Introduction to Inverse Kinematics with Jacobian Transpose Pseudoinverse and Damped Least - PDF document

IntroductiontoInverseKinematicswithJacobianTranspose,PseudoinverseandDampedLeastSquaresmethodsSamuelR.BussDepartmentofMathematicsUniversityofCalifornia,SanDiegoLaJolla,CA92093-0112October7,2009Note:ThisisanintroductionthatwasoriginallywrittenforapaperbyBussandKim[7],butwassubsequentlyseparatedout.Thisreportisbeingmadeavailableviatheinternet|therearenoplanstopublishit.ThisisaintroductiontotheJacobiantransposemethod,thepseudoinversemethod,andthedampedleastsquaresmethodsforinversekinematics(IK).Themathematicalfoundationsofthesemethodsarepresented,withananalysisbasedonthesingularvaluedecomposition.1IntroductionArigidmultibodysystemconsistsofasetofrigidobjects,calledlinks,joinedtogetherbyjoints.Simplekindsofjointsincluderevolute(rotational)andprismatic(translational)joints.Itisalsopossibletoworkwithmoregeneraltypesofjoints,andtherebysimulatenon-rigidobjects.Well-knownapplicationsofrigidmultibodiesincluderoboticarmsaswellasvirtualskeletonsforanimationincomputergraphics.Tocontrolthemovementofarigidmultibodyitiscommontouseinversekinematics(IK).ForIK,itispresumedthatspeci¯edpoints,called\end SupportedinpartbyNSFgrantDMS-0100589.Contactauthor: e®ectors,"onthelinksareareassigned\targetpositions."TosolvetheIKproblem,wemust¯ndsettingsforthejointanglessothattheresultingcon¯gurationofthemultibodyplaceseachende®ectoratitstargetposition.MoregeneralformulationsofIKallowalsoorientationgoals,ordirectionalThereareseveralmethodsforsolvingIKproblems,comingoriginallyfromroboticsapplications.Theseincludecycliccoordinatedescentmeth-ods[43],pseudoinversemethods[45],Jacobiantransposemethods[5,46],theLevenberg-Marquardtdampedleastsquaresmethods[41,34],quasi-Newtonandconjugategradientmethods[43,49,15],andneuralnetandarti¯cialintelligencemethods[19,27,36,38,20,22,40,16].ThepresentpaperfocusesonapplicationsofIKincomputergraphicsandreal-timeanimation.TherehasalreadybeenextensiveuseofIKincomputergraphics[18,26,25,44,23,2,1,17,24,29,39,21,37,12]:themostcommonapplicationsareanimatinghumansorcreaturesbyspecifyingthepositions,andpossiblytheorientations,oftheirhands,feetandhead.Ourinterestslieparticularlyinusingtargetpositionsforende®ectorstoanimateanentiremultibody,andinmethodsthatarerobustandbehavewellinwiderangeofsituations.Aspartoftherobustness,wewanttheende®ectorstotrackthetargetpositionsandtodoareasonablejobevenwhenthetargetpositionsareinunreachablepositions.Inthispaper,weconsideronly¯rstordermethodsandconsiderthefollowinggenericapplication:wepresumeamultibodyhasmultipleende®ectorsandmultipletargetpositions,giveninreal-timeinanonlinefashion,andwanttoupdatethemultibodycon¯gurationsoastodynamicallytrackthetargetpositionswiththeende®ectors.Onemightwonderwhyitisimportanttoallowtargetpositionstobeunreachable.Thereareseveralreasons:First,itmaybedi±culttocompletelyeliminatethepossibilityofunreachablepositionsandstillgetthedesiredmotion.Second,iftargetpositionsarebarelyreachableandcanbereachedonlywithfullextensionofthelinks,thenthesituationisverysimilartohavingunreachabletargets.Unfortunately,thesituationoftargetpositionsinunreachablepositionsisdi±culttohandlerobustly.Manymethods,suchasthepseudoinverseorJacobiantransposemethods,willoscillatebadlyinthissituation;however,(selectively)dampedleastsquaresmethodscanstillperformwellwithunreachabletargetpositions.Theoutlineofthepaperisasfollows.We¯rstintroducethemathemat-icalframeworkfortheIKproblem.WethendiscusstheJacobiantransposemethod,thepseudoinversemethod,thesingularvaluedecomposition,andthedampedleastsquares(DLS)method.ForanextensionoftheDLSmethodstoamethodcalledselectivelydampedleastsquares(SDLS),see