-herent balance between the sometimes conflicting desires forswarm cohesiveness and for following cues from the environ-ment to find food. Such behavior can be seen in, for example,fish schools when a predator attacks the school. In that casethe fish move very fast in all directions away from the predator[55].

Note that the desire of the individuals to “stick together”depends on the inter-individual attraction parameter a and thenumber of individuals M. This is consistent with some biologi-cal swarms, where it has been observed that individuals are at-tracted more to larger (or more crowded) swarms (even thoughthat attraction may not be linearly proportional to the number ofindividuals). In nature the values of the parameters governingthe swarm motion have been tuned for millions of years by theevolutionary process.

One issue to note here is that as M gets large both ε1 andε2 approach constant values. This implies that for a large Mthe individuals will form a cohesive swarm of a constant sizeindependent of the number of the individuals and the character-istics of the profile. Unfortunately, this is not biologically veryrealistic.

The above result is an asymptotic result, i.e., xi(t)→Bε(x(t))as t → ∞. However, from stability theory we know that for anyε∗ > ε, xi(t) will enter Bε∗(x(t)) in a finite time. In other words,it can be shown that the swarm of any size a little larger than εwill be formed in a finite time.

In the following sections, we will analyze the behavior ofthe swarm on different profiles. In particular, we will considerplane, quadratic, Gaussian, and multimodal Gaussian profiles.

IV. MOTION ALONG A PLANE ATTRACTANT/REPELLENTPROFILE

In this section we assume that the profile is described by aplane equation of the form

σ(y) = a>σ y+bσ, (5)

where aσ ∈Rn and bσ ∈R. One can see that the gradient of the

profile is given by∇yσ(y) = aσ

and Assumption 1 holds with σ = ‖aσ‖. However, note alsothat

∇xiσ(xi)− 1M

M

∑j=1

∇x j σ(x j) = 0

for all i, implying that the last term in Eq. (4) vanishes. There-fore, for this profile the bound on the swarm size is given by

ε = εp =b(M−1)

aM<

ba.

Note also that for this case we have

˙x(t) =−aσ,

which implies that the center of the swarm will be moving withthe constant velocity vector −aσ (and eventually will divergetowards infinity where the minimum of the profile occurs).

The motions in this section can be viewed as a model of a for-aging herd that moves in a constant direction (while keeping itscohesiveness) with a constant speed such as the one consideredin [56]. Another view of the system in this section could be as amodel of a multi-agent system in which the autonomous agentsmove in a formation with a constant speed. In fact, transform-ing the system to ei coordinates we obtain

ei =M

∑j=1, j 6=i

g(ei− e j), i = 1, . . . ,M,

which is exactly the model of an aggregating swarm consid-ered in [48]. Therefore, all the results obtained in [48] applyfor ei. In particular, we have ei(t) → 0 as t → ∞. In otherwords, the swarm converges to a constant configuration or aformation (i.e., constant relative positions) that moves with aconstant speed in the direction of −aσ. The only drawback isthat for the given swarm model we cannot a priori specify theformation to be established. However, note that it is possibleto modify the swarm model such that the inter-individual at-traction/repulsion function g(·) are pair dependent i.e., gi, j(·).Then, by appropriate choice of each gi, j(·) one can achieve anydesired formation.

6

V. QUADRATIC ATTRACTANT/REPELLENT PROFILES

In this section, we will consider a quadratic profile given by

σ(y) =Aσ

2‖y− cσ‖2 +bσ, (6)

where Aσ ∈R, bσ ∈R, and cσ ∈Rn. Note that this profile has a

global extremum (either a minimum or a maximum dependingon the sign of Aσ) at y = cσ. Its gradient at a point y ∈ R

n isgiven by

∇yσ(y) = Aσ(y− cσ).

Assume that Aσ > −aM. Then, with few manipulations onecan show that for this profile Assumption 2 holds with strictequality. Therefore, the result of Lemma 1 holds with the bound

εq = ε2 =b(M−1)

aM +Aσ.

Now, let us analyze the motion of the center x. Substitutingthe gradient in the equation of motion of x given in Eq. (3) weobtain

˙x =−Aσ(x− cσ).

Defining the distance between the center x and the extremumpoint cσ as eσ = x− cσ, we have

eσ =−Aσeσ,

which implies that as t → ∞ we have eσ(t)→ 0 if Aσ > 0 andthat eσ(t)→∞ if Aσ < 0 and eσ(0) 6= 0. Therefore, we have thefollowing result.

Lemma 2: Consider the swarm described by the model inEq. (1) with g(·) as given in Eq. (2). Assume that the σ-profileof the environment is given by Eq. (6). As t → ∞ we have

• If Aσ > 0, then x(t) → cσ (i.e., the center of the swarmconverges to the global minimum cσ of the profile), or

• If Aσ < 0 and x(0) 6= cσ, then x(t) → ∞ (i.e., the centerof the swarm diverges from the global maximum cσ of theprofile).

Note that this result holds for any Aσ (i.e., we do not needthe assumption Aσ > −aM). Note also that for the case withAσ > 0 for any finite ε∗ > 0 (no matter how small) it can beshown that ‖x(t)−cσ‖< ε∗ is satisfied in a finite time. In otherwords, ‖x‖ enters any ε∗ neighborhood of cσ in a finite time. Incontrast, for the case with Aσ < 0 and x(0) 6= cσ for any D > 0(no matter how large) it can be shown that ‖x(t)− cσ‖ > D issatisfied in a finite time, implying that ‖x‖ exits any bounded D-neighborhood of cσ in a finite time. If Aσ < 0 and x(0) = cσ, onthe other hand, then x(t) = cσ for all t. In other words, for thiscase the swarm will be either “trapped” around the maximumpoint because of the inter-individual attraction (i.e., desire ofthe individuals to be close to each other) or will disperse in alldirections if the inter-individual attraction is not strong enough(i.e., Aσ <−aM). Note, however, that even if they disperse, thecenter x will not move and stay at cσ. Such a dispersal behaviorcan be seen in fish schools when attacked by a predator [55]. Inother words, for the fish the effect of the presence of a predatorcan be modeled by a large intensity repellent profile .

Here, we did not consider the Aσ = 0 case. This is because ifAσ = 0 then the profile is uniform everywhere and ∇yσ(y) = 0

for all y ∈ Rn. Therefore, the existence of the profile does not

affect the motion of the individuals and stability analysis is re-duced to the one described in [48], where nondrifting aggregat-ing swarms were considered and it was shown that the swarmwill cluster around the stationary center x.

