decentralized swarm coordination: a combined coverage/connectivity approach

Journal of Intelligent and Robotic Systems manuscript No.(will be inserted by the editor)

Decentralized Swarm Coordination: A Combined Coverage / ConnectivityApproach

Yiannis (John) Stergiopoulos · Anthony Tzes

Received: 29 Jun 2010 / Accepted: 22 Dec 2010

Abstract Decentralized motion coordination for coverage optimization purposes in mobile sensor networksis the scope of this paper. Coordination is performed based on spatial Voronoi tessellation, while taking intoconsideration the limited sensing capabilities of the agents. Each node performs an independent optimization inorder to increase network’s area coverage via its motion, while it attains information from its current and futureDelaunay neighbors. A decentralized algorithm is proposedin order to achieve optimal network’s coverage,based on local information. Connectivity issues are analyzed in detail, while a lower bound on the communi-cation radius of the nodes is derived, in order to attain sufficient information for performing the correspondingoptimization. An agent moves inside its region of responsibility in a way that the total area surveyed by thenetwork is a monotonically increasing function of time. Theonline control action makes the network adaptiveto possible changes in the environment.

Keywords Voronoi diagrams· sensor networks· decentralized control· coverage optimization· motioncoordination

1 Introduction

Coordination algorithms for sensor networks have been the center of interest in the last decade due to theirdirect application in coverage/surveillance scenarios, where on–time detection of certain events is crucial [10].Considering high performance sensing coverage of geographic regions of interest, optimum sensors deploy-ment is highly important [7], while mobility abilities of the latter offer adaptation to changes in the surveyedenvironment (as in search–and–rescue missions) or alteration of the network [1,15,24].

Since in most practical scenarios the nodes are not supervised by a global coordinator (due to their largenumber or the communication radius required), connectivity preservation of the network in order to ensure datatransfer from one point to another is essential [7,16,19]. It is clear that such control policies do not achieveoptimum area coverage and this is the trade–off to be balanced.

Corresponding author: Yiannis (John) [email protected]

Y. StergiopoulosDepartment of Electrical & Computer Engineering, University of Patras, 6 Eratosthenous Str., Rio, Achaia GR–26500, GreeceTel.: +30 6972746566, Fax: +30 2610 991812E-mail: [email protected]

A. TzesDepartment of Electrical & Computer Engineering, University of Patras, 6 Eratosthenous Str., Rio, Achaia GR–26500, GreeceTel.: +30 2610 996453, Fax: +30 2610 991812E-mail: [email protected]

2 Yiannis (John) Stergiopoulos, Anthony Tzes

Considering optimum coverage of a certain region of interest by a set of nodes with sensing capabilities,the main problem lays in defining the optimum network configuration such that the sensing overlapping isminimized. Apart from that, since the nodes in most practical scenarios represent mobile robotic vehicles, aset of non–colliding trajectories is required so that the nodes can reach that optimum state. It is well–knownthat centralized swarm–coordination algorithms suffer from computational complexity, while they are charac-terized by the lack of adaptation to possible network alteration. Thus, decentralized control techniques basedon partitioning of the area of interest into smaller ones, known as Voronoi diagrams [2,5], have been studiedby [17]. In such cases each node is responsible for its area ofresponsibility rather than the entire space, tryingto optimize a coverage–based criterion [17,18].

Most of the proposed coordination algorithms for homogeneous sensor networks lay their roots in Cen-troidal Voronoi Tessellation (CVT) theory [8] when the sensing patterns of the nodes are fixed, while optimumpositioning schemes when the nodes’ sensing radii can vary has been studied by [6]. On the other hand, whendealing with heterogeneous networks, modified space–partitioning techniques have been proposed [20,22]when the patterns are symmetric, while anisotropic nodes’ sensories have been examined by [13], [14] and[11].

In this paper coverage optimization is performed by the nodes themselves in an online manner, leading thenetwork toward the optimum state, while algorithmic implementations on the communication range adjustment(considering connectivity issues) are presented, along with corresponding bounds on them for connectivitypreservation with the appropriate subset of nodes. In addition, attention is given in the decentralized manner viawhich optimization is performed, where only local information is obtained by the nodes. Despite the absence ofa centralized coordinator, the proposed scheme is built in away such that the total area covered by the network isincreasing from one step to another. Decision for motion is taken by the nodes autonomously, without the needfor global supervision, while assumptions concerning knowledge of current and future Delaunay neighbors areexamined in detail, providing the corresponding communication radii needed for them to hold.

2 Related work

Area–coverage of a certain region by a set of mobile nodes hasbeen an issue of interest in the last years.This problem has been initially introduced as the “bin packing” or “disc covering” problem. According to thefirst approach, the goal is to find the nodes’ maximum sensing range (considered equal for all members ofthe network) such that there exists a configuration where thenodes are deployed in the interior of a region ofinterest covering the maximum possible area with no overlapping amongst their sensing patterns. On the otherhand, the “disc covering” problem is posed as finding the minimum sensing range of the nodes in a way thatall parts of the region of interest are sensed by at least on node.

The fact is that, in both algorithms, the sensing radius of the nodes is considered as variable and the goal isto determine the nodes’ positions and radius in order to fulfill the appropriate demand. In this paper, however,the nodes’ sensing pattern is considered fixed, and the problem lays in defining (locally) optimal positions suchthat the sensed area of the region of interest is maximized.

The main innovation of this work lays in the fact that the optimization criterion is the total area sensed bythe network, while the control action is not based on Centroidal Voronoi Tessellation (CVT) schemes [17]. Acentralized version of this problem has been examined from an instantaneous coverage point of view by [12],where the agents coordinate their motion such that all points in the environment are surveyed through time byan equal amount. Furthermore the nodes are equipped with radio transceivers with variable range, apart fromthe fixed–range sensors, leading in a hybrid coverage/connectivity solution approach.

