Automatica 53 (2015) 195–207

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Distributed coverage control for concave areas by a heterogeneousRobot–Swarm with visibility sensing constraints✩

Yiannis Kantaros a, Michalis Thanou b, Anthony Tzes b

a Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC-27708, USAb Department of Electrical and Computer Engineering, University of Patras, Rio, Achaia-26500, Greece

a r t i c l e i n f o

Article history:Received 19 May 2013Received in revised form9 November 2014Accepted 7 December 2014

Keywords:Area coverage problemNon-convex environmentMobile robotic networksVisibility-based power diagram

a b s t r a c t

This article addresses the coverage problem of non-convex environments by amobile multi-robot systemcharacterized by omnidirectional ‘field of view’ sensing capabilities. The mobile robotic network isheterogeneous in terms of robots’ sensing ranges. Moreover, each robot is capable of communicatingwith all robots located within a predefined distance from its position. A distributed gradient-basedcoordination scheme is proposed, based on a novel partitioning of the environment, leading the networkto a local optimum of an area-wise criterion. Simulation studies are carried out verifying the efficiency ofthe proposed distributed control law.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The field of swarm-robotics (Martinez, Cortes, & Bullo, 2007)primarily refers to teams of mobile robots responsible for accom-plishing specific tasks. The motivation for the scientific research inthis field stems from the observation of collective behaviors of an-imals, where each individual acts autonomously. In a similar way,swarms of robots can accomplishmissions such as area explorationandmapping (Koveos et al., 2007; Tavakoli, Cabrita, Faria,Marques,& de Almeida, 2012), search and rescue (Nourbakhsh et al., 2005),intruder detection (Howard, Parker, & Sukhatme, 2006), and self-positioning in order to attain desired formations (Zavlanos & Pap-pas, 2007).

The area coverage problem is an attractive research field ofmulti-robot systems. Several distributed coordination algorithmshave been proposed for coordinating groups of robots so that the

✩ This workwas completedwhen the first author was at University of Patras. Thisresearch has been cofinanced by the European Union (European Social Fund—ESF)and Greek national funds EP-1911 through the Operational Program Education andLifelong Learning of the National Strategic Reference Framework (NSRF) ResearchFunding Program: Thalis AUEB DISFER. The material in this paper was partiallypresented at the 2014 IEEE International Conference on Robotics and Automation,May 31–June 7, Hong Kong, China. This paper was recommended for publication inrevised form by Associate Editor Andrey V. Savkin under the direction of Editor IanR. Petersen.

E-mail addresses: yiannis.kantaros@duke.edu (Y. Kantaros),mthanou@ece.upatras.gr (M. Thanou), tzes@ece.upatras.gr (A. Tzes).

http://dx.doi.org/10.1016/j.automatica.2014.12.0340005-1098/© 2014 Elsevier Ltd. All rights reserved.

sensing coverage of a given environment is maximized. Coverageproblems can be classified into dynamic ones such as sweepcoverage problems (Zhai & Hong, 2013) or persistent-awarenesscoverage problems (Song, Liu, Feng, Wang, & Gao, 2013) and staticones. In the field of static coverage problems, several controltechniques have been proposed assuming either convex or non-convex environments.

1.1. Convex area coverage

Cortés,Martinez, and Bullo (2005) propose a distributed controlscheme for a group of mobile robots equipped with sensorsof Euclidean circular footprints that locally maximizes an areaperformance criterion while in Cortés, Martinez, Karatas, andBullo (2004) the authors propose a variation of Lloyd’s algorithmfor minimizing the sensing uncertainty on a given environment,assuming that the performance of the sensors degrades accordingto the square of the distance from the sensor (distortion problem).However, since most footprints of sensing devices are not uniformand, since the sensing range of the robots that comprise a networkmay be unequal, several control schemes extend the above works,assuming either anisotropic (Gusrialdi, Hirche, Hatanaka, & Fujita,2008; Hexsel, Chakraborty, & Sycara, 2011; Stergiopoulos & Tzes,2013) or heterogeneous sensors (Bartolini, Calamoneri, La Porta, &Silvestri, 2011; Lazos & Poovendran, 2006).

196 Y. Kantaros et al. / Automatica 53 (2015) 195–207

1.2. Non-convex area coverage

Design of coordination algorithms for non-convex environ-ments requires to tackle several issues such as signal attenuationor visibility loss because of the obstacles. Different kinds of sens-ing devices have been assumed in previous works classified in Eu-clidean/Geodesic footprint (non-visibility) and visibility sensors inthe next subsections.

1.2.1. Euclidean/geodesic sensing footprintCaicedo-Nunez and Zefran (2008) propose a transformation

of non-convex domains into convex ones that allows them touse the control law presented in Cortés et al. (2004). Then thereal trajectories of the robots are obtained through the inversetransformation. A similar approach is used in Haumann, Listmann,and Willert (2010) for a frontier-based exploration of a non-convex environment, where the latter is transformed into a star-shaped domain. Breitenmoser, Schwager, Metzger, Siegwart, andRus (2010) present a solution based on Lloyd’s algorithm and apath planning method for deploying a group of nodes in a concaveenvironment. However, this method is not effective for all typesof environments, since it maximizes the coverage of the convexhull of the allowable environment rather than the coverage ofthe environment itself. In Howard, Mataric, and Sukhatme (2002),a solution is proposed for the coverage of concave areas, whereeach robotmoves at a direction determined by the repulsive forcesreceived by other robots and/or obstacles. However, this controlscheme may lead to suboptimal topologies. This fact motivatedRenzaglia et al. (2009) to extend the aforementioned scheme viaincorporation of an attractive force to the centroid of the respectiveVoronoi cell. In Savkin, Javed, and Matveev (2012) a distributedblanket coverage algorithm is presented that can be applied in bothconvex and concave environments. Although the algorithm haslow computational cost, since Voronoi partitioning is not required,it is not flexible, since the nodes are allowed to be placed only inspecific, discrete positions.

The geodesic Voronoi diagram is utilized in Pimenta, Kumar,Mesquita, and Pereira (2008), allowing deployment in non-convexenvironments. The algorithm proposed in the aforementionedarticle assumes that the sensing performance degrades accordingto the square of the geodesic, rather than the Euclidean distance. InBhattacharya,Michael, and Kumar (2010), another control strategyis presented,where eachnode is assumed tomove to the projectionof the geometric centroid of its geodesic Voronoi cell onto theboundary of the environment. Additionally, the domain of interestis considered unknown and therefore an entropy metric is usedas a density function that allows the nodes to explore and coverthe area at the same time. Also, in Thanou, Stergiopoulos, andTzes (2013b) the geodesic Voronoi tessellation is utilized for agroup of mobile sensors equipped with sensors characterized bya geodesic range-limited sensing footprint. Then a control law isextracted that guarantees (local) optimization of the area coverage.This paper has been extended by Thanou, Stergiopoulos, and Tzes(2013a) for networks with heterogeneous sensing capabilities.

