STRUCTURAL DYNAMIC MODELING, DYNAMIC STIFFNESS… · Structural Dynamic Modeling, Dynamic Stiffness, and Active Vibration Control of Parallel Kinematic Mechanisms with Flexible Linkages

STRUCTURAL DYNAMIC MODELING, DYNAMIC

STIFFNESS, AND ACTIVE VIBRATION CONTROL OF

PARALLEL KINEMATIC MECHANISMS WITH

FLEXIBLE LINKAGES

By: Masih Mahmoodi

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright 2014 by Masih Mahmoodi

ii

Structural Dynamic Modeling, Dynamic Stiffness, and Active Vibration

Control of Parallel Kinematic Mechanisms with Flexible Linkages

Masih Mahmoodi

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2014

ABSTRACT

This thesis is concerned with modeling of structural dynamics, dynamic stiffness, and

active control of unwanted vibrations in Parallel Kinematic Mechanisms (PKMs) as a

result of flexibility of the PKM linkages.

Using energy-based approaches, the structural dynamics of the PKMs with flexible links

is derived. Subsequently, a new set of admissible shape functions is proposed for the

flexible links that incorporate the dynamic effects of the adjacent structural components.

The resulting mode frequencies obtained from the proposed shape functions are

compared with the resonance frequencies of the entire PKM obtained via Finite Element

(FE) analysis for a set of moving platform/payload masses. Next, an FE-based

methodology is presented for the estimation of the configuration-dependent dynamic

stiffness of the redundant 6-dof PKMs utilized as 5-axis CNC machine tools at the Tool

Center Point (TCP). The proposed FE model is validated via experimental modal tests

conducted on two PKM-based meso-Milling Machine Tool (mMT) prototypes built in the

CIMLab.

For active vibration control of the PKM linkages, a set of PZT transducers are designed,

and bonded to the flexible linkage of the PKM to form a “smart link”. An

electromechanical model is developed that takes into account the effects of the added

mass and stiffness of the PZT transducers to those of the PKM links. The

iii

electromechanical model is subsequently utilized in a controllability analysis where it is

shown that the desired controllability of PKMs can be simply achieved by adjusting the

mass of the moving platform. Finally, a new vibration controller based on a modified

Integral Resonant Control (IRC) scheme is designed and synthesized with the “smart link”

model. Knowing that the structural dynamics of the PKM link undergoes configuration-

dependent variations within the workspace, the controller must be robust with respect to

the plant uncertainties. To this end, the modified IRC approach is shown via a

Quantitative Feedback Theory (QFT) methodology to have improved robustness against

plant variations while maintaining its vibration attenuation capability. Using LabVIEW

Real-Time module, the active vibration control system is experimentally implemented on

the smart link of the PKM to verify the proposed vibration control methodology.

iv

ACKNOWLEDGEMENTS

Firstly, I would like to express my sincere appreciation and gratitude to my supervisors,

Professor James K. Mills and Professor Beno Benhabib for their inspiring guidance, and

encouragement, throughout my thesis program. Through their support and advice, I have

been able to see this program through to its completion.

Also, I would like to thank my colleagues and friends in the Laboratory for Nonlinear

Systems Control and the Computer Integrated Manufacturing Laboratory (CIMLab) for

their assistance. Specially, I would like to thank Dr. Issam M. Bahadur, Mr. Adam Le,

and Mr. Ray Zhao for providing me with invaluable insights and comments in my

research work.

I would also like to acknowledge the Natural Science and Engineering Research

Council of Canada (NSERC)-Canadian Network for Research and Innovation in

Machining Technology (CANRIMT) for financial support of my research project.

Finally, I would like to express my deepest gratitude to my parents and my sister for

their endless support, and patience. Undoubtedly, the constant encouragement and moral

support from my family has helped me become the person I am today.

v

TABLE OF CONTENTS

ABSTRACT……………………………………………………………………………………...ii

ACKNOWLEDGEMENTS…………………………………………………………….……....iv

TABLE OF CONTENTS…………………………………………………………….……..…...v

LIST OF TABLES…………………………………………………………….…………....…...ix

LIST OF FIGURES………………………………………………………………….……..…....x

LIST OF NOMENCLATURES……………………………………………………………....xiv

1 Introduction .................................................................................................................... 1

1.1 Thesis Motivation ................................................................................................. 1

1.2 Literature Review ................................................................................................. 2

1.2.1 Structural Dynamics of PKMs with Flexible Links ...................................... 2

1.2.2 Dynamic Stiffness of Redundant PKM-Based Machine Tools .................... 5

1.2.3 Electromechanical Modeling and Controllability of Piezoelectrically

Actuated Links of PKMs ............................................................................................. 7

1.2.4 Active Vibration Control of PKMs with Flexible Links ............................ 10

1.3 Thesis Objectives ............................................................................................... 12

1.4 Thesis Contributions .......................................................................................... 13

1.5 Thesis Outline .................................................................................................... 15

2 Vibration Modeling of PKMs with Flexible Links: Admissible Shape Functions ...... 17

2.1 Dynamics of the PKM with Elastic Links .......................................................... 17

2.1.1 Modeling of the Elastic Linkages ............................................................... 18

2.1.2 Dynamics of PKM Actuators, Moving Platform, and Spindle/Tool .......... 25

vi

2.1.3 System Dynamic Modeling of the Overall PKM ........................................ 26

2.1.4 Admissible Shape Functions ....................................................................... 30

2.2 Numerical Simulations ....................................................................................... 33

2.2.1 Architecture of the PKM-Based mMT ....................................................... 34

2.2.2 The Accuracy of Admissible Shape Functions as a Function of Mass Ratio

of the Platform/Spindle to Those of the Links .......................................................... 37

2.2.3 Structural Vibration Response of the Entire PKM-Based mMT ................ 39

2.3 Summary ............................................................................................................ 45

3 Dynamic Stiffness of Redundant PKM-Based Machine Tools ................................... 47

3.1 Dynamic Stiffness Definition ............................................................................. 48

3.2 Dynamic Stiffness Estimation ............................................................................ 50

3.2.1 Architecture of the Prototype PKMs ........................................................... 50

3.2.2 FE-based Calculation of the Dynamic Stiffness ......................................... 51

3.2.3 Experimental Verification of the FE-Based Model .................................... 53

3.3 Results and Discussions ..................................................................................... 55

3.3.1 Prototype II and Prototype III ..................................................................... 55

3.3.2 Comparative Analysis of PKM Architectures ............................................ 62

3.3.3 Redundancy ................................................................................................. 64

3.4 Summary ............................................................................................................ 66