Combining the results of Lemmas 1 and 2 together with theabove observations gives us the following result.

Theorem 1: Consider the swarm described by the model inEq. (1) with inter-individual attraction/repulsion function g(·)as given in Eq. (2) with linear attraction and bounded repulsion.Assume that the σ-profile of the environment is given by Eq. (6)and that Aσ >−aM. Then, the following hold

• If Aσ > 0, then for any ε∗ > εq all individuals i = 1, . . . ,M,will enter Bε∗(cσ) in a finite time,

• If Aσ < 0 and x(0) 6= cσ, then for any D < ∞ all individualsi = 1, . . . ,M, will exit BD(cσ) in a finite time.

This result is important because it gives finite time conver-gence (divergence) of all the individuals to nutrient rich (fromtoxic) regions of the profile.

Now, assume that instead of the profile in Eq. (6) we have aprofile which is a sum of quadratic functions. In other words,assume that the profile is given by

σ(y) =N

∑i=1

Aiσ

2‖y− ci

σ‖2 +bσ,

where Aiσ ∈ R, and ci

σ ∈ Rn for all i = 1, . . . ,N, and bσ ∈ R. Its

gradient at a point y is given by

∇yσ(y) =N

∑i=1

Aiσ(y− ci

σ).

Defining

Aσ =N

∑i=1

Aiσ

and

cσ =∑N

i=1 Aiσci

σ

∑Ni=1 Ai

σ

we obtain∇yσ(y) = Aσ(y− cσ),

which is exactly the same as above. The point cσ is the point ofthe unique extremum of the combined profile function. There-fore, the above results will directly transfer without any modi-fication. This also is true because it can be shown that

N

∑i=1

Aiσ

2‖y− ci

σ‖2 =Aσ

2‖y− cσ‖2 +C,

where C is a constant.Quadratic profiles are rather simple profiles and the results

in this section are intuitively expected. However, note also thatmore complicated profiles can be locally modeled (or approx-imated) as quadratic in regions near extremum points. In thefollowing sections, we will consider profiles which are not nec-essarily quadratic or even convex. Moreover, later we will allowthe profile to have multiple extremum points.

7

VI. GAUSSIAN ATTRACTANT/REPELLENT PROFILES

In this section, we consider profiles that are described by aGaussian-type of equation,

σ(y) =−Aσ

2exp(

−‖y− cσ‖2

lσ

)

+bσ, (7)

where Aσ ∈ R, bσ ∈ R, lσ ∈ R+, and cσ ∈ R

n. Note that thisprofile also has the unique extremum (either a global minimumor a global maximum depending on the sign of Aσ) at y = cσ.Its gradient is given by

∇yσ(y) =Aσ

lσ(y− cσ)exp

(

−‖y− cσ‖2

lσ

)

.

Calculating the time derivative of the center of the swarm byusing Eq. (3) one can obtain

˙x =− Aσ

Mlσ

M

∑i=1

(xi− cσ)exp(

−‖xi− cσ‖2

lσ

)

.

Compared to the quadratic case, here we cannot write ˙x as afunction of eσ = x− cσ. This is basically because of the non-linearity of the gradient of the profile. However, intuitively wewould expect that we still should be able to get some resultssimilar to the ones in the preceding section. To this end we notethat Assumption 1 is satisfied with

σ =|Aσ|√

2lσexp(

−12

)

.

Therefore, Lemma 1 holds and we know that as t → ∞ all theindividuals will converge to (and stay within) the

εG = ε1 =(M−1)

aM

[

b+|Aσ|M

√

2lσ

exp(

−12

)]

neighborhood of the (mobile) center x.Now, we have to analyze the motion of x in order to deter-

mine the overall behavior of the swarm.Lemma 3: Consider the swarm described by the model in

Eq. (1) with inter-individual attraction/repulsion function g(·)as given in Eq. (2). Assume that the σ-profile of the environ-ment is given by Eq. (7). Then, as t → ∞ we have

• If Aσ > 0, then ‖eσ(t)‖ ≤maxi=1,...,M ‖ei(t)‖, emax(t),• If Aσ < 0 and ‖eσ(0)‖ > emax(0) (here we assume that

xi(0) 6= x j(0) for all pairs of individuals (i, j), j 6= i,1 ≤i, j ≤M and therefore emax(0) > 0), then ‖eσ‖→ ∞.Proof: To start with, let Vσ = 1

2 e>σ eσ. Then, its derivativealong the motion of the swarm is given by

Vσ = − Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

(xi− cσ)>eσ

= − Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖2

− Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

ei>eσ,

where we used the fact that xi− cσ = ei + eσ.

a) Case 1: Aσ > 0:: Bounding Vσ from above we obtain

Vσ ≤ − Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖2

+Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖ei‖‖eσ‖

≤ − Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖

×

‖eσ‖−∑M

i=1 exp(

− ‖xi−cσ‖2

lσ

)

‖ei‖

∑Mi=1 exp

(

− ‖xi−cσ‖2

lσ

)

≤ − Aσ

Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖[

‖eσ‖− emax

]

,

where emax = maxi=1,...,M ‖ei‖. The above inequality impliesthat as long as ‖eσ(t)‖> emax(t), i.e., the minimum point cσ isoutside the swarm boundary, then the center of the swarm willbe moving toward it. Therefore, as t → ∞ we will asymptoti-cally have ‖eσ(t)‖ ≤ emax(t), i.e., cσ will be within the swarm.

b) Case 2: Aσ < 0:: With analysis similar to the case 1above it can be shown that

Vσ ≥ |Aσ|Mlσ

M

∑i=1

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖[

‖eσ‖− emax

]

,

which implies that we have Vσ > 0. In other words, if ‖eσ‖ >

emax, then ‖eσ‖ will increase. From Lemma 1 we have thatemax is decreasing. Therefore, since by hypothesis ‖eσ(0)‖ >

emax(0) we have that Vσ > 0 holds. Now, given any large butfixed D > 0 and ‖eσ(t)‖ ≤ D we have

exp(

−‖xi− cσ‖2

lσ

)

‖eσ‖[

‖eσ‖− emax

]

≥

exp(

− (D2 + ε23)

lσ

)

D[

D− ε3

]

> 0,

implying that

Vσ ≥|Aσ|lσ

exp(

− (D2 + ε2G)

lσ

)

D[

D− εG

]

> 0

from which using (a corollary to) the Chetaev Theorem [57] weconclude that ‖eσ‖ will exit the D-neighborhood of cσ.

