Non-autonomous Area Coverage and Coordination of a Multi-agent

System for Harbor Protection Applications

Suruz Miah

, Bao Nguyen

, Alex Bourque

and Davide Spinello

Department of Electrical and Computer Engineering, Bradley University, Peoria, Illinois, U.S.A.

Center for Operational Research and Analysis, Defence Research and Development Canada (DRDC),

Department of National Defence, Government of Canada, Canada

Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario, Canada

Keywords:

Harbor Protection, Optimal Control, Area Coverage, Multi-agent Robotic System.

Abstract:

We propose optimal strategies to develop an automated layered defence system for protecting structured envi-

ronments, illustrated here by a typical harbor. The harbor is modeled as a two-dimensional environment, with

a cooperative set of autonomous mobile agents to be deployed as the defence system. Optimal deployment

is measured in a well-known area coverage metric, that encodes agents’ sensing performance and a weight

measure deﬁned on the harbor, which allows to represent a priori and time varying risk ﬁeld. The time-varying

risk ﬁeld allows to model several interesting scenarios. In this work, we consider the case of area surveillance

against moving targets or external threats penetrating through the perimeter. We present a feedback control

law for the platoon of surveying agents that implies the emergency of locally optimal conﬁgurations, that adapt

to time evolving harbor environments.

1 INTRODUCTION

Several applications in military and civilian domains,

such as harbor patrolling (Simetti and Cresta, 2007;

Kitowski, 2012; Miah et al., 2014a), perimeter sur-

veillance (Pimenta et al., 2013; Zhang et al., 2013;

Bishop, 2007), search and rescue missions (Hu et al.,

2013; Allouche and Boukhtouta, 2010), and coope-

rative estimation (Spinello and Stilwell, 2014), have

highly beneﬁted from theoretical and technological

advances in multi-agent robotic systems. On the

theoretical side, this has resulted into developments

of motion control algorithms for networked mobile

agents, that have attracted considerable attention due

to their capabilities to address, in part, various class of

problems in the ﬁeld of multi-agent systems, such as

area coverage (Miah et al., 2015; Miah et al., 2014b;

Leonard and Olshevsky, 2013; Cortes et al., 2004;

Cortes et al., 2005; Pimenta et al., 2008; Caicedo-

Nunez and Zefran, 2008; Kantaros et al., 2015; Zhong

and Cassandras, 2011; Miah et al., 2017a; Miah

et al., 2017b), locational optimization (Guruprasad

and Ghose, 2013), target tracking (Yang et al., 2008),

formation control (Fax and Murray, 2004; Marshall

et al., 2004), environmental tracking and monito-

ring (Porﬁri et al., 2007; Simic and Sastry, 2003), and

coordinated decision and control algorithms (Batalin

and Sukhatme, 2007; Beard et al., 2002; Ferrari et al.,

2009; Shima et al., 2007).

In area coverage applications, a team of mobile

agents spatially conﬁgure themselves over an area

of interest to maximize a coverage metric that typi-

cally encodes agents’ performance and a risk den-

sity that weights points in the area. This theoretical

framework naturally allows to formulate the class of

harbor protection problems considered in this work,

which can be seen as a resource allocation problem.

Speciﬁcally, given a set of cooperative mobile agents

that can sense the environment, and an area (harbor)

with a risk ﬁeld deﬁned on it to quantify the rela-

tive importance or susceptibility of different regions,

one wants to determine an optimal spatial distribu-

tion of the agents to maximize a coverage perfor-

mance index. In the celebrated algorithm proposed

by Lloyd (Lloyd, 1982), it is shown that centroidal

Voronoi tessellations, in which the agents converge

to centroids of Voronoi cells, are optimal with re-

spect to a coverage metric that encodes agents per-

formance with respect to points in the area, and a

scalar risk density function that is a ﬁeld assigning

a weight (that can be interpreted as importance as-

signed to) each point in the domain. Lloyd’s algo-

486

Miah, S., Nguyen, B., Bourque, A. and Spinello, D.

Non-autonomous Area Coverage and Coordination of a Multi-agent System for Harbor Protection Applications.

