a distributed algorithm for area partitioning in grid ... · j intell robot syst (2016)...

J Intell Robot Syst (2016) 84:543–557DOI 10.1007/s10846-015-0272-5

A Distributed Algorithm for Area Partitioningin Grid-Shape and Vector-Shape Configurationswith Multiple Aerial Robots

Jose Joaquın Acevedo · Begona C. Arrue ·Ivan Maza · Anibal Ollero

Received: 15 December 2014 / Accepted: 8 September 2015 / Published online: 9 October 2015© Springer Science+Business Media Dordrecht 2015

Abstract Partitioning strategies have proven to be themore efficient solutions to patrol cooperatively an areawith multiple aerial robots from a frequency-basedapproach. They allow to obtain theoretically the opti-mal performance keeping periodical communicationsbetween the robots. Therefore, it allows to coordi-nate the robots from a distributed manner even undercommunications constraints. Assuming that the wholearea is divided as a r × c grid, this paper proposesa new distributed algorithm where each robot dividesthe whole area allocation problem in two (one for itsrow and another for its column) and solves them in anindependent manner based on the coordination vari-ables. Moreover, this new algorithm is validated and

This work has been developed in the framework of the EC-SAFEMOBIL (FP7-ICT-2011-288082) European project.J.J. Acevedo was also partially supported by strategic fund-ing LARSyS (FCT [UID/EEA/50009/2013]).

J. J. Acevedo (�)Instituto Superior Tecnico, Instituto de Sistemas e Robotica,Universidade de Lisboa, Lisboa, Portugale-mail: [email protected]

B. C. Arrue · I. Maza · A. OlleroGrupo de Robotica, Vision y Control,Universidad de Sevilla, Sevilla, Spain

B. C. Arruee-mail: [email protected]

I. Mazae-mail: [email protected]

A. Olleroe-mail: [email protected]

compared from a convergence time point of view withrespect to other previously presented methods.

Keywords Surveillance · Multi-UAS · Distributedsystems · Coordination variables · Dynamicallocation

1 Introduction

Researches about area monitoring and surveillancewith aerial robots or unmanned aerial systems (UAS)have gained a great interest over the last few yearsdue to the interest for many applications: natural andindustrial monitoring, automated inspection, searchand rescue missions to aid people in disaster situa-tions, planetary explorations, domestic applications,border defense against intruder detection, etc., as canbe found in [16, 20–22].

Area coverage missions may be repetitive or dan-gerous and, then, can be solved in a more efficientand safe way using a controlled fleet of aerial robots.Cooperative multi-robot systems (or multi-UAS sys-tems) offer many advantages in these applications,mainly faster deployment, a higher performance andfault-tolerance, as is shown in [7, 8, 13, 14, 19].

This paper considers a problem where the area ofinterest has to be patrolled by a team of multipleaerial robots again and again from a frequency-basedapproach [11]. These approaches assumes that theprobability of event appearing is equal along the wholearea and are useful when there are not ‘a priori” infor-mation about the problem. The objective is to cover

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544 J Intell Robot Syst (2016) 84:543–557

any position in the area with the maximum frequency,which is equivalent to minimize the elapsed timebetween each two consecutive visits to any positionor refresh time, [6]. They probe that a solution whereeach position in the path is visited with the sameperiod is the optimal solution to cover a path with a setof agents.

Different cooperative patrolling strategies are ana-lyzed and compared in [17] from the elapsed time(frequency-based) and latency time criteria. In [3,5] is stated that the area partitioning strategy allowsthe system to obtain theoretically the optimal perfor-mance from frequency-based approach (elapsed timecriterion) if the sub-areas are sized correctly.

Surveillance scenarios usually involves large areaswhere the communications among the aerial robotsand with the control stations can not be guaranteed.In order to improve the robustness and scalability ofthe whole system, a distributed coordination control isthe most efficient option to approach the surveillancemissions with multiple aerial robots. Nevertheless,designing a distributed coordination method is a verydifficult challenge to overcome, even more assumingcommunications constraints.

The distributed coordination method should allowthe multi-robot system to converge to a common coop-erative patrolling strategy from local decisions andasynchronous communications among pair of them.