1.2.2. Visibility-based sensorsHowever, many articles in the field of area coverage in non-

convex environments consider visibility-based sensors, such ascameras. In Lu, Choi, andWang (2011), the visibility-based Voronoidiagram is introduced and an algorithm based on Lloyd’s oneis implemented for a team of robots with unlimited range, om-nidirectional visibility-based sensors. Although a strict proof forcoverage optimization is not provided, the proposed algorithmachieves good results in simulations. The 3-D case of that prob-lem is addressed in Thanou and Tzes (2014) for a team of UAVsequippedwith omnidirectional visibility sensors of infinite sensing

range. Marier, Rabbath, and Lechevin (2012) utilize non-smoothoptimization techniques for the coordination of a homogeneousrobot-swarmbyminimizing the sensinguncertainty, given that theperformance of the visibility-based sensors is reduced accordingto the square of the distance. In that work, the derived controllaw requires full knowledge of the environment, which is not thecase here. In our previouswork, Kantaros, Thanou, and Tzes (2014),a control law for the distributed coverage of a non-convex envi-ronment is introduced, assuming a homogeneous group of robotsequipped with range-limited, visibility-based sensors. In Ganguli,Cortés, and Bullo (2006b), a gradient-based control law is pre-sented for maximizing the visibility domain of a single robot ina non-convex area while in Ganguli, Cortés, and Bullo (2006a) acoverage approach for an art gallery problem is presented, whererobots are characterized by ‘field of view’ sensing and communica-tion capabilities.

1.3. Contribution

This article examines the sensing problem of a non-convex areaby a group of mobile robots equipped with omnidirectional range-limited visibility-based sensing devices (e.g. cameras) of differentsensing capabilities. Additionally, each robot is assumed to be ableto communicate with all robots located within a fixed distancefrom its position. The contribution of this paper lies in the followingpoints:

• A distributed gradient-based control scheme is presented forthe sensing of a concave environment which leads a group ofheterogeneous robots equipped with omnidirectional range-limited visibility sensors to a local optimum of an area-basedcriterion. The proposed distributed algorithm can be appliedin mobile networks, where the range of the visibility sensorsis either common for all nodes or not, extending our previouswork (Kantaros et al., 2014). Moreover, although in general, weassume range-limited sensors, our algorithm can also accountfor infinite sensing ranges extending Lu et al. (2011), where thesensing radius is supposed to be infinite for all nodes. Further-more, in this paper it is proven that the proposed coordinationscheme (locally) maximizes the area coverage of the network.• Anovel partitioning schemebased on the power diagram is pro-

posed taking into account not only the nodes’ different sensingcapabilities but also the nodes’ visibility domain.

1.4. Structure

The rest of the article is organized as follows: In Section 2the area coverage problem is described and the sensing model ofthe robots is defined, along with the proposed space partitioning.The distributed coordination scheme is presented in Section 3while the simulation results presented in Section 4 illustrate itseffectiveness. The last section provides concluding remarks.

2. Coverage problem formulation

Let A ⊂ R2 be a compact, connected, non-convex in general, setdenoting an environment of interest. Let ∂(·) denote the bound-ary of a set. We assume that the boundary of the environment ∂Ais determined by a simple, concave in general, polygon. The reflex(or concave) vertices of ∂A are the vertices with inner angle greaterthan 180°. Moreover, φ(q) : A→ R+ stands for the density func-tion measuring the a-priory importance of any point q ∈ A. A mo-bile multi-robot network consisting of n robots is responsible forthe sensing coverage of A while the robots’ positions are repre-sented by the set X = {x1, x2, . . . , xn} ⊂ A × A × · · · × A = An.

Y. Kantaros et al. / Automatica 53 (2015) 195–207 197

It should be mentioned that throughout the rest of the paper theterms robot and node will be used interchangeably.

Robots are assumed to move according to the following firstorder differential equation:

xi = ui, ui ∈ R2, xi ∈ A, i = 1, . . . , n, (1)

where ui is the control input associated with the ith robot. Con-sidering coverage purposes, the control input ui should be selectedin such a way so that the nodes’ motion leads to the optimizationof an area-based performance criterion. This performance criterioncan be formulated as follows:

H (X) =

A

maxi=1,...,n

fi(xi, q)φ(q)dq, (2)

where fi : A × A→ R is the performance function measuring thesensing performance of the node i, located at xi, on point q ∈ A.

2.1. Sensing model

A crucial issue in a coverage problem is the selection of nodes’sensing model fi(xi, q). In Cortés et al. (2005), uniform range-limited footprints were utilized for the case of convex envi-ronments. However, this sensing model is not realistic in mostpractical scenarios that involve non-convex domains of interest. Inthese domains, there are often obstacles between the signal sourceand the sensors that result in poor sensing performance. There-fore, a different sensingmodel is proposed by Thanou et al. (2013b)that takes into account the degradation of signals due to the non-convexities of the environment. However, this model cannot de-scribe visibility-based sensors such as cameras.

Visibility-based sensors have been used in many works in thefield of coordination of mobile robotic networks. The authors in Luet al. (2011) have utilized omnidirectional visibility-based sens-ing devices with infinite sensing range. Assuming that the per-formance of a sensing device is independent of the distance issomehow unrealistic, though, since sensing results may be un-reliable after a certain distance. Therefore, in this work, omnidi-rectional range-limited visibility sensors are adopted in order toovercome this issue.

Prior to the mathematical description of the sensor’s model,define the visibility polygon corresponding to an arbitrary node.

Definition 1. The visibility polygon VP i of an arbitrary node i,positioned at xi, is defined as the polygonal region of all pointsq ∈ A that are visible from node i. This can be expressed as follows:

VP i= {q ∈ A|[xi, q] ∈ A}, (3)

where [xi, q] is the line segment connecting xi and q.Taking into account Eq. (3), the selected sensing pattern can be

modeled as a visibility disc defined as follows:

Definition 2. The visibility disc of a node i, denoted by C i, is de-fined as the set of points q ∈ VP i being in distance less than orequal to the sensing radius, ri of the ith node, i.e.:

C i= {q ∈ VP i

∩ Bi}, (4)

where Bi= {q ∈ A| ∥q− xi∥ ≤ ri}.

Fig. 1 shows the visibility polygon and visibility disc corre-sponding to a node located at a non-convex environment.

Assumption 1. In this paper, robots are assumed to be equippedwith omnidirectional visibility sensors, such as cameras, sensingthe points residing in their respective visibility disc as defined in(4). However, it should bementioned that the sensing capability ofthe network is characterized by a heterogeneity in terms of nodes’sensing range.

Fig. 1. Graphical illustration of the visibility polygon (left) and visibility disc (right)corresponding to the black node.

Based on the above assumption the sensors’ performance func-tion can be expressed as:

f (xi, q) =1 if q ∈ C i

∩ A0 otherwise, = 1(q)C i∩A (5)

where 1(q)S is the indicator function in a set S defined as 1(q)S =1, ∀q ∈ S and 1(q)S = 0, ∀q ∈ S.

