Control and coordination of mobile multi-agent systems

TOVEGUSTAVI

Do toral Thesis

ISRNKTH/OPTSYST/DA09/05SE

ISBN 978-91-7415-396-5

RoyalInstituteofTe hnology

SE-10044Sto kholm, SWEDEN

A ademi thesis,whi hwiththeapprovalofRoyalInstituteofTe hnology(KTH),

will be presented for publi review and in partial fulllment of the requirements

foradegreeofDo torofS ien einOptimizationandSystemsTheory. Thepubli

reviewwillbeheldonSeptember4,2009at10.00 inRoomF3,Lindstedtsvägen26,

KTH,Sto kholm, Sweden.

©ToveGustavi,August2009

-ElbertHubbard(1856-1915)

Ameri anwriterandphilosopher

Abstra t

Inthisthesis,various ontrolproblemsoriginatingfromtheeldofmobileroboti sare

onsidered.Inparti ular,thethesisdealswithproblemsthatarerelatedtotheintera tion

and oordinationof multiple mobileunits. Thes ienti ontributionsare presented in

vepapersthattogether onstitutethemainpartofthethesis. Thepapersare pre eded

byalongerintrodu torypart,inwhi hsomeimportantresultsfrom ontrol theory,data

pro essingandroboti sarereviewed.

Intherstoftheappendedpapers,twostabilizing tra king ontrolsareproposedfor

anon-holonomi robotplatformofuni y letype. Toleran etoerrorsandotherproperties

ofthe ontrollersaredis ussed andarea tiveobsta leavoidan e ontrol,that aneasily

bein orporatedwiththeproposedtra king ontrols,issuggested. InPaperB,theresults

fromPaperAareextendedto multi-agent systems. Itisdemonstratedhowthetra king

ontrolsfromPaperA anbeusedas buildingblo kswhenputtingtogetherformations

of robots, in whi h ea h robot maintains a xed position relative itsneighbors during

translation. Inaddition,swit hingbetweenthedierent ontrol fun tionsisshowntobe

robust,implyingthatitispossibleto hangetheshapeofaformationon-line.

In the rst two papers, the tra king problem is fa ilitated by the assumptionthat

theapproximate velo ity of the target/leader is knownto the tra king robot. Paper C

treatsthethe asewherethetargetvelo ityisneitherdire tlymeasurablewiththeavail-

ablesensorsetup,norpossibletoobtainthrough ommuni ationwithneighboringagents.

Straight-forward omputationof the targetvelo ity from available sensor dataunfortu-

natelytend toenhan e measurement errorsand give unreliableestimates. Toover ome

thedi ulties,analternativeapproa hto velo ityestimationis proposed,motivatedby

thelo alobservabilityofthegiven ontrolsystem.

PaperDdealswithanotherproblemati aspe tofdataa quisition.Whenusingrange

sensors, one often obtains a mixeddata set withmeasurementsoriginating from many

dierentsour es. Thisproblemwould,forinstan e,been ounteredbyarobotmovingin

aformation,whereitwassurroundedbyotheragents. Thereexistestablishedte hniques

forsorting mixeddatasets o-line, butfor time-dependingsystemswhere dataneedto

besorted on-lineandonlysmalltime delays anbe tolerated, establishedmethodsfail.

Thesolutionpresentedinthepaperisapredi tion- orre tiontypealgorithm,referredto

asCCIA(Classi ationCorre tionandIdenti ationalgorithm).

Finally,inPaperE, we onsider theproblemofmaintaining onne tivityinamulti-

agentsystem. Often inter-agent ommuni ationabilitiesare asso iated withsomeprox-

imity onstraints,sowhentherobotsmoveinrelationtoea hother, ommuni ationlinks

bothbreak and form. In the paperwepresent a framework for analysis that makesit

possible to ompute a set of onstraints whi h, if satised, are su ient to guarantee

maintained ommuni ationforagivenmulti-agentsystem. Constraintsare omputedfor

twosortsof onsensus-basedsystemsandtheresultsareveriedinsimulations.

Keywords: Mobileroboti s, tra king, obsta le avoidan e, formation ontrol, nonlinear

observers,multi-agent oordination, onne tivitygraphs.

A epting a PhD position at the Division of Optimization and Systems Theory,

formallyapartoftheMathemati sDepartmentatKTH,wasasuddenandrather

unexpe ted movein my areer. Now,several yearslater andwith allfa ts onthe

table, I am glad that I took the leap. I have learned a loton many dierent

levelsandithasbeenmorefunthanI ouldeverhavehopedfor.

Now that I am at the end of this road, I am very mu h aware that I ould

not have rea hed this point on my own. First and foremost, my gratitude goes

to mysupervisor, professor XiaomingHu, whogaveme a han e to makeaturn

in my areer and whohas, with great patien e, guided me throughthe world of

a ademia;boththes ienti andthemorepoliti alpartsofit. :-)Iwouldalsolike

tothankprofessorHenrikI.Christensenwhore ommendedmeforthepositionand

professorAndersLindquistforapprovingmeasastudentatthedepartment. Krille

deservesaspe ialthanksformanyyearsofgood ooperationontheundergraduate

ourses. Allotherfa ultymembersatthedepartment,pastandpresent,havealso

ontributed,inonewayoranother,bothtomyprofessionaldevelopmentandtothe

inspiringenvironmentthatIhaveverymu henjoyed. ForthatIammostindebted.

Clyde,thanksforyouroverwhelminggenerositywheninvitingmetoTexasTe h

and during my stay there. Ifyou are everin Sto kholm and in need of arolling

pin,givemea all!

Tomy o-authorsonPaperE: I hadfun working withyou. I willkeepaneye

onyoufuture areers.

MajamysisterintheA ademi familytree. Thanksforgood ooperationover

theyears! Not onlyhaveweprodu ed several papersand homeworks ofdoubtful

qualitytogether,wehavealsohadsomequite onstru tivemoments. Manythanks

alsoforyour arefulreadingofthisthesis.

Niklas: thankyoueversomu h fortaking thetimeto readand ommentthis

thesis,andformakingmefeelwel omewhenIrst ametotheMathDepartment.

KarinKraft,PerEnqvistandseveralotherpeoplehavealsoassistedindierent

waysduringthewritingofthisthesis. Yourhelphasbeenmostappre iated.

Jakob:thanksagainforen ouragingmetoa eptthePhDpositionatOpt&Syst.

ThoughIwouldprobablyhavea eptedthepositionanyway,Iwouldhavedoneit

withalotlessenthusiasm. :-)

AsfortherestofmyfellowPhD olleagues(+Per),thepostdo sandmaster

thesisstudentsthathavepassedOpt&Syst,Mi ke,Rikardandalltheotherfriendly

mathemati iansintheotherendofthe orridor,MagnusatNADAandthe heerful

bun h at thedepartment formerlyknown as

S ³

^{: Ihaveverymu h enjoyedyour}

ompany! Inparti ularI would liketo mentionAli,Anders B,AndersM,Fredrik

(mylongtimeroommate),JohanK,Mats,Per,Stefan,andMi keF.Youhavebeen

myse ondfamilyforthepast oupleyears,forbetterorworse,quiteliterally. You

have hallengedme intelle tuallyaswellasphysi ally,youhavehelpedmewhenI

havegottenstu kontri kytheoremsorHW:s,youhavedrivenmetotheemergen y

room, you havesharedmytriumphs, ra ed me overthenish line of Vasaloppet

andyouhavemademelaughalmosteverysingleday. SeeyouinOpt&Systalumni!

Tothenewer ontributionsof the division: I wish you allthebest in yourfuture

a ademi al hallenges!

Thanksalsotoallmyotherfriends,newandold,ontheothersideofValhalla-

vägen. Youmeanagreatdealtome,I hopeyouknowthat.

Finally,mumanddad. Thankyouforyourendlesssupportoverthepastthirty-

oneyears.