The article is organized as follows. In section 3 the coverage problem from a group of agents is presented,while a discussion on different policies to deal with the problem follows. The contribution of Voronoi parti-tioning of the space in the coverage problem is examined, while its application in decentralized coordinationschemes is emphasized. In section 4, the main concept of the proposed coverage algorithm is introduced, whilespecial attention is given in its decentralized nature. Theproposed optimization scheme is presented in thesequel, while the monotonicity property of the area coverage is emphasized. In section 5, issues concerning

Decentralized Swarm Coordination: A Combined Coverage / Connectivity Approach 3

inter–agent communication are analyzed, where a lower bound on the communication radius of a node is pro-vided in order to allow information exchange with its current and future Delaunay neighbors. Simulation resultsfollow in section 6 in order to show the efficacy of the proposed hybrid coverage/connectivity algorithm, whileconcluding remarks are provided in the last section.

3 Problem formulation

3.1 Coverage problem setup

Let the region under surveillanceΩ be a convex compact set inR2. Suppose thatn is the number of availablemobile nodes responsible for the sensing coverage ofΩ . Let us define the setIa = i ∈ N : i ≤ a for anya∈N.The agents are considered to move on theR

2 Euclidean configuration space and their positions are denoted asxi ∈ R

2, i ∈ In. The following assumptions are made for the network.

Assumption 1 The nodes are supposed to move in the interior ofΩ ⊂ R2 through two control inputs each,

ui ∈ R2, i ∈ In, while obeying the discrete evolutionary equation (equivalently to [7])

xk+1i = xk

i +uki , ui , xi ∈ R

2, i ∈ In. (1)

The superscript index denotes the corresponding time–step, k= 0,1,2. . . , while only one node is supposed tomove at each step.

Assumption 2 Each node is supposed to have a uniform circular sensing pattern centered at its positionxi

and is limited by a fixed maximum sensing radius r. The latter is the same for all nodes and the networkis considered homogeneous, as far as concerns the nodes’ sensing abilities. Let us denote as Ci the sensingregion of each agent i, i.e.

Ci =

x ∈ R2 : ‖x−xi‖ ≤ r

, i ∈ In. (2)

Assumption 3 Each node is supposed to be equipped with radio transceiversin order to be able to exchangespatial information with other members of the network. The radiation pattern Si of the antennas is consideredas a uniform circular one, centered atxi , i.e.

Si =

x ∈ R2 : ‖x−xi‖ ≤ Ri

, i ∈ In, (3)

while the communication radii of the nodes, Ri , are considered to be adjustable.

As far as concerns Assumption 1, the selection of the node–to–move must be performed by the nodesthemselves, and not by a global supervisor, considering decentralized applications. This can be achieved bydetermining the corresponding node either in a cyclic or in arandom manner. In the first case, an arbitrarynodei moves only at time–stepsk = i + p n, p∈ N, while in the intermediate time–intervals it can be set instandby mode in order to preserve power. Alternatively, thenode that is to perform possible motion can bechosen randomly by the group itself, where the random generators that run on each processor have the sameseed value, so that at each step the node–to–move is unique and same for all members of the network.

The nodes are initially deployed randomly inΩ . Considering coverage optimization scenarios, the goalis to find in an online manner their optimal positions such that the area of the covered region ofΩ by thenetwork is the maximum possible. For a compact polygonal setP⊂ R

2 let ∂P be its boundary. ThenP is fullydefined by the vertices of∂P denoted asp j , j ∈ IN(P), where N(P) is the number of the latter’s vertices. Forany polygonal setP, the area–functionA (·) is then defined as [8]

A (P) =12

∥

∥

∥

∥

∥

∥

∑j∈IN(P)

(p j ×p j+1)

∥

∥

∥

∥

∥

∥

, (4)


where× corresponds to the cross–product of two vectors, the verticesp j are set in counter–clockwise orderandpN(P)+1≡ p1, by convention. The main objective is to position the nodes at certain spatial coordinates suchthat the total region ofΩ surveyed by the network, i.e.Ω ∩

⋃

i∈In Ci , has the maximum possible area (at theoptimal state), while providing at the same time collision–free trajectories for each node to reach that state.

3.2 Control policies for coverage optimization

In order to find the planar coordinates for each nodexi ∈ Ω , i ∈ In so thatA(

Ω ∩⋃

i∈In Ci)

is optimized, anumerical approach will be followed. The aforementioned problem is a standard constrained numerical opti-mization one, the solution of which may converge to possiblelocal extrema. LetX =

(

xT1 ,x

T2 , . . .x

Tn

)Tbe the

vector containing the coordinates of all agents,X ∈ R2n. The constraintxi ∈ Ω , ∀i ∈ In can be cast in a linear

compact form asA X≤B, where the matricesA, B are explicitly defined by the verticesω j of Ω , j ∈ IN(Ω) [3].Thus, the offline version of the coverage problem can be defined as the solution of the following constrainedoptimization

find X :

maximize A

(

Ω ∩⋃

i∈In

Ci

)

subject to: A X≤ B

. (5)

Considering centralized optimization approaches, this isa computationally intensive optimization problemto solve, due to the large number of its local extrema and the time it takes (for even a small number of nodes).Apart from that, one of the most significant disadvantages inthe offline version is its lack of adaptation. Con-sequently, in case the region of interest changes or if a noderuns out of energy, then a new optimization shouldbe performed for defining the “new” optimal positioning of the nodes, which may be disastrous in cases ofemergency. However, despite the centralized nature of the algorithms, global optimal solution is not guaran-teed. Finally, it should be noted that, once the optimum network state is defined, path planning schemes mustbe applied afterwards so that the mobile agents can be able toreach this optimum, while avoiding collisionwith each other.

