Graph methods in Multi Agent Systems Coordination and Social Network Analysis

DOTTORATO DI RICERCA

in Ingegneria Elettronia e Informatia

Cilo XXVI

TITOLO TESI

Graph methods in Multi Agent Systems oordination and

Soial Network Analysis.

Settore sientio disiplinare di aerenza

ING-INF/04 LAutomatia

Presentata da: Daniele Rosa

Coordinatore Dottorato: Prof. Fabio Roli

Tutor: Prof. Alessandro Giua

Ph.D. in Eletroni and Computer Engineering

Dept.of Eletrial and Eletroni Engineering

University of Cagliari

Graph methods in Multi Agent

Systems oordination and

Soial Network Analysis.

Daniele Rosa

Advisor: Prof. Alessandro Giua.

Curriulum: ING-INF/04 Automatia

XXVI Cyle

Marh2013

DanieleRosa gratefullyaknowledges SardiniaRegional Governmentforthe nanial

support of his PhD sholarship (P.O.R. Sardegna F.S.E. Operational Programme of

the Autonomous Region of Sardinia, European Soial Fund 2007-2013 - Axis IV

In this thesis several results on two main topis are olleted: the

oordination of networked multi agents systems and the diusion of

innovationof soialnetworks. The resultsare organized intwo parts,

eah one related with one of the two main topis. The ommon

as-pet of allthe presented problems is thefollowing: allthe system are

represented by graphs.

Two are the main ontributions of the rst part.

•

A formationontrolstrategy, based on gossip,whihleads a set of autonomous vehiles to onverge to a desired spatial

dispo-sition in absene of a ommon referene frame. If the vehiles

haveommondiretion,weprovethatthe proposed algorithmis

robust against noise ondisplaementmeasurement.

•

The formalization of the Heterogeneous Multi Vehile Routing Problem, whih an be desribed as follows: given an

hetero-geneous set of mobile robots, and a set of task to be served

randomly displaed in a 2D environment, nd the optimal task

assignment to minimize the servie ost. We rstly

harater-ize the optimal entralized solution, and then we propose two

distributed algorithms, based on gossip, whih lead the system

to asub-optimal solutionsand are signiantlyomputationally

more eient than the optimal one.

The ontributions of the seond part are the following.

•

Adoptingthe Linear ThresholdModel,weproposean algorithm based on linear programming whih omputes the maximal

solution: the Inuene Maximization Problem in Finite Time

and the diusion of innovationover a targetset.

•

We haraterize the novel Non Progressive Linear Threshold Model,whihextends the lassialLinearThreshold Model. We

formalizethemodelandwegiveaharaterizationofthenetwork

At the end of these three years, I want to thank all the people that

haveshared the importantmomentsof this experiene.

First,I would liketothank allthe peoplewith whom I have

ollabo-rated,thathelpedandsupportedmeinmysientiwork: myadvisor

Prof. Alessandro Giua, Prof. Carla Seatzu, Mauro Franeshelliand

Prof. Franeso Bullo.

Aspeial thankgoestomy Lab-friends: Maro, Stefano,Mehranand

Alessandro. Thank youguys and..."PRIMO!!".

A huge thank to my family, for their enouragement during all these

years.

Finally,I thank Romina...youhave been always newt to me.

Introdution 1

Introdutiontothe thesis . . . 1

Part1: Coordination of multi-agent systems through onsensus . . . . 2

Part2: Diusion of innovation inSoialNetworks . . . 3

I Coordination of Multi-Agent Systems 5 1 Usingonsensustooordinatemulti-agentsystems: introdution and literature overview. 7 1.1 Formationontrol formultivehile systems . . . 8

1.2 The HeterogeneousMulti Vehiles RoutingProblem . . . 10

2 Formation Control Strategy 13 2.1 Preliminaries . . . 13

2.1.1 Coordinatesystems . . . 15

2.2 Formationontrol strategy . . . 16

2.2.1 Position update rule . . . 17

2.2.2 Consensus onthe network entroid . . . 19

2.2.3 Formationontrolstrategy . . . 20

2.2.3.1 Case I: stati topology . . . 21

2.2.3.2 Case II: time-varying topology. . . 22

2.2.4 Charaterizationof the robustness of the approah . . . . 23

2.2.5 Convergene speed . . . 25

2.3 Formationontrol strategy inabsene of a ommonreferene frame 26 2.3.1 Ahieving onsensus ona ommonreferene diretion . . . 27

2.3.3 Formationontrolstrategy . . . 29

2.4 Simulationresults . . . 29

2.4.1 Agentswith a ommonreferene diretion . . . 30

2.4.2 Agentsin absene of ommon referenediretion. . . 31

2.5 Conlusions . . . 33

3 The Heterogeneous Multi Vehile Routing Problem 35 3.1 Problemstatement . . . 35

3.2 Optimalentralizedsolution . . . 37

3.3 Deentralized solution basedon optimalloaltaskassignment . . 42

3.3.1 MILP Gossip algorithm . . . 42

3.3.2 Computationalomplexity of the loaloptimization . . . . 42

3.3.3 Finitetime and almost sure onvergene . . . 47

3.3.4 Performane haraterization of the MILP algorithm . . . 49

3.3.5 Asymptoti behavior . . . 54

3.4 Anheuristi gossip algorithm . . . 55

3.4.1 Computationalomplexity of the loaloptimization . . . . 56

3.4.2 Charaterizationsof the heuristisolution . . . 59

3.5 Numerialsimulations . . . 61

3.6 Conlusionsand future work . . . 65

II Graph methods for diusion of innovation in soial networks 69 4 Mathematial models for the diusion of innovation in soial networks: Introdution and literature overview. 71 5 DiusionofinnovationintheProgressiveLinearThresholdModel 77 5.1 Network representation and referenemodel . . . 77

5.1.1 Network struture. . . 77

5.1.2 Linear threshold model . . . 78

5.1.3 Other mathematialresults . . . 78

6.1 The Inuene Maximizationin FiniteTime Problem (IMFTP). . 87

6.2 Diusionof innovation overa target set . . . 91

6.3 Numerialresults . . . 93

6.4 Conlusions . . . 94

7 A Non-Progressive instane of the Linear Threshold Model 97 7.1 Bakground . . . 97

7.2 Non-Progressive Linear Threshold Model . . . 98

7.2.1 System desription . . . 98

7.2.2 Update rule . . . 98

7.3 Cohesive and Persistent Sets . . . 99

7.4 System's dynami . . . 100

7.4.1 Evolution duringthe seedingtime:

0

≤

t

≤

T

s

. . . 102

7.4.2 Evolution afterthe seedingtime:

t > T

s

. . . 103

7.4.3 Someexamples . . . 106

7.5 Conlusions . . . 110

8 Conlusions 111 A Appendix 115 A.1 Algebraigraph theory . . . 115

Introdution to the thesis

This thesis ollets several results on two main topis: the oordination of

net-worked multi agents systems and the diusion of innovation of soial networks.

Bothtopishave beenwidely studiedinliteratureinreent years andindierent

elds,sine it isevident innature the enormouspowerof the olletivityrespet

to a single individual: the more a group of individuals is organized, the more it

grows up and generate well-beingto eah member. Moreover, ithas always been

evident thatmanytargetan bebetterreahedbyaoordinated groupofpeople

thanasingleindividual,and insome asesooperationisneessary. At thesame

time, there are some phenomena in whih some individuals (or group of them)

have a greater inuene in the ommunity than others. Thus, in the last two

deades, researhers of dierentelds have been attrated by suh onepts:

so-iology,biology,informatis,eletronis,artiialintelligeneand ontroltheory.

