control design for flexible robots using the transfer matrix method · 2013-08-22 · control...
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Control Design for Flexible Robotsusing the Transfer Matrix Method
Ryan KraussPh.D. Thesis Defense
Georgia Institute of Technology
Committee:Dr. Wayne Book, ChairDr. Al FerriDr. Bill SinghoseDr. James Craig (AE)Dr. Dewey Hodges (AE)
June, 12, 2006
Motion Control
Fluid Power
Thanks
“Trust in the Lord with all your heart,and do not lean on your own understanding.
In all your ways acknowledge him,and he will make straight your paths.”
Proverbs 3:5-6
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 2 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Thanks
God
Missy
Dr. Book
Dr. Ferri
Dr. Singhose
Dr. Craig
Dr. Hodges
JD Huggins
Terri Keita
Linda Perry
Olivier Bruls
Davin, LJ, Ben, and Ho
Matt, Amir, Joe, Haihong,and everyone in the IMDL
Classmates and friends
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 3 / 83
Motion Control
Fluid Power
Overview: Four Main Parts
1 IntroductionBackgroundLiterature ReviewContributions
2 Expanding the Modeling Capabilities of the TMMHydraulic ActuatorsNon-collocated FeedbackThree Dimensional Poses
3 Developing the Control Design Capabilities of theTMM
Symbolic TMM AnalysisTwo Approaches to Control Design for SAMII
4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 4 / 83
Motion Control
Fluid Power
Overview: Four Main Parts
1 IntroductionBackgroundLiterature ReviewContributions
2 Expanding the Modeling Capabilities of the TMMHydraulic ActuatorsNon-collocated FeedbackThree Dimensional Poses
3 Developing the Control Design Capabilities of theTMM
Symbolic TMM AnalysisTwo Approaches to Control Design for SAMII
4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 4 / 83
Motion Control
Fluid Power
Overview: Four Main Parts
1 IntroductionBackgroundLiterature ReviewContributions
2 Expanding the Modeling Capabilities of the TMMHydraulic ActuatorsNon-collocated FeedbackThree Dimensional Poses
3 Developing the Control Design Capabilities of theTMM
Symbolic TMM AnalysisTwo Approaches to Control Design for SAMII
4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 4 / 83
Motion Control
Fluid Power
Overview: Four Main Parts
1 IntroductionBackgroundLiterature ReviewContributions
2 Expanding the Modeling Capabilities of the TMMHydraulic ActuatorsNon-collocated FeedbackThree Dimensional Poses
3 Developing the Control Design Capabilities of theTMM
Symbolic TMM AnalysisTwo Approaches to Control Design for SAMII
4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 4 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Software Design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 5 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Outline
1 BackgroundIntroductionProblem Statement
2 Literature Review
3 Contributions
4 Introduction to the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 6 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Areas of Application
Long reach robots
Light weight and/or fast robots
Earthquake engineering
Aerospace applications
Anywhere a control system is interacting with aflexible structure or distributed parameter system
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 7 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Areas of Application
Long reach robots
Light weight and/or fast robots
Earthquake engineering
Aerospace applications
Anywhere a control system is interacting with aflexible structure or distributed parameter system
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 7 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Areas of Application
Long reach robots
Light weight and/or fast robots
Earthquake engineering
Aerospace applications
Anywhere a control system is interacting with aflexible structure or distributed parameter system
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 7 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Areas of Application
Long reach robots
Light weight and/or fast robots
Earthquake engineering
Aerospace applications
Anywhere a control system is interacting with aflexible structure or distributed parameter system
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 7 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Areas of Application
Long reach robots
Light weight and/or fast robots
Earthquake engineering
Aerospace applications
Anywhere a control system is interacting with aflexible structure or distributed parameter system
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 7 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Problem Statement
Need a model for flexible robots that facilitates controldesign
Motion control
Vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 8 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Problem Statement
Need a model for flexible robots that facilitates controldesign
Motion control
Vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 8 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Problem Statement
Need a model for flexible robots that facilitates controldesign
Motion control
Vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 8 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Problem Statement
Need a model for flexible robots that facilitates controldesign
Motion control
Vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 8 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Important Properties to Model
Hydraulic actuators
Continuous elements
3D poses/deflections
Non-collocated feedback
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 9 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Important Properties to Model
Hydraulic actuators
Continuous elements
3D poses/deflections
Non-collocated feedback
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 9 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Important Properties to Model
Hydraulic actuators
Continuous elements
3D poses/deflections
Non-collocated feedback
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 9 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Important Properties to Model
Hydraulic actuators
Continuous elements
3D poses/deflections
Non-collocated feedback
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 9 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Assertion
A new modeling tool is needed.FEA
May be too cumbersome/can have substantiallearning curveNot controls focusedMay not be able to model feedback
Assumed Modes MethodGrows unwieldy as number of links increasesElement connectivity conditions are oftenapproximated
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 10 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Assertion
A new modeling tool is needed.FEA
May be too cumbersome/can have substantiallearning curveNot controls focusedMay not be able to model feedback
Assumed Modes MethodGrows unwieldy as number of links increasesElement connectivity conditions are oftenapproximated
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 10 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Introduction Problem Statement
Claim: TMM can be the Right Tool
Why Use the TMM?