Online decentralized algorithms are faster from an implementation point of view and are adaptive by nature,since optimization is performed online individually by themembers of the network itself. Each node self–organizes its action so that its motion contributes to network coverage, while decision is taken based only onlocal information, without the need of having global knowledge of the whole network’s state. Local information(obtained by radio transceivers attached on the nodes) is the one that poses the inherent adaptive nature of thesealgorithms, while the performed optimizations (executed on the on–board nodes’ processors) are by–far lesscomputationally intensive, making them applicable in real–time scenarios. However, global optimal solutionis similarly not guaranteed, due to the large number of localextrema of the problem. Finally, one of the maindifferences, compared to the offline method, is that the nodes do not focus in defining their final optimalpositions, but their positions at each time–step, such that(if possible) the total area coverage is an increasingfunction of time.

Considering the discussion above, it is clear that, although decentralized algorithm may converge to localextrema of (5), they are closer to real–time applications. What needs to be clearly defined is the term of “localinformation”, along with the coordination algorithm, according to which the nodes should organize their actionin order to converge to a state where the total area covered bythe network is optimal.

3.3 Spatial Voronoi tessellation

Considering the region under surveillanceΩ , a responsibility region can be assigned at each agent basedonits spatial coordinates on the plane. The set of these regions is well–known as a Voronoi diagram [2]. For the


convex compact setΩ and then nodes, the region under surveillance is partitioned inton convex compactsubsetsVi , i ∈ In, which are defined as

Vi =

x ∈Ω : ‖x−xi‖ ≤∥

∥x−x j∥

∥ , ∀ j ∈ In

, i ∈ In. (6)

The setVi is known as the Voronoi cell of nodei. It should be noted that a Voronoi diagram is a full tessellationof Ω ⊂ R

2, since⋃

i∈In Vi = Ω and IntVi ∩ Int Vj = /0, ∀ (i, j) ∈ In× In, i 6= j, where Int· is the interior of theset–argument. A Voronoi cellVi is uniquely characterized by the set of its verticesvi, j , j ∈ IN(Vi).

Two nodes that share an edge of their Voronoi cells (i.e. their Voronoi cells are adjacent) are considered asDelaunay neighbors [2]. The Delaunay neighborsNi of an arbitrary nodei are then defined as

Ni =

j ∈ In : Vi ∩Vj 6= /0 or a , j 6= i

, i ∈ In, (7)

wherea is an arbitrary point inR2. It should be noted that whenVi ∩Vj is a singleton, then the two nodesshare a common Voronoi vertex (instead of a Voronoi edge) andare not considered as Delaunay neighbors.Apart from that, nodei itself is not included into the set of its neighborsNi , as shown by (7). The edges of theVoronoi cell of an arbitrary nodei that do not lay on the boundary ofΩ are then defined as

∆i j =Vi ∩Vj , i ∈ In, j ∈Ni . (8)

In this paper the agents should move in a way to try and cover (ideally, if possible) the whole spaceΩ ,considering their limited sensing capabilities. Thus, theVoronoi tessellation alone is not sufficient for themotion algorithm, since it is based only on the nodes’ positioning, but the sensing regionsCi should be takeninto account. Ther–limited Voronoi cells are then defined as

V ri =Vi ∩Ci, i ∈ In. (9)

An important property of these sets is that, sinceΩ ,Vi ,Ci are convex sets∀i ∈ In, thenVri are all convex sets,

too. However, they do not always consist a full tessellationof Ω .For each node, the unexploited regions of its sensing pattern (parts of the sensing region of the node that

do not contribute to coverage ofVi ) are defined as

Ui =Ci \Vri , i ∈ In. (10)

It should be noted that althoughVi ,Vri ⊆ Ω , ∀i ∈ In by definition, the same does not always hold forUi , as

concluded by (10) and (2). In fact, the set⋃

i∈In Ui corresponds to the parts of the nodes’ sensing regions thatoverlap among each other or lay in the exterior ofΩ . However, the setUi can be decomposed as a union ofsmaller sets which are disjoint amongst each other, as

Ui =UΩi

⋃

j∈Ni

U ji , i ∈ In, (11)

whereUΩi = Ci \Ω , i.e. parts of the sensing region that do not lay inΩ , andU j

i = Ui ∩Vj , i.e. parts of thenode’s unexploited regions that lay in the Voronoi cell of a neighbor node. The aforementioned definition alongwith its decomposition will be proven helpful in section 4.3. The above regions for an arbitrary node are showngraphically in Fig. 1.

Consequently, the area of the total region ofΩ surveyed by the network, can be written as

J = A

(

Ω ∩⋃

i∈In

Ci

)

= ∑i∈In

A (V ri ) . (12)

The main advantage in computation ofJ via (12) is that, since IntV ri ∩ Int Vr

j = /0, ∀ (i, j) ∈ In× In, i 6= j,the area covered by the network can be computed as the summation of the areas of the independentr–limitedVoronoi cells, and thus leads in ease of implementation of decentralized techniques.


Unexploited regions

Sensing region

r-limited Voronoi cell

Voronoi cellx

i

r

Fig. 1 Characterization of the different regions concerning a node and its Voronoi cell.

4 Coverage optimization

4.1 Main concept

Considering sensing coverage applications, the nodes should one–by–one (Assumption 1) move to such spatiallocations in a way that the total covered areaJ of Ω is non–decreasing as time evolves [21]. Taking into accountthat, in most practical scenarios, global knowledge of the network’s state by a node is impossible (since thatwould lead to extremely large communication ranges), each node should have sufficient spatial informationof the nodes in its neighborhood, in order to determine its position at the next step in a way that network’scoverage will increase via its motion.

However, considering (12), in order for the node–to–move todecide if its motion will contribute to coverage–increase at an arbitrary time–step, it should have appropriate knowledge of the state of the nodes whose Voronoicells are to be affected via its motion. In fact, spatial information from these nodes is adequate in order for thenode–to–move to take decision about the spot to move at, in a decentralized concept. In the rest of the article,the indexi will stand for the node–to–move at stepk, and not for an arbitrary node.