4 Electromechanical Modeling and Controllability of PZT Transducers for PKM Links .

................................................................................................................................... 67

4.1 Electromechanical Modeling .............................................................................. 68

4.1.1 Stepped Beam Model .................................................................................. 68

4.1.2 PZT Actuator Constitutive Equations ......................................................... 72

vii

4.1.3 PZT Sensor Constitutive Equations ............................................................ 73

4.1.4 System Modeling of the Combined Beam and PZT Transducers ............... 74

4.2 Controllability .................................................................................................... 75

4.3 Numerical Simulations and Experimental Validation ........................................ 77

4.3.1 Stepped Beam Model Verification .............................................................. 79

4.3.2 Controllability Analysis as a Function of the Tip Mass ............................. 83

4.4 Summary ............................................................................................................ 86

5 Design, Synthesis and Implementation of a Control System for Active Vibration

Suppression of PKMs with Flexible Links ....................................................................... 88

5.1 System Model ..................................................................................................... 88

5.2 Controller Design ............................................................................................... 90

5.2.1 Overview of the Standard Integral Resonant Control (IRC) ...................... 91

5.2.2 Resonance-Shifted IRC ............................................................................... 92

5.2.3 Proposed Modified IRC .............................................................................. 93

5.3 Utilization of the IRC-Based Control Schemes in Quantitative Feedback Theory

(QFT) ............................................................................................................................ 94

5.3.1 Robust Stability ........................................................................................... 95

5.3.2 Vibration Attenuation ................................................................................. 97

5.4 Results and Discussions ..................................................................................... 97

5.4.1 Proof-of-Concept ........................................................................................ 97

5.4.2 Application of the Proposed IRC-Scheme to Vibration Suppression of the

PKM with Flexible Links ........................................................................................ 105

5.5 Summary .......................................................................................................... 110

6 Conclusions and Future Work ................................................................................. 112

viii

6.1. Conclusions ...................................................................................................... 112

6.2. Future Work ..................................................................................................... 115

References ....................................................................................................................... 119

Appendix A ..................................................................................................................... 138

Appendix B ..................................................................................................................... 139

ix

LIST OF TABLES

Table 2.1‏. Dimensions of structural components .............................................................. 36

Table 2.2‏. Physical parameters of the PKM structure ...................................................... 36

Table 2.3‏. Summary of the recommended shape functions for the PKM links with respect

to the mass ratio- error defined by Equation ( 40 .................................................. (2.45‏

Table 2.4‏. Shape functions used for comparison in the simulation set 1. ......................... 41

Table 2.5‏. Shape functions used for comparison in the simulation set 2. ......................... 43

Table 3.1‏. Joint space configurations chosen for prototype II .......................................... 54

Table 3.2‏. Joint space configurations chosen for prototype III ......................................... 55

Table 3.3‏. Mode frequencies corresponding to the peal amplitude FRFs of prototype II 56

Table 4.1‏. Dimensions of the beam and PZT transducer. ................................................. 78

Table 4.2‏. Materials of the beam and PZT transducer. ..................................................... 79

Table 5.1‏. Variation ranges for the beam resonance frequencies and modal residues. .... 98

Table 5.2‏. Four configurations selected for vibration control experiments. ................... 107

x

LIST OF FIGURES

Figure 2.1‏. Schematic of a general PKM with kinematic notations ................................. 18

Figure 2.2‏. Mechanical structure of the example PKM-based mMT ............................... 33

Figure 2.3‏. Schematic of the PKM-based mMT ............................................................... 33

Figure 2.4‏. Elastic displacement component of the linkage for in-plane .......................... 35

Figure 2.5‏. Elastic displacement component of the linkage for out-of-plane ................... 35

Figure 2.6‏. Reaction forces at the spherical joints of the moving platform ...................... 35

Figure 2.7‏. Out-of-plane natural frequencies of the PKM links for the first mode .......... 38

Figure 2.8‏. Out-of-plane natural frequencies of the PKM links for the second mode ...... 38

Figure 2.9‏. In-plane natural frequencies of the PKM links for the first mode .................. 39

Figure 2.10‏. In-plane natural frequencies of the PKM links for the second mode ........... 39

Figure 2.11‏. Tooltip time response for “1st fixed-mass” and “1

st fixed-free” shape

functions for the first out-of-plane mode at ........................................... 42


st fixed-free” shape

functions for the first out-of-plane mode at ................................................ 43

Figure 2.13‏. Tooltip time response for “2nd

fixed-mass” and “1st fixed-pinned” shape

functions for the second out-of-plane mode at . ..................................... 43

Figure 2.14‏. Tooltip time response for “1st and 2nd pinned-mass” and “1st and 2nd

pinned-pinned” shape functions for the first and second in-plane modes at

. ....................................................................................................................... 44


pinned-pinned” shape functions for the first and second in-plane modes at .

................................................................................................................................... 44

xi


pinned-pinned” shape functions for the first and second in-plane modes at

. ....................................................................................................................... 45

Figure 3.1‏. Schematic of a generic PKM .......................................................................... 48

Figure 3.2‏. FRF amplitudes of a PKM for two example configurations .......................... 50

Figure 3.3‏. Prototype II ..................................................................................................... 52

Figure 3.4‏. Prototype III .................................................................................................... 52

Figure 3.5‏. Architecture of PKM prototype II .................................................................. 52

Figure 3.6‏. Architecture of PKM prototype III ................................................................. 52

Figure 3.7‏. Set-up of the experimental modal analysis ..................................................... 53

Figure 3.8‏. FRFxx amplitudes of prototype II for (a) configuration Home, (b)

configuration AA, (c) configuration BB, and (d) configuration CC ......................... 56

Figure 3.9‏. FRFxy amplitudes of prototype II for (a) configuration Home, (b)


Figure 3.10‏. FRFxz amplitudes of prototype II for (a) configuration Home, (b)


Figure 3.11‏.Mode shapes of prototype II at the dominant frequencies for (a) configuration

Home, (b) configuration AA, (c) configuration BB, and (d) configuration CC ....... 58

Figure 3.12‏. FRFxx amplitudes of prototype III for (a) configuration Home, (b)


Figure 3.13‏. FRFxx amplitudes of prototype III for 8 random configurations ................... 59

Figure 3.14‏. FRFzz amplitudes of prototype III for 8 random configurations ................... 60

