Hossein RastgoftarDepartment of Mechanical

Engineering and Mechanics,

Drexel University,

3141 Chestnut Street,

Philadelphia, PA 19104

e-mail: hossein.rastgoftar@mail.drexel.edu

Suhada Jayasuriya1

Department of Mechanical

Engineering and Mechanics,

Drexel University,

3141 Chestnut Street,

Philadelphia, PA 19104

e-mail: sjayasuriya@coe.drexel.edu

Evolution of Multi-AgentSystems as ContinuaIn this paper, a new framework for evolution of multi-agent systems (MAS) based on prin-ciples of continuum mechanics is developed. Agents are treated as mass particles of acontinuum whose evolution (both translation and deformation) is modeled as a homeo-morphism from a reference to the current configuration. Such a mapping assures that notwo mass particles of the continuum occupy the same location at any given time, thusguaranteeing that inter-agent collision is avoided during motion. We show that a specialclass of mappings whose Jacobian is only time varying and not spatially varying hassome desirable properties that are advantageous in studying swarms. Two specific sce-narios are studied where the evolution of a swarm from one configuration to anotheroccurs with no inter-agent collisions while avoiding obstacles, under (i) zero inter-agentcommunication and (ii) local inter-agent communication. In the first case, a desired mapis computed by each agent all knowing the positions of a few leader agents in a referenceand the desired configurations. In the second case, paths of nþ 1 leader agents evolvingin an n-D space are known only to the leaders, while positions of follower agents evolvethrough updates that are based on positions of nþ 1 adjacent agent through local com-munication with them. The latter is based on a set of weights of communication of fol-lower agents that are predicated on certain distance ratios assigned on the basis of theinitial formation of the MAS. Properties of homogeneous maps are exploited to charac-terize the necessary communication protocol. [DOI: 10.1115/1.4026659]

1 Introduction

Formation control of MAS has received considerable attentionduring the past two decades due to their potential applications thatinclude formation flight, air traffic control, environmental sam-pling, gaming, terrain mapping, surveillance, transportation engi-neering, etc. [1]. Some common methods for formation controlare leader–follower [2–5], virtual structures [6–9], behavioralbased [10,11], and artificial potential functions [12–15]. The morerecent partial differential equation (PDE) based approach [16–19]for formation control has some similarity with the work presentedin this paper. In the PDE approach, bulk motion of a swarm iscontrolled by a few agents located at the boundary, called leaderagents. There, every coordinate of motion of all agents is modeledeither as a first order one-dimensional PDE when each agent isconsidered a single integrator or a second order one-dimensionalPDE when each agent is treated as a double integrator, with thepositions of the two boundary (or leader) agents prescribing theboundary conditions of the PDEs. In addition, inter-agent distan-ces among interior agents (called followers) of the swarm are con-trolled by spatially varying parameters of the steady state part ofPDEs. It has been shown that MAS evolution in an n-D space isachieved via 2n leaders with 2n necessary communications forevery follower agent to update its state.

The PDE based techniques have the following limitations. First,since they are boundary controlled, the rate of convergence of theMAS to a desired configuration may be slow [16]. Second,although the PDE based method guarantees that the initial forma-tion of a MAS converges asymptotically to a desired final forma-tion, it is not easy to check whether or not the follower agentsremain in a region trapped by leader agents, during transients.Third, since the distribution of the follower agents is determinedby solving a PDE with spatially varying parameters, it is at times

difficult to solve analytically. Consequently agents’ distribution isonly an approximation of the desired distribution.

Some of the more common limitations in approaches to forma-tion control of swarms may be addressed through the new methodproposed in here that is based on principles of continuum mechan-ics. We consider a MAS as a deformable body whose evolution ismodeled as the deformation of a continuum. In particular, ourfocus is on a specific class of mappings that yields homogenousdeformations. Proposed are two approaches for homogenoustransformation of a MAS. In both approaches, nþ 1 agents of theMAS, called leader agents, can move independently. In the first, an� n Jacobian matrix and a n� 1 rigid body displacement vectorof a transformation are calculated based on evolution of leaderagents which are then made available to the follower agents.Thus, MAS evolution can be achieved with zero inter-agent com-munication when the swarm destination and reference configura-tion are known a-priori. The second approach is communicationbased, where every follower agent learns the Jacobian and therigid body displacement vector of the required mapping throughcommunication with nþ 1 local agents. A special communicationprotocol based on a set of weights that are assigned on the basis ofthe initial distribution of agents of the MAS underlies this learningstrategy.

The paper is organized as follows: In Sec. 2, introduced aredeformation of continua and some useful properties of homoge-nous maps. Swarm evolution through a homogeneous map underzero communication is developed in Sec. 3. Learning of thedesired map by the follower agents through local communicationis in Sec. 4 followed by some simulation results in Sec. 5. Conclu-sions are in Sec. 6.