2.2. Non-convex area tessellation

Several techniques stemming from the Voronoi partitioning(Aurenhammer & Klein, 1999) have been proposed for the tessel-lation of a domain into subsets according to Euclidean or geodesicmetric (Pimenta et al., 2008). However, when the nodes have dif-ferent sensing capabilities, these techniques fail to provide regionsof responsibility amongst the nodes in a ‘fair’ way (Stergiopoulos &Tzes, 2010). Therefore, other tessellation strategies have been pro-posed that take into account the heterogeneity of the network. Oneof these strategies relies on the power diagram, which is defined asfollows:

Definition 3. The power diagram generated by X is the set Vpd =V 1pd, . . . , V

npd

, i = 1, . . . , n, where V i

pd refers to the power cell ofnode i, which is defined as:

V ipd = {q ∈ A | ∥q− xi∥2 − r2i ≤

q− xj2 − r2j , ∀j = i}. (6)

Although the power diagram does not ignore the differentsensing capabilities between the nodes, it ignores their visibilityconstraints. Therefore, the following semi-metric (Marier et al.,2012), based on the Laguerre distance, will be used:

d(q, xi, ri) =∥q− xi∥2 − r2i if q ∈ VP i

∞ otherwise. (7)

Considering the above equation, the visibility-based power dia-gram is introduced in Definition 4,which is a variation of the powerdiagram, that takes into consideration the visibility constraints ofthe nodes.

Definition 4. The visibility-based power diagram generated by Xis the set Vvpd =

V 1

vpd, . . . , Vnvpd

, i = 1, . . . , n, where V i

vpd isthe visibility-based power cell of node i, and contains the pointsthat are visible to node i and closer to it than to any other nodeaccording to Laguerre distance, i.e.

V ivpd = {q ∈ VP i

| d(q, xi, ri) ≤ d(q, xj, rj), ∀j = i}. (8)

In Fig. 2, the power diagram and the visibility-based powerdiagram are applied in a concave environment. It is evident thatthe proposed partitioning (right) is more realistic for visibility-based sensors than the typical power diagram (left). For example,regarding the first node, the power diagram assigns non-visibleparts as well, which is not the case for the visibility-based one.

198 Y. Kantaros et al. / Automatica 53 (2015) 195–207

Fig. 2. Space partitioning according to power diagram (left) and visibility-basedpower diagram (right).

Lemma 1. The visibility-based power diagram, V =V 1

vpd, . . . , Vnvpd

i = 1, . . . , n, provides a full tessellation of the total visible region in-side the domain i.e.:

i=1,...,n V

ivpd =

i=1,...,n VP

i.

Proof. This is a straightforward consequence of the Definition 4,since

i=1,...,n V

ivpd =

i=1,...,n{q ∈ VP i

| ∥q− xi∥2 − r2i ≤q− xj2 − r2j , ∀j = i}which is equal to

i=1,...,n{VP

i}.

In other words, every point inside the union of the visibilityregions of all nodes is assigned to a node. Note, however, thatthe total visible area is a subset of the domain of interest,i.e.

i=1,...,n Vivpd ⊆ A. �

Remark 1. Visibility-based power cellsmay be non-convex and/ordisconnected sets. The disconnected components of the cells arenot discarded as in Lu et al. (2011), but they are still contributingto the control law. A sample case of a disconnected cell is depictedin Fig. 5.

Remark 2. Generators of the visibility-based power diagram maynot lie inside their corresponding cell. For example, node 2 in Fig. 2is located outside its visibility-based power cell. The followinglemma provides the necessary condition that should hold so that anode is located inside its cell.

Lemma 2. The necessary condition for xi ∈ V ivpd is ∥xi − xj∥2 ≥

r2j − r2i , for all j = i for which it holds that xj ∈ VP i.

Proof. From Definition 4, xi is located in V ivpd if d(xi, xi, ri) ≤

d(xi, xj, rj), ∀j = i. If xj ∈ VP i, this expression can be analyzedusing (7) as−r2i ≤ ∥xi−xj∥

2−r2j or∥xi−xj∥2 ≥ r2j −r

2i . On the other

hand, if node j is not visible to i, that is xj ∈ VP i, then the relationd(xi, xi, ri) ≤ d(xi, xj, rj) is written using (7) as d(xi, xi, ri) ≤ ∞,which always holds. �

Lemma 3. If two nodes i and j coincide and ri < rj, then the visibility-based power cell assigned to node i is empty, i.e. V i

vpd = ∅.

Proof. Since nodes i and j coincide, it follows that xi = xj. Thevisibility-based power cell of node i can be written using Defini-tion 4 as:

V ivpd =

d(q, xi, ri) ≤ d(q, xk, rk), k = 1, . . . , n \ {j}, andd(q, xi, ri) ≤ d(q, xj, rj).

(9)

Since xi = xj, it holds that VP i= VP j, and the second relation in

(9) becomes ∥q − xi∥2 − r2i ≤ ∥q − xi∥2 − r2j or ri ≥ rj. Sinceit was assumed that ri < rj, the second relation does not hold forany q resulting in V i

vpd = ∅. The only case where both cells are nonempty is when ri = rj, where V i

vpd = V jvpd. �

Assuming that sensors may have a range-limited visibility do-main, the range-limited visibility-based power diagram can be de-fined as follows:

Fig. 3. Graphical depiction of range-limited visibility-based power diagram in thetopology presented in Fig. 2.

Definition 5. The range-limited visibility-based power diagramgenerated by X is the set Vr,vpd =

V 1r,vpd, . . . , V

nr,vpd

, i = 1, . . . , n,

where V ir,vpd is called the range-limited visibility-based power cell

of node i, and contains the points that belong to both V ivpd and C i,

i.e.:

V ir,vpd = {q ∈ V i

vpd ∩ C i}. (10)

Through the proposed partitioning certain neighboring relation-ships arise between the nodes. Specifically, any two nodes arecalled range-limited visibility-based power Delaunay neighbors, ifthey share an edge (or a part of an edge) of their correspondingrange-limited visibility-basedpower cells. Range-limited visibility-based power Delaunay neighbors corresponding to an arbitrarynode i, are described by the set Dr

i which is defined as:

Dri =

j = i | V i

r,vpd ∩ V jr,vpd = ∅ (non singleton)

. (11)

Fig. 3 shows the range-limited visibility-based power diagramof six nodes. It should be mentioned that the cells V 2

r,vpd and V 6r,vpd

are empty. Therefore, the sensing footprints of nodes 2 and 6 aredesigned with a dashed line so that the sensing capabilities of allthe robots can be easily compared with each other.

For the distributed evaluation of the range-limited visibility-based power diagram, each node i should communicate with allnodes j ∈ Dr

i . For instance, in Fig. 3, node 1must know the positionx3, since Dr

1 = {3}. The minimum communication range requiredfor this purpose is motivated by the following lemma.

Lemma 4. Distributed evaluation of the ith range-limited visibility-based power cell is guaranteed if node i can communicate with allnodes that belong to the set Bi

c , defined as

Bic = {q ∈ A| ∥q− xi∥ ≤ ri + Rmax} , (12)

where Rmax = maxi=1,...,n ri.