BussandKim[7].Nearlyallthepresentpaperisexpository,butnewaspectsincludethepossibilityofformingtheJacobianmatrixwiththetargetpositionsinsteadoftheende®ectorpositions.Weattempttoexplainthemathematicalfoundationsclearlysoastoelucidatethestrengthsandweaknessesofthevariousmethods.Forsimplicityandtokeepthepapershort,wedonotconsideranyaspectsofjointlimitsoravoidingself-collisions;rather,wewillonlyconsiderthe\pure"IKproblemwithoutjointlimitsandwithoutself-collisions.2Preliminaries:forwardkinematicsandJacobiansAmultibodyismodeledwithasetoflinksconnectedbyjoints.Thereareavarietyofpossiblejointtypes.Perhapsthemostcommontypeisarotationaljointwithitscon¯gurationdescribedbyasinglescalaranglevalue.Otherjointtypesincludeprismatic(i.e.,translational,orsliding)joints,screwjoints,etc.Forsimplicity,wewilldiscussonlyrotationaljoints,butthealgorithmsandtheoryallapplytoarbitraryjoints.Thekeypointisthatthecon¯gurationofajointisacontinuousfunctionofoneormorerealscalars;forrotationaljoints,thescalaristheangleofthejoint.Thecompletecon¯gurationofthemultibodyisspeci¯edbythescalarsdescribingthejoints'con¯gurations.Weassumetherearejointsandeachvalueiscalledajointangle(but,aswejustsaid,couldmoregenerallyrepresentavaluewhichisnotanangle).Certainpointsonthelinksareidenti¯edasende®ectors.Ifthereareende®ectors,theirpositionsaredenoted.Eachende®ectorpositionisafunctionofthejointangles.Wewriteforthecolumnvector(;thiscanbeviewedasacolumnvectoreitherwithmanyscalarentriesorwithmanyentriesfromThemultibodywillbecontrolledbyspecifyingtargetpositionsfortheende®ectors.Thetargetpositionsarealsogivenbyavector,whereisthetargetpositionforthethende®ector.Welet,thedesiredchangeinpositionofthethende®ector.WealsoletThejointanglesarewrittenasacolumnvectorasTheende®ectorpositionsarefunctionsofthejointangles;thisfactcanbeexpressedas),or,for).TheIKproblemisto¯ndvaluesforthe'ssothatforall.(1) Unfortunately,theremaynotalwaysbeasolutionto(1),andtheremaynotbeaunique(best)solution.Eveninwell-behavedsituations,theremaybenoclosedformequationforthesolution.Wecan,however,useiterativemethodstoapproximateagoodsolution.Forthis,thefunctionsarelinearlyapproximatedusingtheJacobianmatrix.TheJacobianmatrixisafunctionofthevaluesandisde¯ned Notethatcanbeviewedeitherasamatrixwhoseentriesarevectors,orasmatrixwithscalarentries(withThebasicequationforforwarddynamicsthatdescribesthevelocitiesoftheende®ectorscanbewrittenasfollows(usingdotnotationfor¯rstderivatives):TheJacobianleadstoaniterativemethodforsolvingequation(1).Supposewehavecurrentvaluesfor.Fromthese,theJacobian)iscomputed.Wethenseekanupdatevalue¢forthepurposeofincrementingthejointanglesby¢By(2),thechangeinende®ectorpositionscausedbythischangeinjointanglescanbeestimatedasTheideaisthatthe¢valueshouldchosensothat¢isapproximatelyequalto,althoughitisalsocommontochoose¢sothattheapproximatemovement¢intheende®ectors(partially)matchesthevelocitiesofthetargetpositions(see[45]).Theupdateofthejointanglescanbeusedintwomodes:(i)Eachsimulationstepperformsasingleupdatetothevalueofjointanglesusingequation(3),sothattheende®ectorpositionsapproximatelyfollowthetargetpositions.