2 Deformation of Continuous Bodies

2.1 Kinematics of Motion. As shown in Fig. 1 let a continu-ous body X0 at time t¼ t0 has volume V0 with boundary S0. Ifforces act on X0, it will be deformed into a new configurationwith volume Vt and boundary St at time t> 0. We call X0 a refer-ence configuration and assume that every point in X0 is known.Let Ri be the position vector of a point i in the reference

1Corresponding author.Contributed by the Dynamic Systems Division of ASME for publication in the

JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript receivedAugust 6, 2013; final manuscript received February 2, 2014; published online April8, 2014. Assoc. Editor: Sergey Nersesov.

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configuration with its components called material coordinateswith respect to any orthonormal basis, and let ri be the positionvector of the same material point in the current configuration withits components called spatial coordinates with respect to anotherorthonormal basis. The deformation of X0 into Xt is described bya mapping f : R; tð Þ ! r; tð Þ; R; r�Rn.

We assume that this mapping and its inverse are one-to-one(implying one point in X0 is mapped into one point in Xt and vice-versa) and continuously differentiable in their arguments. Theinverse will exist throughout Xt provided the Jacobian Q(Ri, t) ofthis homeomorphism f is nonvanishing at any point of X0. For acontinuous body, it can be shown that the Jacobian Q(Ri, t) is acontinuous function of t which never vanishes and equals identityat t¼ t0. Hence, Q(Ri, t)> 0 for all Ri and t. Consequently, weassume that the Jacobian of the transformation

Q Ri; tð Þ ¼ @f Ri; tð Þ@Ri

(1)

is a positive definite n� n matrix with

f Ri; t0ð Þ ¼ In�n (2)

where In� n is the identity matrix.

2.2 Homogenous Deformations. These are motions of theform

ri tð Þ ¼ Q tð ÞRi þ D tð Þ (3)

where D and Q are either constant or functions of time only. Sincethese are not spatially dependent, all deformations and associatedstrain measures are independent of the material or spatial coordi-nates. Hence, they are referred to as homogenous deformations.Homogenous deformations are described by linear maps that havesome useful properties:

(i) Points which form planes in the reference configurationdeform into planes; hence two parallel planes willdeform into two parallel planes.

(ii) Points that form straight lines in the reference configu-ration deform into straight lines; hence two parallelstraight lines deform into two parallel straight lines.

(iii) Points that lie on a sphere in the reference configura-tion deform into an ellipsoidal surface.

We note that n2 elements of the Jacobian matrix Q(t) and n ele-ments of the rigid body displacement vector D(t) can be assignedbased on positions of nþ 1 points of the continuum at the currenttime t and initial time t0 when motion evolves as a homogeneoustransformation.

The notion of deformation of continua introduced above can bereadily used to model swarm motions as follows. We can think ofagents of a MAS as points of a deformable body whose boundaryis defined by a few leader agents. For example, suppose the agentsoccupying a region evolve in 2D as shown in Fig. 2.

Such a region occupied by agents in a swarm can be embeddedinside a triangle with three real or virtual agents at A, B, and C asshown. Clearly there are many choices for A, B, and C to embedthe swarm region. If we now treat this triangle ABC as a deforma-ble body and force its motion to occur as a homogeneous transfor-mation, we are guaranteed that all agents will stay within adeformed triangle specified by A0B0C0. This follows immediatelyfrom the fact that points on AB, BC, CA continue to lie on straightlines as the motion follows a homogeneous map. It is easy to seethat in the 1D case, all agents of MAS will lie on a straight lineand only two leader agents at the end points are needed to guidethe MAS. In the case of a 3D swarm, the MAS can be embeddedinside a tetrahedron ABCD by identifying four leader agents thatare either real or virtual. The evolution of the MAS will staybounded within a deforming tetrahedron as long as the motion isgiven by a homogeneous map defined through the positions of theleader agents in X0 and a desired configuration Xt. With the pro-posed embedding of a swarm by an appropriate number of leaderagents (virtual or real), the evolution of the swarm under a homo-geneous map can be captured as in Theorem 1 below.

THEOREM 1. Suppose a swarm of agents evolving in n-D isenclosed by nþ 1 leader agents defining the vertices of a polytopein the reference configuration X0 and that the leader agents definea final desired polytopic shape defined by their final locations asvertices. Then, under a homogenous transformation, all followeragents will stay inside a deformed polytope Xf with the leadersoccupying vertices for all time t� 0; tf½ �.

Proof. Suppose a homogenous map exists such that X0 and Xf

are related by

x tfð Þ ¼ Q tf� �

x t0ð Þ þ D tf� �

(4)

where x(tf) are the final positions of the leaders and x(t0) are theinitial positions of the leaders.