Modular - easy to assemble complicated modelsUseful for controls engineering
Models feedbackOutputs Bode plots naturallyTransfer functions can easily be part of transfermatrices
Element connectivity conditions are handled exactlyand automatically
No discretization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 11 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Outline
1 Background
2 Literature ReviewModeling of Flexible RobotsControl of Flexible RobotsIMDL
3 Contributions
4 Introduction to the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 12 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
Assumed modes:Book (1984): Recursive Lagrangian approachDeLuca & Siciliano (1991): Closed-form modelChalhoub and Chen (1998): Extended the work ofBook (1984) to include large rotations, prismaticjoints, torsional and out-of-plane vibrations, andgeometric foreshorteningSubudhi and Moris (2002): Euler-Lagrange +assumed modes for flexible links and jointsKang and Mills (2002): Lagrange multipliers +assumed modes for parallel robotsBascetta and Rocco (2002): screw theory + assumedmodes for computational efficiency and gyroscopicmotor effects
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 13 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
FEABruls et al. (2004): Nonlinear model reduction, SAMIIand RALFTokhi et al. (2001): Single link flexible robot withexperimental agreementZhou et al. (2000): Component mode synthesis forrigid robot handling flexible payloadNagaraj et al. (1997): FEA + momentum balance forvibrations induced by locking a joint on a flexiblespacecraft
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 14 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
FEABruls et al. (2004): Nonlinear model reduction, SAMIIand RALFTokhi et al. (2001): Single link flexible robot withexperimental agreementZhou et al. (2000): Component mode synthesis forrigid robot handling flexible payloadNagaraj et al. (1997): FEA + momentum balance forvibrations induced by locking a joint on a flexiblespacecraft
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 14 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
FEABruls et al. (2004): Nonlinear model reduction, SAMIIand RALFTokhi et al. (2001): Single link flexible robot withexperimental agreementZhou et al. (2000): Component mode synthesis forrigid robot handling flexible payloadNagaraj et al. (1997): FEA + momentum balance forvibrations induced by locking a joint on a flexiblespacecraft
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 14 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
FEABruls et al. (2004): Nonlinear model reduction, SAMIIand RALFTokhi et al. (2001): Single link flexible robot withexperimental agreementZhou et al. (2000): Component mode synthesis forrigid robot handling flexible payloadNagaraj et al. (1997): FEA + momentum balance forvibrations induced by locking a joint on a flexiblespacecraft
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 14 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Modeling of Flexible Robots
FEABruls et al. (2004): Nonlinear model reduction, SAMIIand RALFTokhi et al. (2001): Single link flexible robot withexperimental agreementZhou et al. (2000): Component mode synthesis forrigid robot handling flexible payloadNagaraj et al. (1997): FEA + momentum balance forvibrations induced by locking a joint on a flexiblespacecraft
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 14 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Control of Flexible Structures
Siciliano and Book (1988): Singular perturbationapproach
Calise et al. (1990): Optimal control of slow and fastsubsystems
Luo (1993): Proof that direct-strain feedback candamp single link flexible robots
Kwon and Book (1994): Feedforward torque forend-point position tracking
Rocco and Book (1996): Extended Siciliano andBook to include contact force
Calise, Yang, and Craig (2002, 2003, 2004):Augmenting adaptive control using neural networksapplied to SAMII and other flexible systems
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 15 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
Progression of work in the IMDL
Dr. Book’s thesis (1974)Transfer matrix methodModeling, design, and control of flexible robotsEffects of flexibility on manipulator design
Majette (1985)Used TMM to model a two flexible beam systemconnected by an actuated jointDeveloped a methodology for TMM→ state spaceUsed modal state variable controlDeveloped a model update procedure
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 16 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Huggins (1988)Modeling of RALF
Linearized Assumed Modes Method (AMM)Finite Element Method (FEM)Experimental
Experimental Actuator Transfer Function
Lee (1990)Symbolic modeling of RALFMode shape determination through component modesynthesisCompared AMM to FEM and experiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 17 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Magee (1996)Optimal Arbitrary Time-Delay Filter (patented)Implemented/investigated on OATF on RALF
Cannon (1996)Design of SAMIICombined command shaping and inertial damping1 DOF / 1 mode
Loper (1998)2 DOF inertial dampingInverse dynamics to calculate interaction force
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 18 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro. Modeling Controls IMDL
IMDL Continued
Obergfell (1998)Experimental identification of RALFEnd-point position sensing and control
George (2002)6 DOF SAMII3 DOF inertial damping controlInvestigated inertia singularities
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 19 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Outline
1 Background
2 Literature Review
3 Contributions
4 Introduction to the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 20 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Background Lit. Review Contributions TMM Intro.
Contributions
ModelingHydraulic actuator model/structural interactionTMM models for three-dimensional poses of flexiblerobotsNon-collocated feedback using the TMM
ControlsSymbolic TMM AnalysisTwo approaches to control design
Software Design and ImplementationCreated an object oriented software package for TMManalysisInvestigated two areas of concern related to thenumeric implementation of the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 21 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Outline
1 Background
2 Literature Review
3 Contributions
4 Introduction to the TMM
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 22 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Finding System Eigenvalues using the TMM
Find transfer matrices for each element type
Multiply element transfer matrices to find systemtransfer matrix
Find sub-matrix whose determinant must go to zero(based on boundary conditions)
Find values of s that cause |subU| = 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 23 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Finding System Eigenvalues using the TMM
Find transfer matrices for each element type
Multiply element transfer matrices to find systemtransfer matrix
Find sub-matrix whose determinant must go to zero(based on boundary conditions)
Find values of s that cause |subU| = 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 23 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Finding System Eigenvalues using the TMM
Find transfer matrices for each element type
Multiply element transfer matrices to find systemtransfer matrix
Find sub-matrix whose determinant must go to zero(based on boundary conditions)
Find values of s that cause |subU| = 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 23 / 83
Motion Control
Fluid Power
Background Lit. Review Contributions TMM Intro.
Finding System Eigenvalues using the TMM
Find transfer matrices for each element type
Multiply element transfer matrices to find systemtransfer matrix
Find sub-matrix whose determinant must go to zero(based on boundary conditions)
Find values of s that cause |subU| = 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 23 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D
Overview: Four Main Parts
1 Introduction and Background2 Expanding the Modeling Capabilities of the TMM3 Developing the Control Design Capabilities of the
TMM4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 24 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D AVS w/SD
Outline
5 Hydraulic ActuatorAngular Velocity Source with Spring/Damper
6 Modeling Feedback
7 Three Dimensional Poses and Deformations
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 25 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D AVS w/SD
Hydraulic Actuator Modeling
Servovalves
d
Ptank PtankPhigh
P1 P2
q2q1x
Input: dOutput: x
Orifice flow:q = k
√∆P
Velocity Source:x
d=
1
s
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 26 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D AVS w/SD
Hydraulic Actuator Modeling
Goal: a simple model that captures the essentialdynamics and integrates cleanly with the structural model
θ =K p v
s(s + p)+
M
cs + k
Uol =
1 0 0 0 0
0 11
cs + k0 Gpv
0 0 1 0 00 0 0 1 00 0 0 0 1
and z =
xθMV1
Gp =
K p
s(s + p)
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 27 / 83
Hydraulic Actuator Feedback 3D AVS w/SD
AVS w/SD Model - Experimental Comparison
joint 2
x
θ
-v Gp-θ
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 28 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Outline
5 Hydraulic Actuator
6 Modeling FeedbackIntroductionAVS w/SDVibration Suppression
7 Three Dimensional Poses and Deformations
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 29 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Non-collocated Feedback Modeling
Why?
May not be practical to precisely collocated sensorsand actuators
This is a limitation of the TMM that should beovercome to expand its use
Two Cases:
Angular velocity source in series with spring/damper
Acceleration feedback for vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 30 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Non-collocated Feedback Modeling
Why?
May not be practical to precisely collocated sensorsand actuators
This is a limitation of the TMM that should beovercome to expand its use
Two Cases:
Angular velocity source in series with spring/damper
Acceleration feedback for vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 30 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Non-collocated Feedback Modeling
Why?
May not be practical to precisely collocated sensorsand actuators
This is a limitation of the TMM that should beovercome to expand its use
Two Cases:
Angular velocity source in series with spring/damper
Acceleration feedback for vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 30 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Non-collocated Feedback Modeling
Why?
