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Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing an Impact Collision


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LYAPUNOV-BASEDCONTROLOFAROBOTANDMASS-SPRINGSYSTEM UNDERGOINGANIMPACTCOLLISION By CHIEN-HAOLIANG ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2007

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2007 Chien-Hao Liang

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TomywifeYen-ChenandsonHsu-Chenwhoconstantly lledmewithloveand joy.

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ACKNOWLEDGMENTS Iwouldliketoexpressmygratitudetomyadvisor,mentor,andfriend,Dr. WarrenE.Dixon,forintroducingmetotheinteresting eldofLyapunov-based control.Asanadvisor,heprovidedthenecessaryguidanceandallowedmetotry somestupidideasduringmyresearch.Asamentor,hehelpedmeunderstandthe highpressureofworkinginaprofessionalenvironmentandwaswillingtogiveme timetolearnandadjust.Ifeelfortunateingettingtheopportunitytoworkwith him. IalsoappreciatemycommitteemembersDr.CarlD.CraneIIIandDr.Gloria J.Wiensforthetimeandhelptheyprovided. Iwouldliketothankallmyfriendsfortheirsupportandencouragement.I especiallythankmyfriendGuoqiangHuforbeingmylistenerandmentorboth intheresearchanddailylifeforthesetwoyears.Iwouldalsoliketothankmy colleaguesKeithStegathandGunLeeforhelpingmeoutonthosedi cultdays whenIwasdoingmyexperiments.Withoutthem,Icannotaccomplishthese experiments.IwouldliketothankmycolleagueDarrenAikenforhiscaringduring my rstyearinU.S.IexpressmygratitudetoKeithDupreeforhelpingmeout onmyresearch.IwouldliketothankSiddharthaMehtaforhelpingmeinthesis editing.IalsoexpressmygratitudetoParagPatreandWillMackuniswho lled mydailylifewithjoy. FinallyIwouldliketothankmyparentsfortheirunconditionalloveand support,mywifeYen-Chenforthesupportandjoyshealwaysbroughttome,and mysonHsu-Chenforhishug,kiss,andsmile.Theirlove,understanding,patience andpersonalsacri cemadethisthesispossible. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv LISTOFFIGURES.... ............................viiABSTRACT.............. ......................ix CHAPTER 1INTRODUCTION.. ............................1 1.1Introduction ...............................1 1.2DynamicModel.............................8 2LYAPUNOV-BASEDCONTROLOFAROBOTANDMASS-SPRING SYSTEMUNDERGOINGANIMPACTCOLLISION...........11 2.1ErrorSystemandControlDevelopment................11 2.1.1ControlObjective.. ......................11 2.1.2Closed-LoopErrorSystem...................12 2.2StabilityAnalysis............................14 2.3ExperimentalResults..........................20 2.4ConcludingRemarks..........................26 3ADAPTIVELYAPUNOV-BASEDCONTROLOFAROBOTANDMASSSPRINGSYSTEMUNDERGOINGANIMPACTCOLLISION.....27 3.1PropertiesandAssumptions......................28 3.2ErrorSystemandControlDevelopment................29 3.2.1ControlObjective.. ......................29 3.2.2Closed-LoopErrorSystem...................30 3.3StabilityAnalysis............................34 3.4ExperimentalResults..........................40 3.5ConcludingRemarks..........................47 4ANIMPACTFORCELIMITINGADAPTIVECONTROLLERFORA ROBOTICSYSTEMUNDERGOINGANONCONTACT-TO-CONTACT TRANSITION.......... ......................48 4.1Properties.... ............................49 4.2ErrorSystemandControlDevelopment................49 4.2.1ControlObjective.. ......................50 v

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4.2.2Closed-LoopErrorSystem...................51 4.3StabilityAnalysis............................52 4.4ExperimentalResults..........................58 4.5ConcludingRemarks..........................64 5CONCLUSIONANDRECOMMENDATIONS...............65 APPENDIX ATHEEXPRESSIONOF 1INSECTION2.1..............67 BTHEEXPRESSIONOF 1INSECTION3.2..............69 REFERENCES....... ............................70 BIOGRAPHICALSKETCH............................74 vi

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LISTOFFIGURES Figure page 1TheMass-SpringRobot(MSR)systemisanacademicexampleofan impactbetweentwodynamicsystems....................7 21Topviewofexperimentaltestbed.......................21 22Sideviewofexperimentaltestbed......................21 2Spring-massandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatestheposition erroroftherobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicates thepositionerrorofthespring-mass(i.e., ( ) )..............24 2Computedcontroltorques ( ) ( ) forthe(a)basemotorand(b)secondlinkmotor..... ............................24 25Appliedcontroltorques ( ) ( ) (solidline)versuscomputedcontrol torques(dashedline)forthe(a)basemotorand(b)secondlinkmotor..25 2Computeddesiredrobottrajectory, 1( ) .................25 2Contact( =1 )andnoncontact( =0 )conditionsfortherobotand mass-springsystem.. ................. ............26 3Mass-springandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatestheposition erroroftherobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicates thepositionerrorofthemass-spring(i.e., ( ) )..............42 3Mass-springandroboterrors ( ) duringtheinitialtwoseconds......43 3Computedcontroltorques ( ) ( ) forthe(a)basemotorand(b)secondlinkmotor..... ............................43 34Appliedcontroltorques ( ) ( ) (solidline)versuscomputedcontrol torques(dashedline)forthe(a)basemotorand(b)secondlinkmotor..44 3Computeddesiredrobottrajectory, 1( ) .................44 36Unitlessparameterestimate ( ) introducedin(313)..........45 vii

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37Estimatefortheunknownconstantparametervector ( ) .(a) 10( )= ,(b) 4( )= ,(c) 1( )=1 ,and(d) 7( )=2 ,where 1, 2 R denotethemassofthe rstandsecondlinkoftherobot, R denotesthemassofthemotorconnectedtothesecondlinkof therobot,and R denotesthemassofthemass-springsystem....45 38Estimatefortheunknownconstantparametervector ( ) .(a) 5( )= ,(b) 2( )=1 ,and(c) 8( )=2 ................46 39Estimatefortheunknownconstantparametervector ( ) .(a) 6( )= ,(b) 3( )= 1,and(c) 9( )= 2...................46 4Mass-springandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatestheposition erroroftherobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicates thepositionerrorofthemass-spring(i.e., ( ) )..............60 42Appliedcontroltorques ( ) ( ) forthe(a)basemotorand(b)second linkmotor....... ............................60 43Appliedcontroltorques ( ) ( ) forthe(a)basemotorand(b)second linkmotorduringthe rst0.8second... .................61 4Computeddesiredrobottrajectory, 1( ) .................61 45Parameterestimate ( ) introducedin(313)...............62 46Estimatefortheunknownconstantparametervector ( ) .(a) 10( )= ,(b) 4( )= ,(c) 1( )=1 ,and(d) 7( )=2 ,where 1, 2 R denotethemassofthe rstandsecondlinkoftherobot, R denotesthemassofthemotorconnectedtothesecondlinkof therobot,and R denotesthemassofthemass-springsystem....62 47Estimatefortheunknownconstantparametervector ( ) .(a) 5( )= ,(b) 2( )=1 ,and(c) 8( )=2 ................63 48Estimatefortheunknownconstantparametervector ( ) .(a) 6( )= ,(b) 3( )= 1,and(c) 9( )= 2...................63 viii

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Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-SPRING SYSTEM UNDERGOING AN IMPACT COLLISION By Chien-Hao Liang May 2007 Chair: Warren E. Dixon Major: Mechanical Engineering The problem of controlling a robot during a noncontact-to-contact transition has been a historically challenging problem that is practically motivated by applications that require a robotic system to interact with the environment. If the c ontact dynamics are not properly modeled and controlled, the cont act forces could result in poor system performance and instabilities. One difficulty in controlling systems subject to noncontactto-contact transition is that the dynamics are different when the system status changes suddenly from the noncontact stat e to a contact state. Anothe r difficulty is measuring the contact force, which can depend on the geom etry of the robot, the geometry of the environment, and the type of contact. The a ppeal of systems with contact conditions is that short-duration effects such as high st resses, rapid dissipation of energy, and fast acceleration and deceleration may be achieved from low-energy sources. Over the last 2 decades, many res earchers have investigated the modeling and control of contact systems. Most controllers ta rget contacts with a static environment for ix

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a fully actuated system. Many researchers also exploit switching or discontinuous controllers to accommodate contact conditions. Motivation exists to explore alternative control strategies because impacts between the robot and the static environment cannot represent all the impact system applications such as the capture of disabled satellites, spaceport docking, manipulation of non-rigi d bodies, and so on, and discontinuous controllers require infinite control frequency (i.e., exhibit chattering ) or yield degraded stability results (i.e., uniformly ultimately bounded). As stated previously, it is necessary to consider the impact control between two dynamic systems. Our study considered a class of fu lly actuated dynamic systems that undergo an impact collision with another dynamic system th at is unactuated. Our st udy is specifically focused on a planar robot colliding with a ma ss-spring system as an academic example of a broader class of such systems. The cont rol objective is defined as the desire to command a planar robot to collide with an unactuated system and regulate the resulting coupled mass-spring robot (MSR) system to a desired compressed state. The collision is modeled as a differentiable impact. Lyapunov-based methods are used to develop a continuous controller that yi elds asymptotic regulation of the mass and robot links. A desired time-varying robot link trajectory is designed that accounts for the impact dynamics and the resulting coupled dynamics of the MSR system. A force controller is then designed to ensure that the robot li nk position tracking erro r is regulated. Unlike some other results in literature, the force co ntroller is continuous and does not depend on measuring the impact force or the measuremen t of other acceleration terms. Experimental results are provided to illustrate the successf ul performance of the controller in each chapter. x

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CHAPTER1 INTRODUCTION 1.1Introduction Theproblemofcontrollingarobotduringanoncontact-to-contacttransition hasbeenahistoricallychallengingproblemthatispracticallymotivatedby applicationsthatrequirearoboticsystemtointeractwiththeenvironment.Ifthe contactdynamicsarenotproperlymodeledandcontrolled,thecontactforcescould resultinpoorsystemperformanceandinstabilities.Onedi cultyincontrolling systemssubjecttononcontact-to-cont acttransitionisthatthedynamicsare di erentwhenthesystemstatuschangess uddenlyfromthenoncontactstatetoa contactstate.Anotherdi cultyismeasuringthecontactforce,whichcandepend onthegeometryoftherobot,thegeometryoftheenvironment,andthetypeof contact.AsstatedbyTornambe[1],theappealofsystemswithcontactconditions isthatshort-duratione ectssuchashighstresses,rapiddissipationofenergy,and fastaccelerationanddecelerationmaybeachievedfromlow-energysources. Overthelast2decades,manyresearchershaveinvestigatedthemodeling andcontrolofcontactsystemsincluding:[1]-[32].2trendsareapparentaftera comprehensivesurveyofcontactsystemsi ncontrolliterature.Mostcontrollers targetcontactswithastaticenvironmentforafullyactuatedsystem.Many researchersalsoexploitswitchingordiscontinuouscontrollerstoaccommodate forthecontactconditions.Theregulationofaone-linkrobotthatundergoes smoothornon-smoothimpactdynamicswasexaminedbyTornambe[1].Aclass ofswitchingcontrollerswereexaminedbyBrogliatoetal.in[2]formechanical systemswithdi erentiabledynamicssubjecttoanalgebraicinequalitycondition andanimpactrulerelatingtheinteractionimpulseandthevelocity.Theanalysis 1

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2 in[2]usedadiscreteLyapunovfunctionthatrequiredtheuseoftheDiniderivative toexaminethestabilityofthesystem.Asimplemechanicalsystemsubjectto nonsmoothimpactsisconsideredbyMeniniandTornambe[3],wherethedesired time-varyingplanarmotionofamassiscontrolledwithinaclosedregionde ned byanin nitelymassiveandrigidcircularbarrier.In[4],Sekhavatetal.utilized LaSallesInvariantSetTheoremtoprovethestabilityofadiscontinuouscontroller thatisdesignedtoregulatetheimpactofahydraulicactuatorwithastatic environmentwherenoknowledgeoftheimpactdynamicsisrequired.Volpeand Khosladevelopedanonlinearimpactcontrolstrategyforarobotmanipulator experiencinganimpactwithastaticenvironment[5].Thecontrollerin[5]was basedontheconceptthatnegativeproportionalforcegains,orimpedancemass ratioslessthanunity,canprovideimpactresponsewithoutbouncing.Tornambe [6]alsoproposedaswitchingcontrollertogloballyasymptoticallyregulatedatwo degree-of-freedom(DOF)planarmanipulatortocontactanin nitelyrigidand massivesurface.PagillaandYu[7]proposedadiscontinuousstabletransition controllertodealwiththetransitionfromanoncontacttoacontactstatewhere explicitknowledgeoftheimpactmodelisnotrequired.AdiscontinuousmodelbasedadaptivecontrollerwasproposedbyAkellaetal.[8]toasymptotically stabilizethecontacttransitionbetweenarobotandstaticenvironment.Tarnet al.[9]proposedasensor-referencedcontrolmethodusingpositiveacceleration feedbackwithaswitchingcontrolstrategyforimpactcontrolforarobotanda constrainedsurface.AswitchingcontrollerwasalsoproposedbyWuetal.in [10]toeliminatethebouncingphenomenaassociatedwitharobotimpactinga staticsurface.Thestructureoftheswitchingcontrollerin[10]wasdependenton impactfeedbackfromaforcesensor.Leeetal.developedahybridbang-bang impedance/time-delaycontrollerthatestablishesastableinteractionbetweena robotwithnonlinearjointfrictionandasti environmentin[11]and[12].Nelson

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3 etal.[13]-[15]proposedanonlinearcontrolstrategythatconsidersforceandvision feedbacksimultaneouslyandthenswitchestopureforcecontrolwhenitisunable toaccuratelyresolvethelocationoftherobotsend-e ectorrelativetothesurface tobecontact.Motivationexiststoexplorealternativecontrolstrategyforthe impactsystemsbecauseimpactsbetweentherobotandthestaticenvironment cannotrepresentalltheimpactsystemapplicationssuchasthecaptureofdisabled satellites,spaceportdocking,manipulationofnon-rigidbodies,andsoon,and discontinuouscontrollersrequirein nitecontrolfrequency(i.e.,exhibitchattering) oryielddegradedstabilityresults(i.e.,uniformlyultimatelybounded).Asstated previously,itisnecessarytoconsidertheimpactcontrolbetweentwodynamic systems. Severalcontrollershavebeendevelopedforunder-actuateddynamicsystems thathaveanimpactcollision.Forexample,afamilyofdead-beatfeedbackcontrol lawswereproposedbyBrogliatoandRio[16]tocontrolaclassofjugglinglikesystems.Oneofthecontributionsin[16]isastudyoftheintermediate controllabilitypropertiesoftheobjectsimpactPoincarmapping.Aproportionalderivative(PD)controllerwasdevelopedbyIndriandTornambe[17]toaddress globalasymptoticstabilizationofunder -actuatedmechanicalsystemssubjectto smoothimpactswithastaticobject.Inourpreviousworkin[18],anonlinear energy-basedcontrollerisdevelopedtogloballyasymptoticallystabilizeadynamic systemsubjecttoimpactwithadeformablestaticmass.Thecontributionin[18]is thattheunder-actuatedstatesarecoupledthroughtheenergyofthesystemasa meanstomitigatethetransientresponseoftheunactuatedstates. Ourstudyconsideredaclassoffullyactuateddynamicsystemsthatundergo animpactcollisionwithanotherdynam icsystemthatisunactuated.Ourstudy isspeci callyfocusedonaplanarrobotcollidingwithamass-springsystemas anacademicexampleofabroaderclassofsuchsystems.Thecontrolobjectiveis

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4 de nedasthedesiretocommandaplanarrobottocollidewithanunactuated systemandregulatetheresultingcoupledmass-springrobot(MSR)systemto adesiredcompressedstate.Thecollisionismodeledasadi erentiableimpact asinrecentworkin[1],[17],andourpreviouse ortsin[18]-[20].Lyapunovbasedmethodsareusedtodevelopacontinuousadaptivecontrollerthatyields asymptoticregulationofthemassandrobotlinks.Adesiredtime-varyingrobot linktrajectoryisdesignedthataccountsfortheimpactdynamicsandtheresulting coupleddynamicsoftheMSRsystem.Thedesiredlinktrajectoryconvergestoa setpointthatequalsthedesiredmasspositionplusanadditionalconstantthatis duetothedeformationofthemass.Aforcecontrolleristhendesignedtoensure thattherobotlinkpositiontrackingerrori sregulated.Unlikesomeotherresults inliterature,thecontinuousforcecontrollerdoesnotdependonmeasuringthe impactforceorthemeasurementofotheraccelerationterms:onlytheposition andvelocitytermsofthespring-masssystemandthejointanglesandtheangular velocitiestermsoftheplanarroboticarmareneededfortheproposedcontroller. Chapter2andourpreliminarye ortsin[19]providea rststepatcontrolling theproposedimpactsystem.ThecontroldevelopmentinChapter2isbasedonthe assumptionofexactmodelknowledgeofthesystemdynamics.ThenonlinearcontinuousLyapunov-basedcontrollerisproventoregulatethestatesofaplanarrobot collidingwiththeunactuatedmass-springsystemandyieldsglobalasymptotic result. InChapter3andourpreliminaryresultsin[20],thedynamicmodelfor thesystemisassumedtohaveuncertainparameters.Thecontrolobjectiveis de nedasthedesiretocommandtheplanarrobottocollidewiththemass-spring systemandregulatetheresultingcoupledmass-springrobot(MSR)systemtoa desiredcompressedstatewhilecompensatingfortheconstant,unknownsystem parameters.Twolinearparameterizati onsaredesignedtoadaptfortheunknown

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5 robotandmass-springparameters.AnadaptivenonlinearcontinuousLyapunovbasedcontrollerisproventoregulatethestatesofthesystemsandyieldsglobal asymptoticregulationresult. WhenthecontrollersinChapter2andChapter3wereimplementedinthe presenceoflargeinitialconditions,violentimpactsbetweentherobotandthe mass-springsystemresulted.Infact,thecontrollerwasarti ciallysaturated(the saturatione ectswerenotconsideredinthestabilityanalysis)toreducethe impactforcessothatthemassde ectionwouldnotdestroyacapacitanceprobe. Variousresearchershaveinvestigatedmethodsthatprescribe,reduce,orcontrol theinteractionforcesduringoraftertherobotimpactwiththeenvironmentsuch as[6]-[15]and[21]-[34]becauselargeinteractionforcescandamageboththerobot and/ortheenvironmentorleadtodegradedperformanceorinstabilities.Walker andGertzetal.exploitedkinematicredundancyofthemanipulatortoreducethe impactforcein[21]-[23].Atwodegree-of -freedom(DOF)planarmanipulatorwas globallyasymptoticallyregulatedtocontactanin nitelyrigidandmassivesurface byTornambe[6]wheretheimpactforcewasestimatedusingareduced-order observer.PagillaandYu[7]proposedastabletransitioncontrollertodealwith thetransitionfromanoncontacttoacontactstatewhichcanimprovetransition performanceandforceregulation.Adisco ntinuousmodel-basedadaptivecontroller wasproposedbyAkellaetal.[8]toasymptoticallystabilizethecontacttransition betweenarobotandstaticenvironment.Thecontrollersforeachcontactstatewere tunedindependentlytoreducecontactforceduringtheprocessofmakingcontact withtheenvironment.Tarnetal.[9],[10]proposedasensor-referencedcontrol methodusingpositiveaccelerationfeedbackwithaswitchingcontrolstrategyfor impactcontrolandforceregulationforarobotandaconstrainedsurfacewherethe peakimpulsiveforceandbouncingcauseddirectlybyovershootingandoscillation ofthetransientforceresponsecanbereduced.Leeetal.developedahybrid

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6 bang-bangimpedance/time-delaycontrollerthatestablishesastablecontactand achievesthedesireddynamicsforcontactornoncontactconditionsin[11]and[12], wheretheforceovershootscanbeminimized.Nelsonetal.[13]-[15]proposeda switchingnonlinearcontrollerthatcombinesforceandvisioncontrol.Whenthe robotsend-e ectorapproachesthetarget,thecontrollerswitchestoforcecontrol tominimizeimpactforceandtoregulatethecontactforce.Bymodelingthe impactdynamicsasastatedependentjumplinearsystem,ChiuandLeewereable toapplyamodi edstochasticmaximumprincipleforstatedependentjumplinear systemstooptimizetheapproachvelocity,theforcetransientduringimpactand thesteadystateforceerroraftercontactisestablished[24].HydeandCutkosky [25]proposedanapproach,basedoninput commandshaping,tosuppressvibration duringthecontacttransitionofswitchingcontrollersbymodifyingfeedforward information.Variousotherapplicationsalsofocusedonthereductionofimpact forcebetweendi erentsystemsduringthecontrolprocesssuchasimpactforce reductionofhoppingrobotconsideredbyShibataandNatori[26]andOhnishiet al.[27],[28],bilateralteleroboticsystemconsideredbyDubeyetal.[29],humanrobotsymbioticenvironmentconsideredbyYamadaetal.[30]andLietal.[31], andspacemanipulatorandfreeyingtargetconsideredbyHuangetal.[32]. Exploringalternativemethodsismotivatedbecausekinematicredundancyisnot alwayspossible,andagainthediscontinuouscontrollersrequirein nitecontrol frequency(i.e.,exhibitchattering)oryielddegradedstabilityresults(i.e.,uniformly ultimatelybounded). TheseresultsprovidethemotivationforthecontroldevelopmentinChapter 4.Speci cally,thefeedbackelementsforthecontinuouscontrollerinChapter 3arecontainedinsideofhyperbolictangentfunctionsasameanstolimitthe impactforcesresultingfromlargeinitialconditionsastherobottransitionsfrom noncontacttocontactstates.Althoughsaturatingthefeedbackerrorisanintuitive

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7 Figure1:TheMass-SpringRobot(MSR)systemisanacademicexampleofan impactbetweentwodynamicsystems. solutionthathasbeenproposedinpreviousliteratureforothertypesofrobotic systemswithlimitedactuation,severalnewtechnicalchallangesariseduetothe impactcondition.Themainchallangeist hattheuseofsaturatedfeedbackdoes notallowsomecouplingtermstobecanceledinthestabilityanalysis,resultingin theneedtodevelopstatedependentupperboundsthatreducethestabilitytoa semi-globalresult(ascomparedtotheglobalresultsinChapter2andChapter3). Thesemi-globalresultisproblematicinthecurrentapplicativecontextbecause certaincontroltermsdonotappearintheclosed-looperrorsystemduringthe noncontactcondition,resultinginauniformlyultimatelyboundedresultuntilthe robotmakescontact.Hence,theresulthingesonnewdevelopmentwithinthesemiglobalstabilityproofforanerrorsystemthatisonlyuniformlyultimatelybounded duringthenoncontactphase.Thisproblemisexacerbatedbythefactthatthe Lyapunovfunctioncontainsradiallyunboundedhyperbolicfunctionsofsomestates thatonlyappearinsideofsaturatedhyperbolictermsintheLyapunovderivative. Newcontroldevelopment,closed-looperrorsystems,andLyapunov-basedstability analysisarguementsareusedtoconcludetheresult.Experimentalresultsare providedtoillustratethesuccessfulperformanceofthecontrollerineachchapter.