Sto kholm,August

ToveGustavi

Introdu tion 1

1 Introdu tion 1

1.1 Roboti s-A histori alsurvey . . . 1

1.2 Presentandfuture appli ations . . . 3

1.3 Reader'sguide . . . 5

2 Ba kground on mobile roboti sand ontrol 13 2.1 Systemar hite ture . . . 14

2.2 Mathemati almodeling . . . 20

2.3 Stabilityanalysis . . . 28

2.4 Somepropertiesof anonlinear ontrolsystem . . . 32

2.5 Data pro essing. . . 39

2.6 Multi-agentsystems . . . 47

Referen es 57 A Robust tra king ontrol and obsta le avoidan e 61 A.1 Introdu tion. . . 61

A.2 Preliminaries . . . 63

A.3 Tra king ontrol . . . 66

A.4 Rea tiveobsta leavoidan e . . . 74

A.5 Simulations . . . 76

A.6 Summary . . . 79

A.7 Referen es . . . 79

B Adaptive formation ontrol formulti-agentsystems 81 B.1 Introdu tion. . . 81

B.2 Preliminaries . . . 82

B.3 Errorpropagation inmulti-agentformations. . . 85

B.4 Formationadaptationandswit hingstability . . . 89

B.5 Summary . . . 93

B.6 Extension1: Cooperativetargettra king . . . 94

B.7 Extension2: MaximumAreaCoverage . . . 98

B.8 Referen es . . . 102

C Estimationof neighborvelo ity in a leader-followernetwork 105 C.1 Introdu tion. . . 105

C.2 Preliminaries . . . 107

C.3 EstimationofNeighborVelo ity . . . 108

C.4 Simulations . . . 114

C.5 Con lusions . . . 123

C.6 Referen es . . . 123

D A Classi ationAlgorithmfor MixedTime-Varying Data Sets 125 D.1 Introdu tion. . . 125

D.2 Preliminaries . . . 127

D.3 Classi ation,Corre tionandIdenti ation . . . 129

D.4 Simulations . . . 133

D.5 Con lusion . . . 138

D.6 Referen es . . . 138

E Maintaining onne tivity ina leader-follower network 141 E.1 Introdu tion. . . 141

E.2 SystemandProblemStatement . . . 142

E.3 CompleteGraphCase . . . 146

E.4 In ompletegraph ase . . . 148

E.5 Simulations . . . 152

E.6 Con lusion . . . 155

E.7 Referen es . . . 156

Thewordrobot wasrstusedin1920,inaplaybyCze hwriterKarelƒapek[2℄,

andwasusedtodenoteme hani allaborersthatwereinpossessionofa ons ious-

ness. Realityhasnotyetrea hedthat far,buttherearetodayma hinesthathave

ahighdegreeofautonomyandthat anindependentlyperform hainsofvery om-

plextasksinwhi htheyintera twiththeenvironmentandmakede isionson-line,

basedonsensoryinput. Mu h oftheprogresshastakenpla eonlyin thepastfew

de ades,during whi h the ontinuousdevelopmentofhardwarehasmadeway for

in reasinglymoreadvan edsoftware.

Traditionallyrobotswerethoughtofashumanoids,buttomostpeopleworking

intheroboti sso ietytoday,whatdenesarobotisnotitsphysi alappearan ebut

rather itsability to fun tion independently. In 1998, professor Ronald C. Arkin,

o-workerandlaterdire tor oftheMobile Robot Laboratory atGeorgiaInstitute

ofTe hnology,stated[1℄:

Definition1.1. Anintelligent robotisama hineabletoextra tinformationfrom

itsenvironmentanduse knowledge about itsworld tomove safely in ameaningful

purposive manner.

Whenthe word robot is used in this thesis, itis understood without saying

thatitisusedina ordan ewithArkin'sdenition.

Today,thes ienti dis iplineknownasroboti s isanestablishedandhighly

a tiveresear hareawithmanysubdis iplines. Inthisthesis,fo usisonthe ontrol

theoreti aspe ts of mobile roboti s. In parti ular, we fo us on ontrol related

problemsthatarisefromtheintera tionbetweentwoormoreautonomousagents.

Tofa ilitateforthereader,aguidetothethesisisaddedintheendofthis hapter.

The rest of the hapter is intended to put the results in the thesis into a larger

ontext. A briefsurvey of the historyof roboti s is given in Subse tion 1.1 and

inSubse tion1.2,somemotivatingexamplesofpresentandfutureappli ationsfor

autonomousma hinesarelisted.

1.1 Roboti s - A histori al survey

The idea of onstru ting ma hines that an intera t with the environment and

evenhelporrepla ehumansin performingdangerousortedioustasks isold. One

exampleisthetwoservantsmadeofgoldthat,a ordingtoAn ientGreekmythol-

ogy, belongedto thegod Hephaestus. Earlyworkin automationand roboti swas

madeby,forinstan e,theArabengineerAl-Jazari(11361206)who,amongother

things, onstru ted a band of me hani al musi ians driven by hydropower [3,4℄.

Later, in the 15th entury, Leonardo da Vin i made drawings for the onstru -

tionof ame hani alknight[5℄. Inthe enturiestofollow,manysimilarideassaw

the light of day, but not until thelast entury has te hnology rea hed thepoint

where the realization of truly intera ting ma hines is possible. In the 1920:sand

30:s, theAmeri an ompanyWestinghouseEle tri Corporationprodu ed aseries

of human-resembling ma hines, some of them whi h ould perform simple tasks

su hasva uum leaning. Noneofthesema hineswereinastri tsenseintera ting

withtheenvironmentbutoneofthem,thehumanoidElektro (1939),hadamong

his other skills (in ludingblowingballoons andsmoking igarettes)the abilityto

distinguishbetweenredandgreenlight[6℄. Therstma hinesthat oulda tually

respondtostimuliare laimedtobeElmerandElsie[7,8℄,twoturtle-likema hines

onwheelsthatweredevelopedbyneurophysiologistWilliamGreyWalteratBurden

Neurologi al Institute, England, in 194849. Elmer and Elsie (names originating

fromELe troME hani alRobot,Light-Sensitive)wereequippedwithlightand on-

ta tsensors whi h they ouldusetoperformbasi navigation. Byatta hinglight

emitting sour es on ea h of the tworobots, they ouldeven be made to intera t

with ea h other, somethingthat was onsidered quite revolutionary at the time.

Soonthereaftertherst ommer ialindustrialrobotsenteredthemarket. Therst

modelswereonlyusedtoperformeasyandrepetitivetasksinstati environments,

su hasforinstan epi kandpla eoperations,painting,welding,et .,butoverthe

yearsindustrialrobotshavebe omein reasinglymoreadvan edandarenowused

in settingswhereahighdegreeofautonomyisrequired.

Essential for thedevelopment within theeld of roboti shas, of ourse, been

the rapidpro esswithin the eld ofele troni s. Therst robots hadverysimple

ontrol ir uitsbasedonele trontubes. In1947, resear hersatBellLaboratories

inventedthetransistor,whi hhadthebenetsofbeingmu hsmallerandrequiring

signi antlylesspowerthantheele trontube. This newdevi esoonrepla ed the

ele trontubein mostappli ations,but therealbreakthroughforroboti sdidnot

ome until thelaun h oftheprogrammable mi ropro essorin the earlyseventies.

Notonlywerethesepro essorssmallenoughtobein orporatedinafreelymovable

body,theyalsomadethe ostfor omputerpowerdropdramati ally.

After the arrival of the mi ropro essor, there was a boost in the market for

industrial andmilitaryrobots. Intheseventiesandearlyeightiesmany ompanies

enteredtheeldofindustrialroboti s,amongthemwere ompaniessu hasGeneral

Ele tri ,GeneralMotors,KUKAandASEA.Several ompaniesinJapanalsojoined

thenewtrendandsoonindustrialrobotsbe amea ommonsightinmanufa turing

industries. However,nothinglikethatwasseenin themarketfor domesti robots

orentertainmentrobotsintendedforprivateorsmalls aleuse. Eventhoughpubli

interest hasalwaysbeensubstantialandexperien edyet anothertopin theearly

eightieswith thereleaseof the StarWars lms and TV seriessu h asStarTrek,

progressin this areahasbeenveryslow. It islikelythat themain reasonforthis

me hani s have not de reased at the same rate as the pri e on omputer power

andare still omparativelyhigh. Hardwarepri eshavelongmade itmoreorless

impossiblefor ompanies that produ e robots for private use to be ommer ially

protable. Another dampingfa toris that ithas turned outto bemore di ult

thanmanyanti ipatedto mimi the amazingabilityseenin animals andhumans

toe ientlyweedoutrelevantinformationfrom apotentiallyverylarge dataset

and to ombine information to drawthe right on lusions. In the attempts to

solvetheseproblems,awiderangeofmoreorlessindependentresear hareashave

evolved, overingdis iplinessu h as omputervision, ltering,spee hre ognition

anddatafusion.

Inre entyears, theattempts to reate Arti ialIntelligen e havestartedto

pay o. Today's robots are more reliable and an handle mu h more omplex

situationsthantheirprede essors. Also,inthelast years,pri esonhardwarehave

startedto go down [9℄. Although many te hni al problems remain to be solved,

robots for ivil use havestarted to appear in a variety of dierentareas. In the

futurewewill mostlikelysee robotsin manynewappli ations. Someexamplesof

areasin whi h roboti shas large potentialand is predi tedto grow are given in

Se tion1.2.

