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A Nonlinear Time Invariant Mechanical System Video 3.1 ...€¦ · Herbert Goldstein. Classical...
Transcript of A Nonlinear Time Invariant Mechanical System Video 3.1 ...€¦ · Herbert Goldstein. Classical...
Robo4x 1.2.2.1.a 1
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit2ANonlinearTimeInvariantMechanicalSystem
Video3.1
Segment1.2.2.1.aRevolute1DoF Physics
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
June,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 2
TheSimplePenduluminGravity
• 1DoF revolutekinematics§ deceptivelyfamiliar§ archetypalnonlinearsystem§ nonlinearspace &dynamics
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Robo4x 1.2.2.1.a 3
OneDoF RevoluteKinematics
• ReviewRobo3Week1§ adoptunitvectorconvention§ bodypositionentries§ denoteunitvector
components• 1DoF revolutekinematics
§ amapfromangles,q§ intothexy-planeb§ lyingonradiusl circle
• Needinfinitesimalkinematicstoo
e1e2
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Robo4x 1.2.2.1.a 4
OneDoF RevoluteEnergies
• Gravitationalpotentialenergy
• Kineticenergy
• Introduceangletangentvector
• Lagrangian
• Totalenergy
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 5
EquationsofMotion:LosslessCase• ReviewRobo3
Week2§ Lagrangian
mechanics• generatedbyEuler-Lagrangeoperator
• appliedtoLagrangianfunction
• balancedbyexternalforces
§ workedoutforourexample• nodampingfornow
• addinshortly• Reference:
HerbertGoldstein.Classicalmechanics.Addison-WesleyWorldStudentSeries,Reading,Mass.:Addison-Wesley,1950
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Robo4x 1.2.2.1.a 6
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit2ANonlnear TimeInvariantMechanicalSystem
Video3.2
Segment1.2.2.1.bRevolute1DoF Physics
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
June,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 7
LosslessOneDoF RevoluteVectorField
• Onceagain,wantVF§ rewrite2nd orderscalar
ODE§ as1st ordervectorODE
• NonlinearVF!§ nomatrices§ noclosedformseries
• Roleoftheory:§ guaranteesflowexists§ givesrigorousmethods
• qualitative(nothingclosedform)reasoning
• formallyguaranteedproperties
Comparetoeqns (2)&(6)insegment1.2.1.2• there:
- closedform(infiniteseries)expressionof- atime-paramtrized family- ofinvertiblenonlinearfunctions
• here:- guaranteedexistenceof- atime-paramtrized family- ofinvertiblenonlinearfunctions
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Robo4x 1.2.2.1.a 8
LosslessMechanismConservesEnergy• Examinepower
§ energychangealongtrajectory• totalenergy(5)• VF(7)
§ asfunctionofstate• Conclude:
§ nopowerdissipation§ energyisconserved
• Nodamping§ webettergetthat
result§ …alsogetforDHO
• whenb:=0• power(Seg.1.2.1.3-eqn1)vanishes
ComparewithSeg.1.2.1.3powercomputation
• justusedcalculus&VF• didn’treallyneedclosedformflow
expression
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Robo4x 1.2.2.1.a 9
ExampleofQualitativeMethod• Energyisconserved
§ hencecansolveconstrainteqn§ e.g.forvelocity§ asafunctionofangle§ andICenergy
• Can’t“get”trajectory§ flowhasnoclosedform
expression§ insteadget“orbit”(geometriccurveonwhichtrajectorymustlie)
q&
q
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Robo4x 1.2.2.1.a 10
Movingahead
• Suchgeometrictechniques• Willbeveryimportant• Althoughusuallymorecomplicated• Asstatespacedimensiongrows
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 11
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit2ANonlnear TimeInvariantMechanicalSystem
Video3.3
Segment1.2.2.2.aStable&UnstableFixedPoints
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 12
SteadyStateNotion
• “steadystate”:acolloquialterm§ intuitively:apersistentoperating
condition§ whatisa“condition”?§ whatis“persistent”?
• thisunit:formalizesimplestcase§ fixedpoint (FP)
• ICwhosesolutionsareconstant
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Robo4x 1.2.2.1.a 13
VFViewofSteadyStateNotion
• thisunit:formalizesimplestcase§ fixedpoint (FP)
• ICwhosesolutionsareconstant• ImplieszerooftheVF
§ notice“condition”means“orbit”(soon:moreinterestingorbits)
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Robo4x 1.2.2.1.a 14
Stability:FirstPassat“Persistence”Notion• Persistenceinstate
§ ifweperturbIC
§ willweremainin“same”operatingcondition?