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8 1.2DynamicModel Thesubsequentdevelopmentismotivatedbytheacademicproblemillustrated inFig.1.Thedynamicmodelforthetwo-linkrevoluterobotdepictedinFig. 11canbeexpressedinthejoint-spaceas ( ) + ( ) + ( )= (11) where ( ) ( ) ( ) R2representtheangularposition,velocity,andacceleration oftherobotlinks,respectively, ( ) R2 2representstheuncertaininertia matrix, ( ) R2 2representstheuncertaincentripetal-Coriolise ects, ( ) [ 1( ) 2( )] R2representsuncertainconservativeforces(e.g.,gravity),and ( ) R2representsthetorquecontrolinputs.TheEuclideanpositionoftheendpointofthesecondrobotlinkisdenotedby ( ) [ 1( ) 2( )] R2,which canberelatedtothejoint-spacethroug hthefollowingkinematicrelationship: = ( ) (12) where ( ) R2 2denotesthemanipulatorJacobian.Theunforceddynamicsof themass-springsysteminFig.11are + ( 0)=0 (13) where ( ) ( ) ( ) R representthedisplacement,velocityandacceleration ofthemass R 0 R representstheinitialundisturbedpositionofthemass, and R representsthesti nessofthespring. Assumption1.1: Weassumethat 1( ) and ( ) canbeboundedas 1( ) ( ) (14) where R isaknownconstantthatisdeterm inedbytheminimumcoordinate oftherobotalongthe 1-axis,and R isaknownpositiveconstant.The

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9 lowerboundassumptionfor 1( ) isbasedonthegeometryoftherobot,andthe upperboundassumptionfor ( ) isbasedonthephysicalfactthatthemassis attachedbythespringtosomeobject,andthemasswillnotbeabletomovepast thatobject. Inthefollowing,thecontactmodelisconsideredasanelasticcontactwith nitesti ness.Animpactbetweenthesecondlinkoftherobotandthespring-mass systemoccurswhen 1( ) ( ) (seeFig.1).Theimpactwillyieldequaland oppositeforcereactionsbetweentherobotandmass-springsystem.Speci cally, theimpactforceactingonthemass,representedby ( ) R ,isassumedto havethefollowingform[1],[17] = ( 1 ) (15) where R representsapositivesti nessconstant,and ( ) R isde ned as = 1 1 0 1 (16) Theimpactforceactingontherobotlinksproducesatorque,denotedby ( ) R2,asfollows: = ( 1 ) 1sin( 1)+ 2sin( 2+ 1) 2sin( 2+ 1) (17) where 12 R denotetherobotlinklengths.Base don(1),(1),and(1)-(1 7),thedynamicmodelfortheMSRsystemcanbeexpressedas ( ) + ( ) + ( ) = + ( 0)= (18)

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10 AfterpremultiplyingtherobotdynamicsbytheinverseoftheJacobian transposeandutilizing(12),thedynamicsin(1)canberewrittenas[19],[20] ( ) + ( ) + ( )+ 0 = (19) + ( 0)= (1) where ( ) ( ) ( ) R2denotesthemanipulatorforce.Thedynamicmodel in(19)and(10)exhibitsthefollowingpropertiesthatwillbeutilizedinthe subsequentanalysis. Property1.1: Theinertiamatrix ( ) issymmetric,positivede nite,and canbelowerandupperboundedas 1|| ||2 2|| ||2 R2(1) where 12 R arepositiveconstants. Property1.2: Thefollowingskew-symmetricrelationshipissatis ed ( 1 2 ( ) ( )) =0 R2 (1)

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CHAPTER2 LYAPUNOV-BASEDCONTROLOFAROBOTANDMASS-SPRINGSYSTEM UNDERGOINGANIMPACTCOLLISION Thischapterandourpreliminarye ortsin[19]providea rststepatcontrollingtheproposedimpactsysteminSection1.2.Theacademicexampleofa planarrobotcollidingwithanunactuatedmass-springsystemisusedtorepresent abroaderclassofsuchsystems.Thecontroldevelopmentinthischapterisbased ontheassumptionofexactmodelknowledgeofthesystemdynamics.Thecontrol objectiveistocommandarobottocollidewithanunactuatedmass-springsystemandregulatethespring-masstoadesiredcompressedstate.Lyapunov-based methodsareusedtodevelopacontinuouscontrollerthatyieldsglobalasymptoticregulationofthespring-massandrobotlinks.Unlikesomeotherresultsin literature,thedevelopedcontinuousforcecontrollerdoesnotdependonsensing theimpact,measuringtheimpactforce,orthemeasurementofotheracceleration terms.Experimentalresultsareprovidedtovalidateouranalysis. Thischapterisorganizedasfollows.Section2.1describestheerrorsystemand controldevelopmentfollowedbythestabilityanalysisinSection2.2.Section2.3 describestheexperimentalsetupandresultsthatindicatethesuccessfulperformanceobtainedbyimplementingtheproposedcontrollerfollowedbyconclusionin Section2.4. 2.1ErrorSystemandControlDevelopment 2.1.1ControlObjective Onegoalinthischapteristoregulatethestatesofadynamicsystem(i.e.,a two-linkplanarrobot)thathasanimpactcollisionwithanotherdynamicsystem (i.e.,amass-spring).Aregulationerror,denotedby ( ) R3,isde nedto 11

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12 quantifythecontrolobjectiveas (21) where ( ) [ 1 2] R2and ( ) R denotetheregulationerrorforthe robotandmass-springsystem,respectively,andarede nedas , (22) In(2), R denotestheconstantknowndesiredpositionofthespring-mass. Thedesiredpositionoftheend-pointofthesecondrobotlink,denotedby ( ) [ 1 2] R2,isselectedsothattherobotwillproducethedesiredspringmasspositionwhileaccountingfortheimpactdynamics.Speci cally, 1( ) (i.e., thedesiredhorizontalEuclideancoordinateinFig.11)isatime-varyingsignal thatissubsequentlydesignedtoaccountfortheimpactconditionandthecoupled dynamicresponseoftheMSRsystem,and 2(i.e.,thedesiredverticalEuclidean coordinateinFig.11)isselectedasaconstant.Filteredtrackingerrors,denoted by ( ) R2and ( ) R ,arede nedas + + (23) tofacilitatethesubsequentcontroldesignandstabilityanalysiswhere R isa positivecontrolparameter. 2.1.2Closed-LoopErrorSystem Bytakingthetimederivativeof ( ) andutilizing(2)and(2),the followingopen-looperrorsystemcanbeobtained: = ( 0) ( 1 )+ (24) wherethespring-massdynamicsin(110)weresubstitutedfor ( ) .Motivated todesignthedesiredrobotlinktrajectorytopositionthespring-mass,(2)isused

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13 torewritetheopen-loopsystemin(24)as = ( 0)+ + 1+ 1 (25) Basedon(25),thedesiredrobotlinkpositionisdesignedas 1, 1 ( + ( 0))+ 1 ( 1+ 2) + (26) 2, where R isanappropriatepositiveconstant(i.e., isselectedsotherobotwill impactthemass-springsystem),and 1and 2 R arepositiveconstantcontrol gains.Aftersubstituting(26)into(25),theclosed-looperrorsystemfor ( ) canbeobtainedas =(1 )( ( 0)+ )+ 1 ( 1+ 2) (27) As ( ) ,(2)and(2)canbeusedtoconcludethat ( ) 0 ( ) 0 ,and ( ) 0 .Hence,(26)canbeusedtoconcludethat 1( ) ( 0)+ (28) Thephysicalmeaningof(2)isthatthedesiredrobotpositionvariesintimeto accountfortheimpactdynamicsandthecoupleddynamicsystem,andthedesired steady-statevalueisaconstantthatequalsthedesiredspring-masspositionplus themassdeformation. Aftertakingthetimederivativeof ( ) andpremultiplyingbytherobot inertiamatrix,thefollowingopen-looperrorsystemisobtained: = + + + ( 1 ) 0 + (29)

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14 where(110)and(22)wereutilized.Basedon(29)andthesubsequentstability analysis,therobotforcecontrolinputisdesignedas + + + ( 1 ) 0 + 3+ + + (2) where 3 R isapositiveconstantcontrolgain.Basedontheuseofthebacksteppingmethod,therobotforcecontrolinputin(20)requiresthe rstandsecond derivativeof ( ) .AsdescribedintheAppendixA,the rstandsecondderivative of ( ) exists(i.e., ( ) iscontinuouslydi erentiable)anddonotdependon accelerationterms.Theclosed-looperrorsystemfor ( ) canbeobtainedafter substituting(2)into(2)as = 3 (2) 2.2StabilityAnalysis Theorem :Thecontrollergivenby(20)ensuresglobalasymptoticstabilityof therobotandspring-massregulationerrorsinthesensethat | ( ) | 0 k ( ) k 0 (2) provided 1 and 2areselectedsu cientlylargeandthefollowinggainconditionissatis ed: 2 4 2(2) Inthefollowingproof,aLyapunovfunctionanditsderivativeareprovided. Theanalysisisthenseparatedintotwocases:contactandnoncontact.Forthe noncontactcase,thestabilityanalysisindicatesthecontrolleranderrorsignalsare boundedandconvergetoanarbitrarilysmallregion.Additionalanalysisindicates thatwithinthisregion,contactmustoccur.Whencontactoccurs,aLyapunov

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15 analysisisprovidedthatillustratestheMSRsystemasymptoticallyconvergesto thedesiredsetpoint. Proof: Let ( ) R denotethefollowingcontinuouslydi erentiable, nonnegative,radiallyunboundedfunction(i.e.aLyapunovfunctioncandidate) 1 2 2 + 1 2 + 1 2 (2) TheLyapunovfunctioncandidatein(214)canbelowerandupperboundedas 1k k2 2k k2(2) where 12 R arepositiveconstants,and ( ) R5isde nedas ( ) Thetimederivativeof(214)canbedeterminedas = ((1 )( ( 0)+ ))+ 1(2) ( 1+ 2) 3 where(112),(23),(2), and(2)wereutilized.Theremainderoftheanalysis isdividedintotwocases:Case1-therobotandmass-springsystemsarenotin contact(i.e., 1and =0 ),andCase2-therobotandmass-springsystems areincontact(i.e., 1 and =1 ). Case1a: Beforetheinitialcontact,themass-springsystemisatrestandthe springisnotcompressed;hence, ( 0)=0 =0 (2) Basedon(217),(216)canbeexpressedas = 3 (2)

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16 Theexpressionsin(214)and(218)canbeusedtoprovethat ( ) ( ) ( ) Landthat ( ) ( ) L2.Basedonthefactthat ( ) ( ) ( ) L,standardsignalchasingargumentscanbeusedtoprovethat ( ) ( ) ( ) ( ) Lalongwithalloftheotherclosed-loopsignals.Since ( ) ( ) L L2andareuniformlycontinuous,BarbalatsLemmacanbeappliedto concludethat k ( ) k 0 k ( ) k 0 (2) Theresultin(2)canbeusedalongwith(2)toconcludethat 1( ) 1 ( 1+ 2) + 0leadingtoanimpactwiththemass-springsystem. Case1b: Afteranimpact,therobotmayloosecontactwiththespring-mass. Inthiscase ( 0) 6=0 = (2) Basedon(220),(216)canbeexpressedas = ( ( 0)+ + ) 3 (2) Forthiscase,theinitialvelocityofthespring-massisdenotedby R andthe initialpositionisdenotedby R .Giventheaforementionedinitialconditions, thesolutionfor ( ) canbeobtainedfrom(1)as ( )= 0+( 0)cos( r )+ r sin( r ) (2) Thetimederivativeof(222)canbeexpressedas ( )= ( 0) r sin( r )+ cos( r ) (2)

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17 Basedon(2)and(2), ( ) canbeexpressedasfollows: =( 0) r sin( r ) cos( r )+ ( ) (2) Theexpressionsin(22)-(224)canbeupperboundedas | | | 0| + | 0| + | | r 1(2) | | | 0| r + | | 2(2) | | | 0| r + | | + ( + 1) 3(2) where 1, 2, 3 R denotepositiveconstants.Afterutilizing(225)-(27),an upperboundfor(2)canbedevelopedas + (2) where R isapositiveconstant,and R isde nedas min( 3 1) 2 Standardtechniquescanbeusedtosolve(228)for ( ) as ( ) (0) exp( )+ (2) Basedon(2),itisclearthatduringthetransientcase(Case1b)that ( ) ( ) ( ) L.Basedonthefactthat ( ) ( ) ( ) L,standardsignal chasingargumentscanbeusedtoprovethat ( ) ( ) ( ) ( ) Lalong withalloftheotherclosed-loopsignals. As ( ) 0 ,eventually .Thepreviousdevelopmentcanbeusedto concludethatforthenoncontactcase andhence, k ( ) k k ( ) k | ( ) | as (2)

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18 Furtheranalysisisrequiredtoprovethatthemanipulatormakescontactwith themass-springsystem.Contactbetweenthemanipulatorandthemass-spring systemoccurswhen 1( ) ( ) .Basedon(2),asu cientconditionfor contactcanbedevelopedas 1 + (2) Afterusing(26),thesu cientconditionin(21)canbeexpressedas 1 ( + ( 0))+ 1 ( 1+ 2) (2) Byusing(23)andperformingsomealgebraicmanipulation,theinequalityin (22)canbeexpressedas ( 1+ 2) + | | + 2 | | + ( + | | + 0) (2) FromAssumption1.1, ( ) canbeupperboundedas (2) where R denotesaknownpositiveconstant.If ( ) 0 ,thenthesu cient conditionin(2)maynotbesatis ed.Theconditionthat ( ) 0 will onlyoccurifanimpactcollisionoccursthatcausesthemasstoovershootthe desiredposition.However,evenifanimpactoccursandthemassovershoots thedesiredposition,thedynamicsofthemass-springsystemwillforcethemass positionerrortoreturntotheinitialconditioninthenoncontactcase.Thatis, ( ) 0 where R denotesaknownpositiveconstant.Basedon (2),(2)andthefactthat ( ) willeventuallybelowerboundedby ina noncontactcondition, ( ) canbeupperandlowerboundedas 3 | 0| r | | + (2)

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19 where R denotesaknownpositiveconstant,providedthefollowinggain conditionissatis ed | 0| r + | | (2) Basedon(2),(2),(2),andthepreviousdevelopment,thecontrol parameter 1and 2canbeselectedaccordingtothefollowingsu cientcondition toensuretherobotandmass-springsystemmakecontact 1+ 2 + + 2 + ( + + 0) (2) Case2: Whentherobotandmass-springsystemsareincontact,(26)can beupperboundedas 1k k2 3k k2 k k2+[ k kk k 2k k2] (2) Aftercompletingthesquaresonthebracketedterms,thefollowingexpressionis obtained: 1k k2 3k k2 2 4 2 k k2 (2) Providedthegainconditiongivenin(2)issatis ed,(2)canbeusedto upperbound(2)as 3 (2) where 3 R isapositiveconstant.Theexpressionin(20)indicatesthatwhile incontact,therobotandspring-masspositionerrorsareexponentiallyregulated, and ( ) ( ) ( ) L.Basedonthefactthat ( ) ( ) ( ) L, standardsignalchasingargumentscanbeusedtoprovethat ( ) ( ) ( ) ( ) Lalongwithalloftheotherclosed-loopsignals.

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20 2.3ExperimentalResults ThetestbeddepictedinFig.2andFig.2wasdevelopedforexperimental demonstrationoftheproposedcontroller.Thetestbediscomposedofamassspringsystemandatwo-linkrobot.Thebodyofthemass-springsystemincludes aU-shapedaluminumplate(item(8)inFig.2)mountedonanundercarriage withporouscarbonairbearingswhichen ablestheundercarriagetoglideonan aircushionoveraglasscoveredaluminumrail.Asteelcorespring(item(1)in Fig.2)connectstheundercarriagetoa naluminumframe,andalinearvariable displacementtransducer(LVDT)(item(2)inFig.21)isusedtomeasurethe positionoftheundercarriageassembly.Theimpactsurfaceconsistsofanaluminum plateconnectedtotheundercarriageassemblythroughasti springmechanism (item(7)inFig.21).Acapacitanceprobe(item(3)inFig.21)isusedto measurethede ectionofthesti springmechanism.Thetwo-linkrobot(items (4-6)inFig.2)ismadeoftwoaluminumlinks,mountedon 240 0 Nm(base link)and 20 0 Nm(secondlink)direct-driveswitchedreluctancemotors.The motorsarecontrolledthroughpowerelectronicsoperatingintorquecontrolmode. Themotorresolversproviderotorpositionmeasurementswitharesolutionof 614400pulses/revolution,andastandardbackwardsdi erencealgorithmisused tonumericallydeterminevelocityfromtheencoderreadings.APentium2.8GHz PCoperatingunderQNXhoststhecontrolalgorithm,whichwasimplemented viaQmotor3.0,agraphicaluser-interface,tofacilitatereal-timegraphing,data logging,andtheabilitytoadjustcontrolgainswithoutrecompilingtheprogram (forfurtherinformationonQmotor3.0,thereaderisreferredto[40]).Data acquisitionandcontrolimplementationwereperformedatafrequencyof2.0kHz usingtheServoToGoI/Oboard.

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21 Figure2:Topviewofexperimentaltestbed. Figure2:Sideviewofexperimentaltestbed

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22 Theparametersforthedynamicmodelin(1)and(1)havethefollowing values: 1=7 10[ ] 2=1 11[ ] =13 6[ ] =10 0[ ] 1=0 37[ ] 2=0 17[ ] =2000[ ] =1 8 106[ ] where 1, 2 R representthemassofthe rstandsecondlink, R isthemassofthecart,and R isthemassofthemotorthatdrivesthe secondlink.Thecontrolgains and 3,de nedasscalarsin(2)and(210), wereimplemented(withnonconsequentialimplicationstothestabilityresult)as diagonalgainmatricestoprovidemore exibilityintheexperiment.Speci cally, thecontrolgainswereselectedas 1=1 2=10000 3= { 250 12 } = { 95 100 90 } Theinitialconditionsfortherobotcoordinatesandthespring-masspositionwere (in[m]) 1(0) 2(0) (0) = 0 0160 4870 203 Theinitialvelocityoftherobotandspring-masswerezero,andthedesiredspringmasspositionwas(in[mm]) =233 Thatis,thetipofthesecondlinkoftherobotwasinitially 217 mmfromthe desiredsetpointand 187 mmfromimpactalongthe 1-axis(seeFig.21). Therefore,oncetheinitialimpactoccurs,therobotisrequiredtodepressthe spring(item(1)inFig.21)tomovethemass 30 mmalongthe 1-axis.

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23 Themass-springandroboterrors(i.e., ( ) )areshowninFig.23.Thepeak steady-statepositionerroroftherobottipalongthe 1-axis(i.e., | 1| )andalong the 2-axis(i.e., | 2| )are 9 6 mand 92 m,respectively.Thepeaksteady-state positionerrorofthespring-mass(i.e., | | )is 7 7 m.The 92 misduetothelack oftheabilityofthemodeltocapturefrictionandslippinge ectsonthecontact surface.Inthisexperiment,asigni cantfrictione ectispresentalongthe 2-axis betweentherobottipandthecontactsurfaceduetoalargenormalspringforce thatisappliedalongthe 1-axis. Theinputcontroltorques(i.e., ( ) ( ) )areshowninFig.2andFig.2. Toconstraintheimpactforcetoalevelthatensuredthealuminumplatedidnot extothepointofcontactwiththecapacitanceprobe,thecomputedtorquesare arti ciallysaturated.Fig.24depictsthecomputedtorques,andFig.2depicts theactualtorques(solidline)alongwiththecomputedtorques(dashedline).The resultingdesiredtrajectoryalongthe 1-axis(i.e., 1( ) )isdepictedinFig.26, andthedesiredtrajectoryalongthe 2-axiswaschosenas 2=368 mm.Fig. 2depictsthevalueof ( ) thatindicatescontact( =1 )andnoncontact ( =0 )conditionsfortherobotandmass-springsystem.Avideooftheexperimentisprovidedathttp://www.mae.u .edu/~dixon/projects/ robman/impact.htm.

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24 0 0.5 1 1.5 2 2.5 3 3.5 4 0 500 (a)[mm] 0 0.5 1 1.5 2 2.5 3 3.5 4 0 200 (b)[mm] 0 0.5 1 1.5 2 2.5 3 3.5 4 0 20 40 (c) [sec][mm]Figure2:Spring-massandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatesthepositionerrorof therobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicatesthepositionerrorof thespring-mass(i.e., ( ) ). 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1000 (a)[Nm] 0 0.5 1 1.5 2 2.5 3 3.5 4 0 50 (b) [sec][Nm]Figure2:Computedcontroltorques ( ) ( ) forthe(a)basemotorand(b) secondlinkmotor.

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25 0 0.2 0.4 0.6 0.8 1 1.2 0 100 200 (a)[Nm] 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 (b) [sec][Nm]Figure2:Appliedcontroltorques ( ) ( ) (solidline)versuscomputedcontrol torques(dashedline)forthe(a)basemotorand(b)secondlinkmotor. 0 0.5 1 1.5 2 2.5 3 3.5 4 215 220 225 230 235 240 [sec][mm]Figure2:Computeddesiredrobottrajectory, 1( )

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26 0 0.5 1 1.5 2 2.5 3 3.5 4 .2 0 0.2 0.4 0.6 0.8 1 [sec]Figure2:Contact( =1 )andnoncontact( =0 )conditionsfortherobotand mass-springsystem. 2.4ConcludingRemarks AnonlinearLyapunov-basedcontrollerisproventoregulatethestatesofa planarrobotcollidingwithanunactuatedmass-springsystem.Thecontinuous controlleryieldsglobalasymptoticregu lationofthespring-massandrobotlinks. Unlikesomeotherresultsinliterature,thedevelopedcontinuousforcecontroller doesnotdependonsensingtheimpact,measuringtheimpactforce,orthemeasurementofotheraccelerationterms.I nnovativeanalysismethodsareusedto provethestabilityofthesystemduringcontactandduringdi erentnoncontact states.Experimentalresultsareprovidedtoillustratethesuccessfulcontroller performance.

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CHAPTER3 ADAPTIVELYAPUNOV-BASEDCONTROLOFAROBOTAND MASS-SPRINGSYSTEMUNDERGOINGANIMPACTCOLLISION SimilartoChapter2,thischapterandourpreliminaryresultsin[20]consider afullyactuatedplanarrobotcollidingwithanunactuatedmass-springsystem, butunlikeChapter2thedynamicmodelforboththemass-springandrobot systemsandtheimpactforceareassumedtohaveuncertainparameters.The controlobjectiveisalsode nedasthedesiretocommandtherobottocollide withanunactuatedsystemandregulatetheresultingcoupledMSRsystemtoa desiredcompressedstatewhilecompensatingfortheconstant,unknownsystem parameters.Speci cally,inthedynamicmodelofSection1.2, ( ) R2 2representstheuncertaininertiamatrix, ( ) R2 2representstheuncertain centripetal-Coriolise ects,and ( ) [ 1( ) 2( )] R2representsuncertain conservativeforces(e.g.,gravity), R representstheunknownmassand R representstheunknownsti nessofthespring,and R representstheunknown positivesti nessconstant. Tocompensatefortheuncertainty,adaptiveLyapunov-basedmethodsareused todevelopacontinuousadaptivecontrollerthatyieldsglobalasymptoticregulation ofthemassandrobotlinks.Twolinearparameterizationsaredesignedtoadapt fortheunknownrobotandmassparameters.Adesiredtime-varyingrobotlink trajectoryisdesignedthataccountsfortheimpactdynamicsandtheresulting coupleddynamicsoftheMSRsystem.Thedesiredlinktrajectoryconvergestoa setpointthatequalsthedesiredmasspositionplusanadditionalconstantthatis duetothedeformationofthemass.Aforcecontrolleristhendesignedtoensure thattherobotlinkpositiont rackingerrorisregulated. 27

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28 Thischapterisorganizedasfollows.TheassociatedpropertiesandassumptionsareprovidedinSection3.1.Section3.2describestheerrorsystemandcontrol developmentfollowedbythestabilityanalysisinSection3.3.Section3.4describes theexperimentalresultsthatindicatethesuccessfulperformanceoftheproposed controllerfollowedbyconclusioninSection3.5. 3.1PropertiesandAssumptions Property3.1: Therobotdynamicsgivenin(1)canbelinearlyparameterizedas ( ) = ( ) + ( ) + ( )+ 0 where Rcontainstheconstantunknownsystemparameters,and ( ) R2 denotestheknownregressionmatrix Assumption3.1: Weassumethatthemassofthemass-springsystemcanbe upperandlowerboundedas where R denoteknownpositiveboundingconstants.Theunknown sti nessconstants and arealsoassumedtobeboundedas (31) where R denoteknownpositiveboundingconstants. Assumption3.2 :Duringthesubsequentcontroldevelopment,weassume thattheminimumsingularvalueof ( ) isgreaterthanaknown,smallpositive constant 0 ,suchthat max {k 1( ) k} isknownapriori,andhence,all kinematicsingularitiesarealwaysavoided.

