RaibertVertical Hopper Video 6.1 Segment 6.0.1 …. S. Knabe, V. Orekhov, M. A. Hopkins, B. Y....
Transcript of RaibertVertical Hopper Video 6.1 Segment 6.0.1 …. S. Knabe, V. Orekhov, M. A. Hopkins, B. Y....
Robo4x 6.0.1 1
edX Robo4MiniMS– LocomotionEngineering
Block1– Week3– Unit1Raibert VerticalHopper
Video6.1
Segment6.0.1Motivation&Agenda
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
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Robo4x 6.0.1 2
Agenda• Historical
§ corecomponentofRaibert’sHoppers
§ inspiredmuchfurtherwork• Conceptual
§ 1DoF vertical“heartbeat”§ organizingrhythmaround
whichotherDoF arecoordinated
• Technical§ firstlookathybriddynamics§ centralfeatureoflocomotion
M.H.Raibert.LeggedRobotsThatBalance.Cambridge:MITPress,1986
https://www.youtube.com/watch?v=XFXj81mvInc
M.Buehler,D.E.Koditschek,andP.J.Kindlmann,“Afamilyofrobotcontrolstrategiesforintermittentdynamicalenvironments,”IEEEControlSystemsMagazine,vol.10,pp.16–22,1990.
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SagittalPlaneMachine• Raibert reducedcomplexity
§ initially,ofcontroltask§ bymechanicalrestriction§ to2dim.workspace§ yields3DoF configurationspace
• ControlIdeasGeneralize§ tospatialmachinein3dim§ with6DoF§ andbeyond figuresfrom:
M.H.Raibert.LeggedRobotsThatBalance.Cambridge:MITPress,1986
videofrom:PlasticPals.“RobotsfromMIT'sLegLab”.YouTubevideo,02:28.Posted[October31,2011].https://www.youtube.com/watch?v=XFXj81mvInc
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• Problemsetting§ 3DoF (RPR) kinematics§ 6dimdynamics
• highlynonlinear(and“hybrid”)• tightlycoupled• underactuated
• Raibert’s controlpolicies§ use(mostly)“proportional-derivative”feedbackloops§ assumeeachDoF isdecoupledfromothers§ “time-share”theactuationresources
• airvalveadjustmentoflegspringinstanceforheightcontrol• hipactuatorcontrolofpitchinstance• hipactuatorselectionoflegangleinflightforfore-aftvelocity
figuresfrom:M.H.Raibert.LeggedRobotsThatBalance.Cambridge:MITPress,1986
RobustModularRobotController
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ControlofHybridDynamics• leggedlocomotionresultsfrom
§ groundreactionforces(GRF)§ impartedtothebodymasscenter§ throughthelimbs
• limbsmake&breakgroundcontact§ introducingimpulses§ changingtheGRFdynamics§ yieldhybriddynamics
• continuous-in-timeforce-driven• discrete-in-timeevent-driven
• controlofhybriddynamics§ byactuationthroughlimbs§ mustbecoordinatedin
continuoustime§ andsensitivetothelimb-contact
events
M.H.Raibert,“LeggedRobots,”Commun.ACM,vol.29,no.6,pp.499–514,Jun.1986.
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Robo4x 6.0.1 6
OtherMachines“Anchor”VerticalHopper• Verticalhopper
§ Canbeusedasa“template”
§ tobe“anchored”§ asthe“heartbeat”
• Inmorecomplexsystems§ 2DoF§ 3DoF§ …andbeyond
Raibert,MITPress,1986,op.cit.
Buehler,etal.IEEEControlSystemsMagazine,1990, op.cit.
A.Rizzi,L.Whitcomb,andD.Koditschek,“Real-TimeControlofaSpatialRobotJuggler,”IEEEComputer,vol.25,no.5,pp.12–24,1992.
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InitialApproachtoVerticalHopper• ModelContinuoustime
flows§ eachmode ofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicalevent interrupts
mode§ locomotion:typically
LO/TD• Study/Expressmodemap• Modelreset map• Compose
§ modemapo resetmap§ furthercomposeeach
compositioninturn• Endupwithreturnmap
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
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edX Robo4MiniMS– LocomotionEngineering
Week6– Unit1Raibert VerticalHopper
Video6.2
Segment6.1.1ContinuousTimeModels
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 6.0.1 9
InitialApproachtoVerticalHopper• ModelContinuoustime
flows§ eachmode ofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicalevent interrupts
mode§ locomotion:typically
LO/TD• Study/Expressmodemap• Modelresetmap• Compose
§ modemapo resetmap§ furthercomposeeach
compositioninturn• Endupwithreturnmap
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
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GravityLoadedHooke’sLawSpring• ReworkSeg.1.2.1.1-eqn(4)
• NotedifferencesfromFDHO§ usingvariablec todenoteverticalextension
• ratherthanby asinpendulumorx:=c - cr asinDHO• tofocusonroleofrelaxedspringpositioncr >0
§ additionofgravitationalforce,FG:=- mg
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AdditionalModelingAssumptions• Simplifyphysics
§ unitmass, m:=1;gravity-matchedspring,m cr = g§ touchdown/liftoff
• springisdisengaged(braked)belowrestlength:ctd = clo =0 <cr• toassuresomespringpotentialineachstance• and(withgravity-matchedspring)zeropotentialforceatc =0
• Yieldsfamiliar1st orderVF(fDHO fromSeg.1.2.1.2)
§ withnewparameters,w,b >0replacingphysicalonesinfDHO§ introducedtoassureunder-dampedsystem
• Butnowworkingwithhybriddynamics§ thisVFonlymodels“passive”stancemodec <0 § actuation,introducednext,yieldsnew“active”mode§ introduceformalguardandresetmapsinnextsegment
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ActiveStanceMode:EnergyInjection• Raibert’s originalhopperhadan“air-spring”
§ pneumaticpumpadjustedpressureinair-chamber§ controlleraddedfixedenergy(viapressure)§ at“bottom”:createsnew“active”modewithownVF
• Presentmodeluses“series-elastic”actuator(SEA)§ forcecontrolviaspringrest-lengthadjustment§ popularizedbyPratt;stillactivelydeveloped,e.g.