Proof. Distributed evaluation of the V ir,vpd requires the knowledge

of xj, ∀j ∈ Dri . Assume that j ∈ Dr

i meaning that V ir,vpd ∩ V j

r,vpd = ∅.Let xc ∈ V i

r,vpd ∩ V jr,vpd. Obviously, it holds that ∥xi − xc∥ ≤ ri andxj − xc

≤ rj ≤ Rmax. Applying the triangle inequality, it yieldsthat

xi − xj ≤ ∥xi − xc∥ +

xj − xc ≤ ri + Rmax entailing that

xj ∈ Bic . Thus, if j ∈ Dr

i then xj ∈ Bic . In case Dr

i = ∅, it is eitherimplied that V i

r,vpd = ∅ or V ir,vpd = Ci. In the latter case, node i does

not need to communicate with any node. As for the first case, it isimplied that i lies in the sensing domain of some other node, j andconsequently, their inter-distance is less than Rmax. Thus, in thiscase, considering the set Bi

c , node i can still communicatewith node

Y. Kantaros et al. / Automatica 53 (2015) 195–207 199

j, so that it can compute that V ir,vpd = ∅ and not equal to Ci (such a

case is the relation between node 2 and 1 in Fig. 3), completing theproof. �

Based on the above lemma, wemake the following assumption:

Assumption 2. Each node is assumed to be able to communicatewith any node j provided xj ∈ Bi

c . In other words, each robot isequipped with an antenna of communication range rci = ri+Rmax.

Note that if the communication range had been selected to begreater than or equal to the diameter of the convex hull of A, theneach robot would trivially know the position of any other robot.However, this is not the case here, since the communication rangerci is only related to the robots’ sensing capabilities and not to theenvironment A. Therefore, each node is only aware of the nodesresiding in the Euclidean disk Bi

c described above.The following algorithm describes how the robots compute

the range-limited visibility-based power diagram in a distributedway, provided that their communication capabilities are thosedescribed by Lemma 4. At the beginning of the algorithm, the ithnode initializes its V i

r,vpd with its visibility-disc (Line 2, 3) and afterthat, the (range-limited) visibility area sharedwith a node j, xj ∈ Bi

cis computed (Line 6). The eighth line implies that this commonvisibility area is tessellated according to the power Voronoidiagram and the region V j

pd(T ) is assigned to node j. Then, the V ir,vpd

is updated subtracting from it the region (V jpd(T )∩ V i

r,vpd) (Line 9).

Algorithm 1 Evaluation of Range-Limited Visibility-based PowerDiagram

At each t > t0, local robot i performs:1: Identify robots’ positions residing in Bi

c2: Evaluate Ci3: Initialize V i

r,vpd ← Ci

4: for j ∈ {{1, . . . , n}, j = i | xj ∈ Bic } do

5: Evaluate Cj6: T ← Ci ∩ Cj7: if T = ∅ then8: Evaluate V j

pd(T )

9: V ir,vpd ← V i

r,vpd \ (V jpd(T ) ∩ V i

r,vpd)10: end if11: end for

An illustrative explanation of Algorithm 1 is presented in Fig. 4.On the upper left part, the light yellow shaded region correspondsto the initial value of V 1

r,vpd. As the algorithm progresses (Fig. 4from left to right and from top to bottom), the parts that belongto V j

pd(T )∩V ir,vpd are subtracted from V 1

r,vpd according to Algorithm1, Line 9. Note that node 4 cannot affect V 1

r,vpd, since it is locatedoutside Bi

c .

Lemma 5. Introducing the set,

S = {X ∈ An|xi = xj and ri = rj for some

i, j ∈ 1, . . . , n, i = j}, (13)

which implies that two or more nodes with equal sensing radii coin-cide, adopting the partitioning policy described in (8) and using As-sumption 1, the area-based performance criterion (2) can be simpli-fied as

H (X) =

ni=1

V ir,vpd

φ(q)dq, (14)

for all X ∈ An\ S.

Fig. 4. Computation of the range-limited visibility-based power cell of node 1. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

Fig. 5. Graphical representation of notations. (For interpretation of the referencesto color in this figure legend, the reader is referred to theweb version of this article.)

Proof. Considering Eq. (8), the performance criterion (2) can bewritten as:

H (X) =

A

maxi=1,...,n

1(q)C iφ(q)dq. (15)

This integral expresses the φ-weighted area of the region coveredby the robots. It can be easily proven that the edges of the visibility-based power diagram pass through regions that are not sensed,or regions that are sensed by two nodes or more. As a result, ifa point q is sensed by a node i, but assigned to j, then it is alsosensed by j. Using this fact and Lemma 1, the result follows forX ∈ An

\S. Now suppose that twonodes, let i and j coincide. If ri = rjthen, using Lemma 3 one of the sets V i

r,vpd, Vjr,vpd will be empty

which implies that the simplification of the coverage objective isvalid. On the other hand, if ri = rj then the visibility-based powercells will coincide, and their (weighted) area will be added twice in(14). However, in the area integral (15) only one of the overlappingregions would be added, resulting that the lemma holds only forX ∈ An

\ S. �

Throughout the rest of the paper, for the sake of simplic-ity, we will use the names of power diagram, range-limitedpower diagram and range-limited Delaunay neighbors instead

200 Y. Kantaros et al. / Automatica 53 (2015) 195–207

of visibility-based power diagram, range-limited visibility-basedpower diagram and range-limited visibility-based power Delaunayneighbors, respectively.

3. Gradient-based Coordination scheme

In this section, the main results of this paper are presented.Particularly, we first develop a distributed controller for solvingthe area coverage problem described in Section 2 that guaranteesmonotonic non-decreasing coverage performance (Theorem 1).Next, considering the proposed controller the validity of the resultpresented in Lemma 5 is examined (Proposition 1) and then theconvergence of the proposed algorithm is shown in Theorem 2.Before presenting the main results let us first describe the basicnotations that will be used in stating Theorem 1.

As mentioned before, ∂V ir,vpd stands for the boundary of the

range-limited power cell of node i. Considering that ∂V ir,vpd con-

sists of arcs and edges, one can denote its arcs by ∂V ir,vpd ∩ ∂Bi

and its edges by ∂V ir,vpd \ ∂Bi. The last mentioned set contains

edges belonging either to: (a) the boundary of the environment∂A, or (b) another node’s range-limited power cell ∂V j

r,vpd, or (c)exclusively to V i

r,vpd. The latter edges can be represented by the setPi = (∂VP i

∩ ∂V ir,vpd) \ (∂A ∪ (

j∈Di

∂V jr,vpd)) and as depicted in

Fig. 5, this set consists of multiple non-intersecting edges. Repre-senting a line segment with endpoints p1 and p2 as [p1, p2], the setPi can be rewritten as:

Pi = (∂VP i∩ ∂V i

r,vpd) \

∂A ∪

j∈Di

∂V jr,vpd

=

pia,1, pib,1

∪pia,2, p

ib,2

∪ · · · ∪

pia,k, p

ib,k

, (16)

where k is the number of the aforementioned edges. Withoutloss of generality, we consider ∥pil,a − xi∥ < ∥pil,b − xi∥, ∀i =1, . . . , n, ∀l = 1, . . . , k.