In this manusriptweaddress dierent problems haraterizedby some

om-mon aspets:

•

alltheonsideredthesystemsaresetsofsimpleautonomoussystems(agents orindividuals),whihare onnetedtogether by a network;

•

in eah system the behaviour of eah agent is inuened by the behaviour of interonneted agents;

•

all the desribed systems an be represented using graphs, thus all the mathematialresults of this thesis are based ongraph theory.

The thesisisorganized intwo parts,eahone fousedononeof the twomain

onsensus

Intherstpartwefousontheoordinationofmultivehilesystems. Givenaset

ofautonomousvehile,whihanexhangeinformationthroughaommuniation

network, we propose several solutionsof problems whih were largely studied in

literature in the reent years. All the results presented in this part are based

ondistributed onsensus algorithm: agentsexhange informationaording to a

ommonprotoolinordertoreahanagreementonaertainquantityofinterest.

In partiular, most of the proposed solutions are based on gossip algorithms,

whihare haraterizedby the following:

•

the ommuniation sheme involve onlya oupleof agents ateah step;

•

the ommuniation steps between ouple of agentsare asynhronous.

The ontributionof the rst part are the following.

(1) Aformationontrolstrategy. Weproposeanoveldeentralizedoordination

strategy, based on gossip, that allows a dynami multi-agent system, in

abseneofaommonrefereneframe,toestimateaommonorientationand

ahieve arbitrary spatial formations with respet to the estimated frame.

We assume that the agents are mobile point-mass vehiles that do not

haveaess toabsolutepositions(GPS). Tothe best ofour knowledgethis

strategyextends thestate ofart sineit simultaneouslysolvestwoproblem

whih are ommonly onsidered separately:

•

the ahievement of an agreement on a ommon referene frame in absene of it;

•

the ahievement of a desiredspatial disposition.

The method is robust against measurement noise of odometry or inertial

navigation.

(2) Distributed solutions for the heterogeneous multi-vehile routing Problem.

We fouson problemsof MTSP (MultiTravellingSalesman Problem),and

problemof MTSPistooptimallyassignnodes, whihhavetobevisited, to

thedierentvehiles,inordertominimizethesumoftheostsofthepaths.

TheproblemofMVRPrepresentsanextensionoftheMTSPinwhihother

variables are taken into aount suh as the apaity of vehiles or osts

assignedtothenodes. Weextendthe stateofartsineweonsider thease

wherea set of heterogeneous tasks arbitrarilydistributed ina planehas to

be servied by a set of mobile robots, eah with a given movement speed

andtask exeutionspeed. Ourgoalistominimizethe maximum exeution

time of robots. We propose two distributed algorithms based on gossip

ommuniation: the rst algorithm is based on a loal exat optimization

and the seond isbased on aloalapproximate greedy heuristi.

Part 2: Diusion of innovation in Soial Networks

Intheseondpartwefousonthediusionofinnovationinsoialnetworks. Whit

the expression soialnetwork we identify a group of people whih are onneted

together by some types of relationship: friendship, love, business. In partiular

we fous on the study of the mehanism whih onvine people to adopt an

idea oran innovation,and how the behaviour ofeah individualis inuened by

the behaviourof the onneted individualsorgroups. Followingthe trend of the

ontrolommunity,westudymehanismsofinnovationspreadinSoialNetworks

in order to foreast, optimize, ontrol some diusion behaviours. Our referene

mathematialmodelisthe so alledlinear threshold model, and the ontribution

of this thesis are the following.

(1) Analysis and ontrol of the diusion of innovation in the Linear Threshold

Model. Weadoptthelassiallinearthresholdmodel,whihisharaterized

asfollows:

•

at eah individual is assigned a threshold value, whih is a value in

[0

,

1]

;

•

a node adopts the innovation as soon as the ratio of its neighbours who have already adopted it isabove itsthreshold value;

•

the innovation isinepted in the network by a seed set of individuals.

Aording to this model, we rstly present an integer programming

prob-lemand aniterativealgorithmbasedonlinear programmingwhihtake as

input the set of innovators and ompute the maximal ohesive set of the

omplement of the seed set. The output of these algorithms an be used

to ompute the set of nal adopters in the network. We extend the state

of art by proposing a way to ompute the maximal ohesive set in a given

soialnetwork,whihwas justdened sofar, tothe best of our knowledge.

Then we introdueand formalizewith integer programmingtwoproblems.

The "inuene maximization in nite time problem (IMFT)" is that of

ndinga seed set of

r

individuals that maximizesthe spreadof innovation

in the network in

k

steps. This problem represent an extension of the

lassial inuene maximizationproblem, whih onsiders an innite time

horizon.

The seondoneis thatof ndingaseed set ofwhose ardinalityisminimal

whih diuses the innovationto adesired set of individual ink steps.

(2) A novelnon-progressive instane of the linear threshold model. The

lassi-al linear threshold model has a progressive nature, i.e., anindividual an

adopt the innovation if it hasn't adopted yet, but one adopted it annot

abandon it. We extend the lassial model by proposing a novel model in

whih eah individual in the soialnetwork is inuened by the behaviour

of its neighbours, and ateah steps it deideseither toadopt, abandon or

maintain the innovation by followinga threshold mehanism.

We assume that the innovation is inepted inthe network by a seed set of

individualswhihare assumed tomaintain theinnovationindependentlyof

the state of their neighbours for a nite time. We identify all the possible

evolution of the network under the proposed model, and we desribe in

details the evolution of the system in terms of two partiular type of

Coordination of Multi-Agent

Using onsensus to oordinate

multi-agent systems: introdution

and literature overview.

Multi Agent Systems (MAS) are a lass of systems haraterized by a set of

entities , agents, whih interat in a shared environment to ahieve a ommon

target. Suh systems have attrated the attention of many researhers from

dif-ferentelds inthe lastdeades: eonomy,soiology , philosophy, and, ofourse,

omputer sieneand automation.

In the ontroltheory ommunity the termagent identify anautonomous

sys-tem, with a simple dynami, whih interat with the environment where it

op-erates and takes autonomous deision to reah a given target. A Networked

Control System (NCS) is a system omposed by a set of agents whih exhange

informationthrough a ommuniation network, and take deisionsinuened by

neighbours to reah a ommon target. These system presents many advantages

with respet toisolated systems.

•

InaMAS agentsan exeuteinparallelsub-tasks ofasingleomplex task: that redues the totalexeution time and the omputationaloasts.

•

Theabsene of asingle deisionenter makesthe system morereliableand robustto failures.

Reently, in literature this onepts have been applied toproblemsuh as:

•

oordination of autonomousvehiles;

•

environmental monitoring;

•

loalizationsystems;

•

oordination of mobilerobots.

TypialmethodsrelatedwithMASarebasedondistributedonsensusalgorithms:

agentsexhangeloalinformationtoreahanagreementonaertainquantityof

interests. These algorithmshave been applied to problems suh as rendez-vous,

oking or intrusion detetion. When the state of the agents onverge to the

average of their initialstates werefer toit asaverage onsensus.

Inthe nexthaptersweapplyonsensus algorithmstotwodierentproblems.

In Chapter 2 wepresent a novel formationontrolstrategy, based on

onsen-sus, whihleads a set of autonomous vehilesto onverge to adesired formation

in absene of a ommon referene frame. In Chapter 3 we use gossip algorithms

tosolvea partiular instane of the MultiVehileRouting Problem.