From property 2 of homogenous maps, we know that all edgesof the polytope will remain straight at the final configuration. Wealso know that at any time t� 0; tf½ �, the positions of the leaders willbe defined by Eq. (4) which remains a homogenous map and it fol-lows that all edges will remain straight proving that the followeragents will continue to be embedded in the polytope Xf corre-sponding to the leader agents’ locations at tf. �

3 Homogenous Transformation Under Zero

Inter-Agent Communication

Let a MAS consist of N agents evolving in Rn with nþ 1 leader(vertex) agents whose position vectors at time t0 and t> t0 pre-scribe the Q(t) and D(t) of a homogenous mapping (Eq. (3)).

Fig. 1 Kinematics of a deforming body

Fig. 2 Homogenous transformation of a planar deformablebody specified by three leader agents

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Let the agents be initially distributed inside the domain X0 attime t¼ t0, with the desired final configuration Xf at time t¼ tf,evolving under a homogenous transformation. By choosing a suit-able Jacobian Q(t) and a rigid body displacement vector D(t)which if available to all follower agents at the beginning, then thewhole MAS is able to achieve the task of moving from X0 to Xf.First, trajectories of the leader agents are planned such that noneof the leaders and the domain enclosed by the leader agents as ver-tices collide with any known obstacles. The followers are initiallyplaced in the interior points of a segment, triangle, or a tetrahe-dron, respectively, for 1D, 2D, and 3D evolution. Figure 3 illus-trates a typical path plan satisfying two boundary configurationswhile avoiding three known obstacles.

In Theorem 2, below, we summarize how Q(t) and D(t) areobtained for an n-D homogenous transformation, based on trajec-tories of nþ 1 leader agents. We assume that N � nþ 1 agentsevolve in Rn, where initial and current position vectors of an agentj are given by

Rj ¼Xn

k¼1

Xkjek (5)

rj tð Þ ¼Xn

k¼1

xkjtð Þek (6)

and leaders’ positions satisfy the rank condition

Rank r1 tð Þ … rnþ1 tð Þ½ � ¼ n; 8t � 0 (7)

THEOREM 2. If leader agents of a MAS evolve in an n� D spacesuch that the rank condition (7) is met, then,

(i) The elements of Jacobian matrix Q(t) and rigid body dis-placement vector D(t) are given by

J ¼ In � L0...

In � 1 nþ1ð Þ�1

h i�1

Pt (8)

where

Jn nþ1ð Þ�1 ¼ Q11 … Q1n … Qn1 … Qnn½ D11 … Dn1 �T

L0 ¼

X11� � � Xn1

..

. . .. ..

.

X1nþ1� � � Xnnþ1

2664

3775

nþ1ð Þ�n

Pt nþ1ð Þ�1¼ x11

tð Þ … x1nþ1tð Þ½ … xn1

tð Þ … xnnþ1tð Þ �T

(ii) The position of every follower agent i (i ¼ 1; 2;…; nþ 1)can be uniquely expressed as

riHTtð Þ ¼

Xnþ1

k¼1

ai;krik tð Þ (9)

subject to

Xnþ1

k¼1

aik ¼ 1 (10)

where ai;k remains invariant during MAS evolution and kdenotes the leader agents 1, 2,…, nþ 1.

Proof. By substituting position vectors of the nþ 1 leaderagents 1, 2,…, nþ 1 at initial time t0 and current time t in Eq. (3),the elements of Q(t) and D(t) follow immediately from the solu-tion to a set of linear equations as given by Eq. (8). The rank con-dition (7) guarantees that the matrix

In � L0...

In � 1 nþ1ð Þ�1

h i

is invertible and that Q(t) and D(t) can be solved for.Furthermore, the position vector of every agent i can be

uniquely expressed as a linear combination of position vectors ofleader agents 1, 2,…, nþ 1 as in Eq. (9) subject to condition (10),if the rank condition (7) is met. Since elements of Q(t) and D(t)are related to the current position vectors of leader agents 1, 2,…,

nþ 1, at time t, via the constant matrix In � L0...

In � 1 nþ1ð Þ�1

h i(where In � L0

..

.In � 1 nþ1ð Þ�1

h iis assigned based on compo-

nents of position vectors of leader agents 1, 2,…, nþ 1, at the ini-tial time t¼ t0), it follows that coefficients ai;k remain constantand can be determined on the basis of the initial positions ofagents i, 1, 2,…, nþ 1. �

Remark 1. Although it is assumed that the leader agents i, 1,2,…, nþ 1 are located on the boundary of a domain, it is not nec-essary to achieve a homogenous motion of a MAS. When theleader agents are at the vertices (boundary), however, a simplecharacterization of the embedding region is possible as a line, tri-angle, or a tetrahedron depending on 1D, 2D, or 3D evolution of aMAS.