(ii)Thejointanglesareupdatediterativelyuntilavalueofisobtainedthatissu±cientlyclosetoasolution.Itisalsopossibletouseahybridof(i)and(ii),thatis,usingasmallnumberofrepeatedupdatesusing(3)soastomoreaccuratelytracktheende®ectorpositions.Therestofthispaperdiscussesstrategiesforchoosing¢toupdatethejointangles.Inlightof(4),oneapproachistosolvetheequation TheentriesintheJacobianmatrixareusuallyveryeasytocalculate.Ifthethjointisarotationaljointwithasingledegreeoffreedom,thejointangleisasinglescalar.Letbethepositionofthejoint,andbeaunitvectorpointingalongthecurrentaxisofrotationforthejoint.Inthiscase,ifanglesaremeasuredinradianswiththedirectionofrotationgivenbytherightruleandifthethende®ectorisa®ectedbythejoint,thenthecorrespondingentryintheJacobianis Ifthethende®ectorisnota®ectedbythethjoint,thenofcourse=@µ=0.Ifthethjointistranslational,theentryintheJacobianmatrixiseveneasiertocompute.Supposethethjointperformstranslationthedirectionoftheunitvector,sothatthethejoint\angle"measuresdistancemovedinthedirection.Thenifthethende®ectorisa®ectedbythethjoint,wehave Formoreinformation,seeOrinandSchrader[35]whodiscusshowtocalculatetheJacobianmatrixentriesfordi®erentrepresentationsofjointsandmultibodies.Thetextbook[6,Ch.12]alsodiscussesarepresentationofrigidmultibodiesandhowtocalculatetheJacobian.CalculatingtheJacobianfor¢.Inmostcases,thisequationcannotbesolveduniquely.Indeed,themaynotbesquareorinvertible,andevenifisinvertible,justsetting¢mayworkpoorlyifisnearlysingular.AnalternateJacobian.Analternatemethodforde¯ningtheJacobianmatrixistolet wherethepartialderivativeiscalculatedusingtheformulafor(=@µsubstitutedfor.Themeaningof=@µisthatthetargetpositionisthoughtofasbeingattachedtothesamelinkasthethend e®ector.TheintuitionisthatwiththisformulationoftheJacobian,wearetryingtomovethetargetpositionstowardstheende®ectors,ratherthantheende®ectorstowardsthetargetpositions.ThealternateJacobianmaybeusedinplaceoftheusualJacobianinanyofthealgorithmsdiscussedbelow.OurexperiencehasbeenthatthisalternatecanimproveontheusualJacobianintermsofreducingoscillationorovershootwhentargetpositionsaretoofarawaytobereachedbytheende®ectors.However,thedrawbackisthatinsomecon¯gurations,thealternativeJacobiancanleadto\jerky"behavior.Thisisparticularlytrueforrotationaljointswhenthemultibody'slinksarefoldedbackoneachothertryingtoreachaclosetargetposition.Settingtargetpositionscloser.Arecurringproblemintrackingtargetpositions,isthatwhenthetargetpositionsaretoodistant,themultibody'sarmsstretchouttotrytoreachthetargetposition.Oncethemultibodyisextendedinthisway,itusuallyisnearasingularity(thatis,theJacobianisverysensitivetosmallchangesinjointangles),andthemultibodywilloftenshakeorjitter,attemptingunsuccessfullytoreachthedistanttarget.Thesee®ectscanbereducedwithDLSandSDLSalgorithms,butaredi±culttoremovecompletely.Onetechniquetoreducethisproblemistomovethetargetpositionsinclosertotheende®ectorpositions.Forthis,wechangethede¯nitionofinsteadofmerelysetting,eachcomponentinthevectoritslengthclampedtoaspeci¯edmaximumvalue.Thatis,wede¯ne=ClampMag(jj· representstheusualEuclideannormof.Thevalueanupperboundonhowfarweattempttomoveanende®ectorinasingleupdatestep.Fordampedleastsquares,clampingthemagnitudesofinthiswaycanreduceoscillationwhentargetpositionsareoutofreach.Thishastheadvantageofallowingtheuseofasmallerdampingconstant;thesmallerdampingconstantallowssigni¯cantlyquickerconvergencetotargetpositions.Whentheende®ectorsaretrackingcontinuouslymovingtarget