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仿形设计之挖掘机设计2020,设计,挖掘机,2020
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KinematicsofaNewStaircaseExcvavtor R. Morales,V.Feliu,A.Gonzalez,andP.Pintado SchoolofIndustrialEngineering,UniversityofCastilla-LaManchaCampusUniversitarios/n13071,CiudadReal,Spain Rafael.Moralesuclm.es 1 Introduction Poweredwheelchairshavebeeninuseformanyyearsnow,anditisunques-tionablethattheygreatlyimprovethemobilityofthehandicapped.Never-theless,architecturalbarriersstillexistinmanycitiesandbuildings,anditisexpensiveandtimeconsuming,ifnotimpossible,toeliminateallofthem.Awheelchairbecomesauselessdevicewhenfacedwithanarchitecturalbarrierand,asaresult,therehavebeenanumberofwheelchairdesignsthatclaimtobeabletoclimbstairs.Theauthorsofthispaperbelievethatmostofthesedesignshaveseveredrawbacksthatimpairtheirwidespreaduse.Draw-backsaregenerallyrelatedtothelackofsafetystemmingfrommechanicallyunstablesituationsduringstaircaseclimbingordescending.Themechanismsdescribedinthispaperhavebeendesignedtoenforcemechanicalstabilitywhilethewheelchairisonthestaircase.Theweightistransferredatalltimestohorizontalsurfaces(thetread),makingitunnecessarytorelyonfrictiontoensuresafety.Ontheotherhand,itwillbeshowninthenextsection,thattheproposedmechanismscanbeeasilymanufacturedusingmostlyoff-the-shelfmechanicalcomponents. Onemayfinddifferentdesignsinthetechnicalliterature.Someofthesedesignsarebasedonseveralwheelsarrangedinarotatinglink1.Thissystemmayworksatisfactorilyforthe“designstaircase”,butproperfunctioningwouldbeimpairedwhendifferenttreadsorstepheightsareencountered. TheiBOT30002isaverycompactdesignthatcanadjusttodiffer-entstepsizes.Whilstthemechanicaldesignisquitesimple,thechairisverysophisticatedsinceitreliesondynamiccontroltomaintaintheuprightposi-tion.Therearemotionphasesduringclimbingordescendwhenthechairisstandingonjusttwowheelswithacommonaxis.Themajordrawbackofthisdesignisthetremendouscostnecessarytomeetreasonablesafetystandards. Thetrackedchairs3have,inouropinion,twoseriousproblemsthatimpairtheirsuccess.Thefirstproblemisthattheweightissupportedbytheverticalcomponentofthecontactforcebetweenthetrackandtheedgeofsteps.Ahighfrictioncoefficientisneededtomaintainequilibrium.Thesecondproblemisthattheentiretrackisforcedtorotatearoundtheedgeofthefirststepwheninitiatingdescend.Sincethereisnosupporttoapplythecorrespondingtorque,themotionmustbeundertakenusingthewheelchairinertia.Thisprocessisbothdifficultanddangerous. 2 Mechanical System Description Thewheelchairdescribedinthispaperisbasedontwodifferentsystems:asystemtoarbitrarilypositionbothaxelswithrespecttotheframetoaccom-modatetheoverallslope(Fig.1),andadevicethatmakestheaxelclimbonestepatatimewhileallowingthewheeltomovearoundthestepeludinginter-feringwithit(Fig.2).Inouropinion,theuseoftwodecoupledmechanismsineachaxleisthekeyfeatureofthemechanicaldesign.Theideamayseemquitesimpleatfirstsight,butitisresponsibleofpermittingmanydifferentclimbingstrategies,makingthedesignextraordinarilyversatile. Inthedesignofthefirstsystem(seeFig.1),boththefrontandtherearaxlesarejoinedtotheframebymeansoffourlinkmechanisms.Eachfourlinkmechanismisdrivenbyanindependentactuator.Thesemechanismsareparallelograms,whichmeansthattheaxledoesntrotatewithrespecttotheframe.Inotherpart,thedesignofthesecondsystem(seeFig.2),eachaxleclimbsthecorrespondingstepbydeployingtherackthatisconnectedtoit.Thisistantamounttosayingthateachaxle“carriesitsownladder”.Allstepsareclimbedwiththesameslope,andtheheightreachedisdeterminedbythestepitself,sincethewheelmovesbacktotheinitialpositionassoonasthestephasbeensurpassed.Thisisachievedbyconnectingthewheeltotheaxle viaafourlinkmechanism.Themechanismmaybedesignedtoguidethewheeltrajectorywithrespecttotheaxle. Thisfourlinkmechanismdesignisquiteversatileandcouldbetunedfordifferentpurposes.Itcouldbeargued,forexample,thatthestabilityconditionisrathersuperfluoussinceitissafertolockthewheelforkinitsrestpositionwheneverneeded. Finally,theprocessdescribedhastobesynchronizedwithaxlepositioning,inordertomaintainverticalityofthechairframe,andinordertogeneratetheappropriatechairframetrajectory.Thissynchronizationispartofthecontrolstrategythatwillbeoutlinedinthenextsection. 3StagesoftheClimbingMotionforthePositioningSystemandSensorialSystem Oncethemechanicalsystemhasbeendescribed,nextwepresentthebasictasksinwhichthestaircaseclimbingprocessisdecomposed: 1 Positioningofthewheelchairwithrespecttothestaircase(previoustotheclimbingprocess).Alignmentbetweentherearwheelsaxisandthefrontofthelowerstep(Fig.3). 2 Climbingoftherearwheelswhilethefrontwheelsremainonthefloor(thefirstfewstepsofthestaircase,Fig.4). 3 Simultaneousclimbingoftherearandfrontwheels(Fig.5). 4 Climbingoffrontwheelswiththerearwheelsremainingontheupperfloor(thelaststepsofthestaircase,Fig.6). Fig. 6. The rear wheels are on the staircase while the front wheels are on the floor Moreinformationaboutstagesoftheclimbingmotionissuppliedin5. Moreoverthetasksdescribedpreviouslymustbecarriedoutinsuchawaythatthecentreofmassofthewheelchairdescribesaspecifiedtrajectory,andtheinclinationisalsoundercontrol,allthisinordertoguaranteethecomfortofthepassenger.Thenarelativelycomplexsensorialsystemisneededtooperatethewheelchair,whichisdetailednext: ? 8 ultrasound sensors (2 per wheel). They are needed to measure the distances of the wheels to the steps. ? 1inclinometerwhichmeasurestheverticalityofthechairanddetectstheinstantatwhichtheclimbingmechanismcontactsthefloorwhendeploying. ? Encodersforthetwojointsofthechairstructure(2),therearwheels(2),andtheracksoftheclimbingmechanism(2). ? 2switchesthatindicatetheinstantatwhichthewheelhasoverpassedthestepbeingclimbed(aspringretractsthewheelwhenitloosescontactwiththestepduringitsraisingprocess,thissuddenmotionbeingdetected). 