Note that the result in Lemma 3 makes intuitive sense. Ifwe have a hole (i.e., a minimum) it guarantees that the indi-viduals will “gather” around it (as expected). If we have a hill(i.e., a maximum) and all the individuals are located on oneside of the hill, it guarantees that the individuals diverge fromit (as expected). If there is a hill, but the individuals are spreadaround it, then we cannot conclude neither convergence nor di-vergence. This is because it can happen that the swarm maymove to one side and diverge or the inter-individual attractionforces can be counterbalanced by the inter-individual repulsioncombined with the repulsion from the hill so that the swarmdoes not move away from the hill.

8

The above result (in Lemma 3) together with the result inLemma 1 allow us to state the following.

Theorem 2: Consider the swarm described by the model inEq. (1) with inter-individual attraction/repulsion function g(·)as given in Eq. (2) with linear attraction and bounded repul-sion. Assume that the σ-profile of the environment is given byEq. (7). Then, as t → ∞ we have

• If Aσ > 0, then all individuals i = 1, . . . ,M, will enter (andstay within) B2εG(cσ),

• If Aσ < 0 and ‖eσ(0)‖ ≥ emax(0), then all individuals i =1, . . . ,M, will exit BD(cσ) for any fixed D > 0.

For the case Aσ > 0 Lemma 1 states that the swarm will havea maximum size of εG, i.e., ‖eσ‖ ≤ εG for all i = 1, . . . ,M, andLemma 3 states that the swarm center will converge to the emaxand therefore to the εG neighborhood of cσ, i.e., ‖eσ‖ ≤ emax ≤εG. Combining these two bounds we obtain the 2εG in the firstcase in Theorem 2.

Theorem 2 is a parallel of Theorem 1. However, here we havea weaker result since for the Aσ > 0 case we cannot guaranteethat x(t)→ cσ. Moreover, the region around cσ in which theindividuals converge is larger (2εG) compared to the region inTheorem 1 (εq).

The drawback of this and the previous case is that both havea single extremum (which is either a minimum or a maximum)and the profile is relatively “uniform.” In the next section, wewill consider a profile for which this is not necessarily the case.

VII. MULTIMODAL GAUSSIAN ATTRACTANT/REPELLENTPROFILES

Now, we will consider a profile which is a combination ofGaussian profiles. In other words, we will consider the profilesgiven by

σ(y) =−N

∑i=1

Aiσ

2exp(

−‖y− ciσ‖2

liσ

)

+bσ, (8)

where ciσ ∈R

n, liσ ∈R

+, Aiσ ∈R for all i = 1, . . . ,N, and bσ ∈R.

Note that since the Aiσ’s can be positive or negative there can

be both hills and valleys leading to a “more irregular” profile.In [23], where social foraging was considered as an optimiza-tion process, a profile of this type was considered and conver-gence to minima of the profile was shown in simulation.

The gradient of the profile at a point y is given by

∇yσ(y) =N

∑i=1

Aiσ

liσ

(y− ciσ)exp

(

−‖y− ciσ‖2

liσ

)

.

Note that for this profile Assumption 1 is satisfied with

σ =N

∑i=1

|Aiσ|

√

2liσ

exp(

−12

)

.

Therefore, from Lemma 1 we have

εmG = ε1 =(M−1)

aM

[

b+1M

N

∑i=1|Ai

σ|√

2liσ

exp(

−12

)

]

as the bound on the swarm size. In other words, as t → ∞ wewill have xi(t)→ BεmG(x(t)), where εmG is as given above.

Using the profile gradient equation we can write the equationof motion of the swarm center x as

˙x =− 1M

N

∑j=1

A jσ

l jσ

M

∑i=1

(xi− c jσ)exp

(

−‖xi− c j

σ‖2

l jσ

)

.

As one can see, it is not obvious from this equation how the cen-ter x will move. Therefore, for this type of profile it is not easyto prove convergence of the individuals to a minimum of theprofile for the general case. However, under some conditionsit is possible to prove convergence to the vicinity of a partic-ular c j

σ (if c jσ is the center of a valley) or divergence from the

neighborhood of a particular c jσ (if c j

σ is the center of a hill).Lemma 4: Consider the swarm described by the model in

Eq. (1) with g(·) as given in Eq. (2). Assume that the σ-profileof the environment is given by Eq. (8). Moreover, assume thatfor some k,1≤ k ≤ N, we have

‖xi(0)− ckσ‖ ≤ hk

√

lkσ

for some hk and for all i = 1, . . . ,M, and that for all j =1, . . . ,N, j 6= k we have

‖xi(0)− c jσ‖ ≥ h j

√

l jσ

for some h j, j = 1, . . . ,N, j 6= k and for all i = 1, . . . ,M. (Thismeans that the swarm is near ck

σ and far from other c jσ, j 6= k.)

Moreover, assume that

Akσ

√

lkσ

hk exp(

−h2k)

>1α

N

∑j=1, j 6=k

|A jσ|

√

l jσ

h j exp(

−h2j)

,

is satisfied for some 0 < α < 1. Then, for ekσ = x− ck

σ as t → ∞we will have

• If Akσ > 0, then ‖ek

σ(t)‖ ≤ εmG +αhk√

lkσ

• If Akσ < 0 and ‖ek

σ(0)‖ ≥ emax(0) + αhk√

lkσ, then

‖ekσ(t)‖ ≥ εmG +αhk

√

lkσ, where emax = maxi=1,...,M ‖ei‖.

Proof: Let V kσ = 1

2 ek>σ ek

σ be the Lyapunov function.Case 1: Ak

σ > 0: Taking the derivative of V kσ along the motion

of the swarm one can show that

V kσ ≤ − Ak

σMlk

σ

M

∑i=1

exp(

−‖xi− ck

σ‖2

lkσ

)

‖ekσ‖

×[

‖ekσ‖− emax−αhk

√

lkσ

]

,

which implies that we have V kσ < 0 as long as ‖ek

σ‖ > emax +

αhk√

lkσ, and from Lemma 1 we know that as t → ∞ we have

emax(t)≤ εmG.Case 2: Ak

σ < 0: Similar to above, for this case it can beshown that

V kσ ≥ |Ak

σ|Mlk

σ

M

∑i=1

exp(

−‖xi− ck

σ‖2

lkσ

)

‖ekσ‖

×[

‖ekσ‖− emax−αhk

√

lkσ

]

,

9

which implies that if ‖ekσ‖ > emax + αhk

√

lkσ, then we have

Vσ > 0. In other words, ‖ekσ‖ will increase. From Lemma 1

we have that emax is decreasing. Therefore, since by hypothe-sis ‖ek

σ(0)‖ > emax(0)+ αhk√

lkσ we have that Vσ > 0 holds at

t = 0. Now, consider the boundary ‖ekσ‖= εmG +hk

√

lkσ. It can

be shown that on the boundary we have

Vσ ≥|Ak

σ|hk(1−α)(

εmG +hk√

lkσ

)

exp(

−h2k

)

√

lkσ

> 0,

from which once again using (a corollary to) the ChetaevTheorem we conclude that ‖ek

σ‖ will exit the εmG + hk√

lkσ-

neighborhood of ckσ.