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29 3.2ErrorSystemandControlDevelopment Thesubsequentcontroldesignisbasedonintegratorbacksteppingmethods.A desiredtrajectoryisdesignedfortherobot(i.e.,avirtualcontrolinput)toensure therobotconvergestoandimpactswiththemass,andtoensurethattherobot regulatesthemasstothedesiredposition.Sincewecannotdirectlycontrolthe robottrajectory,aforcecontrollerisdevelopedtoensurethattherobottracksthe desiredtrajectorydespitethetransitionfromfreemotiontoanimpactcollisionand despiteuncertaintiesthroughouttheMSRsystem.Asistypicalofthebackstepping designmethod,thederivativeofthedesiredrobottrajectoryisrequiredtodevelop theforcecontroller.Takingthederivativeofthedesiredtrajectorycouldleadto unmeasurablehigherorderterms(i.e.,acceleration).Thesubsequentdevelopment exploitsthehyperbolic lterstructuredevelopedin[38]toovercometheproblemof injectinghigherordertermsinthecontrollerandtofacilitatethedevelopmentof su cientgainconditionsusedinthesubsequentstabilityanalysis. 3.2.1ControlObjective Thecontrolobjectiveistoregulatethestatesofanuncertaindynamicsystem (i.e.,atwo-linkplanarrobot)thathasanimpactcollisionwithanotheruncertain dynamicsystem(i.e.,amass-spring ).Aregulationerror,denotedby ( ) R3,is de nedtoquantifythisobjectiveas where ( ) [ 1 2] R2and ( ) R denotetheregulationerrorforthe end-pointofthesecondlinkoftherobotandmass-springsystem(seeFig.1), respectively,andarede nedas , (32)

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30 In(3), R denotestheconstantknowndesiredpositionofthemass,and ( ) [ 1( ) 2] R2denotesthedesiredpositionoftheend-pointofthe secondlinkoftherobot.Thesubsequentdevelopmentisbasedontheassumption that ( ) ( ) ( ) and ( ) aremeasurable,andthat ( ) and ( ) can beobtainedfrom ( ) and ( ) .Tofacilitatethesubsequentcontroldesignand stabilityanalysis, lteredtrackingerrors,denotedby ( ) R and ( ) R2,are de nedas[38] + 1tanh( )+ 2tanh( ) (33) + where 12 R arepositive,constantgains,and ( ) R isanauxiliary lter variabledesignedas[38] 3tanh( )+ 2tanh( ) 1cosh2( ) (34) where 1 R isapositiveconstantcontrolgain,and 3 R isapositiveconstant ltergain. 3.2.2Closed-LoopErrorSystem Bytakingthetimederivativeof ( ) andutilizing(15),(110),(32),and (3),thefollowingopen-looperrorsystemcanbeobtained: = ( 1 )+ 2 cosh 2( ) + 1 cosh 2( ) (35) In(3), ( ) ( ) and .Tofacilitatethesubsequentanalysis,the followingnotationisintroduced[38]: = 1 = (36) = ( )

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31 Afterusing(33)and(3),theexpressionin(35)canberewrittenas = + ( 1 1)+ 1+ 21 (37) where ( ) R isanauxiliarytermde nedas = 1 cosh 2( )( 1tanh( )) 12 cosh 2( )tanh( ) (38) + 2 cosh 2( )( 3tanh( ))+ 2 cosh 2( )( 2tanh( )) Theauxiliaryexpression ( ) de nedin(38)canbeupperboundedas | | 1k k (39) where 1 R isapositiveboundingconstant,and ( ) R3isde nedas = tanh( )tanh( ) (3) Basedon(37)andthesubsequentstabilityanalysis,thedesiredrobotlink positionisdesignedas 1, + + 2tanh( ) 12cosh2( )tanh( ) (3) 2, In(3), R isanappropriatepositiveconstant(i.e., isselectedsotherobot willimpactthemass-springsystemintheverticaldirection), 2 R isapositive constantcontrolgain,andthecontrolgain 1 R isde nedas 1, 1 3+ 12 1 (3) where 1 R isapositiveconstantnonlineardampinggain.Theparameter estimate ( ) R in(3)isgeneratedbytheadaptiveupdatelaw ( ) (3)

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32 In(3), R isapositiveconstant,and ( ) denotesasu cientlysmooth projectionalgorithm[39]utilizedtoguaranteethat ( ) canbeboundedas (3) where R denoteknown,constantlowerandupperboundsfor ( ) respectively. Aftersubstituting(3)into(3),theclosed-looperrorsystemfor ( ) can beobtainedas = ( 1 1)+ ( )+ 12cosh2( )tanh( ) (3) + 2tanh( )+ 21 In(315),theparameterestimationerror ( ) R isde nedas Theopen-looproboterrorsystemcanbeobtainedbytakingthetimederivativeof ( ) andpremultiplyingbytherobotinertiamatrixas = (3) where(19),(32),and(33)wereutilized,and = + + + + + ( 1 ) 0 (3) where ( ) R2 denotesaknownregressionmatrix,and Rdenotesanunknownconstantparametervector.SeeAppendixAforalinearly parameterizableexpressionfor ( ) ( ) thatdoesnotdependonacceleration terms.Basedon(3)andthesubsequentstabilityanalysis,therobotforce

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33 controlinputisdesignedas + + 3 (3) where 3 R isapositiveconstantcontrolgain,and ( ) Risanestimatefor generatedbythefollowingadaptiveupdatelaw ( ) (3) In(3), R isapositivede nite,constant,diagonal,adaptationgain matrix,and ( ) denotesaprojectionalgorithmutilizedtoguaranteethatthe elementof ( ) canbeboundedas where R denoteknown,constantlowerandupperboundsforeachelement of ( ) ,respectively. Theclosed-looperrorsystemfor ( ) canbeobtainedaftersubstituting(38) into(3)as = 3 (3) In(320),theparameterestimationerror ( ) Risde nedas (3) Remark3.1 :Basedon(318),thecontroltorqueinputcanbeexpressedas = + + 3 (3) where ( ) denotesthemanipulatorJacobianintroducedin(12).

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34 3.3StabilityAnalysis Theorem: Thecontrollergivenby(3),(3 13),(3),and(3)ensures globalasymptoticregulationoftheMSRsysteminthesensethat | ( ) | 0 k ( ) k 0 provided 1, 2,and 1areselectedsu cientlylargeandthefollowingsu cient gainconditionissatis ed: 2 max 1 4 2 2 (3) where , ,and arede nedin(14)and(31),respectively. Inthefollowingproof,aLyapunovfunctionanditsderivativeareprovided. TheanalysisisthenseparatedintotwocasesasinChapter2.Forthenoncontact case,thestabilityanalysisindicatesthecontrolleranderrorsignalsarebounded andconvergetoanarbitrarilysmallregion.Additionalanalysisindicatesthat withinthisregion,contactmustoccur.Whencontactoccurs,aLyapunovanalysis isprovidedthatillustratestheMSRsystemasymptoticallyconvergestothedesired setpoint. Proof: Let ( ) R denotethefollowingnonnegative, radiallyunboundedfunction(i.e.aLyapunovfunctioncandidate): 1 2 + 1 2 1 + 1 2 1 (3) + 2[ln(cosh( ))+ln(cosh( ))]+ 1 2 + 1 2 2 Thetimederivativeof(324)canbedeterminedas = + 1 2 + 1 + 1 (3) + 2[tanh( ) +tanh( ) ]+ +

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35 Afterusing(1),(3),(3),(3),(3),(3),and(3)-(3),the expressionin(3)canberewrittenas 3 12tanh2( ) 3 22 23tanh2( ) (3) 12 122 + [ ( 1 1)+ ( )+ ] Theexpressionin(326)willnowbeexaminedundertwodi erentscenarios. Case1-Noncontact: Forthiscase,thesystemsarenotincontact( =0 ) and(3)canberewrittenas 3 12tanh2( ) 23tanh2( ) 3 22 12 122 + [ 1 + ] Rewriting 1( ) andsubstitutingfor ( ) yields 3 12tanh2( ) 23tanh2( ) 2 22 (3) k k2 | |k k 122 12 1k k| | 22 | || 1| Aftercompletingthesquareonthebracketedterms,(327)canbeexpressedas 3 12tanh2( ) 23tanh2( ) 22 (3) 2 2 4 2 + k k2 4 2 1+ 2 ( 1)2 4 2 Provided 1isselectedaccordingtothesu cientcondition 1 1 4 2min n 12 2 32o theexpressionin(3)canbefurtherreducedas 1k k2 3k k2 2 2 4 2 + 2 ( 1)2 4 2 (3)

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36 where 1 R isde nedas 1=min n 12 2 32o 1 4 2 1 Basedon(14)inAssumption1.1,forthenoncontactcase 1 (3) Hence,theexpressionin(39)canbeupperboundedas k k2+ (3) where R isde nedas =min ( 13 2 2 4 !) and ( ) R5and R arede nedas = = 2 2 2(3) where canbemadearbitrarilysmallbymaking 2large.Basedon(324) and(3),either k k2 or k k2.If k k2,thenBarbalats Lemmacanbeusedtoconcludethat ( ) 0 since ( ) islowerbounded, ( ) isnegativesemi-de nite,and ( ) canbeshowntobeuniformlycontinuous.As ( ) 0 ,eventually k k2 .Providedthesu cientgainconditionin(3)is satis ed(i.e., 1 ),then(30),(332),andthefactsthat ( ) and ( ) Lfromtheuseofaprojectionalgorithm,canbeusedtoconcludethat ( ) L; hence, k ( ) k k ( ) k k ( ) k k ( ) k ( ) ( ) ( ) L.Signalchasing argumentscanbeusedtoprovetheremainingclosed-loopsignalsarealsobounded duringthenoncontactcase.Thepreviousdevelopmentcanbeusedtoconclude

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37 thatforthenoncontactcase k ( ) k r andhence, k ( ) k r as (3) Basedon(333),linearanalysismethods(seeLemmaA.19of[37])canbeapplied to(33)toprovethat k ( ) k k (0) k exp( )+ 1 r (1 exp( )) (3) as forthenoncontactcase. Furtheranalysisisrequiredtoprovethatthemanipulatormakescontactwith themass-springsystem.Contactbetweenthemanipulatorandthemass-spring systemoccurswhen 1( ) ( ) .Basedon(3),asu cientconditionfor contactcanbedevelopedas 1 + 1 r (3) Afterusing(311),thesu cientconditionin(335)canbeexpressedas + 2tanh( ) 12cosh2( )tanh( ) 1 r (3) Byusing(32)and(36)andperformingsomealgebraicmanipulation,the inequalityin(336)canbeexpressedas 2tanh( ) 12cosh2( ) 1 r +( + 0) (3) where ( ) and ( ) arede nedin(3).FromAssumption1.1, ( ) canbe upperboundedas (3) where R denotesaknownpositiveconstant.If ( ) 0 ,thenthesu cient conditionin(3)maynotbesatis ed.Theconditionthat ( ) 0 willonly occurifanimpactcollisionoccursthatcausesthemasstoovershootthedesired

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38 position.However,evenifanimpactoccursandthemassovershootsthedesired position,thedynamicswillforcethemasspositionerrortoreturntotheinitial condition.Thatis, ( ) 0 where R denotesaknownpositive constant.Basedon(3)andthefactthat ( ) willeventuallybelowerbounded by inanoncontactcondition,theinequalityin(337)canbesimpli edas 2 tanh( ) 1cosh2( ) 1 r +( + 0) (3) Tofurthersimplifytheinequalityin(339),anupperboundcanbedetermined for ( ) .Theinequalityin(333)alongwi th(3)and(3)canbeusedto concludethatasthemanipulatorapproachesthemass, ( ) willeventuallybe upperboundedas tanh 1 1 r (3) where R isaknownpositiveconstant.Basedon(32)and(30),thecontrol parameter 2canbeselectedaccordingtothefollowingsu cientconditionto ensuretherobotandmass-springsystemmakecontact 21 p +( + 0) tanh( ) 1cosh2( ) (3) where 1ischosenas 1 tanh( ) cosh2( ) Case2-Contact: Providedthesu cientconditionin(3)issatis ed,the robotwilleventuallymakecontactwiththemass.Forthecasewhenthedynamic systemscollide( =1 ),andthetwodynamicsystemsbecomecoupled1,then 1Thedynamicsystemscanseparateafterimpact,howeverthiscasecanstillbe analyzedundertheNoncontactsectionofthestabilityanalysis.

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39 (3)canberewrittenas 3 12tanh2( ) 3 22 23tanh2( ) k k2 | |k k 12 122 1k k| | where(39)wassubstitutedfor ( ) .Completingthesquareonthetwo bracketedtermsyields 3 12 tanh2( ) 32 tanh2( ) (3) 3 22 + 2 2 4 + k k2 4 2 1 A nalboundcanbeplacedon(32)as min n 12 32 2o k k2+ k k2 4 2 1 2 2 2 4 2 3 Because(324)isnon-negativeanditsderivativeisnegativesemi-de nite, ( ) ( ) ( ) ( ) ( ) ( ) and ( ) L Duetothefactthat ( ) ( ) and ( ) L,theexpressionin(3)canbeusedtoconcludethat ( ) L(andhence, ( ) isuniformlycontinuous).Duetothefactthat ( ) L (3)canbeusedtoconcludethat ( ) L Previousfactscan beusedtoprovethat ( ) L andsince ( ) L then ( ) L Due tothefactthat ( ) ( ) ( ) L (3)canbeusedtoconcludethat ( ) L Theexpressionin(3)canbeusedtoconcludethat ( ) L(andhence, ( ) isuniformlycontinuous).Giventhat ( ) ( ) ( ) ,and ( ) L, ( ) L.Since ( ) L,(3)canbeusedtoprovethat ( ) L.Theexpressionin(38)canthenbeusedtoprovethat ( ) L Theexpressionin(30)canbeusedtoconcludethat ( ) L(andhence, ( ) isuniformlycontinuous).Duetothefactthat ( ) ( ) ( ) L2and uniformlycontinuous,BarbalatsLemmacanbeusedtoconcludethat | ( ) | k ( ) k | ( ) | 0 as .Basedonthefactthat k ( ) k 0 as ,

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40 standardlinearanalysismethods(seeLemmaA.15of[37])canthenbeusedto provethat k ( ) k 0 as Asistypicalinliterature,thecontro llerdevelopedin(3),(3),(3), and(3)isbasedontheunderlyingassumptionthatanarbitrarilylarge(but nite)controlauthorityisavailable.Apotentialdisadvantageofthecontroller isthatthegainconditionsdevelopedaboveandin(3)indicatethat 1, 2, and 2,respectively,shouldbeselectedsu cientlylarge.Asdemonstratedbythe subsequentexperimentalresults,thegainsmaybeselectedmuchlowerinpractice (i.e.,thegainsconditionsaretheresultofaconservativeLyapunovanalysis). However,thesubsequentexperimentalsectionalsoillustratesthatevenwhenthe gainconditionsareviolated,largeinitialconditionsandahighsti nesscoe cient resultinahighgaincontrollerthatinitiallysaturatestheactuators.Thecontrol torqueintheexperimentwasarti ciallysaturatedtoreducethemagnitudeofthe impacttoprotectacapacitanceprobefromcontactbyexcessivebendingofthe aluminumrod/springassembly.Researchthatcanlimittherequiredcontroltorque forsystemsthatundergoanimpactcollisionwillbediscussedinChapter4. 3.4ExperimentalResults WiththetestbedinSection2.3,thecontrolgains and 3,de nedasscalars in(33)and(38),wereimplemented(withnonconsequentialimplicationsto thestabilityresult)asdiagonalgainmatricestoprovidemore exibilityinthe experiment.Speci cally,thecontrolgainswereselectedas 1=0 18 2=0 9 3= { 185 170 } 1=45 2=8 3=0 01 = { 60 90 } (3) Thecontrolgainsin(3)wereobtaine dfromatrialanderrorprocessandmay notsatisfythesu cientgainconditionsdevelopedinthetheoremproof.The su cientgainconditionsaretheresultofaconservativestabilityanalysisandwere

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41 usedasaguideinthetuningprocess.Thesubsequentresultsindicatethatthe developedcontrollercanbeapplieddespitethefactthatsomegainconditionsare notsatis ed. Theadaptationgainswereselectedas =90 = { 4 01 1012 1 2 107 0 2 3 3 1012 (3) 6 106 0 1 2 4 1011 7 105 0 1 2 35 1011} Theadaptationgains in(34)areusedtoenabletheadaptiveestimateto su cientlychangerelativetothelargev aluesoftheuncertainparametersin Smalleradaptationgainscouldbeusedtoobtaindi erentresults.Theinitial conditionsfortherobotcoordinatesan dthemass-springpositionwere(in[m]) 1(0) 2(0) (0) = 0 0080 4810 202 Theinitialvelocityoftherobotandmass-springwerezero,andthedesiredmassspringpositionwas(in[mm]) =232 Thatis,thetipofthesecondlinkoftherobotwasinitially 224 [mm]fromthe desiredsetpoint,and 194 [mm]from 0alongthe 1-axis(seeFig.2).Oncethe initialimpactoccurs,therobotisrequiredtodepressthespring(item(1)inFig. 2)tomovethemass 30 [mm]alongthe 1-axis. Themass-springandroboterrors(i.e., ( ) )areshowninFig.4andFig. 32.Thepeaksteady-statepositionerroroftheend-pointofthesecondlinkofthe robotalongthe 1-axis(i.e., | 1( ) | )andalongthe 2-axis(i.e., | 2( ) | )are 0 212 [mm]and 5 77 [ m],respectively.Thepeaksteady-statepositionerrorofthemass (i.e., | ( ) | )is 2 72 [ m].The 0 212 [mm]maximumsteadystateerrorin | 1( ) |

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42 0 5 10 15 0 500 (a)[mm] 0 5 10 15 0 200 (b)[mm] 0 5 10 15 0 50 (c) [sec][mm]Figure3:Mass-springandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatesthepositionerrorof therobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicatesthepositionerrorof themass-spring(i.e., ( ) ). isduetothe ( ) termof 1( ) in(3)where ( ) isapproximately 0 03 [m],and ( ) hasamaximumsteadystatevalueof 0 007 [ ],yielding 0 21 [mm]error.Alloftheothertermsin 1( ) arenegligibleatsteadystate. Theinputcontroltorquesin(3)areshowninFig.3andFig.3. Toconstraintheimpactforcetoalevelthatensuredthealuminumplatedid not extothepointofcontactwiththecapacitanceprobe,thecomputed torquesarearti ciallysaturated.Fig.33depictsthecomputedtorques, andFig.34depictstheactualtorques(solidline)alongwiththecomputed torques(dashedline).Theresultingdesiredtrajectoryalongthe 1-axis(i.e., 1( ) )isdepictedinFig.3,andthedesiredtrajectoryalongthe 2-axis waschosenas 2=370 [mm].Fig.36depictsthevalueof ( ) R andFig.37-Fig.3depictthevaluesof ( ) R10.Theorderofthe curvesintheplotsisbasedontherelativescaleoftheparameterestimates ratherthannumericalorderin ( ) .Avideooftheexperimentisprovidedat http://www.mae.u .edu/~dixon/projects/robman/adaptiveimpact.htm.

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43 0 0.5 1 1.5 2 0 100 200 (a)[mm] 0 0.5 1 1.5 2 0 50 (b)[mm] 0 0.5 1 1.5 2 0 20 40 (c) [sec][mm]Figure3:Mass-springandroboterrors ( ) duringtheinitialtwoseconds. 0 5 10 15 0 500 (a)[Nm] 0 5 10 15 0 200 (b) [sec][Nm]Figure3:Computedcontroltorques ( ) ( ) forthe(a)basemotorand(b) secondlinkmotor.

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44 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 50 (a)[Nm] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 50 (b) [sec][Nm]Figure3:Appliedcontroltorques ( ) ( ) (solidline)versuscomputedcontrol torques(dashedline)forthe(a)basemotorand(b)secondlinkmotor. 0 5 10 15 229 229.5 230 230.5 231 231.5 232 232.5 233 233.5 [sec][mm]Figure3:Computeddesiredrobottrajectory, 1( )

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45 0 5 10 15 .01 0 0.01 0.02 0.03 0.04 0.05 0.06 [sec]Figure36:Unitlessparameterestimate ( ) introducedin(3). 0 5 10 15 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 106 [sec][N/m] (a) (b) (c) (d)Figure3:Estimatefortheunknownconstantparametervector ( ) .(a) 10( )= ,(b) 4( )= ,(c) 1( )=1 ,and(d) 7( )=2 ,where 1, 2 R denotethemassofthe rstandsecondlinkoftherobot, R denotesthemassofthemotorconnectedtothesecondlinkoftherobot,and R denotesthemassofthemass-springsystem.

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46 0 5 10 15 0 200 400 600 800 1000 1200 1400 1600 1800 [sec][N/m] (a) (b) (c)Figure3:Estimatefortheunknownconstantparametervector ( ) .(a) 5( )= ,(b) 2( )=1 ,and(c) 8( )=2 0 5 10 15 0 1 2 3 4 5 6 7 8 9 [sec][Kg] (a) (b) (c)Figure3:Estimatefortheunknownconstantparametervector ( ) .(a) 6( )= ,(b) 3( )= 1,and(c) 9( )= 2.

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47 3.5ConcludingRemarks AnadaptivenonlinearLyapunov-basedcontrollerisproventoregulatethe statesofaplanarrobotcollidingwithanunactuatedmass-springsystem.The continuouscontrolleryieldsglobalasymptoticregulationofthespring-massand robotlinks.Newcontroldesign,errorsystemdevelopmentandstabilityanalysis techniqueswererequiredtocompensateforthefactthatthedynamicschanged fromanuncoupledstatetoacoupledstate.Experimentalresultsareprovided toillustratethesuccessfulperformanceofthecontroller.Su cientconditions developedinthestabilityanalysisindicatethatthecontrolgainsshouldbeselected largeenoughtominimizetheclosed-loopsteadystateerror,buthighgainscould resultinlargetorquesforlargeinitialerrors.Thehighgainproblemisexacerbated inthedevelopedresultbecauseofthepresenceoftheestimatedimpactsti ness coe cient.Theexperimentalresultswereobtainedbyarti ciallysaturatingthe torquetopreventdamagetothecapacitanceprobe.Theseissuespointtoaneed todevelopcontrollersthataccountforlimitedactuationwhichwewilldiscussin Chapter4.

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CHAPTER4 ANIMPACTFORCELIMITINGADAPTIVECONTROLLERFORAROBOTIC SYSTEMUNDERGOINGANONCONTACT-TO-CONTACTTRANSITION SimilartoChapter2and3,theobjectiveofthischapterisalsotocontrol arobotfromanoncontactinitialcondit iontoadesired(contact)positionso thatthemass-springsystemisregula tedtoadesiredcompressedstate.When thecontrollersinChapter2andChapter3wereimplementedinthepresenceof largeinitialconditions,violentimpactsbetweentherobotandthemass-spring systemresulted.Infact,thecontrollerwasarti ciallysaturated(thesaturation e ectswerenotconsideredinthestabilityanalysis)toreducetheimpactforces sothatthemassde ectionwouldnotdestroyacapacitanceprobe.Theseresults providethemotivationforthecontroldevelopmentinthischapter.Speci cally,the feedbackelementsforthecontinuouscontrollerinChapter3arecontainedinside ofhyperbolictangentfunctionsasameanstolimittheimpactforcesresulting fromlargeinitialconditionsastherobot transitionsfromnoncontacttocontact states.Themainchallangeofthisworkisthattheuseofsaturatedfeedbackdoes notallowsomecouplingtermstobecanceledinthestabilityanalysis,resulting intheneedtodevelopstatedependentupperboundsthatreducethestability toasemi-globalresult.Newcontroldevelopment,closed-looperrorsystems,and Lyapunov-basedstabilityanalysisarguementsareusedtoconcludetheresult. Experimentalresultsareprovidedthatsuccessfullydemonstratethecontrol objective. Thischapterisorganizedasfollows.Theassociatedpropertiesareprovided inSection4.1.Section4.2describestheerrorsystemandcontroldevelopment 48

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49 followedbythestabilityanalysisinSection4.3.Section4.4describestheexperimentalresultsthatindicatethesuccessfulperformanceoftheproposedcontroller followedbyconclusioninSection4.5. 4.1Properties Remark4.1 :Toaidthesubsequentcontroldesignandanalysis,wede nethe vector ( ) Rasfollows ( ) [tanh( 1) tanh( )](41) where [ 1] R Property4.1 :Thefollowinginequalitiesarevalidforall =[ 1] R[36] || || || ( ) || || || +1 || || tanh( || || ) (42) || ||2X =1ln(cosh( )) ln(cosh( || || )) (43) ( ) ( ) ( )= || ( ) ||2(44) tanh2( || || ) 4.2ErrorSystemandControlDevelopment Thesubsequentcontroldesignisbasedonintegratorbacksteppingmethods.A desiredtrajectoryisdesignedfortherobot(i.e.,avirtualcontrolinput)toensure therobotconvergestoandimpactswiththemass,andtoensurethattherobot regulatesthemasstothedesiredposition.Sincetherobottrajectorycannotbe controlleddirectly,aforcecontrollerisdevelopedtoensurethattherobottracks thedesiredtrajectorydespitethetransitionfromfreemotiontoanimpactcollision anddespiteuncertaintiesthroughouttheMSRsystem.