4KinematicsModelofControl Themostcriticalpartofthewholecontrolsystemisthesubsystemthatgeneratestherealtimetrajectoriesfortheactuatorsofthewheelchair,insuchwaythatthisvehicleshouldbeabletoclimbanddescendstaircaseskeepingthemaximumpossiblecomfortforthepassenger:smoothmotionsandverylittleinclinationsfromthevertical. Theserealtimetrajectoriesarethereferencesfortheclosed-loopsystems(servocontrols)thatcontroltheanglesofthemotors(actuators)inchargeofmovingtheseveraldegreesoffreedomofourwheelchair.Thistrajectorygeneratorreliesonakinematicmodelthatshouldbe:(a)preciseenoughtodescribethebehaviourofthemechanism,(b)simpleenoughtobeabletobecomputedinrealtime,(c)flexibleenoughtoincludedescriptionsofallthetasksmentionedintheprevioussection,whichincludedifferentchairconfigurationsanddifferentsituationsofcontactwiththeenvironment(floorandstaircases). Figure7showsanschemeofthemechanismofthewheelchair,wherethemostimportantvariablesanddegreesoffreedomaredefined.Agenericfloorprofilef(s)isassumed,thatcanbeeasilyparticularizedtothecasesofflatfloorsorstaircases.Inwhatfollowswewillrepresentpointsintheplanebyusingcomplexnotation:horizontalvariableistherealcomponentandverticalvariableistheimaginarycomponent.Thisnotationgreatlyfacilitatesobtainingkinematicmodelsofourwheelchairbecausewehavefoundthatexpressionofrotationsissimplifiedleadingtomorecompactequations. 5InverseKinematicModel Inversekinematicmodelgivesdevalues(angles)oftheactuatorvariableswhichareneededtoachieveadesiredcentreofmassandinclinationofthewheelchair.Whenthewheelchairclimbthestair,itcanpresentinfourpossibleconfigurations.Nextweshowallthepossibleconfigurationsandweshowtheanalyticalexpressionsoftheinversekinematicmodel. 5.1 FloorFlatwithStepsInverseKinematicModel Weknowthecenterofmass(Pg)andtheinclinationofthewheelchair()andtheheighthbetweenthecentreofthewheelsandweobtainthetwojoints1and2andthepositionofthefrontwheels3.ThisconfigurationisshowinFig.8.Theequationsarepresentednext: l4 A3=Pgx X4 l1cos + cos 22 1 l4 2 l22 cos + (1) 2 22 1 3=A3 l32 B23 (2) R B3=Pgy Y4 l1sin + l4 sin 22 1 l4 2 l22 sin + (3) 2 22 1 = + ang B 3 (4) A3 R3 A=Pgx X4 R3 l1cos + l4 cos( +) 22 1 l2 l4 2 2 cos + (5) 2 22 |2 h2 = A l32 B 2 (6) B=Pgy Y4 h l1sin + l4 sin( +) 22 1 l4 2 l22 sin + (7) 2 22 2 = ang B (8)A |2 h2 5.2 WheelchairStartstoRaisingtheStepswithRearRackandFrontWheelisontheFlatInverseKinematicModel Weknowthetwojoints1and2,thecontactpointPCbetweentherearrackandtheflat,andtheinstantaneouslengthoftherearstem(z),andweobtainthecenterofmass(Pg),andtheinclinationofthewheelchair().ThisconfigurationisshowinFig.9.Theequationsarepresentednext: Fig.9.Wheelchairstarttoraisingthestepswithrearrackandfrontwheelinontheflat 1 l4 2 j( P=Pr+jPi=Pg PC+l7 jl6 l22 e2 +) 22 ej( )(9) Pi 2 = + arcsin (10) l3 l4 1 l4 2 A4=Pgx X4 l1cos2 + 2 cos 2 l22 2 cos + (11) 2 z=Pr l3cos( 2 )(12) B4=Pgy Y4 l1sin + 2 l4 1 l4 2 sin l22 sin + (13) 22 22 1 4=A4 l32 B24 (14) R 1 = + ang B 4 (15) A4 R4 5.3 WheelchairStartstoRaisingtheStepswithFrontRackandRearWheelisontheFlatInverseKinematicModel