Abstract—In this paper, evolution of a multi agent system

(MAS) in n-D space where n+1 or more leader agents located

on the boundary guide the MAS is studied. We consider the

MAS as a deformable body whose motion can be prescribed by

a homogeneous map determined by the initial and current

positions of the leaders. Each follower agent learns this leader

prescribed motion plan by local communication with adjacent

agents. In previous work we assumed that each follower

communicates with n+1 adjacent agent [1-7]. Here we relax

that constraint to include more than 3 adjacent agents by

choosing a polyhedral communication topology where the

vertices are local agents that are adjacent to a follower agent i.

The polytope encloses the follower agent i and is the union of

sub-polyhedra, with one of the n+1 vertices occupied by

agent i. volume weights are defined based on the initial

position of follower i and the set of adjacent agents to . The

motion proceeds by updating the position of every follower

agent such that volume weights in intermediate configurations

of the MAS are close to the initial volume weights. This update

strategy maneuvers the MAS to its final desired formation as a

homogenous map of its initial configuration.

I. INTRODUCTION

The use of boundary agents to manipulate the geometry of

a deployment in a PDE-based multi-agent setting, through

the assignment of boundary conditions has been studied in

[8-11]. They developed a decentralized multi-agent

framework where deployment patterns are generated through

communication with their immediate neighbors [9]. In [1-6,

12] the authors proposed a continuum framework where a

swarm can be deployed directly without solving a PDE using

either zero communication or inter-agent communication

with several nearby neighbors.

Our recent work considered a communication protocol for

evolution of MAS under a homogeneous map [1-6]. In [1

and 12] a homogenous transformation under zero inter-agent

communication was considered for MAS evolution

where a map prescribed by the positions of three leader

H. Rastgoftar is with the Department of Mechanical Engineering and

Mechanics, Drexel University, 3141 Chestnut Street 115 B, Randall Hall Philadelphia, PA 19104-2884, USA (e-mail:

hossein.rastgoftar@mail.drexel.edu).

S. Jayasuriya,, University Distinguished Professor, is Head of Mechanical Engineering and Mechanics, Drexel University, 3141 Chestnut

Street 115 B, Randall Hall Philadelphia, PA 19104-2884, USA (phone: 407-

823-2416; e-mail: sjayasuriya@coe.drexel.edu).

agents of the MAS, becomes available to all the follower

agents. In [1-6], a communication protocol that assures that

the MAS transforms as a homogenous map in was

proposed. There agents of the MAS considered

leaders evolve independently, with the rest of the follower

agents each updating its position through local

communication with neighboring agents. A set of

communication weights is uniquely assigned based on the

initial positions of the agents of the MAS. The

weights of communication are the unique solution of a set of

linear algebraic equations with unknowns.

In this paper, we formulate a communication based

homogenous transformation of a MAS moving in a

space that consists of agents, with leader

agents located at the vertices of a polytope, called the

leading polytope. The leading polytope encloses all the

followers, with every follower agent i communicating with

agents. This leads to , weights of

communications with unknown parameters are to be

determined by a set of n+1 linear algebraic equations [1-6].

Thus, they are not uniquely determined from the initial

positions of agents and as expected communication with

more than agents allows the possibility of meeting

additional requirements. Hence a communication topology is

chosen such that any follower agent i is enclosed by a

communication polytope whose vertices are the agents that

are adjacent to i. The communication polytope is taken to be

covered by polyhedra with the common vertex occupied

by agent i. Then, every follower agent i determines

volume weights based on the initial positions of the agents,

where each volume weight is the ratio of volume of a

tetrahedron inside the communication polytope to the total

volume of the communication polytope corresponding to

agent i. In the ensuing motion the position of every follower

agent i is updated so that a cost function depending on the

initial volume weights, current positions of agent i and

adjacent agents, is minimized. This minimization results in

(i) volume weights of the transient configuration being as

close as possible to the initial volume weights, (ii) volume

weights of the agents in the final formation converge to their

initial weights, and (iii) final configuration of the MAS

remains a homogenous transformation of the initial

formation.