By definition, the edges contained in Pi are parts of the set∂VP i\ ∂A. The latter set defines the frontier of a node’s visibility

region, which consists of line segments having a reflex vertex of∂A on one of their endpoints (Fig. 1). These reflex vertices can bedescribed as the points that block the visibility of a node. Let pir,l bethe reflex vertex that blocks the visibility of node i and results inthe frontier described by [pia,lp

ib,l]. Since pir,l, p

ia,l, p

ib,l belong to the

same frontier of the visibility region of i, they are collinear and itholds that:

∥xi − pir,l∥ ≤ ∥xi − pia,l∥ < ∥xi − pib,l∥

for all i = 1, . . . , n and l = 1, . . . , k. The equality holds if pir,l ≡pia,l. In addition to the above, let ni

o(q) or nio denote the outward

normal unit vector at any q ∈ ∂V ir,vpd and projvu the projection of

a vector u onto v.Fig. 5 illustrates the above notations. The brown (blue) colored

arcs stand for the set ∂V 1r,vpd ∩ ∂B1

∂V 2

r,vpd ∩ ∂B2while the red

[orange] edges represent the set P1 [P2]. One can note that p1r,1 =p2r,1 = p1a,1, which is the reflex vertex of the polygon.

Using these notations, the proposed control law can be formu-lated as follows in the next theorem.

Theorem 1. Let a mobile heterogeneous network consisting of nmobile nodes be equipped with visibility sensors, described by (4).Assuming that the nodes start at X(t0) = X0 ∈ An

\ S, the controlscheme

ui =

proj∂A(ui) if (xi ∈ ∂A) ∧ (ui points outward of A)ui otherwise,

where

ui =

∂V i

r,vpd∩∂Biφ(q)ni

o(q)dq

+

l=1,...,k

−ni

opir,l − xi δlmax

δlmin

φ(Q )δdδ

, (17)

Q = pir,l + δpr,l−xi∥pr,l−xi∥

, δlmin = ∥pa,l − pr,l∥ and δl

max = ∥pb,l − pr,l∥achieves sensing coverage of A, locally maximizing the area-basedcriterion (14) in a non-decreasing way.

Proof. Interested in designing a gradient-ascent controller formaximizing the performance function (14), we first compute thederivative of the coverage objective function H with respect to xii.e.:

∂H

∂xi=

∂

∂xi

V ir,vpd

φ(q)dq+∂

∂xi

j=i

V jr,vpd

φ(q)dq. (18)

Applying the Leibniz integral rule (Flanders, 1973), Eq. (18) canbe simplified as:

∂H

∂xi=

∂V i

r,vpd

(nio)

T ∂q∂xi

φ(q)dq

+

j=i

∂V j

r,vpd

(njo)

T ∂q∂xi

φ(q)dq. (19)

The above equation may be analyzed expanding the set ∂V ir,vpd.

Specifically, as mentioned previously, the boundary of V ir,vpd

consists of edges residing either on ∂A, ∂V ir,vpd ∩ ∂V j

r,vpd, ∂Vir,vpd ∩

∂Bi, or on Pi. Considering this, Eq. (19) can be rewritten as:

∂H

∂xi=

∂V i

r,vpd∩∂Bi(ni

o)T ∂q∂xi

φ(q)dq+Pi(ni

o)T ∂q∂xi

φ(q)dq

+

∂A∩∂V i

r,vpd

(nio)

T ∂q∂xi

φ(q)dq

+

j=i

∂V i

r,vpd∩∂Vjr,vpd

(nio)

T ∂q∂xi

φ(q)dq

+

j=i

∂V i

r,vpd∩∂Vjr,vpd

(njo)

T ∂q∂xi

φ(q)dq. (20)

Considering an infinitesimal motion of the node i, it holds that∂q∂xi= 0, ∀q ∈ ∂A, which means that the third term of (20) is zero.

Furthermore, for the points q ∈ ∂V ir,vpd ∩ ∂V j

r,vpd, it can be easilydeduced that (ni

o)T ∂q

∂xi= −(nj

o)T ∂q

∂xi, since these points move at the

same speed but at opposite directions. Thus, the sums in fourth andthe fifth term of (20) are canceled out. Considering the aforemen-tioned simplifications, Eq. (20) can be expressed as:

∂H

∂xi=

∂V i

r,vpd∩∂Bi(ni

o)T ∂q∂xi

φ(q)dq+Pi(ni

o)T ∂q∂xi

φ(q)dq. (21)

Further simplification of (21) can be attained considering that thepoints q ∈ ∂V j

r,vpd ∩ ∂Bi move along the direction of node i at thesame speed. Therefore, concerning the first integral of (21), it holdsthat:

(nio)

T ∂q∂xi= (ni

o)T , ∀q ∈ ∂V i

r,vpd ∩ ∂Bi. (22)

The second integral of (21) can be further analyzed by determin-ing the equation that describes the points q ∈ Pi. Let q ∈ [pa, pb],

Y. Kantaros et al. / Automatica 53 (2015) 195–207 201

where [pa, pb] is an arbitrary edge contained in the set Pi, and pr bethe associated reflex vertex with this edge. Considering these, thepoints q ∈ Pi can be described as:

q = pr + δnip, (23)

where

• ∥pa − pr∥ ≤ δ ≤ ∥pb − pr∥• ni

p is a unit vector defined as nip ≡

pr−xi∥pr−xi∥

and it holds thatnip⊥n

io.

Let pr,x, pr,y be the x, y coordinates of the vector pr . This nota-tion extends to the rest of the vectors appearing in the rest of thisproof. Taking the partial derivative of (23) w.r.t. to xi and using thefact that ∂pr

∂xi= 0, since pr ∈ A, we have:

∂q∂xi= δ

∂nip

∂xi, (24)

where the symmetric Jacobian matrix∂nip∂xi

is equal to:

∂nip

∂xi=

∂ni

p,x

∂xi,x

∂nip,x

∂xi,y∂ni

p,y

∂xi,x

∂nip,y

∂xi,y

= 1∥pr − xi∥3

m1,1 m1,2m1,2 m2,2

, (25)

wherem1,1 = −pr,y − xi,y

2,m1,2 =pr,y − xi,y

pr,x − xi,x

and

m2,2 = −pr,x − xi,x

2.Considering the property of orthogonality between the unit

vectors nip(q) and ni

o(q), it is deduced for q ∈ [pa, pb] that:

(nip(q))

T· ni

o(q) =pr,x − xi,x, pr,y − xi,y

·nio,x, n

io,y

T= 0. (26)

Combining Eqs. (25) and (26), the following result is derived:

(nio)

T ∂nip

∂xi=

1∥pr − xi∥

−ni

o,x, −nio,y

= −

(nio)

T

∥pr − xi∥. (27)

Using Eqs. (24) and (27), and the fact that the set Pi includes kedges, the second integral in (21) can be rewritten as:

Pi(ni

o)T ∂q∂xi

φ(q)dq

=

l=1,...,k

−(ni

o)Tpir,l − xi δlmax

δlmin

φ(Q )δdδ

, (28)

where δlmin = ∥pa,l − pr,l∥, δl

max = ∥pb,l − pr,l∥ and Q = pir,l+ δ

pr,l−xi∥pr,l−xi∥

. In the above relation, the arc length dq has been ex-pressed as dq = dδ. Moreover, the density function φ(q) can bewritten using (23) as φ(q) = φ

pir,l + δni

p

or equivalently, as

φ(q) = φpir,l + δ

pr,l−xi∥pr,l−xi∥

.