All the presented approahes are based on a speial type of onsensus

al-gorithms, , namely gossip algorithms. Gossip algorithms are haraterized by

an asynhronous pairwise ommuniation sheme: at eah step only two agents

exhange informationindependently of the rest of the agents.

In the next setions weintroduethe two studiedproblems indetails.

1.1 Formation ontrol for multi vehile systems

Multi-agentsystemsonsistinginanetworkofautonomousvehilesbenetgreatly

fromthe globalpositioningsystem (GPS)inthat itallows tolosefeedbak

on-trol loops on estimated positions in a global referene frame ommon to every

vehile, enabling several ontrol tasks suh as surveillane, patrolling,

forma-tion ontrol or searh and resue missions to be performed. Unfortunately suh

a powerful tool may not always be exploited for several reasons: for instane

vulnerable tojamming attaks. Furthermore, if the desired sale of relative

dis-tanes between the vehiles is of the order of meters, the auray provided by

the GPS system might not be enough. The problem of how to oordinate a

network of agents in absene of absolute position information has thus reeived

great attention from the ontrol theory ommunity (1, 2, 3). Furthermore, it is usually assumed that the full network topology is not known by the agents and

that only loal point-to-point ommuniation or sensing are available to model

sensorswithlimitedapabilities. In (4)atheoretialframeworkand amethodto ahieve oking in a multi-agent system is proposed based on the famous three

rules of oking by Reynolds (5) and on loal interation rules based on virtual potentials that allow the ahievement of oking as global emergent behaviour.

In (6, 7, 8,9) the onsensus problem,i.e., the problem of howto make the state ofa set ofagentsonvergetoward aommonvalue, waspresented regarding also

theappliationofmulti-agentoordination. Inpartiularontrolstrategiesbased

ononsensus algorithmswere desribed inthese papers asafundamentaltoolto

ahieve synhronization of veloities, diretions or the attainment of onstant

relativedistanes between the agents.

Inourapproahweassumethateahagentestimatesrelativepositionswithits

neighbours initsown loalreferene frameentered onit. A similarassumption

wasmadein(10),whereaNyquistriteriontodeterminetheeetofthetopology of a multi-agentsystem performing formationontrolwas proposed; in this ase

theagentswere assumedtohaveaommonoordinatesystem butnotaommon

origin. Furthermore we rstly assume that eah agent has anonboard ompass,

whih allows allthe loal frames to have the same orientation. Then we remove

this assumption.

Many formationontrolstrategies arebasedonLeader-basedapproahes (11,

12),whihrequire thenetwork ofvehilestoproperlyfollowoneormore leaders, possibly ontrolled by a pilot, satisfyingeventually some onstraints. Also some

formationontrolstrategies intheliterature take advantage fromthepresene of

leadersexploiting network properties suh asgraph rigidity(13).

InChapter2wedesignaoordinationstrategyforpoint-massagentsinwhih

leadersare not required,and the desiredformationisexpressed with oordinates

againstmeasurementnoise. Theonept ofoverompensationispresented inthe

followingsetions.

In (14) a deentralized algorithm to make a network of agents agree on the loation of the network entroid in absene of ommon referene frames was

presented; the algorithmis based ongossip (onlyrandom asynhronous pairwise

ommuniations) and assumes stati agents displaed in a

3

-d spae. In (15) a deentralized algorithm based on gossip to make a network of agents agree on

a ommon referene point and frame was proposed, assuming stati agents in

a

2

-d plane. Our approah diers from (14, 15) in that we onsider dynami agents that move while the the estimation proess is exeuted, we assume that

all the agents loal referene frames are oriented in the same diretion and that

noise isaeting the relativeposition measurements. Furthermore, the proposed

approah isused to implementformationontrol.

Summarizing, the following are the main ontributions of Chapter2.

•

A novel loal interation protool that ahieves robust estimation of the network entroid robustto parameter unertainties.

•

A method to ahieve provably robust formation ontrol with respet to parameter unertainties inthe agents'dynamis.

•

Anextended methodtoahieverobustformationontrolwithformationsof arbitraryshapebyperformingagreementonaommonrefereneframe. We

providesimulationstoorroboratethedesriptionofthisextended method.

1.2 TheHeterogeneous Multi Vehiles Routing

Prob-lem

The travelling salesman problem (TSP) is a well known topi of researh and

an be stated asfollows: nd the Hamiltonianyle of minimum weight to visit

all the nodes in a given graph. Instrutive surveys an be found in (16, 17, 18). This problem has reeived great attention for both its theoretial impliations

and itsseveral pratial appliations. The Vehile Routing Problem (VRP) is a

of nding

n

tours tovisit allloations inminimum time.

Several extensionsofthe TSPandtheVRPhavebeenproposedtobettersuit

pratialappliationsbyintroduingseveraladditionalonstraintsandobjetives

suh asa variable number of vehiles,a nite load apaity, a ost assoiated to

eah node whih represents the demand of the ostumer, servie time windows

and several more. Numerous extensions are well summarized in (20, 21, 22). Finally, several extensions explore a dynami setting in whih multiple vehiles

serve adynami numberof tasksas disussed in (23).

Multi-vehileroutingproblemshaveaombinatorialnature,asallthepossible

tours must be explored to nd the optimal onguration. Exat algorithmi

formulations are based, for example, on Integer Linear Programming (ILP) as

desribed in (22, 24). General ILP solvers are haraterized by an exponential omputationalomplexity,thusinthe lastdeadesmanyapproximatealgorithms

havebeenproposedwhihareharaterizedbyaloweromputationalomplexity.

Examples of heuristis and approximate algorithmsare presented in (21, 25, 26,

27, 28,29).

Weare interested in aninstane of the VRP, alled the Heterogeneous Multi

Vehile Routing Problem (HMVRP), with the following properties: the number

n

of vehiles is given a priori, a set

K

is given ontaining

k

tasks arbitrarily distributed in a plane, to eah task is assigned a serviing ost, eah vehile is

haraterized by a movementspeed and a taskexeution speed.

Ithas beenshownin(30)thatwhenomparingthelengthofthe optimaltour of one vehile that visits all tasks loations with the multiple vehile ase, the

maximum lengthof the tours for the multiple vehile ase isproportional tothe

tourlength ofthe singlevehilease and proportionallyinverse tothe numberof

vehiles. Both upper and lowerbounds with suhsaling were given.

In Chapter 3 we extend the result in (30) by onsidering exeution times instead of tour lengths to aount for vehiles of dierent speeds, tasks with

arbitrary exeution osts and vehiles with dierent task exeution speeds. We

provideupperand lowerboundsto theoptimal solutionasfuntion ofthe single

vehile optimaltourlengthtoputinevidenehowtheperformane isaetedby

the number of vehiles.

between pairs of vehiles(31), the seond one is based onloaltask exhange of assignedtasks, oneby one,betweenouplesof vehiles(32). Forbothalgorithms weprovidedeterministibounds totheirperformane. The proposedapproahes

to the HMVRP are distributed algorithms easy to implement in a networked

system and have favorable omputational omplexity with respet to the ratio

k/n

between the number of tasks and vehiles instead of

k

as inthe entralized approah.

Notethat the onsidered probleman alsobeseen asa partiularinstane of

a min/max VRP problemwhose main feature is the heterogeneity of the speed

and the tasks exeution speed of the vehiles. Related works on the min/max

VRP probleminlude (33, 34, 35).

Summarizing, the following are the main ontributions of Chapter3.

•

We formalize the entralized problem in terms of a mixed integer linear programming(MILP)problemandextend thebounds in(30)forthe multi TSP to the HMVRP.

