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


Robo4x 1.2.2.1.a 3

OneDoF RevoluteKinematics

• ReviewRobo3Week1§ adoptunitvectorconvention§ bodypositionentries§ denoteunitvector

components• 1DoF revolutekinematics

§ amapfromangles,q§ intothexy-planeb§ lyingonradiusl circle

• Needinfinitesimalkinematicstoo

e1e2


Robo4x 1.2.2.1.a 4

OneDoF RevoluteEnergies

• Gravitationalpotentialenergy

• Kineticenergy

• Introduceangletangentvector

• Lagrangian

• Totalenergy


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


Robo4x 1.2.2.1.a 6


Block1– Week2– Unit2ANonlnear TimeInvariantMechanicalSystem

Video3.2

Segment1.2.2.1.bRevolute1DoF Physics



June,2017


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


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


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


Robo4x 1.2.2.1.a 10

Movingahead

• Suchgeometrictechniques• Willbeveryimportant• Althoughusuallymorecomplicated• Asstatespacedimensiongrows


Robo4x 1.2.2.1.a 11



Video3.3

Segment1.2.2.2.aStable&UnstableFixedPoints



July,2017


Robo4x 1.2.2.1.a 12

SteadyStateNotion

• “steadystate”:acolloquialterm§ intuitively:apersistentoperating

condition§ whatisa“condition”?§ whatis“persistent”?

• thisunit:formalizesimplestcase§ fixedpoint (FP)

• ICwhosesolutionsareconstant


Robo4x 1.2.2.1.a 13

VFViewofSteadyStateNotion

• thisunit:formalizesimplestcase§ fixedpoint (FP)

• ICwhosesolutionsareconstant• ImplieszerooftheVF

§ notice“condition”means“orbit”(soon:moreinterestingorbits)


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


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


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”)


Robo4x 1.2.2.1.a 17

RecallDampedHarmonicOscillatorVF• DHO(Seg.1.2.1.2)

§ insteadofphysicalcoordinates

§ userealcanonicalcords

§ writeoutconjugatematrixrepresentation

§ assumeeithers orwnonzero

§ concludeFPonlyatorigin


Robo4x 1.2.2.1.a 18

StabilityofDampedHarmonicOscillator• formalizeforFPcondition

§ perturborbitthroughICatorigin

§ closedformperturbedtrajectory

§ easytocomputemagnitude§ conclude:stabilityimplies nonpositive s


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


Robo4x 1.2.2.1.a 21



Video3.4

Segment1.2.2.2.bStable&UnstableFixedPoints



July,2017


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.


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”


Robo4x 1.2.2.1.a 24

PredictDifferentStabilityProperties?

• Showedpowerisnonpositive

• ButFPmayormaynotbestable• Totalenergy

§ norm-likeatbottom

§ “saddle”attop


Robo4x 1.2.2.1.a 25

SimulationsSuggestWhat’sHappening• totalenergyat“bottom”

§ levelcurvesencloseneighborhoods(norm-like)

§ stableFP§ withbasinofattraction

• totalenergyat“top”§ levelcurvesrunoff§ unstableFP§ nobasin

DD


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?


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”


A Nonlinear Time Invariant Mechanical System Video 3.1 ...€¦ · Herbert Goldstein. Classical...

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