Figure 3.15‏. Mode shapes of prototype III at configuation Home for (a) 1st mode at 85 Hz,

and (b) 2nd

mode at 157 Hz ....................................................................................... 60

xii

Figure 3.16‏. Variation of FRF peak amplitudes for 8 configurations using (a) original,

and (b) simplified FE model ..................................................................................... 61

Figure 3.17‏. Compared 6-dof PKMs (a) the Eclipse PKM, (b) the Alizade PKM, (c) the

Glozman PKM, and (d) the proposed PKM .............................................................. 62

Figure 3.18‏. FRF for all PKMs along the (a) xx, (b) yy, and, (c) zz directions ................. 63

Figure 3.19‏. Three redundant configurations for a given platform pose. ......................... 65

Figure 3.20‏. FRFxx of three redundant configurations for a given platform pose. ............ 65

Figure 4.1‏. Schematic of the beam and the PZT actuator pairs ........................................ 69

Figure 4.2‏. Euler-Bernoulli beam model for 2N+1 jumped discontinuities. .................... 69

Figure 4.3‏. PZT transducer configuration of the smart link ............................................. 78

Figure 4.4‏. FRFs of the PZT transducer pair obtained from experiments, uniform model,

and stepped beam mode for (a) 1st pair, (b) 2

nd pair, and (c) 3

rd pair ........................ 80

Figure 4.5‏. First three mode shapes of the beam with PZT transducer pairs: (a) 1st

mode,

(b) 2nd

mode, and (c) 3rd

mode .................................................................................. 82

Figure 4.6‏. First three modal strain distributions along the beam with PZT transducer

pairs: (a) 1st mode, (b) 2

nd mode, and (c) 3

rd mode ................................................... 83

Figure 4.7‏. Variation of the mode shapes as a function of the tip mass for (a) 1st mode, (b)

2nd

mode, and (c) 3rd

mode ........................................................................................ 85

Figure 4.8‏. Variation of the controllability indices of the individual PZT pairs based on

(a) state controllability (b) output controllability ...................................................... 86

Figure 5.1‏. (a) IRC scheme proposed in [81], and (b) its equivalent representation. ....... 91

Figure 5.2‏. Resonance-shifted IRC scheme in [84]. ......................................................... 92

Figure 5.3‏. Proposed modified IRC scheme ..................................................................... 93

Figure 5.4‏. Equivalent representation of the proposed modified IRC scheme ................. 94

Figure 5.5‏. Open-loop FRFs for variable tip mass. ........................................................... 98

xiii

Figure 5.6‏. closed-loop FRFs of the proof-of-concept for 1X for (a) strandard IRC, (b)

resonance-shifted IRC, and (c) proposed modified IRC schemes .......................... 100

Figure 5.7‏. FRF magnitudes of the proof-of-concept for open-loop and with (a) standard

IRC, (b) resonance-shifted IRC, and (c) proposed IRC. ......................................... 102

Figure 5.8‏. Plant template in the QFT design environment. ........................................... 103

Figure 5.9‏. QFT robust stability of the compared control schemes. ............................... 104

Figure 5.10‏. QFT disturbance attenuation of the compared control schemes. ............... 105

Figure 5.11‏. PZT transducers bonded on flexible link of a PKM. .................................. 106

Figure 5.12‏. Diagram of the active vibration control system. ........................................ 107

Figure 5.13‏. Open-loop FRF pf the PKM link for four example configurations. ........... 108

Figure 5.14‏. FRF of the flexible PKM link with and without controller for (a)

configuation AA, (b) configuation BB, (c) configuration CC, and (d) configuration

Home. ...................................................................................................................... 109

Figure 5.15‏. Time-response of the PKM link for configuration Home. ......................... 110