•

We propose a rst distributed algorithm, based on gossip ommuniation and on the solution of loal MILP, to solve the HMVRP and haraterize

some of itsproperties.

•

We propose a seond distributed algorithm to solve HMVRP, based on gossip ommuniation and on loaltask exhanges, haraterized by a low

omputationalomplexity.

•

We provide simulations that show that the proposed algorithms attain a onstant fator approximation of the optimal solution with respet to the

number of vehiles. A detailedomparison amongthe performanes of the

Formation Control Strategy

This hapter is organized as follow. In Setion 2.1 we present the onsidered

systemandthesetofassumptionsadopted. InSetion2.2weproposeaformation

ontrol strategy whih is haraterize by a parallel appliation of two dierent

deentralizedalgorithms: aloaldisplaementontrolrulewhihmoveeahagent

toward atargetpoint andaonsensus algorithmwhihallowsagentstoreahan

agreementonaommonrefereneframe. Theoneptofoverompensationishere

presented. InSetion2.2.4therobustnessoftheproposedstrategyisinvestigated

and anoptimal hoie of the algorithmparameters is disussed.

2.1 Preliminaries

Leta network of agents be desribed by a time-varying undireted graph

G

(

t

) =

{

V,

E

(

t

)

}

,where

V

=

{

1

, . . . , n

}

is the set of nodes (agents),

E

⊆ {

V

×

V

}

isthe set ofedges

e

ij

representing point-to-pointbidiretionalommuniationhannels available to the agents,

E

(

t

) :

R

+

→

E

is the set of edges being ative at time

t

.

Given a time interval

T

, the jointgraph

G

([

t, t

+

T

))

is the union of graphs

G

(

t

)

inthe time interval

[

t, t

+

T

)

dened as

G

([

t, t

+

T

)) =

{

V,

E

([

t, t

+

T

)))

}

, where

E

([

t, t

+

T

)) =

E

(

t

)

[

E

(

t

+ 1)

[

. . .

[

E

(

t

+

T

)

A node

u

∈

V

is said to be reahable from

v

∈

V

if there exists a path in the graphfrom

v

to

u

. Node

u

∈

V

is saidto bea enter node if itisreahable from any node in

V

. In a onneted undireted graph all the nodes are enter nodes.

A node

u

∈

V

is said to be aperiodi if the greatest ommon divisor of all the possible path length from

u

to

u

is

1

.

The state of eah agent

i

is haraterized by its absolute position

x

i

, an estimation of the origin of the ommon referene frame

s

i

∈

R

2

and an angle

θ

i

whih represents the orientation of the

x

-axis of the loal referene frame with

respet to the

x

-axisof the global referene frame.

Let

N

i

(

t

) =

{

j

:

e

ij

(

t

)

∈

E

(

t

)

}

be the set of agents that send and reeive informationtoagent

i

attime

t

,these agentsare alled neighbors of agent

i

. We

denethedegree ofagent

i

as

δ

i

(

t

) =

|

N

i

(

t

)

|

where

|

N

i

(

t

)

|

denotesthe ardinality ofset

N

i

(

t

)

. Theelementsof theLaplaianmatrix

L

ofgraph

G

(

t

)

are dened as

l

ij

=











−

1

,

if

(

i, j

)

∈

E

(

t

)

δ

i

(

t

)

.

if i

=

j

0

otherwise

Given a generi square matrix

M

n

×

n

, the assoiated graph

G

M

=

{

V

M

, E

M

}

is omposed asfollow:

•

G

M

has

n

nodes, with index

i

∈

[1

, n

]

,so

V

M

=

{

1

, . . . , n

}

;

•

G

M

has anedge

e

ij

if the entry

m

ij

∈

M

isnonzero, so

E

M

=

{

(

i, j

)

|

m

i,j

6

=

0

}

If

M

has non zero diagonal entry

m

ii

, than node

i

∈

G

M

has a self loop. If

M

is symmetri then

G

M

isan undireted graph. For atime-varying square matrix

M

(

t

)

the assoiated graphis denoted as

G

M

(

t

) =

{

V

M

, E

M

(

t

)

}

.

A square matrix

A

is stohasti if its elements are non-negative and the row

sumsequalsone. Astohastimatrixsaidtobeergodi if

rank

lim

k

→∞

A

k

= 1

. An ergodimatrix

A

isSIA (stohasti, indeomposableand aperiodi) if

lim

k

→∞

A

k

=

1

n

π

T

,

where

π

isthe left eigenvetor of

A

orresponding to the unitary eigenvalue and

1

n

is the n-element vetor of ones. Given two matries

A

(m

×

n)

and

B

(p

×

q)

, the Kroneker produt is denoted as

A

⊗

B

(mp

×

nq)

.

In our disussion weonsider the followingworking assumptions: i. Agents

are modelled by disrete time single integrators; ii. Neighboring agents

know the oordinatesystem of others.

2.1.1 Coordinate systems

A

2

-dreferene frame

Σ

′

= (

o

′

, θ

′

)

is anorthogonal oordinate system

harater-ized by an origin

o

′

∈

R

2

and orientation of the

x

-axis

θ

′

∈

[0

,

2

π

)

respet to a

global oordinatesystem

Σ

dened by

o

= (0

,

0)

and

θ

= 0

. We deal with three kindsof oordinate systems, whih are showed inFig. 2.1.

Figure 2.1: Coordinate systems.

•

Globaloordinatesystem: istherefereneframeusedtodesribethesystem from the point of view of an external observer. We denote it with

Σ

, and the urrent position of agent

i

speied in

Σ

is

x

i

∈

R

2

.

•

Loal oordinate system: eah agent owns a loal referene frameentered onit. The loaloordinatesystem of agent

i

is denoted with

Σ

i

= (

x

i

, θ

i

)

,

where

x

i

is the position of agent

i

in

Σ

and

θ

i

is the angle between the x-axis of

Σ

and the x-axis of

Σ

i

. We denotethe position of ageneri point

j

with respet to

Σ

i

as

x

i

j

. Therefore, the positionof

j

is

x

j

=

R

i

x

i

j

+

x

i

where

R

i

=

"

cos

θ

i

−

sin

θ

i

sin

θ

i

cos

θ

i

#

is arotationmatrix assoiated to the angle

θ

i

.

•

Estimated oordinate system: eah agent keeps a loal estimation of the ommon referene frame. With respet to

Σ

the estimated ommon ref-erene frame by agent

i

is denoted with

Σ

i,es

= (

s

i

, θ

i

)

, where

s

i

is the estimated refereneenter and

θ

i

isthe estimated anglebetween the x-axis of the ommon referene frame and the x-axis of

Σ

. Note that the orien-tation ofthe loalestimated referene frameisthe same asthe orientation

od

Σ

i

. We denote the position of a generipoint

j

with respet to

Σ

i,es

as

x

i,es

j

. The position of agent

j

in frame

Σ

i

is:

x

i

j

=

x

i,es

j

+

s

i

i

.

2.2 Formation ontrol strategy

Inthissetionwepresentadeentralizedontrolstrategywhihallowsanetwork

of mobile agents in a

2

-D spae to reah an agreement on a ommon referene frame and simultaneously onverge to a desired formation. Here we assume

that allthe agents have aompass onboard, whihallows themhave aommon

referene diretion. In partiular,we assume that

∀

i

∈

V

,

θ

i

= 0

. The state of

i

-thagent is haraterized by a position

x

i

∈

R

2

and a variable

s

i

∈

R

2

whih represents the estimated enter of the ommon referene frame. When referring tothe state of the agent inits own referene frame

Σ

i

we denote

itsurrentestimation as

s

i

i

∈

R

2

.