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50 4.2.1ControlObjective AsinChapter2and3,thecontrolobjectiveistoregulatethestatesofthe mass-spingsystemviaatwo-linkplanarrobotthattransitionsfromnoncontactto contactwiththemass-springthroughanimpactcollision.Anadditionalobjective inthischapteristolimittheimpactforcetopreventdamagetotherobotor theenvironment(i.e.,themass-springsystem).Aregulationerror,denotedby ( ) R3,isde nedtoquantifythisobjectiveas where ( ) [ 1( ) 2( )] R2and ( ) R denotetheregulationerrorforthe end-pointofthesecondlinkoftherobotandmass-springsystem(seeFig.1), respectively,andarede nedas , (45) In(4), R denotestheconstantknowndesiredpositionofthemass,and ( ) [ 1( ) 2] R2denotesthedesiredpositionoftheend-pointofthe secondlinkoftherobot.Thesubsequentdevelopmentisbasedontheassumption that ( ) ( ) ( ) and ( ) aremeasurable,andthat ( ) and ( ) can beobtainedfrom ( ) and ( ) .Tofacilitatethesubsequentcontroldesignand stabilityanalysis, lteredtrackingerrors,denotedby ( ) R and ( ) R2,are de nedas[38] + 1tanh( )+ 2tanh( ) (46) +

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51 where 12 R arepositive,constantgains,and ( ) R isanauxiliary lter variabledesignedas[38] 3tanh( )+ 2tanh( ) 1cosh2( ) (47) where 1 R isapositiveconstantcontrolgain,and 3 R isapositiveconstant ltergain. 4.2.2Closed-LoopErrorSystem Inthissection,theclosed-looperrorsystemfor isexactlythesameasin (3-5)-(3-15)inChapter3.Theopen-looproboterrorsystemcanbeobtainedby takingthetimederivativeof ( ) andpremultiplyingbytherobotinertiamatrix as = (48) where(19),(45),and(46)wereutilized,and + + + + + ( 1 ) 0 where ( ) R2 denotesaknownregressionmatrix,and Rdenotesanunknownconstantparametervector.Bymakingsubstitutions fromthedynamicmodelandthepreviouserrorsystems, ( ) canbeexpressed withoutadependanceonaccelerationterms.Basedon(4)andthesubsequent stabilityanalysis,therobotforcecontrolinputisdesignedas + ( )+ 3 ( ) (49) where 3 R isapositiveconstantcontrolgain,and ( ) Risanestimatefor generatedbythefollowingadaptiveupdatelaw ( ) (4)

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52 In(4), R isapositivede nite,constant,diagonal,adaptationgain matrix,and ( ) denotesaprojectionalgorithmutilizedtoguaranteethatthe elementof ( ) canbeboundedas (4) where R denoteknown,constantlowerandupperboundsforeachelement of ( ) ,respectively. Theclosed-looperrorsystemfor ( ) canbeobtainedaftersubstituting(4) into(48)as = ( ) 3 ( ) (4) In(412),theparameterestimationerror ( ) Risde nedas (4) 4.3StabilityAnalysis Theorem: Thecontrollergivenby(3),( 3),(4),and(4)ensures semi-globalasymptoticregulationoftheMSRsysteminthesensethat | ( ) | 0 k ( ) k 0 providedthefollowingsu cientgainconditionsaresatis ed: 1 1 4 2min n 12 32 2o (4) 2 max 1 4 2 2 (4) 3 r 2 1+1 !2(4)

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53 where R isde nedas max 1k (0) k2+ 5 1(tanh 1( r ))2+ R isaknownpositiveconstant,and 1, 1, 2, 3, 1, 2, 3, 1, , ,and arede nedin(1),(111),(31),(31),(32),(4),(47),and (4). Asinthepreviouschapters,thesubsequentanalysisisseparatedintotwo cases:contactandnoncontact.Forthenoncontactcase,thestabilityanalysis indicatesthecontrolleranderrorsignalsareboundedandconvergetoanarbitrarily smallregionwherecontactmustoccur.Whencontactoccurs,aLyapunovanalysis isprovidedthatillustratestheMSRsystemasymptoticallyconvergestothedesired setpoint. Proof :Let ( ) R denotethefollowingnon-negative,radiallyunbounded function(i.e.aLyapunovfunctioncandidate): 1 2 + 1 2 1 + 1 2 1 + + 1 2 2 (4) + 2[ln(cosh( ))+ln(cosh( ))]+ln(cosh( 1))+ln(cosh( 2)) where(1)and(4)canbeutilizedtobound ( ) as 1 2 || ||2 1k k2+ (4) where 1 R and R7arede nedas 1, max { 2 2 3 2 2 } 1 2 2 1+ 1 2 max{ 1 }k k2 [ ](4) where max{} R denotesthemaximumeigenvalueofamatrix,and k k aretheknownupperboundsof ( ) and ( ) ,respectively.Afterusing(112),

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54 (3),(3),(3),(4),(4),(4 7),(4),and(4),thetimederivative of(4)canbedeterminedas 3tanh2( k k ) 12tanh2( ) 32tanh2( ) (4) +2 2 3 22 12 122 tanh2( k k ) + [ ( 1 1)+ ( )+ ] Theexpressionin(420)willnowbeexaminedundertwodi erentscenarios. Case1-Noncontact: Forthiscase,thesystemsarenotincontact( =0 ) and(4)canberewrittenas 3tanh2( k k ) 12tanh2( ) 32tanh2( ) (4) +2 2 3 22 12 122 tanh2( k k ) + [ 1 + ] Basedon(3),(3),and(4),theex pressionin(4)canberewrittenas 3tanh2( k k ) 12 tanh2( ) 32 tanh2( ) (4) 2 22 tanh2( k k ) k k2 | |k k 12 122 1k k| | 22 | || 1| [ k k2 2 k kk k ] Aftercompletingthesquareinthebracketedterms,(42)canbeexpressedas k k2 tanh2( k k ) 2 2 4 2 (4) + 2 ( 1)2 4 2 3 k k2 tanh2( k k ) tanh2( k k ) where R isde nedas =min n 12 32 2o 1 4 2 1 0

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55 provided 1isselectedaccordingto(414).Provided 2isselectedaccordingto thesu cientconditionin(4),then(1),(4),(4),andthefactthat 1 (4) forthenoncontactcase,canbeusedtorewrite(43)as k k2 tanh2( k k )+ (4) 3 1 r 2 1+1 !2 tanh2( k k ) In(4), R isaknownpositivearbitrarilysmallconstantthatisde nedas 2 2 2 Providedthefollowingsu cientconditionissatis ed 3 r 2 1+1 !2 (4) theexpressionin(45)canbeexpressedas k k2+ (4) where isaknownconstantand R5isde nedas tanh( k k )tanh( k k ) (4) In(4), canbemadearbitrarilysmallbymaking 2large.Basedon(4 17)and(4),if k ( ) k2,thenBarbalatsLemmacanbeusedtoconclude that ( ) 0 since ( ) islowerbounded, ( ) isnegativesemi-de nite,and ( ) canbeshowntobeuniformlycontinuous.As ( ) 0 ,eventually k ( ) k2 While k ( ) k2,(4),(4),and(4)canbeusedtoconcludethat

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56 ( ) L andthesu cientconditiongivenin(46)canbeexpressedas 3 s 2 1k (0) k2+ 1+1 2 (4) Provided 2 2 2 theneventually k ( ) k2 .Basedon(3)and (4),thefactthat k ( ) k2 canbeusedtoconcludethat ( ) ( ) ( ) ( ) ( ) L.Since ( ) and ( ) Lfromtheuseofaprojection algorithm,thepreviousfactscanbeusedtoconcludethat ( ) L.Signal chasingargumentscanbeusedtoprovetheremainingclosed-loopsignalsarealso boundedduringthenoncontactcaseprovided(426)issatis ed. Iftheinitialconditionsfor (0) arelargeenoughthat k ( ) k2,then theconditionin(429)issu cient.However,iftheinitialconditionsfor (0) are insidetheregionde nedby ,then ( ) cangrowlargeruntil k ( ) k2 Therefore,furtherdevelopmentisrequiredtodeterminehowlarge ( ) cangrow sothatthesu cientconditionin(4)canbesatis ed.When (0) isinsidethe regionde nedby ,then k ( ) k r (4) Theexpressionin(40)canbeusedalon gwith(3),(4),and(4)to concludethat k ( ) k 5tanh 1( r ) (4) Theinequalityin(4)canbeusedalongwith(4)and(4)torewritethe su cientconditionin(4)as 3 s 2 5 1(tanh 1( p ))2+ 1+1 2 (4) Hence,the nalsu cientconditionfor(426)isgivenby(416).Thatis,provided 1, 2,and 3areselectedlargerthanknownconstants(thatdependsonthe

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57 initialconditionsofthestates)accordingto(414)-(46)thentheallthestates convergetoanarbitrarilysmallneighborhood.Developmenthasbeenprovided(see Section3.3)toprovethatcontactwiththemass-springsystemisensuredwithin thisneighborhood. Case2-Contact: Forthecasewhenthedynamicsystemscollide( =1 )and thetwodynamicsystemsbecomecoupled1,then(420)canberewrittenas 3tanh2( k k ) 12 tanh2( ) 32 tanh2( ) 3 22 tanh2( k k ) k k2 | |k k 12 122 1k k| | [ k k2 2 k kk k ] where(31)wassubstitutedfor and(3)wassubstitutedfor ( ) Completingthesquareonthethreebracketedtermsyields k k2 tanh2( k k ) 2 2 4 2 (4) 3 k k2 tanh2( k k ) tanh2( k k ) Because(4)isnon-negative,as longas(4)-(4)aresatis ed,(4)is negativesemi-de nite,and ( ) ( ) ( ) ( ) ( ) ( ) and ( ) L Duetothefactthat ( ) ( ) and ( ) L,theexpressionin(46)can beusedtoconcludethat ( ) L(andhence, ( ) isuniformlycontinuous (UC)).Duetothefactthat ( ) L ( ) L Basedon(14), ( ) L Previousfactscanbeusedtoprovethat ( ) L andsince ( ) L then ( ) L Duetothefactthat ( ) ( ) ( ) L (47)canbeusedto concludethat ( ) L Theexpressionin(35)canthenbeusedtoconclude 1Thedynamicsystemscanseparateafterimpact,howeverthiscasecanstillbe analyzedundertheNon-Impactsectionofthestabilityanalysis.

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58 that ( ) L(andhence, ( ) isUC).Basedon(46)andthefactthat ( ) and ( ) L ( ) L Also,basedon(34)andthefactthat ( ) ( ) ( ) ( ) and ( ) L theexpressionin(31)canbeusedtoprovethat ( ) L Basedonthefactthat ( ) and ( ) L theexpressionin (4)canbeusedtoprovethat ( ) L Giventhat ( ) ( ) ( ) ( ) ( ) ,and ( ) L, ( ) L.Theexpressionin(4),(411)canthenbe usedtoprovethat ( ) L Theexpressionin(4)canbeusedtoconclude that ( ) L(andhence, ( ) isUC).Since ( ) and ( ) areUCwhich implies tanh( ( )) and tanh( k ( ) k ) arealsoUC,andthefactthat tanh( ( )) tanh( k ( ) k ) and ( ) aresquareintegrable,BarbalatsLemmacanbeusedto concludethat tanh( ( )) tanh( k ( ) k ) | ( ) | 0 as ,whichalsoimplies | ( ) | k ( ) k 0 as .Basedonthefactthat k ( ) k 0 as standardlinearanalysismethods(seeLemmaA.15of[37])canthenbeusedto provethat k ( ) k 0 as 4.4ExperimentalResults WiththetestbedinSection2.3,thecontrolgains and 3,de nedasscalars in(46)and(4),wereimplemented(withnonconsequentialimplicationsto thestabilityresult)asdiagonalgainmatricestoprovidemore exibilityinthe experiment.Speci cally,thecontrolgainswereselectedas 1=0 28 2=0 855 3= { 110 10 } 1=60 2=2 8 3=0 06 = { 40 8 } Theadaptationgainswereselectedas

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59 =0 06 = { 1 8 1011 1 5 105 0 2 7 1010 7 104 0 1 1 1010 1 104 0 01 2 8 109} Theinitialconditionsfortherobotcoordinatesandthemass-springpositionwere (in[m]) 1(0) 2(0) (0) = 0 0700 2850 206 Theinitialvelocityoftherobotandmass-springwerezero,andthedesiredmassspringpositionwas(in[mm]) =236 Thatis,thetipofthesecondlinkoftherobotwasinitially 166 mmfromthe desiredsetpointand 136 mmfromtheinitialimpactpointalongthe 1-axis (seeFig.21).Therefore,oncetheinit ialimpactoccurs,therobotisrequired todepressthespring(item(1)inFig.21)tomovethemass 30 mmalongthe 1-axis. Themass-springandroboterrors(i.e., ( ) )areshowninFig.41.Thepeak steady-statepositionerroroftherobottipalongthe 1-axis(i.e., | 1( ) | )and alongthe 2-axis(i.e., | 2( ) | )are 0 855 mmand 0 179 mm,respectively.The peaksteady-statepositionerrorofthemass-spring(i.e., | ( ) | )is 0 184 mm. Theinputcontroltorques(i.e., ( ) ( ) )areshowninFig.4andFig.4. Theresultingdesiredtrajectoryalongthe 1-axis(i.e., 1( ) )isdepictedinFig. 4,andthedesiredtrajectoryalongthe 2-axiswaschosenas 2=357 5 mm. Fig.4depictsthevalueof ( ) R andFig.4-Fig.4depictthevalues of ( ) R10.Theorderofthecurvesintheplotscomesfromtheirscalesrather thantheirnumericalorderin ( ) .

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60 0 2 4 6 8 10 0 200 400 (a)[mm] 0 2 4 6 8 10 0 100 (b)[mm] 0 2 4 6 8 10 0 50 (c) [sec][mm]Figure4:Mass-springandroboterrors ( ) .Plot(a)indicatesthepositionerror oftherobottipalongthe 1-axis(i.e., 1( ) ),(b)indicatesthepositionerrorof therobottipalongthe 2-axis(i.e., 2( ) ),and(c)indicatesthepositionerrorof themass-spring(i.e., ( ) ). 0 2 4 6 8 10 0 100 (a)[Nm] 0 2 4 6 8 10 0 5 (b) [sec][Nm]Figure4:Appliedcontroltorques ( ) ( ) forthe(a)basemotorand(b)secondlinkmotor.

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61 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 100 (a)[Nm] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 5 (b) [sec][Nm]Figure4:Appliedcontroltorques ( ) ( ) forthe(a)basemotorand(b)secondlinkmotorduringthe rst0.8second. 0 2 4 6 8 10 220 240 260 280 300 320 340 360 380 400 420 [sec][mm]Figure4:Computeddesiredrobottrajectory, 1( )

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62 0 2 4 6 8 10 2 3 4 5 6 7 8 x 10 [sec][]Figure4:Parameterestimate ( ) introducedin(3). 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 106 [sec][N/m](a) (b) (c) (d)Figure4:Estimatefortheunknownconstantparametervector ( ) .(a) 10( )= ,(b) 4( )= ,(c) 1( )=1 ,and(d) 7( )=2 ,where 1, 2 R denotethemassofthe rstandsecondlinkoftherobot, R denotesthemassofthemotorconnectedtothesecondlinkoftherobot,and R denotesthemassofthemass-springsystem.

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63 0 2 4 6 8 10 0 200 400 600 800 1000 1200 1400 1600 [sec][N/m](a) (b) (c)Figure4:Estimatefortheunknownconstantparametervector ( ) .(a) 5( )= ,(b) 2( )=1 ,and(c) 8( )=2 0 2 4 6 8 10 0 2 4 6 8 10 12 [sec][kg](a) (b) (c)Figure4:Estimatefortheunknownconstantparametervector ( ) .(a) 6( )= ,(b) 3( )= 1,and(c) 9( )= 2.

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64 4.5ConcludingRemarks Inthischapter,weconsideratwolinkplanarroboticarmthattransitions fromfreemotiontocontactwithanunactuatedmass-springsystem.Anadaptive nonlinearLyapunov-basedcontrollerwithboundedtorqueinputamplitudes isproventoregulatethestatesofthesystem.Thefeedbackelementsforthe controllerinthischapterarecontainedinsideofhyperbolictangentfunctionsasa meanstolimittheimpactforcesresultingfromlargeinitialconditionsastherobot transitionsfromnoncontacttocontact.Thecontinuouscontrolleryieldssemi-global asymptoticregulationofthespring-massandrobotlinks.Experimentalresultsare providedtoillustratethesuccessfulperformanceofthecontroller.

PAGE 75

CHAPTER5 CONCLUSIONANDRECOMMENDATIONS Motivatedbythefactthatimpactsbetweentherobotandthestaticenvironmentcannotrepresentalltheimpactsystemapplications,anddiscontinuouscontrollersrequirein nitecontrolfrequency(i.e.,exhibitchattering)oryielddegraded stabilityresults(i.e.,uniformlyultimatelybounded),continuousLyapunov-based controllersforafullyactuateddynamicsystemsundergoinganimpactcollision withanotherunactuateddynamicsystemaredeveloped.Lyapunov-basedmethods areusedtoprovetheasymptoticregulationsofthemassandrobotlinks.Unlike someotherresultsinliterature,thecont inuousforcecontrollerdoesnotdependon measuringtheimpactforceorthemeasurementofotheraccelerationterms:only thepositionandvelocitytermsofthespring-masssystemandthejointanglesand theangularvelocitiestermsoftheplanarroboticarmareneededfortheproposed controller. Chapter2providesa rststepatcontrollingtheproposedimpactsystem. ThecontroldevelopmentinChapter2isbasedontheassumptionofexactmodel knowledgeofthesystemdynamics.Thecontrollerisproventoregulatethestates ofthesystemandyieldsglobalasymptoticresult.InChapter3,thedynamicmodel forthesystemisassumedtohaveuncertainparameters.Thecontrolobjectiveis de nedasthedesiretobothregulatethesystemtoadesiredcompressedstateand compensatefortheconstant,unknownsystemparameters.Twolinearparameterizationsaredesignedtoadaptfortheunknownrobotandmass-springparameters. Thecontrollerisproventoyieldglobalasymptoticregulationresult.Anextension ofthedevelopedregulationcontrollerofChapter3ispresentedinChapter4where 65

PAGE 76

66 thefeedbackelementsforthecontrollerinthischpaterarecontainedinsideofhyperbolictangetfunctionsasameanstolimittheimpactforcesresultingfromlarge initialconditionsastherobottransitionsfromnoncontacttocontactstates.The controlleryieldssemi-globalasymptotic regulationofthesystem.Experimental resultsareprovidedtoillustratethesuccessfulperformanceofthecontrollerineach chapter. Futuree ortscanfocusonutilizingnon-modelbasedcontrolmethodssuchas neuralnetworks,fuzzylogic,orrobustcontrolmethodstocompensatethemore complexuncertaintiesofthesystem.Thedevelopedmodelcanalsobemodi ed tocontroltheimpactbetweenanactuateddynamicsystemandanunactuated dynamicsystemwithnonlinear exibilities.Futureworkcouldalsofocusonthe speci capplicationofthedevelopedmethodsforapplicationssuchasdockingspace vehicles,walkingrobots,etc.E ortscouldalsofocusonobtainingaglobalresultof thefeedbacksaturatedcaseinChapter4.

PAGE 77

APPENDIXA THEEXPRESSIONOF 1INSECTION2.1 Duetothefactthattakingthetimederivativeofanyexpressionwith except for ( 1 ) isunde ned,somecarehastobetakenintakingthetime derivativeof 1( ) .Thisisdoneby rsttakingthetimederivativeof(2-6)and utilizing(2-3)and(2-7),toobtainthefollowingexpression: 1= [(1 )( ( 0)+ )]+ [ 1 ( 1+ 2) ] (A1) 2 ( )+ +1 + 1 ( 1+ 2)(1 ) ( 0) + 1 ( 1+ 2)(1 ) + 1 ( 1+ 2) [ 1 ( 1+ 2) ] Inordertoderive 1( ) thesecondtimederivativeof(2-6)istakenratherthan the rsttimederivativeof(A1)toobtain 1= 1 ( ... + +( 1+ 2) )+ (A2) Theexpressionfor ( ) canbeobtainedbyrewriting(2-3)as = (A3) Di erentiating(A)yieldsthefollowingexpressionsfor ( ) and ... ( ) : = (A4) ... = (A5) Byusing(2-7),theexpressionfor ( ) canbewrittenas = 1 [(1 )( ( 0)+ )]+ 1 [ 1 ( 1+ 2) ] (A6) 67

PAGE 78

68 Thetimederivativeof(2-4)isgivenby = 1 ( ( 1 )+ ) (A7) Thedynamicsforthemass-springsystem ( ) canbewrittenas = 1 ( ( 1 ) ( 0)) (A8) Byusing(A)and(A5),thefollowingexpressioncanbeobtained: 1= ( )+( +1) + ( 1+ 2) (A9) Substituting(A4)and(A7)-(A9)yields 1= ( ( 1 )+ ) 2 + 3 (A) +( +1) 1 ( ( 1 ) ( 0)) + ( 1+ 2) [ ( 1 )+ ] Afterusing(A)and(A6),asimpli edexpressionfor 1( ) canbeobtainedas follows: 1= ( ( 1 )) (A) +( +1) 1 ( ( 1 ) ( 0)) + ( 1+ 2) [ ( 1 )] + ( 1+ 2) (1 )( ( 0)) + ( 1+ 2) 1 ( 1+ 2)2

PAGE 79

APPENDIXB THEEXPRESSIONOF 1INSECTION3.2 Since 2isaconstant,thesubsequentdevelopmentisonlyfocusedon determining 1( ) .Afterusing(3-2),(3-4),(3-11),and(3-13),the rsttime derivativeof 1( ) canbedeterminedas 1= ( ( ))+ +1 2cosh 2( ) (B) 12 sinh2( )+cosh2( ) ( 3tanh( )+ 2tanh( ) 1cosh2( ) ) Basedonthefactthattheprojectionalgorithmfor ( ) isdesignedtobesu cientlysmooth[39],theexpressionsin(3-13)and(B1)canbeusedtodetermine thesecondtimederivativeof 1( ) as 1= ( ( )) + ( ( )) +2 ( ) (B) 2 2cosh 3( )sinh( ) 2 + +1 2cosh 2( ) 4 12(sinh( )cosh( )) 2 12 sinh2( )+cosh2( ) 3cosh 2( ) 2 1cosh( )sinh( ) + 12 sinh2( )+cosh2( ) 2cosh 2( ) + 1cosh2( ) Aftersubstituting(3-4)and(3-5)into(B)for ( ) and ( ) ,respectively,and substituting(1-5)and(1-8)into(B)for ( ) ,theexpressionfor ( ) ( ) inthelinearparameterizationin(3-17)canbedeterminedwithoutrequiring accelerationmeasurements. 69

PAGE 80

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71 [11]E.Lee,ForceandImpactControlforRobotManipulatorsUsingTime Delay, IEEEInternationalSymposiumonIndustrialElectronics, 1999,pp. 151-156. [12]E.Leeetal.,Bang-BangImpactControlUsingHybridImpedanceTimeDelayControl, IEEE/ASMETransactionsonMechatronics ,Vol.8,No.2, pp.272-277,2003. [13]B.J.Nelson,J.D.Morrow,andP.K.Khosla,FastStableContactTransitionswithasti ManipulatorUsingForceandVisionFeedback, IEEE/RSJ InternationalConferenceonIntelligentRobotsandSystems, Pittsburgh,PA, 1995,Vol.2,pp.90-95. [14]Y.Zhou,B.J.Nelson,andB.Vikramaditya,FusingForceandVision FeedbackforMicromanipulation, ProceedingsoftheIEEEInternational ConferenceonRoboticsandAutomation ,Leuven,Belgium,1998,pp.12201225. [15]G.YangandB.J.Nelson,MicromanipulationContactTransitionControl bySelectiveFocusingandMicroforceControl, ProceedingsoftheIEEE InternationalConferenceonRoboticsandAutomation ,Taipei,Taiwan,2003, pp.3200-3206. [16]B.BrogliatoandA.ZavalaRio,OntheControlofComplementary-Slackness JugglingMechanicalSystems, IEEETransactionsonAutomaticControl Vol.45,No.2,pp.235-246,2000. [17]M.IndriandA.Tornambe,Controlo fUnder-ActuatedMechanicalSystems SubjecttoSmoothImpacts, ProceedingsoftheIEEEConferenceonDecision andControl ,Atlantis,ParadiseIsland,B ahamas,2004,pp.1228-1233. [18]G.Hu,W.E.Dixon,andC.Makkar,Energy-BasedNonlinearControlof Under-ActuatedEuler-LagrangeSystemsSubjecttoImpacts, Proceedings oftheIEEEConferenceonDecisionandControl ,Seville,Spain,2005,pp. 6859-6864. [19]K.Dupree,W.E.Dixon,G.Hu,andC.Liang,Lyapunov-BasedControlofa RobotandMass-SpringSystemUndergoingAnImpactCollision, Proceedings oftheIEEEAmericanControlConference, Minneapolis,MN,2006,pp. 3241-3246. [20]K.Dupree,C.Liang,G.Hu,andW.E.Dixon,GlobalAdaptiveLyapunovBasedControlofaRobotandMass-SpringSystemUndergoinganImpactCollision, ProceedingsoftheIEEEConferenceonDecisionandControls, San Deigo,California,2006,pp.2039-2044. [21]I.D.Walker,TheUseofKinematicRedundancyinReducingImpactand ContactE ectsinManipulation, ProceedingsoftheIEEEInternational

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72 ConferenceonRoboticsandAutomation, Cincinnati,OH,1990,vol.1,pp. 434-439. [22]M.W.Gertz,J.O.Kim,andP.K.Khosla,ExploitingRedundancyto ReduceImpactForce, IEEE/RSJInternationalWorkshoponIntelligent RobotsandSystemsIROS, Osaka,Japan,1991,pp.179-184. [23]I.D.Walker,ImpactCon gurationsandMeasuresforKinematicallyRedundantandMultipleArmedRobotSystems, IEEETransactionsonRobotics andAutomation, Vol.10,No.5 pp346-351,1994. [24]D.ChiuandS.Lee,RobustJumpImpactControllerforManipulators, ProceedingsoftheIEEE/RSJInternationalConferenceonIntelligentRobots andSystems ,Pittsburgh,Pennsylvania,1995,Vol.1,pp.299-304. [25]J.M.HydeandM.R.Cutkosky,ContactTransitionControl:AnExperimentalStudy, IEEEControlSystem, Vol.14,No.1,pp.25-31,1994. [26]M.ShibataandT.Natori,ImpactForceReductionforBipedRobotBased onDecouplingCOGControlScheme, ProceedingsoftheIEEEInternational WorkshoponAdvancedMotionControl ,Nagoya,Japan,2000,pp.612-617. [27]E.OhashiandK.Ohnishi,MotionControlintheSupportPhaseforaOneLeggedHoppingRobot, ProceedingsoftheIEEEInternationalWorkshopon AdvancedMotionControl ,Kawasaki,Japan,2004,pp.259-262. [28]Y.Sato,E.Ohashi,andK.Ohnishi, ImpactForceReductionforHopping Robot, ConferenceofIEEEIndustrialElectronicsSociety, 2005,pp.18211826. [29]R.V.Dubey,T.F.Chan,andS.E.Everett,VariableDampingImpedance ControlofaBilateralTeleroboticSystem, IEEEControlSystems, Vol.17, Issue1,pp.37-45,1997. [30]Y.Yamada,Y.Hirasawa,S.Huang,Y.Umetani,andK.Suita,HumanRobotContactintheSafeguardingSpace, IEEE/ASMETransactionson Mechatronics, Vol.2,No.4,pp.230-236,1997. [31]Z.Li,A.Ming,N.Xi,Z.Xie,J.Gu,andM.Shimojo,Collision-Tolerant ControlforHybridJointbasedArmofNonholonomicMobileManipulatorin Human-RobotSymbioticEnvironments, ProceedingsoftheIEEEInternationalConferenceonRoboticsandAutomation ,Barcelona,Spain,2005,pp. 4037-4043. [32]P.Huang,Y.Xu,andB.Liang,ContactandImpactDynamicsofSpace ManipulatorandFree-FlyingTarget, IEEE/RSJInternationalConferenceon IntelligentRobotsandSystems, 2005,pp.1181-1186.