4.2 Local Delaunay graph alteration

Considering now the Voronoi space–partitioning defined in (6), the selected node should first define the regionof responsibility (own Voronoi cell) that is assigned to it.The nodes are supposed to be capable of exchanginginformation concerning their spatial coordinates (Assumption 3). Taking into account the fact that no othernode (apart fromi) moves at that time–step, the only part of the network that alters (considering coverageperformance) are the Voronoi cells of the union of Delaunay neighbors of nodei before and after its motion,along with the Voronoi cell of the moving node itself.

Let us denote asN ki andN

k+1i the Delaunay neighbors of nodei at stepsk andk+1, which correspond

to the time–instances before and after the motion of the latter, respectively. Figure 2 shows such a scenario,where a node moves in the interior of its Voronoi cell, resulting in alteration of its Delaunay neighbors. The reddot represents the node–to–move to the spot denoted with the× sign in Fig. 2(a). Consider that Fig. 2(a) and2(b) correspond to time–stepsk andk+1, respectively. The green dots correspond to the Delaunay neighborsof the red node in each case, i.e.N k

i (Fig. 2(a)) andN k+1i (Fig. 2(b)). The grey sign at Fig. 2(b) corresponds

to the position of the red node before its motion, i.e.xki .

Each Delaunay neighbor of the red node is reflected on an edge of the latter’s Voronoi cell. It is easy to seethat if a nodej ∈N k

i is to leave the setNi at the next step, i.e.j ∈N ki \N

k+1i , then the edge of the moving

node’s Voronoi cell that corresponds to that node degenerates into a vertex in the next step. In a similar manner,


(b)

Fig. 2 Alteration of the Delaunay neighbors of a node caused by the motion of the latter.

if a “new” node j /∈N ki is to enter the set of the moving node’s Delaunay neighbors after motion of the latter is

performed, i.e.j ∈Nk+1

i \N ki , then a vertex ofVi will evolve into two vertices after its motion, adding∆i j into

the set of its Voronoi edges. Important is the fact that the rest of the network’s state (considering Voronoi cellalteration), apart from the setN k

i ∪Nk+1

i ∪i, does not alter at all. Indeed, if we suppose that the Voronoicell of a node that does not belong inN k

i ∪Nk+1

i ∪i alters, this means that a Delaunay neighbor of thatnode has moved and thus has perturbed a Voronoi edge. But, since the only node that moves is nodei, then theaforementioned node should belong to the above set.

Suppose now that the possible motion of nodei at stepk is restricted in a convex compact subset ofVki

denoted asWki ⊂Vk

i , containingxki , i.e.xk

i ,xk+1i ∈Wk

i . Let us define the setF ki as

Fki = N

ki ∪i

⋃

x k+1i ∈Wk

i

Nk+1

i , (13)

which corresponds to the union of the current (stepk) Delaunay neighbors of nodei, the moving node itself,along with the union of all possible Delaunay neighbors at the next step for all possible node’s motions inWk

i .

Lemma 1 The setF ki contains the nodes of the network whose Voronoi cell is possibly affected by the motion

of node i, given the restriction ofx k+1i ∈Wk

i .

Proof The proof is straightforward. ut

Assumption 4 The node–to–move i is considered to be able to exchange information at each time–step k withthe nodes in the setF k

i for a given subset Wki ⊂Vki .

Information acquisition from a node’s Delaunay neighbors is a non–restrictive assumption, which has ap-peared repeatedly in the existing literature [8,13]. The main difference between the above assumption and theexisting ones lays in the fact that the corresponding node–to–move needs connectivity with its current, alongwith its possible future Delaunay neighbors. Communication issues concerning Assumption 4 are to be exam-ined in detail in section 5. It should be noted that the notation F k

i concerns motion at time–stepk, while it isparameterized by the setWk

i in which possible motion of nodei is to be performed, i.e.xk+1i ∈Wk

i .

4.3 Coordination algorithm

Let the notationq` |I stand for the evaluation of the arbitrary variableq at step` based on information fromnodes in the setI ⊆ In. In the above notation it is implied that evaluation is performed by nodei (i.e. node–to–move) at stepk (i.e. before motion of the latter is performed). It should benoted that when = k+1, it is


supposed that motion of nodei is to be performed at a specifiedxk+1i . The aforementioned point is specified

for evaluation purposes at stepk and might not necessarily be the point at which nodei decides to move at.Considering Assumption 4, ther–limited Voronoi cell of a nodej ∈ F k

i as evaluated by nodei at stepk is denoted asV r

j |F ki. Considering coverage optimization scenarios, nodei should move at a pointxk+1

i

such that network’s coverage will be increased at the maximum possible rate. Consequently, in order for thenode–to–move to be able to evaluate its contribution to coverage–increase (via its motion), it should be able toevaluateJk+1−Jk (for a givenxk+1

i ). However, since the latter has spatial information only ofthe nodes inF ki ,

network’s coverage–increase from its own point of view is provided asJk+1 |F k

i−Jk |

F ki. The above become

clear via the following example.Consider the homogeneous network shown in Fig. 3(a). The node to movei at the current stepk is denoted

(a)

x

(b)

x

Fig. 3 Illustrative example for definition ofVr,kj |F k

isets.

by the red dot, while its supposed position at the next step (for evaluation purposes) is marked with an×sign. The Delaunay neighbors of nodei are denoted by green dots, while the yellow dot corresponds to thefuture Delaunay neighbor of the latter at the next step (if the motion is performed at that point). ConsideringAssumption 4, nodei is informed about existence of both yellow and green nodes, while connectivity issuesfor satisfying that assumption will be discussed later. Ther–limited Voronoi cells of the nodesVr

j are seen inFig. 3(a) as the section of their Voronoi cells with the corresponding sensing patterns. However, since nodeidoes not have any knowledge of existence of the rest network’s nodes (apart fromF k

i ), it can evaluate theirr–limited Voronoi cells via Fig. 3(b). The grey dots in the latter denote the rest of the network’s nodes that arenot taken into account during the computation of the aforementioned cells. Similar procedure is followed bythe moving node in order to evaluateVr,k+1

j |F k

i; supposing that its position is that in the×mark (for evaluation

purposes only), it should compute the corresponding Voronoi tessellation and then follow the same procedureas before. Before proceeding to the main theorem, let us firstprove the following lemma.