May not be practical to precisely collocated sensorsand actuators
This is a limitation of the TMM that should beovercome to expand its use
Two Cases:
Angular velocity source in series with spring/damper
Acceleration feedback for vibration suppression
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 30 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Position Feedback/Motion Control
-θd+ g - Gθ
-v
Gp
actuator/structureinteraction qθ - Gflexb
accelerationresponse
-x
(θ feedback loop)
6−
v = Gθ (θd − θ)
θ =GθGpθd
1 + GθGp
+M
(1 + GθGp) (cs + k)
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 31 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
TMM Closed-Loop Model
θ =GθGpθd
1 + GθGp
+M
(1 + GθGp) (cs + k)
Ucl =
1 0 0 0 0
0 1 1(1+GθGp)(cs+k)
0 GθGp θd
1+GθGp
0 0 1 0 00 0 0 1 00 0 0 0 1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 32 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
θ Feedback - Experimental Results
θ/θd x/θd
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 33 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Vibration Suppression
-θd+ g -
θd + g - Gθ-
vGp
actuator/structureinteraction qθ - Gflexb
accelerationresponse
x
� (x feedback loop)Ga
?−
(θ feedback loop)
6−
θd = Gas2xbeam + θd
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 34 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Vibration Suppression Continued
θ =GaGθGps
2xbeam
1 + GθGp
+GθGpθd
1 + GθGp
+M
(cs + k) (1 + GθGp)
Uacc =
1 0 0 0 0
0 1 0 0 GaGθGps2xbeam
GθGp+1
0 0 1 0 00 0 0 1 00 0 0 0 1
za = UclUacczb
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 35 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Problem: xbeam is not available
ztip = Ulink2UclUaccUlink1Ujoint1Ulink0UbeamUbasezbase
zlink1 = Ulink1Ujoint1Ulink0zbeam
zbeam = (Ulink1Ujoint1Ulink0)−1 zlink1
xbeam = axlink1 + bθlink1 + cMlink1 + dVlink1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 36 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Problem: xbeam is not available
ztip = Ulink2UclUaccUlink1Ujoint1Ulink0UbeamUbasezbase
zlink1 = Ulink1Ujoint1Ulink0zbeam
zbeam = (Ulink1Ujoint1Ulink0)−1 zlink1
xbeam = axlink1 + bθlink1 + cMlink1 + dVlink1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 36 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Problem: xbeam is not available
ztip = Ulink2UclUaccUlink1Ujoint1Ulink0UbeamUbasezbase
zlink1 = Ulink1Ujoint1Ulink0zbeam
zbeam = (Ulink1Ujoint1Ulink0)−1 zlink1
xbeam = axlink1 + bθlink1 + cMlink1 + dVlink1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 36 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Final Acceleration Feedback Model
xbeam = axlink1 + bθlink1 + cMlink1 + dVlink1
Uacc =
1 0 0 0 0
aGaGθGps2
1+GθGp
bGaGθGps2
1+GθGp+ 1 cGaGθGps2
1+GθGp
dGaGθGps2
1+GθGp0
0 0 1 0 00 0 0 1 00 0 0 0 1
ztip = Ulink2UclUaccUlink1Ujoint1Ulink0UbeamUbasezbase
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 37 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Accel. Feedback - Experimental Results
θ/θd x/θd
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 38 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction AVS w/SD Vibration Suppression
Accel. Feedback - Effectiveness
Effectiveness
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 39 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Outline
5 Hydraulic Actuator
6 Modeling Feedback
7 Three Dimensional Poses and DeformationsIntroductionNumerical Verification
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 40 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Arbitrary Three Dimensional Poses
Still considering serial connections of onedimensional links
Arbitrary posesThree main components:
1 Beams2 Rigid links3 Joints
Three dimensional deformations of one dimensionalbeam elements
Bending about two axesTorsion and axial vibration
Careful attention to detail
Numerical verification by FEA comparison
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 41 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
FEA Comparison for an L-Shaped Structure
Natural FrequenciesMode FEA TMM %
(Hz) (Hz) Diff.1 3.0934 3.0937 -0.008232 3.7973 3.7976 -0.009223 25.794 25.804 -0.03764 31.339 31.356 -0.05535 53.422 53.482 -0.1126 75.434 75.495 -0.08017 99.203 99.505 -0.3048 143.73 144.04 -0.229 171.25 171.93 -0.392
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 42 / 83
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Mode 1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 43 / 83
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Mode 2
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 44 / 83
Motion Control
Fluid Power
Hydraulic Actuator Feedback 3D Introduction Numerical Verification
Mode 3: Animation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 45 / 83
Motion Control
Fluid Power
Controls
Overview: Four Main Parts
1 Introduction and Background2 Expanding the Modeling Capabilities of the TMM3 Developing the Control Design Capabilities of the
TMM4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 46 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Outline
8 Control DesignSymbolic TMM ImplementationBode-Based OptimizationPole Placement/Optimization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 47 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Symbolic TMM Analysis
Closed-form expressions for the closed-loop systemresponse without discretization
Use Python analysis package like a higher levellanguage to generate symbolic transfer matrices
Transparent to the user
Facilitates control design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 48 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Symbolic TMM Analysis
Closed-form expressions for the closed-loop systemresponse without discretization
Use Python analysis package like a higher levellanguage to generate symbolic transfer matrices
Transparent to the user
Facilitates control design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 48 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Symbolic TMM Analysis
Closed-form expressions for the closed-loop systemresponse without discretization
Use Python analysis package like a higher levellanguage to generate symbolic transfer matrices
Python Maxima FORTRAN
Python
Compiled
Transparent to the user
Facilitates control design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 48 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Symbolic TMM Analysis
Closed-form expressions for the closed-loop systemresponse without discretization
Use Python analysis package like a higher levellanguage to generate symbolic transfer matrices
Python Maxima FORTRAN
Python
Compiled
Transparent to the user
Facilitates control design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 48 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Symbolic TMM Analysis
Closed-form expressions for the closed-loop systemresponse without discretization
Use Python analysis package like a higher levellanguage to generate symbolic transfer matrices
Python Maxima FORTRAN
Python
Compiled
Transparent to the user
Facilitates control design
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 48 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Bode-Based Optimization
joint 2
x
θ
-θd+ e -
θd+ e - Gθ-v Gp
qθ- Gflexb-+ e x
�Ga
?−
6−
d6
+
Gθ = Kθ
Ga =Kaω
2c
s2 + 2ζωcs + ω2c
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 49 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Bode-Based Optimization
joint 2
x
θ
-θd+ e -
θd+ e - Gθ-v Gp
qθ- Gflexb-+ e x
�Ga
?−
6−
d6
+
Gθ = Kθ
Ga =Kaω
2c
s2 + 2ζωcs + ω2c
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 49 / 83
Controls Symbolic Bode Pole
Bode Cost Functions
Gp
cost = 100− fco + penalty ifphase margin < 60◦
Gflexb
cost = 100− peak1 + penalty ifphase margin < 60◦ or Ka > 20
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 50 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Bode Design Results
θ/θd
SequentialKθ = 1.684Ka = 20.00fc = 1.352
x/θd
SimultaneousKθ = 1.204Ka = 19.99fc = 1.953
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 51 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Placement/Optimization
Direct variation of control gains
Track pole locations as gains are varied
Use optimization to choose gains that minimize somecost
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 52 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Placement/Optimization
Direct variation of control gains
Track pole locations as gains are varied
Use optimization to choose gains that minimize somecost
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 52 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Placement/Optimization
Direct variation of control gains
Track pole locations as gains are varied
Use optimization to choose gains that minimize somecost
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 52 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Compare to Method of Book and Majette
Discretization
Ackermann
Iteration
x = Ax + Bu
y = Cx
A and C are known
Curve fit to find B
Control affects mode shapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 53 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Compare to Method of Book and Majette
Discretization
Ackermann
Iteration
x = Ax + Bu
y = Cx
A and C are known
Curve fit to find B
Control affects mode shapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 53 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Compare to Method of Book and Majette
Discretization
Ackermann
Iteration
x = Ax + Bu
y = Cx
A and C are known
Curve fit to find B
Control affects mode shapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 53 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Compare to Method of Book and Majette
Discretization
Ackermann
Iteration
x = Ax + Bu
y = Cx
A and C are known
Curve fit to find B
Control affects mode shapes
Kp
Kd
desired eigenvalues=−0.3± 3j
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 53 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Comparison
Book and Majette Symbolic TMMKp 0.94519830 0.94521384Kd 0.22833651 0.22834017
Eigenvalue -0.300003+2.999992j -0.300000+3.000000jEvaluationTime (sec)
3.887 0.091
Direct control design based on a symbolic implementationof the TMM is faster and avoids modal discretization.