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73 [33]K.Youcef-ToumiandD.A.Guts,ImpactandForceControl, Proceedingsof theIEEEInternationalConferenceonRoboticsandAutomation, Scottsdale, AZ,1989,vol.1,pp.410-416. [34]M.IndriandA.Tornambe,ImpactModelandControlofTwoMulti-DOF CooperatingManipulators, IEEETransactionsonAutomaticControl ,Vol. 44,No.6,pp.1297-1303,1999. [35]K.Dupree,C.Liang,G.Hu,andW.E.Dixon,GlobalAdaptiveLyapunovBasedControlofaRobotandMass-SpringSystemUndergoinganImpact Collision, IEEETransactionsonRobotics, submitted,2006. [36]W.E.Dixon,M.S.deQueiroz,D.M.Dawson,andF.Zhang,Tracking ControlofRobotManipulatorswithBoundedTorqueInputs, Robotica, Vol. 17,pp.121-129,1999. [37]W.E.Dixon,A.Behal,D.M.Dawson,andS.Nagarkatti, NonlinearControl ofEngineeringSystems:ALyapunov-BasedApproach ,BirkhuserBoston, 2003. [38]W.E.Dixon,E.Zergeroglu,D.M.Dawson,andM.W.Hannan,Global AdaptivePartialStateFeedbackTrackingControlofRigid-LinkFlexible-Joint Robots, Robotica ,Vol.18.No3.pp.325-336,2000. [39]Z.Cai,M.S.deQueiroz,andD.M.Dawson,ASu cientlySmoothProjection Operator, IEEETransactionsonAutomaticControl ,Vol.51,No.1,pp.135139,2006. [40]M.Lo er,N.Costescu,andD.Dawson,Qmotor3.0andtheQmotorRobotic Toolkit-AnAdvancedPC-BasedReal-TimeControlPlatform, IEEEControl SystemsMagazine ,Vol.22,No.3,pp.12-26,2002.

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BIOGRAPHICALSKETCH Chien-HaoLiangwasborninTaipei,Taiwan.HecompletedhisBachelor inOceanEngineeringintheyear2001fromNationalTaiwanUniversity,Taipei. After nishinghismilitaryservice,hejoinedthenonlinearcontrolsandrobotics researchgroupintheUniversityofFloridaforMasterofScienceinmechanicaland aerospaceengineeringintheyear2005. 74


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LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-SPRING SYSTEM
UNDERGOING AN IMPACT COLLISION
















By
CHIEN-HAO LIANG


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2007



































O 2007 Chien-Hao Liang





















To my wife Yen-Chen and son Hsu-Chen who constantly filled me with love and

joy.















ACKNOWLEDGMENTS

I would like to express my gratitude to my advisor, mentor, and friend, Dr.

Warren E. Dixon, for introducing me to the interesting field of Lyapunov-based

control. As an advisor, he provided the necessary guidance and allowed me to try

some stupid ideas during my research. As a mentor, he helped me understand the

high pressure of working in a professional environment and was willing to give me

time to learn and adjust. I feel fortunate in getting the opportunity to work with

him.

I also appreciate my committee members Dr. Carl D. Crane III and Dr. Gloria

J. Wiens for the time and help they provided.

I would like to thank all my friends for their support and encouragement. I

especially thank my friend Guoqiang Hu for being my listener and mentor both

in the research and daily life for these two years. I would also like to thank my

colleagues K~eith Stegath and Gun Lee for helping me out on those difficult days

when I was doing my experiments. Without them, I cannot accomplish these

experiments. I would like to thank my colleague Darren Aiken for his caring during

my first year in U.S. I express my gratitude to K~eith Dupree for helping me out

on my research. I would like to thank Siddhartha Mehta for helping me in thesis

editing. I also express my gratitude to Parag Patre and Will Mackunis who filled

my daily life with joy.

Finally I would like to thank my parents for their unconditional love and

support, my wife Yen-Chen for the support and joy she always brought to me, and

my son Hsu-Chen for his hug, kiss, and smile. Their love, understanding, patience

and personal sacrifice made this thesis possible.


















TABLE OF CONTENTS

page

ACK(NOWLEDGMENTS ................... .......... iv

LIST OF FIGURES ................... ............. vii

ABSTRACT .................. .................. i

CHAPTER

1 INTRODUCTION ................... ........... 1

1.1 Introduction ................... ............ 1
1.2 Dynamic Model ................... .......... 8

2 LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-SPRING
SYSTEM UNDERGOING AN IMPACT COLLISION .. . 11

2.1 Error System and Control Development .. .. .. .. .. .. .. 11
2.1.1 ControlObjective .................. ..... 11
2.1.2 Closed-Loop Error System .. .. .. . .. 12
2.2 Stability Analysis.................. ......... 14
2.3 ExperimentalResults ................... ....... 20
2.4 Concluding Remarks ................... ....... 26

3 ADAPTIVE LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-
SPRING SYSTEM UNDERGOING AN IMPACT COLLISION .. .. 27

3.1 Properties and Assumptions .. .. .. .. .. .. .. .. .. .. 28
3.2 Error System and Control Development .. .. .. .. .. .. .. 29
3.2.1 ControlObjective.................. ..... 29
3.2.2 Closed-Loop Error System .. .. .. . .. 30
3.3 Stability Analysis.................. ......... 34
3.4 ExperimentalResults ................... ....... 40
3.5 Concluding Remarks ................... ....... 47

4 AN IMPACT FORCE LIMITING ADAPTIVE CONTROLLER FOR A
ROBOTIC SYSTEM UNDERGOING A NONCONTACT-TO-CONTACT
TRANSITION ................... ............. 48

4.1 Properties ................... ............. 49
4.2 Error System and Control Development .. .. .. .. .. .. .. 49
4.2.1 ControlObjective.................. ..... 50











4.2.2 Closed-Loop Error System .. .. .. .. .. .. .. .. .. 51
4.3 Stability Analysis ................... ......... 52
4.4 ExperimentalResults ................... ....... 58
4.5 Concluding Remarks ................... ....... 64

5 CONCLUSION AND RECOlillr~DAIENDAIOS .. .. .. .. .. .. 65

APPENDIX

A THE EXPRESSION OF x,d, IN SECTION 2.1 .. .. .. .. .. .. 67

B THE EXPRESSION OF x,dl IN SECTION 3.2 .. .. .. .. .. .. 69

REFERENCES ................... ................ 70

BIOGRAPHICAL SKETCH ................... ......... 74
















LIST OF FIGURES

Figure page

1-1 The Mo-I~-Spring Robot (MSR) system is an academic example of an
impact between two dynamic systems. .. .. .. .. .. .. .. 7

2-1 Top view of experimental testbed. .. .. .. .. .. .. .. .. .. .. 21

2-2 Side view of experimental testbed .. .. .. .. .. .. .. .. .. .. 21

2-3 Spring-mass and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., ery(t)), (b) indicates the position
error of the robot tip along the X2-8XiS (i.e., er2(t)), and (c) indicates
the position error of the spring-mass (i.e., em(t)). .. .. .. .. 24

2-4 Computed control torques J (q)F(t) for the (a) base motor and (b) sec-
ond link motor.................... ............ 24

2-5 Applied control torques J (q)F(t) (solid line) versus computed control
torques (dashed line) for the (a) base motor and (b) second link motor. .25

2-6 Computed desired robot trajectory, xay (t). .. .. .. .. .. .. .. .. 25

2-7 Contact (A = 1) and noncontact (A = 0) conditions for the robot and
mass-spring system. ................... ........... 26

3-1 Mass-spring and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., er,(t)), (b) indicates the position
error of the robot tip along the X2-8XiS (i.e., er2(t)), and (c) indicates
the position error of the mass-spring (i.e., em (t)). .. .. .. .. 42

3-2 Mass-spring and robot errors e(t) during the initial two seconds. .. .. 43

3-3 Computed control torques J (q)F(t) for the (a) base motor and (b) sec-
ond link motor.................... ............ 43

3-4 Applied control torques J (q)F(t) (solid line) versus computed control
torques (dashed line) for the (a) base motor and (b) second link motor. .44

3-5 Computed desired robot trajectory, Xas(t). . . . . 44

3-6 Unitless parameter estimate Odk (t) introduced in (3-13). .. .. .. .. 45










3-7 Estimate for the unknown constant parameter vector Orttl- (a) Brlo(t)
K,, (b) Or4 1)= m, C) ,1(t) Km, and (d) 6,7(t)= -m, where
mi, m2 ERI denote the mass of the first and second link of the robot,
m, ERI denotes the mass of the motor connected to the second link of
the robot, and m ERI denotes the mass of the mass-spring system. .. 45

3-8 Estimate for the unknown constant parameter vector Or (f (a) Grs (t) =
",(b) r2~(t)- s-, and (c) 8,s(t)- s. ............... 46

3-9 Estimate for the unknown constant parameter vector Or (1) (a) Gre (t)
m,, (b) 0r3(t) = 1, and (c) 0r9@t) m2. .. .. .. .. .. .. .. .. 46

4-1 Mass-spring and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., ery(t)), (b) indicates the position
error of the robot tip along the X2-8XiS (i.e., er2(t)), and (c) indicates
the position error of the mass-spring (i.e., em(t)). .. .. .. .. 60

4-2 Applied control torques J (q)F(t) for the (a) base motor and (b) second
link motor. ................... ............. 60

4-3 Applied control torques J (q)F(t) for the (a) base motor and (b) second
link motor during the first 0.8 second. .. .. .. . . 61

4-4 Computed desired robot trajectory, Xay (t). . . . . 61

4-5 Parameter estimate Odk (t) introduced in (3-13). .. .. .. .. .. .. 62

4-6 Estimate for the unknown constant parameter vector Orttl- (a) Brlo(t)
K,, (b) Or4 1)= m, C) ,1(t) Km, and (d) 6,7(t)= -m, where
mi, m2 ERI denote the mass of the first and second link of the robot,
m, ERI denotes the mass of the motor connected to the second link of
the robot, and m ERI denotes the mass of the mass-spring system. .. 62

4-7 Estimate for the unknown constant parameter vector Or (f (a) Grs (t) =
",(b) r2~(t)- s-, and (c) 8,s(t)- s. ............... 63

4-8 Estimate for the unknown constant parameter vector Or (1) (a) Gre (t)
ms, (b) 0r3t) = 1, and (c) 0r9@) m2. .................. 63
















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-SPRING SYSTEM
UNDERGOING AN IMPACT COLLISION

By

Chien-Hao Liang

May 2007

Chair: Warren E. Dixon
Major: Mechanical Engineering

The problem of controlling a robot during a noncontact-to-contact transition has

been a historically challenging problem that is practically motivated by applications that

require a robotic system to interact with the environment. If the contact dynamics are not

properly modeled and controlled, the contact forces could result in poor system

performance and instabilities. One difficulty in controlling systems subj ect to noncontact-

to-contact transition is that the dynamics are different when the system status changes

suddenly from the noncontact state to a contact state. Another difficulty is measuring the

contact force, which can depend on the geometry of the robot, the geometry of the

environment, and the type of contact. The appeal of systems with contact conditions is

that short-duration effects such as high stresses, rapid dissipation of energy, and fast

acceleration and deceleration may be achieved from low-energy sources.

Over the last 2 decades, many researchers have investigated the modeling and

control of contact systems. Most controllers target contacts with a static environment for









a fully actuated system. Many researchers also exploit switching or discontinuous

controllers to accommodate contact conditions. Motivation exists to explore alternative

control strategies because impacts between the robot and the static environment cannot

represent all the impact system applications such as the capture of disabled satellites,

spaceport docking, manipulation of non-rigid bodies, and so on, and discontinuous

controllers require infinite control frequency (i.e., exhibit chattering) or yield degraded

stability results (i.e., uniformly ultimately bounded). As stated previously, it is necessary

to consider the impact control between two dynamic systems.

Our study considered a class of fully actuated dynamic systems that undergo an

impact collision with another dynamic system that is unactuated. Our study is specifically

focused on a planar robot colliding with a mass-spring system as an academic example of

a broader class of such systems. The control objective is defined as the desire to

command a planar robot to collide with an unactuated system and regulate the resulting

coupled mass-spring robot (MSR) system to a desired compressed state. The collision is

modeled as a differentiable impact. Lyapunov-based methods are used to develop a

continuous controller that yields asymptotic regulation of the mass and robot links. A

desired time-varying robot link traj ectory is designed that accounts for the impact

dynamics and the resulting coupled dynamics of the MSR system. A force controller is

then designed to ensure that the robot link position tracking error is regulated. Unlike

some other results in literature, the force controller is continuous and does not depend on

measuring the impact force or the measurement of other acceleration terms. Experimental

results are provided to illustrate the successful performance of the controller in each

chapter.















CHAPTER 1
INTRODUCTION

1.1 Introduction

The problem of controlling a robot during a noncontact-to-contact transition

has been a historically challenging problem that is practically motivated by

applications that require a robotic system to interact with the environment. If the

contact dynamics are not properly modeled and controlled, the contact forces could

result in poor system performance and instabilities. One difficulty in controlling

systems subject to noncontact-to-contact transition is that the dynamics are

different when the system status changes suddenly from the noncontact state to a

contact state. Another difficulty is measuring the contact force, which can depend

on the geometry of the robot, the geometry of the environment, and the type of

contact. As stated by Tornambe [1], the appeal of systems with contact conditions

is that short-duration effects such as high stresses, rapid dissipation of energy, and

fast acceleration and deceleration may be achieved from low-energy sources.

Over the last 2 decades, many researchers have investigated the modeling

and control of contact systems including: [1]-[32]. 2 trends are apparent after a

comprehensive survey of contact systems in control literature. Most controllers

target contacts with a static environment for a fully actuated system. Many

researchers also exploit switching or discontinuous controllers to accommodate

for the contact conditions. The regulation of a one-link robot that undergoes

smooth or non-smooth impact dynamics was examined by Tornambe [1]. A class

of switching controllers were examined by Brogliato et al. in [2] for mechanical

systems with differentiable dynamics subject to an algebraic inequality condition

and an impact rule relating the interaction impulse and the velocity. The analysis










in [2] used a discrete Lyapunov function that required the use of the Dini derivative

to examine the stability of the system. A simple mechanical system subject to

nonsmooth impacts is considered by Menini and Tornambe [3], where the desired

time-varying planar motion of a mass is controlled within a closed region defined

by an infinitely massive and rigid circular barrier. In [4], Sekhavat et al. utilized

LaSalle's Invariant Set Theorem to prove the stability of a discontinuous controller

that is designed to regulate the impact of a hydraulic actuator with a static

environment where no knowledge of the impact dynamics is required. Volpe and

K~hosla developed a nonlinear impact control strategy for a robot manipulator

experiencing an impact with a static environment [5]. The controller in [5] was

based on the concept that negative proportional force gains, or impedance mass

ratios less than unity, can provide impact response without bouncing. Tornambe

[6] also proposed a switching controller to globally asymptotically regulated a two

degree-of-freedom (DOF) planar manipulator to contact an infinitely rigid and

massive surface. Pagilla and Yu [7] proposed a discontinuous stable transition

controller to deal with the transition from a noncontact to a contact state where

explicit knowledge of the impact model is not required. A discontinuous model-

based adaptive controller was proposed by Akella et al. [8] to asymptotically

stabilize the contact transition between a robot and static environment. Tarn et

al. [9] proposed a sensor-referenced control method using positive acceleration

feedback with a switching control strategy for impact control for a robot and a

constrained surface. A switching controller was also proposed by Wu et al. in

[10] to eliminate the bouncing phenomena associated with a robot impacting a

static surface. The structure of the switching controller in [10] was dependent on

impact feedback from a force sensor. Lee et al. developed a hybrid bang-bang

impedance/time-delay controller that establishes a stable interaction between a

robot with nonlinear joint friction and a stiff environment in [11] and [12]. Nelson










et al. [13]-[15] proposed a nonlinear control strategy that considers force and vision

feedback simultaneously and then switches to pure force control when it is unable

to accurately resolve the location of the robot's end-effector relative to the surface

to be contact. Motivation exists to explore alternative control strategy for the

impact systems because impacts between the robot and the static environment

cannot represent all the impact system applications such as the capture of disabled

satellites, spaceport docking, manipulation of non-rigid bodies, and so on, and

discontinuous controllers require infinite control frequency (i.e., exhibit chattering)

or yield degraded stability results (i.e., uniformly ultimately bounded). As stated

previously, it is necessary to consider the impact control between two dynamic

systems.

Several controllers have been developed for under-actuated dynamic systems

that have an impact collision. For example, a family of dead-beat feedback control

laws were proposed by Brogliato and Rio [16] to control a class of juggling-

like systems. One of the contributions in [16] is a study of the intermediate

controllability properties of the object's impact Poincari: mapping. A proportional-

derivative (PD) controller was developed by Indri and Tornambe [17] to address

global asymptotic stabilization of under-actuated mechanical systems subject to

smooth impacts with a static object. In our previous work in [18], a nonlinear

energy-based controller is developed to globally asymptotically stabilize a dynamic

system subject to impact with a deformable static mass. The contribution in [18] is

that the under-actuated states are coupled through the energy of the system as a

means to mitigate the transient response of the unactuated states.

Our study considered a class of fully actuated dynamic systems that undergo

an impact collision with another dynamic system that is unactuated. Our study

is specifically focused on a planar robot colliding with a mass-spring system as

an academic example of a broader class of such systems. The control objective is










defined as the desire to command a planar robot to collide with an unactuated

system and regulate the resulting coupled mass-spring robot (MSR) system to

a desired compressed state. The collision is modeled as a differentiable impact

as in recent work in [1], [17], and our previous efforts in [18]-[20]. Lyapunov-

based methods are used to develop a continuous adaptive controller that yields

asymptotic regulation of the mass and robot links. A desired time-varying robot

link trajectory is designed that accounts for the impact dynamics and the resulting

coupled dynamics of the MSR system. The desired link trajectory converges to a

setpoint that equals the desired mass position plus an additional constant that is

due to the deformation of the mass. A force controller is then designed to ensure

that the robot link position tracking error is regulated. Unlike some other results

in literature, the continuous force controller does not depend on measuring the

impact force or the measurement of other acceleration terms: only the position

and velocity terms of the spring-mass system and the joint angles and the angular

velocities terms of the planar robotic arm are needed for the proposed controller.

Chapter 2 and our preliminary efforts in [19] provide a first step at controlling

the proposed impact system. The control development in Chapter 2 is based on the

assumption of exact model knowledge of the system dynamics. The nonlinear con-

tinuous Lyapunov-based controller is proven to regulate the states of a planar robot

colliding with the unactuated mass-spring system and yields global asymptotic

result.

In Chapter 3 and our preliminary results in [20], the dynamic model for

the system is assumed to have uncertain parameters. The control objective is

defined as the desire to command the planar robot to collide with the mass-spring

system and regulate the resulting coupled mass-spring robot (MSR) system to a

desired compressed state while compensating for the constant, unknown system

parameters. Two linear parameterizations are designed to adapt for the unknown










robot and mass-spring parameters. An adaptive nonlinear continuous Lyapunov-

based controller is proven to regulate the states of the systems and yields global

asymptotic regulation result.

When the controllers in Chapter 2 and Chapter 3 were implemented in the

presence of large initial conditions, violent impacts between the robot and the

mass-spring system resulted. In fact, the controller was artificially saturated (the

saturation effects were not considered in the stability analysis) to reduce the

impact forces so that the mass deflection would not destroy a capacitance probe.

Various researchers have investigated methods that prescribe, reduce, or control

the interaction forces during or after the robot impact with the environment such

as [6]-[15] and [21]-[34] because large interaction forces can damage both the robot

and/or the environment or lead to degraded performance or instabilities. Walker

and Gertz et al. exploited kinematic redundancy of the manipulator to reduce the

impact force in [21]-[23]. A two degree-of-freedom (DOF) planar manipulator was

globally asymptotically regulated to contact an infinitely rigid and massive surface

by Tornambe [6] where the impact force was estimated using a reduced-order

observer. Pagilla and Yu [7] proposed a stable transition controller to deal with

the transition from a noncontact to a contact state which can improve transition

performance and force regulation. A discontinuous model-based adaptive controller

was proposed by Akella et al. [8] to asymptotically stabilize the contact transition

between a robot and static environment. The controllers for each contact state were

tuned independently to reduce contact force during the process of making contact

with the environment. Tarn et al. [9], [10] proposed a sensor-referenced control

method using positive acceleration feedback with a switching control strategy for

impact control and force regulation for a robot and a constrained surface where the

peak impulsive force and bouncing caused directly by overshooting and oscillation

of the transient force response can be reduced. Lee et al. developed a hybrid










bang-bang impedance/time-delay controller that establishes a stable contact and

achieves the desired dynamics for contact or noncontact conditions in [11] and [12],

where the force overshoots can be minimized. Nelson et al. [13]-[15] proposed a

switching nonlinear controller that combines force and vision control. When the

robot's end-effector approaches the target, the controller switches to force control

to minimize impact force and to regulate the contact force. By modeling the

impact dynamics as a state dependent jump linear system, Chiu and Lee were able

to apply a modified stochastic maximum principle for state dependent jump linear

systems to optimize the approach velocity, the force transient during impact and

the steady state force error after contact is established [24]. Hyde and Cutkosky

[25] proposed an approach, based on input command shaping, to suppress vibration

during the contact transition of switching controllers by modifying feedforward

information. Various other applications also focused on the reduction of impact

force between different systems during the control process such as impact force

reduction of hopping robot considered by Shibata and Natori [26] and Ohnishi et

al. [27], [28], bilateral telerobotic system considered by Dubey et al. [29], human-

robot symbiotic environment considered by Yamada et al. [30] and Li et al. [31],

and space manipulator and free-flying target considered by Huang et al. [32].

Exploring alternative methods is motivated because kinematic redundancy is not

always possible, and again the discontinuous controllers require infinite control

frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly

ultimately bounded).