Our strategy involvesthree loal state updaterules:

•

A rule to update the position of the agents;

Eahagent ismodeledby disrete time single integratordynamis

x

i

(

t

+ 1) =

x

i

(

t

) +

qu

i

(

t

)

,

(2.1)

where

x

i

∈

R

2

is the agent position,

u

i

∈

R

2

is the ontrolation representing a

displaement and

q

∈

R

+

is a gain. Eah agent has to reah a onstant target

position

D

i

∈

R

2

with respet to its estimated ommon referene frame. The

targetposition

d

i

i

(

t

)

with respet to

Σ

i

attime

t

an beomputed as

d

i

i

(

t

) =

s

i

i

(

t

) +

D

i

.

In the ommon refereneframe

Σ

the target position of agent

i

is

d

i

(

t

) =

x

i

(

t

) +

d

i

i

(

t

) =

x

i

(

t

) + (

s

i

i

(

t

) +

D

i

)

.

(2.2) Therefore, eah agent drives itself toward its target position

d

i

i

(

t

)

with the following state update

x

i

(

t

+ 1)

−

x

i

(

t

) =

q

(

d

i

(

t

)

−

x

i

(

t

))

(2.3)

with respet to

Σ

. By replaing equation (2.2) in (2.3) we nd the following position update rule:

x

i

(

t

+ 1) = (1

−

q

)

x

i

(

t

) +

q

(

s

i

(

t

) +

D

i

)

(2.4)

Therefereneframeof agent

i

thusmovingrigidlywithit,displaeitsurrent

estimation of the ommon referene point. Therefore, the agent attempts to

ompensate this displaement by updating its estimation of the position of the

ommonreferenepointasfollowsInotherwords,beausetheagents'loalframe

is entered on

x

i

and moves rigidlywith it, eah agent

i

needsto update

s

i

i

, and onsequently

d

i

i

.

s

i

i

(

t

+ 1) =

s

i

i

(

t

)

−

q s

i

i

(

t

) +

D

i

(2.5) whih, with respet to referene frame

Σ

, keeps the absolute position of the estimated point onstantin time

To implement these updates, however, a perfet knowledge of parameter

q

is

required whih orresponds toan exat measurement ofthe movementor

atua-tors with perfet preision.

Sine measurements may beaeted by disturbaneand atuators subjeted

to malfuntioning, we introdue a dierent state update rule, whih we prove is

robust against unertainties in the parameter

q

of any agent. We all this state

update as overompensation beause it eetively moves the urrent estimation

further away than neessary, as follows:

s

i

i

(

t

+ 1) =

s

i

i

(

t

)

−

k s

i

i

(

t

) +

D

i

(2.6)

Equation (2.6)represents aoverompensation of agentdisplaementbasedon

parameter

k

,whihontrolshowmuhtheagentsompensatetheirdisplaement.

Using equation (2.4) and equation (2.6) in terms of

s

i

(

t

)

, we an express the generalupdate rule asfollow:

(

x

i

(

t

+ 1) =

x

i

(

t

) +

q

((

s

i

(

t

) +

D

i

)

−

x

i

(

t

))

s

i

(

t

+ 1) =

s

i

(

t

)

−

k

((

s

i

(

t

) +

D

i

)

−

x

i

(

t

)) +

q

((

s

i

(

t

) +

D

i

)

−

x

i

(

t

))

(2.7)

Wean set

h

=

k

−

q

and rewriteequation (2.7) asfollows:

(

x

i

(

t

+ 1) =

x

i

(

t

) +

q

((

s

i

(

t

) +

D

i

)

−

x

i

(

t

))

s

i

(

t

+ 1) =

s

i

(

t

)

−

h

((

s

i

(

t

) +

D

i

)

−

x

i

(

t

))

(2.8)

(

x

i

(

t

+ 1) = (1

−

q

)

x

i

(

t

) +

qs

i

(

t

) +

qD

i

s

i

(

t

+ 1) = (

h

)

x

i

(

t

) + (1

−

h

)

s

i

(

t

) + (

−

h

)

D

i

(2.9) Note that:

•

if

h

=

−

q

(

k

= 0

) the distane vetor

d

i

(

t

)

−

x

i

(

t

)

isonstant, thusthere is noompensation;

•

if

−

q < h <

0

(

0

< k < q

),

d

i

(

t

)

translate in the same diretion of

x

i

(

t

)

and

|

d

i

(

t

+ 1)

−

x

i

(

t

+ 1)

|

<

|

d

i

(

t

)

−

x

i

(

t

)

|

, thus there is only a partial ompensation;

•

if

h

= 0

(

k

=

q

)the targetposition

d

i

(

t

)

isonstant,thusthe ompensation is perfet;

•

if

h >

0

, (

k > q

)

d

i

(

t

)

moves toward

x

i

(

t

)

, thus an overompensation is made.

Eahagent has a loalestimate

s

i

i

(

t

)

whih onsiders as the enter of a ommon estimated frame. By exhanging this loal informationwith neighbours, agents

are abletoreahanagreementonaommonrefereneenter, whihmeansthat:

∀

i, j

∈

V,

lim

t

→∞

k

s

i

(

t

)

−

s

j

(

t

)

k

= 0

At eah time step agent

i

reeives the value

s

j

j

from eah agent

j

∈

N

i

(

t

)

. In Figure 2.2 it is shown how agent

i

is able to determine the orret value

s

i

j

of agent

j

with respet to

Σ

i

by only knowing

x

i

j

and the reeived value

s

j

j

. The

Figure 2.2: Informationexhange between agent

i

and

j

.

update rule for the loalestimate is:

s

i

i

(

t

+ 1) =

s

i

i

(

t

) +

ε

X

j

∈

N

i

(t)

(

s

j

j

(

t

) +

x

i

j

(

t

)

−

s

i

i

(

t

))

(2.10)

with

0

< ε

≤ |

N

i

(

t

)

|

. The same rule ould be writtenwith respet to

Σ

:

s

i

(

t

+ 1) =

s

i

(

t

) +

ε

X

j

∈

N

i

(t)

With respet to

Σ

the overall estimate update rule ould be expressed as follow:

s

(

t

+ 1) = (

P

(

t

)

⊗

I

2

×

2

)

s

(

t

)

(2.12) where

P

(

t

)

∈

P

is a time-varying matrix whih depends on network topology at time

t

and

ε

, and

P

is the set of all possible matries representing the system update dened in (2.11). Due to the update rule denition all matries

P

(

t

)

∈

P

are stohasti. Note that equation (2.12) an represent both deterministi synhronous onsensus algorithms and randomized gossip algorithms. At eah

t

, algorithm(2.12) an be represented by the assoiated graph

G

P

(

t

)

. If

∀

t >

0

there exists a

T >

0

suh that

G

P

([

t, t

+

T

))

is onneted, than

lim

t

→∞

s

1

(

t

) =

. . .

= lim

t

→∞

s

n

(

t

)

, where

G

P

([

t, t

+

T

))

isthe union of graphs

G

P

(

t

)

in the time interval

[

t, t

+

T

)

(7)(8).