Now, as in the preceding section we can combine this result(i.e., Lemma 4) together with Lemma 1 to obtain the followingtheorem.

Theorem 3: Consider the swarm described by the model inEq. (1) with inter-individual attraction/repulsion function g(·)as given in Eq. (2) with linear attraction and bounded repul-sion. Assume that the σ-profile of the environment is given byEq. (8). Assume that the conditions of Lemma 4 hold. Then, ast → ∞ all individuals will

• Enter the hyperball Bε5(ckσ), where ε5 = 2εmG + αhk

√

lkσ,

if Akσ > 0, or

• Leave the hk√

lkσ-neighborhood of ck

σ, if Akσ < 0.

The only drawback of the above result is that we need

2εmG +αhk

√

lkσ < hk

√

lkσ

in order for the result to make sense. This implies that we need

εmG <

(

1−α2

)

hk

√

lkσ

which sometimes may not be easy to satisfy. However, oneissue to note is that εmG is a very conservative bound. In real-ity, the actual size of the swarm is typically much smaller thanthe bound. Therefore, effectively, εmG can be replaced withemax(∞) < εmG and it may be easier to satisfy the above condi-tion.

VIII. TIGHTER BOUNDS, UNBOUNDED REPULSION, ANDCOLLISION AVOIDANCE

The bounds on the swarm size obtained in Lemma 1 arerather conservative. In this section, we will show that for atleast some of the cases it is possible to obtain a tighter boundon the root-mean-square of the distances of the individuals totheir center. Moreover, we will allow for unbounded repulsionfunctions which could be used to guarantee collision avoidanceor, in other words, will prevent two individuals to occupy thesame space (an issue that was overlooked before).

The repulsion functions that we consider are assumed to sat-isfy [50](i) as ‖xi − x j‖ → 0, we have gr(‖xi − x j‖)‖xi − x j‖ → ∞

(hence unbounded repulsion),(ii) gr(‖xi− x j‖)≤ b

‖xi−x j‖2 , for some b > 0.

Note that provided that xi(0) 6= x j(0) for all pairs (i, j), j 6= i,condition (i) above will guarantee that xi(t) 6= x j(t) for all pairs(i, j), j 6= i and for all t ≥ 0. Condition (ii), on the other hand,brings some restrictions on the growth of the repulsion and willbe useful in the Lyapunov analysis.

Now, consider the Lyapunov function

V =M

∑i=1

Vi +MVσ,

where Vi are the terms quantifying cohesion and are given byVi = 1

2‖ei‖2 and Vσ represents the distance to a minimum of theprofile and is

• Vσ = 0 for a plane profile,• Vσ = 1

2‖eσ‖2 for quadratic and Gaussian profiles, and• Vσ = 1

2‖ekσ‖2 for some k,1 ≤ k ≤ N for the multimodal

Gaussian profile.Taking the derivative of V we obtain

V = −aMM

∑i=1‖ei‖2 +

M

∑i=1

M

∑j=1, j 6=i

gr(‖xi− x j‖)(xi− x j)>ei

−M

∑i=1

[

∇xi σ(xi)− 1M

M

∑j=1

∇x j σ(x j)

]>

ei

−M

∑i=1

∇xi σ>(xi)eσ.

Consider the third term on the right hand side (denoted withA below). With few manipulations it can be shown that we have

A =M

∑i=1

[

∇xi σ(xi)− 1M

M

∑j=1

∇x j σ(x j)

]>

ei

=M

∑i=1

∇xi σ>(xi)(xi− cσ)−M

∑i=1

∇xi σ>(xi)eσ.

Now, consider the second term on the right hand side of theabove equation (denoted with B below). After some manipula-tions it can be shown that we have

B =M

∑i=1

M

∑j=1, j 6=i

gr(‖xi− x j‖)(xi− x j)>ei

=12

M

∑i=1

M

∑j=1, j 6=i

gr(‖xi− x j‖)‖xi− x j‖2.

Substituting the values of A and B in the V equation and aftersome cancellations we have

V = −aMM

∑i=1‖ei‖2 +

12

M

∑i=1

M

∑j=1, j 6=i

gr(‖xi− x j‖)‖xi− x j‖2

−M

∑i=1

∇xi σ>(xi)(xi− cσ).

Then, from condition (ii) above we have

V ≤−aMM

∑i=1‖ei‖2 +

12

M(M−1)b−M

∑i=1

∇xiσ>(xi)(xi− cσ).

10

Now, note that the third term on the right hand side of the aboveequation (which we denote with C below) depends on the (gra-dient of the) profile and can be both negative and positive. Fordifferent profiles we have

• plane:

∇xi σ>(xi) = 0⇒C =M

∑i=1

∇xi σ>(xi)(xi− cσ) = 0,

• quadratic:

∇xi σ>(xi) = Aσ(xi− cσ)>⇒C = Aσ

M

∑i=1‖xi− cσ‖2

,

• Gaussian:

∇xi σ>(xi) =Aσ

lσexp(

−‖xi− cσ‖2

lσ

)

(xi− cσ)>⇒

C =Aσ

lσ

M

∑i=1‖xi− cσ‖2 exp

(

−‖xi− cσ‖2

lσ

)

,

• multimodal Gaussian:

∇xi σ>(xi) =N

∑j=1

A jσ

l jσ

exp

(

−‖xi− c j

σ‖2

l jσ

)

(xi− c jσ)>⇒

C =M

∑i=1

(xi− ckσ)>

N

∑j=1

A jσ

l jσ

(xi− c jσ)exp

(

−‖xi− c j

σ‖2

l jσ

)

,

for some 1≤ k ≤ N.Consider the cases in which we have Aσ > 0 for the quadratic

and Gaussian profiles, and Akσ > 0 for the multimodal Gaussian

profile (i.e., the attractant profile cases). Then, the last term inthe V equation is non-positive for the first three profiles, namelythe plane, quadratic and the gaussian profiles. For the multi-modal Gaussian profile after some tedious but straightforwardmanipulation it is guaranteed to be non-positive provided thatthe condition

Akσ

lkσ‖xi− ck

σ‖exp(

−‖xi− ck

σ‖2

lkσ

)

≥

N

∑j=1, j 6=k

|A jσ|

l jσ‖xi− c j

σ‖exp

(

−‖xi− c j

σ‖2

l jσ

)

, (9)

is satisfied for all i. Note that this condition is very similar tothe condition in Lemma 4. Then, for these cases we can deducethat as t → ∞ we will have

1M−1

M

∑i=1‖ei‖2 ≤ b

2a.