This paper is organized as follows: In section II,

properties of homogenous transformations are described. In

section III, communication based homogenous

transformation of a MAS based on preserving volume

Continuum Evolution of Multi Agent Systems under a

Polyhedral Communication Topology

Hossein Rastgoftar and Suhada Jayasuriya

2014 American Control Conference (ACC)June 4-6, 2014. Portland, Oregon, USA

978-1-4799-3274-0/$31.00 ©2014 AACC 5115

weights is developed. Dynamics of MAS evolution under the

new communication protocol is formulated in section IV.

Simulation results are in section V followed by concluding

remarks in section VI.

II. KEY PROPERTIES OF HOMOGENEOUS

TRANSFORMATIONS

Homogenous transformation of a MAS in an space

is expressed by

( ) ( ) ( ) ( )

where ( ) is a nxn positive definite Jacobian matrix, ( ) is nx1 rigid body displacement vector,

∑

( )

is the material coordinate of agent i, i.e., position of agent i

at the initial formation, and

( ) ∑ ( )

( )

is the position of agent i at the current time . Shown in

Fig. 1 is homogenous transformation of a polygonal domain

deforming in a plane.

Fig. 1 Schematic of homogenous transformation

The following are some unique features of a homogenous

deformation:

1- It is a linear map that transforms every straight line in

the initial configuration to another straight line in the

current configuration.

2- An ellipsoid in the reference configuration is mapped to

another ellipsoid in the current configuration.

3- If a polytope is considered as the union of

sub-polyhedra , , …, , where there is no

intersection between any two sub-polyhedra inside ,

then, the volume ratio of every sub-polyhedron j ,

( ) to the volume of the polytope is

preserved.

We develop a new communication topology for evolution

of a MAS in an space based on above property (3) of

homogenous mappings. Suppose that is a convex

polytope enclosing follower agent at any time during

MAS evolution, and agents , , …, are the vertices of

that are positioned at , , …, , respectively, and

satisfy following rank condition:

{ } ( )

Additionally the vertices of the sub-polyhedron

are the agent and the adjacent agents , , …, . The

volume of the sub-polyhedron , , can then be

calculated as

|

|

|

| ( )

Remark: We note that must always be a positive

quantity. This can easily be accomplished by interchanging

two of the rows of the determinant as needed.

Let us define the transient volume weight ( ) as the

ratio:

( )

where

∑

( )

is the net volume of the polytope at a time . Then, under

homogenous mapping of the polytope , the transient

volume weight ( ) remains invariant at

any time during deformation. Therefore, can be

calculated based on initial configuration of the polytope ,

( )

where and are initial volume of the sub-

polyhedron and the polytope , respectively. We call

the volume weight.

Deformation: As shown in Fig. 2, the polygon , constructed with vertices , ,…, , is the union of

triangles with initial areas , ,…, , where

denotes the total area of the

polygon.

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Fig. 2 Schematic of a polygon as the union of sub-

triangles

For 2D deformation, the ratio

( )

is called area weight of the triangle j with respect to polygon

, that remains constant if the polygon transforms under

a homogenous mapping. It is obvious that

∑

( )

We note that area of a sample triangle j can be calculated

as

|

| ( )

where ( ), ( ), and ( ), are initial

positions of agents i, , and , respectively.

III. HOMOGENOUS TRANSFORMATION BASED

COMMUNICATION PROTOCOL FOR MAS EVOLUTION

A. Communication Topology

Let a MAS consist of N agents moving in a n-D space

where (i) agents 1, 2,…, l are leader agents located at the

vertices of a leading polytope, that is transformed under a

homogenous mapping, and (ii) rest of the agents of the MAS

are followers that update their positions through local

communication with some neighboring agents. We note that

every follower communicates with local agents

where the rank condition (4) is satisfied.

Deformation: Shown in Fig. 3 is a typical

communication topology for a MAS moving in a plane,

where circle nodes represent leader agents that are located

at the vertices of the leading polygon and numbered by 1, 2,

…, 5. Furthermore, follower agents shown as squares, are

placed inside the leading polygon and numbered 6, , …, 14.