AlsoforSDLS,[7]havefoundthatclampingthemagnitudesofinthiswaycane®ectivelyreduceoscillationwhentargetpositionsareoutofreach. positions,thedistanceshouldbeatleastseveraltimeslargerthananende®ectormovesinasingleupdatestep.Inourexperience,settingtobeapproximatelyhalfthelengthofatypicallinkworkswell.Fortargetpositionsthatmayjumpdiscontinuously,wehaveusedseparatemaximumvaluesforeach.Afteradiscontinuousmovementofthetargetpositions(orwhenbeginningasimulationofacontinuouslymovingtarget),weinitiallysettoin¯nity.Afterthe¯rstsimulationstep,weletbetheamountbywhichtheprevioussimulationstepmovedthethende®ectorclosertoitstargetposition.Then,welet,andusetoclampthemagnitudeof3TheJacobiantransposemethodTheJacobiantransposemethodwas¯rstusedforinversekinematicsby[5,46].Thebasicideaisverysimple:usethetransposeofinsteadoftheinverseof.Thatis,weset¢equaltoforsomeappropriatescalar.Now,ofcourse,thetransposeoftheJacobianisnotthesameastheinverse;however,itispossibletojustifytheuseofthetransposeintermsofvirtualforces.Moregenerally,itcanbeshownthatthefollowingtheoremholds[5,46].Theorem1ForallProofTheproofistrivial:Theapproximation(4)impliesthat,forsu±cientlysmallupdatingtheanglesbyequation(3)using¢willchangetheende®ectorpositionsbyapproximately®JJ.ByTheorem1,thishasthee®ectofreducingthemagnitudeoftheerrorvectorissmallenough.Itremainstodecidehowtochoosethevalueof.Onereasonablewaytotrytominimizethenewvalueoftheerrorvectoraftertheupdate.Forthis,weassumethatthechangeinende®ectorpositionwillbeexactly®JJ,andchoosesoastomakethisvalueascloseaspossibletoThisgives;JJ ;JJ 4ThepseudoinversemethodThepseudoinversemethodsetsthevalue¢equaltowheretheisthe,alsocalledtheMoore-Penroseinverse.Itisde¯nedforallmatrices,evenoneswhicharenotsquareornotoffullrowrank.Thepseudoinversegivesthebestpossiblesolutiontotheequationinthesenseofleastsquares.Inparticular,thepseudoinversehasthefollowingniceproperties.Let¢bede¯nedbyequation(6).First,supposeisintherange(i.e.,thecolumnspan)of.Inthiscase,;furthermore,¢istheuniquevectorofsmallestmagnitudesatisfying.Second,supposethatisnotintherangeof.Inthiscase,isimpossible.However,hasthepropertythatitminimizesthemagnitudeofthedi®erence.Furthermore,¢istheuniquevectorofsmallestmagnitudewhichminimizes,orequivalently,whichminimizesThepseudoinversetendstohavestabilityproblemsintheneighborhoodsofsingularities.Atasingularity,theJacobianmatrixnolongerhasfullrowrank,correspondingtothefactthatthereisadirectionofmovementoftheende®ectorswhichisnotachievable.Ifthecon¯gurationisatasingularity,thenthepseudoinversemethodwillnotattempttomoveinanimpossibledirection,andthepseudoinversewillbewell-behaved.However,ifthecon¯gurationisclosetoasingularity,thenthepseudoinversemethodwillleadtoverylargechangesinjointangles,evenforsmallmovementsinthetargetposition.Inpractice,roundo®errorsmeanthattruesingularitiesarerarelyreachedandinsteadsingularityhavetobedetectedbycheckingvaluesforbeingnear-zero.Thepseudoinversehasthefurtherpropertythatthematrix(performsaprojectionontothenullspaceof.Therefore,forallvectors.Thismeansthatwecanset¢foranyvectorandstillobtainavaluefor¢whichminimizesthevalue.Thisnullspacemethodwas¯rstexploitedLi¶egeois[28],whousedittoavoidjointlimits.Bysuitablychoosing,onecantrytoachievesecondarygoalsinadditiontohavingtheende®ectorstrackthetargetpositions.Forinstance,mightbechosentotrytoreturnthejointanglesbacktorestpositions[18]:thiscanhelpavoidsingularcon¯gurations. Anumberofauthors(see[4])haveusedthenullspacemethodtohelpavoidsingularcon¯gurationsbymaximizingYoshikawa'smanipulabilitymeasure[48,47].MaciejewskiandKlein[30]usedthenullspacemethodforobstacleavoidance.Amoresophisticatednullspacemethod,calledtheextendedJacobianmethod,wasproposedbyBaillieul[4]:intheextendedJacobianmethodalocalminimumvalueofafunctionistrackedasasecondaryobjective.Thenullspacemethodhasalsobeenusedtoassigndi®erentprioritiestodi®erenttasks(see[10,3]).Analgorithmforthepseudoinversemethodcanbederivedasfollows:Fromequation(5),wegetthenormalequationThenweletandsolvetheequationNowitcanbeshownthatisalwaysintherangeof,henceequation(8)alwayshasasolution.Inprinciple,rowoperationscanbeusedto¯ndthesolutionto(8)withminimummagnitude;however,intheneighborhoodofsingularities,thealgorithmisinherentlynumericallyunstable.hasfullrowrank,thenisguaranteedtobeinvertible.Inthiscase,theminimummagnitudesolution¢toequation(8)canbeexpressedasToprovethis,notethatif¢satis¯es(9),then¢isintherowspanof.Equation(9)cannotbeusedifdoesnothavefullrowrank.Ageneralformulaforthepseudoinversefornotoffullrowrankcanbefoundin[6].Thepseudoinversemethodiswidelydiscussedintheliteraturebutitoftenperformspoorlybecauseofinstabilitynearsingularities.The(selectively)dampedleastsquaresmethodshavemuchsuperiorperformance.5DampedleastsquaresThedampedleastsquaresmethodavoidsmanyofthepseudoinversemethod'sproblemswithsingularitiesandcangiveanumericallystablemethodofselecting¢.ItisalsocalledtheLevenberg-Marquardtmethodandwas¯rstusedforinversekinematicsbyWampler[41]andNakamuraandHanafusa[34]. Thedampedleastsquaresmethodcanbetheoreticallyjusti¯edasfollows(see[42]).Ratherthanjust¯ndingtheminimumvector¢givesabestsolutiontoequation(5),we¯ndthevalueof¢thatminimizesthequantityisanon-zerodampingconstant.ThisisequivalenttominimizingthequantityThecorrespondingnormalequationisThiscanbeequivalentlyrewrittenasItcanbeshown(bythemethodsofsection6below)thatnon-singular.Thus,thedampedleastsquaressolutionisequaltoNowisanmatrix,whereisthenumberofdegreesoffreedom.Itiseasytoshowthat(Thus,Theadvantageofequation(11)over(10)isthatthematrixbeinginvertedisisthedimensionofthespaceoftargetpositions,isoftenmuchlessthanAdditionally,(11)canbecomputedwithoutneedingtocarryoutthematrixinversion,insteadrowoperationscan¯ndsuchthat(andthenisthesolution.Thedampingconstantdependsonthedetailsofthemultibodyandthetargetpositionsandmustbechosencarefullytomakeequation(11)numericallystable.Thedampingconstantshouldlargeenoughsothatthesolutionsfor¢arewell-behavednearsingularities,butifitischosentoolarge,thentheconvergencerateistooslow.Therehavebeenanumberofmethodsproposedforselectingdampingconstantsdynamicallybasedonthecon¯gurationofthearticulatedmultibody[34,14,15,31,11,13,8,9,33,32]. 6SingularvaluedecompositionThesingularvaluedecomposition(SVD)providesapowerfulmethodforanalyzingthepseudoinverseandthedampedleastsquaresmethods.Inaddition,weshallusetheSVDtodesignaselectivelydampedleastsquaresmethodin[7].LetbetheJacobianmatrix.AsingularvaluedecompositionconsistsofexpressingintheformUDVareorthogonalmatricesandisdiagonal.If,and.Theonlynon-zeroentriesinthematrixarethevaluesalongthediagonal.Wehenceforth.Withoutlossofgenerality,¸¢¢¢¸Notethatthevaluesmaybezero.Infact,therankofisequaltothelargestvaluesuchthat=0.For=0.Weusetodenotethethcolumnsof.Theorthogonalityofimpliesthatthecolumnsof(resp.,of)formanorthonormalbasisfor(resp.,for).Thevectorsareanorthonormalbasisforthenullspaceof.Thesingularvaluedecompositionofalwaysexists,anditimpliesthatcanbewrittenintheformThetranspose,,ofisthediagonalmatrixwithdiagonalentries.Theproductisthematrixwithdiagonalentries.Thepseudoinverse,),ofisthematrixwithdiagonalentries0if=0.ThepseudoinverseofisequaltoThus, ThedampedleastsquaresmethodisalsoeasytounderstandwiththeSVD.ThematrixisequaltoUDVThematrixisthediagonalmatrixwithdiagonalentriesClearly,isnon-singular,anditsinverseisthematrixwithnon-zeroentries(.Then,VEUisthediagonalmatrixwithdiagonalentriesequalto