Control Design for Flexible Robots using the Transfer Matrix Method · 2013-08-22 · Control...

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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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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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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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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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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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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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

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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

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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

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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

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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

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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

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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

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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

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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

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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 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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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

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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

]

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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

]

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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

]

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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

]

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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

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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