Lemma 2 For each node j∈F ki , it holds that

A

(

Vr,`j

)

= A

(

Vr,`j |F k

i

)

− ∑m∈N `

j \Fki

A

(

Um,`j

)

,

where` ∈ k,k+1.

Proof Let us first examine the case` = k. Considering (10)–(11) and Fig. 1, for an arbitrary nodej ∈F ki , it

holds thatA (Cj) = A

(

Vr,kj

)

+∑m∈N kjA

(

Um,kj

)

+A

(

UΩ ,kj

)

=


= A

(

V r,kj

)

+∑m∈N kj ∩F k

iA

(

Um,kj

)

+∑m∈N kj \F

kiA

(

Um,kj

)

+A

(

UΩ ,kj

)

.

Furthermore,A (Cj) can be written as

A (Cj) = A

(

Vr,kj |F k

i

)

+∑m∈N kj ∩F k

iA

(

Um,kj |

F ki

)

+A

(

UΩ ,kj |

F ki

)

=

= A

(

Vr,kj |F k

i

)

+∑m∈N kj ∩F k

iA

(

Um,kj

)

+A

(

UΩ ,kj |

F ki

)

,

since the nodesm∈N kj ∩F k

i are already known to nodei (which performs the evaluation), i.e.N kj ∩F k

i ⊆

F ki .

Combining the above expressions forA (Cj) results in

A

(

Vr,kj

)

= A

(

V r,kj |F k

i

)

−∑m∈N kj \F

kiA

(

Um,kj

)

+(

A

(

UΩ ,kj |

F ki

)

−A

(

UΩ ,kj

))

.

However, sinceUΩ ,kj =Cj \Ω is dependent only on the sensing pattern and the region of interest, it holds that

A

(

UΩ ,kj |

F ki

)

= A

(

UΩ ,kj

)

,

and the result is proven.The proof is totally identical for the case`= k+1. ut

Theorem 1 If the node–to–move i has spatial information of the nodes inF ki , then evaluation of the network’s

coverage–increase by ignoring the rest nodes in the networkis the same as if the latter had been evaluated sup-posing existence of all the network’s nodes, i.e.

Jk+1 |F k

i−Jk |

F ki= Jk+1−Jk. (14)

Proof Considering (12), it holds that

Jk+1−Jk = ∑ j∈In

[

A

(

V r,k+1j

)

−A

(

V r,kj

)]

=

= ∑ j∈F ki

[

A

(

V r,k+1j

)

−A

(

V r,kj

)]

+∑ j∈In\F ki

[

A

(

Vr,k+1j

)

−A

(

Vr,kj

)]

.

But the second summation in the expression above is equal to zero, according to Lemma 1. Furthermore,

substitution ofA(

Vr,kj

)

,A(

V r,k+1j

)

from the result of Lemma 2 results in

Jk+1−Jk = ∑ j∈F ki

[

A

(

Vr,k+1j |

F ki

)

−A

(

Vr,kj |F k

i

)]

−∑ j∈F ki

∑m∈N kj \F

ki

[

A

(

Um,k+1j

)

−A

(

Um,kj

)]

.

However, the second term is consisted of the alteration in the unexploited regions of the nodes inF ki that lay

in the Voronoi cells of nodes that belong inN kj \F

ki . Thus, since the∆ jm edges are not altered

(

m /∈F ki

)

, theaforementioned term is zero.Consequently, it holds that


[

A

(

Vr,k+1j |

F ki

)

−A

(

V r,kj |F k

i

)]

.

At this point, without loss of generality, since nodei does not have information of existence of the nodesIn \F k

i , their evaluated Voronoi cells can be arbitrarily be set to empty–sets, which results in


[

A

(

Vr,k+1j |

F ki

)

−A

(

Vr,kj |F k

i

)]

+

∑ j∈In\F ki

[

A

(

Vr,k+1j |

F ki

)

−A

(

Vr,kj |F k

i

)]

= Jk+1 |F k

i−Jk |

F ki,

and the result is proven. ut

At this point, one can select the regionWki as a circular one as

Wki =

x ∈ R2 :∥

∥

∥x−xk

i

∥

∥

∥≤ αd

(

xki ,V

ki

)

, (15)

where 0< α 1 andd(x,M) is the distance of pointx from the setM defined as

d(x,M) := inf ‖x−y‖ : y ∈M . (16)


It is clear thatWki expresses a disc centered at the agent’s position. The advantage obtained by the selection of a

circular region forWki is its symmetry aroundxk

i . It is easy to see that sincexk+1i ∈Wk

i , then the node–to–moveperforms its motion in the interior of its Voronoi cell, leading so in collision free paths for the nodes until thenetwork reaches its optimum state.

Consequently, the destination point at which nodei should move, i.e. the optimalx k+1i ∈Wk

i denoted asx? k+1

i , can be is obtained numerically via solving at each stepk the following optimization

find xk+1i ∈Wk

i :

maximize

Jk+1 |F k

i−Jk |

F ki

subject to : Jk+1 |F k

i> Jk |

F ki

, (17)

either via gradient–based nonlinear constrained optimization schemes [23] or griding onWki . Considering The-

orem 1 and (17), the area of the region surveyed by the networkwill be increased at each step in the maximumpossible rate (due to the node’s motion). The control actionin (1) can be then selected as

u ki = x? k+1

i −x ki . (18)

It is clear that in the case where any admissible motion of thenode inWki results in less or equal coverage area

than that in the current step, then the node stays idle, sincex ki ∈Wk

i . Via the proposed scheme, it is guaranteednot only that the total area is increased in a monotonic manner, but with the highest possible rate, too.