1.2 Present and future appli ations

With today's te hnology, robots annot be expe ted to handle unforeseen situa-

tionsasgoodashumans. Still,theyhavemanyqualitiesthatmakethemextremely

useful, bothin ivil and non- ivil appli ations. For example, pi k-and-pla e ma-

hines anhandleverysmall omponentsandoperatewithanimpressivea ura y

withwhi hahuman annotpossibly ompete. Anotheradvantageisthatarobot

anbesetto performrepetitivetasksforalongtimewithoutlosingattentionand

gettingbored. As te hnology progresses,the omplexityof thetasks that anbe

performed by autonomous agents in rease. Some examples of emerging areas in

roboti saregivenbelow:

Domesti appli ations: It was now a long time sin e the dishwasher made

its entran e in the homesof ordinarypeople, but untilnow, ma hines have

only beenable to take overtasks where the a tion is ompletely determin-

isti . Withmoresophisti atedma hines, more omplextasks thatrequirea

higherdegreeof autonomy ould be performed withouthumaninterferen e.

In fa t,mu h of theevery-day household work is verywell suited forbeing

takenoverbyma hinesasitisrelativelymonotonous. Few ompletely-out-of-

the-ordinaryeventshappenandama hine an opewithmostsituationsby

varyingbetweenafewdierentbehaviors/modes. Alreadyexistinghousehold

robots in lude lawn mowers(su h asAutomowerfrom Husqvarna), va uum

leaners(su hasTrilobitefromEle trolux)andma hinesintendedfor lean-

ingofrainguttersandsewers. Otherdomesti tasksthat inthefuture ould

ingand wateringof pot-plantsand owerbeds, window leaning, andsnow

shoveling.

Assistan etodisabled/elderly:Robotsthatfa ilitatehouseworkand horesin

thehome anin reasetheindependen e forpeoplewithphysi aldisabilities.

In addition to the domesti robots just mentioned, more spe ialized robots

are now beingdeveloped to meet the needs of people with dierent sort of

handi aps. Oneofmanyexamplesisthesemi-automati eatingaidavailable

from Besti AB. Anotherimportant ontribution toin reased life qualityis

improvedgeographi almobility. Te hnologyfrommobileroboti s anbeused

to programsemi-automati wheel hairsand guidan e robots that anavoid

obsta les,followasidewalkandstopat redlightevenifthe ontrolinputis

sometimesde ient[10℄.

Surveillan e: Mobile surveillan e robots an be used as an alternative or as

a omplement to stati surveillan e systems and human guards in private

residents, shopping malls, o ial buildings, museums or military fa ilities

[11℄. Mobilerobotshavetheadvantageoverxedsensorsthatthey aneasily

adaptto hanges in theenvironmentorthemission. Forexample, lo ations

anbe added and removed from the list of pla es to patrol and the riskof

beingsetoutoffun tionasaresultofbeingblo kedbybadlypla editemsin

theenvironmentis signi antlyredu ed. Another advantageis that mobile

robots are less predi table as they an be set to vary their patrol rounds

randomly. Withtoday'ste hnologyitisstillnotadvisableto ompletelytake

humansoutoftheloop,butmobilerobots anprovideex ellentassistan eto

humanguardsby overingmostoftheroutinetasks. An exampleofarobot

designedforpatroltasksisshowninFigure1.

Workinhighriskareas: Bylettingrobotsinsteadofhumansperformmissions

inhazardousenvironments,manyinjuries anbeavoidedandevenlives an

bespared. Robots ouldbeused,forexample,to learanareaofminesorto

assist reghters andres ue rew in sear h and res ue missions in burning

houses,avalan hezones and buildingsthat are under riskof ollapsing[12℄.

Notonly anrobotsbeusedtominimizeriskofhumansgettinghurtorkilled

in dangerousenvironments; they analsobeset to operatein environments

wherehumans annotgoatalldue to onditionssu hasveryhightemper-

atures,poisonousgasesorradiation.

S ien e/Exploration: Not only anrobots help us in dailylife; they anhelp

us getabetterunderstanding of theworld aroundus. Robots havealready

been of great importan e in s ien e, mainly as olle tors of data in areas

that an not be rea hed by humans. Some examples are spa e (a re ent

example is the NASA's Mission-to-Mars proje t), the deep-seas, avesand

vol anoes. Forsu h missions itis oftenusefulto haveteams of autonomous

Figure1: Patrolrobot( ourtesyofMobileRobotsIn ).

thesametime. Su h teams an,forinstan e,beusedto dete t urrentsand

temperaturegradientsintheo eanorradiationfromdistantstarsorrea tions

inspa e.

1.3 Reader's guide

Thethesisathand onsistsoftwoparts. Inadditiontoashortintrodu tory hap-

ter,therstpart ontainsalongerandmore omprehensive hapterthatprovides

morespe i ba kgroundmaterialonmobileroboti sand ontrol.These ondand

mostimportantpartofthethesis onsistsofves ienti papersthattreatvarious

problems in vehi le ontrol, data pro essing and multi-agent oordination. Al-

thoughsomeofthepapersare loselyrelated,allpapers anbereadindependently

ofea hother.

The ba kground hapter is meant to fa ilitate the reading of the appended

papers, but any reader that sowishes an skip theba kground hapter and read

these ond partdire tlywithoutmissingtheessentialsofthethesis.

1.3.1 Summaryof papers

Thepapersin ludedinthisthesisarebasedonworkthathasbeeneitherpublished

tional onferen esorin established journals. Forthesake of presentation, partly

overlappingpapershavebeenmergedtogether. Also,somesupplementary results

and parts that,due to page limitations, ould nott in thepublished versionsof

the papershave beenadded. A summary of the papers follows below. For ea h

paper,abriefdes riptionofthe onsideredproblemsisgivenandthemain results

and limitationsare ommented.

Paper A : Robusttra king ontrolandobsta leavoidan efornon-holonomi mo-

bileagents, oauthoredwith X.Hu.

Inthispaperwe onsidertheproblemofdesigningfeedba k ontrolfun tions

for amobile robot platform of uni y le type. More pre isely, the aim is to

design ontrolalgorithmsthatensuresafetra kingofeitherpre-plannedtra-

je toriesor ofother mobile agents. It turns outthat thereferen evalue for

the tra kingrobot's angle-to-target has a dire t ee t on the properties of

the ontrolsystem. Tohandlethetwos enariosofinterestwetreatthetwo

orresponding ontrolproblemsseparately. The ontrolproblemsare ompli-

atedbythenon-holonomi motion onstraintsasso iatedwith theuni y le

platform. Furthermore,itisassumedthatthetra kingrobotonlyhasa ess

to lo al information about the environment and that sensor data are on-

taminated with inherent noise. Due to the limited amount of information

availabletothetra kingrobot,thereisanimminentriskforen ounterswith

unexpe tedobsta les. Toavoid ollisions,itis re ommendedthatthetra k-

ingalgorithmsare ombinedwitharea tiveobsta leavoidan e ontroller. To

omplete the paper, an obsta le avoidan e ontrol that an easily be inte-

gratedwiththepresentedtra kingalgorithms,issuggested.

Main ontributions:

ˆ Themain ontributionofPaperAisalo allystablefeedba k ontrolfor

paralleltra king,whi h showsbetterperforman ethan theestablished

serial ontroldoesforangles loseto

π 2

^.

Limitations:

ˆ Theassumptions on available sensor data and asso iated noisemodels

arehighlysimplied.

PaperAisbasedonthefollowingpubli ations:

A1: X.Hu,D.F. Alar ónand T.Gustavi,Sensor-BasedNavigation Coordi-

nationfor Mobile Robots, Pro . of 42ndIEEE Conferen eon De ision

andControl,Maui,Hawaii,De ember,2003.

A2: T.GustaviandX. Hu,Formation Control for Mobile Robotswith Lim-

ited Sensor Information, Pro . of IEEE International Conferen e on

A3: T.GustaviandX.Hu,RobustFormationAdaptationfor MobileRobots,

Pro . ofIEEE/RSJ InternationalConferen eonIntelligentRobots and

Systems,Beijing,China,O tober,2006.

Paper B: Adaptive formation ontrol for non-holonomi multi-agent systems,

oauthoredwithX.Hu.

Inthispaper,theresultsfromPaperAareextendedto multi-agentsystems.

Itisshownhowthetwotra king ontrolspresentedin PaperAtogether an

be used as base fun tions for multi-agent leader-followingformations. Ro-

bustness to noise and propagation of positioning errors in line formations

are studied in simulations. In addition, it is shown that swit hing between

dierent tra king angles and tra king ontrols anbe performed with high

safety. Theresultsonswit hingstabilityimpliesthatitispossibleto hange

the stru ture of atra king-basedformation on-line. Forinstan e, it ispos-

sible to develop s hemes for oordinated rea tive obsta le avoidan e. The

usefulnessoftheresultspresentedin thepaperisdemonstratedin twomore

omprehensiveexamplesthatareaddedto thepaperasextensions.