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 54 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Comparison
Book and Majette Symbolic TMMKp 0.94519830 0.94521384Kd 0.22833651 0.22834017
Eigenvalue -0.300003+2.999992j -0.300000+3.000000jEvaluationTime (sec)
3.887 0.091
Direct control design based on a symbolic implementationof the TMM is faster and avoids modal discretization.
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 54 / 83
Controls Symbolic Bode Pole
Pole Optimization Design for SAMII
Top Level:cost =(abs(pr − dpr))
2
+(abs(p1 − dp1))2
+ penalties
Problem:How does thealgorithm find thepoles and tell themapart?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 55 / 83
Controls Symbolic Bode Pole
Pole Optimization Design for SAMII
Top Level:cost =(abs(pr − dpr))
2
+(abs(p1 − dp1))2
+ penalties
Problem:How does thealgorithm find thepoles and tell themapart?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 55 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
A pole finding algorithm finds the four poles for thecurrent values of the control gains
Uses Newton’s method to find the polesRequires an initial guess for each pole
What if Newton’s method converges to the “wrong”pole? Or fails to converge?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 56 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
A pole finding algorithm finds the four poles for thecurrent values of the control gains
Uses Newton’s method to find the polesRequires an initial guess for each pole
What if Newton’s method converges to the “wrong”pole? Or fails to converge?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 56 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
A pole finding algorithm finds the four poles for thecurrent values of the control gains
Uses Newton’s method to find the polesRequires an initial guess for each pole
What if Newton’s method converges to the “wrong”pole? Or fails to converge?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 56 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
A pole finding algorithm finds the four poles for thecurrent values of the control gains
Uses Newton’s method to find the polesRequires an initial guess for each pole
What if Newton’s method converges to the “wrong”pole? Or fails to converge?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 56 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
A pole finding algorithm finds the four poles for thecurrent values of the control gains
Uses Newton’s method to find the polesRequires an initial guess for each pole
What if Newton’s method converges to the “wrong”pole? Or fails to converge?
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 56 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A
B
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A
B
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A
B
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A → B: -10.62+72.05j
A
B
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
Define a point C:Kθ=0.979, Ka=20.82, ωc=13.95
A
B
C
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
Define a point C:Kθ=0.979, Ka=20.82, ωc=13.95
A → C: -11.13+71.56j A
B
C
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A
B
CD
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A → D: -10.90+14.85j
D → C: -10.25+13.77j
C → E: -9.614+12.72j
E → B: -9.013+11.71j
A
B
CD
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A → D: -10.90+14.85j
D → C: -10.25+13.77j
C → E: -9.614+12.72j
E → B: -9.013+11.71j
A
B
CD
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A → D: -10.90+14.85j
D → C: -10.25+13.77j
C → E: -9.614+12.72j
E → B: -9.013+11.71j
A
B
CD
E
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Adaptive Interpolation Algorithm
Automated process
At point A:Kθ=1.06, Ka=23.36, ωc=15.23pfA=-11.56+15.96j
At point B:Kθ=0.897, Ka=18.28, ωc=12.67pfB=?
A → D: -10.90+14.85j
D → C: -10.25+13.77j
C → E: -9.614+12.72j
E → B: -9.013+11.71j
A
B
CD
E
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 57 / 83
Motion Control
Fluid Power
Controls Symbolic Bode Pole
Comparing Pole and Bode Optimizations
θ/θd
BodeKθ = 1.204Ka = 19.99fc = 1.953
x/θd
PoleKθ = 0.813Ka = 20.00fc = 2.28
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 58 / 83
Motion Control
Fluid Power
Software Design Numerical
Overview: Four Main Parts
1 Introduction and Background2 Expanding the Modeling Capabilities of the TMM3 Developing the Control Design Capabilities of the
TMM4 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 59 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Outline
9 Software DesignIntroductionSystem ID
10 Numerical Issues
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 60 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 61 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
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Software Design Numerical Introduction System ID
Software Design
Goal: make the full power of the TMM accessible toengineers who are not TMM experts
Implementation: user-extensible, object-orientedframeworkLanguage: Python
Open sourceCross platformClean syntaxMany existing modules
Scipy, Matplotlib, iPython, Mayavi, . . .
Two Primary Classes1 TMMElement2 TMMSystem
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Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Capabilities
System natural frequencies and mode shapes
Bode analysis
Symbolic analysis
Control design
Integrated system ID
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Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Capabilities
System natural frequencies and mode shapes
Bode analysis
Symbolic analysis
Control design
Integrated system ID
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 62 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Capabilities
System natural frequencies and mode shapes
Bode analysis
Symbolic analysis
Control design
Integrated system ID
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 62 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Capabilities
System natural frequencies and mode shapes
Bode analysis
Symbolic analysis
Control design
Integrated system ID
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 62 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Capabilities
System natural frequencies and mode shapes
Bode analysis
Symbolic analysis
Control design
Integrated system ID
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 62 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
SAMII System Schematic
Joint 1
Link 2
Link 1
Link 0
Beam
Base Spring
Joint 2/
Actuator
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Software Design Numerical Introduction System ID
SAMII Modeldef olsami imodel ( ) :
basespr ing=TorsionalSpringDamper4x4 ({ ’k’:166358.0 ,’c’ :468.789} )
beam=samiiBeam ( )l i n k 0 =sami iL ink0 ( )j 1 s p r i n g =TorsionalSpringDamper4x4 ({ ’k’ :4028.28 ,
’c’ : 6.3058} )l i n k 1 =sami iL ink1 ( )avs=AngularVeloc i tySource4x4 ({ ’K’ :0 .435489 ,’tau
’ :173.833} )j 2 s p r i n g =TorsionalSpringDamper4x4 ({ ’k’ :1900.49 ,
’c’ :21 .6805} )l i n k 2 =sami iL ink2 ( )
...
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Software Design Numerical Introduction System ID
SAMII Model Continued
bodeout1=bodeout ( i n pu t =’j2v’ , ou tput=’a1’ , type=’abs’ , ind=beam, post=’accel’ , dof =0 ,gain =0.35 , gainknown=False )
bodeout2=bodeout ( i n pu t =’j2v’ , ou tput=’j2a’ ,type=’diff’ , ind =[ j2sp r ing , l i n k 1 ] , post=’’ , dof =1 , gain =180.0/ p i )
r e t u r n ClampedFreeTMMSystem ( [ basespring , beam,l i nk0 , j 1sp r ing , l i nk1 , avs , j2sp r ing ,l i n k 2 ] , bodeouts =[ bodeout1 , bodeout2 ] )
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Software Design Numerical Introduction System ID
System Identification
Can be a labor intensive and error prone process
Existing system ID tools do not have TMM modelingcapabilitiesTwo main points:
How system ID capabilities have been integrated intothe softwareSystem ID approach for SAMII
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 68 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
System Identification
Can be a labor intensive and error prone process
Existing system ID tools do not have TMM modelingcapabilitiesTwo main points:
How system ID capabilities have been integrated intothe softwareSystem ID approach for SAMII
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 68 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
System Identification
Can be a labor intensive and error prone process
Existing system ID tools do not have TMM modelingcapabilitiesTwo main points:
How system ID capabilities have been integrated intothe softwareSystem ID approach for SAMII
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 68 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
System Identification
Can be a labor intensive and error prone process
Existing system ID tools do not have TMM modelingcapabilitiesTwo main points:
How system ID capabilities have been integrated intothe softwareSystem ID approach for SAMII
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 68 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
System Identification
Can be a labor intensive and error prone process
Existing system ID tools do not have TMM modelingcapabilitiesTwo main points:
How system ID capabilities have been integrated intothe softwareSystem ID approach for SAMII
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 68 / 83
Motion Control
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Software Design Numerical Introduction System ID
Software Integration
Specify unknown parameters for each element
def o l sami imode l w i th ig ( ) :basespr ing=TorsionalSpringDamper4x4 ({ ’k’
:166358.0 ,’c’ :468.789} , symlabel=’base’ ,unknownparams =[ ’k’ ,’c’ ] )
avs=AngularVeloc i tySource4x4 ({ ’K’ :0 .435489 ,’tau’ :173.833} , symlabel=’act’ ,unknownparams =[ ’K’ ,’tau’ ] )
...