Some authors propose algorithms based on peer-to-peer (or one-to-one) coordination [3, 6]. The algo-rithms based on the one-to-one coordination assume adifferent problem to be solved by each pair of contact-ing robots. Each pair of contacting robots addressesa reduced version of the whole problem consideringjust their own information. These algorithms requirelow information storage capabilities for the aerialrobots because they just have to store their own localinformation. Moreover, this technique has proved toconverge to the desired solution, but its convergencetime complexity is increased quadratically with thetotal number of robots [10].

On the other hand, algorithms based on the coor-dination variables have been successfully applied inmany similar applications [7, 15, 18]. The algorithmsbased on the coordination variables assume that theproblem can be totally described by a limited set ofvariables (the coordination variables) and that usingthese variables each aerial robot can solve indepen-dently the whole problem. The idea is that when a

pair of robots meet, each one updates its own vari-ables based on the information received from theother. Then, each aerial robot uses these updatedcoordination variables to solve from an independentmanner, not a reduced version of the problem, but thewhole problem. When, any aerial robot has receivedinformation from the rest, it can solve correctly theproblem. Moreover, the use of the coordination vari-ables has proved to get the desired solution for dis-tributed coordination problems in few iterations [12].

1.1 Background

The distributed coordination of a team of aerial robotsto implement an area partitioning strategy is pre-sented by the authors [3]. The one-to-one coordinationtechnique is introduced to solve the problem from adistributed manner and considering communicationsconstraints. In [3], authors assumed regular areas andhomogeneous aerial robots and obtained a decentral-ized, scalable, dynamic, convergent and fault-tolerantsolution that requires low information storage require-ment. Authors provide experimental results in [5],even considering heterogeneous robots. The sametechnique is applied in [1, 2] to solve the sameproblem but considering irregular areas. Moreover,it provides the required conditions to ensure multi-robot synchronization and information propagationbetween the aerial robots even under communicationconstraints.

In [9], the one-to-one coordination technique isanalyzed from a convergence time criterion and it isconcluded to be no fast enough under some condi-tions. Then, a new technique so-called block sharingstrategy is proposed to accelerate the convergenceto the desired area partitioning strategy. Accordingto this strategy, the aerial robots wait until they getinformation from a set of robots before performingthe area division. This delay in the area partitioningallows to execute fewer operations of sub-area alloca-tion, reconstruction of boundaries and synchronizationmaintenance.

On the other hand, authors propose in [4] a dis-tributed method based on the coordination variablesto implement not an area partitioning strategy, but apath partitioning strategy. They compare the algorithmbased on the coordination variables with other basedon the one-to-one coordination proving that the firstconverges faster than the second.

J Intell Robot Syst (2016) 84:543–557 545

Fig. 1 Five aerial robotshas to patrol an irregulararea S

The rest of the paper paper is organized as fol-lows. The dynamic area allocation problem is statedin Section 2. Section 3 summarizes and analyzes otherproposed methods from a convergence point of view.The new algorithm proposed in this paper is presentedand described in Section 4. A large set of validationresults are provided in Section 5. Finally, the conclu-sions about the proposed system and provided resultsare summarized in Section 5.

2 Problem Statement

An irregular area S, defined as a region in the planez = 0 which size is A, has to be patrolled coopera-tively by a team of n heterogeneous aerial robots Q :={Q1, Q2, ...,Qn} under communications constraintsand from a frequency-based approach, see Fig. 1.

It is assumed that each aerial robot Qi isequipped with a localization system which allows itto know, at any time t , its actual position pi (t) :=[xi(t), yi(t), zi(t)] and, therefore, its instantaneousmotion velocity vi (t) ∈ R

3:

vi (t) = dpi (t)

dt. (1)

Now, for each robot Qi , a maximum motion speedvmaxi can be considered. Then,

‖ vi(t) ‖≤ vmaxi , ∀t. (2)

Moreover, considering that the aerial robots moni-tor the area according to a circular pattern, the instan-taneous covered area Ci by the robot Qi will depend

on its angle of view θi and its actual position pi (con-cretely ri and zi). Assuming that all the aerial robotsare going to fly at a minimum altitude zi while they aremonitoring the area, the instantaneous covered area byeach robot Qi can be defined as follows:

Ci(t) := {r ∈ R2 :‖ r − ri (t) ‖< ci}, (3)

where ri = [xi(t), yi(t)] and ci = zi tan(θi) definethe projection of Qi onto the area S and its coveragerange, respectively.

