DesignandModelingofaRedundantOmni-directionalRoboCupGoalieLanceWilson,CraigWilliams,JustinYance,JaeLew,RobertL.WilliamsIIDepartmentofMechanicalEngineeringOhioUniversity,Athens,Ohio,USAPaoloGallinaDepartmentofInnovationinMechanicsandManagementUniversityofPadova,ItalyAbstract.Thispapercoversthemechanicaldesignandmodelingofaredundantomni-directionalmobilerobotdevelopedfortheOhioUniversityRoboCupgoalie.Thegoalierobotisactuatedredundantlywithfourwheelsforgoodmobility.Inthepaper,firsttheoverallmechanicaldesignofthesystemisdiscussed.Second,theinversekinematicssolutionisobtainedusingtheprincipleofvirtualwork.Third,dynamicequationsofmotionofthesystemarederivedinasymbolicformresultingincouplednonlineardifferentialequations.Finally,asimulationdemonstratestheinversekinematicssolutionandsimpleindependentPDwheelcontrolofourredundantomni-directionalgoalierobot.1.INTRODUCTIONOmni-directionaldriveistheabilitytomoveinanydirection(0to360degrees)atanygivenorientation.Usingthistypeofdrivesystemenablesarobottomoveinagivendirectionwhilebeingabletorotateduringlineartravel.TimeissavedbyeliminatingtheneedtorotatetherobotbeforetranslatingfrompointAtopointB.Omni-directionaldrivegivesthegoalierobottheabilitytoalwaysfacetheballwhilebeingabletomoveinanydirection.Inaddition,itprovidessimplerinversekinematicssolutionforpathplanning.Forthesereasons,manyresearchgroupsarestudyingomni-directional(holonomic)mobilerobotsandvehicles1-5.TheOhioUniversity(OU)RoboCupTeamincorporatestheomni-directionaldriveintotheirmobilerobots.TheOURoboCupTeamconsistsof4playersand1goalie；theplayersandgoalieeachhaveseparatemechanical,electrical,andsoftwaredesigninordertomeetdifferentperformancespecifications.Forexample,theplayersaredesignedtooptimizeforwardmotion,whilethegoalieisdesignedtooptimizeside-to-sidemotion.Thispaperwillfocusonthemechanicaldesign,dynamicmodeling,andsimulationoftheredundant,omni-directionalRoboCupgoalie.2.DESIGNThedesignforthegoaliehastwomainfunctions.Themajorfunctionistohavetheabilitytoquicklymoveside-to-sidetoguardthegoal.Thishasbeenaccomplishedbyhavingthewheelsalignedhorizontallyforquickmovement.Theotherfunctionisthekickingoftheball.Thecurrentdesignnotonlyallowsforastrongkickbutalsotheabilitytocatchingballundercontrol.2.1ConstraintsInordertocompeteintheRobocuptournament,robotshavetomeetspecifiedsizeconstraints.Thegoalierobotcanbenotallerthan22.5cmiflocalvisionistobeused,andmustfitwithina18cmdiametercylinder.Anypartthatmayextendoutwardontherobotmustbefullyextendedwhenplacedinsidethe18cmcylinder.Thesesizeconstraintsgreatlylimitthearrangementofdriveandkickingsystems.Kickinganddrivesystemsarerectangularinshapeanddesigningalayouttomeettherobotsizeconstraintsresultsin“wastedspace”aroundthesesystems.2.2.PerformanceSpecificationThroughtestingvariousradiocontrolledtoycars,experimentswereperformedtobettervisualizewhatkindofperformancewedesireforthegoalierobot.Highvaluewasplacedonmakingtherobot“zippy”whichmeansitcanmoveorchangedirectionquicklybutdoesnthaveahighmaximumvelocity.Anaccelerationof2m/sec2andamaximumvelocityof1m/secweretheperformancespecificationsdecideduponforthegoalieslocomotion.Thekickingmechanismusedonthegoalieneededtobepowerfulenoughtopropeltheballatleasthalfthelengthoftheplayingfield.Controllingthekickingpowerisaccomplishedbyusingavariablespeedmotoronthekickingmechanism.Oneadvantageofthegoalieistheabilitytocapturetheballandcompletelyencloseit.Enclosingtheballandhidingitfromopposingrobotsoperatingontheglobalvisionsystemcanleadtoconfusioniftheycannotlocatetheball.2.3LayoutThegoalielayoutconsistsofthedrivesystemandkickingmechanism(Figure1).Tomaintaintractiononthedrivewheels,therobotcenterofmassshouldbelocatedonthecentralaxesofthedrivewheels.Fourwheelsarebeingusedtodrivetherobotomni-directionally.Keepingthegoaliescenterofmassneartheseaxesbecomesdifficultoncethekickermechanismisaddedtothesystem.Thekickermechanismshiftsthecenterofmasstowardthecenteroftherobot,whichcancausetherearwheeltoslip.Shiftingthecenterofmasstocompensatefortheweightofthekickercanbeaccomplishedthroughshiftingbatterypositionstowardtherearoftherobot.Figure1:OUGoalie(a)CADmodel(b)HardwarePhotographThereare6motorsutilizedintherobotassembly.Fourareplacedinthewheelassemblytodrivewheels.Oneisinkickerassemblyandrotatesthedrivebar.Thereisalsoaservomotorassociatedwiththekickerassembly.2.4Omni-DirectionalDriveTheKornylakCorporationmanufacturestheOmniwheelthatappearsinFigure2(a).Tominimizewheelwidth,theoriginalOmniwheelismodifiedbyremovingmaterialfromthecenterandouterfaces.Removalofmaterialalsoallowedroomtoinsertafabricatedhubtobeusedforgearmounting.Figure2(b)showsthemodifiedwheelmountedtoanaluminumhub.Figure2:(a)KornylakOmniwheel(b)ModifiedOmniwheel2.5Kicking/Dribbling/CapturingMechanismThekickingmechanism,Figure3,isstructuredaroundonebasicprinciple,capturingtheball.Ithastheabilitytorotatearoundapivotpointwiththisrockingmotioncontrolledbyaservomotor.Theservomotorislocatedonthegoaliebaseandthusisstationarywithrespecttothekicker.Duringthegamethegoaliewillbelocatednearthegoalwithitsdrivebarrotatingbackwards.Thedrivebarwillbeencasedinrubbertubing.Initiallythekickingmechanismwillbeatitshighestpointinthearc.Whentheballgetsnearthekickerthedrivebarwillcapturetheballviabackspin.Atthistimetheballwillberotatingbetweenthedrivebarandtheidlerbar,locatedunderneaththerobot.Beingunderneaththerobotwillcausetheballtobehiddenfromtheglobalcamera.Thedrivebarwillthenreversedirectionandcausetheballtospinintheoppositedirectionbetweenthetwobars.Aftertheballgainsasufficientamountofforwardrotationthekickingmechanismwillberockedbackedtheservomotor.Thiswillallowtheballtobereleasedwhilehavingtopspin.Afterreleasingtheballthekickingmechanismwillbebroughtuptoitshighestpositionintherotatingarcbytheservomotortowaitforthenextball.Figure3:Close-upofKickerAssembly3.ModelingThedynamicequationsforthefour-wheeledOhioUniversityRoboCupgoalieisderivedfromthemodelshowninFigure4,similarto1.Itisassumedthemobilerobotismovingonahorizontalplayingfield.Asitisshown,theinertialcoordinateframeXwYwisfixedontheplaneandthemovingcoordinateframeXmYmisattachedtothemasscenterofthemobilerobot.Liisthedistancefromthemass-centertothecenterofthecorrespondingwheel.fistheorientationofthemobilerobotwithrespecttowS.Aspreviouslystatedthisrobotisomni-directional,givingusnononholonomicconstraintequations.Also,itisimportanttopointoutthattheOURoboCupgoaliehasthreedegreesoffreedom,buthasfouractuatorsmeaningthatthesystemisredundantlyactuated.First,wedefinethepositionandforcevectorsofthemass-centerfortheOURoboCupGoalieintheabsolutecoordinateframeas:TGyxwTwwwMFFFyxS=f(1)Figure4:Kinematic/Free-BodyDiagramwhereMGisthetorqueappliedtotherobot.ThefollowingtwoequationsrelateCartesianposeandwrenchforthemovingandinertialframes:fRFsRSmwmwmwmw=(2),(3)wherethecoordinaterotationmatrixRwmrepresentstheinertialframewithrespecttotheorientationofthemovingframe.Theposeandwrenchvectorsofthegoalieintermsofthemovingcoordinatesare:TGyxmTmmmMfffyxs=fApplyingNewtons2ndLaw:=SMFww(4)whereMisapositive-definitediagonalmatrixwiththemobilerobotsmassMonthediagonal.Equation(4)canberewrittenintermsofthemovingcoordinatesusingEquations(1)-(3):fssRRMmmmwmTwm=+(5)SimplifyingEquation(5),thedynamicequationsoftherobotare:GmymmxmmMIfxyMfyxM=+=fff(6),(7),(8)whereImisthemassmomentofinertiaforthemobilerobot.TocomputetherelationshipbetweentherobotCartesiancoordinatesandthewheelangles,theprincipleofvirtualworkisused.Summingtheforcesandmomentsgives:(9)whereQistheJacobianmatrixbaseduponthesystemgeometry,andTiisthetractionforcefromeachwheel.Si