DOI: 10.5220/0006926804860492

In Proceedings of 8th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2018), pages 486-492

ISBN: 978-989-758-323-0

rithm can be interpreted as an iterative search for

ﬁxed points of maps deﬁned on the coverage me-

tric (Liu et al., 2009). Considerable efforts in the

literature of area coverage control have been devo-

ted to developing this idea, and generalize it to diffe-

rent multi-agent system scenarios, with time-varying

and/or space distributed network topology (that is,

inter-agent communication structures). The majority

of the work focuses on the case of time-invariant risk

density function (eventually non-uniform), to which

the classical Lloyd algorithm apply, and can, there-

fore, be used to generate optimal trajectories. Howe-

ver, there are several interesting scenarios that emerge

and are implied by time-varying risk density (Cortes

et al., 2002; Lekien and Leonard, 2009; Lee and Eger-

stedt, 2013; Lee et al., 2015), where the problem of

optimality with non-autonomous performance index

(coverage metric) has been addressed within some

restrictive hypotheses. In recent work (Miah et al.,

2015; Miah et al., 2017a) it was proposed a gene-

ral feedback control law that ensures asymptotic local

optimality with time varying environment, and there-

fore coverage metric. In technical terms, the proof of

asymptotic optimality cannot be addressed by using

classic LaSalle arguments, that apply to autonomous

metrics, but it is required the use of Barbalat’s Lemma

which imposes stronger conditions. The general con-

trol feedback law has been applied to simulate area

coverage scenarios with time varying environments,

both considering an area surveillance case with inﬁl-

tration of external targets, or an environmental moni-

toring problem with diffusion of some substance (i.e.

oil) through the boundary.

In this paper we apply these ideas to model and

simulate a harbor protection scenario with a realistic

harbor geometry. The optimal deployment of a set

of autonomous agents is time varying, to react to a

change in the risk function deﬁned on the harbor en-

vironment. The risk ﬁeld function models in a distri-

buted sense an a priori distribution of importance (sta-

tic), superimposed to a time varying part that evolves

due to the penetration of external threats through the

boundary. Locally optimal solution correspond to the

set of agents converging to the centroids of a genera-

lized Voronoi partition of the environment, with the

challenge of time varying centroids due to the evol-

ving environment. The control law proposed in (Miah

et al., 2017b) allows to react to the evolution of the

environment, with agents tracking the trajectories of

the centroids. Simulation results show the application

of the theoretical framework, and reveal important as-

pects in terms of applicability.

2 HARBOR GEOMETRY AND

SYSTEM DESCRIPTION

A high-level architecture of a harbor defence system

is depicted in Fig. 1. A high value unit is located on

the left side (area with patterns), introducing a non-

uniformity in the risk ﬁeld function deﬁned in the

domain. Sensors (sonars, for example) are mounted

on the outermost trip-wire boundary. The purpose

of the trip-wire sensors is to detect and track attac-

kers while they are in the detect and track zones (see

Fig. 1), therefore providing boundary conditions in-

formation to the system of agents deployed in the har-

bor. In the scenario illustrated in Fig. 1, four agents

are deployed inside the area (inner reaction zone),

with threats that may enter from several directions

(underwater or sideways, for instance). We consider

a decentralized architecture in which there is no cen-

tral control unit, as opposed to the scenario presented

in (Strode et al., 2009); therefore, in our case each

agent locally performs relevant computations, based

on local information and shared information with the

rest of the platoon.

HVU

(danger zone)

Autonomous agents

Trip−wire sensors (e.g., sonars)

Detect and track zone

Harbour entrance

Damage contour (innermost boundary)

2.5 km

Reference point

20 m

100 m

2.0 km

Underwater Barrier (UWB)

Detect and track zone

Trip−wire sensors (e.g., sonars)

Outermost boundary

Inner Reaction Zone (IRZ)

Island

Target

Bridge

1.5 km

Targets

Figure 1: Harbor protection system architecture.

2.1 Modeling Agents

Formally, we consider the inner defence system to

be a group of n homogeneous mobile agents where

the motion of the ith, i ∈ {1,2,...,n} ≡ I , agent is

described by a simple integrator as

[i]

(t) = u

[i]

(t),

where p

[i]

(t) =

[i]

(t), y

[i]

(t)

is the 2–D position

and u

[i]

(t), u

[i]

(t)

is its velocity vector in-

put at time t ≥ 0. The state of the agents’ therefore

collectively evolves as

p(t) = u(t), (1)

Non-autonomous Area Coverage and Coordination of a Multi-agent System for Harbor Protection Applications

487

where p(t) =

[1]

(t),...,p

[n]

(t)

∈ R

is the

state of agents’ group at time t and u(t) =

[1]

(t),...,u

[n]

(t)

∈ R

the corresponding velo-

city vector.