xiv

LIST OF NOMENCLATURES

Latin Symbols

system matrix of the smart link in state-space representation

coefficient of the in-plane shape function of the PKM link

location of the jth

PZT sensor pair along the smart link

coefficient of the out-of-plane shape function of the PKM link

rth

mode modal residue of the plant transfer function

maximum r

th mode modal residue of the plant transfer function

minimum r

th mode modal residue of the plant transfer function

input matrix of the smart link in state-space representation



b width of the beam and the PZT transducers

output matrix of the smart link in state-space representation

equivalent damping matrix of the PKM at the TCP



modal damping matrix of the PKM smart links

capacitance of the PZT sensor

( ) modal matrix of Coriolis and centrifugal effects of the PKM links

matrix of the Coriolis and centrifugal forces of the actuators,

xv

moving platform, and spindle/tool

( ) transfer function of the compensator

( ) equivalent transfer function of the compensator

constant feed-through term

disturbance input signal



transverse piezoelectric strain constant

vertical position of the prismatic actuator column of prototype II

vertical position of the linear prismatic joints for i

th chain of the

PKM

linear position of the radial actuators of prototype III

E Young’s modulus

{ } moving frame attached at the platform center point

( ) flexural rigidity of the ith

segment of the smart link

Young's modulus of the PZT transducers

dynamic‏applied force vector at the TCP

modal coupling force vector of the PKM

vector of active joint forces

vector of passive joint forces

vector of gravity and Coriolis/centrifugal forces of active joints

xvi

vector of gravity and Coriolis/centrifugal forces of passive joints

modal electromechanical coefficients matrix of the PZT actuator

vector of generalized modal external forces applied on the PKM

links

vector of generalized forces other than external actuator/platform,

spindle/tool forces

(.) unknown functions of the reaction forces at the distal end of the

PKM links for in-plane motion

(.) unknown functions of the reaction forces at the distal end of the

PKM links for out-of-plane motion

natural frequencies corresponding to a selected shape function

natural frequencies corresponding to the realistic mode shapes of

the PKM links

( ) transfer function of the smart link with variable tip mass

( ) modified transfer function of the smart link with variable tip mass

gravitational acceleration

vector of gravity forces of the actuators, moving platform, and

spindle/tool

vector of modal gravity forces of the PKM links

GM gain margin

( ) Heaviside function

( ) equivalent transfer function of the plant in the resonance-shifted

xvii

IRC scheme

( ) equivalent transfer function of the plant in the proposed IRC

scheme

kinematic constraints of the ith

closed-loop chains

and identity matrices

in-plane area moment of inertia of the PKM links

out-of-plane area moment of inertia of the PKM links

imaginary operator

Jacobian matrix of the entire PKM

matrix of the derivative of kinematic constrains with respect to

active joints

transformation matrix from the joint velocities of the i

th PKM

chain to Cartesian velocity of an arbitrary point

in-plane component of the mass moment of inertia of the effective

portion of the platform and spindle/tool

out-of-plane component of the mass moment of inertia of the

effective portion of the platform and spindle/tool

matrix of the derivative of kinematic constrains with respect to

passive joints

partitioned stiffness matrix of the PKM for active joint, and modal

coordinates

xviii

PZT actuator coefficient for the j

th PZT transducer pair

dynamic stiffness matrix of the PKM at the TCP

modal stiffness matrix of the PKM with smart links

modal stiffness matrix of the PKM links

generalized modal stiffness matrix of the entire PKM

PZT sensor coefficient for the j

th PZT transducer pair

static stiffness matrix of the PKM at the TCP

integral compensator gain

feed-forward/feedback compensator gain

L PKM link length

( ) loop gain for kth

control scheme

length of the tool

l number of the closed kinematic chains in the PKM

and position of the discontinuity of the i

th segment with respect to link

origin

structural mass matrix of the PKM at the TCP

total mass of the moving platform and spindle/tool

bending moment created by the jth

PZT actuator pair

inertia matrix of the PKM partitioned for active joint/modal

coordinates

xix

inertia matrix of the i

th sub-chain actuator

in-plane component of the bending moment at the distal end of the

ith

link

upper bound on the robust stability of the closed-loop system

out-of-plane component of the bending moment at the distal end

of the ith

link

modal mass matrix of the PKM with smart link

inertia matrix of the moving platform

modal inertia matrix of the PKM links

inertia matrix of the actuators, moving platform, and spindle/tool

generalized modal mass/inertia matrix of the entire PKM

inertia matrix of the spindle/tool

mass of each link

mass of each actuator

mass per unit length of the ith


mass of the moving platform

mass of the spindle/tool

n number of serial sub-chains in a generic PKM

number of truncated modes of the smart link

N number of jump discontinuities in the smart link

xx

{O} inertial frame

pole of the compensator

PM phase margin

reaction force vector acting on the i

th link at

reaction force vector acting on the i

th link at

peak amplitude of the FRF for configuration AA

peak amplitude of the FRF for configuration BB

state controllability index

output controllability index

p number of PZT transducer pairs

vector of the complete set of generalized coordinates of the PKM

structure

( ) joint-space position vector of the actuated joints of the i

th chain

vertical component of the i

th actuator position vector

( )

mth

modal coordinate

vector of modal coordinates for the i

th link

vector of modal coordinates for all n sub-chains

vector of the generalized coordinates of the PKM with smart link

( ) joint-space position vector of the passive joints of the i

th chain

xxi

vector of the rigid-body motion coordinates of the entire n sub-

chains

vector of all dependent rigid coordinates

vector of total generalized coordinates of the PKM

initial joint-space configuration vector

initial modal coordinates vector

( ) rth

modal coordinate of the smart link

ratio of the effective mass of the moving platform and spindle to

the mass of the link

absolute Cartesian position vector of an arbitrary point on PKM

link

Number of truncated modes

vertical component of the position vector

radius of the circular base platform

( ) reference input signal

radius of the moving platform

Laplace transform variable

( ) distribution function of the input voltage over the j

th PZT actuator

pair

transformation matrix from the passive joint velocities to active

joint velocities

xxii

( ) closed loop transfer function of unity-feedback system from

reference input to plant output for kth

control scheme

transformation matrix from the modal velocities to the elastic

displacements at point

the total kinetic energy of the PKM links

total kinetic energy of the actuators, the moving platform, and the

spindle/tool

time

beam thickness

PZT transducer thickness

the total kinetic energy of the PKM links

vector of input PZT actuator voltage

( ) input signal to the open-loop plant

input voltage to the j

th PZT actuator pair

in-plane component of the shear force for the ith

link

out-of-plane component of the shear force for the ith

link

input voltage to the j

th PZT sensor pair

output controllability Grammian matrix

state controllability Grammian matrix

( ) local vector of the two elastic lateral displacements of the ith

chain

xxiii

state vector in state-space representation

Cartesian position of the circular prismatic joints for ith

chain

Cartesian position of the spherical joint for i

th chain‏

Cartesian position of the vertical prismatic joints

( ) Cartesian task-space position and orientation (pose) of the

platform and spindle center of mass

local position of an arbitrary point along the link of the ith

chain

( ) plant output signal

vector of output PZT sensor voltage

characteristic matrix of the smart link

vertical distance of the mass center of the moving platform from

the base platform

Greek Symbols

upper bound on the vibration attenuation of the closed-loop system

eigenvalue solution of the in-plane natural frequencies

eigenvalue solution of the out-of-plane natural frequencies

variation of the total kinetic energy of the links

variation in the Cartesian coordinate of the position vector

xxiv

Cartesian x-component of vector at the boundaries

Cartesian y-component of vector at the boundaries

Cartesian z-component of vector at the boundaries

variation of the total potential energy of the links

virtual external forces done on the links

damping ratio of the rth

mode

damping ratio of the k

th mode

( ) rth

mode shape of the smart link

( )

mode shape of the ith


angular position of the actuator column of prototype II

angular position of the circular prismatic joints for ith

chain

angular position of the curvilinear prismatic joints of prototype III

vector of Lagrange multipliers

eigenvalues of the state controllability Grammian matrix

eigenvalues of the output controllability Grammian matrix

mass per unit length of the PKM links

mass density of the beam

xxv

mass density of the PZT transducer

external generalized input forces on the actuators, the platform and

spindle/tool system

angular position of the passive revolute joints for ith

chain‏

[ ] and [ ]

eigenvectors of the entire PKM at the TCP

( ) in-plane admissible shape functions of the PKM link

( ) out-of-plane admissible shape functions of the PKM link

frequency of the applied external forces at the TCP

natural frequency of the combine link and PZT transducers

frequency set of interest

shifted resonance frequencies of the equivalent plant in resonance-

shifted IRC scheme

natural frequencies of the PKM link for in-plane motion

natural frequencies of the PKM link for out-of-plane motion

kth

mode natural frequency

r

th mode pole of the plant

resonance frequency of the rth

mode of the smart link

maximum r

th mode natural frequency

minimum r

th mode natural frequency

xxvi

r

th mode zero of the plant

Acronyms

3-PPRS 3-“P” Prismatic, “R” Revolute, “S” Spherical

3-PRR 3-“P” Prismatic, “R” Revolute

AMM Assumed Mode Method

CMS Component Mode Synthesis

DAE Differential-Algebraic-Equation

DAQ data acquisition

dof degrees-of-freedom

EMA Experimental Modal Analysis

FE Finite Element

FEA Finite Element Analysis

FRF Frequency Response Function

IMSC Independent Modal Space Control

IRC Integral Resonant Control

LQG Linear Quadratic Gaussian

LQR Linear Quadratic Regulator

xxvii

mMT meso-Milling Machine Tool

ODE Ordinary Differential Equation

PKM

Parallel Kinematic Mechanism

PPF Positive Position Feedback

PZT Piezoelectric

QFT Quantitative Feedback Theory

SRF Strain Rate Feedback

TCP Tool Center Point

1

1 Chapter

Introduction

This chapter provides the motivation of this thesis, followed by a review of the state-of-

the-art of the literature on the topic. Subsequently, the thesis objectives, and contributions

are given, followed by a brief discussion of the thesis outline.