These results provide the motivation for the control development in Chapter

4. Specifically, the feedback elements for the continuous controller in Chapter

3 are contained inside of hyperbolic tangent functions as a means to limit the

impact forces resulting from large initial conditions as the robot transitions from

noncontact to contact states. Although saturating the feedback error is an intuitive



















X
I~ r91 ~I
I


Figure 1-1: The Me I~-Spring Robot (MSR) system is an academic example of an
impact between two dynamic systems.


solution that has been proposed in previous literature for other types of robotic

systems with limited actuation, several new technical challenges arise due to the

impact condition. The main challenge is that the use of saturated feedback does

not allow some coupling terms to be canceled in the stability analysis, resulting in

the need to develop state dependent upper bounds that reduce the stability to a

semi-global result (as compared to the global results in Chapter 2 and Chapter 3).

The semi-global result is problematic in the current applicative context because

certain control terms do not appear in the closed-loop error system during the

noncontact condition, resulting in a uniformly ultimately bounded result until the

robot makes contact. Hence, the result hinges on new development within the semi-

global stability proof for an error system that is only uniformly ultimately bounded

during the noncontact phase. This problem is exacerbated by the fact that the

Lyapunov function contains radially unbounded hyperbolic functions of some states

that only appear inside of saturated hyperbolic terms in the Lyapunov derivative.

New control development, closed-loop error systems, and Lyapunov-based stability

analysis arguments are used to conclude the result. Experimental results are

provided to illustrate the successful performance of the controller in each chapter.









1.2 Dynamic Model

The subsequent development is motivated by the academic problem illustrated

in Fig. 1-1. The dynamic model for the two-link revolute robot depicted in Fig.

1-1 can be expressed in the joint-space as


M~(q)q + C(q, q)q + h(q) = 7, (1-1)

where q(t), q(t), q(t) E RW2 represent the angular position, velocity, and acceleration

of the robot links, respectively, M~(q) E RW2x2 represents the uncertain inertia

matrix, C(q, q) E RW2x2 represents the uncertain centripetal-Coriolis effects, h(q) n

[hi (q), h2(~l E I2 repreSentS uHCertain COHSerVatiVe forCeS (e.g., gravity), and

7(t) E RW2 represents the torque control inputs. The Euclidean position of the end-

point of the second robot link is denoted by x,(t) n [x,l(t), x(tr2T )F 6 2, which

can be related to the joint-space through the following kinematic relationship:


Xr = J(q)q, (1-2)


where J(q) E RW2x2 denotes the manipulator Jacobian. The unforced dynamics of

the mass-spring system in Fig. 1-1 are


mim + ks (Xm Xo) = 0, (1-3)


where xm(t), xm(t), xm(t) ERI represent the displacement, velocity and acceleration

of the mass m E R, so ERI represents the initial undisturbed position of the mass,

and ks ERI represents the stiffness of the spring.

Assumption 1.1: We assume that x,z (t) and xm(t) can be bounded as


<~i I Xr(t) wZm7 < Xm(t)l < Om (1-4)


where P0, ERI is a known constant that is determined by the minimum coordinate

of the robot along the X1-axis, and pm ERI is a known positive constant. The









lower bound assumption for x,z (t) is based on the geometry of the robot, and the

upper bound assumption for xm(t) is based on the physical fact that the mass is

attached by the spring to some object, and the mass will not be able to move past

that object.

In the following, the contact model is considered as an elastic contact with

finite stiffness. An impact between the second link of the robot and the spring-mass

system occurs when xwe(t) > xm(t) (see Fig. 1-1). The impact will yield equal and

opposite force reactions between the robot and mass-spring system. Specifically,

the impact force acting on the mass, represented by Fm(x,, xm) E RW, is assumed to

have the following form [1], [17]


Fm = Kc,A(xrl Xm), (1-5)

where K, ERI represents a positive stiffness constant, and A(x,, xm) ERI is defined


A = 1 we > Xm(16
0=( X,1 < Xm-1~

The impact force acting on the robot links produces a torque, denoted by

7d(X,, Xm, q) E RW2, aS follows.


7a = KIAn (sX, Xm) I I sin(ql) + 12 sinq 92 ql 17


where IIl, 12 ER denote the robot link lengths. Based on (1-1), (1-3), and (1-5)-(1-

7), the dynamic model for the MSR system can be expressed as

M~(q)q + C(q, q)q + h(q) 7d = 7
(1-8)
mxm + k~s(Xm Xo) = Fm-









After premultiplying the robot dynamics by the inverse of the Jacobian

transpose and utilizing (1-2), the dynamics in (1-8) can be rewritten as [19], [20]


M~ (x,) x, + C (x,, x,) x, + h(x,) + Fm = F (1-9)


mxm + ks (Xm Xo) = Fm, (1-10)

where F(t) n J-T(q)7(t) E RW2 denotes the manipulator force. The dynamic model

in (1-9) and (1-10) exhibits the following properties that will be utilized in the
subsequent analysis.

Property 1.1: The inertia matrix M~(x,) is symmetric, positive definite, and
can be lower and upper bounded as


alI1 2 I TM(E < a8I12, ~ E2 211

where al, a2 ERI are positive constants.

Property 1.2: The following skew-symmetric relationship is satisfied


T( 12I(XT) C(XT, XT))( = 0 V( e RW2. (1-12)















CHAPTER 2
LYAPUNOV-BASED CONTROL OF A ROBOT AND MASS-SPRING SYSTEM
UNDERGOING AN IMPACT COLLISION

This chapter and our preliminary efforts in [19] provide a first step at con-

trolling the proposed impact system in Section 1.2. The academic example of a

planar robot colliding with an unactuated mass-spring system is used to represent

a broader class of such systems. The control development in this chapter is based

on the assumption of exact model knowledge of the system dynamics. The control

objective is to command a robot to collide with an unactuated mass-spring sys-

tem and regulate the spring-mass to a desired compressed state. Lyapunov-based

methods are used to develop a continuous controller that yields global asymp-

totic regulation of the spring-mass and robot links. Unlike some other results in

literature, the developed continuous force controller does not depend on sensing

the impact, measuring the impact force, or the measurement of other acceleration

terms. Experimental results are provided to validate our analysis.

This chapter is organized as follows. Section 2.1 describes the error system and

control development followed by the stability analysis in Section 2.2. Section 2.3

describes the experimental set up and results that indicate the successful perfor-

mance obtained by implementing the proposed controller followed by conclusion in

Section 2.4.

2.1 Error System and Control Development

2.1.1 Control Ob jective

One goal in this chapter is to regulate the states of a dynamic system (i.e., a

two-link planar robot) that has an impact collision with another dynamic system

(i.e., a mass-spring). A regulation error, denoted by e(t) E RW3, is defined to









quantify the control objective as

e= E e" em~ (2-1)


where e,(t) n [ery,er2FT E I2 and em(t) ERI denote the regulation error for the

robot and mass-spring system, respectively, and are defined as


e, = Xwe X, em, = sma Xm. (2-2)


In (2-2), sma ERI denotes the constant known desired position of the spring-mass.

The desired position of the end-point of the second robot link, denoted by xwa(t)

a [X,dl, Xrd2] I2, is selected so that the robot will produce the desired spring-

mass position while accounting for the impact dynamics. Specifically, xay (t) (i.e.,

the desired horizontal Euclidean coordinate in Fig. 1-1) is a time-varying signal

that is subsequently designed to account for the impact condition and the coupled

dynamic response of the MSR system, and Zrd2 (i.e., the desired vertical Euclidean

coordinate in Fig. 1-1) is selected as a constant. Filtered tracking errors, denoted

by r,(t) E RW2 and rm(t) E RW, are defined as


r, = e, + ae, rm = em + aem (2-3)


to facilitate the subsequent control design and stability analysis where a ERI is a

positive control parameter.

2.1.2 Closed-Loop Error System

By taking the time derivative of mrm(t) and utilizing (2-2) and (2-3), the

following open-loop error system can be obtained:


m~im = k(xm xo) KA(x,l xm) + amem (2-4)


where the spring-mass dynamics in (1-10) were substituted for mxm(t). Motivated

to design the desired robot link trajectory to position the spring-mass, (2-2) is used









to rewrite the open-loop system in (2-4) as


m~im = k(xm xo) + amem + Knery + Knxm KAXar. (2-5)


Based on (2-5), the desired robot link position is designed as

1 1
wa (amem ,+ k(xm ro)) + ( +k)m+ m(2-6)
Xd K K

Xrd2 =


where seREI is an appropriate positive constant (i.e., e is selected so the robot will

impact the mass-spring system), and ki and k2 ERI are positive constant control

gains. After substituting (2-6) into (2-5), the closed-loop error system for rmit)
can be obtained as


m~im = (1 A) (k(xm xo) + amem) + Kners A (ki + k2) rm. (2-7


As xm(t) z ma, (2-2) and (2-3) can be used to conclude that rm(t) 0,

em(t) 0, and em(t) 0 Hence, (2-6) can be used to conclude that


Xay (t) K ,(Xm Xo) + zma. (2-8)


The physical meaning of (2-8) is that the desired robot position varies in time to

account for the impact dynamics and the coupled dynamic system, and the desired

steady-state value is a constant that equals the desired spring-mass position plus

the mass deformation.

After taking the time derivative of r,(t) and premultiplying by the robot

inertia matrix, the following open-loop error system is obtained:


MFi, = M~xre F + Cx, + h + KA(x,l xm) + al~e,, (2-9)









where (1-10) and (2-2) were utilized. Based on (2-9) and the subsequent stability

analysis, the robot force control input is designed as


F n M~Xre + CXre + h + + k37r x) S11 + a MeC, + Ce,.) + e, (2 10)


where k3 ERI is a positive constant control gain. Based on the use of the backstep-

ping method, the robot force control input in (2-10) requires the first and second

derivative of Xrd(t). As described in the Appendix A, the first and second derivative

of X,d(t) exists (i.e., x,d(t) is continuously differentiable) and do not depend on

acceleration terms. The closed-loop error system for rr (t) can be obtained after

substituting (2-10) into (2-9) as


M~i, = k3Tr Crr Br (2-11)


2.2 Stability Analysis

Ti;,~ ..;~, n: The controller given by (2-10) ensures global asymptotic stability of

the robot and spring-mass regulation errors in the sense that


I, (t)| 0 ||e,(t)||I 0 as t oo (2-12 )


provided a, ky, and k2 are selected sufficiently large and the following gain condi-
tion is satisfied:
K2
a > (2-13)

In the following proof, a Lyapunov function and its derivative are provided.

The analysis is then separated into two cases: contact and noncontact. For the

noncontact case, the stability analysis indicates the controller and error signals are

bounded and converge to an arbitrarily small region. Additional analysis indicates

that within this region, contact must occur. When contact occurs, a Lyapunov









analysis is provided that illustrates the MSR system asymptotically converges to

the desired setpoint.

Proof: Let V(rm, Tr, e,) ERI denote the following continuously differentiable,

nonnegative, radially unbounded function (i.e. a Lyapunov function candidate)

1 1 1
VA mr2 + -r,"l~r, + -e 'er. (2-14)
2"2 2

The Lyapunov function candidate in (2-14) can be lower and upper bounded as


11 ||-1< V < yz : 2-15)


where yl, y2 ERI are positive constants, and z(t) E R"5 is defined as





The time derivative of (2-14) can be determined as







where (1-12), (2-3), (2-7), and (2-11) were utilized. The remainder of the analysis

is divided into two cases: Case 1-the robot and mass-spring systems are not in

contact (i.e., wel < xm and A = 0), and Case 2-the robot and mass-spring systems

are in contact (i.e., wel > xm and A = 1).

Case la: Before the initial contact, the mass-spring system is at rest and the

spring is not compressed; hence,


k(xm xo) = 0 em = 0. (2-17)

Based on (2-17), (2-16) can be expressed as


V = -k3r,~ -- 0, Br.


(2-18)









The expressions in (2-14) and (2-18) can be used to prove that e, (t), r, (t),

rm(t) E C, and that e,(t), r,(t) E 2. Based on the fact that e,(t), r,(t),

rm(t) E C,, standard signal chasing arguments can be used to prove that e,(t),

r,(t), Xwa(t), F(t) E Coo along with all of the other closed-loop signals. Since e,(t),

r, (t) E C, n 2 and are uniformly continuous, Barbalat's Lemma can be applied to
conclude that

|| r, (t)||I 0 ||e,(t)||I 0 as t oo. (2-19)

The result in (2-19) can be used along with (2-6) to conclude that


X,dl(t) (ki1 + k2 tem 0 x

leading to an impact with the mass-spring system.

Case lb: After an impact, the robot may loose contact with the spring-mass.

In this case

k(xm xo) / 0 em, = -j:m. (2-20)

Based on (2-20), (2-16) can be expressed as




For this case, the initial velocity of the spring-mass is denoted by ( ERI and the

initial position is denoted by xm E RW. Given the aforementioned initial conditions,

the solution for am (t) can be obtained from (1-10) as


xm(t) = zo + (xm xo) cos(l -t) + (iI sin(l -t). (2-22)

The time derivative of (2-22) can be expressed as


j:m(t) = (xm xo) II- sin(l t)+(o( -t.(-3
Sm m m23









Based on (2-22) and (2-23), rm(t) can be expressed as follows:


rm = (xm xo) ii- sin(l -t) ( cos(l -t) + max axm(t). (2-24)
Vm m Vm

The expressions in (2-22)-(2-24) can be upper bounded as


|zm| < |zoI + |Im Xo| + |(|A I < (1 (2-25)


|sml < Im xo i + IEl P2 (2-26)
Sm

|rm| < |I m Xo| i + |C| + a~ (xmm + (1) < P3 (2-27)
Sm

where (1, (2, P3 ERI denote positive constants. After utilizing (2-25)-(2-27), an

upper bound for (2-21) can be developed as


V I V + 6 2-28>


where 6 ERI is a positive constant, and p ERI is defined as

8 min(k3, a, 1)


Standard techniques can be used to solve (2-28) for V(t) as


V~t < (0)- e~xp(--t)f + -. (2-29)


Based on (2-29), it is clear that during the transient case (Case lb) that e, (t),

r,(t), rm(t) E C,. Based on the fact that e,(t), r,(t), rm(t) E C,, standard signal

chasing arguments can be used to prove that e,(t), i,(t), x,d(t), F(t) E C, along

with all of the other closed-loop signals.

As V(t) 0 eventually PV < 6. The previous development can be used to
conclude that for the noncontact case


V and hence, || r, (t) || || e, (t) || |rm(t) | as t oo. (2-30)









Further analysis is required to prove that the manipulator makes contact with

the mass-spring system. Contact between the manipulator and the mass-spring

system occurs when xmi(t) > xm(t). Based on (2-30), a sufficient condition for

contact can be developed as



After using (2-6), the sufficient condition in (2-31) can be expressed as

1 1 6
(amem + k(xm ro)) + (k 2 m' .(2-32)
K K P

By using (2-3) and performing some algebraic manipulation, the inequality in

(2-32) can be expressed as

6K
(ki + k2) rm '> + m Tm, I 2m | + k~(Xma + I'. + Xo). (2-33)


From Assumption 1.1, em (t) can be upper bounded as


em < m (2-34)


where, Em ER denotes a known positive constant. If em(t) < 0, then the sufficient

condition in (2-33) may not be satisfied. The condition that em(t) < 0 will

only occur if an impact collision occurs that causes the mass to overshoot the

desired position. However, even if an impact occurs and the mass overshoots

the desired position, the dynamics of the mass-spring system will force the mass

position error to return to the initial condition in the noncontact case. That is,

em(t) zma~ Xo > Em where mERI denotes a known positive constant. Based on

(2-24), (2-27) and the fact that em,(t) will eventually be lower bounded by m in a

noncontact condition, rm(t) can be upper and lower bounded as


P3 > m m 0 X O I a, m r (2-35)
Sm









where 8, ERI denotes a known positive constant, provided the following gain

condition is satisfied

|zm Xo i + I(
a > m (2-36)
-m
Based on (2-33), (2-34), (2-35), and the previous development, the control

parameter ki and k2 can be selected according to the following sufficient condition

to ensure the robot and mass-spring system make contact

ki~" + k2 t + 2mem + kC(Xma + Em + Xo) (-7


Case 2: When the robot and mass-spring systems are in contact, (2-16) can

be upper bounded as


V' < -kI ||rm 2l k3 Ilr 2l Q lr 12 + [K ||rm|| ||e7|| ki2 Il, 12]. (2-38)


After completing the squares on the bracketed terms, the following expression is

obtained:

V <-k |rm||2s i 3T 2-0- 4k r 2 .i (2-39)

Provided the gain condition given in (2-13) is satisfied, (2-15) can be used to

upper bound (2-39) as

Vi < YV (2-40)

where y3 ERI is a positive constant. The expression in (2-40) indicates that while

in contact, the robot and spring-mass position errors are exponentially regulated,

and e,(t), r,(t), rm(t) E C,. Based on the fact that e,(t), r,(t), rm(t) E m,,

standard signal chasing arguments can be used to prove that e,(t), i,(t), X,d(t),

F(t) E C, along with all of the other closed-loop signals.









2.3 Experimental Results

The testbed depicted in Fig. 2-1 and Fig. 2-2 was developed for experimental

demonstration of the proposed controller. The testbed is composed of a mass-

spring system and a two-link robot. The body of the mass-spring system includes

a U-shaped aluminum plate (item (8) in Fig. 2-1) mounted on an undercarriage

with porous carbon air bearings which enables the undercarriage to glide on an

air cushion over a glass covered aluminum rail. A steel core spring (item (1) in

Fig. 2-1) connects the undercarriage to an aluminum frame, and a linear variable

displacement transducer (LVDT) (item (2) in Fig. 2-1) is used to measure the

position of the undercarriage assembly. The impact surface consists of an aluminum

plate connected to the undercarriage assembly through a stiff spring mechanism

(item (7) in Fig. 2-1). A capacitance probe (item (3) in Fig. 2-1) is used to

measure the deflection of the stiff spring mechanism. The two-link robot (items

(4-6) in Fig. 2-1) is made of two aluminum links, mounted on 240.0 Nm (base

link) and 20.0 Nm (second link) direct-drive switched reluctance motors. The

motors are controlled through power electronics operating in torque control mode.

The motor resolvers provide rotor position measurements with a resolution of

614400 pulses/revolution, and a standard backwards difference algorithm is used

to numerically determine velocity from the encoder readings. A Pentium 2.8 GHz

PC operating under QNX hosts the control algorithm, which was implemented

via Qmotor 3.0, a graphical user-interface, to facilitate real-time graphing, data

logging, and the ability to adjust control gains without recompiling the program

(for further information on Qmotor 3.0, the reader is referred to [40]). Data

acquisition and control implementation were performed at a frequency of 2.0 kHz

using the ServoToGo I/O board.

















































Figure 2-1: Top view of experimental testbed.


Figure 2-2: Side view of experimental testbed


1.* `


e*lr'. ~
:~q









The parameters for the dynamic model in (1-9) and (1-10) have the following
values :


mi = 7.10 [kg] m2 = 1.11 [kg]

m =13.6 [kg] m, = 10.0 [kg]

lI = 0.37 [m] 12 = 0.17 [m]

k = 2000 [N/m] K = 1.8 x 106 [N/m] ,

where mi, m2 ERI represent the mass of the first and second link, m ERI

is the mass of the cart, and m, ERI is the mass of the motor that drives the

second link. The control gains a and k~3, defined as scalars in (2-3) and (2-10),

were implemented (with nonconsequential implications to the stability result) as

diagonal gain matrices to provide more flexibility in the experiment. Specifically,
the control gains were selected as


ki= k = 10000 k3 = ding {250, 12} a = ding {95, 100, 90} .

The initial conditions for the robot coordinates and the spring-mass position were

(in [m])

[r,(0) i;r2(0) 21m(0) ]= 0016 0.487 0.203 -

The initial velocity of the robot and spring-mass were zero, and the desired spring-

mass position was tin [mml

Xma = 233.

That is, the tip of the second link of the robot was initially 217 mm from the

desired setpoint and 187 mm from impact along the X1-axis (see Fig. 2-1).

Therefore, once the initial impact occurs, the robot is required to depress the

spring (item (1) in Fig. 2-1) to move the mass 30 mm along the X1-axis.









The mass-spring and robot errors (i.e., e(t)) are shown in Fig. 2-3. The peak

steady-state position error of the robot tip along the X1-axis (i.e., |erl|) and along

the X2-aXiS (i.e., er2 |) are 9.6 pm and 92 pm, respectively. The peak steady-state

position error of the spring-mass (i.e., '~. |) is 7.7 pm. The 92 pm is due to the lack

of the ability of the model to capture friction and slipping effects on the contact

surface. In this experiment, a significant friction effect is present along the X2-8XiS

between the robot tip and the contact surface due to a large normal spring force

that is applied along the X1-axis.

The input control torques (i.e., J (q)F(t)) are shown in Fig. 2-4 and Fig. 2-5.

To constrain the impact force to a level that ensured the aluminum plate did not

flex to the point of contact with the capacitance probe, the computed torques are

artificially saturated. Fig. 2-4 depicts the computed torques, and Fig. 2-5 depicts

the actual torques (solid line) along with the computed torques (dashed line). The

resulting desired trajectory along the X1-axis (i.e., xar (t)) is depicted in Fig. 2-6,

and the desired trajectory along the X2-axis was chosen aS Xrd2 = 368 mm. Fig.

2-7 depicts the value of A(x,, xm) that indicates contact (A = 1) and noncontact

(A = 0) conditions for the robot and mass-spring system. A video of the experi-

ment is provided at http://www.mae.ufl.edu/~dixon/projects/robaimcthm
































































-1000

-2000

0 05 1 15 2 25 3 35 4

(b)
50







0 05 1 15 2 25 3 35 4


-500
0


05 1


15 225
(b)


3 35 4


0
-20
005s1


15 225
(c)


3 35 4


S20


200 05 1 15 2 25 3 35 4
[sec]



Figure 2-3: Spring-mass and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., en (t)), (b) indicates the position error of

the robot tip along the X2-aXiS (i.e., er2(t)), and (c) indicates the position error of

the spring-mass (i.e., em(t))-










(a)


Figure 2-4: Computed control torques J (q)F(t)
second link motor.


for the (a) base motor and (b)


























02 04 06 08

(b)


02 04 06
[sec]


Figure 2-5: Applied control torques J (q)F(t) (solid line) versus computed control
torques (dashed line) for the (a) base motor and (b) second link motor.













240


235


230


15 2
[sec]


3 35 4


Figure 2-6: Computed desired robot trajectory, xay (t).
















08

06

04

02


-0-
-00 05 1 15 2 25 3 35 4
[sec]


Figure 2-7: Contact (A = 1) and noncontact (A = 0) conditions for the robot and
mass-spring system.


2.4 Concluding Remarks

A nonlinear Lyapunov-based controller is proven to regulate the states of a

planar robot colliding with an unactuated mass-spring system. The continuous

controller yields global asymptotic regulation of the spring-mass and robot links.

Unlike some other results in literature, the developed continuous force controller

does not depend on sensing the impact, measuring the impact force, or the mea-

surement of other acceleration terms. Innovative analysis methods are used to

prove the stability of the system during contact and during different noncontact

states. Experimental results are provided to illustrate the successful controller

performance.















CHAPTER 3
ADAPTIVE LYAPUNOV-BASED CONTROL OF A ROBOT AND
MASS-SPRING SYSTEM UNDERGOING AN IMPACT COLLISION

Similar to Chapter 2, this chapter and our preliminary results in [20] consider

a fully actuated planar robot colliding with an unactuated mass-spring system,

but unlike Chapter 2 the dynamic model for both the mass-spring and robot

systems and the impact force are assumed to have uncertain parameters. The

control objective is also defined as the desire to command the robot to collide

with an unactuated system and regulate the resulting coupled MSR system to a

desired compressed state while compensating for the constant, unknown system

parameters. Specifically, in the dynamic model of Section 1.2, M\(q) E RW2x2

represents the uncertain inertia matrix, C(q, q) E RW2x2 represents the uncertain

centripetal-Coriolis effects, and h (q) n [hi (q), h 2 9)l E 6 2 repreSentS uHCertain

conservative forces (e.g., gravity), to ERI represents the unknown mass and ks ERI

represents the unknown stiffness of the spring, and K, ERI represents the unknown

positive stiffness constant.

To compensate for the uncertainty, adaptive Lyapunov-based methods are used

to develop a continuous adaptive controller that yields global asymptotic regulation

of the mass and robot links. Two linear parameterizations are designed to adapt

for the unknown robot and mass parameters. A desired time-varying robot link

trajectory is designed that accounts for the impact dynamics and the resulting

coupled dynamics of the MSR system. The desired link trajectory converges to a

setpoint that equals the desired mass position plus an additional constant that is

due to the deformation of the mass. A force controller is then designed to ensure

that the robot link position tracking error is regulated.









This chapter is organized as follows. The associated properties and assump-

tions are provided in Section 3.1. Section 3.2 describes the error system and control

development followed by the stability analysis in Section 3.3. Section 3.4 describes

the experimental results that indicate the successful performance of the proposed

controller followed by conclusion in Section 3.5.