5 Issues concerning spatial information exchange

As mentioned earlier, in this paper it is considered that node i should be able to receive spatial informa-tion from the set of nodesF k

i (Assumption 4). This section is dedicated into deriving a lower bound on thecommunication radius of the node–to–move in order for the assumption to hold. It is known that, in mostpractical scenarios, the maximum possible communication range of a node’s transceiver is limited due to phys-ical/manufacturing restrictions. Posing, though, such a constraint is beyond the scope of this article.

5.1 Connectivity with current Delaunay neighbors

Considering communication issues, nodei should be able to acquire information at stepk from the nodes of thesetF k

i , given the subsetWki ⊂V k

i , in order to be able to apply the coverage algorithm presented in the previoussection. Ignoring the trivial case of receiving information from the node itself, as far as concerns the first partof the aforementioned union set, the node’s transceiver should first have adequate communication range so thatit can communicate with its current Delaunay neighborsN k

i .Let us denote byRk

i (I) andRki (I)wcs the minimum communication radius of nodei at stepk required in

order to exchange information with a set of nodesI ⊂ In, from a centralized and decentralized point of view,respectively. It is easy to see that, from a centralized aspect, the minimum range required for guaranteeingconnectivity of nodei with N k

i at stepk is equal to

Rki

(

Nk

i

)

= 2max

d(

xki ,∆k

i j

)

: j ∈Nk

i

= max∥

∥

∥xk

i ,xkj

∥

∥

∥: j ∈N

ki

, (19)

where∆i j andd(x,M) are defined in (8) and (16), respectively. Although, expression (19) provides indeedthe minimum range required, it cannot be used as a bound, froman independent point of view. In fact, nodeineeds to increase even more its communication range until itis ensured that its Voronoi cell will not be affectedfurther, even if another node falls in range.

Considering algorithmic implementations of communication range adjustment, the main concept lays ingradual increase of the latter until sufficient informationfrom neighbor nodes is obtained [8]. In fact, node


i gradually increases its range and updates its Voronoi cell according to the nodes that fall in its range. Theprocedure ends when eitherSk

i ⊇ Ω or the node’s range becomes twice the distance between the node and itsfarthest Voronoi cell vertex, i.e.

Rki

(

Nk

i

)

wcs= 2max

∥

∥

∥xk

i −vki, j

∥

∥

∥: j ∈ IN(Vk

i )

, (20)

since from that time on, any other node identified does not alter its Voronoi cell. The notation N(·) stands forthe number of the argument’s vertices, as stated already in section 3.1. It should be noted that (20) seems toinclude some kind of recursiveness in its body; indeed, in order to compute the minimum communication rangein order to exchange information withN k

i , one needsVki (considering the second part), whereN k

i is neededfor its evaluation. However, (20) depictsRk

i

(

N ki

)

wcs from ananalysispoint of view. The iterative algorithmicprocedure for the latter’s numerical evaluation is shown inTable 1.

Table 1 Decentralized Communication Range Adjustment Algorithm for Connectivity withN ki

Goal: Identify current Delaunay neighbors and Voronoi cell

Rki ← 0, ˆN k

i ← /0, Vki ←Ω , Sk

i ← /0

while Rki ≤ 2max

∥

∥

∥xk

i − vki, j

∥

∥

∥: j ∈ IN(Vk

i )

and Ski ⊂Ω

increaseRki

updateSki

if node j detectedˆN ki ←

ˆN ki ∪ j

updateVki

end ifend whileVk

i ← Vki

isolateN ki from the set ˆN k

i

Note also thatRki

(

N ki

)

wcs is larger than the centralized bound, due to the decentralized nature of thescheme. During the rest of the analysis that follows in this section, nodei will be considered to obtain informa-tion from its current Delaunay neighborsN k

i via decentralized proper adjustment of its communication radiusat Rk

i

(

N ki

)

wcs.

5.2 Guaranteeing connectivity with all possible future Delaunay neighbors

Considering (13), what needs to be further ensured is connectivity of node i with the set⋃

xk+1i ∈Wk

iN

k+1i , given

Wki . Suppose an arbitrary pointxk+1

i ∈Wki ⊂ Vk

i . The goal is to find the minimum communication radius ofnodei at stepk in order to guarantee connectivity at that step withN

k+1i , from a decentralized point of view. At

stepk, nodei, positioned atxki with communication rangeRk

i

(

N ki

)

wcs, has information about the coordinatesof the nodesN k

i of the network. Thus, at that time, the first can evaluate its future Voronoi cell, supposingthat its motion is to be performed atxk+1

i , by taking into account only the nodes inN ki and itself (i.e. ignoring

the rest of the network, since it does not have knowledge of existence of the rest nodes, yet). Let us denote asVk+1

i |N k

ithe aforementioned evaluated Voronoi cell. At this point, node i can be aware if a nodej ∈N k

i isabout to leave the set of its Delaunay neighbors at the next step, supposing that its motion will be performed atxk+1

i , via simple evaluation ofVk+1i |

N ki

. What is unknown to the node–to–move yet are possible nodes of the

network that may enterN k+1i .