Completely automated process: symbolic Bodefunctions, cost functions, curve-fitting scripts, andinitial guess vectors
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 70 / 83
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Software Design Numerical Introduction System ID
Software Integration
Specify unknown parameters for each element
def o l sami imode l w i th ig ( ) :basespr ing=TorsionalSpringDamper4x4 ({ ’k’
:166358.0 ,’c’ :468.789} , symlabel=’base’ ,unknownparams =[ ’k’ ,’c’ ] )
avs=AngularVeloc i tySource4x4 ({ ’K’ :0 .435489 ,’tau’ :173.833} , symlabel=’act’ ,unknownparams =[ ’K’ ,’tau’ ] )
...
Completely automated process: symbolic Bodefunctions, cost functions, curve-fitting scripts, andinitial guess vectors
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 70 / 83
Software Design Numerical Introduction System ID
System ID Data
One input: v Two outputs: θ and xθ/v x/v
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Software Design Numerical Introduction System ID
Accuracy Quantification
How well can the resulting model be used for controldesign?
Fit open-loop data
Measure accuracy from open and closed-loop Bodeplots
Provides quantitative measure of accuracy tocompare different models or system ID approaches
Best results come from respecting the logarithmicnature of the Bode plots
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Software Design Numerical Introduction System ID
Accuracy Quantification
How well can the resulting model be used for controldesign?
Fit open-loop data
Measure accuracy from open and closed-loop Bodeplots
Provides quantitative measure of accuracy tocompare different models or system ID approaches
Best results come from respecting the logarithmicnature of the Bode plots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 72 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Accuracy Quantification
How well can the resulting model be used for controldesign?
Fit open-loop data
Measure accuracy from open and closed-loop Bodeplots
Provides quantitative measure of accuracy tocompare different models or system ID approaches
Best results come from respecting the logarithmicnature of the Bode plots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 72 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Accuracy Quantification
How well can the resulting model be used for controldesign?
Fit open-loop data
Measure accuracy from open and closed-loop Bodeplots
Provides quantitative measure of accuracy tocompare different models or system ID approaches
Best results come from respecting the logarithmicnature of the Bode plots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 72 / 83
Motion Control
Fluid Power
Software Design Numerical Introduction System ID
Accuracy Quantification
How well can the resulting model be used for controldesign?
Fit open-loop data
Measure accuracy from open and closed-loop Bodeplots
Provides quantitative measure of accuracy tocompare different models or system ID approaches
Best results come from respecting the logarithmicnature of the Bode plots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 72 / 83
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Software Design Numerical Introduction System ID
Accuracy Quantification Continued
θ/θd x/θd
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Software Design Numerical Introduction System ID
Results
θ/θd x/θd
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Software Design Numerical Floating Point Repeated Roots
Outline
9 Software Design
10 Numerical IssuesFloating PointRepeated Roots
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Software Design Numerical Floating Point Repeated Roots
Floating Point: Pinned-Pinned Beam
Characteristic Determinant: big number ± smallnumber
subU =
(sinh β + sin β) L
2 β−(sinh β − sin β) L3
2 β3 EIβ (sinh β − sin β) EI
2 L−(sinh β + sin β) L
2 β
|subU| = (sinh β − sin β)2 L2
4 β2− (sinh β + sin β)2 L2
4 β2
sinh β ± sin β = sinh βwhen β ≥ 37.429948
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Software Design Numerical Floating Point Repeated Roots
Floating Point: Pinned-Pinned Beam
Characteristic Determinant: big number ± smallnumber
subU =
(sinh β + sin β) L
2 β−(sinh β − sin β) L3
2 β3 EIβ (sinh β − sin β) EI
2 L−(sinh β + sin β) L
2 β
|subU| = (sinh β − sin β)2 L2
4 β2− (sinh β + sin β)2 L2
4 β2
sinh β ± sin β = sinh βwhen β ≥ 37.429948
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Software Design Numerical Floating Point Repeated Roots
Floating Point: Pinned-Pinned Beam
Characteristic Determinant: big number ± smallnumber
subU =
(sinh β + sin β) L
2 β−(sinh β − sin β) L3
2 β3 EIβ (sinh β − sin β) EI
2 L−(sinh β + sin β) L
2 β
|subU| = (sinh β − sin β)2 L2
4 β2− (sinh β + sin β)2 L2
4 β2
sinh β ± sin β = sinh βwhen β ≥ 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 76 / 83
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Software Design Numerical Floating Point Repeated Roots
Characteristic Determinant vs. β
|subU| = −sin β sinh β L2
β2
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Software Design Numerical Floating Point Repeated Roots
Characteristic Determinant vs. β
|subU| = −sin β sinh β L2
β2
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Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
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Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 78 / 83
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Fluid Power
Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 78 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 78 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 78 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 78 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Repeated Roots 0...0
= subU(s)zbase
Find values of s that cause subU(s) to have a nullspace ⇒ eigenvalues of the system
Find null space of subU(s) ⇒ mode shapesThree approaches to finding the null space:
RREFEigenvalues/eigenvectorSVD
All 3 approaches include means to check forrepeated or nearly repeated roots
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Motion Control
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Software Design Numerical Floating Point Repeated Roots
Nearly Repeated Roots
RREF:
subU(s1) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
Eigenvalues: abs of second smallest eig → 0
SVD: abs of second smallest sv → 0
SVD handles ortho-normalization automatically
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Software Design Numerical Floating Point Repeated Roots
Nearly Repeated Roots
RREF:
subU(s1) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
Eigenvalues: abs of second smallest eig → 0
SVD: abs of second smallest sv → 0
SVD handles ortho-normalization automatically
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 79 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Nearly Repeated Roots
RREF:
subU(s1) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
Eigenvalues: abs of second smallest eig → 0
SVD: abs of second smallest sv → 0
SVD handles ortho-normalization automatically