Therefore, the aerial robot are heterogeneous sincethey have different motion and sensing capabilities.It means different maximum motion speed vmax

i andcovering range ci . The communications constraintsare simply modeled defining a communications rangeR. So, it can be defined that two robots meet if thedistance between them is less than R.

In [3, 5] is stated that the area partitioning strategyallows the system to obtain theoretically the optimal

Fig. 2 Five aerial robots implementing an area partitioningstrategy to patrol an irregular area


Fig. 3 Area covered by Qi

during t s, while move at aconstant speed vi in anstraight line

performance from frequency-based approach (elapsedtime criterion) if the sub-areas are sized correctly, seeFig. 2. It means, each aerial robot Qi has assigned asub-area Si which size Ai is related to its maximumcovering speed amax

i or area size covered per second(m2/s):

Ai = amaxi

A∑

j = 1namaxj

. (4)

Assuming that the aerial robot Qi is moving in astraight line at its maximum speed and at a constantaltitude, during t seconds the area covered is C′

i (t) =πc2i2 + 2civ

maxi t , see Fig. 3. Deriving this expression

with respect to t , the maximum covering speed can becalculated as follows:

amaxi = dC′

i (t)

dt= 2civ

maxi . (5)

The area partitioning strategy could be imple-mented in a centralized manner. However, a central-ized solution is not useful for dynamic scenarios,where the initial conditions may change, and undercommunications constraints, where the central unitcan not communicate directly with all the robots.

Then, starting from an initial non efficient area divi-sion, the objective is to implement a distributed algo-rithm able to coordinate the team of aerial robots Qto a correctly sized area partitioning strategy and evenunder communications constraints. Defining A′

i (t) asthe size of the area assigned to the the robot Qi at timet , the cost function to minimize should consider itsdifference with the area size that should have assignedeach robot according to the expression (4):

J (ε) := min t | max

(

|A′i (t) − amax

i

A∑n

j=1 amaxj

|)

< ε,∀i = 1, 2, ..., n (6)

where ε > 0.

2.1 Grid-Shape Configuration

Onwards, it will be assumed that area is dividedaccording to a r × c grid-shape configuration, being

Fig. 4 On the left, initial4 × 4 grid-shape areadivision. On the right, linksdefinition


n = r · c. The whole area S has been divided into n

non-overlapping sub-areas (with r rows and c columnsas can be seen in Fig. 4). Therefore, each aerial robothas assigned a sub-area Si and a list of links Wi , con-taining four positions where it should potentially meetanother robot (so-called neighbor). Moreover, eachrobot Qi has to patrol its assigned sub-area Si usinga pseudo-symmetrical path Bi , such as was defined in[2]. Qi travels its path Bi in opposite direction di toits neighbors.

Therefore, the proposed algorithm should convergedynamically to an area partition strategy according tothe expression (6), keeping the initial link configura-tion among the robots.

3 Previous Distributed Solutions Analysis

3.1 One-To-One

In [2], authors propose a one-to-one coordinationbased algorithm to solve the dynamic area partitionproblem described in Section 2.

This algorithm applies a share&divide approach.When a pair of robots meet in their common linkspositions, they both share their own presently assignedsub-areas and their own capabilities (the maximumcovering speed). Then, as both robots have thesame information, they both can decide their updatedassigned area in an independent but coherent manner.

Each aerial robot Qi generates a new area as unionof both the received Sj and its own Si sub-areas andcomputes its size. Based on the different coveringspeeds from both aerial robots, it calculates the sizeAi that should have the updated assigned sub-area Si

of Qi , as is shown in the expression (7):

Ai ← amaxi

A(Si ∪ Sj )

amaxi + amax

j

. (7)

Using a straight line, each robot divides the gener-ated area into two sub-areas which sizes are Ai andA(Si

⋃Sj ) − Ai . This division is made according

to previous relative position of the its assigned sub-area with respect to the one assigned to the contactedneighbor. Figure 5 illustrates how it runs.