3.1 Properties and Assumptions

Property 3.1: The robot dynamics given in (1-9) can be linearly parameter-
ized as


Y(XT, in, XT)0 = M~ (XT) XT + C (XT, XT) XT + h(XT) + Fm


where 8 E R~p contains the constant unknown system parameters, and Y(X,, in,

XT) E RW2xp denotes the known regression matrix.

Assumption 3.1: We assume that the mass of the mass-spring system can be

upper and lower bounded as

mi < m
where mi, m, ERI denote known positive bounding constants. The unknown

stiffness constants K, and ks are also assumed to be bounded as


-,
where ( i:,1 (K~ Cks ERI denote known positive bounding constants.

Assumption 3.2: During the subsequent control development, we assume

that the minimum singular value of J(q) is greater than a known, small positive

constant 6 > 0, such that max {||J-l(q)||} is known a priori, and hence, all

kinematic singularities are always avoided.









3.2 Error System and Control Development

The subsequent control design is based on integrator backstepping methods. A

desired trajectory is designed for the robot (i.e., a virtual control input) to ensure

the robot converges to and impacts with the mass, and to ensure that the robot

regulates the mass to the desired position. Since we can not directly control the

robot trajectory, a force controller is developed to ensure that the robot tracks the

desired trajectory despite the transition from free motion to an impact collision and

despite uncertainties throughout the MSR system. As is typical of the backstepping

design method, the derivative of the desired robot trajectory is required to develop

the force controller. Taking the derivative of the desired trajectory could lead to

unmeasurable higher order terms (i.e., acceleration). The subsequent development

exploits the hyperbolic filter structure developed in [38] to overcome the problem of

injecting higher order terms in the controller and to facilitate the development of

sufficient gain conditions used in the subsequent stability analysis.

3.2.1 Control Ob jective

The control objective is to regulate the states of an uncertain dynamic system

(i.e., a two-link planar robot) that has an impact collision with another uncertain

dynamic system (i.e., a mass-spring). A regulation error, denoted by e(t) E RW3, iS

defined to quantify this objective as


e= I em e"


where e,(t) n [ery,er2FT E I2 and em(t) ERI denote the regulation error for the

end-point of the second link of the robot and mass-spring system (see Fig. 1-1),

respectively, and are defined as


e, = Xwe X, em = sma xm.


(3-2)









In (3-2), sma ERI denotes the constant known desired position of the mass, and

X~d(t) n [X,ax(t), Xrd2] I2 denotes the desired position of the end-point of the
second link of the robot. The subsequent development is based on the assumption

that q(t), q(t), xm(t), and j:m(t) are measurable, and that x,(t) and j:,(t) can

be obtained from q(t) and q(t). To facilitate the subsequent control design and

stability analysis, filtered tracking errors, denoted by rm (t) ERI and r, (t) E RW2, are

defined as [38]


rlm = em + al tanh(em) + a2 tanh(ef) (3-3)

Tr e, + ae,,

where a, al, a2 ERI are positive, constant gains, and ef(t) ERI is an auxiliary filter

variable designed as [38]


ef = -a3 tanh(ef) + a2 tanh(em) ki cosh2 6f )rm,(34

where ki ERI is a positive constant control gain, and a3 ERI is a positive constant

filter gain.

3.2.2 Closed-Loop Error System

By taking the time derivative of mrlm(t) and utilizing (1-5), (1-10), (3-2), and

(3-3), the following open-loop error system can be obtained:

milm = Yabla KAi (sX, 2m) + r2m cosh-2 (ep)i-f f olmcosh-2(em)em,. (3-5)

In (3-5), Yd(Xm) n (Xm Xo) and Odn A k. To facilitate the subsequent analysis, the
following notation is introduced [38]:


Ydla = YaKK,-'94 = Ydkcedk (3-6)

= K,,, (m o









After using (3-3) and (3-4), the expression in (3-5) can be rewritten as


milm = Ydea + K, (Xway Az,2) + Kc,Amm KIsX~d + X a2mk10lm, (3-7)

where X(em, ef, rim, t) ERI is an auxiliary term defined as


S=aim cosh-2( 6m) trm 01tanh(em,)) tit~m cosh-2(em) tanh(ef) (3-8)

+ Ll2mn cosh-2 6/~i) (--3 tanh(e7~i)) + az2m cosh-2 6/.) (L2~ tanh(em)) -

The auxiliary expression X(em, ef, rim, t) defined in (3-8) can be upper bounded as


IX | (1 | ,3-9)

where (1 ERI is a positive bounding constant, and z(t) E RW3 is defined as


z = Im tanh(em,) tanh(e7) (3-10)

Based on (3-7) and the subsequent stability analysis, the desired robot link

position is designed as

Ild ~ ddk + 1Cm + k2z tanh(em) kik2L~ cosh2(eyi) tanh(ef) (3-11)

Xrd2 = -~


In (3-11), seREI is an appropriate positive constant (i.e., e is selected so the robot
will impact the mass-spring system in the vertical direction), k2 ERI is a positive

constant control gain, and the control gain ki ERI is defined as

S1
kiC -^ (3 + k,l1(f) (3 12)

where kni ERI is a positive constant nonlinear damping gain. The parameter

estimate Odk (t) ERI in (3-11) is generated by the adaptive update law


Ojdk npTOjTFY / ).


(3-li









In (3-13), r ERI is a positive constant, and proj(-) denotes a sufficiently smooth

projection algorithm [39] utilized to guarantee that Odk (t) can be bounded as


-, dk e Od 8dkc (3-14)

where Odkc 8dkc ERI denote known, constant lower and upper bounds for Odkct)

respectively.

After substituting (3-11) into (3-7), the closed-loop error system for rlm(t) can

be obtained as


mrilm = KI (Zpar Azl,l) + KI (Amm, Ism) + Klklik2 cosh2 (e ) tanh(e y) (3-15)

+ Ydkcedk KI~k2 tanh(em) + X a2mklrlm-


In (3-15), the parameter estimation error Odk (t) ERI is defined as

edkc edk idk-


The open-loop robot error system can be obtained by taking the time deriva-

tive of r, (t) and premultiplying by the robot inertia matrix as


M~i, = Y8, Cr, F, (3-16)


where (1-9), (3-2), and (3-3) were utilized, and


Y,, = M~xre + al~e, + h + C~rd + aCzC~r + KI~ m aCx,, (3-17)


where Y(x,, x,, xm, im, ef, rlm, t) E RW2xP denotes a known regression matrix, and 8,

E RW~denotes an unknown constant parameter vector. See Appendix A for a linearly

parameterizable expression for M~ (x,) Xrd(t) that does not depend on acceleration

terms. Based on (3-16) and the subsequent stability analysis, the robot force









control input is designed as


FA n ,, + e, + k37r, 38

where k3 ERI is a positive constant control gain, and 8,(t)l E RWP is an estimate for

8, generated by the following adaptive update law


8, T rjFYr) (3-19)


In (3-19), r, E RWPxP is positive definite, constant, diagonal, adaptation gain
matrix, and proj(-) denotes a projection algorithm utilized to guarantee that the

i th element of Or (t) can be bounded as




where 8,i, 8,i ERI denote known, constant lower and upper bounds for each element

of 8,(t)l, respectively.

The closed-loop error system for rr (t) can be obtained after substituting (3-18)

into (3-16) as

M~i, = Y,0, k~37r Cr, er. (3-20)

In (3-20), the parameter estimation error Or t)l E RWP is defined as

8, 8, 8,. (3-21)

Remark 3.1: Based on (3-18), the control torque input can be expressed as

7 = JT (Y, + e, + k37r (3-2)


where J(q) denotes the manipulator Jacobian introduced in (1-2).









3.3 Stability Analysis

Ti;,~ ..;~, u: The controller given by (3-11), (3-13), (3-18), and (3-19) ensures

global asymptotic regulation of the MSR system in the sense that


I, (t)| 0 ||Ie,(t)| 0I~ as t 00o


provided ky, k2, and km1 are selected sufficiently large and the following sufficient

gain condition is satisfied:


a2 > maX1 -, PK (3-23)


where pGm, Pir, PK, and a are defined in (1-4) and (3-1), respectively.

In the following proof, a Lyapunov function and its derivative are provided.

The analysis is then separated into two cases as in Chapter 2. For the noncontact

case, the stability analysis indicates the controller and error signals are bounded

and converge to an arbitrarily small region. Additional analysis indicates that

within this region, contact must occur. When contact occurs, a Lyapunov analysis

is provided that illustrates the MSR system asymptotically converges to the desired

setpoint .

Proof: Let V(r,, e,, em, ef, rlm, 8r, Odk, t) ERI denote the following nonnegative,

radially unbounded function (i.e. a Lyapunov function candidate):


V n -r,"1R, + Of r',le + 1 k,,Fc-'edk (3-24)
2 T1l y12 2
1 1
+ k2K, [ln (cosh(em)) + In (cosh (ef))] + 2ee 2m .

The time derivative of (3-24) can be determined as


Vj = r,TMi, + ~rr"Mr, + 6T~, C'0 + OfkK;Fl-ledke (3-25)

+ kK, tan~em4m+ tanh(ef)ef] + e, i: + mml~m-j,









After using (1-12), (3-3), (3-4), (3-12), (3-13), (3-15), and (3-19)-(3-21), the

expression in (3-25) can be rewritten as



- k,~I1(f 1'/. 2 ae, er + rim [K, (X1


'7' 1 /,,~ k2Kt3 tanh2(ei)

dl Az,,) + K, (Amm Xm) + x] -


(3-26)





0)


The expression in (3-26) will now be examined under two different scenarios.

Case 1-Noncontact: For this case, the systems are not in contact (A =

and (3-26) can be rewritten as

T~< -k3TrTTr 0lk2K, tanh2(em) k2Kyt3~ tanh2(ef) 3 ~11/



Rewriting Xgaz(t) and substituting for X(em, ef, rim, t) yields

<~j -kTrTr 01k2Kl tanh2(em) k2Kyt3a tanh2(ef) 2 1 't 3

[0 ||e,||2" CK- Ijm, Br||] [knla5t2 l ~ 1 I Om17n1


(3-27)


After completing the square on the bracketed terms, (3-27) can be expressed as


T~< -k3TrTTr 0lk2Ky; tanh2(em) k2Kyt3a tanh2(eyf) n y;1:: (3-28)

CK 2 I PK(Z, m rl
(2 02 2
4a m 4a2kni 4a2

Provided kni is selected according to the sufficient condition


4a2 min (a1k2( ,' k2 ,3,0

the expression in (3-28) can be further reduced as


As | ||m -~ k3C~ Fr 2 2 K 02


(3-29)









where Az ERI is defined as


X1 = min arl k2 K, k2i K3, na2

Based on (1-4) in Assumption 1.1, for the noncontact case




i'< 1 <~l sm <, m.3-30




where A ERI is defined as

A = min A z, kF3, P2_K


and y(t) E R"5 and e, ERI are defined as


[ T 2z r" ex = K s~2m (3-32)
r 0

where e, can be made arbitrarily small by making a2 large. Based on (3-24)

and (3 31), either A ||y||2 E Or X 1 2 1 Er- If 212 > c,, then Barbalat's
Lemma can be used to conclude that Vilt) 0 since V(t) is lower bounded, V/(t)

is negative semi-definite, and V/(t) can be shown to be uniformly continuous. As

V'(t) 0- eventually A ||y||2 E. Pr-ovided the sufficient gain condition in (3-23) is
satisfied (i.e., e, < 1), then (3-10), (3-32), and the facts that 8,(t) and Odkct 00Co

from the use of a projection algorithm, can be used to conclude that V(-) E foo;
hence, ||y(t)||, | :(t)||, ||rr(t)||, ||e,(t)||, r1m(t), ef(t), em(t) E foo. Signal chasing

arguments can be used to prove the remaining closed-loop signals are also bounded
during the noncontact case. The previous development can be used to conclude









that for the noncontact case


||y(t)||' and hence, ||r,(t)|| ~A as t oo. (3-33)

Based on (3-33), linear analysis methods (see Lemma A.19 of [37]) can be applied

to (3-3) to prove that


Ilr (t)l 6 Ie(0) || exp(-at) + -I (1 exp(-at)) (3-34)

as t 00o for the noncontact case.

Further analysis is required to prove that the manipulator makes contact with

the mass-spring system. Contact between the manipulator and the mass-spring

system occurs when xmi(t) > xm(t). Based on (3-34), a sufficient condition for
contact can be developed as


Xwdl > Xm + r1. (3-35)


After using (3-11), the sufficient condition in (3-35) can be expressed as


Ydeldk + kF2 tanh(em,) kik2 cosh2(ef) tanh(ef) '> (3-36)


By using (3-2) and (3-6) and performing some algebraic manipulation, the

inequality in (3-36) can be expressed as


ki2 tanh(em) kik2h~ cosh2(r)> mCd dk + em 0o edk (3 37

where ~dk(t) and Odkc(t) are defined in (3-14). From Assumption 1.1, em(t) can be

upper bounded as

em < m (3-38)

where, Em ER denotes a known positive constant. If em(t) < 0, then the sufficient

condition in (3-37) may not be satisfied. The condition that em(t) < 0 will only

occur if an impact collision occurs that causes the mass to overshoot the desired









position. However, even if an impact occurs and the mass overshoots the desired

position, the dynamics will force the mass position error to return to the initial

condition. That is, em(t) ama~ xo > Em where ,Em ER denotes a known positive

constant. Based on (3-38) and the fact that em(t) will eventually be lower bounded

by m, in a noncontact condition, the inequality in (3-37) can be simplified as


k2~ (tanh(Em) ki cosh2~f) _-r md xdkd m 0e ) ~dk. (3-39)

To further simplify the inequality in (3-39), an upper bound can be determined

for ef(t). The inequality in (3-33) along with (3-10) and (3-32) can be used to

conclude that as the manipulator approaches the mass, ef(t) will eventually be

upper bounded as

ef < tanh- ( e I < ef, (3-40)

where af yER is a known positive constant. Based on (3-32) and (3-40), the control

parameter k2 can be selected according to the following sufficient condition to

ensure the robot and mass-spring system make contact


k2> > V (3-41)
tanh(em) kcoh2lOh(f

where ki is chosen as
tanh(Em)
cosh2(f
Case 2-Contact: Provided the sufficient condition in (3-41) is satisfied, the

robot will eventually make contact with the mass. For the case when the dynamic

systems collide (A = 1), and the two dynamic systems become coupled then



1 The dynamic systems can separate after impact, however this case can still be
analyzed under the Noncontact section of the stability analysis.









(3-26) can be rewritten as

V/ < -k~3,T~r 01k2KI tanh2(em) 3a029@ k2Kyt3 tanh2 (f

-[(y ||e,||2 K ljOm Br||~] -[k,1s(Y2 29 1 O

where (3-9) was substituted for X(em, ef, rim, t). Completing the square on the two

bracketed terms yields


V' < -k3,7,~ 011k2eK tanh2 6m) 03k2~ Ktanh2(eyS) (3-42)

s 4a 4a2kni

A final bound can be placed on (3-42) as


1'<-min alk2 KZ,; 3k2 K, ~ l n 2a | |2 2 _K 7 3T

Because (3-24) is non-negative and its derivative is negative semi-definite,

Tr(t), 8r(1), 8dk 1t), er(t), em(t), ef(t), and rlm(t) E C,. Due to the fact that em(t),

ef(t), and rlm(t) E Coo, the expression in (3-3) can be used to conclude that

em(t) E C, (and hence, em(t) is uniformly continuous). Due to the fact that

em(t) E C,, (3-2) can be used to conclude that xm(t) E C,. Previous facts can
be used to prove that X,d(t) E Coo, and since e,(t) E C,, then x,(t) E C,. Due

to the fact that ef(t), em(t), rlm(t) E C,, (3-4) can be used to conclude that

ef(t) E Co. The expression in (3-5) can be used to conclude that ilm(t) E C,

(and hence, rlm(t) is uniformly continuous). Given that r,(t), em(t), ef(t), and

rlm(t) E Coo, Yr() E C,. Since e,(t) E C,, (3-21) can be used to prove that

8,(t) E C. The expression in (3-18) can then be used to prove that F(t) E C,-
The expression in (3-20) can be used to conclude that ri, (t) E C, (and hence,

r,(t) is uniformly continuous). Due to the fact that em(t), Tr(t), rlm(t) E C2 and
uniformly continuous, Barbalat's Lemma can be used to conclude that t)|

|| r, (t) || | Om (t)l | 0 as t oo. Based on the fact that || r, (t) || 0 as t oo,









standard linear analysis methods (see Lemma A.15 of [37]) can then be used to

prove that || ez,(t)l || 0 as t oo.

As is typical in literature, the controller developed in (3-11), (3-13), (3-18),

and (3-19) is based on the underlying assumption that an arbitrarily large (but

finite) control authority is available. A potential disadvantage of the controller

is that the gain conditions developed above and in (3-23) indicate that kn1, k2,

and a2, respectively, should be selected sufficiently large. As demonstrated by the

subsequent experimental results, the gains may be selected much lower in practice

(i.e., the gains conditions are the result of a conservative Lyapunov analysis).

However, the subsequent experimental section also illustrates that even when the

gain conditions are violated, large initial conditions and a high stiffness coefficient

result in a high gain controller that initially saturates the actuators. The control

torque in the experiment was artificially saturated to reduce the magnitude of the

impact to protect a capacitance probe from contact by excessive bending of the

aluminum rod/spring assembly. Research that can limit the required control torque

for systems that undergo an impact collision will be discussed in Chapter 4.

3.4 Experimental Results

With the testbed in Section 2.3, the control gains a and k~3, defined as scalars

in (3-3) and (3-18), were implemented (with nonconsequential implications to

the stability result) as diagonal gain matrices to provide more flexibility in the

experiment. Specifically, the control gains were selected as

ki = 0.18 k2 = 0.9 k3 = diag {185, 170}
(3-43)
or = 45 02 8 a3 = 0.01 0 = diag {60, 90} .

The control gains in (3-43) were obtained from a trial and error process and may

not satisfy the sufficient gain conditions developed in the theorem proof. The

sufficient gain conditions are the result of a conservative stability analysis and were









used as a guide in the tuning process. The subsequent results indicate that the

developed controller can be applied despite the fact that some gain conditions are

not satisfied.

The adaptation gains were selected as


r = 90

r, = diag {4.01 x 1012, 1.2 x 10 0.2, 3.3 x 1012, (3-44)

6 x 106, 0.1, 2.4 x 101, 7 x 105, 0.1, 2.35 x 1011)


The adaptation gains r, in (3-44) are used to enable the adaptive estimate to

sufficiently change relative to the large values of the uncertain parameters in Or -

Smaller adaptation gains could be used to obtain different results. The initial

conditions for the robot coordinates and the mass-spring position were (in [m])


[r,(0) iir2(0) 21m(0) ]= 0008 0.41 0.202 -

The initial velocity of the robot and mass-spring were zero, and the desired mass-

spring position was tin [mml

Xma = 232.

That is, the tip of the second link of the robot was initially 224 [mm] from the

desired setpoint, and 194 [mm] from xo along the X1-axis (see Fig. 2-1). Once the

initial impact occurs, the robot is required to depress the spring (item (1) in Fig.

2-1) to move the mass 30 [mm] along the X1-axis.
The mass-spring and robot errors (i.e., e(t)) are shown in Fig. 4-1 and Fig.

3-2. The peak steady-state position error of the end-point of the second link of the

robot along the X1-axis (i.e., |erl(t)|) and along the X2-aXiS (i.e., ler2(t)|) are 0.212

[mm] and 5.77 [pm], respectively. The peak steady-state position error of the mass

(ie,.(t)|) is 2.72 [pm]. The 0.212 [mm] maximum steady state error in |erl(t)|






42




soo

500 5 10 15
200


200 5 10 15




50 5 10 15
[sec]


Figure 3-1: Mass-spring and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., en (t)), (b) indicates the position error of
the robot tip along the X2-aXiS (i.e., er2(t)), and (c) indicates the position error of
the mass-spring (i.e., em(t))-


is due to the Ydeldk(t) term of XAdl(t) in (3-11) where Yd(Xm) is approximately 0.03

[m], and Odkc(t) has a maximum steady state value of 0.007 [ '], yielding 0.21

[mm] error. All of the other terms in erl(t) are negligible at steady state.

The input control torques in (3-22) are shown in Fig. 3-3 and Fig. 3-4.

To constrain the impact force to a level that ensured the aluminum plate did

not flex to the point of contact with the capacitance probe, the computed

torques are artificially saturated. Fig. 3-3 depicts the computed torques,

and Fig. 3-4 depicts the actual torques (solid line) along with the computed

torques (dashed line). The resulting desired trajectory along the X1-axis (i.e.,

Xay (t)) is depicted in Fig. 3-5, and the desired trajectory along the X2-aXiS

was chosen aS Xrd2 = 370 [mm]. Fig. 3-6 depicts the value of Odkct W

and Fig. 3-7 Fig. 3-9 depict the values of 8,(t)l E RIo1. The order of the

curves in the plots is based on the relative scale of the parameter estimates

rather than numerical order in 8,(t). A video of the experiment is provided at

http: //www.mae.ufl.edu/~-dixon/proj ects /robman/adaptiveimpact .htm.
































































0 5 10 1

(b)
200



-200

-400


200
\
E100
o
0



-50
-100
-is


40
E 20
EO


Figure 3-2: 1\ass-spring and robot errors e(t) during the initial two seconds.


S-500H


J (q)F(t) for the (a) base motor and (b)


Figure 3-3: Computed control torques
second link motor.



















(a)
50





-100~ -i,
0 005 01 015 02 025 03 035 04

(b)


005 01 015 02
[sec]


025 03 035 04


Figure 3-4: Applied control torques J (q)F(t) (solid line) versus computed control

torques (dashed line) for the (a) base motor and (b) second link motor.















2335

232

2325


231 5
E
231


Figure 3-5: Computed desired robot trajectory, xay (t).















0 06

0 05

0 04

0 03

0 02

0 01


Figure 3-6: Unitless parameter estimate Odk (t) introduced in (3-13).


1


Figure 3-7: Estimate for the unknown constant parameter vector Or (1) (a)
erio(t) K,, (b) Or4 1) Km, (C) ,1(t) m, and (d) ,7(t) lm, where
mi, m2 E RW denote the mass of the first and second link of the robot, m, E RW
denotes the mass of the motor connected to the second link of the robot, and m
ERI denotes the mass of the mass-spring system.




























Figure 3-8: Estimate for the unknown constant parameter vector Or (f ) (a) Brs (t)
", (b) Or2(t) ~1, and (c) B,s(t)- -


parameter vector Or (1) (a) Gre (t)


Figure 3-9: Estimate for the unknown constant
m,, (b) Or3 1) = 1, and (c) Or9 1) = 2-










3.5 Concluding Remarks

An adaptive nonlinear Lyapunov-based controller is proven to regulate the

states of a planar robot colliding with an unactuated mass-spring system. The

continuous controller yields global asymptotic regulation of the spring-mass and

robot links. New control design, error system development and stability analysis

techniques were required to compensate for the fact that the dynamics changed

from an uncoupled state to a coupled state. Experimental results are provided

to illustrate the successful performance of the controller. Sufficient conditions

developed in the stability analysis indicate that the control gains should be selected

large enough to minimize the closed-loop steady state error, but high gains could

result in large torques for large initial errors. The high gain problem is exacerbated

in the developed result because of the presence of the estimated impact stiffness

coefficient. The experimental results were obtained by artificially saturating the

torque to prevent damage to the capacitance probe. These issues point to a need

to develop controllers that account for limited actuation which we will discuss in

Chapter 4.















CHAPTER 4
AN IMPACT FORCE LIMITING ADAPTIVE CONTROLLER FOR A ROBOTIC
SYS TEM UNDERGOING A NONCONTACT- TO- CONTACT T TRANSITION

Similar to Chapter 2 and 3, the objective of this chapter is also to control

a robot from a noncontact initial condition to a desired (contact) position so

that the mass-spring system is regulated to a desired compressed state. When

the controllers in Chapter 2 and Chapter 3 were implemented in the presence of

large initial conditions, violent impacts between the robot and the mass-spring

system resulted. In fact, the controller was artificially saturated (the saturation

effects were not considered in the stability analysis) to reduce the impact forces

so that the mass deflection would not destroy a capacitance probe. These results

provide the motivation for the control development in this chapter. Specifically, the

feedback elements for the continuous controller in Chapter 3 are contained inside

of hyperbolic tangent functions as a means to limit the impact forces resulting

from large initial conditions as the robot transitions from noncontact to contact

states. The main challangfe of this work is that the use of saturated feedback does

not allow some coupling terms to be canceled in the stability analysis, resulting

in the need to develop state dependent upper bounds that reduce the stability

to a semi-global result. New control development, closed-loop error systems, and

Lyapunov-based stability analysis arguments are used to conclude the result.