1.1 Thesis Motivation

Parallel Kinematic Mechanisms (PKMs) have been used in many industries that require

high accuracy, e.g. precision optics, nano-manipulation, medical surgery, and machining

applications [1]. The demands on high accuracy in such industries require the PKMs to

be built highly stiff, and massive. However, massive PKMs are not the best design

solution in terms of efficient power consumption and limited footprint for the PKMs.

Given the trend to be more efficient in terms of power consumption, modern PKMs

employ lightweight moving links, making a flexible structure that will exhibit unwanted

structural vibrations.

The structural vibration of PKMs decreases accuracy of operation, and can even damage

the PKM structural parts. The unwanted structural vibration in PKMs is either caused by

external forces applied on the PKM structure, or by the inertial forces due to

acceleration/deceleration motion of the PKM. In the former case, it is expected that

structural vibration would have the most undesirable effect on the PKM when the

frequency of the external forces applied on the PKM is close to one of the natural

frequencies of the PKM structure. For example, for PKM-based machine tools, structural

vibrations could have a significant undesirable effect when the cutting force frequency is

close to the natural frequencies of the machine tool structure [2], [3].

In order to avoid excessive vibration in general, the unwanted structural vibrations of

PKMs need to be accurately predicted, measured, and controlled. Specifically, the PKM

2

structural components with the largest compliance (e.g. flexible links) must be detected

and accurately modeled as the first step. Once an accurate model is developed, it must be

used for real-time control system synthesis to suppress the unwanted structural vibrations.

Moreover, an accurate structural vibration model can be used to estimate and compare

dynamic stiffness characteristics of the PKM-based machine tools at the Tool Center

Point (TCP) with an aim to enhance the structural design of PKM-based machine tools.

This thesis is focused on modeling of the structural dynamics and active vibration control

of PKMs with flexible links using piezoelectric (PZT) actuators and sensors. A

methodology is also presented for estimation and comparison of the dynamic stiffness of

various PKM-based machine tools at the TCP, which provides a basis for possible design

improvements of machine tools, as well as optimization of the TCP trajectory for

maximized stiffness. Section 1.2 provides the state-of-the-art of research on related topics

covered in this thesis.

1.2 Literature Review

1.2.1 Structural Dynamics of PKMs with Flexible Links

The development of accurate structural vibration models for PKMs with flexible linkages

has been the subject of a number of works. Among them, various modeling

methodologies such as lumped parameter modeling [4], [5], [6], Finite Element (FE)

method [7], [8], [9], [10], [11], Component Mode Synthesis (CMS) [12], and Kane’s

method [13] have been proposed. Specifically, the lumped parameter approach

approximates the dynamics of the distributed-parameter flexible links of PKMs with a

number of lumped masses along the link. Due to such approximations, the lumped

parameter method might lead to results with limited accuracy. The FE-based approaches

have higher accuracy compared to the lumped parameter modeling approach, however,

FE models usually involves a large number of degrees of freedom (i.e. a large number of

equations of motion) which leads to computationally expensive approach, and hence is

not suitable for real-time control.

3

Analytical dynamic modeling methods can provide relatively accurate and time-efficient

tools that can be further used to synthesize real-time controllers. In this regard, a

recursive Newton-Euler approach was developed for a flexible Stewart platform in [4].

Using the Newton-Euler approach, the internal joint forces and moments of the PKM can

be determined. However, it is often difficult to express explicit relationships in terms of

acceleration joint variables for forward dynamics, a property of the dynamic model which

is required for real-time model based control methods. To address this limitation, the use

of energy-based methods for flexible links of the PKM along with Assumed Mode

Method (AMM) provides an elegant and systematic approach for deriving the structural

dynamic matrices in explicit closed-form [14]. Specifically, Lagrange’s formulation with

AMM was used to model the structural dynamics of a 3-PRR PKM with flexible

intermediate links in [1], [15] and [16].

While the focus of this research includes the structural dynamic modeling of PKMs with

flexible links, the dynamics of rigid-link PKMs is worth mentioning here. Despite the

numerous works reported on the dynamic modeling of rigid link PKMs, the

generalization of the available methods on rigid-body modeling of PKMs to those with

flexible links is not trivial. The issue arises due to the presence of unknown boundary

conditions for the flexible links of the PKMs. There have also been numerous works on

theoretical formulation, numerical simulation and experimental implementation of

structural dynamics of serial mechanisms and especially single flexible links e.g. [17],

[18], [19], [20], [21]. The methodologies developed for structural dynamic modeling of

flexible serial mechanisms can be applied to PKM linkages. However, exact structural

dynamic modeling of the entire PKM requires the use of additional methodologies related

to the incorporation of closed-kinematic chain in the PKM structure [22]. The presence of

closed kinematic chains in PKMs generally results in the existence of passive joints in

conjunction with active (or actuated) joints and modal coordinates. In most PKM

configurations, there exists no explicit expressions describing passive joint variables in

terms of active joint variables and modal coordinates and most of the existing models on

PKM structural dynamics are established based on dependent coordinates and are non-

4

explicit formulations. Due to the presence of closed chains, the resulting structural

dynamics of PKMs form a set of Differential-Algebraic-Equations (DAEs) which

represent differential equations with respect to the generalized coordinates and algebraic

equations with respect to Lagrange multipliers. Authors in [22] proposed various

approaches for dynamic representation of closed-chain multibody systems (e.g. PKMs) in

terms of dependent or independent coordinates. From a control design viewpoint, it is

desirable to develop the structural dynamic model of PKMs in terms of active joints and

modal coordinates only.