2.2.3 Formation ontrol strategy

Let us dene olumn vetors

x

(

t

) =

{

x

1

(

t

)

, . . . , x

n

(

t

)

}

,

s

(

t

) =

{

s

1

(

t

)

, . . . , s

n

(

t

)

}

and

D

=

{

D

1

, . . . , D

n

}

. Note that

D

represents the desired formationrespet to a ommon enter. By summing the ontributions of equations (2.8) and (2.12)

the overall formation ontrolstrategy ouldbeexpressed as follow:

"

x

(

t

+ 1)

s

(

t

+ 1)

#

= (

M

(

t

)

⊗

I

2

×

2

)

"

x

(

t

)

s

(

t

)

#

+

"

qD

−

hD

#

(2.13) where

M

(

t

) =

"

(1

−

q

)

I

n

×

n

qI

n

×

n

hI

n

×

n

(

P

(

t

)

−

hI

n

×

n

)

#

(2.14)

For all

t

,

M

(

t

)

∈

M

, where

M

is the set of all possible matries of type (2.14) orresponding to dierent

P

(

t

)

∈

P

. A given formation is onsidered to be ahieved if

•

x

(

t

) =

s

(

t

) +

D

;

• ∀

i, j

∈

V,

k

s

i

(

t

)

−

s

j

(

t

)

k

= 0

Lemma 2.2.1 Considersystem (2.13). If

lim

t

→∞

(

M

(1)

M

(2)

. . . M

(

t

)

⊗

I

2

×

2

)

"

x

(0)

s

(0)

#

=

c

1

2n

(2.15)

•

lim

t

→∞

x

(

t

) =

s

(

t

) +

D,

• ∀

i, j

∈

V,

lim

t

→∞

k

s

i

(

t

)

−

s

j

(

t

)

k

= 0

.

Thus the desired formation is asymptotially ahieved.

Proof: Condition (2.15) implies that system (2.13) is stable. At the

equilib-rium

x

(

t

+ 1) =

x

(

t

)

and

s

(

t

+ 1) =

s

(

t

)

. From the rst equation of (2.13) we nd:

(1

−

q

)

Ix

(

t

) +

qIs

(

t

) +

qID

=

x

(

t

)

x

(

t

) =

s

(

t

) +

D

By substituting in the seond equation:

P s

(

t

) = (

I

−

εL

)

s

(

t

) =

s

(

t

)

whih implies

s

(

t

) =

c

1

, where

c

∈

R

isa onstant.

Convergene of the proposed strategy toward the desired formation an thus be

addressed by studying the stability of the followinglinear time-varying system

"

x

(

t

+ 1)

s

(

t

+ 1)

#

= (

M

(

t

)

⊗

I

2

×

2

)

"

x

(

t

)

s

(

t

)

#

(2.16)

2.2.3.1 Case I: stati topology

Ifthe network topology isstati and onneted, than

M

(

t

) =

M,

∀

t

.

Lemma 2.2.2 (Lin,(36)) A stohasti matrix M is SIA if and only if the asso-iated graph

G

M

has a entre node whih isaperiodi.

Nowwe are able to prove the followingresult.

Theorem 2.2.1 Consider a network of agents with a stati onneted topology.

Given system (2.16) with

M

(

t

) =

M

, if

0

≤

h

≤

1

−

εδ

max

(2.17)

where

δ

max

=

max

{

δ

1

,

· · ·

, δ

n

}

represents the maximum degree for the network, then

lim

t

→∞

"

x

(

t

)

s

(

t

)

#

=

c

1

2n

,

where

c

∈

R

isa onstant.

Proof If ondition (2.17) holds

M

is stohasti as all entries are non negative and row sums are equal to 1. Now we have to prove that

M

is SIA. We an

representsystem (2.16)usingaundiretedgraph

G

M

assoiatedtomatrix

M

. In this graph eahagent

i

isrepresented by two nodes:

•

one assoiated tothe agent position

x

i

,that we allpositionnode;

•

one assoiated tothe agent estimate

s

i

, that we allestimatenode.

Foreahagentthetwoassoiatednodesareonnetedtogetherbyabidiretional

edge, as the position update depends on the position estimate and vie versa.

The onnetions between agents depend on matrix

P

−

hI

. In partiular, given a ouple of agents

(

i, j

)

there exists an edge between their estimation nodes if the

p

ij

entry of

P

is non zero. As the network is onneted and undireted by assumption, the graph

G

M

is onneted as well and eah node is a enter node. More, as all diagonal entries in

(1

−

q

)

I

are nonzero, eah position node in the assoiatedgraphhas aselfloop,so

G

M

isaperiodi. ItfollowsfromLemma 2.2.2

that matrix

M

is SIA, so

lim

t

→∞

M

t

"

x

(0)

s

(0)

#

=

c

1

2n

where

c

is aonstant.

2.2.3.2 Case II: time-varyingtopology.

Inordertoprovetherobustnessof(2.16)weneedrsttopresentsomepreliminary

notions.

Lemma 2.2.3 (Jadbabaieetal.,(8)) Let

{

M

1

, M

2

, . . . , M

m

}

beasetof stohasti matriesofthesameordersuhthatthejointgraph

{

G

(

M

1

)

S

G

(

M

2

)

S

. . .

S

G

(

M

m

)

}

Lemma 2.2.4 (Wolfowitz,(37)) Let

{

M

1

, M

2

, . . . , M

m

}

be a set of ergodi ma-tries with thepropertythatforeahsequene

M

i

1

, M

i

2

, . . . , M

i

j

of positivelength

j

the matrix produt

M

i

1

M

i

2

. . . M

i

j

is ergodi. Then for eah innite sequene

M

i

1

, M

i

2

, . . .

there existsa rowvetor

c

suhthat

lim

j

→∞

M

i

1

M

i

2

. . . M

i

j

=

1c

.

Nowwe an state the following theorem

Theorem 2.2.2 Consider a network of agents with time-varying topology

de-sribed by (2.16). Let us assume that

∀

t >

0

there exists a

T >

0

suh that

G

P

([

t, t

+

T

))

isonneted. Thefollowingondition is suientfor the systemto onverge to the desired formation:

0

≤

h

≤

1

−

εδ

max

(2.18)

Proof Let

M

c

bethe set ofallpossible produt matriesin

M

of length

T

suh that the joint graph

G

P

([

t, t

+

T

))

is onneted. In the theorem we assume that for eah time interval

[

t, t

+

T

)

the matrix

M

(

t

)

M

(

t

+ 1)

. . . M

(

t

+

T

)

∈

M

c

Thus we an represent the evolution of the system as a produt of matries

M

c

(

t

)

∈

M

c

. Ifondition (2.18)holds,then allmatries

M

(

t

)

∈

M

are stohasti asshowed inthe proofofTheorem2.2.1,and itfollowsfromLemma2.2.3thatall

matries

M

c

(

t

)

∈

M

c

are ergodiaswell asall produtsin

M

c

. Finally it follows fromLemma 2.2.4 that:

lim

t

→∞

(

M

c

(1)

M

c

(2)

. . . M

c

(

t

)

⊗

I

2

×

2

)

"

x

(0)

s

(0)

#

=

c

1

2n

2.2.4 Charaterization of the robustness of the approah

The proposed oordination strategy desribed in setion 2.2 an be aeted by

errorsduetotheodometryorinertialnavigationsystem. Inpartiularthedesired

displaementthatthegeneriagent

x

i

(

t

)

shouldahievewithinonesampleoftime isas follows

wherethe time-varyingparameter

q

i

(

t

) =

q

+ ∆

i

(

t

)

modelsarandomerrorinthe position update attime

t

.