Plugging equations (22) and (28) into (21) yields the followingresult for the gradient of the coverage objective function:

∂H

∂xi=

∂V i

r,vpd∩∂Binioφ(q)dq

+

l=1,...,k

−ni

opir,l − xi δlmax

δlmin

φpir,l + δni

p

δdδ

. (29)

However, the gradient direction ∂H∂xi

, given by (29), may pointoutward of the environment A. This implies that if a node i is lo-cated at the boundary of A and it moves along a gradient direction

Fig. 6. Projection of the gradient vector ui to the boundary of the environment ∂A.

pointing outwards of A, then the node will move outside the envi-ronment A. Such a case is depicted in Fig. 6. This can be avoidedby projecting the gradient direction ∂H

∂xito the boundary of the

environment A, in case the node i is about to leave A. Hence, theproposed controller can be expressed as:

ui =

proj∂A(ui) if (xi ∈ ∂A) ∧ (ui points outward of A)ui otherwise. (30)

Moreover, note that the projected gradient can be defined as:

proj∂A(ui) =∂H

∂xicos θi, (31)

where θi ≤ π/2 is the acute angle between ∂H∂xi

and the tangentof ∂A at xi (Fig. 6).

Considering the above along with that dHdt =

ni=1

∂H∂xi

dxidt =n

i=1∂H∂xi

ui, we conclude that:

dH

dt=

ni=1

∂H

∂xi

2 cos θi ≥ 0

if (xi ∈ ∂A) ∧ (ui points outward of A)

ni=1

∂H

∂xi

2 ≥ 0 otherwise,

(32)

proving that the proposed control law (17) increases monotoni-cally the coverage performance, which completes the proof. �

Remark 3. For the proposed distributed control scheme thefollowing points are worth mentioning:• The controller (17) can be applied both to convex and non-

convex domains. In the case of convex ones, there are novisibility limitations, implying that the set Pi is empty. Thus,the second part of (17) is zero meaning that the control law isidentical to that presented in Cortés et al. (2005):

ui =

∂V i

r,vpd∩∂Biφ(q)ni

odq. (33)

• The proposed control scheme is also applicable in mobilenetworks where robots are assumed to be equipped withvisibility-based sensors of unlimited sensing range as in Lu et al.(2011). In these cases, the sets ∂V i

r,vpd ∩ ∂Bi, are empty for alli = 1, . . . , n and the proposed control scheme is simplified asfollows:

ui =

proj∂A(ui,b)if (xi ∈ ∂A) ∧ (ui,b points outward of A)

ui,b otherwise,(34)

where ui,b =

l=1,...,k

−niopir,l−xi

δlmaxδlmin

φ(Q )δdδ.

202 Y. Kantaros et al. / Automatica 53 (2015) 195–207

Fig. 7. Nodes’ placement at t∗−ϵ; red arcs denote parts ofW∩Z . (For interpretationof the references to color in this figure legend, the reader is referred to the webversion of this article.)

• The control scheme (17) can be characterized as distributed,since for the evaluation of the ith range-limited visibility-basedpower cell only communication with a limited set of nodes isrequired (Lemma 4).• Full knowledge of A is not required, since for the computation of

the range-limited cells of the nodes (V ir,vpd) only the nodes that

reside in Bic are required (Algorithm 1).

Next we prove that if X0 ∈ S and the network evolves accordingto the controller (17) then it is guaranteed that X(t) ∈ S, ∀t > t0.In doing so, we show that the result of Lemma 5 is valid duringnetwork evolution.

Proposition 1. Assuming amobile heterogeneous network, under theassumptions of Theorem 1, that evolves under the controller (17), itis guaranteed that if X0 ∈ S then X(t) ∈ S, ∀t > t0.

Proof. The proof is based on contradiction. Assume that there is at∗ ∈ (t0,∞) such that xi(t∗) = xj(t∗) for some i = j and ri = rj.Then, ∃ϵ > 0 : ∀t ∈ (t∗ − ϵ, t∗),

xj ∈ VPi

∧j ∈ Dr

i

. Let

the distance between these two nodes at t∗ − ϵ be d (Fig. 7) andconsider the unitary vector υ =

xj−xi∥xj−xi∥

. The control vector ui canbe decomposed into two vectors as ui = ui,a + ui,b where ui,a [ui,b]refers to the direction determined by the arcs [the set Pi] of therange-limited Voronoi cell, as shown in Fig. 8.

Assume that the first term of ui,a leads the node i toward j.Moreover, suppose that node i does not need to move along theprojected gradient direction. Defining the sets Z = {q ∈ V i

r,vpd|(q−xi)T · υ > 0} andW = ∂V i

r,vpd ∩ ∂Bi (see also Fig. 7), the vector ui,acan be written as

ui,a =

W∩Z

φ(q)nio(q)dq+

W\Z

φ(q)nio(q)dq, (35)

where the two terms in (35) result in vectors pointing toward andaway from node j, respectively.

Since ui,a leads the node i to j, it holds that ui,a ·υ > 0, resultingin

W∩Zφ(q)ni

o(q)dq cosω1 >

W\Z

φ(q)nio(q)dq

cosω2, (36)

where ω1 is the angle between υ and the vector obtained from thefirst integral in (35) and ω2 is the angle between−υ and the resultof the second integral in (35). It can be shown that ω1, ω2 < π/2.Since we assumed that node i moves toward j, it holds that d→ 0and m(W ∩ Z) → 0, where m(·) denotes the measure of a set. In

Fig. 8. Graphical explanation of the proposed control law. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version ofthis article.)

order for (36) to be valid, m(W \ Z)→ 0, resulting in m(W )→ 0.This contradicts the fact that the direction of ui,a results in maxi-mizing the area covered by node i. Therefore, ui,a cannot drive nodei toward node j. In case node i needs to move along the projectedgradient direction, the same logic proves our assertion. Note thatthe only difference would occur in the inequality (36), where bothof its sideswould bemultiplied by |cos θi| ≥ 0 (with θi correspond-ing to the projection angle as defined in the proof of Theorem 1). If|cos θi| > 0, then this term can be eliminated from this inequalityand the rest of the proof is the same as above. The remaining case iscos θi = 0, which does not satisfy the aforementioned inequality.Therefore, in neither case, node i is driven toward j. Consideringthe above, we deduce that if ri = rj the direction stemmed fromthe arcs of ∂V i

r,vpd cannot drive node i toward j, since the coverageobjective function is decreased. Therefore, the gradient directionover the arcs must deviate at least 90° from the direction υ .