Considering the challenges regarding the closed-loop kinematic chain of PKMs with

flexible links, a significant issue that has not been yet addressed in the literature is the

accuracy of the “admissible shape functions” utilized to approximate the exact “mode

shapes” of the PKM flexible links. Specifically, assuming the utilization of energy-based

methodologies for the dynamic model development, “admissible shape functions” are

typically used in the AMM as an approximation of the unknown exact “mode shapes” of

the PKM links. The exact mode shapes are typically unknown since the analytical

determination of the exact mode shapes and natural frequencies requires the solution of

the frequency equation, which is very complex in the case of multilink mechanisms such

as PKMs [23]. This complexity results from the existence of non-homogeneous natural

(or dynamic) boundary conditions that must be satisfied for the shear force/bending

moment of PKM links at the end joints. The shear force and bending moments at the end

joints of the PKM links are dependent on the mass/inertia properties of the adjacent

structural components. Hence, the frequency equation, mode shapes and natural

frequencies in general, are dependent on the relative mass/inertia properties of the

flexible intermediate links of the PKM and their adjacent structural components [24].

To avoid the complexities of solution of the exact frequency equation for flexible link

mechanisms, admissible shape functions based on “pinned”, “fixed”, or “free” boundary

conditions are typically used in the AMM in the literature to approximate the natural

frequencies and mode shapes. Furthermore, the accuracy of the admissible shape

5

functions has been investigated for single link and two link manipulators in [25], [26]

with the exact (or unconstrained mode) solution for a range of beam-to-hub and beam-to-

payload ratios.

Generally, the adjacent structural components connected to the PKM links include the

moving platform and the payload mounted on it. Considering a PKM with flexible links

as a simple mass-spring system from a practical point of view, it is expected that the

natural frequencies of the PKM decrease if the platform/payload mass is increased.

Therefore, such intuitive effects of the platform/payload mass on the natural frequencies

of the entire PKM must be seen in its structural dynamic model. However, the use of the

existing admissible shape functions based on “pinned”, “fixed”, or “free” boundary

conditions does not take into account the effects of the inertia of adjacent structural

components on the natural frequencies and mode shapes of the PKM links.

Thus, a crucial issue is to determine the accuracy of a set of admissible functions in

approximation of the realistic behavior of the flexible links in the context of a full PKM

structure considering the ratio of the mass of the links to the mass of the platform and

spindle [27]. Specifically, no work has been reported so far to examine the accuracy of

the use of admissible shape functions for flexible intermediate links of PKMs for a given

range of moving platform and payload mass to link mass ratios.

1.2.2 Dynamic Stiffness of Redundant PKM-Based Machine Tools

PKM-based machine tools generally provide higher stiffness characteristics than their

serial counterparts which make PKMs suitable for machining applications [28]. In PKM-

based machine tools, the TCP is expected to follow a desired path in the workspace with

a required accuracy. The machining accuracy is directly related to the dynamic stiffness

of the PKM-based machine tool structure at the TCP [29], [30].

It is known that the resulting change of joint-space configuration, due to the TCP motion,

causes the structural dynamic behavior of the PKMs to experience configuration-

dependent variations within the workspace [31]. Knowledge of the configuration-

dependent structural dynamic characteristics of the PKM can provide an insight into

6

trajectory planning of the TCP in the workspace in order to avoid regions/directions of

excessive structural vibration [31]. Moreover, the excessive vibration at the TCP at a

given configuration can lead to process instability of the machine tool. Motivated by

prediction of the dynamic stability of the milling processes for machine tools, the

Frequency Response Functions (FRFs) of the machine tool structure at the TCP has been

calculated in [32], [33] for multiple configurations of the machine.

Moreover, knowledge of the configuration-dependent structural dynamic characteristics

can also be used in the design of effective closed-loop controllers to damp out unwanted

structural vibrations. In this regard, the effect of the resulting change of linkage axial

forces of a 3-dof (degree-of-freedom) flexible PKM due to its configuration change on

the natural frequencies of the PKM has been investigated in [16]. The experimental FRFs

of a flexible 3-dof PKM have been compared for a set of PKM configurations [34] for

subsequent controller design. Furthermore, the analytical and experimental, and

numerical study of the configuration-dependent natural frequencies and FRFs of flexible

PKMs are given in [7], [29], [35], [36] and [37].

Although the configuration-dependent structural dynamic behavior of the PKMs has been

examined, little work has been reported to investigate the variation of the dynamic

stiffness for kinematically redundant PKM-based machine tools such as 6-dof PKMs

utilized for 5-axis CNC machining [38]. The issue with the kinematically redundant

PKM-based machine tools is that in addition to the configuration-dependent stiffness of

the PKM for various position and orientation (pose) of the moving platform, the stiffness

at the TCP varies for a given (i.e. fixed) pose of the platform. The reason is because in

kinematically redundant PKMs, there exist infinitely many joint-space configurations

associated with a given platform pose for the PKM. Therefore, the stiffness at the TCP

can vary depending on the joint-space configuration of the robot. The use of such

kinematically redundant PKM-based machine tools have been proposed in numerous

works to improve upon the stiffness, and to reduce kinematic singularity (i.e. increase

operational workspace) of the robot, with examples given in [39], [40], [41], [42], [43],

[44].

7

Therefore, to estimate the dynamic stiffness of PKM-based at the TCP, the model should

capture both the configuration-dependent behavior of the robot within the workspace and

the configuration-dependency related to a given platform pose due to the redundancy of

the PKM. To this end, the use of FE-based calculations along with experimental

measurements can provide accurate and reliable results. Specifically, the results could be

accurate when the CAD model to be used for the FE incorporates detailed geometrical

features of PKM structure, and the kinematic joints and bolted connections are

maintained as they represent the realistic PKM structure [45].

1.2.3 Electromechanical Modeling and Controllability of

Piezoelectrically Actuated Links of PKMs

Once the structural vibration model of the PKMs with flexible links is developed, the

model must be used in a vibration control methodology to suppress the unwanted

vibrations of the PKM. To this end, various passive vibration suppression methods have

been proposed to attenuate the unwanted vibrations by developing robot links made from

composite materials with inherently superior stiffness and damping characteristics [46],

[47], [48]. However, as passive vibration suppression methods rely on the structural

properties of the robot, they are sensitive to variations in the structural dynamics of the

robot, a property which is significant in PKMs. Consequently, the vibration suppression

method to be used for PKM links must have robust characteristics with minimized

sensitivity against variations in the in the structural dynamics of the PKM.