Weknowthecenterofmass(Pg)andtheinclinationofthewheelchair(),theheighthbetweenthecentreofthewheels,andthecontactpointPCbetweenthefrontrackandtheflat,andweobtainthetwojoints1and2,thepositionofthefrontwheels3andtheinstantaneouslengthofthefrontstem(z).ThisconfigurationisshowinFig.10.Theequationsarepresentednext: j( l4 P=Pr+jPi=Pg PC l1e2 +) ej 2 1 l4 2 j( j( ) l2 e2 +) e(16) 2 2 2 z=Pr l3cos(1+ )(17) A=Pgx PCx zcos(+) l1cos + + l4 cos( +) 22 1 l2 l4 2 2 cos + (18) 2 22 Pi 1 = + + arcsin (19) l3 3 =1 PC + zej(+)+ X4 jY4 jh (20) R 2 = ang B 3 (21) A3 R3 5.4 WheelchairStartstoRaisingtheStepswithFrontandRearRackInverseKinematicModel Weknowthecenterofmass(Pg),theinclinationofthewheelchair(),andthecontactpointsPC1andPC2betweenthefrontandrearracksandtheflat,andweobtainthetwojoints1and2andtheinstantaneouslengthsofthefrontandrearstems(z1andz2).ThisconfigurationisshowinFig.11.Theequationsarepresentednext: j( l4 F=Fr+jFi=ej( )Pg PC1 l1e2 +) ej 2 2 1l4 + l2 e (22) 2 2 2 z1=Fr l3cos(1+ )(23) G=Gr+jGi=Pg PC2+l7 jl6+l5 1 l22 l4 2 ej( ej( )(24) 2+) 22 Fi 1 = + arcsin (25) l3 Gi 2 = + arcsin (26) l3 z2=Gr l3cos( 2 )(27) 6 Movement Conditions ThetrajectorydesiredforthepointPgisdesignedbylinkingtwoarcsofcircumferenceandonestraighttrajectory.Also,themovementmustbecom-fortableforthepassenger.Thisconstraintimpliesthatwecantexceedthemaximumaccelerationandthemaximumvelocitydefined,andthatvertical-ityoftheseatmustbemaintained.InFig.12,wecanseetheglobaltrajectory T1 2T1t Fig.13.Profilevelocityinthepartofthecircumferencetrajectory desiredbythewheelchair.AndinFig.13,wecanseetheprofileofvelocitydesired.WeassumethatthevelocityisV=atintheinterval(0,T1),whereaisthetangentialacceleration.Theresultsare: a x(t)=x0+Rsint2 (28) 2R a y(t)=y0+R Rcost2 (29) 2R a Vx(t)=atcost2 (30) 2R Vy(t)=atsin2aRt2 (31) a a2t2 a ax = a cos t2 sin t2 (32) 2RR 2R a a2t2 a ay = a sin t2 + cos t2 (33) 2RR 2R IfweassumethatthevelocityofPgint=T1isVmax,theconditionsoftangencialaccelerationa,curvatureradiusR,andtimeT1toachieveVmaxare: a = amax (34) 1+(2)2 V 2 R = max (35) 2 a 2R T1 = (36) a Intheanotherpartsofthetrajectory,themovementismadewithamax. 7 Simulation Results Thisisoneexampleoftheclimbingstairmovementofthewheelchair.Wesupposeonestairwitheightsteps.Thedimensionsofthestepsare150mm(height)250mm(width).Thetrajectorymustbecomfortableforthehand-icapped,thismeansthataccelerationsandvelocitiesmustbeshorterthanthemaximumaccelerationofcomfortandmaximumvelocityofcomfort,andtheinclinationofthewheelchairmustbenull(=0).Theseconditionsin-volvethatourmovementwillbecomposedoftwomovements(oneofthemtoacceleratethewheelchairandtheothertodeccelerateit).Theresultsofthesimulationare: Inthissimulation,wehaveobtainedoneimportantresult:60%offullthetimeneededtoclimbthestaircaseisadeadtime.Oneofthefutureworkwillbethedesignofnewtrajectoriesthatminimizetheexistenceofdeadtimes. 8 Conclusions Anewstaircaseclimbingwheelchairhasbeendesignedwhosemaincharac-teristicisbeingespeciallystable.Moreover,itsadditionaldegreesoffre
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