So far, we have shown that swarm motion under a homogenousmap with zero inter-agent communication is possible provided theswarm tasks and obstacles are known a-priori. On the other hand,when a task has to be altered while in motion as in the case of anobstacle appearing suddenly in the motion field, zero communica-tion approach will not work. So to deal with such situations,where on-line, real time path planning must be done, an alterna-tive pathway to incorporate some sort of communication betweenleader agents and followers is necessary. In what follows such anapproach based on a communication protocol that allows the fol-lower agents to learn a new map that is determined by the leaderagents by communicating with a few local agents is developed.

4 Communication Based Homogenous Maps

The basic framework developed thus far for determining the ho-mogeneous map on the basis of the initial and final locations ofthe leader agents applies in this case as well. Specifically, we sim-ply replace the reference configuration with the current configura-tion and the final configuration with a new desired intermediateconfiguration where the latter is chosen to work around a pop-upobstacle or change of the swarm (team) plan. In essence all thatone has to know is two configurations over which to specify aQ(t) and D(t). Once these parameters are determined by the vertexor the leader agents, the followers will learn the map over time bycommunicating with some of their neighbors according to a spe-cial protocol. A step-by-step process for accomplishing this is pre-sented next by developing certain preliminaries first andcollecting them in the form of Theorem 3.Fig. 3 Typical path plan for a MAS in the plane

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4.1 Numbering of the Agents. Every agent of the MAS isidentified by an index i� 1; 2;…;Nf g. The leader agents are desig-nated by the indices 1, 2,…, nþ 1, and the follower agents areindexed nþ 2, nþ 3,…, N.

4.2 Designing Communication Topology. The proposedcommunication network has the following two features:

(i) Every follower agent communicates with nþ 1 local agentsto update its position.

(ii) Each leader agent’s position is known to a single nearbyfollower agent.

The communication topology of the system has a graphG ¼ u[ @u, with the nodes representing the leader agentsbelonging to the boundary @u of G, and the nodes representingthe follower agents belonging to the subgraph u. So, nodes of @uare indexed 1, 2,…, nþ 1, and nodes of u are indexed nþ 2,nþ 3,…, N. Communication between two follower agents isshown by a nondirected edge, which implies bidirectional com-munication between the two. Furthermore, communicationbetween a leader and a follower is shown by an arrow terminatingon the follower agent implying that any leader agent can move in-dependently with its position tracked by one of the followeragents. A sample communication graph of a MAS consisting of20 agents with three leaders and 17 followers evolving in R2 isshown in Fig. 4. As illustrated, every follower agent (denoted bycircles) communicates with three local agents, with the positionsof three leader agents (denoted by squares) tracked by three fol-lower agents.

4.3 Assigning Initial Weight Matrix W0. We define theelements W0ij of the ðN � n� 1Þ�N weight matrix W0 as follows[20–22]:

W0ij¼

0 < w0iþnþ1;j< 1 If iþ nþ 1 � j

�1 If iþ nþ 1 ¼ j

0 Otherwise

8><>: (11)

where the symbol� denotes adjacency between the follower agentiþ nþ 1 and agent j. We call w0iþnþ1;j

the weight of communica-tion of follower iþ nþ 1 with agent j. Weights of communicationof any follower agent iþ nþ 1 are chosen to satisfy the condition

XN

j ¼ 1

and

j � i

w0iþnþ1;j¼ 1 (12)

Condition (12) implies that the sum of weights of communica-tion of every follower agent iþ nþ 1 is 1 and is convenient topartition the weight matrix W0 as

W0 ¼ B0 N�n�1ð Þ�ðnþ1Þ...

A0 N�n�1ð Þ� N�n�1ð Þ

��(13)

which has the following properties [20–22]:

(i) W0 is a zero-sum row stochastic matrix.(ii) Except diagonal elements of A0 that are all� 1, there are

nþ 1 positive entries on every row of the weight matrixW0.

(iii) Although A0 is not necessarily symmetric, if A0ij6¼ 0,

then, A0ji6¼ 0, and if A0ij

¼ 0, then, A0ji¼ 0.

4.4 Initial Distribution of the Agents. The leader agents areinitially located at nþ 1 points of the boundary of an n-D convexdomain, such that rank condition (7) is satisfied.

Next, we arbitrarily choose initial weights of communication,for every follower agent i, satisfying Eqs. (11) and (12).