Thus,thedampedleastsquaressolutioncanbeexpressedintheform Comparisonofequations(13)and(14)makescleartherelationshipbetweenthepseudoinverseanddampedleastsquaresmethods.Inbothis\inverted"byanexpression.Forpseudoinverses,thevalueisjust(setting0=0);whereasfordampedleastsquares,).Thepseudoinversemethodisunstableasapproacheszero;infact,itisexactlyatsingularitiesthat'sareequaltozero.Forvaluesofwhicharelargecomparedto,thedampedleastsquaresmethodisnotverydi®erentfromthepseudoinversesinceforlarge.But,whenisofthesameorderofmagnitudeasorsmaller,thenthevalues)diverge.Indeed,forany0as0.Thus,thedampedleastsquaresmethodtendstoactsimilarlytothepseudoinversemethodfromsingularitiesande®ectivelysmoothsouttheperformanceofpseudoinversemethodintheneighborhoodofsingularities.7SelectivelydampedleastsquaresForthematerialthatusedtobeinthissection,seeBussandKim[7]. Figure1:TheYanddouble-Yshapes.Theende®ectorsareattheendsofthebranches;theredballsindicatethetargetpositions.8ExperimentalresultsandrecommendationsFortherestofthematerialthatusedtobeinthissection,plustheresultsofadditionalexperiments,seeBussandKim[7].TocomparetheIKalgorithms,weimplementedthe\Y"-shapedand\double-Y"shapedmultibodiespicturedin¯gure1.The¯rsthassevenlinkswithtwoende®ectorsandthelatterhas16linkswith4ende®ectors.Weletthetargetpositions(theredballsinthe¯gures)moveinsinusoidallyvaryingcurvesinandoutofreachofthemultibodies.Thetargetpositionsmovedinsmallincrements(justlargeenoughtostilllookvisuallysmooth),andineachtimestepweupdatedthejointanglesonce.Sincejointangleswereupdatedonlyoncepertimestep,theende®ectorstrackedthetargetpositionsonlyapproximately,evenwhenthetargetpositionswerewithinreach.Wevisuallyinspectedthesimulationsforoscillationsandtrackingabilities.Wealsomeasuredtheaccuracyofthetrackingoveraperiodofhundredsofsimulationsteps.TheJacobiantransposehadtheadvantageofbeingfast,butofpoorquality.ItsqualitywaspoorfortheYshapeandextremelypoorforthedouble-Yshape.However,inothersimultations,wehaveseentheJacobiantransposemethodworkwellforasystemwithasingleende®ector.Thepseudoinversemethodworkedverypoorlywheneverthetargetpositionswereoutofreach,andwedonotrecommenditsuseunlessjoint Alloursoftware,includingsourcecode,isavailablefromthewebpage.ShortmovieclipsoftheJacobiantranspose,thepurepseudoinversemethod,theDLSandtheSDLSmethodsarealsoavailablethere. anglesareseverelyclampedwithClampMaxAbs.ThedampedleastsquaresmethodworkedsubstantiallybetterthantheJacobiantransposemethod,althoughitissomewhatslower.Weattemptedtosetthedampingconstantsoastominimizetheaverageerroroftheende®ectors'spositions,butatthepointwheretheerrorwasminimized,therewasalotofoscillationandshaking.Thus,wehadtoraisethedampingconstantuntilunwantedoscillationbecameveryrare(butatthecostofaccuracyintrackingthetargetpositions).WealsoimplementedaversionoftheDLSmethodwhichusestheClampMagmethodtoclampthecomponentsofthevector:thismethodiscalledDLS.TheadvantageoftheDLSmethodisthattheclampingofreducesoscillationandshaking,andthusalowerdampingconstantcanbeused.Thelowerdampingconstantallowsthemultibodytomoreagressivelymovetowardsthetargetpositions.Theruntimesfortwodi®erentmethodsaredescribedinthetablebelow.Runtimesareinmicrosecondsandweremeasuredwithcustomcodeona2.8GHzPentium.TheDLSruntimeisnotreported,butisveryclosetothatofDLS.FortheY-shape,theJacobianmatrixis67,forthedouble-Y,itis1216. 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