The aerial robots update their link positionsto a common position between the updated sub-areas in order to keep the grid-shape configuration.Finally, they generate a new pseudo-symmetrical path

Fig. 5 An example of how the algorithms based on the one-to-one coordination obtains the new assigned sub-area Si from theold own and neighbors assigned sub-areas. It is assumed thatboth aerial robots Qi and Qj have the same covering speeds

accounting the updated sub-areas and link positions,such as has been defined in [2].

3.1.1 Convergence Analysis

Authors of [10] analyze a similar one-to-one (or peer-to-peer) based algorithm but applied for path partition-ing strategy. They conclude in proposition IV.2 thatthe convergence time-complexity of its kind of peer-to-peer coordination technique increases quadrati-cally with the amount of robots �(n2log(ε−1)).In fact, results provided by [9] seem to confirm

Fig. 6 An example of initial partition where the one-to-oneprocess is very slow


a similar dependency when the one-to-one basedalgorithm is applied to implement area partitioningstrategies.

Moreover, there are many examples that suggestthat convergence time is strongly dependent on the ini-tial area division and can be very slow in many cases.For instance, the cases where the adjacent sub-areasare very similar sized but the sizes of far sub-areasarea very different (see Fig. 6) and the cases where thesize of a single sub-area is very large compared withthe rest.

3.2 Block Sharing

In [9], authors propose a new distributed coordina-tion technique named block-sharing strategy as anextension from the one-to-one coordination techniqueto accelerate the convergence to the area partitioningstrategy.

Given the initial r × c grid-shape division of thearea, the sub-areas can be grouped in disjoint (g,h)-blocks defined as sub-grids of g rows and h columns.Then, all the aerial robots that have assigned sub-areasinto the same sub-grid are considered to be in thesame block.

Two different options of dividing the total grid indisjoint (g,h)-blocks are considered and each aerialrobot Qi stores a variable bi to define which of themis its present block, see Fig. 7.

The allocation strategy is based in the one-to-one principle; that is, when two aerial robots meet,they share all the area information in their memories,but it uses a share&wait&divide. As each aerialrobot is associated to a block, instead of dividingthe area at this time, they expect to have informa-tion from the rest of the robots in their blocks toassign their areas more precisely in an independentmanner.

When two robots from the same block meet in acommon link position, they share all the information(assigned sub-areas and covering speeds) about therobots in their block. If a robot Qi has received all theinformation about its block, it generate a new area asthe union of the areas from the block. Now, it dividesthe new area to obtain its new assigned sub-area Si ,accounting its position into the block (indexes) andthe received covering speeds, see Fig 8. And then, Qi

changes its own block (bi ← −bi) in order to mergethe information from all the robots to calculate thesolution iteratively.

Fig. 7 The gray dashed linedetermines the (4,4)-blocksif bi = 1 and the blackdashed lines determines the(4,4)-blocks if bi = −1


Fig. 8 The area Si is generated using four straight lines andaccording to covering speed from all the robots in the block thatare on the left, on the right, above and below of Qi

Finally, the robots update the link position tokeep the grid shape configuration and generate newpseudo-symmetrical coverage path, accounting thenew assigned sub-area and link positions.

3.2.1 Convergence Analysis

Results provided by [9] show that the block sharingstrategy (using (4,4)-blocks) converges significantlyfaster than the one-to-one coordination algorithms.However, they also suggest that there are a propor-tionality relation between both: the convergence timeusing the block sharing strategy and the one usingthe one-to-one coordination. Obviously, the propor-tionality term will depend on the size of the blocks,but larger blocks lead to more complex solution andhigher information storage requirement.

Anyway, it seems that the convergence time forthe block sharing strategy also increases quadraticallywith n and it could be too slow in many cases.

