Dynamic Optimization of a Rimless Wheel with an Actuated Pendulum

107 Pages·2016·1.36 MB·English

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Theses
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Benjamin Thomas Mihevc
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Dynamic Optimization of a Rimless
Wheel with an Actuated Pendulum
by
Benjamin Thomas Mihevc
AThesisSubmittedinPartialFulfillmentoftheRequirementsforthe
DegreeofMasterofScience
inComputerEngineering
Supervisedby
JuanC.Cockburn
DepartmentofComputerEngineering
KateGleasonCollegeofEngineering
RochesterInstituteofTechnology
Rochester,NewYork
August 2015
Approvedby:
JuanC.Cockburn,AssociateProfessor
ThesisAdvisor,DepartmentofComputerEngineering
MarioW.Gomes,AssistantProfessor
Co-advisor,DepartmentofMechanicalEngineering
AgamemnonL.Crassidis,AssociateProfessor
CommitteeMember,DepartmentofMechanicalEngineering
AndresKwasinski,AssociateProfessor
CommitteeMember,ComputerEngineering
ii
Abstract
DynamicOptimizationofaRimlessWheelwithanActuatedPendulum
BenjaminThomasMihevc
As the demand for mobile robots that work alongside humans increases,
theamountofenergythattheseco-robotsconsumewillbecomeacritical
limiting factor in their deployment. This need is clearly captured in one
ofthefifteenmaingoalsofthe2009RoadmapforUSRoboticswhichisto
createarobotthatcanwalkwithhalftheenergyconsumptionofahuman
being. At this point, the most energy-efficient walking robot is about as
energyefficientasahuman.
Energy efficient bipedal motion is an active area of research. It has been
proven that it is theoretically possible to design a robot with intermittent
support,oneofthemostfundamentalattributesofleggedlocomotion,to
haveazero-energycostcollisionlessgait.
Optimal control has been used by a number of researchers to study the
generation of periodic gaits for walking robots. However little research
exists demonstrating walkers with energy efficient collisionless motion.
The research that does exist demonstrates that a significant amount of
iii
theenergylosttothesystemwhenwalkingisfromlossesduetostepcol-
lisions.
In this work energy efficient locomotion of a prototype actuated rimless
wheel on level ground is explored using numerical optimal control. The
actuated rimless wheel has an internal pendulum driven by a DC motor.
The locomotion problem is posed as an optimal control problem. Dif-
ferent cost functions and initial configurations are investigated and the
corresponding gait trajectories analyzed and assessed based on their use
ofenergyandthepotentialforcollisionlessmotion.
Theresultsofthisworkwillprovidethefoundationforthedesignandim-
plementation of more energy efficient actuated rimless wheel prototype
withnearcollisionlessmotion.
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Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 LiteratureReview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 TheActuatedRimlessWheel . . . . . . . . . . . . . . . . . . . . . . 9
3.1 Newton-EulerEquations . . . . . . . . . . . . . . . . . . . . . . 9
3.2 LagrangianDynamics . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 ActuatorDynamics . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 CollisionDynamics . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.5 OverviewofOptimalControlTheory . . . . . . . . . . . . . . 30
4 ExperimentsandResults . . . . . . . . . . . . . . . . . . . . . . . . 33
4.1 OverviewofImplementation . . . . . . . . . . . . . . . . . . . 33
4.2 SystemParameters . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 ProgramSetup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
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4.4 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.4.1 SingleCollidingStep . . . . . . . . . . . . . . . . . . . . 40
4.4.2 SingleCollisionlessStep . . . . . . . . . . . . . . . . . . 45
4.4.3 TwoPerfectlyElasticCollidingSteps . . . . . . . . . . . 50
4.4.4 ThreePerfectlyElasticCollidingSteps . . . . . . . . . . 67
4.4.5 APlasticCollision . . . . . . . . . . . . . . . . . . . . . . 70