Let us consider again the worst case scenario. Suppose thatm∈ In is an extra node to possibly enter the setN

k+1i if motion is performed atxk+1

i , where existence ofm is unknown to nodei yet. Considering (6), lethi j

stand for the line that equally divides the space into two halfplanes between two arbitrary nodes of the network,i.e.

hi j =

x ∈ R2 : ‖x−xi‖=

∥

∥x−x j∥

∥

, (i, j) ∈ In× In, i 6= j. (21)

The critical case for nodem to enter the setN k+1i is whenhk+1

im marginally crosses the farthest vertex ofVk+1

i |N k

i. Let vk+1

i, j∗ |N ki

denote the farthest vertex ofVk+1i |

N ki

, where the indexj∗ is given as

j∗ = argmax

∥

∥

∥xk+1

i −vk+1i, j |N k

i

∥

∥

∥: j ∈ I

N

(

Vk+1i |

N ki

)

. (22)

It is well–known that, given two pointsa,b∈R2 and a family of straight linesL , whereb∈ `, ∀`∈L , thefarthest line froma (whered(a, `) is defined via (16)) is the one that is perpendicular to the line that connectsa andb. Considering the above, one can conclude that the worst casescenario for the position of nodem iswhen it lays along the line that connectsxk+1

i andvk+1i, j∗ |N k

i, at a distance fromxk+1

i equal to twice that of the

aforementioned points. This case is depicted graphically in Fig. 4. The red dot representsxk+1i , while the blue

...x

i

k+1

m

m m

m

m

Fig. 4 Worst case scenario for the existence of a nodem that is to enterN k+1i .

line connects that node with the farthest vertex of its evaluated Voronoi cellVk+1i |

N ki

. The blue dot representsthe worst case scenario (comparing to the other possible cases denoted by grey color) for the existence of nodem. The rest of the nodes in the network are omitted for visualization purposes.

Considering Fig. 4, the worst–case for the position of nodem is

xk+1m = xk+1

i +2(

vk+1i, j∗ |N k

i−xk+1

i

)

.

Thus, nodei atxki should have adequate communication range so that it can at least exchange information with

that node,m, provided as

Rki

(

Nk+1

i

)

wcs=∥

∥

∥xk

i −(

xk+1i +2

(

vk+1i, j∗ |N k

i−xk+1

i

))∥

∥

∥

=∥

∥

∥

(

xki +xk+1

i

)

−2vk+1i, j∗ |N k

i

∥

∥

∥,

(23)

where the indexj∗ is defined in (22). It should be noted thatxki appears in the norm–argument of the upper

part of (23), and notxk+1i , since we are interested in finding the appropriate communication range of nodei


at stepk in order to communicate with nodem. Furthermore, special attention should be given to the factthatthe range defined in (23) is not the minimum required radius for guaranteeing connectivity withN k+1

i froma centralized point of view, since the first depicts the worstcase scenario (wcs). However, it is considered asthe optimum range from a decentralized point of view, considering that at that time, nodei has knowledge ofexistence for the nodesN k

i .The above become clearer via the following example. Consider the network depicted in Fig. 5(a). Node

Fig. 5 Communication radius of the node–to–move (red color) required in order to guarantee connectivity: (a) withN ki , (b) with

Nk+1

i , for a given node’s motion (worst case scenario).

i is depicted with the red dot, while the time–step is considered ask. The blue circle specifies the minimumcommunication radius of nodei (from a decentralized point of view) required in order to communicate withN k

i , which are the green nodes of the figure. One can observe that the corresponding radius allows informationexchange with some nodes of the network, other than the greenones.

Consider now that nodei evaluates its future Voronoi cell, as if the network was consisted only of the nodesN k

i , i.e.Vk+1i |

N ki

, as depicted in Fig. 5(b). The red dot denotes the possible destination of nodei (depicted

with the× mark at Fig. 5(a)), while the grey one is its real position at stepk. The farthest vertex ofVk+1i |

N ki

is needed to determine the worst–case for the position of a possible nodem. In Fig. 5(b),xk+1i is connected

with the imaginaryxk+1m via the blue dashed line passing fromvk+1

i, j∗ |N ki

. One can see that no node lays in the

worst–case position of nodem. Thus, at this pointRki

(

Nk+1

i

)

wcs can be computed via (23). It should be notedthat the circle is centered at the grey sign (current node’s position) and not at the red one (possible future node’sposition). Apart from that, the most right green node that will enter the set of Delaunay neighbors of nodei, ifthe latter moves atxk+1

i , is not taken into account into the computation ofVk+1i |

N ki

, since that is the node tobe identified. In fact, the only nodes in the network that playa role to its evaluation are the green nodes of Fig.5(a). Finally, one can observe thatRk

i

(

Nk+1

i

)

wcs in Fig. 5(b) is not the minimum radius required to exchangeinformation with the green nodes of Fig. 5(b), since it corresponds to the worst case scenario.

Considering the above, the minimum communication radius ofnodei at stepk required in order to exchangeinformation with the nodes in the set

⋃

xk+1i ∈Wk

iN

k+1i is

Rki

⋃

xk+1i ∈Wk

i

Nk+1

i

wcs

= sup

Rki

(

Nk+1

i

)

wcs: xk+1

i ∈Wki

, (24)

whereRki

(

Nk+1

i

)

wcs is defined in (23). To summarize, the algorithm followed for evaluatingRki

(

F ki

)

wcs isprovide in Table 2. An issue of major importance is the fact that, after identification ofN k

i via Table 1,


only evaluations are needed throughout the body of the algorithm procedure, and no communication rangeadjustment is demanded untilRk

i

(

F ki

)

wcs is defined.

Table 2 Decentralized Communication Range Adjustment Algorithm for Connectivity withF ki

Goal: Identify nodes whose Voronoi cells are possibly affected

Rki

(

⋃

xk+1i ∈Wk

iN

k+1i

)

wcs← 0

identify N ki andVk

i via Table 1perform gridding onWk

ifor eachxk+1

i ∈Wki

evaluateVk+1i |

N ki

evaluatevk+1i, j∗ |N k

i

evaluateRki

(

Nk+1

i

)

wcs

Rki

(

⋃

xk+1i ∈Wk

iN

k+1i

)

wcs←max

Rki

(

⋃

xk+1i ∈Wk

iN

k+1i

)

wcs,Rk

i

(

Nk+1

i

)

wcs

end forRk

i

(

F ki

)

wcs←max

Rki

(

N ki

)

wcs,Rki

(

⋃

xk+1i ∈Wk

iN

k+1i

)

wcs

updateSki

identify F ki

Corollary 1 The communication radius of node i at step k in order to guarantee connectivity with both currentand all possible future Delaunay neighbors, should be at least

Rki

(

Fki

)

wcs= max

Rki

(

Nk

i

)

wcs, Rk

i

⋃

xk+1i ∈Wk

i

Nk+1

i

wcs

, (25)

where the corresponding radii are given by (20) and (24).

6 Simulation results

Simulation studies are presented in this section in order toshow the efficacy of the proposed scheme. TheregionΩ to be surveyed is a convex set inR2. During network evolution the covered area is increasing untilit converges to an extremum solution, while the agents’ kinematics are described by (1). The control action isbased on the coordination scheme proposed in section 4.3. Two series of simulations follow: one considering asparse network and another considering a congested one. By the term sparse/congested, it is implied that thereexists/does not exist configuration such thatΩ is covered in the maximum possible ratio with no overlappingamong the nodes’ sensing patterns or with the boundary ofΩ .

The number of agents in the first case (i.e. sparse network) isset ton= 18, while in the second scenariothe latter is set ton = 10. In both cases, the latter are deployed randomly inΩ . The critical sensing radiusof the agents’ patterns is considered equal tor = 1.5m andr = 3m, respectively. The maximum theoretically(if possible) achievable sensing area, supA

(⋃

i∈In Ci)

= nπr2, corresponds to 127.23m2 and 282.74m2 ineach case, while the area of the region of interest isA (Ω ) = 226.37m2. The convex area under surveillanceis that presented in [6] and [7]. Ideally, if possible, the agents should be able to cover the region withoutany overlapping, and thus coveringnπr2

A (Ω) = 56.2% and 100% ofΩ , respectively, where the latter maximum


possible percentage ratio is trimmed at 100%. The value ofα in (15) was set toα = 0.1. Simulation is stoppedwhen all nodes’ motions are unable to further increase coverage.

Considering the sparse–network case, the agents’ initial positions, their evolution through time, along withthe final network’s state, when the control scheme presentedin section 4.3 is applied, are shown in the top partof Fig. 6, in this order. As far as concerns the final nodes’ state, it is obvious that the agents have self–positioned

0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

k

Acov

(%)

Fig. 6 Sparse–network case study: (a) Initial network configuration. (b) Network evolution through time. The black circles (bluedots) represent the nodes’ final (initial) positions. (c) Final network optimum state. (d) Percentage of covered area w.r.t. time. Theblue line represents the maximum possible coverage ratio.

themselves in a way that there is no overlapping between their sensing patterns or the boundary of the regionunder surveillance, while attaining optimum coverage, i.e. J = nπr2.

The way in which the area of the covered region increases is shown in the bottom part of Fig. 6. The bluestraight line represents the maximum possible coverage ratio, which in this case is 56.2%. The latter, startingfrom an initial value of 18.66% (dependent on the initial network configuration) , increases as time passes byin a monotonic way, until it reaches its optimum value in lessthan 2000 steps.

Similarly, considering a congested network, the network’sinitial configuration, the nodes’ evolution throughtime, along with their optimum configuration are shown in thetop part of Fig. 7. It is obvious that in this casethere exists overlapping among the nodes’ sensories. Contrary to the previous case, the maximum possible cov-erage percentage of 100% is not reached exactly, though approached at 97.33% in less than 1000 steps, startingfrom an initial value of 32.45%, as also seen by the bottom part of Fig. 7. In fact, the only way to guaranteeexistence of a configuration for complete coverage ofΩ is via global optimization techniques, something thatis beyond the scope of this paper. Either way, the state that the network converged to is an extremum of (12),while the coverage attained by the nodes is more than satisfactory.

Overall, one can see that the less the number of nodes in the network, the faster optimum coverage isreached. Indeed, since one node moves at a time and the nodes are selected in a random (or even in cyclic)manner, the less the number of nodes, the less possible is to come for a node that cannot contribute to coverage


0 200 400 600 800 1000 1200 1400 1600 1800 20000

10

20

30

40

50

60

70

80

90

100

k

Acov

(%)

Fig. 7 Congested–network case study: (a) Initial network configuration. (b) Network evolution through time. The black circles(blue dots) represent the nodes’ final (initial) positions.(c) Final network optimum state. (d) Percentage of covered area w.r.t. time.The blue line represents the maximum possible coverage ratio.

to move. Comparing this work to previous ones, it is seen thatthe network tries to optimize the total area cov-ered by itself, while it does not base its action on CVT coordination schemes or nearest–neighbor rules [6,17,9,4]. The fact that only one node moves at a time parts the maindifference, while optimal area achievement isguaranteed in a monotonic manner by sufficient knowledge of anode’s current and future Delaunay neighbors.

7 Conclusions and future work

In this paper a decentralized control strategy for sensing coverage optimization by a group of mobile agents thatconsist a homogeneous sensor network was presented. The coordination scheme was based on spatial Voronoitessellation, where the nodes self–organize their action in a way that the total area covered by the network is anincreasing function of time. Each node plans its current motion according to information obtained by its currentand future Delaunay neighbors, which is guaranteed by proper communication range adjustment. Simulationsconfirmed the efficacy of the proposed scheme. Possible extensions of this work consist of posing constrainton the maximum communication range of the nodes, while letting more than one node move at a time.

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decentralized swarm coordination: a combined coverage/connectivity approach

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