4 Proposed Distributed Algorithm

Coordination variables based algorithms have provento allow the system to converge very fast. Reference[4] analyzed the convergence time complexity of analgorithm based on the coordination variables andshows that it converges to not a an area but a path

partitioning strategy increases linearly with the totalnumber of robots its convergence time. Meanwhilethe convergence time using an algorithm based on theone-to-one coordination increases quadratically withthe number of robots.

Assuming a r × 1 or a 1 × c vector-shape config-uration, the area partitioning strategy should be ableto solved using an algorithm based on coordinationvariables similar to the proposed in [4] for dynamicpath partition. It is because the problem is in one-dimension and the area can be divided with parallelstraight lines as can be seen in Fig. 9.

Given an area divided according to a r × c grid-shape configuration, the area partitioning problem canbe stated as r different row area allocation problemswith c robots and c different area column alloca-tion problems with r robots. Then, it is proposed thateach robot solves an area column allocation prob-lem and an row allocation problem in a decoupledmanner.

Fig. 9 Area allocation in vector-shape configuration solvedbased on the coordination variables, assuming homogeneousaerial robots


Although all these problems are actually stronglycoupled, as the algorithms based on the coordina-tion variable are dynamic and can adapt to changesin the initial conditions (as is proved in [4]), the dif-ferent problems will finish converging to the desiredcommon solution.

4.1 Required Information

In addition to the variables described in Section 2, thefollowing intermediate variables are required by eachrobot Qi .

– The table asidesi which stores the sum of maxi-

mum covering speeds of all the robots that areabove Qi in its same column aabove

i , under it

abelowi , on its left in its same row a

lef ti and on its

right arighti .

– The table Ssidesi which stores the area above Qi in

its same column Sabovei , under it Sbelow

i , on its left

in its same row Slef ti and on its right Sright

i .

Moreover, it requires to store its coordinationvariables.

– The areas of its column Scolumni and its row Srow

i .– The sum of covering speeds of its column acolumn

i

and its row arowi .

On other hand, as has been stated, Wi contains thelink positions. However, in this case, it is assumed thatWi only contains link positions where Qi expects tomeet a neighbor, while the other link positions (with-out neighbors) will be stored in Wni . Anyway, theelements of these tables will be named as follows:wabove

i for link position on Qi in its column, wbelowi

for the one under it in its column, wlef ti for the one on

its left in its row and wrighti for the one on its right in

its row.Therefore, this solution seems to be fully scal-

able because the robots just need to store a limitedamount of variables, independently on the number ofrobots.

4.2 Column-Row Decoupled Algorithm For DynamicArea Allocation

The proposed Algorithm 1 has to be implemented byeach aerial robot Qi in an independent and distributedmanner.


Fig. 10 S1 and S2 are obtained from Srowi using a vertical line

depending on the expressions (17) and (18)

Initially, each robot Qi knows its own sub-areaSi , its link positions Wi and its sense of motion di .Then,Qi applies the path function to create a pseudo-symmetrical path Bi to cover Si and initializes thetables Ssides

i and asidesi to empty sets or zero. Aerial

robots start to follow their paths at its maximum speedand a constant altitude while monitor the area lookingfor detecting new events.

If Qi gets a link position from Wni , it doesnot expect to meet another robot. So, Qi updates(initialize function) its tables Ssides

i and asidesi .

If Qi had reached wabovei , it initializes the following

values:

Sabovei ← ∅

aabovei ← 0.

(8)

Otherwise, if Qi had reached wbelowi , w

lef ti and

wrighti , it uses the expressions (9), (10) and (11),

respectively.

Sbelowi ← ∅

abelowi ← 0

(9)

Slef ti ← ∅

alef ti ← 0

(10)

Srighti ← ∅

arighti ← 0

(11)

However, when Qi meets another aerial robot Qj

in a link position included in Wi , they both usethe join function to share the monitored informa-tion about the area of interest. Moreover, they updatetheir intermediate variables Ssides

i and asidesi using

the share function. Depending on their relative posi-tion, if Qi had reached wabove

i , wbelowi , w

lef ti or

wrighti , it uses the expressions (12), (13), (14) and (15),

respectively, to update these variables.