4.4.6 SingleStepExtensions . . . . . . . . . . . . . . . . . . . 76
4.5 AdditionalAnalysisofResults . . . . . . . . . . . . . . . . . . . 82
5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1 ForwardWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.1.1 DynamicModelRevision . . . . . . . . . . . . . . . . . 86
5.1.2 PhysicalConstruction . . . . . . . . . . . . . . . . . . . 87
5.1.3 FeedbackControl . . . . . . . . . . . . . . . . . . . . . . 88
5.2 ConcludingRemarks . . . . . . . . . . . . . . . . . . . . . . . . 89
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
vi
List of Figures
1.1 DiagramofaRimlessWheel . . . . . . . . . . . . . . . . . . . . 2
3.1 DiagramofaPendulumCoupledRimlessWheel . . . . . . . 10
3.2 FreeBodyDiagramsofthePendulumCoupledRimlessWheel 10
3.3 GeneralizedCircuitofDCMotor . . . . . . . . . . . . . . . . . 22
3.4 SystemBefore(Left)andAfter(Right)anInstantaneousCol-
lision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1 SingleCollidingStepwithVoltageCost . . . . . . . . . . . . . 43
4.2 SingleCollidingStepwithaMultiStateCost . . . . . . . . . . 44
4.3 SingleCollidingStepEnergy . . . . . . . . . . . . . . . . . . . . 45
4.4 SingleCollisionlessStep . . . . . . . . . . . . . . . . . . . . . . 47
4.5 SingleCollisionlessStepSimulationVerification . . . . . . . . 48
4.6 SingleCollisionlessStepEnergy . . . . . . . . . . . . . . . . . 49
4.7 TwoCollidingStepswithLowWheelPositionCost . . . . . . 54
4.8 TwoCollidingStepswithLowWheelPositionCostVerification 54
4.9 TwoCollidingStepswithLowWheelPositionCostEnergy . . 55
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4.10 TwoCollidingStepswithLowWheelCosts . . . . . . . . . . . 57
4.11 TwoCollidingStepswithLowWheelCostsVerification . . . . 57
4.12 TwoCollidingStepswithLowWheelCostsEnergy . . . . . . . 58
4.13 TwoCollidingStepswithLowPositionCosts . . . . . . . . . . 59
4.14 TwoCollidingStepswithLowPositionCostsVerification . . . 60
4.15 TwoCollidingStepswithLowPositionCostsEnergy . . . . . 61
4.16 TwoCollidingStepswithHighWheelVelocityCost . . . . . . 62
4.17 TwoCollidingStepswithHighWheelVelocityCostVerifica-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.18 TwoCollidingStepswithHighWheelVelocityCostEnergy . . 64
4.19 TwoCollidingStepswithHighWheelVelocityCost . . . . . . 65
4.20 TwoCollidingStepswithHighWheelVelocityCostVerifica-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.21 TwoCollidingStepswithHighWheelVelocityCostEnergy . . 66
4.22 ThreeCollidingStepsGPOPS-II . . . . . . . . . . . . . . . . . . 68
4.23 TwoCollidingStepswithHighWheelVelocityCostVerifica-
tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.24 TwoCollidingStepswithHighWheelVelocityCostEnergy . . 69
4.25 TwoStepsCollisionDynamicsGPOPS-II . . . . . . . . . . . . 74
4.26 TwoStepswithCollisionDynamicsVerification . . . . . . . . 75
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4.27 TwoCollidingStepswithHighWheelVelocityCostEnergy . . 76
4.28 SingleStepExtensiontoMulti-StepResult,Experiment1 . . 77
4.29 SingleStepExtensiontoMulti-StepResult,Experiment1 . . 78
4.30 SingleStepExtensiontoMulti-StepResult,Experiment1 . . 79
4.31 FirstSingleStepExtensiontoMulti-StepResult,Experiment5 80
4.32 Second Single Step Extension to Multi-Step Result, Experi-
ment5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.33 SingleStepExtensiontoMulti-StepVerification,Experiment
1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.34 SingleStepExtensiontoMulti-StepEnergy,Experiment1 . . 82
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List of Tables
1 ListofVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . x
4.1 NominalParameterValues . . . . . . . . . . . . . . . . . . . . 37
4.2 MotorParameters . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3 Experiment1StateConstraints . . . . . . . . . . . . . . . . . . 41
4.4 Experiment2StateConstraints . . . . . . . . . . . . . . . . . . 46
4.5 Experiment3StateConstraints . . . . . . . . . . . . . . . . . . 51
4.6 PhysicalParameterValues . . . . . . . . . . . . . . . . . . . . . 71
4.7 Experiment5StateConstraints . . . . . . . . . . . . . . . . . . 71