Thus, the proposed loalinteration rule beomes











x

i

(

t

+ 1) = (1

−

q

i

(

t

))

x

i

(

t

) +

q

i

(

t

)

s

i

(

t

)

s

i

(

t

+ 1) =

h

(

t

)(

x

i

(

t

)

−

s

i

(

t

))

+(

s

i

(

t

) +

ε

P

j

∈

N

i

l

ij

(

s

j

(

t

)

−

s

i

(

t

)))

(2.20) where

h

i

(

t

) =

h

−

∆

i

(

t

)

.

Let

Q

(

t

)

and

H

(

t

)

be

n

×

n

diagonalmatrieswhere

Q

ii

=

q

i

(

t

)

and

H

ii

(

t

) =

h

i

(

t

)

. The global system dynamisare thusdesribed by

"

x

(

t

+ 1)

s

(

t

+ 1)

#

= (

M

∆

(

t

)

⊗

I

2

×

2

)

"

x

(

t

)

s

(

t

)

#

(2.21)

M

∆

(

t

) =

"

I

−

Q

(

t

)

Q

(

t

)

H

(

t

)

P

(

t

)

−

H

(

t

)

#

(2.22)

Forall

t

,

M

∆

(

t

)

∈

M

∆

,where

M

∆

isainnitesetofmatries

M

∆

(

t

)

haraterized by dierent values of

q

(

t

)

,

h

(

t

)

and

P

(

t

)

. Now we haraterize the robustness of the proposed strategy with respet tomeasurement noise.

Theorem 2.2.3 Consider a system as in eq. (2.21). Let us assume that

∀

t >

0

there exists a

T >

0

suh that

G

P

([

t, t

+

T

))

is onneted . If the measurement noise

∆

i

(

t

)

is bounded by

h

+

εδ

max

−

1

≤

∆

i

(

t

)

≤

min

{

h,

(1

−

q

)

}

,

∀

i, t

(2.23) then

lim

t

→∞

"

x

(

t

)

s

(

t

)

#

=

c

1

2n

where

c

is a onstant.

Proof The diagonalentries of the matries

I

−

Q

(

t

)

and

P

(

t

)

−

H

(

t

)

are

[

I

−

Q

(

t

)]

ii

= 1

−

q

−

∆

i

(

t

)

We an assumethat

q >

∆

i

(

t

)

. Ifondition (2.23)hold, then allmatriesin

M

∆

are stohasti, beause all entries are non negative and row sums equal to one.

Thus, the proof follows asin theorem 2.2.2.

Notethat

∆(

t

)

ould be positive ornegative.

We now disuss what is the best parameter hoie to ahieve maximum

ro-bustness. Given a xed value of

q

, the optimum value of

h

is the one whih

maximizesthe following objetive funtion:

max

h

{

min

{

h,

(1

−

q

)

,

|

h

+

εδ

max

−

1

|}}

Bysubstitution it holds

•

If

1

−

εδ

max

2

≤

(1

−

q

)

theoptimumvalue of

h

is

h

=

1

−

δ

max

2

thusthebound (2.23)beomessymmetri

−

1

−

εδ

max

2

≤

∆

i

(

t

)

≤

1

−

εδ

max

2

,

∀

i, t

•

If

1

−

εδ

max

2

>

(1

−

q

)

the optimumvalue of

h

is

h

= (1

−

q

)

. It holds

εδ

max

−

q

≤

∆

i

(

t

)

≤

1

−

q,

∀

i, t

2.2.5 Convergene speed

Wenowharaterizethe onvergene speed ofthe proposedstrategy inthe

time-invariantase

M

(

t

) =

M

and

P

(

t

) =

P

. Let

Λ

M

bethe set of the

2

n

eigenvalues of

M

. As

M

isSIA,

λ

= 1

isasimple eigenvalue of

Λ

M

,andallothereigenvalues have moduleless than

1

. The onvergene speed of (2.16)dependsontheseond biggest module eigenvalue

λ

2

∈

Λ

, whih is alled algebrai onnetivity. By knowing the eigenvalues of

P

,

Λ

M

an be determined.

Theorem 2.2.4 Let

M

be a

2

×

2

blok matrix as in eq. (2.16). Let

Λ

P

=

{

λ

p1

, λ

p2

, . . . , λ

pn

}

be theset of the

n

eigenvalues of

P

. The

2

n

eigenvaluesof

M

are funtion of the eigenvalues of

P

as follows:

λ

m

i

1

,

2

=

(

λ

p

+ 1

−

h

−

q

)

2

±

p

(

λ

p

+ 1

−

h

−

q

)

2

−

4((1

−

q

)

λ

p

−

h

)

2

(2.24)

Where

λ

m

i

1

,

2

∈

Λ

M

are the two eigenvalues orresponding to

λ

pi

∈

Λ

P

Proof Followingthe work in(38)on howtoompute the determinant of

2

×

2

blokmatriesasfuntionofthebloks,weomputetheeigenvaluesof

M

solving

det

(

M

−

λ

m

I

) = 0

.

Sine

(1

−

q

)

hI

=

h

(1

−

q

)

I

det

(

M

−

λ

m

I

) =

det

((1

−

q

−

λI

)(

P

−

hI

−

λI

)

−

hqI

)

,

by some manipulations

det

(

λ

2

I

−

λ

(

P

−

hI

−

qI

+

I

) + (1

−

q

)

P

−

hI

)

= 0

,

putting

(1

−

q

−

λ

)

in evidene:

(1

−

q

−

λ

)

n

det

((

λ

2

I

−

λ

(1

−

h

−

q

)

I

−

hI

)

1

−

q

−

λ

+

P

))) = 0

for

(1

−

q

−

λ

)

6

= 0

,

det

((

λ

2

I

−

λ

(1

−

h

−

q

)

I

−

hI

)

1

−

q

−

λ

+

P

))) = 0

.

Now, let

λ

p

=

−

λ

2

−

λ(1

−

h

−

q)

−

h)

1

−

q

−

λ

. Sine

λ

p

is the solution of

det

(

λ

p

−

P

) = 0

, the eigenvalues of

M

as funtion of the eigenvalues of

P

are, after trivial

manipula-tions, the solutionsof

λ

2

−

λ

(1

−

h

−

q

+

λ

p

) + (1

−

q

)

λ

p

−

h

= 0

whose solutionsare (2.24).

2.3 Formation ontrol strategy in absene of a

ommon referene frame

In the previous parts we have assumed that all the agents have a ompass on

board,whihallowsthemtomaintainaommonorientationoftheloalreferene

frame. In this setion we remove this assumption, thus eah agent

i

belongs to

the agent

i

and

θ

i

is the orientation of the x-axes with respet to the x-axes of the global referene frame. Under this new assumption the state of eah agent

i

is desribed by the three state variables

{

x

i

, s

i

, θ

i

}

. We modify the formation ontrolstrategyproposedinsetion2.2whihisnotsuitableanymoretoorretly

ontrolthe system, byintroduinganalgorithmwhihleadsthe agenttoreah a

ommonreferenediretion. Thenew formationontrolstrategyisharaterized

by:

(1) arule to ahieve agreement ona ommonreferene diretion;

(2) arule toupdate the position of the agents;

(3) arule to ahieve agreement ona ommonreferene point.

Alltheresultsinthissetionarepresented withrespetoftheglobalreferene

frame

Σ

, and we assume that the agents are able toexhange loalinformation. An interesting method whih allows the agent to exhange loal estimates of

pointsanddiretionsinabsene ofaommonrefereneframeispresentedin(14), thusweanassumethatthe agentsexhangeinformationbyusingit. Underthis

assumption, we don't need to modify the onsensus algorithm on the network

entroid, whilethepositionupdateruleneedstotakeintoaountthe variability

of the target point due to the variability of the orientation of the orientation of

the loal referene frame.