In other words, for the root-mean-square of the distances of theindividuals to the center we have

erms =

√

1M

M

∑i=1‖ei‖2 ≤

√

b2a

,

which is smaller than the bounds found in Lemma 1. Therefore,we have proved the following result.

Lemma 5: Consider the swarm described by the model inEq. (1) with an attraction/repulsion function g(·) as given inEq. (2) with linear attraction and (possibly unbounded) repul-sion satisfying gr(‖xi− x j‖) ≤ b

‖xi−x j‖2 , for some b > 0. As-sume that the σ-profile is one of the following

• A plane profile in Eq. (5),• A quadratic profile in Eq. (6) with Aσ > 0, or• A Gaussian profile in Eq. (7) with Aσ > 0, or• A multimodal Gaussian profile in Eq. (8) with the condi-

tion in Eq. (9) satisfied for some k with Akσ > 0 and for all

i.Then, as t → ∞ we will have

erms ≤√

b2a

.

One issue to note here is that the above result holds for all re-pulsion functions satisfying condition (ii) above. Therefore, itholds also for the bounded repulsion functions considered ear-lier. The advantage of having unbounded repulsion functionsis that they guarantee the avoidance of collision, as mentionedbefore.

Note also that if we had kept the negative third term in thederivative of the Lyapunov equation above, then the conditionobtained for the negative definiteness of V would suggest abound on the distance of the center x to the minimum cσ (orck

σ in the multimodal Gaussian case). However, this bound isnot necessarily small. Nevertheless, all the results for differentprofiles obtained in the preceding sections still hold (with ap-propriate modifications on the bounds on the swarm size) evenwith the new unbounded repulsion functions.

Finally, note that for the cases excluded in the theorem wecannot guarantee the same bounds. However, it is possible toderive bounds (which may be slightly larger) for these casestoo. For example, one can show that for the Gaussian case withAσ < 0 we will have

εrms =

√

b2a

+|Aσ|aM

exp(−1).

IX. ANALYSIS OF INDIVIDUAL BEHAVIOR IN A COHESIVESWARM

The results in the previous sections specify whether theswarm will diverge or converge, and if it converges they specifyin which regions of the profile it will converge, together withbounds on the swarm size. The results above do not provideinformation about the ultimate behavior of the individuals. Inother words, they do not specify whether the individuals willeventually stop moving or will end up in oscillatory motionswithin the specified regions. In this section, we will investigatethe ultimate behavior of the individuals. In particular, we willanalyze the ultimate behavior of the individuals in a quadraticprofile with Aσ > 0, a Gaussian profile with Aσ > 0, and in amultimodal Gaussian profile with the conditions of Lemma 4for the Ak

σ > 0 case satisfied. To this end, first, we define thestate x of the system as the vector of the positions of the swarm

11

members x = [x1>, . . . ,xM>]>. Let the invariant set of equilib-rium points be

Ωe = x : x = 0.

We will prove that for the above mentioned cases as t → ∞ thestate x(t) converges to Ωe, i.e., eventually all the individualsstop moving.

Theorem 4: Consider the swarm described by the model inEq. (1) with an attraction/repulsion function g(·) as given inEq. (2) with linear attraction and bounded repulsion. Assumethat the σ-profile is one of the following

• A quadratic profile in Eq. (6) with Aσ > 0, or• A Gaussian profile in Eq. (7) with Aσ > 0, or• A multimodal Gaussian profile in Eq. (8) with conditions

of Lemma 4 for the Akσ > 0 case satisfied.

Then, as t → ∞ we have the state x(t)→Ωe.Proof: Choose the generalized Lyapunov function J(x) as

the total artificial potential energy E in the system. In otherwords, choose J(x) as

J(x) =M

∑i=1

σ(xi)+12

M−1

∑i=1

M

∑j=i+1

[

a‖xi− x j‖2− Jr(‖xi− x j‖)]

whose gradient at xi is

∇xi J(x) = ∇xi σ(xi)+M

∑j=1, j 6=i

(xi−x j)[

a−gr(‖xi− x j‖)]

=−xi,

and its time derivative is given by

J(x) = [∇xJ(x)]> x =M

∑i=1

[∇xi J(x)]> xi =−M

∑i=1‖xi‖2 ≤ 0,

for all t ≥ 0. Now, note that for all the cases in the hypothesisof the theorem, we have J(x) bounded from below and the set

Ωc = x : J(x)≤ J(x(0))

is compact and positively invariant with respect to the motionsof the system. Then, we can apply the LaSalle’s InvariancePrinciple from which we conclude that as t →∞ the state x con-verges to the largest invariant subset of the set Ω ⊂ Ωc definedas

Ω = x : J(x) = 0= x : x = 0= Ωe.

Since each point in Ωe is an equilibrium, Ωe is an invariant setand this proves the result.

One issue to note here is that for the cases excluded in theabove theorem, i.e., for the plane profile, quadratic profile withAσ < 0, Gaussian profile with Aσ < 0, and the multimodal Gaus-sian profile for Aσ < 0 case or not necessarily satisfying the con-ditions of Lemma 4, the set Ωc may not be compact. Therefore,we cannot apply the LaSalle’s Invariance Principle. Moreover,since they are (possibly) diverging, intuitively we do not expectthem to stop their motion. Furthermore, note that for the planeprofile we have Ωe = /0. In other words, there is no equilibriumfor the swarm moving in a plane profile.

X. SIMULATION EXAMPLES

In this section we will provide some simulation examples toillustrate the theory developed in the preceding sections. Wechose an n = 2 dimensional space for ease of visualization ofthe results and used the region [0,30]× [0,30] in the space. Inall the simulations performed below we used M = 11 individ-uals. As parameters of the attraction/repulsion function g(·) inEq. (2) we used a = 0.01, b = 0.4, and c = 0.01 for most ofthe simulations and a = 0.1 for some of them. We performedsimulations for all the profiles discussed in this article.