Main ontributions:

ˆ Errorpropagationinleader-followerformationsbasedonthetwotra k-

ing ontrolspresentedinPaperAismethodi allyexamined.

ˆ It is shown, both in theory and in simulations, that it is possible to

performsafeswit hingbetweenthetwotra kingalgorithms.

Limitations:

ˆ In this paper, as well as in Paper A, the assumptions on sensors and

availabledataaresimplied. Whenextendingtheresultstomulti-agent

formations,theee tsofina uratemodelingmaybeenhan ed. Exper-

imentalresultsareneededtosupportthesimulationsandthetheoreti al

results.

PaperBisbasedonthefollowingpubli ationsandpresentations:

B1: T.GustaviandX. Hu,Formation Control for MobileRobotswith Lim-

ited Sensor Information, Pro . of IEEE International Conferen e on

Roboti sandAutomation,Bar elona,Spain,April,2005.

B2: T.Gustavi,X.HuandM.Karasalo,FormationAdaptationwithLimited

SensorInformation,Pro . of

16 ^th

^{IFACWorldCongress,Prague,Cze h}

Republi ,July,2005.

B3: T.Gustavi,X.HuandM.Karasalo,Formationadaptationformaximum

area overage, presentedat Reglermöte2006,Sto kholm, Sweden,May,

B4: T.GustaviandX.Hu,RobustFormationAdaptationforMobile Robots,

Pro . ofIEEE/RSJ InternationalConferen eonIntelligentRobotsand

Systems,Beijing,China,O tober,2006.

Paper C: Estimation of neighbor velo ity in a leader-follower network, oau-

thoredwithX.Hu.

In Paper A and B, it was assumed in the implementation of the tra king

ontrollers that the velo ity of the leader agent was known through dire t

ommuni ationbetweentheagents. Thissort of ommuni ationmaynotbe

possiblebetweenless-advan edandinexpensiveagentswithoutadequate om-

muni ationequipmentor,forthatmatter,betweenmoreadvan ed agentsin

appli ationswhere radio ommuni ationforsomereasonshould beavoided.

Theoreti ally the desired velo ity ould be omputed from distan e mea-

surements,but asthose omputationswouldinvolveevaluation ofnumeri al

derivativestheoutputwouldbeextremelysensitiveformeasurementnoise. In

the paper, twodierentapproa hesto velo ity estimation are implemented

and evaluated. First, the standard Extended Kalman Filter is used to ob-

tainanestimateofthedesiredvelo ity. Then,alo alnonlinearobserverfor

theunknownvariable is onstru tedandtheresultsof thetwomethodsare

ompared.

Main ontributions:

ˆ Twononlinearstateobserversarepresentedinthepaper. Theobservers

are developed to re onstru t the target velo ity in atra king appli a-

tionand theyare shown to stabilize the tra king ontrolspresentedin

PaperA.

Limitations:

ˆ Althoughtheobserverapproa hitselfisgeneral,thepresentedobservers

areonlyvalid fortheexa tsetup onsidered inthepaper.

ˆ Inthe paper, the observerapproa h for the onsidered system is om-

paredtotheEKF-approa h. Itmaybearguedthatthemodiedsystem

onwhi h the lterwasset to operate ould havebeen modeled dier-

ently, orthat the parameters ould have been better tuned, and that

the omparisonisthereforeunfair. This isindeed possibleastheEKF,

unlike the linear Kalman lter, is based on heuristi s. However, the

EKF-lter was implemented with the intention of obtaining the best

possiblequalityoftheoutput.

PaperCisbasedonthefollowingpubli ations:

C1: T. Gustavi and X. Hu, Stable Target Tra king using Observer Based

Velo ityEstimation, Pro . of

17 ^th

^{IFACWorldCongress,Seoul, Korea,}

C2: T.GustaviandX.Hu,ObserverBasedLeader-FollowingFormationCon-

trolUsingOn-boardSensors,IEEETransa tionsonRoboti s,De ember,

2008.

PaperD: A lassi ationalgorithmformixedtime-varyingdatasets, oauthored

withM.Karasalo,X. HuandC.Martin.

Inthispaper,we onsidertheproblemofretrievinginformationfromadata

set where ea h data point is known to be the out ome of exa tly one of a

nite number of possible sto hasti variables with time-varying mean. The

problemismotivatedbyamobileroboti sappli ation,whereamobilerobot

isexploringitsenvironmentusingarangesensor. Iftherangesensorsimulta-

neouslydete tsseveral dierentobje ts,andthemeasurementsfurthermore

areobstru ted bynoise,itis notpossibletoknowforsurewhi hdistan e a

spe i measurementrepresents. Thedete tedobje ts ouldtypi allyin lude

bothstati stru turesintheenvironmentandothermobilerobotsmovingin

thesamearea. Inorderto retrievetheunderlying urves,thedatasetmust

somehowbesortedbeforestandardlteringmethods anbeapplied. Similar

problems have been studied both in statisti s and in system identi ation.

What hara terizesthe onsideredproblem isthat:

1. Theidenti ationoftheunderlying urvesmustbemadeon-lineasthe

measurements are obtained. As the robot in the intended appli ation

reliesontheresultfornavigation,onlysmalltimedelays anbetolerated.

2. Thedete ted urvesaretime-varying. Asaresult,severalknownmeth-

odsfordata lassi ation an notbeapplied intheirstandardform.

Todealwiththese onstraints,ahybridmethodisproposed.

Main ontributions:

ˆ Analgorithmforon-linesortingofmixeddatawithtime-varyingexpe -

tationvalues.

Limitations:

ˆ The method is heuristi and do notguarantee that the sorting of the

datapointsis orre t. Ifthe orre tion partof thealgorithmfailsthen

unreasonableresultsmaybeobtained.

Extensions:

ˆ Improvementofthe orre tionpartofthealgorithm anbemadeinorder

toin reasereliability. This in ludesdevelopmentofmoresophisti ated

methodsfordete tingwhenthesortingfails.

ˆ Forsimpli ity,thelteringofthesorteddatainthepaperisdonewith

asometimedelay. However,theintentionistodosimultaneous lassi-

the implementation should be quite straight forward asthe method is

designedforthispurpose.

ˆ Itwould bedesirabletoextendthemethodsothat thenumberofdata

sets ouldbeautomati allydete ted.

PaperD isbasedonthefollowingpaper:

D1: M.Karasalo,T.Gustavi,X.HuandC.Martin, Re ursiveIdenti ation

of aHybrid System,Pro . of EuropeanControl Conferen e,Budapest,

Hungary,August,2009.

Paper E: Su ient onditions for maintaining onne tivityin aleader-follower

network, oauthoredwithD. Dimarogonas,M.EgerstedtandX.Hu.

Awellknownprobleminmulti-agentappli ationsisthatthe ommuni ation

betweenthe agents an onlybemaintainedif theagentsremain su iently

lose. The ommuni ationstru ture in amulti-agentnetwork isoften mod-

eled as agraph, where the nodes representrobots and the edges represent

a tive ommuni ationlinks. In roboti s literature there exists an extensive

amountof resear h dealing with design of ontrol fun tions and algorithms

that oordinatethemotionofateamofrobots. Manyofthese ontributions

providesomeformof onvergen eresultthatrelyontheassumptionthatthe

robotsareableto ontinuouslyshareinformationthrougha tive ommuni a-

tion. Still,in many asesthequestionofwhether the ommuni ationgraph

remains onne tedovertimewiththeproposed ontrolisnot onsideredasit

wouldmaketheproblemtoo ompli ated. Inthispaperwetakea loserlook

atthisproblem. Inparti ular,westudythe ommuni ationgraphinamulti-

agentsystemin whi h oordinationis obtainedusing thepopular onsensus

equation.

Main ontributions:

ˆ Inthepaper,wederivesetof onstraintsthataresu ienttoguarantee

onne tivityinaleader-follower onsensus-basedmulti-agentsystem.

Limitations:

ˆ Someofthe onstraintsarevery onservative.

Extensions:

ˆ Anobviousextensionistoinvestigateifsimilarbounds anbeobtained

for networks where the agents use other typesof oordinating motion

ontrols.

E1: D. Dimarogonas,T.Gustavi,M.Egerstedtand X.Hu,On the Number

ofLeaders NeededtoEnsureNetworkConne tivity,Pro . of47thIEEE

Conferen eonDe isionandControl,Can ún,Mexi o,De ember,2008.