• FormalizeforFPcondition§ e.g.,1dimVF
§ fromSeg.1.2.1.1
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Robo4x 1.2.2.1.a 15
Stability:FirstPass“Persistence”Example• Persistenceinstate
§ ifweperturbIC§ willweremainin“same”operatingcondition?
• e.g.,1dimVFfromSeg.1.2.1.1§ checkflow
§ imposeFPassumption
§ torealize:“same”(ornear)impliesl nonpositive
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Robo4x 1.2.2.1.a 16
NextExample:DampedHarmonicOscillatorThis image cannot currently be displayed.
• RecallfromSeg.1.2.1.2§ 2nd orderscalarODE(twoscalartimetrajectories)§ 1st ordervectorODE(planartrajectorylieson“orbit”)
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Robo4x 1.2.2.1.a 17
RecallDampedHarmonicOscillatorVF• DHO(Seg.1.2.1.2)
§ insteadofphysicalcoordinates
§ userealcanonicalcords
§ writeoutconjugatematrixrepresentation
§ assumeeithers orwnonzero
§ concludeFPonlyatorigin
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Robo4x 1.2.2.1.a 18
StabilityofDampedHarmonicOscillator• formalizeforFPcondition
§ perturborbitthroughICatorigin
§ closedformperturbedtrajectory
§ easytocomputemagnitude§ conclude:stabilityimplies nonpositive s
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Robo4x 1.2.2.1.a 19
CapturingPlanarStabilityviaNorm• Seg.1.2.1.3exercises:
§ totalenergyisanorm
§ inrealcanonicalcoordinates• nowcomputepower
§ alsoanorm PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 20
AScalarViewofVectorFlowStability• poweralsoanorm
• soenergysatisfiesscalarODE§ stabilityfromscalarLTIODE§ inheritedbyvectorflow
• neighborhoodsof0inR• defineneighborhoodsof0inR2
§ againconclude:stabilityimpliess nonpositive
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Robo4x 1.2.2.1.a 21
edX Robo4MiniMS– LocomotionEngineering
Block1– Week2– Unit2ANonlnear TimeInvariantMechanicalSystem
Video3.4
Segment1.2.2.2.bStable&UnstableFixedPoints
DanielE.Koditschekwith
Wei-HsiChen,T.TurnerToppingandVasileiosVasilopoulosUniversityofPennsylvania
July,2017
PropertyofPennEngineeringandDanielE.Koditschek
Robo4x 1.2.2.1.a 22
DampedPendulum:Stable&UnstableFP
• LookatPendulumagain§ thistimewith
damping§ previously
• “top”IC• unstableFP
§ damping• stabilizesbottom• “top”FPstillunstable
This image cannot currently be displayed.
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Robo4x 1.2.2.1.a 23
MultipleFP:DifferentStabilityProperties• DPVF(Seg.1.2.2.1)
• DPPower(Exr.1.2.2.1)
• NLsystems§ canhavemultipleFP§ pendulum:
• “bottom”&“top”
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Robo4x 1.2.2.1.a 24
PredictDifferentStabilityProperties?
• Showedpowerisnonpositive
• ButFPmayormaynotbestable• Totalenergy
§ norm-likeatbottom
§ “saddle”attop
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Robo4x 1.2.2.1.a 25
SimulationsSuggestWhat’sHappening• totalenergyat“bottom”
§ levelcurvesencloseneighborhoods(norm-like)
§ stableFP§ withbasinofattraction
• totalenergyat“top”§ levelcurvesrunoff§ unstableFP§ nobasin
DD
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Robo4x 1.2.2.1.a 26
NeedaQualitativeTheory• DampedHarmonicOscillator§ LTI:closedformsolutions§ totalenergy:norm-like
• anexplicitnorm• inthe“right”coordinates• yieldsscalarLTIenergyODE
• DampedPendulum§ NLTI:noclosedform
solutions§ totalenergy:sometimes
norm-like• whatisthe“norm-like”property?
• howtogetrigorousconclusionswithoutascalarODE?
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Robo4x 1.2.2.1.a 27
MovingAhead• Linearization
§ locally(e.g.nearFP)§ oftencanuseTaylorapproximationofNLVF§ togetTaylorapproximationofNLflow
• Lyapunov Functions§ locally(e.g.nearFP)§ stabilitypropertiesformallycharacterized§ bygeneralized“totalenergy”
• Beyondthiscourse§ notionof“global”Lyapunov function§ gives“fundamentaltheoremofdynamicalsystems”
PropertyofPennEngineeringandDanielE.Koditschek