Finally, every follower agent i is located such that its positionvector, Ri, is a linear combination of position vectors of the adja-cent agents which are prescribed by the communication topologyas

Riþnþ1 ¼XN

j ¼ 1

and

j � i

w0iþnþ1;jRj (14)

Equation (14) can also be rewritten as

Riþnþ1 ¼XN

j¼1

w0iþnþ1;jRj (15)

where

w0iþnþ1;j> 0 iþ nþ 1 � j

w0iþnþ1;j¼ 0 Otherwise

((16)

Thus, the initial positions of agents satisfy

A0Z0 þ B0U0 ¼ 0 (17)

where Eq. (15) is the row i of Eq. (17), U0 ¼ R1 … Rnþ1½ �Tis the initial positions of the leader agents, Z0

¼ Rnþ2 … RN½ �T is the initial positions of the follower agents,and A0 and B0 are partitions of the initial weight matrix W0. InTheorem 3, we will show that A0 is negative definite from whichit follows that

Z0 ¼ �A�10 B0U0:

4.5 Interpretation of Weights of Communication. To getsome insight on what the communication protocol does, supposethat the follower agent i communicates with agents i1, i2,…, inþ 1,that are initially placed at Ri, Ri1, Ri2,…, Rinþ 1, respectively,with w0i; i1, w0i; i2,…, w0i; inþ1 the corresponding weights ofcommunication. The position vector Rj (j can take on values i, i1,i2,…, inþ 1) can be expressed in Cartesian coordinates as inEq. (5). From Eqs. (14) and (12), the weights of communicationcan be obtained by solving the following set of nþ 1 linear alge-braic equations:Fig. 4 Sample communication topology for a 2D swarm

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X1i1X1i2

� � � X1 inþ1

X2i1X2i2

� � � X2inþ1

..

. ... ..

.

Xni1Xni2

� � � Xninþ1

1 1 � � � 1

2666666664

3777777775

w0i;i1

w0i;i2

..

.

w0i;in

w0i;inþ1

26666666664

37777777775¼

X1i

X2i

..

.

Xni

1

2666666664

3777777775

(18)

The rank condition

Rank Ri1;Ri2 ;…;Rinþ1

� �¼ n (19)

guarantees that Eq. (18) has a unique solution for the weights ofcommunication [20–22].

For the case of a 2D MAS, where an agent i communicateswith agents i1, i2, and i3, initially located at ðXi; YiÞ; ðXi1 ;Yi1Þ;ðXi2 ; Yi2

Þ; and ðXi3 ;Yi3Þ, respectively, Eq. (18) reduces to

Xi1 Xi2 Xi3

Yi1 Yi2 Yi3

1 1 1

264

375

w0i;i1

w0i;i2

w0i;i3

2664

3775 ¼

Xi

Yi

1

264

375 (20)

Equation (20) requires that agents i1, i2, and i3 are not aligned,and the following ratios are obtained as the weights of communi-cation w0i;i1

, w0i;i2, and w0i;i3

:

w0i;i1¼ Xi3

� Xi2ð Þ Yi � Yi2ð Þ � Yi3 � Yi2ð Þ Xi � Xi2ð ÞXi3 � Xi2ð Þ Yi1 � Yi2ð Þ � Yi3 � Yi2ð Þ Xi1 � Xi2ð Þ (21)

w0i;i2¼ Xi1

� Xi3ð Þ Yi � Yi3ð Þ � Yi1 � Yi3ð Þ Xi � Xi3ð ÞXi1 � Xi3ð Þ Yi2 � Yi3ð Þ � Yi1 � Yi3ð Þ Xi2 � Xi3ð Þ (22)

w0i;i3¼ Xi2

� Xi1ð Þ Yi � Yi1ð Þ � Yi2 � Yi1ð Þ Xi � Xi1ð ÞXi2 � Xi1ð Þ Yi3 � Yi1ð Þ � Yi2 � Yi1ð Þ Xi3 � Xi1ð Þ (23)

Equations (21)–(23) imply that each weight ratio can be posi-tive, zero, or negative depending on the initial placement of agentsi, i1, i2, and i3. Suppose we partition the plane of evolution of theMAS into seven subregions that are based on the initial locationsof agents i, i1, i2, and i3 (see Fig. 5). It is then clear that the

weights of communication are all positive if agent i is locatedinside the triangle formed by agents i1, i2, and i3 at the vertices.

4.6 MAS Evolution Dynamics. It is assumed that the leaderagents move independently with their positions to be acquired bythe follower agents through the communication topology. If we letthe follower agent iþ nþ 1, (i¼ 1, 2, N� n� 1) update its posi-tion according to

vi ¼ _ri tð Þ ¼ g ridtð Þ � ri tð Þð Þ (24)

where g is a positive control parameter, and

rid tð Þ ¼XN

j¼1

w0iþnþ1;jrj tð Þ (25)

is the desired position of the follower agent i at time t, then, Eq.(24) corresponds to the row i of the following state space repre-sentation [20–22]:

_Z ¼ g A0Z þ B0Uð Þ (26)

where U ¼ r1 tð Þ … rnþ1 tð Þ½ �T and Z ¼ rnþ2 tð Þ … rN tð Þ½ �T areleader and follower position vectors, respectively.