E2: T.Gustavi,D.Dimarogonas,M.EgerstedtandX.Hu,Topology-indu ed

onne tivity bounds in leader-follower networks, provisionallya epted

forpubli ationinAutomati a.

1.3.2 Work Division

Ex ept for Subse tion A.3.1 in Paper A, whi h is based on earlier work by Hu,

and Egerstedt[13,14℄, the rsttwo papersof this thesis arein all essentialsdue

to therstauthor. These ond author,who took theroll ofadvisor, helpedwith

theproofs and ontributed with valuableideas and omments. The third author

onPaperBwasinvolvedin the originalworkon whi h Extension2is based,but

wasnotinvolvedin anyotherpartsofthepaper.

Thevelo ityobserversin PaperCwere in theiroriginal versions suggestedby

therstauthor.These ondauthorhelpedformalizingtheideawiththeintegrating

velo ityobservertotastandard ontrolsetting. Parts ofthestabilityproofsare

alsoduetothese ondauthor.

PaperD waswritten in lose ooperation with, espe ially, the se ond author.

Themain ontributions of therst author lies in thesorting algorithm while the

main ontributions of the se ond author lies in theadaptiveltering. The third

author took the roll of main advisor, while the fourth author a ted as riti al

reviewer,providingmanyusefulsuggestionsand omments.

PaperEwaswrittenin ollaborationwiththese ondauthor,althoughtherst

author provided the basi ideas and had the main responsibility for the deriva-

tionof Theorems E.2and E.3. Thetwoother o-authors ontributed with ideas,

onstru tive ommentsandproof-reading.

1.3.3 Remark on notation

Notethatnotational ollisionsmayo urinthethesis. Intheappendedpapers,no-

tationisintrodu edseparatelyinea hpaper. InChapter2,Ba kgroundon mobile

roboti s and ontrol, standardnotations from literature have, whenever possible,

beenkept. In aseanotationisusedmorethanon e,theintendedmeaningshould

be learfromthe ontext.

Throughoutthethesis

|| • ||

^{isusedto denotetheEu lideannorm ofave tor.}

ontrol

Thefo us in thisthesisis on ontrol and systemstheoryin mobile roboti s. Mo-

bile roboti s is the subdis ipline of roboti s that deals with the displa ement of

autonomousvehi lesmovingin

R ⁿ

. Centralissueswithinthisdis iplineofroboti s are navigation and manipulation of rigid bodies with onstraintson motion and

availableinformation. Intheappli ationsdes ribedinSe tion1.2itispossiblefor

theattentivereaderto identify anumberof sub-tasksthatfall within the eld of

mobileroboti s. These tasksin lude:

ˆ Transportationfromapoint

A

^toapoint

B

^{(fet hingthemail).}

ˆ Tra kingofaxedpath/referen etraje tory(snowshoveling).

ˆ Tra kingofamovinggoal/target(tra kingofintrudersin aguardedarea).

ˆ Coordination ofmotion in amulti-agent formation(underwaterexploration

anddataa quisition).

ˆ Coverageof onnedorunboundedareas(explorationofunknownterrainfor

s ienti purposes,sear h-res uemissions, lawnmowing,et .).

To exe ute any one of these tasks a robot would have to solve a large number

of subproblems, ranging from high leveltask planning to ontrol of urrents and

potentialsin ele tri al ir uits. Inthisthesis, solutionsto someoftheseproblems

areproposed. Morespe i ally,thefo usofthisthesisisthemathemati alaspe ts

ofrobot ontrolanddatapro essing. Sin enoexperimentalresultsarepresented,it

shouldbepointedoutthatthetheoreti alresultsarenotne essarilyimplementable

inreallife in theexa tform theyarepresentedinthethesis. Rather,theyshould

be onsideredasguidelinesforfuture experimentalwork.

The onsideredproblems,thesettingsandtheresultsarepresentedinPapersA-

E. The intentionof the urrent hapter is to provide someba kground to those

problems. Hopefully, this hapter an both serve as an introdu tion to roboti s

for readers with some previous mathemati al experien e and asan introdu tion

to mathemati al ontrol theory for readers with a ba kground in experimental

roboti s. Theoutlineofthis hapter isasfollows.

First, in Se tion 2.1, an overview of the integrated systemthat onstitutes a

mobile robot is given. In Se tion 2.2, it is dis ussed how su h a system should

best be des ribed in mathemati al terms. In parti ular, the obje tiveis to write

the system onthe form of astandardnonlinear ontrol problem. InSe tions 2.3

and 2.4, we take a loser look at the mathemati al aspe ts of nonlinear ontrol.

Se tion2.5treatsmethodsforpro essingandmanipulationofsensordata. Finally,

Se tion2.6isdedi atedtomulti-agentsystemsandthespe i onsiderationsthat

mustbetakeninto a ountwhentwoormoreautonomousagentsintera t.

2.1 System ar hite ture

What hara terizesamobilerobotis,besidesitsabilitytotransportitself,itsability

tointera twithapartlyunpredi tableenvironmentandtoindependently arryout

taskswithout ontinuousguidan efromexternalsour es. Tobetterunderstandthe

hallengesthatareen ounteredinmobileroboti s, onsiderthefollowings enario:

Example2.1. AnarbitraryPhDstudentwantstoprogramamobilerobottoo a-

sionallymovefromherdesktothe oeema hinedownthehalltofet hmore oee.

A olleague of the PhD student ontributesto the proje t bylending aroboti arm

that anbemountedonthemobileplatform,therebysolving theproblemof pouring

the oee. All the PhD student has to do is make the robot move ba k and forth

tothe right positioninfront ofthe ma hine. Thepositionof the oeema hine is

knownandtherobotisequippedwithodometrysensorswhi hit anuseto ompute

its own position inside the o e. In addition, the robot isequipped with avision

systemandrangesensorsthatenableittomeasuredistan estosurroundingobje ts.

Howshouldthe PhDstudentprogramtherobotin ordertomakeitsu eedwith its

task?

Itis easyto seethat eveniftheproblem ofpouringthe oee ouldbesolved

and the requirementon smoothmotion (to preventthe oee from beingspilled)

is negle ted,theproblem ofprogrammingtherobotishighly non-trivial. It isnot

possibletoforeseeandindividuallytreatallpossibles enariosthattherobot ould

en ounteronitswaybetweenthedeskandthe oeema hine;someonemaypla e

a bag or a waste-paperbasket su h that the way be omes partly blo ked, there

may be people moving aroundthat haveto beavoided, someone may bump into

therobotand auseittoloseorientationet . To opewiththissortof hallenges,

itisne essaryfortherobottohaveatleastsomeintera tionwiththesurrounding

world. This, in turn,meansthat severalsubsystems withdierentresponsibilities

must be made to ooperate. Together these subsystems must be able to extra t

andinterpretinformationfromtheenvironment,buildupastrategyon-line,predi t

the out ome ofpossible a tions, omputethe me hani sand nally translate the

planned a tion into motor input. The intera tion and data ows between the

dierentsubsystemsaretogetherreferredtoasthesystemar hite ture. Thedesign

Motion Planning

−compute appropriate motion

Task Planning

−decide strategy

Sensors

Motor Control

−execute motion

Vehicle Physical system

Figure2.1: Exampleofsystemar hite tureinamobilerobot. Thearrowsrepresent

informationow.

the omplexity of the system. As the ar hite ture denes the interfa es between

dierent subsystems, it is possible with a good design to prevent many future

implementational problems that otherwise tend to show up when, for instan e,

addingnewhardwareor hangingthefun tionalitiesofthesystem. Togetabetter

overview of the over-all system it is helpful to draw a diagram overthe system

ar hite ture. Figure 2.1 shows as hemati draft of how the system ar hite ture

ouldlookin the oeema hineexample. Notethatthes hemeinFigure2.1only

representsoneof manypossibledivisions into subsystemsthat ouldbemade for

thesamesystem.

2.1.1 Planning vs. rea tive approa h to robot ontrol

Navigationandmotion ontrolforautonomousmobilerobotshasbeenanindepen-

dentresear hareasin ethelateeighties. Alreadyintheearlyyearsitwaspossible

inliteraturetodistinguishthetwodierentapproa hesthathavebeendominating

theeld eversin e,namelypre-planningandrea tive ontrol.

Formany years, pre-planning wasthedominating approa h in theroboti so-

iety. Chara teristi forwork basedon theplanningapproa h isthe existen e of

a high level ontrol system that plan ahead and makelong term de isions. For

mobilerobots,tasksthat aretypi allyperformedbythehigh level ontrolin lude

navigationandpathplanning. Thepre-planningapproa hmakesitpossibletotake

future eventsand risksinto onsiderationat anearly stage,so unless unforeseen

eventso ur,the over-all behaviorof thesystemis usuallyverygood. Long-term

Planning approa h Rea tive approa h

Pros

ˆ Long-time plans give

smooth over-all mo-

tion.