In this regard, the use of feedback control along with PZT materials for sensing and

actuation have received growing attention. Specifically, PZT materials have many

advantageous properties such as small volume, large bandwidth, and efficient conversion

between electrical and mechanical energies. Moreover, PZT transducers can be easily

bonded or embedded with various metallic and composite structures [49].

Various methodologies employing piezoelectric (PZT) transducers have been proposed

for vibration suppression of PKMs with flexible links [50], [51], [52], [53], [54]. The

PZT transducers have been bonded or embedded within the PKM links to form a “smart

8

link”. Moreover, depending on the PKM architecture, the PZT transducers have been

employed in various configurations such as PZT stack actuators/sensors for suppression

of axial vibrations of PKM linkages [55], [56], [57] and PZT patch actuators/sensors for

bending vibrations of PKM linkages [9], [58].

Having designed and built a smart link, an electromechanical model that relates the input

voltage to the PZT actuators to the voltage output from the PZT sensors must be

developed. Accurate development of such electromechanical model enables successful

synthesis and implementation of the control algorithm in the closed-loop system. To this

end, several works have been proposed to model the electromechanical behavior by

developing the constitutive equations of the smart links of the PKM. The methods used in

the reported works focused on suppression of bending (or transverse) vibration and fall

into two main categories:

1) Methods that neglect the effects of the added mass and stiffness of the PZT actuators

and sensors on the dynamics of the linkages. These models develop the dynamic

models of the links using “uniform beam model”, and the structural dynamic model

of the beam with the PZT actuators and sensors attached is identical to that of a

simple beam. The effects of the added PZT actuators and sensors are accounted for in

the “uniform beam model” through incorporation of an external bending moment,

caused by the PZT actuators, to the structural dynamic model of the beam.

Furthermore, the composite beam mode shapes obtained in this approach are identical

to those of a simple beam as if no PZT actuator and sensors were attached. Namely, it

is assumed that the addition of PZT actuator and sensors to a beam does not change

its mode shapes. This approach is easy to implement, yet, the results are subject to

debate especially when the thickness of the PZTs are not negligible compared to that

of the beam. The “uniform beam model” has been used in works such as: [59], [60],

[61].

2) Methods that take into account the effects of the added mass and stiffness of the PZT

actuators and sensors to those of the host structure (i.e. flexible link) [61], [62], [63].

9

These methods utilized the “stepped beam model”. The “stepped beam model” takes

into account the effects of the added mass and stiffness of the PZT transducers to

those of the beam by adopting a discontinuous beam model (Euler-Bernoulli in [61],

[62], [63] or Timoshenko in [64]) with jump discontinuities. Using this modeling

approach, the mode shapes obtained from the composite beam structure are no longer

similar to those of a simple beam. Hence, the structural dynamics and the subsequent

controller design of the flexible links is different compared to that of the uniform

beam model. In this thesis, the “stepped beam model” is used to model the combined

dynamics of the beam and PZT transducers.

In addition to the issues related to the electromechanical modeling of PZT transducers, it

is known that effective vibration control of the smart structures for a number of modes

can be achieved through proper placement of the PZT transducers [65], [66]. Generally,

the effectiveness of the vibration suppression from a PZT actuator is quantified by the

“controllability”. In this regard, several performance indices have been defined and

reported to represent the controllability of a smart cantilever beam with PZT actuators.

For instance, the controllability of a smart beam for vibration suppression is defined

based on singular values of controllability matrices in [67], [68], [69]. The norm of

the transfer function of the control system is utilized in [70], and the eigenvalues of the

controllability Grammian matrix [71] to represent the controllability. The controllability

considered in the above mentioned works was based on “state controllability” which, in

the case of flexible smart structures becomes the “modal controllability”. The “output

controllability” is used in [72] as a performance index to maximize the actual elastic

displacement that can be achieved by PZT actuators. These indices have been typically

utilized for subsequent optimization of the location, (and length and thickness) of a set of

PZT actuators to maximize controllability [73].

While several works have been reported on the optimization of the location (and

dimension) of the PZT actuators for effective vibration control of cantilever beams and

plates, little work has been done to examine the controllability of PZT-actuated links of

10

the PKMs. Specifically, it is known that the mode shapes of PKM links vary as a function

of the moving platform mass. Therefore, it might be possible to achieve the desired

controllability with a given PZT-actuated PKM link, by adjusting the mass of the

platform.

1.2.4 Active Vibration Control of PKMs with Flexible Links

Once the smart link is designed, a vibration control algorithm must be designed and

synthesized with flexible link of the PKM to suppress the unwanted vibrations. To

achieve this objective, various control schemes have been proposed in the literature.

Examples of the control schemes utilized for vibration suppression of smart structures

include the Strain Rate Feedback (SRF) [74], the Positive Position Feedback (PPF) [75],

and the Independent Modal Space Control (IMSC) [76]. Recently, a nonlinear/adaptive

controller with state observers was implemented on a PKM undergoing high

acceleration/decelerations [77]. The SRF and IMSC methods were subsequently used in

vibration suppression of PKM links in [48], and [78], respectively. The use of SRF while

increases the bandwidth, leads to a reduced robustness for the closed-loop system, and

the PPF method, and the IMSC was noted in [78] to lack robustness against variations in

the structural dynamics of the PKM links with the configuration. Such configuration-

dependent structural dynamic properties poses a significant challenge in the vibration

control of PKMs with flexible links [79]. Therefore, the variable structural dynamics of

the PKM links requires a control system design that is robust to variations in the

resonance frequencies and mode shapes of the PKM links. Also, while the control system

design is generally based in the a nominal model of the PKM link dynamics, it is

expected that in the typical use of the PKM, the vibration frequencies, and mode

amplitudes vary as a results of changes in the physical parameters of the PKM such as

added masses/payloads to the moving platform. Hence, an improvement in the robust

performance is very important. These variations in the structural dynamic characteristics

and physical parameters of the PKM are typically treated as plant uncertainties in the

11

design of the robust controller. The current status of research which addresses this issue

is briefly summarized here:

An -based robust gain scheduling controller was proposed for a segmented robot

workspace in [80]. The controller was implemented on a piezoelectric (PZT) actuated rod

of a PKM to suppress the axial vibrations of the robot links. To account for variation in

the modal frequencies of the PKM, an controller was proposed [56], [55] and was

implemented on a PZT stack transducer mounted on the robot links. In [51], [52], Linear

Quadratic Regulator (LQR)-based controllers were used in conjunction with Integral

Force Feedback and -based robust controllers to suppress the axial vibrations of the

PKM link. The above-mentioned model-based robust control techniques are shown to be

able to suppress the configuration-dependent resonance frequencies of the PKM links.