THEOREM 3. Given that MAS consists of N agents that are (i) ini-tially distributed according to Eq. (17), with A0 and B0 as the par-titions of the initial weight matrix W0 (see Eqs. (11)–(13)), (ii)leader agents evolve independently, such that, rank condition (7)is met, and (iii) they come to rest at the final time, and (iv) everyfollower agent i updates its position according to Eqs. (24) and(25), then it follows that the initial formation of the MAS asymp-totically converges to the final configuration, as a homogenoustransformation, between the two configurations.

Proof. Condition (i) yields positive weights of communicationfor follower agents, and the partition A0 of the initial weight ma-trix W0 can be written as

A0 ¼ � I � D0ð Þ (27)

where I is the identity matrix and D0 is a non-negative matrix.Since A0 results from the elimination of the first nþ 1 columns ofW0, and W0 is zero-sum row stochastic, the sum of nþ 1 rows ofA0 is negative and the sum of the rest of the rows of A0 is zero[20–22]. Then, by invoking the Perron Ferobenius theorem [23],we conclude that the spectrum q(D0)< 1 and the matrix A0 is neg-ative definite [20–22]. Consequently, if conditions (ii) and (iii) aresatisfied, the initial formation of the MAS asymptotically con-verges to a final configuration given by

Zf ¼ �A�10 B0Uf (28)

where Uf and Zf are the leader and follower position vectors,respectively. Thus both initial and final formations of the MASsatisfy Eq. (17) which in turn implies that weights of communica-tion of every follower agent in the final and initial formations isthe same. Therefore from Theorem 2, it now follows that the finalformation of the MAS is a homogenous transformation of the ini-tial configuration. �

Remark 2. Since A0 is negative definite,�A0�1B0 can be

obtained by applying some row algebraic operations on W0. SinceW0 is zero-sum row-stochastic and A0 is negative definite (nonsin-gular), by applying appropriate row operations on W0, we canobtain a matrix

S ¼ A�10 B0

� �N�n�1ð Þ� nþ1ð Þ

..

.I N�n�1ð Þ� N�n�1ð Þ

� �(29)Fig. 5 Seven subregions based on the signs of weights of

communication

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which is zero-sum row stochastic as well. Hence, the sum of everyrow of�A�1B is 1 and the desired state prescribed by the homog-enous mapping can be expressed by

ZHT ¼ �A�10 B0U (30)

where U ¼ r1 tð Þ … rnþ1 tð Þ½ �T and Zt ¼ rnþ2HTtð Þ … rNHT

tð Þ½ �T .Note that row i� n� 1 of Eq. (30) represents the state of the fol-lower agent i, which is of the form given in Eq. (9) and satisfiesthe properties of a homogeneous mapping.

4.7 Control Parameter g. So far, we have shown that if thecontrol parameter g is positive and position of every followeragent is updated according to Eqs. (24) and (25), then, asymptoticconvergence of the initial configuration to a final formation is ahomogenous transformation of the initial distribution of theagents. However, in a transient state of the formation of the MAS,the follower agents deviate from the state defined by the homoge-nous transformation of Eq. (9). To minimize this deviation, con-trol parameter g can be made appropriately large. Increasing thecontrol parameter g leads to an increased control force for guidingthe motion of every follower agent.

Since the focus of this paper is on developing the frameworkfor a continuum treatment, we only focused on MAS evolutionunder a constant control gain g [20–22]. Thus, for 1D, 2D, and 3DMAS evolution, a g> gmin is chosen such that none of the fol-lowers leave the domain bounded by the leader agents while intransition from the initial segment, triangle, or the tetrahedron, totheir corresponding final configurations. We assign gmin numeri-cally by trial and error such that the actual transient position vec-tor of every follower agent i (i¼ nþ 2, nþ 3,…, N) at time t,given by

~ri tð Þ ¼Xnþ1

k¼1

ai;k tð Þ~rk tð Þ (31)

satisfies

ai;k tð Þ � 0; 8t � t0 (32)

We note that rk tð Þ on the right hand side of Eq. (31) denotes thetransient position vectors of the leader agents.

5 Simulation Results

In this section, we simulate a scenario where a swarm of 20agents is to negotiate a narrow channel in the plane to highlightthe main attributes of the proposed continuum framework. Since,MAS evolves in 2D in this scenario, three agents of the MAS areconsidered leaders with the rest treated followers. As shown inFig. 6, agents are initially distributed inside the triangle P whosevertices are placed at P1 �6:5;�6:5ð Þ, P2 �5:5; 6ð Þ, and P3 6; 5ð Þ.