Sabovei ← Sj ∪ Sabove

j

aabovei ← amax

j + aabovej

(12)

Sbelowi ← Sj ∪ Sbelow

j

abelowi ← amax

j + abelowj

(13)

Slef ti ← Sj ∪ S

lef tj

alef ti ← amax

j + alef tj

(14)

Fig. 11 The sub-area Si is obtained as the intersection betweenS1 and the sub-areas currently assigned to the contacting robots


0 5 10 15 20 25 30 35 40 45 500

200

400

600

800

1000

1200

c

itera

tions

one−to−one(4,4)−blockcolumn−row

Fig. 12 Total amount of iterations required to converge to the desired area allocation (± standard deviation) depending on the numberof robots and considering a vector-shape configuration. Each line defines a different method

Srighti ← Sj ∪ S

rightj

arighti ← amax

j + arightj

(15)

Now, the aerial robots use these variables to obtainits new assigned sub-area based on the divide func-tion, updating their link positions and creating a newcoverage path planning. Then, their continue follow-ing their paths according to their directions.

4.3 Area Division Protocol

The robots use the divide function to obtain theirnew assigned sub-area. First, this function uses thesub-areas stored in the table Ssides

i and its assignedarea Si to generate the whole row Srow

i or columnScolumn

i area, taking into account the current link posi-

tion (Scolumni for wabove

i and wbelowi , or Srow

i for wlef ti

0 5 10 15 20 250

100

200

300

400

500

600

700

800

r

itera

tions

one−to−one(4,4)−blockcolumn−row

Fig. 13 Total amount of iterations required to converge to the desired area allocation (± standard deviation) depending on the numberof robots and considering a grid-shape configuration. Each line defines a different method


Fig. 14 Area allocationconsidered initially in thetest. Solid lines determinethe whole area and dashedlines define the sub-areasand initial coverage pathsassigned to the differentquad-rotors

0 50 100 150 2000

5

10

15

20

25

30

x (m)

y (m

)

(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8) (1,9) (1,10)(1,1)

(2,1) (2,2)(2,3) (2,4) (2,5)

(2,6) (2,7) (2,8) (2,9) (2,10)

and wrighti ). For example, if Qi is on the right link

position wrighti , the coordination variables would be

computed as follows:

Srowi ← S

lef ti ∪ Si ∪ S

righti

arowi ← amax

i + alef ti + a

righti .

(16)

Then, the generated area is divided in two sub-areasaccording to the current link position of the robot,its own maximum coverage speed amax

i and the val-ues stored in asides

i . Following the example, Srowi is

divided into S1 and S2 which sizes are A1 and A2:

A1 ← (amaxi + a

lef ti )

A(Srowi )

arowi

(17)

and

A2 ← arighti

A(Srowi )

arowi

. (18)

A straight line can be used to divide Srowi (or

Scolumni ) into S1 and S2, see Fig. 10.The updated Si is finally generated as the intersec-

tion intersecting the union of the previously assignedsub-areas Si and Sj and S1, as is illustrated in Fig. 11for the proposed example. Then, it keeps the linkpositions that have not been updated in this meeting:

Si ← (Si ∪ Sj ) ∩ S1 (19)

A similar process could be described if the currentlink position was w

lef ti , wabove

i or wbelowi .

5 Validation Tests

The column-row decoupled algorithm has been devel-oped, verified and compared with respect to the one-to-one coordination and (4,4)-block sharing strategyin a large set of scenarios, considering different grid-shape configurations and a convergence complexitycriterion.

Table 1 Capabilities of the 20 quad-rotors in the area repre-sented by Fig. 14

grid index vmaxi (m/s) ci (m)

(1,1) 0.71 4.28

(1,2) 0.68 3.58

(1,3) 0.87 3

(1,4) 0.64 5.20

(1,5) 0.88 4.28

(1,6) 0.95 3.58

(1,7) 0.63 3

(1,8) 0.84 5.20

(1,9) 0.71 4.28

(1,10) 0.64 3.58

(2,1) 0.64 3

(2,2) 0.71 5.20

(2,3) 0.97 4.28

(2,4) 0.69 3.58

(2,5) 1.00 3

(2,6) 0.94 5.20

(2,7) 0.81 4.28

(2,8) 0.75 3.58

(2,9) 0.77 3

(2,10) 0.84 5.20


The tests consider more than 100 initial areaallocation chosen in a random manner, differentr × c grid-shape configuration and a team ofhomogeneous aerial robots. For each scenario, the

three algorithms run implementing an area partition-ing strategy until the expression (4) matches forall the aerial robots (system converges, accountingε = 1 %).