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 79 / 83
Motion Control
Fluid Power
Software Design Numerical Floating Point Repeated Roots
Nearly Repeated Roots
RREF:
subU(s1) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
Eigenvalues: abs of second smallest eig → 0
SVD: abs of second smallest sv → 0
SVD handles ortho-normalization automatically
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 79 / 83
Motion Control
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Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
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Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Modeling Contributions
TMM hydraulic actuator modelCaptures essential dynamicsProvides quantitative agreement with experiment
Derived transfer matrices to model arbitrarythree-dimensional poses of flexible robots
Numerically verified these matrices through FEAcomparison
Developed techniques for modeling non-collocatedfeedback using the TMM
Quantitative agreement between model andexperiment
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 80 / 83
Motion Control
Fluid Power
Contributions
Controls Contributions
Developed two approaches to control design forflexible robots including multiple feedback loops
Simultaneous optimization of multiple Bode plotsOptimized closed-loop pole-locationsDeveloped a robust pole-tracking algorithm to beused with the optimization technique
Implemented the TMM symbolically, allowingclosed-form expressions for the closed-loop systemresponse to be found
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 81 / 83
Motion Control
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Contributions
Controls Contributions
Developed two approaches to control design forflexible robots including multiple feedback loops
Simultaneous optimization of multiple Bode plotsOptimized closed-loop pole-locationsDeveloped a robust pole-tracking algorithm to beused with the optimization technique
Implemented the TMM symbolically, allowingclosed-form expressions for the closed-loop systemresponse to be found
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 81 / 83
Motion Control
Fluid Power
Contributions
Controls Contributions
Developed two approaches to control design forflexible robots including multiple feedback loops
Simultaneous optimization of multiple Bode plotsOptimized closed-loop pole-locationsDeveloped a robust pole-tracking algorithm to beused with the optimization technique
Implemented the TMM symbolically, allowingclosed-form expressions for the closed-loop systemresponse to be found
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 81 / 83
Motion Control
Fluid Power
Contributions
Controls Contributions
Developed two approaches to control design forflexible robots including multiple feedback loops
Simultaneous optimization of multiple Bode plotsOptimized closed-loop pole-locationsDeveloped a robust pole-tracking algorithm to beused with the optimization technique
Implemented the TMM symbolically, allowingclosed-form expressions for the closed-loop systemresponse to be found
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 81 / 83
Motion Control
Fluid Power
Contributions
Controls Contributions
Developed two approaches to control design forflexible robots including multiple feedback loops
Simultaneous optimization of multiple Bode plotsOptimized closed-loop pole-locationsDeveloped a robust pole-tracking algorithm to beused with the optimization technique
Implemented the TMM symbolically, allowingclosed-form expressions for the closed-loop systemresponse to be found
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 81 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Contributions
Software Design and ImplementationContributions
Created an object-oriented software package forTMM analysis
Provides a user-extensible frameworkCreated the TMMElement and TMMSystem classes anddemonstrated their usefulness as softwareabstractions of physical things
Investigated two areas of concern related to thenumeric implementation of the TMM
Demonstrated ways to recognize floating-point errorproblems and showed that symbolic TMM analysismay avoid these problemsCompared three approaches to dealing with repeatedsystem eigenvalues and finding the associated modeshapes
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method 82 / 83
Motion Control
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Contributions
Overview
Modeling
HydraulicActuators3D PosesNon-collocatedFeedback
Control Design
BodeOptimization
Pole-PlacementPole-Tracking
SymbolicTMM
Software Design
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Intro Modeling Controls SW+Num. TMM Intro
Outline
11 IntroTMM Intro
12 Modeling
13 Controls
14 Software Design and Implementation
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Intro Modeling Controls SW+Num. TMM Intro
What is the TMM?
m
k
F (t)
Spring analysis
F1 = F0
x1 =F0
k+ x0
[x1
F1
]=
[1 1/k0 1
] [x0
F0
]
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
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Intro Modeling Controls SW+Num. TMM Intro
What is the TMM?
m
k
F (t)x0
F0
x1
F1
x2F2
Spring analysis
F1 = F0
x1 =F0
k+ x0
[x1
F1
]=
[1 1/k0 1
] [x0
F0
]
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
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Intro Modeling Controls SW+Num. TMM Intro
What is the TMM?
m
k
F (t)x0
F0
x1
F1
x2F2
Mass analysis
x2 = x1
F2 = F1 +ms2x1
[x2
F2
]=
[1 0
ms2 1
] [x1
F1
]
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
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Intro Modeling Controls SW+Num. TMM Intro
TMM Introduction
[x1
F1
]=
[1 1/k0 1
] [x0
F0
][
x2
F2
]=
[1 0
ms2 1
] [x1
F1
][
x2
F2
]=
[1 0
ms2 1
] [1 1/k0 1
] [x0
F0
][
x2
F2
]=
[1 1/k
ms2 ms2/k + 1
] [x0
F0
]
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
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Intro Modeling Controls SW+Num. TMM Intro
Free Response
x0 = 0
F2 = 0[x2
0
]=
[1 1/k
ms2 ms2/k + 1
] [0F0
](
ms2
k+ 1
)F0 = 0
ms2
k+ 1 = 0 ⇒ s2 = − k
m⇒ s = j
√k
m
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
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Intro Modeling Controls SW+Num. TMM Intro
Free Response
x0 = 0
F2 = 0[x2
0
]=
[1 1/k
ms2 ms2/k + 1
] [0F0
](
ms2
k+ 1
)F0 = 0
ms2
k+ 1 = 0 ⇒ s2 = − k
m⇒ s = j
√k
m
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMM Intro
Free Response
x0 = 0
F2 = 0[x2
0
]=
[1 1/k
ms2 ms2/k + 1
] [0F0
](
ms2
k+ 1
)F0 = 0
ms2
k+ 1 = 0 ⇒ s2 = − k
m⇒ s = j
√k
m
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. Feedback
Outline
11 Intro
12 ModelingNon-collocated Feedback
13 Controls
14 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. Feedback
Open Loop Model
-v
Gp
actuator/structureinteraction
-θ
Gflexb
accelerationresponse
-x
Uol =
1 0 0 0 00 1 1
cs+k0 Gpv
0 0 1 0 00 0 0 1 00 0 0 0 1
θa = θb +
M
cs + k+ Gpv
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. Feedback
Determining xbeam
a = −L0m1r1s2 − c1s− k1
c1s + k1
b = − Nb
c1s + k1
Nb =(L0m1r
21 − L0L1m1r1 + Iz1L0
)s2 + . . .