ˆ Can give optimal or

nearoptimalbehavior.

ˆ Canberunwithamin-

imumof omputational

resour es.

ˆ Fast response to sud-

den events.

Cons

ˆ Slow response to sud-

denevents.

ˆ Requires more ompu-

tationalresour es.

ˆ Con urrentlya tivebe-

haviors an an elea h

other.

ˆ Do not take long-term

benetsintoa ount.

Table2.1: Summaryofprosand onsforthetwodominatingapproa hestoroboti s

behaviordesign.

ingand omputation. Oneofthemaindrawba kswiththepre-planningapproa h

is onsequentlytheslowresponsetonewinformationandsuddenevents,whi h an

neverbe ompletelyavoidedin the ever- hangingreal world. Inthemid eighties,

s ientistwerestartingtoseetheadvantagesinlettingpartsofthesystembemore

lessdeliberateandmorerea tive[15℄.

In1986Brookspublishedapaperthatwastobe omeoneofthemostinuential

papersin modern roboti s [16℄. Inthe paper Brooks advo ateda new approa h

to roboti sbehaviordesign. Somewhatsimplied, Brook'smain ideawasthat the

omputational resour esshould notbeusedto model therealworld, astheworld

already is its own best model. Instead, the a tions should be dire t responses

to the robot's dete tion of the urrent state of the system. With no need for

long-time planning, response time at sudden events was dramati ally shortened.

Computational apa ity ouldbede reasedandtherobots ouldbemadesimpler

and heaper. Anotherfeature oftherea tiveapproa hwasthat severalbehaviors

ouldbeallowedtorun on urrentlyinsteadofonlysequentially.

A summary of the most important pros and ons of the two approa hes is

found in Table 2.1. At the time of publi ation, the 1986 paper by Brooks was

onsideredquite ontroversial.Todaybothmethodsarea eptedandareoftenused

to omplementea hother. Whilethelong-termplanninginasystemistaken are

of byahigh level ontrol system,suddenmaneuvers anbehandled byarea tive

sub- ontroller. Thissortofhybridar hite tureisusedinthe urrentworkaswell.

moreadvan edleaderagents,ontheotherhand,performbothnavigationandlong-

termplanningbeforetheya t,buttheyalsohavetheabilitytorespondrea tively,

forinstan e toavoid ollisionwithunexpe tedobsta les.

2.1.2 Ar hite ture in the urrent work

Whiletheremainingse tionsin this hapter providemoredetailed des riptionsof

theparts ofthe systemthat aremostrelevant toPapersA-E, wedevotetherest

ofthe urrentse tion to an overviewof the systemas awhole. Togive abetter

understanding of the ontext in whi h the resultsin Papers A-E should be seen,

thedierentparts ofthesystemaredes ribed.

InFigure2.2,asimplieddraftofthear hite tureusedfortheleaderagentsin

PapersA,B,CandEisshown. Apartfromthefa tthatthes hemein Figure2.2

representsa robot in a multi-agent system, the main features in the system an

bere ognizedfromthes hemeinFigure2.1. Thehigherlayersinthear hite ture

dealswithtaskssu hasplanningandde isionmaking,i.e., thingsthatliemainly

in thedomain of roboti ists. The middlelayersare signi antly moreinteresting

from a ontrol point of view,while the bottom layersare loselyintegratedwith

thehardware of the system. When des ribing the fun tionalitiesof the dierent

subsystemsbelow,weusethesystemin Figure2.2asstartingpoint.

Strategy: In the aseof multi-agentsystems,theover-all strategyisoften, but

notne essarily, de idedon a entral levelby amain omputer oramanual

operator. The main unit has superior omputational skills and may have

a ess to information that is not known by the individual agents at lo al

level. Inthebeginningofanewmission,themission isdivided,ona entral

level, into sub-tasks that are distributed among the individual agents. The

divisionoftasksmayalsobemadein ollaborationwiththeagentsoritmay

beleft fortheagentsto de ideamong themselves. Atthesametime asthe

tasksaredivided,moredetailedinstru tionsaboutthemissionaresentout,

eithertoalloftheagentsortoasele tfew,oftenreferredtoasleaders. Itmay

notbepossibleforthemainunit to ommuni atewiththeagentsduringthe

missionsoafterre eivingtheirinstru tions,themobileagentsareassumedto

independently arryonwiththeirassignedtasksuntilgivennewinstru tions.

Task Planning: Having been assigned a task, an agent has to de ide, based

on available information, sensor data, et ., whi h a tions to take in order

to a omplishthe given task. The three main responsibilities for the Task

planningpartofthissystemare:

1. Tonavigateusingsensordata.

2. Tode idewhi hone,orwhi hones,outofanumberofpossiblemotion

ontrol algorithms that should be a tivated (possible hoi es ould be

Strategy

(i+1) Agent

Agent (i−1) Agent i

Vehicle

Planning Task

Vehicle

Physical system Planning

Task

Vehicle

Planning Task

Planning Motion

Processing Data

Motor Control

Sensors

PSfragrepla ements

Task

Θ ref

u

U,I,R,...

Figure 2.2: Exampleof systemar hite tureinamulti-agentnetwork.

3. To provide the ontrol algorithms with referen e data su h as desired

velo ity,safety-distan etoobsta lesor oordinatesofthetraje torythat

shouldbefollowed.

Note that none of these assignments requiredetailed knowledge about the

physi alrobotplatformthatisbeingused. Infa t,thispartofthesystemis

modelindependent. Assurroundingsare hangingandthereisa onstantin-

owofnewinformation,de isionsandreferen edatamustbere-evaluatedand

re- omputedregularly. Some parts ofthese pro esses require omparatively

heavy omputations, introdu ing anon-negligibletime-delayin the de ision

making. Forthat reason itis suitablefor sometime- riti al de isionsto be

madeatlowerlevelsinthesystem(aswasdis ussedin Subse tion2.1.1).

MotionPlanning: Unlikehigher ontrollayers,thispartofthesystemdepends

bothonthepropertiesofthespe i robotplatformthat isusedandonthe

mathemati almodelthatis hosentodes ribeit. Typi ally,themodelofthe

systemisontheform

˙x = f (x) + g(x)u, y = h(x),

where

x

^{representsthestateofthesystemand}

y

^{representssensordata. The}

modelisdesignedtotake,forinstan e, onstraintsonmotionindu edbythe

platformintoa ount(seeSe tion2.2). Theobje tiveofthemotionplanning

partofthesystemisto omputeafeedba k ontrol,

u(y)

^{,thatgivethea tion}

requestedbytheabove ontrollayer. Foragivenplatform,thesetof ontrol

parameters,

u

^{, is xed. The quantities that anbe ontrolled on a ground}

vehi letypi allyin ludevelo ity,angularvelo ity,orientationofawheelpair

relativetothe hassiset .

Motor Control: Themotor ontrol isthelink betweenhardwareand software.

When aset of ontrol parametersis re eived, the task of this ontrol is to

translate thein oming parametersto urrents, voltages, et ., that give the

desiredmotion. This isfarfrom trivial. Therelationbetweentheee t put

into a motor and the resulting output motion is not ne essarily linear. A

feedba k ontrol loop is more or less ne essary in order to ompensate for

fa torssu hasdynami s,slip,fri tionet .

Vehi le: Byvehi leweheremeanonlythemobileplatformthat determinesthe

dynami softhesystemandnotthepro essorsorthesensorsmountedonit.

Thephysi alvehi le ouldtakemanydierentshapes. Somedierentvehi le

typesaredis ussedin Se tion2.2.

Sensors: Thereisawiderangeofsensorsindierentpri e ategoriesonthemar-

ket today,manyof themsuitableforroboti sappli ations. Some ommonly

usedsensorstypesare:

ˆ Rotaryen oders: Sensesrotationofthewheelsand anbeusedtoesti-

matetranslation(odometry);

ˆ GPS:Givesposition oordinates;

ˆ Ultrasound: Usedfordistan emeasurements;

ˆ IRsensors: Usedfordistan emeasurements;

ˆ Vision: Canbeused to distinguishobje ts,estimatedistan es and an-

gles,et .;

ˆ Bumpers: Conta tsensors.

Itisbeyondthes opeofthisthesistogointo detailsonperforman e,range

andother hara teristi sthatare spe i to ea h sensortype. Forthis type

ofinformation,thereaderisreferredtoliteratureonexperimental roboti s.