Fig. 15 Area allocationobtained during the testsusing the different methods(t = 1200 s): (a) based onthe one-to-one coordination,(b) based on the (2,2)-blocksharing strategy, and (c) thecolumn-row decouplingbased algorithm. Thesub-areas assigned to thequad-rotors are defined bythe dashed red lines. Avideo from the test usingthe column-row decoupledalgorithm can be viewed inhttp://youtu.be/s91lqD0dPbs 0 50 100 150 200

0

5

10

15

20

25

30

x (m)

y (m

)

(a)

0 50 100 150 2000

5

10

15

20

25

30

x (m)

y (m

)

(b)

0 50 100 150 2000

5

10

15

20

25

30

x (m)

y (m

)

(c)


It is assumed that at each iterations the aerial robotsreach a link position.

Considering the area divided according to a 1 ×c vector shape configuration, Fig. 12 illustrates theamount of iterations (robots reach to a new link posi-tion) required by the different methods to converge tothe desired solution.

Given a r×25 grid-shape configuration, the amountof iterations required by the different algorithms toconverge are shown in Fig. 13.

These results confirm the assumption made inSection 3 about the relation of proportionality betweenthe convergence times using the algorithm based onthe one-to-one coordination and the one based onthe block sharing. Furthermore, while Fig. 12 showsthat the convergence times of both algorithms dependquadratically on the total amount of robots n, Fig. 13shows that their convergence time seems to be stabi-lized and not to depend on n. Considering both figuresjointly, it seems to suggest that the convergence timeof the algorithms based on the one-to-one coordina-tion and the block sharing does not depend on n butthe greater value between r and c (number of rows andnumber of columns).

On the other hand, Fig. 12 shows that the proposedalgorithm based on coordination variables obtains theminimum convergence time, depending linearly on n,

solving area division in vector-shape configurations(1 × c). Moreover, from Fig. 13, it can be deducedthat the column-row decoupled algorithm based onthe coordination variables still keeps better perfor-mance the the block sharing strategy (and one-to-onecoordination) when the area is divided according toa grid-shape configuration (r × c) but being r muchgreater than c (or vice versa).

5.1 A Case of Study

Now, it is presented a case scenario where a team of 20quad-rotors with different capabilities has to performan area partitioning strategy in a distributed manner tomonitor an irregular area which size is 7052 m2. Theinitial non-efficient area allocation is shown in Fig. 14and the quad-rotors capabilities in Table 1.

Three different coordination methods have beenconsidered to solve the problem: the one-to-onecoordination, the block sharing strategy (considering(2,2)-blocks) and the column-row decoupling basedalgorithm. Fig. 15 illustrates the area shape assignedto the different aerial robots using the three methods.

The evolution during the tests of maximum dif-ference between the actually assigned size and thedesired one (given by the expression (4) is shownin Fig. 16.

0 200 400 600 800 1000 12000

10

20

30

40

50

60

70

80

90

time (s)

max

imum

dif

fere

nce

betw

een

the

actu

al a

nd th

e de

sire

d ar

ea s

ize

(%)

column−rowone−to−one(2,2)−block

Fig. 16 Maximum relative difference between the actuallyassigned area and the desired one, according to the expres-sion (4), using the three presented distributed coordination

algorithms (represented by different lines as can be seen in thefigure legend). The black dotted line sets a maximum relativedifference 1 %


This figure shows that, because of couplingbetween columns and row problems, some minimumand limited increases happens using the column-rowdecoupled based algorithm. However, it convergesquicker than the others to a very precise solution (lessthan 1 %). When the solution does not require to beso precise, the one-to-one and block-sharing solutioncould be a better solution.