Leader agents are initially located at the vertices of triangle P sothat all follower agents lie inside P. It is desired that triangle Pdeforms in to a triangle Q, with vertices at Q1 18;�3ð Þ, Q2 18; 3ð Þ,and Q3 22; 0ð Þ.

Leader Trajectories: A set of trajectories for the leader agents 1,2, and 3, are arbitrarily chosen as shown in Fig. 7. Also, shown aretransient leader triangles at t¼ 12 s and t¼ 20 s with the leaderscoming to rest in 30 s. We note that although these leader trajecto-ries may be designed using actuator constraints and/or kinematicconstraints, for the present case those were not considered.

Initial Distribution of Follower Agents: Follower agents of theMAS are initially distributed according to the development inSec. 4. The inter-agent communication graph in Fig. 4 is usedwhere weights are determined based on initial positions of theagents using Eqs. (21)–(23) as listed in Table 1. Components ofinitial positions of the agents are also given in Table 1.

Final (Desired) Configuration of Follower Agents: Positionsreached by the follower agents, in the final configuration, can beobtained using Eq. (28), and are listed in Table 2.

Final configuration of the MAS is also illustrated in Fig. 8.Having set up the complete initial configuration and the final

desired configuration of the leaders and the initial locations of thefollowers shown below are the time evolutions of MAS under (i)zero inter-agent communication and (ii) local inter-agentcommunication.

Fig. 6 Planar motion field; initial and desired (final) configura-tions of MAS

Fig. 7 Paths of leader agents 1, 2, and 3

Table 1 Initial weights of communications and Initial positionsof followers

Initial weights of communication w0i,j x0 y0

F 4 w04;1¼ 0:65, w04;7

¼ 0:20, w04;8¼ 0:15 � 5.4776 �4.5886

F 5 w05;2¼ 0:65, w05;9

¼ 0:21, w05;10¼ 0:14 �4.7379 4.9601

F 6 w06;3¼ 0:72, w06;11

¼ 0:15, w06;12¼ 0:13 4.8628 4.6781

F 7 w07;4¼ 0:29, w07;13

¼ 0:36, w07;15¼ 0:35 �3.1087 �1.0772

F 8 w08;4¼ 0:28, w08;15

¼ 0:29, w08;16¼ 0:43 �4.2060 �0.9880

F 9 w09;5¼ 0:31, w09;16

¼ 0:42, w09;17¼ 0:27 �4.1478 2.4466

F 10 w010;5¼ 0:35, w010;18

¼ 0:23, w010;19¼ 0:42 �2.0849 3.9020

F 11 w011;6¼ 0:33, w011;13

¼ 0:20, w011;20¼ 0:47 2.0673 3.6233

F 12 w012;6¼ 0:22, w012;19

¼ 0:41, w012;20¼ 0:37 1.7904 4.1125

F 13 w013;7¼ 0:32, w013;11

¼ 0:33, w013;14¼ 0:35 �1.0684 1.2105

F 14 w014;13¼ 0:42, w014;15

¼ 0:27, w014;18¼ 0:31 �2.1595 1.0272

F 15 w015;7¼ 0:41, w015;8

¼ 0:34, w015;14¼ 0:25 �3.2445 �0.5208

F 16 w016;8¼ 0:35, w016;9

¼ 0:36, w016;17¼ 0:29 �4.0265 1.0415

F 17 w017;9¼ 0:27, w017;16

¼ 0:43, w017;18¼ 0:30 �3.6591 1.7465

F 18 w018;10¼ 0:29, w018;14

¼ 0:34, w018;17¼ 0:37 �2.6927 2.1270

F 19 w019;10¼ 0:32, w019;12

¼ 0:42, w019;20¼ 0:26 0.4589 3.9924

F 20 w020;11¼ 1=3, w020;12

¼ 1=3, w020;19¼ 1=3 1.4389 3.9094

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5.1 Scenario I: Zero Inter-Agent Communication. Here,the follower agents have full information about the Jacobian ma-trix Q(t) and rigid body displacement vector D(t), obtained fromEq. (8) of Theorem 2, using initial and the current positions of theleaders. Elements of Q(t) and D(t) of a 2D homogenous transfor-mation are obtained from solving

Q11 tð ÞQ12 tð ÞQ21 tð ÞQ22 tð ÞD11 tð ÞD21 tð Þ

266666666664

377777777775¼

X1 Y1 0 0 1 0

X2 Y2 0 0 1 0

X3 Y3 0 0 1 0

0 0 X1 Y1 0 1

0 0 X2 Y2 0 1

0 0 X3 Y3 0 1

266666666664

377777777775

�1 x1 tð Þx2 tð Þx3 tð Þy1 tð Þy2 tð Þy3 tð Þ

266666666664

377777777775

(33)

In Eq. (33), (Xi, Yi) denotes the initial position of leader i,i¼ 1,2,3.