This setion isorganized as follows: inthe rst part we haraterize rule (1),

thenwe haraterizerule (2), by modifyingtherule presented insetion2.2,and

we point out the dependene of these rules from (1). Finally we desribe the

globalformation ontrolstrategy.

2.3.1 Ahieving onsensus ona ommonreferene diretion

Inordertolead theagentstoreahaonsensus onaommonreferenediretion,

weuseAlgorithm1,originallyproposedin(39),whihallows thesystemtoreah aglobalsynhronizationonaommonheading. Algorithm1isbasedonaGossip

ommuniation sheme: ateah

t

a ouple of nodes

(

i, j

)

suh that

(

i, j

)

∈

E

(

t

)

is randomlyseleted, and the seleted nodes synhronize the orientationof their

(i) Attime

t

arh

(

i, j

)

∈

E

(

t

)

is randomly seleted.

(ii) Agents

i

and

j

updatetheorientationoftheirloalrefereneframeasfollows:

•

if

max

{

θ

i

(

t

)

, θ

j

(

t

)

} −

min

{

θ

i

(

t

)

, θ

j

(

t

)

} ≤

π

2

θ

i

(

t

+ 1) =

θ

i

(

t

+ 1) =

θ

i

(

t

) +

θ

j

(

t

)

2

•

if

max

{

θ

i

(

t

)

, θ

j

(

t

)

} −

min

{

θ

i

(

t

)

, θ

j

(

t

)

}

>

π

2

θ

i

(

t

+ 1) =

θ

i

(

t

+ 1) =

θ

i

(

t

) +

θ

j

(

t

)

2

+

π

2

•

Foreah

a

∈

V

suh that

a

6

=

i

and

a

6

=

j

:

θ

a

(

t

+ 1) =

θ

a

(

t

)

(39) a onvergene analysis of Algorithm 1is alsoprovided: applying Algorithm

1 the set of agents globallyasymptotiallysynhronize with probability

1

.

2.3.2 Position update rule

The positionupdaterule proposedinsetion2.2doesn't onsiderthe orientation

of the loal referene frame

θ

i

(

t

)

for eah agent

i

, whih may hange among the time aording to Algorithm1. For eah

i

∈

V

, the estimated target point

d

i

(

t

)

andof theestimatedommonrefereneenter

s

i

(

t

)

inglobaloordinates, attime

t

, depend on

θ

i

(

t

)

as follows:

s

i

(

t

) =

x

i

(

t

) +

R

i

(

θ

i

(

t

))

s

i

i

(

t

)

(2.25) and

d

i

(

t

) =

x

i

(

t

) +

R

i

(

θ

i

(

t

))(

s

i

i

(

t

) +

D

i

) =

s

i

(

t

) +

R

i

(

θ

i

(

t

))

D

i

(2.26) where

R

i

(

θ

i

(

t

)) =

"

cos(

θ

i

(

t

))

−

sin(

θ

i

(

t

))

sin(

θ

i

(

t

))

cos(

θ

i

(

t

))

#

.

(2.25)and (2.26), we obtain the followingposition update rule:

(

x

i

(

t

+ 1) = (1

−

q

)

x

i

(

t

) +

qs

i

(

t

) +

qR

i

(

θ

i

(

t

))

D

i

s

i

(

t

+ 1) = (

h

)

x

i

(

t

) + (1

−

h

)

s

i

(

t

) + (

−

h

)

R

i

(

θ

i

(

t

))

D

i

(2.27)

2.3.3 Formation ontrol strategy

Letus denenow the olumn vetor

D

(

θ

)

asfollows:

D

(

θ

) =







R

1

(

θ

1

(

t

))

D

1

. . .

R

n

(

θ

n

(

t

))

D

n







as the vetor of the target point, whih depend on the orientations of the loal

frames. The new formationontrolstrategy an beexpressed as follows:

"

x

(

t

+ 1)

s

(

t

+ 1)

#

= (

M

(

t

)

⊗

I

2

×

2

)

"

x

(

t

)

s

(

t

)

#

+

"

qD

(

θ

)

−

hD

(

θ

)

#

(2.28)

Aording tothe assumptions madein this setion, agiven formation is

on-sideredto be ahieved if

• ∀

i, j

∈

V

,

θ

i

(

t

) =

θ

j

(

t

)

•

x

(

t

) =

s

(

t

) +

D

;

• ∀

i, j

∈

V

,

k

s

i

(

t

)

−

s

j

(

t

)

k

= 0

The onvergene of the agents to the desired formation depends on the

onver-gene of Algorithm 1: a given formation annot be ahieved until all the loal

frames onverge toaommonorientation. Insetion 2.4weprovideaset of

sim-ulations whih are useful to understand the behaviour of the system under the

assumption madein this setion.

2.4 Simulation results

Inthispartwepresenttheresultsofsomesimulationswithtwopurpose: validate

the analytial results obtained in setions 2.2 and 2.2.4, and introdue some

onjetures about thebehavior ofthe system inthe senariodesribed insetion

−4

−3

−2

−1

0

1

2

3

4

−4

−3

−2

−1

0

1

2

3

4

(a)Initial positions (b)Evolution

−4

−3

−2

−1

0

1

2

3

4

−4

−3

−2

−1

0

1

2

3

4

() Finalformation

Figure 2.3: Exampleof formation

2.4.1 Agents with a ommon referene diretion

In Fig 2.3 an example of ahievement of a desired formation using formation

ontrol strategy (2.13) is presented. The system is omposed by a set of agents

with a ommon referene diretion, that are initially randomly sattered in a

2-D spae as in Fig 2.3a. They exhange loal information through a gossip

ommuniation sheme, and for all of them

q

= 0

.

1

and

h

= 0

.

05

. The agents reah the desired formation (a rux shape) by following the trajetories showed

in Fig 2.3b. The red lines represent the trajetories of the estimated ommon

ontrol strategy (2.13), in a system of agent with a ommonreferene diretion,

thedesiredformationisreahedforeahvalueof

h

in

−

q < h <

0

,i.e., forvalues of

h

that donot respet ondition (2.18). In other words, a smallompensation isenough for the system to onverge to the desired formation.

−0.05

−0.045

−0.04

−0.035

−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

Value of h

Value of |

λ

2

|

n=10

n=100

n

Figure 2.4: Value of

|

λ

2

|

for

q

= 0

.

15

,

−

0

.

15

< h <

0

and

n

∈

[10

,

100]

Fig 2.5 shows the the value of

|

λ

2

|

, i.e., the module of the seond largest eigenvalue of the matrix

M

, of a system of agents with

q

= 0

.

15

, omputed for

−

q < h <

0

and

n

∈

[10

,

100]

. For all the simulations the topology of the networkisonneted andrandomlygenerated. Itan beobserved that inase of

no ompensation, i.e., for

h

=

−

q

,

|

λ

2

|

= 1

, and the system is not stable, while for

−

q < h <

0

the seond largest eigenvalue of

M

has a module smaller than one, and the system onverge to the desiredformation.

2.4.2 Agents in absene of ommon referene diretion

Let us now onsider the ase of absene of ommon referene frame. In

Se-tion 2.3 we have haraterized the algorithmwhih lead the agents to reah the

targetformation. Wehavesupposedthattheagentsloallyinteratandexhange

informationusing the methodproposed in(14)whihisbased onthe determina-tionoftherelativepositions,i.e.,relativedistaneandangles,