DataPro essing: Typi ally,outputfromthesensorsisusedasinputtothetask

andmotionplanningpartsofthesystem. Dependingontypeandqualityof

thesensors,theoutputmayhavetobepro essedbeforeit anbeused. Data

pro essing ishereusedasa ommontermformanydierentoperationsthat

lteringandnoiseredu tionte hniquesareveryoftenusedtoimproveresults

eveniftheyareperhapsnotalwaysstri tlyne essary. Anotherexampleisa

vision system, where the information ontainedin the ameraimages must

be extra ted and translated to a data lass that an be interpreted by the

planningand ontrol partsofthesystem.

With thear hite tureinFig 2.2inmind, itis easierto seehowtheproblems on-

sideredin PapersA-E arerelated. Thetra kingproblem onsideredin PaperAis

atypi almotionplanningproblem,whilePaperBdealswithbothmotionplanning

(formation ontrol) and task planning (formation adaptation). The problems in

PapersCandD anbothbe ategorizedasdatapro essing. Theproblem onsid-

eredin Paper Eis moredi ultto lassify asit on erns thea tualsetup of the

system. It aneither beseenasapartof the strategyin alarger systemorasa

problemfortheoperator,meaningthatitfallsoutsidethestru tureofthedened

ar hite ture.

2.2 Mathemati al modeling

Asmentioned,animportantpartofthe urrentthesisdealswithnonlinear ontrol.

For any ontrol problem, the rst step towards a su essful solution is to ome

up with agood mathemati al des ription of the physi al system at hand. More

pre iselyonewouldliketoexpressthedynami softhesystemasastandardtime-

invariantnonlinear ontrolproblem

˙x = f (x) + g(x)u

^(2.1)

y = h(x),

where

x ∈ R ⁿ

^{representsthe statevariables,}

u ∈ R ^m

^{representsthe ontrol input}

and

y ∈ R ^p

^{istheoutput from thesystem. The possibly nonlinearfun tions}

f, g

and

h

^{are dened su h that}

f : R ⁿ → R ⁿ

^,

g : R ⁿ → R ^n×m

^and

h : R ⁿ → R ^p

^.

In this ase of amobile robot appli ation, the state variables ould typi ally be

angles,distan es,position oordinatesandotherquantitiesthatdenetherelation

betweentherobotand theenvironment. Theinput

u

^{ontainsquantitiesthat an}

bedeliberatelysteered,forexamplewheelorientation,andtheoutput,

y

^,represents

theavailable measurements.

On ethemodelistransferredtotheaboveform,wellknownresultsfromsystems

analysis and ontrol an be used in the attempts to design a ontrol fun tion

u

whi hwill produ ethedesiredbehavior. Wewillreturntothesystem(2.1)many

timesinthefollowingse tions. Inthisse tionwe on entrateonhowtoobtainan

appropriatemathemati alrepresentationof thesystem.

Whenmodeling aphysi al system, oneof thebiggest hallengesis to nd the

rightlevelof abstra tion. A mathemati al model of areal systemwill alwaysbe

asimpli ationofthea tualsystem. Inthe aseofamobileplatformitwouldbe

virtually impossibleto takeallfor es thata t onamovingobje tinto onsidera-

omplexto analyze. Thus, ratherthannding amodelthat asa uratelyaspos-

sibledes ribesthereal systemoneshould fo uson ndingamodelthat des ribes

the system su iently well with respe t to the intended appli ation. With this

onsiderationin mindwedis uss, in Subse tion2.2.1, when to in lude dynami al

onstraintsin the model and whennotto. In Subse tion2.2.2, someofthe most

ommonly used mathemati al models for wheeled ground vehi les are presented,

andsomeimportantpropertiesofthemodelsaredis ussed.

2.2.1 Kineti vs. kinemati models

In lassi alme hani s,adistin tionismadebetweenkineti (dynami )modelsand

kinemati models. Assomereadersmayre all,kineti models onsider onstraints

originatingfromNewtons famousse ondlaw,

F = ma

^{,whilekinemati modelsdo}

not. Kineti modelsgiveamorerealisti des riptionoftherealworld,buttheypay

thepri eofhighermathemati al omplexitywhi hfollowsfromtakingse ondorder

derivatives(a eleration) into a ount. Kinemati models are generallyeasier to

workwith,butshouldonlybeusedtodes ribesystemswithslowtimeprogression

where dynami onstraintshave littleimpa ton the systembehavior. Inthe ap-

pli ations onsideredinthisthesis,thevehi les areassumedto bedrivingwithout

slipat omparativelylowspeeds. Underthese onditionsitissu ientto onsider

onlykinemati onstraints, but it is neverthelessimportant for thereader to un-

derstandthedieren es betweenkinemati modelsandkineti (dynami )models.

Ifthesettingsare hanged,forinstan eifvelo ityisin reased,slipissigni antor

ifthemassofthevehi leislargeinrelationtoma hinepower,thenthekineti sof

thesystemhasto betakenintoa ountas well.

Remark2.1. Notethatitispossibletorewrite onstraintsonse ondorderderiva-

tives

¨

x = ˆ f (x, ˙x) + ˆ g(x, ˙x)u

totthe model (2.1) usinga simple mathemati al tri k. Dene

w 1 = x, w 2 = ˙x

^.

Then

˙

w 1 = w 2

˙

w 2 = f (w ˆ 1 , w 2 ) + ˆ g(w 1 , w 2 )u.

2.2.2 Vehi le models

Intheproblems onsideredinthisthesis,themobileplatformitselfisanimportant

partof the physi al system. Thus, animportant partof the modeling pro ess is

tomathemati allydes ribethekinemati sasso iatedwiththeplatform. Although

someoftheresultsthatarereviewedinthissubse tionarevalidinarbitrarydimen-

sions,thefo ushereisongroundvehi les,i.e., vehi lesmovingona2Dmanifold.

Vehi les that ould t this des riptionin lude dierentsorts of wheeled vehi les,

te hni allyadvan edplatforms,su haswalkingrobotsandsnake-likerobots, ould

theoreti ally t into thesames heme, but theyare not onsidered in the urrent

ontextasthemathemati al omplexityoftheirmotionistoohighfortheintended

appli ations. Theobje tiveoftheplatforms onsideredhereis mainlyto serveas

arriersforequipment,sensorsandmaterialonleveledground.

Single integrator and doubleintegrator models

Thesimplestmodelthat is onsideredhereistheso alled singleintegrator model.

Unlike the other models presented in this se tion, the single integrator model is

not restri ted to ground vehi les. The model assumes that the vehi le an be

des ribed asa point mass in

R ⁿ

whose state is ompletely dened by a set of

n

spa e oordinates whi h are here referred to as

x = (x 1 , x 2 , ..., x n ) ∈ R ⁿ

^{. It is}

assumed that thevehi le anmovefreely in anydire tion and that itsvelo ityis

dire tly ontrolledviaa ontrolinput

u ∈ R ⁿ

^:

˙x = u.

^(2.2)

Thisverysimplekinemati modelofavehi leissometimesextendedtotheso alled

double integrator model. Thedouble integrator diers from the single integrator

onlyin thatthe ontrolinput

u ∈ R ⁿ

^{ae ts the}a elerationofthevehi lerather thanthevelo ity:

¨

x = u.

^(2.3)

Sin e boththesingleanddoubleintegratormodel assumethatavehi leisfreeto

moveinanydire tionwithoutrestri tionstheydes ribethefeaturesofrealvehi les

ratherpoorly. Still,themodelsarewidelyusedin ontrolsin ethesimpli ityofthe

modelsmakesitpossibletoprodu etheoreti alresultsin aseswheremorerealisti

modelsgiveresultsthataretoo omplextobethoroughlyanalyzed. Also,thesingle

anddoubleintegratormodelsareveryusefulwhen onsideringhigherlevel ontrol

problems (su h as, for instan e, navigation problems) where it is reasonable to

assumethatlowerlevelmotion ontrolistaken areofbyaseparatesub- ontroller.

An exampleofthis istreatedinPaperE.

Thesingle/doubleintegratormodelmaybeuseful,but inmanyproblemsmore

realisti modelsareneeded. Oneobviousawinthesingle/doubleintegratormodel

is that it does not allow for a vehi le to have an orientation. In real life, most

vehi leshaveawell-denedorientationandsubsequent onstraintsontheirmotion.

Wheeled ground vehi les are onstrained to roll in the dire tion dened by the

orientation of their wheels and airplanes need to maintain a omparatively high

forwardvelo ityin order to stayin theairbut annot moveeither ba kwardsor

dire tly sideways. Inthe mathemati alformulation, the des ribedrestri tions on

motiongiverisetoso allednon-holonomi onstraints,whi harefurtherdis ussed

in Subse tion2.2.3. Unfortunately,non-holonomi onstraintstendtoin reasethe

PSfragrepla ements

(x, y)

φ v

x y

Figure2.3: Uni y le

Uni y le model

Amodelthatisoftenusedtodes ribenon-holonomi vehi lesistheuni y lemodel.