Also, assume that ui,b directs node i toward j at time instantt∗ − ϵ, which equivalently means that there is a set of edges Pithat causes this direction. Consequently, this implies that node jhas ‘larger’ visibility over at least one of the halfspaces, defined bythe line

q ∈ A |

q− xj2 = ∥q− xi∥2

, than node i. Let denote

this halfspace by Hj (see, for instance, the halfspace H1 in Fig. 8).This equivalently means that there are no points q ∈ Hj for whichit holds q ∈ VP i and q ∈ VP j. Moreover, assuming that ui,b directsnode i toward j means that there are edges in the set Pi that drivenode i toward j. Consequently, these edges belong to the aforemen-tioned halfspace and are not visible fromnode j, by definition of theset Pi, as well. However, this is impossible considering the afore-mentioned visibility property of theHj. Thus, our initial assumptionis invalid. Note that this reasoning also excludes the directions thatdeviate at most 90° from the unitary vector υ . Specifically, if thegradient direction over the set Pi deviates from the unitary vectorυ less than 90°, then this implies that the latter direction can alsoincrease visibility with the same implications as described above,leading to a contradiction. A similar argument can be used if the ithnode belongs to the boundary of the region and moves accordingto the projected gradient; if the node i moves along the projectedgradient having the same direction with υ , then the previous logicleads to a contradiction. Therefore, if that robot needs to move ac-cording to the projected gradient, it willmove toward the direction−υ .

Combining the aforementioned results, we conclude that thevector sum of ui,a and ui,b cannot also drive the ith node towardthe jth node, completing our proof. �

The convergence of the switching closed loop system as definedby (1)–(17) is examined in the next theorem.

Y. Kantaros et al. / Automatica 53 (2015) 195–207 203

Fig. 9. Simulation study 1-Proposed control scheme is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from originating (red) to converged (green) dot-positions. (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Theorem 2. Assume a mobile heterogeneous network under theassumptions of Theorem 1 that evolves according to the controllaw ui described by (17). Then the dynamical system (1) convergesasymptotically to the set E = {X ∈ An

\ S | ∥ui∥ = 0, ∀i =1, . . . , n}.

Proof. The convergence of the dynamical system (1) when thecontrol law (17) is applied can be proven considering the switchedsystem

xi = (1− ρi)∂H

∂xi+ ρi

proj∂A

∂H

∂xi

, i = 1, . . . , n (37)

where ρi = {0, 1} is the switching index associated with systemi and ρi = 1 if the gradient direction ui as defined in Theorem 1points outward of A and xi ∈ ∂A. Otherwise, we have that ρi = 0.

Let V (X) ≡ H (X)−1 be a candidate Lyapunov function for thesystem (37), which is positive definite, since H (X) > 0, for allX ∈ An

\ S. The time derivative of function V (X) can be written as

dV (X)

dt=

1,...,n

∂V (X)

∂xi

∂xi∂t

= −

i=1,...,n

(1− ρi)

∂H

∂xi

2 + ρi

∂H

∂xi

2 cos θi

. (38)

Since V (X) ≤ 0 for i = 1, . . . , n and ρi = {0, 1}, it follows thatV (X) is a common weak Lyapunov function for system (37). Notealso that A is a compact set by assumption and weakly invariant,since according to the control law (17) if x(t0) ∈ A then x(t) ∈ A,∀t > t0. Then, according to the generalization of the LaSalle’sinvariance principle for switched systems, proposed by Bacciottiand Mazzi (2005), and Proposition 1, the system converges to theset E = {X ∈ An

\S | ∥ui∥ = 0 or θi = π/2, ∀i = 1, . . . , n} = {X ∈An\ S | ∥ui∥ = 0,∀i = 1, . . . , n}, provided that X0 ∈ An

\ S. �

In Fig. 8, the proposed control law is illustrated. The set ∂V 1r,vpd∩∂B

1

is representedwith brown color. Also the set P1 contains two edges,depicted with red and blue colors. The brown vector stands for thefirst term of (17), while the red and blue ones represent the resultof the second term of (17). Since ∥p1a,1 − xi∥ < ∥p1a,2 − xi∥ and∥p1b,1 − p1a,1∥ > ∥p1b,2 − p1a,2∥ the blue vector has larger magnitudethan the red one (assuming uniform density function). The vectorsum of the aforementioned vectors (black vector) results in theproposed control action, u1.

4. Simulation studies

This section provides simulation studies for a mobile roboticnetwork that is coordinated using the proposed control law. Inall simulations non-convex environments A with regions of equalimportance will be considered. Therefore, we assume a uniformdensity function φ defined over the area A, i.e. φ(q) = 1

A dq .In the first simulation study, a concave environment with

normalized areaA dq = 1 should be sensed by a heterogeneous

network of n = 30 nodes. The sensing radii of the nodes are suffi-ciently large to cover the concave environment, since

30i=1

Bi dq

= 1.199. The robotic swarm is divided into two clusters locatedas shown in the left part of Fig. 9. The robots’ trajectories and theconverged robot positioning and sensing appear in the middle andright parts of Fig. 9, respectively. The performance criterion duringthe network evolution is shown in Fig. 11.

In the second study, the same configuration of nodes isused, but their sensing radii is decreased by 30%, following that30

i=1

Bi dq = 1, 199 × 0, 72

= 0, 587. The initial positioning,robots’ trajectories and converged state appear in the left, mid-dle and right part of Fig. 10, respectively. The area covered in thesteady state does not reach themaximum limit, since several nodeshave been ‘blocked’ in the right part of the environment. In otherwords, the network has reached a local, rather than the globalmax-imum of H . Note also that the covered area increases monotoni-cally (Fig. 11), as expected. Regarding the trajectories of the nodesin the first two simulations, we notice that the nodes tend to movein the middle of the corridors, mostly due to the second term ofthe control law (17), which causes them to rotate around a reflexvertex of the environment.

In the third simulation study, we compare our coverage algo-rithm with the control scheme presented in Thanou et al. (2013a).Note that the latter assumes totally different types of sensors, de-scribed as geodesic discs rather than visibility-based ones. How-ever, this comparison allows us to make interesting remarks.Specifically, Fig. 12 shows the initial state, the evolution and thefinal state of the network of n = 10 nodes, when the control law(17) is applied. On the other hand, Fig. 13 shows the same networkconfiguration regarding the initial positions and the sensing radiiof the nodes, but as mentioned before, their sensing model corre-sponds to geodesic discs and they are also governed by the controllaw presented in Thanou et al. (2013a). Comparing the initial statesin Figs. 12, 13 the difference in the two sensing patterns is obvious,since a geodesic circlemay include non-visible parts aswell. There-fore, a visibility-based disc is always a subset of the geodesic one.

204 Y. Kantaros et al. / Automatica 53 (2015) 195–207

Fig. 10. Simulation study 2-Proposed control scheme is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from originating (red) to converged (green) dot-positions. (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. Coverage performance H (t) for the first (red) and second (green)simulation study. The dotted lines represent the maximum coverage that can beachieved in each simulation. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

In both cases, a node remains idle throughout the simulation, sinceits very small sensing range does not contribute to the total cover-age. Regarding the trajectories and the final state, although bothalgorithms achieve good results in the final state, the nodes havebeen placed in different positions. As for the total coverage perfor-mance (Fig. 14), the algorithm in Thanou et al. (2013a) achievesslightly better performance, which was expected, since the sen-sors cover non-visible parts as well. However, the geodesic sens-ing footprint is not as realistic as the range-limited field of view,since the first one is an approximation of attenuation of signals.