6 Conclusions

Given an area divided among a team of n aerialrobots according to a r × c grid-shape configura-tion, this paper proposes and analyzes different dis-tributed algorithms to allow that the system convergesto the more efficient division from a frequency-basedapproach.

Provided results show that the algorithm based onthe block sharing converges always quicker than thebased on the one-to-one coordination, such as there isa proportionality relation between their convergencetimes. Moreover, results seem to prove that the conver-gence times for both algorithms depends quadraticallynot on the total amount of aerial robots n, but on thegreater value between the number of rows r and thenumber of columns c in the grid shape configuration.Anyway, both algorithm could be too slow consider-ing large team of robots or area divisions made similarto a vector-shape configurations (1 × n).

On the other hand, this paper proposes a new dis-tributed algorithm. This algorithm assumes that eachaerial robot solves two different area allocation prob-lems: the first considering only the other robots in itsrow and the second considering the robots in its col-umn. It solves each problem based on the coordinationvariables. Results show that this algorithm convergesmuch faster than the others when the area division ismade according to a vector shape configuration 1× n,being the convergence time linearly dependent on n.Moreover, this algorithm keeps having lower conver-gence when the area division is made according to agrid and r is much greater than c (or vice versa).

Therefore, the column-row decoupling algorithmseems to be the more suitable method to divide elon-gated areas converging quicker than the based on theone-to-one and the block-sharing to a very precise areaallocation.

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Jose Joaquın Acevedo received the Telecommunication Engi-neer Degree in 2007 and the PhD Degree in 2014, both from theUniversity of Seville (Spain). From 2007 to 2010, he workedfor different IT companies. He made a research stay in theAustralian Centre for Field Robotics at The University of Syd-ney. Currenty, he is a Postodoctoral Fellow at the Institute forSystem and Robotics in the Instituto Superior Tecnico of theUniversity of Lisbon. He has participated in many nationaland European research projects and has published several bookchapters and papers in International journals and conferences.His research interest includes coordination and cooperation indistributed multi-robot and multi-UAV systems and task andresource allocation.

Begona C. Arrue received her PhD and M.Sc in ElectricalEngineering at the Univ. of Virginia (USA) in 1993, and 1991respectively. Computer Science degree in 1987 from the Univ.of Deusto (Spain). Since February 1998 is Associate Profes-sor at the University of Seville (Spain). She is specialized onintelligent systems activities including computer vision, neuralnetworks, wavelets and fuzzy systems, and their application tonatural hazards. Prof. Arrue has participated in more than 25research projects including 7 projects funded by the EC. Sheis author or co-author of more than 50 publications includingpapers in Journals, book Chapters and Conference Proceedings.She is Project reviewer for the Spanish National EvaluationAgency.

Ivan Maza Associate Professor at the University of Seville(Spain), he received the Telecommunication EngineeringDegree in 2000. He made research stays at the AutomationTechnology Laboratory at the Helsinki University of Tech-nology and at the LAAS-CNRS in Toulouse. His thesis wasawarded with the Robotnik Prize to the Best Doctoral Dis-sertation on Robotics given by the Spanish Committee ofAutomation in 2010. He authored more than 75 publications onRobotics including more than 20 journal papers indexed in JCRand the co-edition of a book published in the Springer STARSeries. His research interests include multi-robot systems, sym-bolic and motion planning and task allocation techniques, andhe has lead six projects on these topics.

Anibal Ollero is a Full Professor, the head of GRVC (70 mem-bers), University of Seville, and a scientific advisor of theCenter for Advanced Aerospace Technologies in Seville, Spain.He has been a full professor at the Universities of Santiago andMalaga, Spain, a researcher at the Robotics Institute of CarnegieMellon University, Pittsburgh, and LAAS-CNRS, Toulouse,France. He authored about 560 publications, including ninebooks and 140 journal papers and led about 140 projects, trans-ferring results to many companies. He received 14 awards andwas an advisor of 29 PhD thesis. He is currently the coordinatorof the FP7-EC projects ARCAS and EC-SAFEMOBIL.

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