RaibertVertical Hopper Video 6.1 Segment 6.0.1 …. S. Knabe, V. Orekhov, M. A. Hopkins, B. Y. Lattimer, and D. W. Hong, “Two configurations of series elastic actuators for linearly

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

PropertyofPennEngineeringandDanielE.Koditschek

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.


Robo4x 6.0.1 3

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


Robo4x 6.0.1 4

• 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


Robo4x 6.0.1 5

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.


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.


Robo4x 6.0.1 7

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.


Robo4x 6.0.1 8


Week6– Unit1Raibert VerticalHopper

Video6.2

Segment6.1.1ContinuousTimeModels



July,2017


Robo4x 6.0.1 9


flows§ eachmode ofcontact§ governedbydifferentVF



LO/TD• Study/Expressmodemap• Modelresetmap• Compose





Robo4x 6.0.1 10

GravityLoadedHooke’sLawSpring• ReworkSeg.1.2.1.1-eqn(4)

• NotedifferencesfromFDHO§ usingvariablec todenoteverticalextension

• ratherthanby asinpendulumorx:=c - cr asinDHO• tofocusonroleofrelaxedspringpositioncr >0

§ additionofgravitationalforce,FG:=- mg


Robo4x 6.0.1 11

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


Robo4x 6.0.1 12

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


Robo4x 6.0.1 13

ActiveStanceModeControl• SEAforcecontrol:an“inversestatics”controlpolicy

§ assumeactuatorcanmovespringrestlength,cr• exactprecisioninpositionandtiming• knowninversestaticsforgravity-loadedHooke’sspring

§ givenadesiredreferenceforceprofile,• actuatorrestlengthcommand

• injectsexactlytherightforceattherighttime

• Getnewdynamics


Robo4x 6.0.1 14

ConstantThrustControlPolicy• Stancecontrolstrategy

§ holdrestlengthfixedatnominal• untilcompressionendsat“bottom”• definedbyvanishingofguard function

§ atwhichpointinjectconstantupwardthrustforce

§ forbrieffixedtimeintervalafterbottom

• YieldsdampedconstantthrustVF


Robo4x 6.0.1 15

BallisticFlightMode

• Decompressioncontinuesafterendofthrust§ restlengthagainfixedatnominal

• untildecompressionendsat“liftoff”• definedbyvanishingofguard function

§ atwhichpointgroundreactionforcescease• onlyforceisduetogravity• yieldsflightmodeVF


Robo4x 6.0.1 16

• ModelContinuoustimeflows§ eachmode ofcontact§ governedbydifferentVF



LO/TD§ hadtointroduceextra

mode• decompression• modelastimedevent• tosimplifyanalysis

• Nextup:§ study/expressmodemaps§ modelthereset maps

MovingAhead


Thrust

Decomp


Robo4x 6.0.1 17


Week6– Unit1Raibert VerticalHopper

Video6.3

Segment6.1.2ContinuousTimeModeMaps



July,2017


Robo4x 6.0.1 18


flows§ eachmodeofcontact§ governedbydifferentVF

• Modelnaturalguardconditions§ physicaleventinterrupts


LO/TD• Study/Expressmode map• Modelreset map• Compose




Thrust

DecompressionMap

FlightMap

CompressionMap


Robo4x 6.0.1 19

BallisticFlightModeFlowMapDefinition• Exercisesforthissegmentshow

• FlightendsatTouchdown§ which,recall,occurswhenc = ctd :=0 § representedbythevanishingofguard

• Nowsolvethevanishingconditionfort

• togetimplicitfunction,

• yieldingmodeflowmap


Robo4x 6.0.1 20

GravityLoadedHooke’sLawSpringFlow• RecallfromSeg. 1.2.1.2 &exercisesRCcoordinates

• Reveals“oscillatory”(underdamped)GLHflow

• Simplifystillfurthervianonlinear(“polar”)CCD

(don’tconfusewithSeg.1.2.2.1vector!)• Exercises:decoupledconjugateVFwithsimpleflow


Robo4x 6.0.1 21

CompressionFlowMapDefinition• Compressionstartsattouchdown,• Compressionendsat“bottom”

§ which,recall,occurswhen§ representedbythevanishingofguardpreferablyexpressedinpolarRCcoordinates

• Solvevanishingconditionfort inpolarRCcoords

• Yieldsmodeflowmap


Robo4x 6.0.1 22

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


Robo4x 6.0.1 23


flows§ eachmodeofcontact§ governedbydifferentVF

• Modelnaturalguardconditions§ physicaleventinterrupts


LO/TD• Study/Expressmode map• Modelreset map• Compose




Thrust

DecompressionMap

FlightMap

CompressionMap


Robo4x 6.0.1 24

• ModelContinuoustimeflows§ eachmode ofcontact§ governedbydifferentVF



LO/TD§ hadtointroduceextra

mode• decompression• modelastimedevent• tosimplifyanalysis

• Nextup:§ study/expressmodemaps§ modelthereset maps

MovingAhead


Thrust

Decomp


Documents

RaibertVertical Hopper Video 6.1 Segment 6.0.1 …. S. Knabe, V. Orekhov, M. A. Hopkins, B. Y. Lattimer, and D. W. Hong, “Two configurations of series elastic actuators for linearly