The first plot shown in Figure 1 is for a plane profile withaσ = [0.1,0.2]> for the plots on the left and aσ = [0.5,1]> forthose on the right. One easily can see that in both of the cases,

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

Fig. 1. The response for a plane profile.

as expected, the swarm moves along the negative gradient −aσexiting the simulation region toward unboundedness. Note thatinitially for the case aσ = [0.1,0.2]> some of the individualsmove in a direction opposite to the negative gradient. This isbecause the inter-individual attraction is much stronger than theintensity of the profile. In contrast, for the aσ = [0.5,1]> case,the intensity of the profile is high enough to dominate the inter-individual attraction. This, of course, does not mean that theswarm will not aggregate. As they move they will eventuallyaggregate as was shown in the preceding sections. We alsoshow the plots of the swarm centers. Note that the motion ofthe centers is similar for both of the cases as expected from theanalysis in the preceding sections.

The next result is for the quadratic profile as shown in Fig-ure 2. We chose a profile with extremum at cσ = [20,20]>

and magnitude Aσ = ±0.02. The two plots on the left of thefigure show the paths of the individuals and the center of theswarm for the case Aσ > 0, whereas those on the right are forthe Aσ < 0 case. Once more, we observe that the results supportthe analysis of preceding sections. Note also that the center xof the swarm converges to the minimum of the profile cσ forthe Aσ > 0 case and diverges from the maximum for the Aσ < 0case.

12

0 10 20 300

5

10

15

20

25

30

The paths of the individuals (Aσ > 0)

0 10 20 300

5

10

15

20

25

30

The path of the center (Aσ > 0)

0 10 20 300

5

10

15

20

25

30

The paths of the individuals (Aσ < 0)

0 10 20 300

5

10

15

20

25

30

The path of the center (Aσ < 0)

Fig. 2. The response for a quadratic profile.

Results of a similar nature were obtained also for the Gaus-sian profile as shown in Figure 3. Once more we chose

0 10 20 300

5

10

15

20

25

30

The paths of the individuals (Aσ > 0)

10 15 20 25 3010

15

20

25

30

The path of the center (Aσ > 0)

0 10 20 300

5

10

15

20

25

30

The paths of the individuals (Aσ < 0)

5 10 15 20 25 3010

15

20

25

30

The path of the center (Aσ < 0)

Fig. 3. The response for a Gaussian profile.

cσ = [20,20]> as the extremum of the profile. The other pa-rameters of the profile were chosen to be Aσ =±2 and lσ = 20.Note that for the Aσ > 0 case, even though in theory we couldnot prove that x(t)→ cσ, in simulations we observe that this isapparently the case. This was happening systematically in allthe simulations that we performed.

In the simulation examples for the multimodal Gaussian pro-file we used the profile shown in Figure 4, which has severalminima and maxima. The global minimum is located at [15,5]>

with a magnitude of 4 and a spread of 10. The plot in Figure 5shows two example runs with initial member positions nearbya local minimum and show convergence of the entire swarmto that minimum. The attraction parameter a was chosen tobe a = 0.01 for this case. Figure 6, on the other hand, illus-

Fig. 4. The multimodal Gaussian profile.

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

Fig. 5. The response for a multimodal Gaussian profile (initial positions closeto a minimum).

trates the case in which we increased the attraction parameterto a = 0.1. You can see that the attraction is so strong thatthe individuals climb gradients to form a cohesive swarm. Forthis and similar cases, the manner in which the overall swarmwill behave (where it will move) depends on the initial posi-tion of the center x of the swarm. For these two runs the centerhappened to be located on regions which caused the swarm todiverge. For some other simulation runs (not presented here)with different initial conditions the entire swarm converges toeither a local or global minima. Figure 7 shows two runs forwhich we decreased the attraction parameter again to a = 0.01and initialized the swarm member positions all over the region.For both of the simulations you can see that the swarm fails toform a cohesive cluster since the initial positions of the indi-viduals are such that they move to a nearby local minima andthe attraction is not strong enough to “pull them out” of thesevalleys. This causes formation of several groups or clusters of

13

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

Fig. 6. The response for a multimodal Gaussian profile.

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

0 10 20 300

5

10

15

20

25

30The paths of the individuals

0 10 20 300

5

10

15

20

25

30The path of the center

Fig. 7. The response for a multimodal Gaussian profile.

individuals at different locations of the space. For these reasons,the center x of the swarm does not converge to any minimum.Note, however, that this is expected since for this case the con-ditions of Theorem 3 are not satisfied. Nevertheless, the resultof Lemma 1 still holds. The only issue is that εmG is large andcontains all the region within which all the individuals eventu-ally remain. Finally, note also that during their motion to thegroups, the individuals try to avoid climbing gradients and thisresults in motions resembling the motion of individuals in realbiological swarms.

XI. CONCLUDING REMARKS

In this article we developed a simple model of swarming inthe presence of an attractant/repellent profile and analyzed itsstability properties for different profiles. Our model can beviewed as a model for stable social foraging of swarms in a pro-file of nutrients. We showed collective convergence to more fa-

vorable regions of the profile (i.e., the regions with higher con-centration of nutrients) and diverge from unfavorable regions(i.e., the regions with higher concentration of toxic substances).The bounds on the swarm size and the distance of the swarmcenter from the minimum points of the profile illustrate the ba-sic concepts in foraging theory of balance between the desire tostay with the swarm and the desire to find more food. Note thatthe model that we presented here directly addresses the problemof coordination of agents and interactions with the environmentbased on simple potentials. Therefore, even though we get ourinspiration from biology, we believe that our work is a contribu-tion to the multi-agent coordination literature. In fact, note thatin some applications, such as undersea explorations by a groupof robots, the agents may need to follow the gradient of somesubstance [36], which is the problem considered here, and theresults of this article are directly applicable.

Our model is essentially a kinematic model illustrating andproviding proof for multi-agent aggregation. It can serve as astarting point for engineers who need to design a multi-agentsystem that posesses such characteristics. For example, the tra-jectories generated by our model can be used as reference tra-jectories for robots (or other agents) to follow/track if there isa need for a group of robots to perform aggregating behaviorin an environment containing targets and threats/obstacles. Forthis case, the threats (to be avoided) are analogous to the toxicsubstances and the targets (to be moved towards) are analogousto food. Then, the problem of the system designer is reduced todeveloping a controller to guarantee trajectory following giventhe specific (actual) robot dynamics. For this reason, our resultsin this article serve as a “proof of concept” for such operation- an aggregating behavior of agents moving in an environmentwith targets and obstacles. Note also that the swarm model dis-cussed here can be viewed as performing distributed optimiza-tion using a distributed gradient method.

As pointed out in [23] foraging has important relations tocontrol and automation. Foraging strategies in biological crea-tures have been “designed” and “tested” by evolution for mil-lions of years. By studying, understanding, and modelingsuch behavior we may be able to gain ideas for develop-ing distributed coordination and control strategies for cooper-ative team behavior of multi-agent robotic systems such as un-manned undersea, land or air vehicles. For example, there is agrowing interest in the development of distributed coordinationand control strategies for uninhabited autonomous air vehicles(UAAVs) [23] moving on a landscape where there are mobilethreats/targets each with “target priority” and “thread severity.”Such a problem can clearly be viewed as a foraging problem(although not of the type discussed in this article), and the ideasfrom social foraging of biological creatures can be helpful forsolving it. In the light of this, the results in this article can beviewed as an initial step towards developing a comprehensivetheory for stable social foraging of swarms as well as transfer-ring the ideas from social foraging to the control literature.

Possible future extensions of the work here could be exten-sion to the case of time-varying attractant/repellent profiles.In engineering context this will correspond to a dynamicallychanging environment or environment containing moving tar-gets or threats (a very relevant and important problem). Another

14

possible extension is to consider swarm members (or agents)which have limited sensing range and analyze stability (cohe-sion) under these conditions. For such a model it may be possi-ble to prove (under some conditions) local stability (cohesion)results. The emergent behavior of such a model will (probably)be biologically more realistic.

REFERENCES

[1] D. Grunbaum, “Schooling as a strategy for taxis in a noisy environment,”in Animal Groups in Three Dimensions, J. K. Parrish and W. M. Hamner,Eds., pp. 257–281. Cambridge Iniversity Press, 1997.

[2] D. Grunbaum, “Schooling as a strategy for taxis in a noisy environment,”Evolutionary Ecology, vol. 12, pp. 503–522, 1998.

[3] C. M. Breder, “Equations descriptive of fish schools and other animalaggregations,” Ecology, vol. 35, no. 3, pp. 361–370, 1954.

[4] K. Warburton and J. Lazarus, “Tendency-distance models of social co-hesion in animal groups,” Journal of Theoretical Biology, vol. 150, pp.473–488, 1991.

[5] A. Okubo, “Dynamical aspects of animal grouping: swarms, schools,flocks, and herds,” Advances in Biophysics, vol. 22, pp. 1–94, 1986.

[6] D. Grunbaum and A. Okubo, “Modeling social animal aggregations,” inFrontiers in Theoretical Biology, vol. 100 of Lecture Notes in Biomathe-matics, pp. 296–325. Springer-Verlag, New York, 1994.

[7] A. Mogilner and L. Edelstein-Keshet, “A non-local model for a swarm,”Journal of Mathematical Biology, vol. 38, pp. 534–570, 1999.

[8] R. Durrett and S. Levin, “The importance of being discrete (and spatial),”Theoretical Population Biology, vol. 46, pp. 363–394, 1994.

[9] S. Gueron and S. A. Levin, “The dynamics of group formation,” Mathe-matical Biosciences, vol. 128, pp. 243–264, 1995.

[10] J. K. Parrish, S. V. Viscido, and D. Grunbaum, “Self-organized fishschool: An examination of emergent properties,” Biol. Bull., vol. 202,pp. 296–305, June 2002.

[11] J. K. Parrish and W. M. Hamner, Eds., Animal Groups in Three Dimen-sions, Cambridge University Press, 1997.

[12] S. Camazine, J.-L. Deneubourg, N. R. Franks, J. Sneyd, G. Theraulaz,and E. Bonabeau, Self-Organization in Biological Systems, PrincetonUniversity Press, April 2001.

[13] L. Edelstein-Keshet, Mathematical Models in Biology, Brikhauser Math-ematics Series. The Random House, New Yourk, 1989.

[14] J. D. Murray, Mathematical Biology, Springer-Verlag, New Yourk, 1989.[15] T. Vicsek, A. Czirok, E. Ben-Jacob, I. Cohen, and O. Shochet, “Novel

type of phase transition in a system of self-driven particles,” PhysicalReview Letters, vol. 75, no. 6, pp. 1226–1229, August 1995.

[16] A. Czirok, E. Ben-Jacob, I. Cohen, and T. Vicsek, “Formation of complexbacterial colonies via self-generated vortices,” Physical Review E, vol. 54,no. 2, pp. 1791–1801, August 1996.

[17] A. Czirok, H. E. Stanley, and T. Vicsek, “Spontaneously ordered motionof self-propelled partciles,” Journal of Physics A: Mathematical, Nuclearand General, vol. 30, pp. 1375–1385, 1997.

[18] A. Czirok and T. Vicsek, “Collective behavior of interacting self-propelled particles,” Physica A, vol. 281, pp. 17–29, 2000.

[19] N. Shimoyama, K. Sugawa, T. Mizuguchi, Y. Hayakawa, and M. Sano,“Collective motion in a system of motile elements,” Physical ReviewLetters, vol. 76, no. 20, pp. 3870–3873, May 1996.

[20] H. Levine and W.-J. Rappel, “Self-organization in systems of self-propelled particles,” Physical Review E, vol. 63, no. 1, pp. 017101–1–017101–4, January 2001.

[21] D.W. Stephens and J.R. Krebs, Foraging Theory, Princeton Univ. Press,Princeton, NJ, 1986.

[22] E. Bonabeau, M. Dorigo, and G. Theraulaz, Swarm Intelligence: FromNatural to Artificial Systems, Oxford Univ. Press, NY, 1999.

[23] K.M. Passino, “Biomimicry of bacterial foraging for distributed opti-mization and control,” IEEE Control Systems Magazine, vol. 22, no. 3,pp. 52–67, June 2002.

[24] J. Kennedy and R. C. Eberhart, Swarm Intelligence, Morgan KaufmannPublisher, 2001.

[25] M. Clerc and J. Kennedy, “The particle swarm—explosion, stability, andconvergence in a multidimensional complex space,” IEEE Trans. on Evo-lutionary Computation, vol. 6, no. 1, pp. 58–73, February 2002.

[26] M. M. Polycarpou, Y. Yang, and K. M. Passino, “Cooperative controlof distributed multi-agent systems,” to appear in IEEE Control SystemsMagazine.