A s hemati drawingof a uni y leis shown in Figure 2.3. The uni y le hastwo

independentlydrivenwheelswithxedorientation(paralleltoea hother)thatare

usedtodrivetheplatformandoneadditional astorwheelforbalan e. Be auseof

theindependentwheels,theuni y le anrotate onthespot, butthetranslational

motionis onned to thedire tion of its main wheels. Thestate of auni y le is

ompletelydened byaset of spa e oordinates,

(x, y)

^{, whi h} orrespondsto the mid-point of the wheel axis, and an orientation,

φ

^{, whi h is given relative some}

xed oordinatesystem. Insomerespe ttheuni y le anbeseenasthesimplest

possibleextensionofthesingleintegrator.Mathemati allytheuni y leisdes ribed

bythefollowingequations:

˙x = v cos φ

˙y = v sin φ

^(2.4)

φ ˙ = ω,

where

v

^and

ω

^denotethetranslationalandrotationalvelo itieswhi hareassumed to be the ontrol input of thesystem, i.e.,

u = (v, ω)

^{. The groupof robot plat-}

forms that an be said to have uni y le dynami s in lude many ommonly used

experimental type platforms, su h as Khepera from K-team and PowerBotfrom

MobileRobots In . The three-wheel onguration is impra ti al in rough terrain

androbots of uni y letype are thereforemostly used in ontrolled environments

(labs, fa tory oors, gardens, homes et .), but the mathemati al uni y le model

has a wider range of appli ations sin e it an also be used to approximate the

PSfragrepla ements

(x 1 , x 2 ) φ

θ

l

Figure2.4: Single-tra kmodel(simpliedkinemati modelofa ar-likevehi le).

Single-tra k model

A betterapproximationofa ar-likevehi leisgivenbythesingle-tra k model. In

a standard ar, the front wheel-pair is used to steer while the rear wheel-pairis

xed and alignedwith the ar. In thesingle-tra kmodel, ea h of thetwowheel-

pairsismodeledasonesinglewheello atedatthemidpointofthewheelaxle(see

Figure 2.4). Thestatevariables onsist of apair of spa e oordinates,

(x 1 , x 2 ) ∈ R ²

, representingtheposition of therearwheel(midpoint ofthe rearwheelaxle), one variable,

φ

^{, that representsthe} orientation of the hassisrelative somexed oordinateframeandonevariable,

θ

^{,thatgivestheangleofthefrontwheelswith}

respe tto the hassis,i.e. thesteeringangle. The ontrolinput isassumedto be

the translational velo ity of the rear wheel (midpoint of the rear wheel axle),

v

^,

and therotational velo ityof thefront(steering) wheel,

ω

^{. Ifwelet}

l

^{denote the}

distan e betweenthetwowheel-pairsweget:

˙x 1 = v cos φ

˙x 2 = v sin φ φ ˙ = v

l tan θ

^(2.5)

˙θ = ω.

Remark2.2. Atrstglan e,thesingle-tra kmodelmayappearusefulinmodeling

the motion of abike, butin fa t itisnot very useful for that purpose. Re allthat

the single-tra k model is only a kinemati model. When riding a bike at normal

speed, dynami s plays an important role. For instan e, turningwith a bike isnot

only a matter of steering with the front wheel but also about the rider tilting the

bike usinghis orherbody weight.

Notethatallthevehi lemodelsdes ribedinthisse tionarealreadyexpressed

on thegeneralform(2.1), butin mostappli ations thevehi leis onlyone partof

theequations des ribing the rest of the system. That way, properties asso iated

withthevehi lemodel, for examplenon-holonomi onstrains,will dire tly ae t

the ontrol propertiesofthefullsystemaswell(seeSe tion2.4).

Of the des ribed models, the uni y leis the onemost frequently used in this

thesis. The uni y le model is used in Papers A -C, where fo us is on nonlinear

ontrol. Theproblems onsidered inthose papersaremotivatedbyreal lifeappli-

ationsandin orderto obtainsu ient a ura y,non-holonomi onstraintshave

tobe onsidered. InPaperEontheother hand,fo usis on on eptualideasand

higher level formation ontrol. In that paper the more appealingat least from

a omputational point of viewsingle integrator model is used to represent the

mobileagents.

2.2.3 Non-holonomi onstraints

Thekinemati models above an be seenas onstraintson themotion. Ea h one

ofthemodelsgivesrisetoaset of onstraintswhi h anbewritten as

a i (q) ˙q = 0, i = 1, ..., m,

where

q ∈ R ⁿ

^{isave tor ontainingthestatevariablesand}

a i : R ⁿ → R

^,

i = 1, ..., m

^.

Constraintsonthisformare alledPfaan onstraints.

Example2.2. Considerthe uni y lemodel fromSubse tion2.2.2,

˙x = v cos φ

˙y = v sin φ φ ˙ = ω.

Therelationbetweenthe statevariables an beexpressedasaPfaan onstraint,

 sin φ − cos φ 0 





˙x

˙y φ ˙



 = 0.

APfaan onstraint anbe lassiedasholonomi ornon-holonomi in a or-

dan ewiththefollowingdenitions.

Definition2.1. A onstraint

a(q) ˙q = 0, q ∈ R ⁿ , a : R ⁿ → R

^{that an}equivalently be expressed as

h(q) = 0

^{for some fun tion}

h : R ⁿ → R

^{is alled a holonomi}

onstraint.

Definition2.2. A onstraint

a(q) ˙q = 0, q ∈ R ⁿ , a : R ⁿ → R

^{thatisnotholonomi}

is alleda non-holonomi onstraint.

Aset of holonomi onstraintsrestri tsthestatevariablesto asmoothhyper-

surfa ein state spa e. On the a essiblehyper-surfa e thevariables an be on-

A set of non-holonomi onstraintsalso restri ts the motionof a system to a

smoothhyper-surfa e, but in this ase, therestri tion is only valid lo ally. If all

onstraintsare non-holonomi ,the setof globally a essiblestatesforthe system

willnotberedu ed. Thenon-holonomi onstraints anbethoughtofas onstraints

that limitthethepossiblewaysto rea haspe i state.

Example 2.3. A real-life ontrol problem with non-holonomi onstraints is the

problem of parking a ar. Even though a ar, whi h represents a non-holonomi

system, an not move dire tly sideways, it an be maneuvered into an empty spot

between two other ars parked along the side of a street through repeated ba king

and forwardmotion (dis ontinuous ontrol input).

Fromamathemati alpointofview,non-holonomi ontrolsystemsaregenerally

moredi ult todealwith thanholonomi systems. The ee tsof non-holonomi

onstraintsonpropertiessu has ontrollabilityandstabilizabilityofa ontrolsys-

temwillbefurther treatedinSe tion 2.4. Intheremainingpartofthisse tionwe

fo usonhowtoestablishwhetherasetof onstraintsisholonomi ,partlyholonomi

(a mixtureofholonomi andnon-holonomi onstraints),ornon-holonomi .

Letusrst onsiderasinglePfaan onstraint

a(q) ˙q = 0,

^(2.6)

where

a : R ⁿ → R

^{. Notethatforanarbitraryfun tion}

h : R ⁿ → R

^,

h(q) = 0 ∀q = ⇒

X n j=1

∂h(q)

∂q j

˙q = 0 ∀q.

^(2.7)

Thus,itiseasytoseethatthePfaan onstraintisholonomi if

a(q)

^isthederiva-

tiveof somefun tion

h(q)

^{. Infa t,it anbeenshownthat aPfaan onstraintis}

holonomi ifandonlyifthereexistsafun tion

α(q)

^{,oftenreferredtoas}integrating fa tor,su hthat

α(q)a(q) = X n j=1

∂h(q)

∂q j

.

^(2.8)

IfasystemhasseveralPfaan onstraints,theanalysisbe omesmore ompli ated.

Eveniftheindividual onstraintsarenotintegrable,thesystemmaybeholonomi ,

or at least partly holonomi , if it is possible to nd linear ombinations of the

onstraints,

X m i=1

α i (q)a i (q) ˙q = 0

^(2.9)

thatareintegrable. Abetterwaytoestablishwhetherornotasetof onstraintsis

holonomi ,i.e., ifthe onstraintsareintegrable,istouseFrobenius theorem[17℄.

Theorem 2.1. (Frobenius)A regulardistribution isintegrable if andonly if it is