Apart from this, the computation of the geodesic power diagramproposed in Thanou et al. (2013a) is computationally more expen-sive than the tessellation process described in this paper.

In the fourth simulation, we consider a network that consistsof 15 nodes, initially deployed in the upper left part of a non-convex environment, depicted in Fig. 15. Regarding the final state,the proposed algorithm achieves good results (see Fig. 16) despitethe complexity of the environment. For example, notice that twonodes have traversed the narrow passage and reached the upperright part of the domain. The blue dashed lines depict the visibility-based power diagram for the initial and the final state of thenetwork. Apparently, in the initial topology, the area A is not fullypartitioned, since it holds that

15i=1 VP

i⊂ A in contrast to the final

topology where it holds that A \15

i=1 VPi= ∅.

In the fifth simulation study, two nodes with unlimitedsensing capabilities are considered. Fig. 17 (left) shows the initialpositioning of the nodes and their visibility domains, which arerepresented by different colors. Since the sensing radii of the twonodes are considered infinite (in practice greater then the diameterof the convex hull of A), then the first term of the robot’s controllaw (17) is zero, as indicated by the second point of Remark 3.As a result, the nodes roughly tend to move around the reflexvertices of the environment that block their visibility (Fig. 17)(center). Regarding the final state of the network (Fig. 17 (right))the nodes have achieved full sensing coverage of the environment.It is worth mentioning that the narrow yellow strip on the rightpart of the environment is a region assigned to the upper nodeby the tessellation, since it is not visible by the lower one. Inaddition, the covered area throughout the simulation (Fig. 18) ismonotonically increasing, converging to the maximum limit.

Additional simulation studies were carried out when the robotsare deployed as in the left part of Fig. 12 for different sets of

Fig. 12. Simulation Study 3a-Proposed control scheme is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from the originating positions (red dots) to theconverged ones (green dots). (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

Y. Kantaros et al. / Automatica 53 (2015) 195–207 205

Fig. 13. Simulation Study 3b-Control law presented in Thanou et al. (2013a) is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from the originating positions(red dots) to the converged ones (green dots). (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article.)

Fig. 14. Coverage performance H (t) for the third simulation study when theproposed algorithm (green) and the algorithm in Thanou et al. (2013a) (blue) isapplied. The red line represents the maximum coverage that can be achieved. (Forinterpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

sensing radii. Particularly, denoting by r the set of radii selected inthe third simulation study, simulation studies were conducted as-suming that the robots’ set of sensing radii is kr with k ∈ [0.5, 1.5].Initially we selected k = 0.5 and then at each subsequent casestudy k was increased by 0.05. The set of radii corresponding tothe third simulation study is r ∈ {0.18, 0.42, 0.47, 0.48, 0.5, 0.55,0.58, 0.62, 0.67, 0.74} and the total area of the environment is

Fig. 16. Coverage performance H (t) for the fourth simulation study.A dq = 8. The achieved coverage performance of the network for

each case is depicted in Fig. 19 along with the maximum coverage

performance which is defined as Cmaxp = min

Ni=1

Bi dq

A dq , 1. Also,

in Fig. 19, we notice that the final coverage provided by the net-work increases monotonically with respect to the robots’ sensingradii until it reaches a maximum limit.

Next we consider again the initial network configuration ofthe third simulation study when additional nodes are deployed(of sensing range equal to 0.5) in the environment which results

Fig. 15. Simulation study 4-Proposed control scheme is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from the originating positions (red dots) to theconverged ones (green dots). (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

206 Y. Kantaros et al. / Automatica 53 (2015) 195–207

Fig. 17. Simulation study 5-Proposed control scheme is applied: (Left) Initial network state. (Middle) Nodes’ trajectories from the originating positions (red dots) to theconverged ones (green dots). (Right) Final network state. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

Fig. 18. Coverage performance H (t) for the fifth simulation study.

Fig. 19. Comparison of maximum coverage performance (red) and achievedperformance (blue) for various sensing radii. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

in an increase of the coverage performance as shown in Table 1.Note that when additional nodes are positioned in A, the othernodes’ positions remain the same. In all these cases, the proposedcontrol schemeachieves quite good results, since the final coverageperformance is satisfactory considering that it is always quite closeto the maximum performance Cmax

p .

5. Conclusions

A distributed gradient-based approach for the coverage prob-lem of concave domains by heterogeneous mobile robotic net-works was presented in this paper. Each robot was assumed to be

Table 1Achieved and maximum coverage performance vs. N .

r-sensing radii case, varying N Cmaxp % H (X)%

12 100% 95%14 100% 97%18 100% 100%

equippedwith omnidirectional range-limited visibility-based sen-sor, such as a camera while sensing capabilities were not assumedto be the same for all the nodes. Considering that, a novel parti-tioning technique of the concave areawas proposed, extending thepower diagram into the visibility-based power diagram. The pro-posed distributed coordination scheme guarantees maximizationof the covered area in a non-decreasing way. Simulation studieswere conducted illustrating the efficacy of the proposed controlscheme.

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Yiannis Kantaros received the Diploma Engineering de-gree in Electrical and Computer Engineering in 2012 fromthe University of Patras, Greece. He is currently workingtoward the Ph.D. degree in the Department of MechanicalEngineering and Materials Science, Duke University. Hiscurrent research interests include distributed control, dis-tributed optimization, multi-agent systems and robotics.He was a recipient of a Best Student Paper Award at the2nd IEEEGlobal Conference on Signal and Information Pro-cessing in 2014.

Michalis Thanou received the Diploma Engineering de-gree in Electrical and Computer Engineering in 2011 fromthe University of Patras, Greece. He was then admitted tothe Ph.D. program and his research interests focused in thefield of cooperative control of networked robots. Hepassedaway in March 2014 at the age of 27.

Anthony Tzes is Professor of the Electrical & Computer En-gineering Department of the University of Patras (UPAT)in Greece. He is a graduate of UPAT (1985) and receivedhis doctorate from the Ohio State University (1990). From1990 till 1999 he was with Polytechnic Institute of NYU.His research interests include cooperative control of net-worked mobile robots, surgical robots, and control engi-neering applications.

He has received research funding from various orga-nizations including NASA, the National (US) Science Foun-dation, and the European Union (FP6, H2020). He has been

a member of the Greek committee of the European Control Association (EUCA),member at several committees of the International Federation of Automatic Control(IFAC), and until 2008 Greece’s national representative to EU’s FP7’s thematic area‘‘Regions of Knowledge, Research Potential and Coherent Development of Policies’’.He has served in various positions (Program Chairman (MIM’00), Organizing Com-mittee Chairman (ECC’07), Chairman (MED2011)), and as IPC-member at several in-ternational conferences. He is the leader and principal investigator of the ‘‘AppliedNetworked Mechatronics Systems group’’. He has authored more than 70 (200) pa-pers published in international technical journals (conferences) and served in theeditorial board of several journals.