forthmotions.Morse-basedexactcellulardecompositionsareareliableframewo - PDF document

forthmotions.Morse-basedexactcellulardecompositionsareareliableframeworkforgeneratingcoveragepaths[4].Thismethodguaranteesfullcoverageofthetargetspaceandalsoallowsforusingdifferentcoveragepatterns.Moreover,itcanbeappliedtoanyn-dimensionalspace.Efciencyisveryimportantinmanycoverageapplica-tions.Huangpresentedanoptimalline-sweepbasedmethodforcellulardecompositionalgorithms[5].Thisapproachpro-ducesanoptimallengthcoveragepathbyallowingdifferentsweepdirectionsinthelawnmowerpathsusedtocovereachcellinordertominimizethenumberofturns.However,veryfewCPPproposalsintheliteratureaddressthecostofthepathgeneratedtocoverthegivenarea.Tosweeptheareaofinterest,mostexistingcoveragealgorithmsproducesomesortoflawnmowerorboustrophe-don—thewordcomingfromGreek,“thewayanoxdragsaplough”.Whencoveringaplanaroreffectivelyplanarsurface,asinaoorcleaningtask,theinter-lapspacingoftheback-and-forthmotionsisdeterminedbytherobot'ssensorcoveragerange.However,whencoveringavariable-heightsurfaceliketheseaoorwhilenavigatingaboveitatconstantdepth,thesensor'sFOVchangesdependingontheheightofthebottomsurface.Therefore,toensurecompletecoverageofthetargetarea,theinter-lapspacingallowedbytheshallowestdepth(i.e.themaximumheight)inthetargetsurfacemustbeused,producinganundesired,considerableamountofsensorcoverageoverlappingamongthepathlaps.Uptodate,noresearchhasaddressedthisissue.ItisalsoworthnoticingthatmostoftheresearchinCPPhasbeenintendedformobileroboticsapplications,whileveryfewhasaddressedtheparticularitiesofmarineenvironments.Inthiswork,weproposeanextensiontothecellulardecompositionapproachtogenerateacoveragepaththatcompletelycoversa2.5Dsurfaceofinterestontheseaoorbynavigatinginaconstant-depthplaneaboveit.TheproposedmethodminimizesthecoverageoverlappingbysegmentingthetargetsurfaceintoregionsofsimilardepthfeaturesandaddressingthemasindividualCPPproblems.Acelldecompositioncoveragemethodisappliedtoeachregion.Thesurfacegradientisusedtodeterminethebestsweeporientationineachcell,andtheinter-lapspacingintheboustrophedonpathsusedtocovereachcellismaximizedonalap-by-lapbasis,henceobtainingashorter,moreefcientcoveragepath.Theproposalisvalidatedinsimulationexperimentsconductedwithareal-worldbathymetricdatasetthatshowasignicantincreaseonpathefciency.Theremainingsectionsinthispaperareorganizedasfollows.Sec.IIdiscussestheMorse-basedcellulardecom-positionforplanarsurfacesthatwillbeusedbyourmethodtoensurecompletecoverage.Next,Sec.IIIdescribestheproposedseabedcoveragemethod.InSec.IVwevalidatetheproposedmethodwithsimulationexperimentsconductedwithareal-worlddataset.Finally,concludingremarksaregiveninSec.V.II.MORSE-BASEDCELLULARDECOMPOSITIONCOVERAGEInthiswork,weusetheoff-lineMorse-basedcellulardecompositionmethodtoachievecompletecoverage[4].Thismethod,asanexactcellulardecompositionmethod,dividestherobot'sfreespaceintosimpleregions(cells)suchthattheunionoftheregionsllsthefreespace.Twocellsareadjacentiftheyshareacommonboundary.Anadjacencygraphisusedtoencodethecelldecomposition,whereanoderepresentsacellandanedgerepresentsanadjacencyrelationshipbetweentwocells.Giventhatnoobstacleslieinsideacell,tocovereachcellisconsideredatrivialtask,asitcanbecoveredusingsimplemotions,suchaslawnmower-likeback-and-forthmotions.Therefore,ndingapaththatvisitseachcellonce(i.e.ndinganexhaustivewalkthroughtheadjacencygraph)isequivalenttosolvingthecoverageproblem.TheMorse-basedcelldecompositionmethodusescriticalpointsontherestrictionofaMorsefunctiontotheobstacleboundariestodeterminethecelldecomposition.AMorsefunctionisonewhosecriticalpointsyarenondegenerate[6].Practicallyspeaking,thismeansthatcriticalpointsareisolated.Changesontheconnectivityofthefreespace(thespacefreeofobstacles)occuratthesecriticalpointsontheobstacleboundaries.Thus,criticalpointscanbeusedtodeterminethecellulardecompositionofthefreespace.Todeterminethecelldecomposition,asliceissweptthroughthetargetspace.Thissliceisdenedintermsofthepreimageofareal-valuedMorsefunction,h:WS!R,whereWSistherobot'sworkspace(i.e.thespacethatneedstobecovered).ChoosingdifferentMorsefunctionsproducesdifferentsliceshapesandhencedifferentcelldecompositionpatterns.Forsimplicity,wewilldescribetheMorse-basedboustrophedondecomposition[7],whichhappensintheplane.Intheboustrophedondecomposition,averticalslice,denedintermsoftheMorsefunctionh(x;y)=x,issweptfromlefttorightintheworkspace.Thus,theverticalsliceisdeterminedbythepreimageofthisMorsefunction,WS=h�1().Increasingthevalueofthesliceparameter,,sweepstheslicefromlefttorightthroughtheworkspace.Astheslicesweepsthespaceitintersects(orstopsintersecting)obstacles,whichseveritintosmallerpiecesastheslicerstencountersanobstacle,thatis,theconnectivityofthesliceinthefreespaceincreases.Also,immediatelyafterthesliceleavesanobstacle,smallerslicepiecesaremergedintolargerpieces(theconnectivityofthesliceinthefreespacedecreases).Thepointswheretheseconnectivitychangesoccurarethecriticalpoints.(Noticethatcriticalpointsarealwayslocatedontheobstacleboundaries.)Thus,atcriticalpoints,thesliceisusedtodeterminethecellsinthedecomposition.Onecanseethatwithinacell,thesliceconnectivityremainsconstant.Fig.1showshow,attheyRecallthatinthecaseofafunctionofasinglerealvariable,f(x),acriticalpointisavaluex0inthedomainoffwhereeitherthefunctionisnotdifferentiableoritsderivativeis0,f0(x0)=0. ensurethattheregionsaresimplyconnected.Wedothisbecause,ratherthananaccuratesegmentationthatwouldproduceregionswithabruptbordershencedifculttocover,wefosterthedivisionoftheterrainintosmoothregionsthatallowtominimizethepath'scoverageoverlapping.Smoothregionsareobtainedthankstothedilateanderodeoperations.Fig.6showsthesegmentationofthebathymetricmapshowninFig.4withn=3regions. Fig.6.SegmentationofthebathymetricmapshowninFig.4withn=3regions,withtheworkspaceobstaclesshowninblackE.CoveragePathGenerationforEveryRegionOncethesurfacesegmentationisobtained,weactuallytacklendifferentCPPproblems,oneforeachregion.Aswementioned,thiscontributestominimizethecoverageoverlappingalongthegeneratedpath.Then,whatwedointhisstepistoplanacoveragepathforeverysurfaceregionobtainedinthesegmentation.Eachindividualcoveragepathinaregionisplannedbyadheringtothefollowingsteps.First,theMorse-basedbous-trophedoncelldecompositionmethoddiscussedinSec.IIisappliedtoeachregiontoobtainitscellulardecomposition.Second,thesweeporientationineachcellisdeterminedbythegradientoftheunderlyingsurface.Third,usingthedeterminedsweepdirections,aboustrophedonpathisgeneratedtocovereachcellwheretheinter-lapspacingismaximizedonalap-by-lapbasis.Fourthandlast,theindividualpathsineachregionareconcatenatedtoobtainthenalcoveragepath.1)RegionCellDecomposition:Thecellulardecompo-sitionofeachregionisobtainedbyapplyingtheMorse-basedboustrophedoncelldecompositionmethod.Fig.7showsthecellulardecompositionforregion2ofthesurfacesegmentationshowninFig.6.Anexhaustivewalkthroughtheadjacencygraphassociatedtothedecompositioniscom-puted.Theexhaustivewalkdeterminestheorderinwhichthecellsarecovered.2)SweepOrientation:Oncethecelldecompositionoftheregionisobtained,wecomputethesweeporientationoftheindividualboustrophedonpathsusedtocovereachcelltobeperpendiculartothemainseaoorsurfacegradientunderthecell.Wedothisbecausenavigatingperpendicularlytothesurfacegradientallowsformaximizingtheinter-lapspacing,assteepsurfaceascendsordescendsunderalapareavoidedandhencethedifferencebetweenthelowest Fig.7.Celldecompositionofregion2ofthesegmentationshowninFig.6andhighestsurfaceelevationunderalapissmaller.Thus,wecomputethemeansurfacegradientunderthecellasameasureofitsmaininclineorientation.Then,wemakethelapsperpendiculartotheangledeterminedbythemeangradient.3)VariableInter-LapSpacingBoustrophedonPaths:Next,wegeneratetheindividualboustrophedonpathstocovereachcellwheretheinter-lapspacingvariesaccordingtotheminimumdistancetothebottomsurfaceundereachlap.Thatis,thespacingbetweenthecurrentlapandthenextlapisdeterminedbythehighestpointontheseaoorsurfaceunderthecurrentlap,wheretheminimumsensorFOVwidthoccurs.Byadheringtothisvariableinter-lapspacingstrategyweguaranteefullcoverageofthecellwhileminimizingthecoverageoverlappingamonglaps,henceobtainingashorter,moreefcientpath.Fig.8showsthegeneratedcoveragepathsforeachcellinthedecompositionshowninFig.7. Fig.8.CoveragepathsforeachcellofthecellulardecompositionshowninFig.74)FinalCoveragePath:Thethreestepswejustdescribedareappliedtoalltheregionsinthesurfacesegmentation.Finally,weconcatenatetheindividualcoveragepathscom-putedforeachregioninthesurfacesegmentation.Thewell-knownstart-to-goalpathplannerA*[8]isusedtocomputeapathgoingfromthelastpointofacellcoveragepathtotherstpointofthenextcell'scoveragepath.Fig.9showsthegeneratednalcoveragepath.IV.RESULTSTovalidatethemethodproposedinthiswork,wecompareouradaptiveinter-lapspacingapproachtotheMorse-basedboustrophedondecompositionforplanarspaces.Itisworth Fig.9.FinalcoveragepathoftheworkspaceshowninFig.5overlappedonthetargetbathymetricmapnoticingthatthelattermethodisintendedforcoveringplanarspacesbyapplyingbothmethodstocovertheseabedareaneartheFormiguesislandsdescribedaboveinIII-A.Seekingafaircomparison,weusetheminimumpossiblesensorFOVwidthineachcelltodeterminetheinter-lapspacing(thatis,theFOVwidthwhenthevehicleislocatedoverthehighestpointontheseaoorsurface).Ourcomparisonistwofold.Ononehand,wequantitativelymeasurebothpathsintermsofpathlengthandrunningtime.Ontheotherhand,weprovidecoveragedensitymapsgeneratedbyfollowingthepathscomputedbybothmethodsandtakingintoaccounttheFOVofthevehicle'ssensorineverypoint.Inotherwords,acoveragedensitymapshows,foragivenpathontheworkspace,howmanytimeshaseverypointontheworkspacebeencovered.TableIshowsthequantitativecomparisonofbothmeth-ods.ThevehiclenavigatesatdepthD=�8:5mandthetargetseaoorsurfaceissegmentedinn=3regions.TheFOVangleissetto=50andthevehiclespeedis2m/s.TABLEIPATHLENGTHCOMPARISON