• https://www.youtube.com/watch?v=wJwKwLUUTjcC.S.Knabe,V.Orekhov,M.A.Hopkins,B.Y.Lattimer,andD.W.Hong,“Twoconfigurationsofserieselasticactuatorsforlinearlyactuatedhumanoidrobotswithlargerangeofmotion,”inHumanoidRobots(Humanoids),201414thIEEE-RASInternationalConferenceon,2014,pp.1096–1096
§ presentuse:introduceconstantupwardthrustforce• initiatedat“bottom”asbefore(controlledtoperformpositivework)• restoresenergylosttodampingeachsuccessivestance
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ActiveStanceModeControl• SEAforcecontrol:an“inversestatics”controlpolicy
§ assumeactuatorcanmovespringrestlength,cr• exactprecisioninpositionandtiming• knowninversestaticsforgravity-loadedHooke’sspring
§ givenadesiredreferenceforceprofile,• actuatorrestlengthcommand
• injectsexactlytherightforceattherighttime
• Getnewdynamics
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ConstantThrustControlPolicy• Stancecontrolstrategy
§ holdrestlengthfixedatnominal• untilcompressionendsat“bottom”• definedbyvanishingofguard function
§ atwhichpointinjectconstantupwardthrustforce
§ forbrieffixedtimeintervalafterbottom
• YieldsdampedconstantthrustVF
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BallisticFlightMode
• Decompressioncontinuesafterendofthrust§ restlengthagainfixedatnominal
• untildecompressionendsat“liftoff”• definedbyvanishingofguard function
§ atwhichpointgroundreactionforcescease• onlyforceisduetogravity• yieldsflightmodeVF
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• ModelContinuoustimeflows§ eachmode ofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicalevent interrupts
mode§ locomotion:typically
LO/TD§ hadtointroduceextra
mode• decompression• modelastimedevent• tosimplifyanalysis
• Nextup:§ study/expressmodemaps§ modelthereset maps
MovingAhead
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
Thrust
Decomp
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edX Robo4MiniMS– LocomotionEngineering
Week6– Unit1Raibert VerticalHopper
Video6.3
Segment6.1.2ContinuousTimeModeMaps
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 6.0.1 18
InitialApproachtoVerticalHopper• ModelContinuoustime
flows§ eachmodeofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicaleventinterrupts
mode§ locomotion:typically
LO/TD• Study/Expressmode map• Modelreset map• Compose
§ modemapo resetmap§ furthercomposeeach
compositioninturn• Endupwithreturnmap
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
Thrust
DecompressionMap
FlightMap
CompressionMap
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Robo4x 6.0.1 19
BallisticFlightModeFlowMapDefinition• Exercisesforthissegmentshow
• FlightendsatTouchdown§ which,recall,occurswhenc = ctd :=0 § representedbythevanishingofguard
• Nowsolvethevanishingconditionfort
• togetimplicitfunction,
• yieldingmodeflowmap
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GravityLoadedHooke’sLawSpringFlow• RecallfromSeg. 1.2.1.2 &exercisesRCcoordinates
• Reveals“oscillatory”(underdamped)GLHflow
• Simplifystillfurthervianonlinear(“polar”)CCD
(don’tconfusewithSeg.1.2.2.1vector!)• Exercises:decoupledconjugateVFwithsimpleflow
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CompressionFlowMapDefinition• Compressionstartsattouchdown,• Compressionendsat“bottom”
§ which,recall,occurswhen§ representedbythevanishingofguardpreferablyexpressedinpolarRCcoordinates
• Solvevanishingconditionfort inpolarRCcoords
• Yieldsmodeflowmap
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DampedConstantThrustModeFlow
• Raibert’s strategyofconstantthrustfrombottom• Resultsinfamiliar2nd orderODE• Yields1st orderVF
• WithfamiliarflowderivedfromfD inSeg.1.2.1.1• Resultingin
D.E.KoditschekandM.Bühler.Analysisofasimplifiedhoppingrobot.TheInternationalJournalofRoboticsResearch,10(6):587–605,Dec1991
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InitialApproachtoVerticalHopper• ModelContinuoustime
flows§ eachmodeofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicaleventinterrupts
mode§ locomotion:typically
LO/TD• Study/Expressmode map• Modelreset map• Compose
§ modemapo resetmap§ furthercomposeeach
compositioninturn• Endupwithreturnmap
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
Thrust
DecompressionMap
FlightMap
CompressionMap
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• ModelContinuoustimeflows§ eachmode ofcontact§ governedbydifferentVF
• Modelnaturalguardconditions§ physicalevent interrupts
mode§ locomotion:typically
LO/TD§ hadtointroduceextra
mode• decompression• modelastimedevent• tosimplifyanalysis
• Nextup:§ study/expressmodemaps§ modelthereset maps
MovingAhead
figuremodifiedfromRaibert,Commun.ACM,1986,op.cit.
Thrust
Decomp
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