2.2 Time–varying Risk Field

Let Ω be the inner harbor region. A time–varying den-

sity deﬁned in Ω weights each point with a measure

of risk. In area protection problems, the risk quan-

tiﬁes the relative importance of different regions in

Ω, dictating how to distribute resources to protect the

area. In this work, the risk density ﬁeld is affected by

the motion of m homogeneous mobile targets that at

some point entered the inner harbor domain. Targets’

kinematics is simply described by

[ j]

= v

[ j]

, (2)

where s

[ j]

(t) and v

[ j]

are the 2–D position and velocity

vectors, respectively, for jth target, j = 1,...,m. We

assume that the risk density φ is twice differentiable in

Ω, and consists of a time invariant component which

can be considered as a priori independent of the tar-

gets, and of a time-varying component associated to

the motion of the targets:

φ(q,t) =

φ(q) +

∑

j=1

(q,t), (3)

where q ∈ Ω is a point in the area,

φ(q) > 0 represents

the time-invariant density in the absence of any target,

and φ

(q,t) is given by

(q,t) = exp







−



q − s

[ j]

(t)



2σ







(4)

where we have adopted Gaussian functions centered

at targets’ positions, to reﬂect the assumption that

each target contribution to the risk distribution φ is

maximal at its own position, and it decreases with the

relative distance from it. The choice of σ > 0 deter-

mines how narrow is the distribution of φ

around s

[ j]

Note that the choice of the function φ

is not unique,

but it is informed by modeling considerations. Anot-

her suitable choice is the Lognormal function, which

is often used in geological studies. The function

φ models the a priori risk, and intuitively is higher

around the high value unit, so that protection agents

allocate more resources in protecting the related area,

rather than protecting more remote regions. The a pri-

ori risk ﬁeld for this work is given by

φ(q) = exp

−

− ¯q

)

− ¯q

)

(5)

where the highest risk is in the region centered around

( ¯q

, ¯q

), which is a point characterizing the high value

area in the harbor.

2.3 Voronoi Tessellation

For optimal spatial placements and area coverage,

agents partition the area to be covered using Voro-

noi tessellations (Okabe et al., 2000), where the op-

timality has to be intended as local since the objective

function is in general nonconvex (Schwager et al.,

2011). Following (Guruprasad and Ghose, 2013),

the area Ω is partitioned in terms of Voronoi cells

V (p) = (V

(p),...,V

(p)), where the ith agent ope-

rates in the ith Voronoi cell, V

(p), deﬁned by

(p) =



q ∈ Ω : f (r

) ≥ f (r

), ∀ j ∈ I \ {i}



(6)

∀i ∈ I , r

= kq − p

[i]

k is the Euclidean distance bet-

ween the point q ∈ Ω and the ith agent position, and

f (·) is the agent’s sensor performance function, which

is differentiable in its argument. When f is the iden-

tity function, we have the classical Voronoi tessella-

tion based on the Euclidean distance r

. Here, we

adopt the Gaussian form

f (r

) = α

exp



−β



(7)

which models sensors with degrading performance

with the distance, modulated by the shape parame-

ters α

and β

. Therefore, the generators of the Voro-

noi partition are the states (p

[1]

,...,p

[n]

). For sim-

plicity, V

(p) will be denoted by V

throughout the

paper. Intuitively, V

represents an area where each

point is better sensed by the ith agent than to all other

agents. The mass and the centroid of the ith Voro-

noi cell with respect to the density φ are respectively

deﬁned as M

,t) =

φ(q,t)dΩ and C

,t) =

,t)

qφ(q,t)dΩ.

3 HARBOR PROTECTION AS AN

OPTIMAL CONTROL

PROBLEM

3.1 Problem Formulation

Consider the time-varying density map φ : Ω× R

→

as a density function representing the likelihood

that some events take place over Ω at time t. This

leads to a non-uniform time-varying distribution of

agents, where more (less) agents are deployed with

higher (lower) values of the measure φ(q,t), ∀q ∈ Ω.

Furthermore, we assume that the sensing performance

DMSS 2018 - Defence and Military

488

function f (r

) is Lebesgue measurable, homogene-

ous, and that it is strictly decreasing with respect to

the Euclidean distance r

(Okabe et al., 2000). Moti-

vated by the locational optimization problem (Okabe

et al., 2000), we consider the total coverage metric

H(p,V ,t) =

∑

i=1

f (r

)φ(q,t)dΩ, (8)

The model (8) encodes how rich the coverage in Ω is.