[27] F. Giulietti, L. Pollini, and M. Innocenti, “Autonomous formation flight,”IEEE Control Systems Magazine, vol. 20, no. 6, pp. 34–44, December2000.

[28] T. Balch and R. C. Arkin, “Behavior-based formation control for multi-robot teams,” IEEE Trans. on Robotics and Automation, vol. 14, no. 6,pp. 926–939, December 1998.

[29] I. Suzuki and M. Yamashita, “Distributed anonymous mobile robots: For-mation of geometric patterns,” SIAM Journal on Computing, vol. 28, no.4, pp. 1347–1363, 1999.

[30] H. Yamaguchi, “A cooperative hunting behavior by mobile-robot troops,”The International Journal of Robotics Research, vol. 18, no. 8, pp. 931–940, September 1999.

[31] J. H. Reif and H. Wang, “Social potential fields: A distributed behavioralcontrol for autonomous robots,” Robotics and Autonomous Systems, vol.27, pp. 171–194, 1999.

[32] J. P. Desai, J. Ostrowski, and V. Kumar, “Controlling formations ofmultiple mobile robots,” in Proc. of IEEE International Conference onRobotics and Automation, Leuven, Belgium, May 1998, pp. 2864–2869.

[33] M. Egerstedt and X. Hu, “Formation constrained multi-agent control,”IEEE Trans. on Robotics and Automation, vol. 17, no. 6, pp. 947–951,December 2001.

[34] P. Ogren, M. Egerstedt, and X. Hu, “A control Lyapunov function ap-proach to multi-agent coordination,” in Proc. of Conf. Decision Contr.,Orlando, FL, December 2001, pp. 1150–1155.

[35] N. E. Leonard and E. Fiorelli, “Virtual leaders, artificial potentials and co-ordinated control of groups,” in Proc. of Conf. Decision Contr., Orlando,FL, December 2001, pp. 2968–2973.

[36] R. Bachmayer and N. E. Leonard, “Vehicle networks for gradient descentin a sampled environment,” in Proc. of Conf. Decision Contr., Las Vegas,Nevada, December 2002, pp. 112–117.

[37] K. Jin, P. Liang, and G. Beni, “Stability of synchronized distributed con-trol of discrete swarm structures,” in Proc. of IEEE International Confer-ence on Robotics and Automation, San Diego, California, May 1994, pp.1033–1038.

[38] G. Beni and P. Liang, “Pattern reconfiguration in swarms—convergenceof a distributed asynchronous and bounded iterative algorithm,” IEEETrans. on Robotics and Automation, vol. 12, no. 3, pp. 485–490, June1996.

[39] Y. Liu, K. M. Passino, and M. Polycarpou, “Stability analysis of one-dimensional asynchronous swarms,” in Proc. American Control Conf.,Arlington, VA, June 2001, pp. 716–721.

[40] Y. Liu, K. M. Passino, and M. Polycarpou, “Stability analysis of one-dimensional asynchronous mobile swarms,” in Proc. of Conf. DecisionContr., Orlando, FL, December 2001, pp. 1077–1082.

[41] V. Gazi and K. M. Passino, “Stability of a one-dimensional discrete-timeasynchronous swarm,” in Proc. of the joint IEEE Int. Symp. on Intelli-gent Control/IEEE Conf. on Control Applications, Mexico City, Mexico,September 2001, pp. 19–24.

[42] Dimitri P. Bertsekas and John N. Tsitsiklis, Parallel and Distributed Com-putation: Numerical Methods, Athena Scientific, Belmont, MA, 1997.

[43] J. Bender and R. Fenton, “On the flow capacity of automated highways,”Transport. Sci., vol. 4, pp. 52–63, February 1970.

[44] D. Swaroop, J. K. Hedrick, C. C. Chien, and P. Ioannou, “A comparison ofspacing and headway control laws for automatically controlled vehicles,”Vehicle System Dynamics, vol. 23, pp. 597–625, 1994.

[45] S. Darbha and K. R. Rajagopal, “Intelligent cruise control systems andtraffic flow stability,” Transportation Research Part C, vol. 7, pp. 329–352, 1999.

[46] Y. Liu, K. M. Passino, and M. M. Polycarpou, “Stability analysis ofm-dimensional asynchronous swarms with a fixed communication topol-ogy,” in Proc. American Control Conf., Anchorage, Alaska, May 2002,pp. 1278–1283.

[47] Y. Liu, K. M. Passino, and M. M. Polycarpou, “Stability analysis ofm-dimensional asynchronous swarms with a fixed communication topol-ogy,” IEEE Trans. on Automatic Control, vol. 48, no. 1, pp. 76–95, Jan-uary 2003.

[48] V. Gazi and K. M. Passino, “Stability analysis of swarms,” in Proc.American Control Conf., Anchorage, Alaska, May 2002, pp. 1813–1818.

[49] V. Gazi and K. M. Passino, “Stability analysis of swarms,” IEEE Trans.on Automatic Control, vol. 48, no. 4, pp. 692–697, April 2003.

[50] V. Gazi and K. M. Passino, “A class of attraction/repulsion functions forstable swarm aggregations,” in Proc. of Conf. Decision Contr., Las Vegas,Nevada, December 2002, pp. 2842–2847.

[51] O. Khatib, “Real-time obstacle avoidance for manipulators and mobilerobots,” The International Journal of Robotics Research, vol. 5, no. 1,pp. 90–98, 1986.

[52] E. Rimon and D. E. Koditschek, “Exact robot navigation using artificialpotential functions,” IEEE Trans. on Robotics and Automation, vol. 8, no.5, pp. 501–518, October 1992.

[53] V. Gazi and K. M. Passino, “Stability analysis of swarms in an envi-

15

ronment with an attractant/repellent profile,” in Proc. American ControlConf., Anchorage, Alaska, May 2002, pp. 1819–1824.

[54] V. Gazi and K. M. Passino, “Stability analysis of social foraging swarms:Combined effects of attractant/repellent profiles,” in Proc. of Conf. Deci-sion Contr., Las Vegas, Nevada, December 2002, pp. 2848–2853.

[55] B. L. Partridge, “The structure and function of fish schools,” ScientificAmerican, vol. 245, pp. 114–123, 1982.

[56] S. Gueron, S. A. Levin, and D. I. Rubenstein, “The dynamics of herds:From individuals to aggregations,” Journal of Theoretical Biology, vol.182, pp. 85–98, 1996.

[57] H. K. Khalil, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ,second edition, 1996.

16