(c1L1 + c1L0) s + k1L1 + k1L0
c =L0
c1s + k1
d =L0L1
c1s + k1
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. Majette Pole Tracking
Outline
11 Intro
12 Modeling
13 ControlsBook and MajettePole Tracking Algorithm
14 Software Design and Implementation
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. Majette Pole Tracking
Method of Book and Majette - Results
Iteration Kp Kd Eigenvalue Error (abs)0 0.01000000 0.00000000 -0.000000+0.485676j 2.5321581 0.38861147 0.02565663 -0.103581+2.340975j 0.6876732 0.70769571 0.07418619 -0.159152+2.733151j 0.3017393 0.90009130 0.12972375 -0.200068+2.888061j 0.1500554 0.98011847 0.17500927 -0.232876+2.957729j 0.0793255 0.99570142 0.20407931 -0.258310+2.989046j 0.0431056 0.98606356 0.21957749 -0.276411+3.001490j 0.0236367 0.97177247 0.22654284 -0.288012+3.004937j 0.0129658 0.96026864 0.22905671 -0.294675+3.004685j 0.0070939 0.95280668 0.22959107 -0.298086+3.003363j 0.003870
10 0.94857000 0.22940842 -0.299611+3.002071j 0.00210711 0.94642341 0.22906731 -0.300167+3.001134j 0.00114612 0.94546725 0.22877188 -0.300286+3.000553j 0.00062313 0.94511758 0.22856918 -0.300245+3.000234j 0.00033814 0.94504021 0.22844810 -0.300167+3.000077j 0.00018415 0.94506385 0.22838349 -0.300099+3.000010j 0.00010016 0.94511097 0.22835283 -0.300053+2.999988j 0.00005417 0.94515241 0.22834040 -0.300025+2.999984j 0.00002918 0.94518107 0.22833667 -0.300010+2.999988j 0.00001619 0.94519830 0.22833651 -0.300003+2.999992j 0.000009
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Intro Modeling Controls SW+Num. Majette Pole Tracking
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
Inside the black box: vary s to find poles given Kθ,Ka, and ωc (treated as constants)
pole:1
TF (s)= 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Intro Modeling Controls SW+Num. Majette Pole Tracking
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
Inside the black box: vary s to find poles given Kθ,Ka, and ωc (treated as constants)
pole:1
TF (s)= 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Intro Modeling Controls SW+Num. Majette Pole Tracking
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
Inside the black box: vary s to find poles given Kθ,Ka, and ωc (treated as constants)
s, Kθ, Ka, ωcBodeFunction TF (s)
pole:1
TF (s)= 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Intro Modeling Controls SW+Num. Majette Pole Tracking
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
Inside the black box: vary s to find poles given Kθ,Ka, and ωc (treated as constants)
s, Kθ, Ka, ωcBodeFunction TF (s)
pole:1
TF (s)= 0
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Intro Modeling Controls SW+Num. Majette Pole Tracking
Pole Tracking Algorithm
Kθ
KaBlackBox
p1
pr
pf
p2fc
Top level algorithm seeks to optimize the polelocations by varying the control gains Kθ, Ka, and fc
Inside the black box: vary s to find poles given Kθ,Ka, and ωc (treated as constants)
s, Kθ, Ka, ωcBodeFunction TF (s)
pole:1
TF (s)= 0
si
Kθ, Ka, ωcNewton pi
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Outline
11 Intro
12 Modeling
13 Controls
14 Software Design and ImplementationTMMElement and TMMSystemTMMElementTMMSystemNumerical IssuesNewton
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMElement Class
Represents one element in a TMM system
Primary purpose is to return transfer matrix U(s)
Primary means of user extensibility
User must derive from the class and override GetMat,GetMaximaString, and GetHT
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMElement Class
Represents one element in a TMM system
Primary purpose is to return transfer matrix U(s)
Primary means of user extensibility
User must derive from the class and override GetMat,GetMaximaString, and GetHT
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMElement Class
Represents one element in a TMM system
Primary purpose is to return transfer matrix U(s)
Primary means of user extensibility
User must derive from the class and override GetMat,GetMaximaString, and GetHT
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMElement Class
Represents one element in a TMM system
Primary purpose is to return transfer matrix U(s)
Primary means of user extensibility
User must derive from the class and override GetMat,GetMaximaString, and GetHT
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Inheritance Example
define GetMat
define GetHT
c lass TorsionalSpringDamper ( TMMIHTElement ) :def GetMat ( s e l f , s , sym=False ) :
N= s e l f . maxsizei f sym :
myparams= s e l f . symparamselse :
myparams= s e l f . paramsk=myparams [ ’k’ ]c=myparams [ ’c’ ]spr ing term =1 / ( k [ 0 ] + c [ 0 ] ∗ s )
...
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Inheritance Example
define GetMat
define GetHT
c lass TorsionalSpringDamper ( TMMIHTElement ) :def GetMat ( s e l f , s , sym=False ) :
N= s e l f . maxsizei f sym :
myparams= s e l f . symparamselse :
myparams= s e l f . paramsk=myparams [ ’k’ ]c=myparams [ ’c’ ]spr ing term =1 / ( k [ 0 ] + c [ 0 ] ∗ s )
...
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Inheritance Example
define GetMat
define GetHT
c lass TorsionalSpringDamper ( TMMIHTElement ) :def GetMat ( s e l f , s , sym=False ) :
N= s e l f . maxsizei f sym :
myparams= s e l f . symparamselse :
myparams= s e l f . paramsk=myparams [ ’k’ ]c=myparams [ ’c’ ]spr ing term =1 / ( k [ 0 ] + c [ 0 ] ∗ s )
...
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Inheritance Example
define GetMat
define GetHT
c lass TorsionalSpringDamper ( TMMIHTElement ) :def GetMat ( s e l f , s , sym=False ) :
N= s e l f . maxsizei f sym :
myparams= s e l f . symparamselse :
myparams= s e l f . paramsk=myparams [ ’k’ ]c=myparams [ ’c’ ]spr ing term =1 / ( k [ 0 ] + c [ 0 ] ∗ s )
...
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Inheritance Continuedi f sym :
maxlen= len ( spr ing term ) +10matout=eye (N, dtype=’f’ )matout=matout . astype ( ’S%d’%maxlen )
e lse :matout=eye (N, dtype=’D’ )
matout [1 ,2 ]= spr ingtermi f max( shape ( k ) )>1 and s e l f . maxsize >=8:
matout [ 5 , 6 ] = 1 / ( k [ 1 ] + c [ 1 ] ∗ s )i f max( shape ( k ) )>2 and s e l f . maxsize >=12:
matout [ 9 , 1 0 ] = 1 / ( k [ 2 ] + c [ 2 ] ∗ s )r e t u r n matout
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMSystem Class
Primary means for TMM analysis
Made up of a list of TMMElements, system boundaryconditions, and bode outputs
Methods for finding the system transfer matrix,eigenvalues, mode shapes, Bode responses,performing symbolic analysis, . . .
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMSystem Class
Primary means for TMM analysis
Made up of a list of TMMElements, system boundaryconditions, and bode outputs
Methods for finding the system transfer matrix,eigenvalues, mode shapes, Bode responses,performing symbolic analysis, . . .
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
TMMSystem Class
Primary means for TMM analysis
Made up of a list of TMMElements, system boundaryconditions, and bode outputs
Methods for finding the system transfer matrix,eigenvalues, mode shapes, Bode responses,performing symbolic analysis, . . .