In other words, higher H implies that the correspon-

ding distribution of agents achieves better coverage

of the area Ω. Hence, the problem can be stated as

follows: Given the time-varying density function (3),

ﬁnd a distribution of agents such that the coverage H

is maximized, i.e.,

max

H(p,V ,t), subject to (1) as t → ∞. (9)

The coverage metric is in general non-convex, and

therefore the equilibrium conﬁgurations p correspond

to local maxima (Du et al., 2006).

3.2 Coordination Control of the

Autonomous Robots

The coverage metric (8) encodes robots’ performan-

ces and the risk ﬁeld φ, which describes the envi-

ronment to which the agents react to reach a suita-

ble spatial distribution corresponding to a partition of

the domain Ω in which each region is assigned to an

agent. When the density φ is time-invariant, geome-

tric center laws (Bullo et al., 2009, Ch. 5) on planar

agents provide a well-established solution to the co-

verage optimization problem (9), where each agent

asymptotically converge to its Voronoi centroid (or

critical point). By considering a time-varying density,

Voronoi centroids being time-varying no longer de-

ﬁne the invariant set of the agents’ trajectories. In this

case, the solution of problem (9) relies on the deﬁni-

tion of a non-autonomous feedback control law for the

velocity vector u(t) so that agents track time-varying

Voronoi centroids. A general non-autonomous feed-

back law for area coverage problems (9) has been pro-

posed in (Miah et al., 2017b), with the proof that the

trajectories generated by the feedback law maximize

the coverage metric (8) by tracking time varying ge-

neralized Voronoi centroids. Therefore, the feedback

law ensures that the group of agents can react to a time

varying environment described by a smooth risk ﬁeld

φ, and optimally redistribute spatially with respect to

the metric (8). The control is distributed in the sense

that each feedback law depends only on local infor-

mation; however, the computation requires the pre-

liminary determination of a Voronoi partition of Ω,

which in turn requires a shared knowledge of the state

of the other mobile robots, as well as an estimation of

the function φ. In a distributed scenario, this know-

ledge depends on information sharing among agents.

Asymptotic convergence with respect to random com-

munication failures has been studied in (Miah et al.,

2015).

Since the system is non-autonomous, the techni-

cal proof of asymptotic convergence requires Barba-

lat’s lemma, which poses a stronger condition that La-

Salle’s principle (uniform continuity of the second de-

rivative of the Lyapunov function.). Here we just re-

port the main result without repeating the proof, that

can be consulted in detail in (Miah et al., 2017b).

Consider the following non-autonomous feedback

control law

[i]

(t) =



∂H/∂p

[i]





∂H/∂p

[i]







− p

[i]

(t)



−

∂H

∂t



(10)

with κ > 0. In view of (1), u

[i]

is the velocity input for

the ith agent, and

and H

are deﬁned by

= −2

∂ f (r

)

∂r

φ(q,t) (11)

˜m

dΩ (12)

˜m

dΩ (13)

f (r

)

∑

j=1

[ j]

· (q − s

[ j]

)φ

(q,t)dΩ (14)

where m is the number of external targets in the ith

Voronoi cell. The following Proposition establishes

that the trajectories deﬁned by (10) asymptotically

maximize the non-autonomous coverage metric H.

Proposition 1. (From (Miah et al., 2017b)) Con-

sider the kinematic model of a team of mobile

agents (1), and the sensory performance function

f (r

) to be strictly decreasing with respect to its ar-

gument. Then the feedback law (10) maximizes the

non-autonomous coverage metric H deﬁned in (8).

4 SIMULATION

In this section we present simulations of harbor pro-

tection operations consisting of the autonomous de-

ployment and operation of a platoon of mobile ro-

bots in a harbor environment, and their time va-

rying redistribution to react to the motion of mo-

bile targets penetrating the inner protection area.

Non-autonomous Area Coverage and Coordination of a Multi-agent System for Harbor Protection Applications

489

(a) (b)

Figure 2: Evolution of density due to six mobile targets at

time (a) 20s, (b) 40 s, (c) 60 s, and (d) 100s.