Method PathLength Constantinter-lapspacing 15846.08m Variableinter-lapspacing 10349.63m Theseresultsshowthat,forthereal-worldenvironmentpresentedinthispaper,ourmethodperformsapproximately34%betterthantheboustrophedonapproachforplanarspacesintermsofpathlengthandrunningtime.Fig.10showsthecomparisonofthecoveragedensitymapsobtainedusingthe“naive”boustrophedondecomposi-tionmethodandthenovelapproachproposedinthispaper.Ideally,eachpointinthetargetspaceshouldbecoveredonceandhenceberepresentedinadarkbluecolor.Thecoveragedensitymapcorrespondingtothe“naive”boustrophedonmethod(Fig.10(a))showsahighamountofoverlapping,denotedbyyellowandredcolors.Ontheotherhand,thecoverageoverlappingishighlyreducedwithourmethod,asdenotedbythebluecolorsonFig.10(b). (a)“Naive”boustrophedonmethod (b)OurmethodFig.10.CoveragedensitymapcomparisonV.CONCLUSIONSAnovelalgorithmforcoverageofseabedsurfacesfromanoverlyingplanarsurfacehasbeenpresented.Theproposedmethodminimizesthecoverageoverlappingbysegment-ingthetargetsurfaceinregionsofsimilardepthfeaturesandaddressingthemasindividualCPPproblems.Acelldecompositioncoveragemethodisappliedtoeachregion.Thesurfacegradientisusedtodeterminethebestsweeporientationineachcell,andtheinter-lapspacingintheboustrophedonpathsusedtocovereachcellismaximizedonalap-by-lapbasis,henceobtainingashorter,moreef-cientcoveragepath.Thevalidityofthealgorithmhasbeendemonstratedinsimulationexperimentsconductedusingareal-worldbathymetricdataset,whereasignicantincreaseinthecoveragepathefciencyhasbeenshown.REFERENCES[1]Y.GabrielyandE.Rimon,“Spiral-stc:anon-linecoveragealgorithmofgridenvironmentsbyamobilerobot,”inProc.IEEEInt.Conf.RoboticsandAutomationICRA'02,vol.1,2002,pp.954–960.[2]Y.-H.Choi,T.-K.Lee,S.-H.Baek,andS.-Y.Oh,“Onlinecompletecoveragepathplanningformobilerobotsbasedonlinkedspiralpathsusingconstrainedinversedistancetransform,”inProc.IEEE/RSJInt.Conf.IntelligentRobotsandSystemsIROS2009,2009,pp.5788–5793.[3]H.Choset,K.Lynch,S.Hutchinson,G.Kantor,W.Burgard,L.Kavraki,andS.Thrun,PrinciplesofRobotMotion:Theory,Algorithms,andImplementation.TheMITPress,2005.[4]E.U.Acar,H.Choset,A.A.Rizzi,P.N.Atkar,andD.Hull,“Morsedecompositionsforcoveragetasks,”InternationalJournalofRoboticsResearch,vol.21,no.4,pp.331–344,2002.[5]W.H.Huang,“Optimalline-sweep-baseddecompositionsforcoveragealgorithms,”inProc.ICRARoboticsandAutomationIEEEInt.Conf,vol.1,2001,pp.27–32.[6]J.Milnor,MorseTheory.PrincetonUniversityPress,1963.[7]H.Choset,“Coverageofknownspaces:theboustrophedoncellulardecomposition,”AutonomousRobots,vol.9,no.3,pp.247–253,2000.[8]S.J.RussellandP.Norvig,ArticialIntelligence:AModernApproach.PearsonEducation,2003.[Online].Available:http://portal.acm.org/citation.cfm?id=773294