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外文翻译--用区间法解决正向运动学中Gough类型的并联机构问题.pdf

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外文翻译--用区间法解决正向运动学中Gough类型的并联机构问题.pdf

J.P.MerletINRIASophiaAntipolis,FranceSolvingtheForwardKinematicsofaGoughTypeParallelManipulatorwithIntervalAnalysisAbstractWeconsiderinthispaperaGoughtypeparallelrobotandwepresentanefficientalgorithmbasedonintervalanalysisthatallowsustosolvetheforwardkinematics,i.e.,todetermineallthepossibleposesoftheplatformforgivenjointcoordinates.Thisalgorithmisnumericallyrobustasnumericalroundofferrorsaretakenintoaccounttheprovidedsolutionsareeitherexactinthesensethatitwillbepossibletorefinethemuptoanarbitraryaccuracyortheyareflaggedonlyasapossiblesolutionaseitherthenumericalaccuracyofthecomputationdoesnotallowustoguaranteethemortherobotisinasingularconfiguration.Itallowsustotakeintoaccountphysicalandtechnologicalconstraintsontherobotforexample,limitedmotionofthepassivejoints.Anotheradvantageisthat,assumingrealisticconstraintsonthevelocityoftherobot,itiscompetitiveintermofcomputationtimewitharealtimealgorithmsuchastheNewtonscheme,whilebeingsafer.KEYWORDSforwardkinematics,parallelrobot1.Introduction1.1.RobotGeometryInthispaperweconsiderasixdegreesoffreedom6DOFparallelmanipulatorFigure1consistingofafixedbaseplateandamobileplateconnectedbysixextensiblelinks.LegiisattachedtothebasewithaballandsocketjointwhosecenterisAiwhileitisattachedtothemovingplatformwithauniversaljointwhosecenterisBi.ThelengthofthelegsthedistancebetweenAiandBiwillbedenotedbyρi.AreferenceframeA1,x,y,zisattachedtothebaseandamobileframeB1,xr,yr,zrisattachedtothemovingplatform.TheInternationalJournalofRoboticsResearchVol.23,No.3,March2004,pp.221235,DOI10.1177/0278364904039806©2004SagePublications1.2.TheForwardKinematicsProblemTheforwardkinematicsFKproblemmaybestatedasbeinggiventhesixleglengths,findthecurrentposeScoftheplatform,i.e.,theposeoftherobotwhentheleglengthshavebeenmeasured.Althoughitmayseemthatthisproblemhasbeenaddressedinnumerousworks,ithasneverbeenfullysolved.Indeed,aswewillsee,allauthorshaveaddressedasomewhatdifferentalthoughrelatedproblemPbeinggiventhesixleglengths,findallthenpossibleposesS1{S1,...,Sn}oftheplatform.ItmaybeacceptedthatsolvingPisthefirststepforsolvingtheFKproblemassoonassomemethodallowsustodeterminewhichsolutionSjinthesolutionsetofPisthecurrentposeScoftherobot.Unfortunately,nosuchmethodisknowntodate,evenforplanarparallelrobots.ThispaperwillalsoaddressthePproblem,althoughwewillbeabletotakeintoaccount,duringthecalculation,realisticconstraintsontherobotmotionthatmayreducethenumberofsolutions.ProblemPisnowconsideredasaclassicalprobleminkinematicsandisalsousedinothercommunitiesasadifficultbenchmark.Raghavan1991andRongaandVust1992werethefirsttoestablishthattheremaybeupto40complexandrealsolutionstoPwhileHusty1996succeededinprovidingaunivariatepolynomialofdegree40thatallowsustodetermineallthesolutions.Dietmaier1998exhibitedconfigurationsforwhichtherewere40realsolutionposes.1.3.SolvingMethodfortheForwardKinematicsThemethodstraditionallyusedtosolvePmaybeclassifiedastheeliminationmethodthecontinuationmethodtheGröebnerbasismethod.221222THEINTERNATIONALJOURNALOFROBOTICSRESEARCH/March2004A1A2A3A4A5A6B1B2B3B4B5B6COxyzyrzrxrFig.1.Goughplatform.Allthesemethodsassumeanalgebraicformulationoftheproblemwithnunknowns,x1,...,xn.Thesemethodswillbedescribedintuitivelywithouttryingtoberigorous.IntheeliminationmethodInnocenti2001LeeandShim2001aeachequationofthesystemisexpressedasalinearequationintermofmonomialsproducttextxi11...xinnincludingtheconstantmonomial1inwhichoneoftheunknowns,xk,issupposedtobeconstanti.e.,thecoefficientsoftheequationsarefunctionsofxk.AdditionalequationsareobtainedbymultiplyingtheinitialequationsbyamonomialuntilweobtainasquaresystemoflinearequationsthatcanbeexpressedinmatrixformasAxkX01whereXisasetofmonomialsincludingtheconstantmonomial1.Duetothisconstantmonomial,theabovesystemhasasolutiononlyif|Axk|0,whichisaunivariatepolynomialPeinxk.Aftersolvingthispolynomial,abacktrackmechanismallowsustodeterminealltheotherunknownsforeachrootofthepolynomialPe.Themainweaknessofthismethodisthecalculationof|A|usuallyAisaratherlargematrixanditsdeterminantcannotbecalculatedinclosedform.Mostauthorsproposetouseanumericalmethodtoevaluatethecoefficientsofthepolynomial|A|thedeterminantofordern,whichisalinearfunctionofthepolynomialcoefficients,iscalculatednumericallyforn1valuesofxkandthereforethecoefficientscanbeobtainedbysolvingasystemofn1linearequations.However,suchaprocedureisnumericallyunstableandhencethereisnoguaranteeofthecorrectnessofthesolutions.AneliminationmethodhasbeenusedbyHusty1996toobtaina40thorderpolynomialbutusingonlysymboliccomputationandacarefuleliminationprocessthatguaranteethatweobtaintheexactpolynomialcoefficientsunfortunately,thisprocedureseemstobedifficulttoautomate.TosolveasystemofequationsFX0,thecontinuationmethodRaghavan1991SreenivasanandNanua1992LiuandYang1995Wampler1996usesanauxiliarysystemGXF1−λF1−F0,whereF1isasystemsimilartoF,inthesensethatithasatleastthesamenumberofsolutionsasF,ofwhichallthesolutionsareknownandλisascalar.Whenλisequalto0,GF1andconsequentlythesolutionsofGareknown.Thesesolutionsareusedasaninitialguesstosolve,usingaNewtonscheme,anewversionofGobtainedforλepsilon1whereepsilon1hasasmallvalue.Thisprocessisrepeatedforλ2epsilon1usingthesolutionsofthepreviousrunasaninitialguessandsoonuntilλ1forwhichGF.Inotherwords,startingfromasystemwithknownsolutionswefollowthesolutionbranchesofasystemthatslowlyevolvestowardF.ThemainweaknessofthisapproachisthatitisnecessarytofollowalargenumberofbranchestofindallthesolutionsofF.Inourcase,F1hastohaveatleast40solutionsandhence40brancheswillbefollowed,someofwhichwillvanishiftheFKproblemhaslessthan40solutions.Furthermore,numericalrobustnessisdifficulttoensureifasingularityisencounteredwhenfollowingthebranches.IntheGröebnerbasisapproach,thepropertyisusedthatthesolutionsofanyalgebraicsystemFarealsosolutionsofvariousothersystemsofequationsinsomeotherunknownsyi.Amongallthesesystems,oneofthemhasthepropertyofbeingtriangular,i.e.,thesystemhasafirstequationinoneunknowny1,thesecondequationhasonlyy1,y2asunknownsandsoon,untilthelastequationwithunknownsy1,...,yn.Henceallthesolutionsofthissystemcanbeobtainedbysolvinginsequencethefirstequation,thenthesecondandsoon.SuchatriangularsystemcanbeobtainedbyusingtheBuchbergeralgorithmLazard1992FaugèreandLazard1995.Althoughthismethodiscurrentlythefastesttosolveinaguaranteedmanner,theFKproblemusingtheFGbandtheRealSolvingalgorithmsofFaugère1andRouillier1995,2003thisapproachcanonlybeusedwhenthecoefficientsoftheequationsarerationalinwhichcasetheresultsarecertifiedanditsimplementationinvolvestheuseoflargeintegers.2.SolvingwithIntervalAnalysis2.1.IntervalAnalysisIntervalanalysisisanalternativenumericalmethodthatcanbeusedtodetermineallthesolutionstoasystemofequationsandinequalitiessystemswithinagivensearchspace.1.Seehttp//wwwcalfor.lip6.fr/∼jcf/index.html.Merlet/SolvingtheForwardKinematics223AnintervalXisdefinedasthesetofrealnumbersxverifyingx≤x≤x.ThewidthwXofanintervalXisthequantityx−xwhilethemidpointMXoftheintervalisxx/2.ThemignitudemagnitudeofanintervalXisthesmallesthighestvalueof|x|,|x|.ApointintervalXisobtainedifxx.Aboxisatupleofintervalsanditswidthisdefinedasthelargestwidthofitsintervalmembers,whileitscenterisdefinedasthepointwhosecoordinatesarethemidpointoftheranges.LetfbearealvaluedfunctionofnunknownsX{x1,...,xn}.AnintervalevaluationFoffforgivenranges{X1,...,Xn}fortheunknownsisanintervalYsuchthat∀X{x1,...,xn}∈X{X1,...,Xn}Y≤fX≤Y.2Inotherwords,Y,YarelowerandupperboundsforthevaluesoffwhentheunknownsarerestrictedtoliewithintheboxX.TherearenumerouswaystocalculateanintervalevaluationofafunctionHansen1992Moore1979.ThesimplestisthenaturalevaluationinwhichallthemathematicaloperatorsinfaresubstitutedbytheirintervalequivalenttoobtainF.Forexample,theclassicaladditionissubstitutedbyanintervaladditiondefinedasX1X2x1x2,x1x2.Intervalequivalentsexistforalltheclassicalmathematicaloperatorsandhenceintervalarithmeticsallowsustocalculateanintervalevaluationformostnonlinearexpressions,whetheralgebraicornot.Forexample,iffxxsinx,thentheintervalevaluationoffforx∈1.1,2canbecalculatedasF1.1,21.1,2sin1.1,21.1,20.8912,11.9912,3.Intervalevaluationexhibitsinterestingproperties,asfollows.1.If0negationslash∈FX,thenthereisnovalueoftheunknownsintheboxXsuchthatfX0.Inotherwords,theequationfXhasnorootintheboxX.2.TheboundsoftheintervalevaluationFusuallyoverestimatetheminimumandmaximumofthefunctionovertheboxX,buttheboundsofFareexactlytheminimumandmaximumifthereisonlyoneoccurrenceofeachunknowninfPropertyA.3.Intervalarithmeticscanbeimplementedtakingintoaccountroundofferrors.Forexample,theintervalevaluationoffx/3whenXisthepointinterval1,1willbetheintervalα1,α2whereα1,α2aretheclosestfloatingpointnumbers,respectivelylowerandgreaterthan0.3333....Therearenumerousintervalarithmeticspackagesimplementingthisproperty.OneofthemostwellknownisBIAS/Profil2usingtheCdoubleforintervalrepresentation.However,apromisingnewpackageisMPFIRevolandRouillier2002,basedonthemultiprecisionsoftwareMPFRdevelopedbytheSPACESproject3,inwhichtheintervalisrepresentedbyanumberwithanarbitrarynumberofdigits.2.2.BasicSolvingAlgorithmIntervalanalysisbasedalgorithmshavebeenusedinroboticsforsolvingtheinversekinematicofserialrobotsKiyoharu,Ohara,andHiromasa2001Tagawaetal.1999andparallelrobotsFKCastellet1998Didrit,Petitot,andWalter1998Jaulinetal.2001,workspaceanalysisChablat,Wenger,andMerlet2002Merlet1999,singularitydetectionMerletandDaney2001,evaluatingthereliabilityofparallelrobotsCarrerasetal.1999,optimalplacementofrobotsTagawaetal.2001,mobilerobotlocalizationBouvetandGarcia2001andtrajectoryplanningPiazziandVisioli1997.However,intervalanalysisisamorecomplexmethodthanmaybethoughtatafirstglanceandwewillpresentinthispapervariousimprovementsthathaveadrasticinfluenceontheefficiency.Westartwiththemostbasicsolvingalgorithmthatmaybederivedfromthepropertiesofintervalarithmetics.LetB0{X1,...,Xn}beaboxandf{f1X,...,fnX}asetofequationstobesolvedwithinB0.AlistLwillcontainasetofboxesandinitiallyL{B0}.Anindexi,initializedto0,willindicatewhichboxBiinLiscurrentlybeingprocessed,whilenwilldenotethenumberofboxesinthelist.TheintervalevaluationofthefunctionfjfortheboxBiwillbedenotedFjBi.Athresholdepsilon1willbeusedand,ifthewidthoftheintervalevaluationofallthefunctionsforaboxBiislowerthanepsilon1andincludes0,thenBiwillbeconsideredasasolutionofthesystem.Thealgorithmproceedalongthefollowingsteps.1.Ifin,thenreturntothesolutionlist.2.IfatleastoneFjBiexistssuchthat0negationslash∈FjBi,thenii1andgoto1.3.If∀j∈1,n0∈FjBiandwFjBi≤epsilon1,thenstoreBiinthesolutionlist,ii1andgoto1.4.SelecttheunknownkwhoseintervalhasthelargestwidthinBi.BisectitsintervalinthemiddlepointandcreatetwonewboxesfromBias{X1,...,Xk−1,Xk,XkXk/2,...,Xn}and{X1,...,Xk−1,XkXk/2,Xk,...,Xn}.StorethesetwoboxesasBn1,Bn2,nn2,ii1andgoto1.2.http//www.ti3.tuharburg.de/Software/PROFILEnglisch.html.3.http//www.mpfr.org.224THEINTERNATIONALJOURNALOFROBOTICSRESEARCH/March2004Notethatthestoragemethodusedherefortheboxesisnotveryefficientasfarasmemorymanagementisconcerned.AfirstimprovementistosubstitutetheboxBibyoneofthetwoboxesthatarecreatedwhenbisectingit.Asecondimprovement,denotedadepthfirststoragemode,istostorethesecondboxatpositioni1afterashiftoftheexistingboxes.ThisensuresthatthewidthofBiisalwaysdecreasinguntileithertheboxiseliminatedorasolutionisfound.Inthismode,forasystemofnequationsinnunknowns,thewidthofBiisatleastdividedby2afternbisection.IfthewidthoftheinitialboxB0isw0thenumberNofboxesthatareneededissuchthat2K/nw0/epsilon1andhenceNnlogw0/epsilon1/log2.Forexample,ifn9,w010andepsilon110−6,weobtainthatthenumberofboxesofLshouldbe210towhichwemustaddthememorytostorethesolutions.Hence,evenwithahighaccuracyforthesolutionandalargeinitialsearchspacethenecessarymemorystorageissmall.Asamatteroffact,thedescribedalgorithmwillusuallynotbeveryefficient,buttherearenumerouswaystoimproveitaswillbeshownlateron.However,notethatthereisaneasywaytoimprovethecomputationtimeofthebasicalgorithmindeed,wemaynoticethateachboxinLissubmittedtoaprocessingthatdoesnotdependupontheotherboxes.Henceitispossibletoimplementthealgorithminadistributedmanner.Amastercomputerwillsendtonslavecomputersthefirstnboxesinthelist.TheseslavecomputerswillindividuallyperformafewiterationsofthebasicalgorithmandwillsendbacktothemastertheremainingboxesinitsLlistifanyandthesolutionsithasfoundifany.ThemasterwillmanageagloballistLandperformafewiterationsofthebasicalgorithmifalltheslavesarebusy.WewilldiscusstheefficiencyofthedistributedimplementationintheExamplesections.3.EquationsfortheForwardKinematicsLetAiandBibetheattachmentpointsofthelegionthebaseandontheplatform,respectively.ThecoordinatesofAiinthereferenceframewillbedenotedxai,yai,zaiwhilethecoordinatesofBiinthesameframearexi,yi,zi.Withoutlackofgeneralitywemaychoosexa1ya1za10andya2za20.Notethatforagivenrobotandgivenleglengthsitisalwayspossibletochangethenumberingoftheleglengthsandwewillseethatthishasaninfluenceonthecomputationtimeofouralgorithm.TherearenumerouswaystowritetheequationsoftheinversekinematicswhichconstitutethesystemofequationstobesolvedfortheFKproblemaccordingtotheparametersthatareusedtorepresenttheposeoftheplatform.InthispaperaposeoftheplatformwillbedefinedeitherbythecoordinatesofthethreepointsB1,B2,B3assumedtobenotcollinearsuchatripletcanalwaysbefoundotherwisetherobotisalwayssingulariftheplatformisplanar,orbythecoordinatesofthefourpointsB1,B2,B3,B4inthegeneralcase.Thechosenpointswillbedenotedthereferencepointsofthesystem,andtheassociatedlegsthereferencelegs.Ifm,m∈3,4pointsareusedfordefiningtheposeoftheplatformthenforanyjinm1,6wehaveOBjkmsummationdisplayk1αkjOBk,3whereαkjareknownconstantssuchthatsummationtextkmk1αkj1.Afirstsetofequationsisobtainedbyexpressingtheleglengthsforthemchosenreferencelegsxj−xaj2yj−yaj2zj−zaj2ρ2j,j∈1,m,4whereρjistheknownleglength.Asecondsetofequationsisobtainedbywritingtheleglengthsforthelegsm1to6,usingthecoordinatesoftheBjpointsdefinedineq.3parenleftBiggimsummationdisplayi1αijxi−xajparenrightBigg2parenleftBiggimsummationdisplayi1αijyi−yajparenrightBigg2parenleftBiggimsummationdisplayi1αijzi−zajparenrightBigg2ρ2j,j∈m1,6.5ThethirdsetofequationsisobtainedbywritingthatthedistancebetweenanycoupleofreferencepointsB1,...,Bmisaknownquantityxi−xj2yi−yj2zi−zj2d2ij,i,j∈1,m,inegationslashj,6wheredijisthedistancebetweenthepointsBiandBj.Itmaybenotedthateqs.4,5and6areasetofdistanceequationswhichdescribethedistancebetweeneitherpointsinthethreedimensional3Dspaceorvirtualpoints,i.e.,pointswhosecoordinatesarealinearcombinationofreferencepointsherepointsBm1,...,B6arethevirtualpoints.Weendupwithasystemofn3mequationsinthe3munknownsxi,yi,zi.Itappearsthatforeachequationinthesystem4,5and6thereisonlyoneoccurrenceofeachunknown.Consequently,accordingtoPropertyAtheintervalevaluationofeachequationgivestheexactminimumandmaximumvaluesoftheequationsandthismotivatestheuseofsuchrepresentationoftheposeoftheplatform.Itmustbenoted,however,thatifm4wehavenotaminimalparametrizationofthesystemasonlythreepointsareneededtodefinetheposeoftheplatform.IndeedforpointBkwithkin4,6wehaveB1Bkµk1B1B2µk2B1B3µk3B1B2B1B3,7

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