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Numerical Experiment
sinh β ± sin β = sinh β
m=1.0e=1.0wh i le m+e>m:
m=m∗2.0
m = 9.00719925e + 15252 = 4.50359963e + 15
sinh β = 9.00719925e + 15β = 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Numerical Experiment
sinh β ± sin β = sinh β
m=1.0e=1.0wh i le m+e>m:
m=m∗2.0
m = 9.00719925e + 15252 = 4.50359963e + 15
sinh β = 9.00719925e + 15β = 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Numerical Experiment
sinh β ± sin β = sinh β
m=1.0e=1.0wh i le m+e>m:
m=m∗2.0
m = 9.00719925e + 15252 = 4.50359963e + 15
sinh β = 9.00719925e + 15β = 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Numerical Experiment
sinh β ± sin β = sinh β
m=1.0e=1.0wh i le m+e>m:
m=m∗2.0
m = 9.00719925e + 15252 = 4.50359963e + 15
sinh β = 9.00719925e + 15β = 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Numerical Experiment
sinh β ± sin β = sinh β
m=1.0e=1.0wh i le m+e>m:
m=m∗2.0
m = 9.00719925e + 15252 = 4.50359963e + 15
sinh β = 9.00719925e + 15β = 37.429948
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Finding Natural Frequencies
Mode Theoretical Symbolic Numeric Symbolic NumericNumber β Analysis Determinant Error (%) Error (%)
1 3.1416 3.1416 3.1416 1.414e-14 8.447e-52 6.2832 6.2832 6.2832 1.414e-14 -7.469e-53 9.4248 9.4248 9.4248 -1.885e-14 -2.164e-54 12.5664 12.5664 12.5664 -1.414e-14 -0.00023385 15.7080 15.7080 15.7080 1.131e-14 -0.00023386 18.8496 18.8496 18.8496 0 -0.00023387 21.9911 21.9911 21.9911 -3.231e-14 0.00022098 25.1327 25.1327 25.1327 0 0.0001649 28.2743 28.2743 28.2745 0 -0.0005875
10 31.4159 31.4159 31.4160 -3.393e-14 -0.000233811 34.5575 34.5575 34.5485 0 0.026112 37.6991 37.6991 3.1416 -1.885e-14 91.6713 40.8407 40.8407 3.1416 0 92.31
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Recommendations
Not practical to make general claims about accuracyUse symbolic analysis
Pinned-pinned beam: |subU| = −sin β sinhβ L2
β2
Inspect form of characteristic determinant
Plot characteristic determinant vs. β or ω
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Recommendations
Not practical to make general claims about accuracyUse symbolic analysis
Pinned-pinned beam: |subU| = −sin β sinhβ L2
β2
Inspect form of characteristic determinant
Plot characteristic determinant vs. β or ω
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Recommendations
Not practical to make general claims about accuracyUse symbolic analysis
Pinned-pinned beam: |subU| = −sin β sinhβ L2
β2
Inspect form of characteristic determinant
Plot characteristic determinant vs. β or ω
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Recommendations
Not practical to make general claims about accuracyUse symbolic analysis
Pinned-pinned beam: |subU| = −sin β sinhβ L2
β2
Inspect form of characteristic determinant
Plot characteristic determinant vs. β or ω
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Recommendations
Not practical to make general claims about accuracyUse symbolic analysis
Pinned-pinned beam: |subU| = −sin β sinhβ L2
β2
Inspect form of characteristic determinant
Plot characteristic determinant vs. β or ω
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
RREF
RREF (subU(s1)) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
RREF (subU(s1)) =
0.29614 0 0 1
0 1 −0.29614 00 0 O(ε) 00 0 0 O(ε)
(1.0)
EI2/EI1 ‖row1‖ ‖row2‖ ‖row3‖ ‖row4‖1.5 1.0429 1.0243 4.8878e-2 2.2204e-161.1 1.0429 1.0378 1.1818e-2 2.2204e-16
1.01 1.0429 1.0424 1.2384e-3 2.2204e-161.001 1.0429 1.0429 1.2444e-4 2.2204e-16
1.0 1.0429 1.0429 5.5511e-17 2.2204e-16Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
RREF
RREF (subU(s1)) =
0.29614 0 0 1
0 1 −0.22185 00 0 0.048878 00 0 0 O(ε)
(1.5)
RREF (subU(s1)) =
0.29614 0 0 1
0 1 −0.29614 00 0 O(ε) 00 0 0 O(ε)
(1.0)
EI2/EI1 ‖row1‖ ‖row2‖ ‖row3‖ ‖row4‖1.5 1.0429 1.0243 4.8878e-2 2.2204e-161.1 1.0429 1.0378 1.1818e-2 2.2204e-16
1.01 1.0429 1.0424 1.2384e-3 2.2204e-161.001 1.0429 1.0429 1.2444e-4 2.2204e-16
1.0 1.0429 1.0429 5.5511e-17 2.2204e-16Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Eigenvalues/Eigenvectors
Av = λv
Av = 0
EI2/EI1 λ1 λ2 λ3 λ4
1.5 3.0378 2.5627 1.2746e-1 -1.8182e-161.1 3.0378 2.9125 3.0360e-2 -3.3795e-16
1.01 3.0378 3.0243 3.1764e-3 -1.8182e-161.001 3.0378 3.0364 3.1912e-4 -1.8182e-16
1.0 3.0378 3.0378 -8.8037e-17 -1.8182e-16
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Eigenvalues/Eigenvectors
Av = λv
Av = 0
EI2/EI1 λ1 λ2 λ3 λ4
1.5 3.0378 2.5627 1.2746e-1 -1.8182e-161.1 3.0378 2.9125 3.0360e-2 -3.3795e-16
1.01 3.0378 3.0243 3.1764e-3 -1.8182e-161.001 3.0378 3.0364 3.1912e-4 -1.8182e-16
1.0 3.0378 3.0378 -8.8037e-17 -1.8182e-16
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Eigenvalues/Eigenvectors
Av = λv
Av = 0
EI2/EI1 λ1 λ2 λ3 λ4
1.5 3.0378 2.5627 1.2746e-1 -1.8182e-161.1 3.0378 2.9125 3.0360e-2 -3.3795e-16
1.01 3.0378 3.0243 3.1764e-3 -1.8182e-161.001 3.0378 3.0364 3.1912e-4 -1.8182e-16
1.0 3.0378 3.0378 -8.8037e-17 -1.8182e-16
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
SVD
EI2/EI1 sv1 sv2 sv3 sv4
1.5 5.5788 5.3280 6.1306e-2 3.1063e-171.1 5.5788 5.5096 1.6049e-2 1.4325e-16
1.01 5.5788 5.5712 1.7242e-3 3.1063e-171.001 5.5788 5.5781 1.7371e-4 3.1063e-17
1.0 5.5788 5.5788 3.1063e-17 3.1063e-17
Ortho-normalization is handled automatically
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
SVD
EI2/EI1 sv1 sv2 sv3 sv4
1.5 5.5788 5.3280 6.1306e-2 3.1063e-171.1 5.5788 5.5096 1.6049e-2 1.4325e-16
1.01 5.5788 5.5712 1.7242e-3 3.1063e-171.001 5.5788 5.5781 1.7371e-4 3.1063e-17
1.0 5.5788 5.5788 3.1063e-17 3.1063e-17
Ortho-normalization is handled automatically
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Repeated Roots Summary
Three valid approaches to finding the null space ofthe sub-matrix
RREFEigenvalues/eigenvectorsSVD
Each approach includes a means to check forrepeated or nearly repeated roots
SVD automatically handles ortho-normalization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Repeated Roots Summary
Three valid approaches to finding the null space ofthe sub-matrix
RREFEigenvalues/eigenvectorsSVD
Each approach includes a means to check forrepeated or nearly repeated roots
SVD automatically handles ortho-normalization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Repeated Roots Summary
Three valid approaches to finding the null space ofthe sub-matrix
RREFEigenvalues/eigenvectorsSVD
Each approach includes a means to check forrepeated or nearly repeated roots
SVD automatically handles ortho-normalization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Repeated Roots Summary
Three valid approaches to finding the null space ofthe sub-matrix
RREFEigenvalues/eigenvectorsSVD
Each approach includes a means to check forrepeated or nearly repeated roots
SVD automatically handles ortho-normalization
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Newton’s Method
For a system of equations:
∆x = −J−1f(x)
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Newton’s Method
For a system of equations:
∆x = −J−1f(x)
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method
Motion Control
Fluid Power
Intro Modeling Controls SW+Num. TMMElement TMMSystem Numerical Newton
Main Contents
1 Introduction2 Modeling3 Controls4 Software Design and Implementation5 Appendix
Ryan Krauss Control Design for Flexible Robots using the Transfer Matrix Method