Non-autonomous Area Coverage and Coordination of a Multi-agent
System for Harbor Protection Applications
Suruz Miah
1
, Bao Nguyen
2
, Alex Bourque
2
and Davide Spinello
3
1
Department of Electrical and Computer Engineering, Bradley University, Peoria, Illinois, U.S.A.
2
Center for Operational Research and Analysis, Defence Research and Development Canada (DRDC),
Department of National Defence, Government of Canada, Canada
3
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
Copyright © 2018 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
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
s
20 m
100 m
2.0 km
r
d
Underwater Barrier (UWB)
Detect and track zone
Trip−wire sensors (e.g., sonars)
Outermost boundary
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
˙
p
[i]
(t) = u
[i]
(t),
where p
[i]
(t) =
h
x
[i]
(t), y
[i]
(t)
i
T
is the 2–D position
and u
[i]
=
h
u
[i]
x
(t), u
[i]
y
(t)
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) =
h
p
[1]
(t),...,p
[n]
(t)
i
T
R
2n
is the
state of agents’ group at time t and u(t) =
h
u
[1]
(t),...,u
[n]
(t)
i
T
R
2n
the corresponding velo-
city vector.
2.2 Time–varying Risk Field
Let be the inner harbor region. A time–varying den-
sity defined in weights each point with a measure
of risk. In area protection problems, the risk quan-
tifies the relative importance of different regions in
, 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
˙
s
[ 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
, 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) +
m
j=1
φ
j
(q,t), (3)
where q is a point in the area,
¯
φ(q) > 0 represents
the time-invariant density in the absence of any target,
and φ
j
(q,t) is given by
φ
j
(q,t) = exp
q s
[ j]
(t)
2
2σ
2
(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
j
around s
[ j]
.
Note that the choice of the function φ
j
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
1
2
(q
x
¯q
x
)
2
σ
2
x
+
(q
y
¯q
y
)
2
σ
2
y
!!
(5)
where the highest risk is in the region centered around
( ¯q
x
, ¯q
y
), 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 is partitioned in terms of Voronoi cells
V (p) = (V
1
(p),...,V
n
(p)), where the ith agent ope-
rates in the ith Voronoi cell, V
i
(p), defined by
V
i
(p) =
q : f (r
i
) f (r
j
), j I \ {i}
,
(6)
i I , r
i
= kq p
[i]
k is the Euclidean distance bet-
ween the point q 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
2
i
. Here, we
adopt the Gaussian form
f (r
i
) = α
i
exp
β
i
r
2
i
(7)
which models sensors with degrading performance
with the distance, modulated by the shape parame-
ters α
i
and β
i
. Therefore, the generators of the Voro-
noi partition are the states (p
[1]
,...,p
[n]
). For sim-
plicity, V
i
(p) will be denoted by V
i
throughout the
paper. Intuitively, V
i
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
i
(V
i
,t) =
R
V
i
φ(q,t)d and C
i
(V
i
,t) =
1
M
i
(V
i
,t)
R
V
i
qφ(q,t)d.
3 HARBOR PROTECTION AS AN
OPTIMAL CONTROL
PROBLEM
3.1 Problem Formulation
Consider the time-varying density map φ : × R
+
R
+
as a density function representing the likelihood
that some events take place over 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 .
Furthermore, we assume that the sensing performance
DMSS 2018 - Defence and Military
488
function f (r
i
) is Lebesgue measurable, homogene-
ous, and that it is strictly decreasing with respect to
the Euclidean distance r
i
(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) =
n
i=1
Z
V
i
f (r
i
)φ(q,t)d, (8)
The model (8) encodes how rich the coverage in is.
In other words, higher H implies that the correspon-
ding distribution of agents achieves better coverage
of the area . 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
p
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 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 ,
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
u
[i]
(t) =
H/p
[i]
T
H/p
[i]
2
κ
˜
c
i
p
[i]
(t)
2
H
i
t
,
(10)
with κ > 0. In view of (1), u
[i]
is the velocity input for
the ith agent, and
˜
c
i
and H
i
are defined by
˜
φ
i
= 2
f (r
i
)
r
2
i
φ(q,t) (11)
˜m
i
=
Z
V
i
˜
φ
i
d (12)
˜
c
i
=
1
˜m
i
Z
V
i
q
˜
φ
i
d (13)
H
i
=
Z
V
i
f (r
i
)
m
j=1
v
[ j]
· (q s
[ j]
)φ
j
(q,t)d (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
i
) 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)
(c) (d)
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
x
, ¯q
y
) = (50,85)m. Therefore the
a priori risk field
¯
φ(q), q , around the high-valued
unit is modeled using (5) with the shape parame-
ters σ
x
= σ
y
= 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
2
, 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 using
the generalized Voronoi partitioning model (6) with
the agents’ sensory performance function defined by
f (r
i
) = α exp(βr
2
i
), 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
0
20
40
60
80
100
120
140
160
X [m]
Y [m]
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
(a)
0 10 20 30 40 50 60 70 80 90 100
0
0.2
0.4
0.6
0.8
1
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
H
L
(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 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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