Communication between a follower and a leader is shown by

an arrow, where it is unidirectional since leaders move

independently. The leader positions however need to be

tracked by some follower agents. In addition, non-directed

edges represent bidirectional communication between two

follower agents.

Fig. 3 A sample communication topology

We note that every follower agent i is required to

communicate with at least 3 local agents, where adjacent

agents are not aligned in the initial configuration (So the

rank condition (4) is satisfied.). Also, every follower agent i

is initially placed inside a communication polygon

constructed by vertex agents that are adjacent to i.

B. State of Homogenous Transformation

Let the leader agents of the MAS be transformed as a

homogenous map and

( )

∑( ( ) ( ))

( )

be the cost function characterizing the system state

deviation of an agent i from the one corresponding to the

desired homogenous transformation. Then, if the whole

MAS (followers and leaders) transforms under that

homogenous mapping, all agent cost functions ( ) ( ) will vanish simultaneously at any

[ ] where denotes the time for the leaders to settle.

In other words, the states prescribed by the homogenous

map for all agents of the MAS can be achieved if

[ ] ( ) ( )

When the desired homogenous mapping is only known to

the leader agents then followers can acquire the mapping

through local communication with the adjacent agents where

each follower agent updates its current position to

minimize the cost function ( )

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Theorem 1:

Suppose leader agents are transformed under a

homogenous mapping, and

( ) ∑

( )

is the volume of the j-th sub-polyhedron lying inside the

communication polytope , at time , where and

are dependent on positions of the adjacent agents (See

eqn. (5).), then the local minimum of the cost function ( ) is

( ) ∑ ( )

( )

where ( )s are given by the solution of

{

}

[ ∑

∑

∑

∑

∑

∑

∑

∑

∑

]

{

∑

∑∑

∑

∑∑

∑

∑∑

}

( )

Proof:

Since the total volume ( ) of the communication

polytope at any time , does not depend on position

of agent i, we have

∑

and

( ) ∑ ( )

∑ ( )

( )

which simplifies the local cost function ( ) to

( ) ∑(∑

∑ ( )

)

( )

Thus, the local minimum of each cost function ( ) can

be simply obtained as eqn. (8) if

are met. ■

IV. MAS EVOLUTION DYNAMICS

As developed in section III the leader agents of the MAS

move independently and the follower agents update their

positions though local communication. Suppose the position

of follower agent i is updated as follows

( )

where

( ) ( )

is a positive constant control parameter and is

obtained through eqns. (15 and 16).

Theorem 2:

If (i) position of every follower agent is updated according

to eqn. (19 and 20), where is obtained as in eqns. (15

and 16), volume weights are all positive and assigned based

on initial positions of the agents, and (ii) leader agents are

initially placed at the vertices of a leading polytope which

transforms homogenously during transition, then, the final

formation of the MAS is achieved asymptotically, once

leader agents settle. Furthermore, the final formation of the

MAS is a homogenous transformation of the initial

configuration.

Fig. 4 Level curves of the local cost function ( )

Proof:

According to eqns. (19 and 20) the velocity of every

follower agent i is directed toward the minimum of the local

cost function ( ), at any time , as shown in Fig. 4. Since

( ) is convex, then, position of every follower agent i

gets updated such that the corresponding local cost decreases. As leader agents eventually occupy the vertices

of a leading polytope, that is a homogenous map of the

initial leading polytope, the local minimum of every

follower agent i asymptotically converges to zero. Therefore,

for every follower agent i, actual position tends to thus

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making the final formation of the MAS a homogenous

transformation of the initial configuration. ■

V. SIMULATION RESULTS

Shown in Fig. 3, is a MAS that consists of 14 agents.