The non-autonomous optimal coverage of the

agents’ nonuniform deployment in the harbor is illus-

trated by simulating a 2D inner reaction zone of the

harbor with its boundary vertices at (0, 0) m, (150,

0) m, (150, 170) m, and (0, 170) m. The high-valued

unit is located at ( ¯q

, ¯q

) = (50,85)m. Therefore the

a priori risk ﬁeld

φ(q), q ∈ Ω, around the high-valued

unit is modeled using (5) with the shape parame-

ters σ

= σ

= 20m. The trajectories of six mobile

targets (m = 6) affect the time-varying risk density

φ(q,t) deﬁned in (3) with σ = 7.5 m. The evolution of

time-varying targets’ positions s

[ j]

(t) is described by

the kinematic model of Newtonian accelerated parti-

cles, with initial speeds kv

[ j]

k = 0.5 m s

−1

, and con-

stant accelerations 0.01 m/s

, for j = 1, 2, . ..,6. Ini-

tial positions of targets are (0, 70) m, (0, 85) m, (70,

180) m, (160, 100) m, (160, 0) m, and (170, 180) m.

Figure 2 shows four snapshots of the evolving risk

ﬁeld by six mobile targets at time t = 20 s, t = 40 s,

t = 60 s, and t = 100 s. As can be seen, the risk den-

sities are the highest at the locations of targets and

around the high value unit. Five agents (n = 5) are

initially placed at positions (45, 130) m, (80, 110) m,

(75, 90) m, (60, 70) m, and (45, 50) m. Each agent

computes its Voronoi partition in the area Ω using

the generalized Voronoi partitioning model (6) with

the agents’ sensory performance function deﬁned by

f (r

) = α exp(−βr

), where α = 5 and β = 10

−2

. The

performance metric used in this simulation study is

the non-autonomous coverage metric deﬁned in (8),

which we aim to maximize. The performance of

the proposed feedback control law (10) in maximi-

zing the coverage metric, H, is summarized in Fig. 3

with feedback gain κ = 0.3. The ﬁgure visualizes

the agents approaching time varying centroids, which

0 50 100 150

100

120

140

160

X [m]

Y [m]

0.2

0.4

0.6

0.8

(a)

0 10 20 30 40 50 60 70 80 90 100

0.2

0.4

0.6

0.8

1.2

T im e [S e c ]

N o r m a l i z e d C ov e rage M e tric s

(b)

Figure 3: Performance of the proposed non-autonomous

controller using target driven risk density: (a) Agents’ opti-

mal conﬁguration at time t = 100 s (solid arrow: agent; di-

amond: centroid) and (b) coverage metric H and Lyapunov

function versus time.

evolve due to the motion of targets in the domain. The

vertical color bar in Fig. 3(a) represents the level of

density in Ω with 0 (1) corresponding to lowest (hig-

hest) density for the time span of the simulation. The

coverage metric (normalized), H, resulting from the

proposed strategy is shown in Fig. 3(b) (solid blue

line). The normalized coverage H = 0 when agents

are initially placed in the harbor environment. As the

mobile targets evolve, agents move towards their op-

timal conﬁgurations asymptotically yielding the nor-

malized coverage metric H = 1. Note that the das-

hed red line represents the Lyapunov function (used to

prove the proposed feedback law (10)) that is decre-

asing, as expected. Therefore, the feedback law (10)

places all agents in the optimal conﬁgurations asymp-

totically yielding (local) maximum coverage.

5 SUMMARY AND

CONCLUSIONS

We have applied a recently proposed non-autonomous

state-feedback control law for maximizing non-

autonomous area coverage metrics to an harbour pro-

tection problem. The non-autonomous feedback law

applies to the general class of twice differentiable risk

densities deﬁned in closed domains. This feedback

DMSS 2018 - Defence and Military

490

law allows the agents to solve the optimal control

problem of time varying formations in area coverage

problems with coordinated platoons of autonomous

mobile robots, by maximizing a non-autonomous co-

verage that encodes the platoon’s performance and

an evolving environment. Simulation results modeled

by a geometrically realistic harbor environment with

non-uniform risk deﬁned on the inner reaction zone

illustrate the applicability and effectiveness of the op-

timal control framework to address harbor protection

problems.

A fundamental assumption in this work is the

mass particle kinematic model for the robots, which

needs to be generalized to include the inertia a more

sophisticated kinematics. Moreover, simulations are

based on the assumption that all vehicles have a com-

mon knowledge of the environment (risk function φ)

and the knowledge of the state of all the other robots

in the platoon, so that Voronoi cells can be computed

at every iteration. These assumptions need to be re-

laxed in order to adhere to realistic scenarios. Current

work includes the application of reinforcement lear-

ning techniques to estimate the risk function φ. The

effect of random drop of communication data packets

to share information about robots’ positions has been

studied in (Miah et al., 2015).

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