Elements of Q and D versus time are shown in Figs. 9 and 10.Note that Q(0)¼ I and D(0)¼ 0, ensuring that Eq. (2) is satisfiedas required.

As seen from Fig. 11, Q(t) remains positive definite during theMAS evolution.

We emphasize that when there is zero communication everyfollower agent is assumed to have full knowledge of Q(t) andD(t). Shown in Fig. 12 are the configurations of the MAS, att¼ 0 s, t¼ 10 s, t¼ 20 s, t¼ 25 s, and t¼ 30 s.

5.2 Scenario II: Communication Based HomogenousTransformation. In this case, every follower agent communi-cates with three local agents, according to the topology shown inFig. 4. Specifically, a follower i uses the weights of communica-tion as listed in Table 1 to update its position at any time t basedon positions of adjacent agents. Initial positions and the finalreached positions of followers are listed in Tables 1 and 2,respectively.

By trial and error, we obtained a value of gmin¼ 30.2632 whichguarantees that none of the followers leave the triangle defined bythe three leaders during transition. We used g¼ 32> gmin for sim-ulation of the MAS evolution.

In Fig. 13, x and y coordinates of the actual position vectors(ri tð Þ ¼ xi tð Þex þ yi tð Þey) of follower agent 18 are shown. Also, xand y coordinates of transient states corresponding to the

Table 2 Final (desired) configuration of follower agents

xf yf xf yf

F 4 18.3003 �2.2718 F 13 19.6631 �0.3467F 5 18.2920 2.3169 F 14 19.2913 �0.2005F 6 21.6161 0.0874 F 15 18.9593 �0.7343F 7 19.0216 �1.0407 F 16 18.6460 0.2129F 8 18.6401 �0.7575 F 17 18.7534 0.4837F 9 18.5653 0.9382 F 18 19.0767 0.4626F 10 19.2376 1.2133 F 19 20.1138 0.7046F 11 20.6796 0.1710 F 20 20.4546 0.4501F 12 20.5704 0.4747

Fig. 8 Final distribution of follower agents

Fig. 9 Elements of Jacobian matrix Q(t)

Fig. 10 Elements of rigid body displacement vector D(t)

Fig. 11 Eigenvalues of the Jacobian matrix Q(t)

Fig. 12 MAS configurations at five sample times t 5 0 s,t 5 10 s, t 5 20 s, t 5 25 s, and t 5 30 s

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homogenous map (riHT tð Þ ¼ xiHT tð Þex þ yiHT tð Þey) defined viathe initial and final configurations are illustrated, showing thedeviation between the actual paths followed and that given by thehomogenous transformation.

Also shown are the configurations of the MAS at four sampletimes t¼ 5 s, t¼ 10 s, t¼ 15 s, t¼ 20 s, and t¼ 25 s in Figs. 14–18.As expected, the follower agents do not leave the transient triangledefined by the leader agents during their time evolution.

6 Conclusion

In this paper, we proposed a framework for treating swarm evo-lution as a deformable continuum. In particular, motion evolutionin an n-dimensional space under a special class of maps leading tohomogeneous deformations was developed. Two basic approacheswere presented to deal with motion evolution under no inter-agentcommunication and that with local inter-agent communication. Inthe case of no communication, the reference configuration and thefinal desired configuration defined by the vertex positions of theleaders are known to all agents enabling each follower agent toquickly acquire the key matrices defining the necessary homoge-neous map. This scenario works well when a motion plan is fullyscripted ahead of time. When inter-agent communications areallowed a specially constructed communication protocol providesa mechanism for updating each agent’s evolution that guaranteesasymptotic convergence of the swarm configuration to the finaldesired configuration. The update law uses the position informa-tion of some of the neighboring agents. This algorithm is based ona specific homogenous transformation that is derived from the

Fig. 13 x and y coordinates of r18(t) and r18HT(t) of follower 18

Fig. 14 Configuration of agents at t 5 5 s

Fig. 15 Configuration of agents at t 5 10 s

Fig. 16 Configuration of agents at t 5 15 s

Fig. 17 Configuration of agents at t 5 20 s

Fig. 18 Configuration of agents at t 5 25 s

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initial and final locations of the leader agents defining the verticesof the domain.

Acknowledgment

This work has been supported in part by the National ScienceFoundation under Award Nos. 1134669 and 1250280 from theENG/CMMI division, and Drexel University.

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