Leader agents 1, 2, …, 5 move independently where they are

initially at (0,0), (10,4), (9,12), (2,15), and (-3,9) which

eventually settle at (45,35), (60,49), (69,42), (65.75,28.33),

and (53.25,24) at time t=20s. In Fig. 5, leaders’ trajectories

and position of leading polygon at sample time t=0s, t=7.5s,

t=11s, and t=20s are shown. Initial positions of the follower

agents and corresponding area weight are all listed in Table

1 and Table 2, respectively, where in Table 2, represent

index of a follower agent, and numbers followed by “:”

denote that are the index numbers of the adjacent agents.

In addition, the communication topology is shown in Fig. 3.

Fig. 5 Trajectories of the leader agents; position leading

polygon at sample time , , , and

Table 1 Initial position of the follower agents

X (m) Y (m) X (m) Y (m)

F6 1.8474 4.5152 F11 -1.3275 8.8683

F7 7.6986 4.8704 F12 5.2544 7.0789

F8 6.6566 10.5475 F13 4.6015 9.3031

F9 2.8193 12.6743 F14 2.7536 9.1197

F10 1.3242 10.5533

Table 2 Area weights of follower agents Fi Area Weight F6 1: 0.2745 7: 0.1469 12: 0.1424 14:0.1978 11:0.2384

F7 2: 0.1304 12:0.6087 6:0.2609

F8 3: 0.3386 9:0.2933 13:0.1727 12:0.1954

F9 8: 0.2291 4:0.1663 10:0.1651 14:0.4395

F10 9: 0.2143 11:0.4286 14:0.3571

F11 5:0.2549 6:0.6275 10:0.1176

F12 7: 0.2894 8:0.1346 13:0.2313 6:0.3448

F13 8: 0.1667 14:0.3667 12:0.4667

F14 13: 0.2345 9:0.1851 10:0.2819 6:0.2984

The desired final formation of the MAS is a homogenous

transformation of the initial configuration, where the

Jacobian and rigid body displacement vector are

assigned based on positions of the leader agents 1, 2, and 3

[1-6]. Desired final positions of the follower agents are listed

in Table 3.

Every follower agent i updates its position according to

eqns. (19 and 20) where the control parameter is

chosen. In Fig. 6 x and y coordinates of actual position

vector of sample agents 14 are shown by continuous curves,

respectively. In addition, the state corresponding to that from

the homogenous transformation of follower agents 14 are

illustrated by dotted curves, respectively, in Fig. 6.

Table 3 Desired final position of the follower agents

X (m) Y (m) X (m) Y (m)

F6 45.3479 46.0583 F11 31.0512 57.1917

F7 62.2211 46.3848 F12 52.5070 52.1127

F8 52.7240 60.6093 F13 48.0639 57.7088

F9 38.9896 66.2664 F14 42.8157 57.4285

F10 36.9743 61.1316

Fig. 6 x and y coordinates of actual and desired position

vector of agent 14

In addition, formations of the MAS at different sample

times , , , , and are

shown in Fig. 7.

Fig. 7 Formation of the MAS at sample times ,

, , , and

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In Fig. 8, transient area weights for follower agent 14 are

given as a function of time. As it is seen the initial and final

area weights are identical thus validating theorem 2.

Fig. 8 Transient area weights of agent 14 versus time

It is also noted that the transient area weight ( ) is

calculated as

( ) ( )

( ) ( )

where

∑ ( )

( )

VI. CONCLUSION

A new protocol for communication is proposed for the

homogenous transformation of a MAS where the number of

inter-agent communications and leader agent are not

necessarily restricted to be as in [1-6]. We showed

that homogenous transformation of a MAS moving in a

plane can be achieved under local communication, where the

mapping is prescribed by leader agents with every

follower agent i communicating with

neighboring agents. Each follower agent updates its position

such that certain volume weights that are assigned based on

initial positions of adjacent agents are preserved during

MAS evolution. To make followers preserve volume

weights, a cost function depending on the deviation from

state of homogenous mapping is imposed on each follower

agent. Thus, every follower tries to minimize its own cost

that result in a final formation which is a homogenous

mapping of the initial configuration.

ACKNOWLEDGEMENT

This work has been supported in part by the National

Science Foundation under award nos. 1134669, 1250280

from the ENG/CMMI division and Drexel University.

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