Experimental results are provided that successfully demonstrate the control

objective.

This chapter is organized as follows. The associated properties are provided

in Section 4.1. Section 4.2 describes the error system and control development









followed by the stability analysis in Section 4.3. Section 4.4 describes the experi-

mental results that indicate the successful performance of the proposed controller

followed by conclusion in Section 4.5.

4.1 Properties

Remark 4.1: To aid the subsequent control design and analysis, we define the

vector Tanh(-) E RW" as follows


Tanh(G) n [tanh(bl), ..., tanh(6,)]" (4-1)

where 6 n [61, ..., 6,]T E R".

Property 4.1: The following inequalities are valid for all ( = [(1, ..., s]T E RW"

[36]

||(|| > ||Tanh(()||

| |( | +1 >(4-2)
Stanh(|| ||)

||(|2 > In(osh((4)) > In(cosh(||(|)) (4-3)
i=1


( Tanh(() > Tank (()Tanh(() = ||Tanh(()||2 (4-4)

> tanh2(11)


4.2 Error System and Control Development

The subsequent control design is based on integrator backstepping methods. A

desired trajectory is designed for the robot (i.e., a virtual control input) to ensure

the robot converges to and impacts with the mass, and to ensure that the robot

regulates the mass to the desired position. Since the robot trajectory can not be
controlled directly, a force controller is developed to ensure that the robot tracks

the desired trajectory despite the transition from free motion to an impact collision

and despite uncertainties throughout the MSR system.









4.2.1 Control Ob jective

As in Chapter 2 and 3, the control objective is to regulate the states of the

mass-sping system via a two-link planar robot that transitions from noncontact to

contact with the mass-spring through an impact collision. An additional objective

in this chapter is to limit the impact force to prevent damage to the robot or

the environment (i.e., the mass-spring system). A regulation error, denoted by

e(t) E RW3, is defined to quantify this objective as

e 4 em e" ,'


where e,(t) n [erz(t),er2(~l I2 and em(t) ERI denote the regulation error for the

end-point of the second link of the robot and mass-spring system (see Fig. 1-1),

respectively, and are defined as


e, A go X, em = sma xm. (4-5)


In (4-5), sma ERI denotes the constant known desired position of the mass, and

X~d(t) n [X,ax(t), Xrd2] I2 denotes the desired position of the end-point of the
second link of the robot. The subsequent development is based on the assumption

that q(t), q(t), xm(t), and xm(t) are measurable, and that x,(t) and x,(t) can

be obtained from q(t) and q(t). To facilitate the subsequent control design and

stability analysis, filtered tracking errors, denoted by rm (t) ERI and r, (t) E RW2, are

defined as [38]


rlm = em ,+ az tanh(em,) + a2 tanh(ef)(46

Tr e, + ae,,









where a, al, a2 ERI are positive, constant gains, and ef(t) ERI is an auxiliary filter

variable designed as [38]


ey a -a3 tanh(ep) + a2 tanh(em) kl cosh26f )llm, (4~-7

where ki ERI is a positive constant control gain, and a3 ERI is a positive constant

filter gain.

4.2.2 Closed-Loop Error System

In this section, the closed-loop error system for rlm is exactly the same as in

(3-5)-(3-15) in Chapter 3. The open-loop robot error system can be obtained by
taking the time derivative of r, (t) and premultiplying by the robot inertia matrix



M~i, = Y,, Cr, F, (4-8)

where (1-9), (4-5), and (4-6) were utilized, and


Y,, n M~xre + ale, + h + C~rd + aCzrd + I aCX,,


where Y(x,, x,, xm, im, ef, rlm, t) E RW2xP denotes a known regression matrix, and

8, ER Idenotes an unknown constant parameter vector. By making substitutions

from the dynamic model and the previous error systems, xra(t) can be expressed

without a dependance on acceleration terms. Based on (4-8) and the subsequent

stability analysis, the robot force control input is designed as


FA n r, + Tanh(e,) + k3T Thr,), (4-9)

where k3 ERI is a positive constant control gain, and 8,(t)l E RWP is an estimate for

8, generated by the following adaptive update law


8r A proj(r, ,r,).


(4-10)









In (4-10), r, E RWPxP is positive definite, constant, diagonal, adaptation gain
matrix, and proj(-) denotes a projection algorithm utilized to guarantee that the

i th element of Or (t) can be bounded as


Ori < ari < Or, (41-11)

where 8,i, 8,i ERI denote known, constant lower and upper bounds for each element

of 8,(t)l, respectively.

The closed-loop error system for rr (t) can be obtained after substituting (4-9)

into (4-8) as

M~i, = Y,, Cr, Tanh(e,) k3T Thr,). (4-12)

In (4-12), the parameter estimation error Or t)l E RWP is defined as

8, 8, 8,. (4-13)

4.3 Stability Analysis

Ti;,~ ..;~, u: The controller given by (3-11), (3-13), (4-9), and (4-10) ensures

semi-global asymptotic regulation of the MSR system in the sense that


S.(t)|I 0 ||e,(t)||I 0 as t 00o

provided the following sufficient gain conditions are satisfied:

k,2 > 1(4-14)
402 mint a1k2 CG3k29K ,02


02 ma (4-15)


k3a > 12V 2~ (4-16)









where V, ERI is defined as


V, max Asl |(0)||2 +(o, 5Az(tanh-'( ))2V

A ERI is a known positive constant, and al, a, al, a2, a3, k,1, k2, k~3, X1, P$cm) PCc,

-K. and (Ki are defined in (1-4), (1-11), (3-1), (3-11), (3-12), (4-6-), (4-7), and
(4-9).
As in the previous chapters, the subsequent analysis is separated into two

cases: contact and noncontact. For the noncontact case, the stability analysis

indicates the controller and error signals are bounded and converge to an arbitrarily

small region where contact must occur. When contact occurs, a Lyapunov analysis

is provided that illustrates the MSR system asymptotically converges to the desired

setpoint .
Proof: Let V(t) ERI denote the following non-negative, radially unbounded

function (i.e. a Lyapunov function candidate):

1 1" 1" 1
V n -r,"Air, + ,"0,-1, + -~ "k,-d 6B m (4-17)
2, 2iref~~~ 2 2l,

+ k2KI [ln (cosh(em)) + In (cosh (ef))] + In(cosh(ery)) + In(cosh(er2)

where (1-11) and (4-3) can be utilized to bound V(t) as

4811,1 2 < V < Az | rl|| + Co (4-18)


where Az,P Co eR and E IR are defined as

a2 m,
Az Amax{ 3, ", kF2 K
2 2
CO KC5d a p O (4~-19)

z[TrT ep lm em, ey]T

where Amx,( { E denotes the maximum eigenvalue of a matrix, and Peodkl Pa r

are the knownI uIpTper bolunds of dk (t) anrd H,(t) resp~ectively. After usiing (1 12),









(3-12), (3-13), (3-15), (4-4), (4-6), (4-7), (4-10), and (4-12), the time derivative

of (4-17) can be determined as


V~ < -k3 tanh2 lr, I) 0lk2K, tanh2 6m) 03sk2K, tanh2(ef)

+ 2e r, -2ae, e, 3 1 I,I -k,1(fI 11/y, atanh2( l erI

+ rlm [K, (Xway Az,,) + K(n, (A- Xm) + x] -


(4-20)


The expression in (4-20) will now be examined under two different scenarios.

Case 1-Noncontact: For this case, the systems are not in contact (A = 0)

and (4-20) can be rewritten as


Vj < -kJ3 tanh2( Tr, ) 01k2K~1tanh2 (em) -

+ 2e r, 2ae e, 3a292, k,lp(f t24,

+ rlm [Kp~rdl Kymm + X] -


(4-21)


Based on (3-1), (3-9), and (4-5), the expression in (4-21) can be rewritten as


V' < -k3 tanh2( Tr ) 01k2CKze tanh2 6m,) 03k2CK- tanh2(ey) (4-22a)

2 oa /,I a tanh2( li, r

[0 ||e,||2" Kl 77m1 Br||.] [k,~1(f;a21 t29 I 1 111m1


After completing the square in the bracketed terms,


(4-22) can be expressed as


4(


-23)


l' < --8 | ||I -- a tanh2(le Ir 02 _~ K ,,
4a
-2 2C2~
40X,- X12 a tanh2( I ,IIITr 1 lh i


where ,8 ERI is defined as


:3 = min asrk2zli K~Eari 3k2 ) K 2 >


03k2K~ltanh2(ef)

-a tanh2( IIerI









provided kni is selected according to (4-14). Provided a2 is selected according to
the sufficient condition in (4-15), then (1-4), (4-2), (4-18), and the fact that

<~i I Xr < Xm < pOm (4-24)

for the noncontact case, can be used to rewrite (4-23) as

V < -,8 | || a tanh2 Br ||l)~ + (4~-25)

3_ 2~3 tanh2( Fr, -


In (4-25), ,Ex ER is a known positive arbitrarily small constant that is defined as




Provided the following sufficient condition is satisfied


k~t > 1 2 (4-26)

the expression in (4-25) can be expressed as

V' < A 12 + E 4-27)

where A is a known constant and ye R is defined as

= z' tanh(||e,|| tanh(|r,|| .j (4-28)

In (4-27), e, can be made arbitrarily small by making a2 large. Based on (4-
17) and (42-27), if A ||y(t)||2" > c,, then Bar2bala~t's T;Lemma can be uLsed to conclude~C
that Vj(t) 0 since V(t) is lower bounded, Vj(t) is negative semi-definite, and Vj(t)
can be shown to be uniformly continuous. As 11'(t) 0, eventually A ||y(t)||2 < x-
Wi~hile X ||y(t)||2 > Ez, (4-17), (4-18), and (4-27) canl be used to conclude that









V(t) E Em, and the sufficient condition given in (4-26) can be expressed as


k~t > 2 (AI |(0)||2 + 1 (4-29)


Provided a2 >~ C thlen eventually: A ||y(t)||2 < E, < A. Based on (3-10) anld

(4-28), the fact, that A||yl(t)||2 < e, < A can be: used to conclude: that em,(t), eyS(t),

e,(t), r,(t), flm(t) E C,. Since e,(t) and Odkct E a frOm the use of a projection
algorithm, the previous facts can be used to conclude that V(t) E Em. Signal
chasing arguments can be used to prove the remaining closed-loop signals are also

bounded during the noncontact case provided (4-26) is satisfied.

If the initial conditions for V(0) are large enough that A ||y(t)||2" > --,, then
the condition in (4-29) is sufficient. However, if the initial conditions for V(0) are

inside the region defined by e,, then V(t) can growv larger unttil A ||yl(t)||2 ex-.

Therefore, further development is required to determine how large V(t) can grow

so that the sufficient condition in (4-26) can be satisfied. When V(0) is inside the

region defined by e,, then

||y(t)|| < .J (4-30)

The expression in (4-30) can be used along with (3-10), (4-19), and (4-28) to
conclude that

|7(t)|
The inequality in (4-31) can be used along with (4-18) and (4-19) to rewrite the
sufficient condition in (4-26) as


~a > 2 (5Az(tanh- ())2 +~)(-2


Hence, the final sufficient condition for (4-26) is given by (4-16). That is, provided

k,1, a, a2, and kF3 are selected larger than known constants (that depends on the









initial conditions of the states) according to (4-14)-(4-16) then the all the states

converge to an arbitrarily small neighborhood. Development has been provided (see

Section 3.3) to prove that contact with the mass-spring system is ensured within

this neighborhood.

Case 2-Contact: For the case when the dynamic systems collide (A = 1) and

the two dynamic systems become coupled then (4-20) can be rewritten as


Vi < -k~3 tanh2 Flrl> 01k2 K tanh2 6m) 03k2CK tanh2(f






where (3-1) was substituted for KI and (3-9) was substituted for X(em, ef, rim, t)-

Completing the square on the three bracketed terms yields


1' < |I || a tanh2( Ilr P 2_K 9-3





Because (4-17) is non-negative, as long as (4-14)-(4-16) are satisfied, (4-33) is

negative semi-definite, and r,(t), 8,ttel(t), Od 1, Bt em t ef(t), and rlm(t) E Co-

Due to the fact that em(t), ef(t), and rlm(t) E C,, the expression in (4-6) can

be used to conclude that em(t) E C, (and hence, em(t) is uniformly continuous

(UC)). Due to the fact that em(t) E C,, zim(t) E C,. Based on (1-4), xm(t) E C,-
Previous facts can be used to prove that x,d(t) E C,, and since e,(t) E C,, then

x,(t) E C,. Due to the fact that ef(t), em(t), rlm(t) E C,, (4-7) can be used to
conclude that e7 (t) E Co. The expression in (3-5) can then be used to conclude



1 The dynamic systems can separate after impact, however this case can still be
analyzed under the Non-Impact section of the stability analysis.









that ilm(t) E C, (and hence, rlm(t) is UC). Based on (4-6) and the fact that r,(t)

and e,(t) E C,, e,(t) E C,. Also, based on (3-14) and the fact that xm(t), em(t),

ef (t), rlm(t), and xm(t) E C,, the expression in (3-11) can be used to prove that

X,d(t) E Co. Based on the fact that e,(t) and x,d(t) E C,, the expression in

(4-5) can be used to prove that x,(t) E C,. Given that x,(t), x,(t), xm(t), xm(t),

ef(t), and rlm(t) E Coo, Yr() E C,. The expression in (4-9), (4-11) can then be
used to prove that F(t) E C,. The expression in (4-12) can be used to conclude

that i,(t) E C, (and hence, r,(t) is UC). Since em(t) and r,(t) are UC which

implies tanh(em(t)) and tanh(||r,(t)||)> are also UC, and the fact that tanh(em(t)),

tanh(||r, (t)||), and rlm(t) are square integrable, Barbalat's Lemma can be used to
conclude that tanh(em(t)), tanh(||r,(t)||), |9m,(t)| 0 as t oo, which also implies

S. (t)|, ||r,(t)|| 0 as t oo. Based on the fact that ||r,(t)|| 0 as t oo,
standard linear analysis methods (see Lemma A.15 of [37]) can then be used to

prove that || e, (t)l || 0 as t oo.
4.4 Experimental Results

With the testbed in Section 2.3, the control gains a and k~3, defined as scalars

in (4-6) and (4-9), were implemented (with nonconsequential implications to
the stability result) as diagonal gain matrices to provide more flexibility in the

experiment. Specifically, the control gains were selected as


ki = 0.28 k2 = 0.855 k~3 = ding {110, 10}

al = 60 a2 = 2.8 a3 = 0.06 a = diag {40, 8} .
The adaptation gains were selected as












r = 0.06

r, = diag {1.8 x 1011, 1.5 x 105, 0.2, 7 x 1010,

7 x 104, 0.1, 1 x1010, 1 x104, 0.01, 2.8 x 109)


The initial conditions for the robot coordinates and the mass-spring position were

(in [m])

[r,(0) i;r2(0) 21m(0) ]= 0070 0.285 0206 -

The initial velocity of the robot and mass-spring were zero, and the desired mass-

spring position was tin [mml

Xma = 236.

That is, the tip of the second link of the robot was initially 166 mm from the

desired setpoint and 136 mm from the initial impact point along the X1-axis

(see Fig. 2-1). Therefore, once the initial impact occurs, the robot is required
to depress the spring (item (1) in Fig. 2-1) to move the mass 30 mm along the

X1-axis .

The mass-spring and robot errors (i.e., e(t)) are shown in Fig. 4-1. The peak

steady-state position error of the robot tip along the X1-axis (i.e., |er, (t)|i) and

along the X2-aXiS (i.e., lr2 (t) |) are 0.855 mm and 0.179 mm, respectively. The

peak steady-state position error of the mass-spring (i.e., I~. (t) |) is 0.184 mm.

The input control torques (i.e., J (q)F(t)) are shown in Fig. 4-2 and Fig. 4-3.

The resulting desired trajectory along the X1-axis (i.e., Xar (t)) is depicted in Fig.

4-4, and the desired trajectory along the X2-axis was chosen aS xrd2 = 357.5 mm.

Fig. 4-5 depicts the value of Odk (t) ERI and Fig. 4-6 Fig. 4-8 depict the values

of 8,(t)l E RIo1. The order of the curves in the plots comes from their scales rather

than their numerical order in 8,(t).


















































(a)
100



100

200
0 24 6 8 10


0 24 6 8 10
100

0i--

-100I
0 24 6 8 10


Figure 4--1: Mass-spring and robot errors e(t). Plot (a) indicates the position error
of the robot tip along the X1-axis (i.e., en (t)), (b) indicates the position error of
the robot tip along the X2-aXiS (i.e., er2(t)), and (c) indicates the position error of
the mass-spring (i.e., em(t))-


-10



Figure 4--2: Applied control torques
ond link motor.


[sec]


J (q)F(t) for the (a) base motor and (b) sec-























o




-200
0


57


01 02 03 04 05 06 07

(b)


01 02 03 04
[sec]


05 06 07 08


Figure 4-3: Applied control torques J"(q)F(t) for
ond link motor during the first 0.8 second.


the (a) base motor and (b) sec-






































8 10


420

400

380

360

340

320

300

280

260

240

220
0


Figure 4-4: Computed desired robot trajectory, xay (t).















x 10
8






6


05


4


3


[sec]


Figure 4-5: Parameter estimate Odk (t) introduced in (3-13).


[sec]


Figure 4-6: Estimate for the unknown constant parameter vector Or (1) (a)
orio(t) Kr,,(b) Or4 1) K", C) ,1(t) "', and (d) 6,7(t) I", where
mi, m2 E RW denote the mass of the first and second link of the robot, m, E RW
denotes the mass of the motor connected to the second link of the robot, and m
ERI denotes the mass of the mass-spring system.






















S800

600

400


Figure 4-7: Estimate for the unknown constant parameter vector Or (1) (a) As ()
"", (b) Or2(t) m, and (c) 8,s(t)= -s


Figure 4-8: Estimate for the unknown constant parameter vector Or (1) (a) Gre t)
m,, (b) Or3(t = m1, and (c) Or9 1) = 2-










4.5 Concluding Remarks

In this chapter, we consider a two link planar robotic arm that transitions

from free motion to contact with an unactuated mass-spring system. An adaptive

nonlinear Lyapunov-based controller with bounded torque input amplitudes

is proven to regulate the states of the system. The feedback elements for the

controller in this chapter are contained inside of hyperbolic tangent functions as a

means to limit the impact forces resulting from large initial conditions as the robot

transitions from noncontact to contact. The continuous controller yields semi-global

asymptotic regulation of the spring-mass and robot links. Experimental results are

provided to illustrate the successful performance of the controller.















CHAPTER 5
CONCLUSION AND RECOli lli lENDATIONS

Motivated by the fact that impacts between the robot and the static environ-

ment cannot represent all the impact system applications, and discontinuous con-

trollers require infinite control frequency (i.e., exhibit chattering) or yield degraded

stability results (i.e., uniformly ultimately bounded), continuous Lyapunov-based

controllers for a fully actuated dynamic systems undergoing an impact collision

with another unactuated dynamic system are developed. Lyapunov-based methods

are used to prove the asymptotic regulations of the mass and robot links. Unlike

some other results in literature, the continuous force controller does not depend on

measuring the impact force or the measurement of other acceleration terms: only

the position and velocity terms of the spring-mass system and the joint angles and

the angular velocities terms of the planar robotic arm are needed for the proposed

controller.

Chapter 2 provides a first step at controlling the proposed impact system.

The control development in Chapter 2 is based on the assumption of exact model

knowledge of the system dynamics. The controller is proven to regulate the states

of the system and yields global asymptotic result. In Chapter 3, the dynamic model

for the system is assumed to have uncertain parameters. The control objective is

defined as the desire to both regulate the system to a desired compressed state and

compensate for the constant, unknown system parameters. Two linear parameteri-

zations are designed to adapt for the unknown robot and mass-spring parameters.

The controller is proven to yield global asymptotic regulation result. An extension

of the developed regulation controller of Chapter 3 is presented in Chapter 4 where










the feedback elements for the controller in this chpater are contained inside of hy-

perbolic tanget functions as a means to limit the impact forces resulting from large

initial conditions as the robot transitions from noncontact to contact states. The

controller yields semi-global asymptotic regulation of the system. Experimental

results are provided to illustrate the successful performance of the controller in each

chapter.

Future efforts can focus on utilizing non-model based control methods such as

neural networks, fuzzy logic, or robust control methods to compensate the more

complex uncertainties of the system. The developed model can also be modified

to control the impact between an actuated dynamic system and an unactuated

dynamic system with nonlinear flexibilities. Future work could also focus on the

specific application of the developed methods for applications such as docking space

vehicles, walking robots, etc. Efforts could also focus on obtaining a global result of

the feedback saturated case in Chapter 4.















APPENDIX A
THE EXPRESSION OF x,d, IN SECTION 2.1

Due to the fact that taking the time derivative of any expression with A except

for KA(x,l xm) is undefined, some care has to be taken in taking the time

derivative of Xar (t). This is done by first taking the time derivative of (2-6) and

utilizing (2-3) and (2-7), to obtain the following expression:


Xsdl = K[(1 A) (k(xm xo) + amem)] + KA [Kers (ki + k2) rm] (A-1)
K KmK

1 1
+ (ki + k2) (1 A) amrem + (i+k)A[ey-(i+k)E
mK mK

In order to derive X,4, (t), the second time derivative of (2-6) is taken rather than

the first time derivative of (A-1) to obtain


Xwdl = ,(am'e m + kim + (ki + k2)rm) + xm. (A-2)


The expression for em(t) can be obtained by rewriting (2-3) as


em = rm II. (A-3)


Differentiating (A-3) yields the following expressions for em(t) and 'e mi):


em = im ... (A-4)


em = m .. .(A-5)

By using (2-7), the expression for im(t) can be written as

1 1
rm [(1 A) (k(xm xo) + amem)] + [Knery A(ki + k2)rm] (A-6)
m m









The time derivative of (2-4) is given by


m


KA(xry xm) + amem).


(A-7)


The dynamics for the mass-spring system xm(t) can be written as


xm (KA(x,l xm) k(xm ro)).
m

By using (A-2) and (A-5), the following expression can be obtained:

am k(i+k
Xwdl = (Tm aem) + ( + 1)xm + Em.
K K

Substituting (A-4) and (A-7)-(A-9) yields


(A-8)


(A-9)


Q20
in1 xm) + amim ) im, +
SK



KA (xin xm) + amem] -


Q3m
K


Xwdl = (km, -- KA (j
k1
+ ( + 1) (KA (
K m
(ki + k2)
+ [kim -
Km


After using (A-4) and (A-6), a simplified expression for Argy (t) can be obtained as

follows:


Xay = (kim
K


KA (xin xm))


k1
K m
(ki + k2)
+ [kim K
Km
(ki + k2) a
+ (1 A)
Km
(ki + k2) 0
+ Aery -
m 1


- Xm) k(Xm Xo))

A (X,1 Xm)]

(k(xm Xo))


(A-10)


(A-11)













APPENDIX B
THE EXPRESSION OF x,d, IN SECTION 3.2
Since Xrd2 is a constant, the subsequent development is only focused on
determining xrdl(t). After using (3-2), (3-4), (3-11), and (3-13), the first time
derivative of Xar (t) can be determined as


nral = Y (prloj (rY, t; )) + Hdk+1 -k2 coshl(-2 6m im (B-1)

kik~2 (sinh2(ep) + cosh2 By)) (-a3 tanh(ei) + cy2 tanh(e~m)- ki cosh2 6f )rlm

Based on the fact that the projection algorithm for Odkc(t) is designed to be suffi-
ciently smooth [39], the expressions in (3-13) and (B-1) can be used to determine
the second time derivative of Xar (t) as

8(poj(Y(t.) 8(proj(rY, ,.)) .r,r: )X B2

-2k2 cosh-3(r m) sinhjem)i r + Hdk+ 1 0S-k2( cos-2 m

4kik~2 (sinh(ef) cosh(eyf)) e)~

lkzk (sinh2(ef) + cosh2 By)) (-a3 cosh-2(ef) 2kil cosh(ej) sinh(ef)7lm) eyS

+ lkzk (sinh2(ef) + cosh2 By)) (-n2 cosh-2(em)em, + ki cosh2 6f il) m

After substituting (3-4) and (3-5) into (B-2) for ef(t) and rjm(t), respectively, and
substituting (1-5) and (1-8) into (B-2) for xm(t), the expression for M~ (x,) X44(t)
in the linear parameterization in (3-17) can be determined without requiring
acceleration measurements.















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BIOGRAPHICAL SKETCH

Chien-Hao Liang was born in Taipei, Taiwan. He completed his Bachelor

in Ocean Engineering in the year 2001 from National Taiwan University, Taipei.

After finishing his military service, he joined the nonlinear controls and robotics

research group in the University of Florida for Master of Science in mechanical and

aerospace engineering in the year 2005.