However, the implementation of such control techniques on flexible robotics is often

problematic due to the mathematical complexity of the dynamic models.

The Quantitative Feedback Theory (QFT) is another control methodology that directly

incorporates the plant uncertainty in the controller design. Generally, the QFT approach

accommodates the frequency-domain response of a set of possible plants that fall within

the predefined parameter ranges, called the plant templates. The control scheme is

designed such that all possible closed-loop systems satisfy the performance requirements.

The QFT approach has been applied for active vibration control of a five-bar PKM [81],

and flexible beams equipped with piezoelectric actuators and sensors [82], [83], [84],

[85]. Current design methodology of the controller scheme in the QFT is based on loop-

shaping, which is a heuristic procedure [86].

The Integral Resonant Control (IRC), originally introduced in [87], is a relatively simple

method to suppress vibration of flexible structures equipped with collocated transducers.

Specifically, the application of the IRC approach leads to a lower order controller when

compared with other control schemes (e.g. H2, H∞, and LQG). The IRC scheme was

proved to perform well in vibration suppression of flexible beams [87] and single-link

manipulators [88]. Furthermore, the robustness of the IRC scheme to variations of the

12

resonance frequencies of a flexible beam was also examined in [87] and [89] by

increasing the tip mass of the cantilever beam and obtaining the closed-loop response in

the presence of the added mass.

Motivated by increasing the bandwidth of the IRC scheme, and its ability to maintain its

robustness with respect to plant uncertainties, a resonance-shifting IRC scheme was

recently introduced in [90]. The underlying concept of the resonance-shifting IRC in [90]

was to add a unity-feedback loop around the plant with a constant gain compensator in

the feed-forward path. The resulting closed-loop system was then combined with a

standard IRC control scheme to impart damping (and tracking capability) to the system.

The unity-feedback loop with constant compensator gain shifted the resonance

frequencies of the plant forward to higher frequencies, leading to an increase in the

system bandwidth.

Given the above discussion, the current literature lacks a simple control scheme with

high-bandwidth that is robust to configuration-dependent structural dynamics of PKM

links. Improvement of the controller robustness while maintaining its vibration

attenuation characteristics is a significant step that must be taken to suppress the

unwanted vibration of the configuration-dependent PKM links.

1.3 Thesis Objectives

The overall objective of this thesis is to develop an active-vibration-control system for

suppression of configuration-dependent vibration modes of PKMs with flexible links

using PZT transducers. To achieve the overall objective, the four sub-objectives that must

be attained are presented herein:

1) To develop a structural dynamic model that can accurately predict the PKM natural

frequencies and link mode shapes.

2) To develop a methodology for estimation of the configuration-dependent dynamic

stiffness of the redundant PKM-based machine tools.

13

3) To develop an electromechanical model of the PKM links with PZT actuators and

sensors and to examine the controllability of the PKM links as a function of the platform

mass.

4) To design, synthesize, and implement a robust active-vibration-control system for

suppression of the configuration-dependent vibration of flexible links of the PKMs.

1.4 Thesis Contributions

The contributions achieved in this thesis include:

1) An analytical structural dynamic model of the PKM with flexible links has been

proposed that determines the most accurate “admissible shape function” (i.e. the closest

one to the realistic mode shape) to be used for the modeling of the flexible links of the

PKMs, depending on the relative mass of the moving platform to the mass of the links.

It is known that the mode shapes in mechanisms with flexible links vary as a function of

the mass/inertia of the adjacent structural components [24]. For example, the mode

shapes of a two flexible link mechanism with revolute joints vary as a function of the tip

mass and hub inertia [24]. As exact determination of the exact mode shapes is complex

in flexible link mechanisms, admissible shape functions have been typically used in the

literature to address the vibration behavior of the links. However, the use of such shape

functions does not incorporate the mass/inertia effects of the adjacent structural

components such as the platform mass. The presented shape functions for the flexible

links of the PKM in this thesis are able to approximate the realistic behavior of the link

mode shape by taking into account the effects of the adjacent structural components to

the flexible links of a PKM such as the platform/payload system. Using the presented

shape function for the flexible links, the structural dynamic model of the entire PKM is

developed.

2) An FE-based methodology for estimation of the configuration-dependent dynamic

stiffness of kinematically redundant PKMs within the workspace has been developed.

The model developed to estimate the dynamic stiffness of PKM-based at the TCP, is able

to capture both the configuration-dependent behavior of the robot within the workspace

14

and the configuration-dependency related to a given platform pose due to the redundancy

of the PKM. The model enables the designer to select the configuration with maximum

stiffness among infinitely many possible PKM configurations for a given tool pose. The

method has been applied on multiple random configurations of the PKM architectures

and the results have been verified via Experimental Modal Analysis (EMA). The

configuration-dependent dynamic stiffness results obtained from the methodology can be

potentially used in an emulator (e.g. Artificial Neural Network) for fast prediction of the

dynamic stiffness which could be used in an on-line optimization algorithm to select the

configuration of the redundant PKM with the highest dynamics stiffness.

In addition, there is always a need to improve the design of the PKM through presenting

new architectures that exhibit enhanced stiffness. The same methodology presented

herein to estimate the configuration-dependent dynamic stiffness of a given PKM

architecture has been used to analyze new PKM architectures and to compare them with

other design alternatives.

3) A methodology for electromechanical modeling of a set of bender piezoelectric (PZT)

transducers for vibration suppression PKM links is presented. The proposed model takes

into account the effects of the added mass and stiffness of the PZT transducers to those of

the PKM link. The developed electromechanical model is subsequently utilized in a

methodology to obtain the desired controllability for a proof-of-concept cantilever beam

by adjusting the tip mass where it can represent a portion of the platform/payload mass.

Given the mode shapes of the PKM links depend on the platform mass, the methodology

proposed for the controllability analysis is directly applicable to the PKM links.

Specifically, the methodology can be used in the design of the platform and its mass so as

to adjust the controllability of the PKM with flexible links to a desired value. In addition,

the results can be used for an estimation of the relative control input for each PZT

actuator pair.

4) A new modified IRC-based control scheme has been proposed in order to suppress the

structural vibration resulting from the flexible links of the PKM. Typically, the resonance

15

frequencies and response amplitudes of the structural dynamics of the PKM links

experience configuration-dependent variation within the workspace. Such configuration-

dependent behavior of the PKM links requires a vibration controller that is robust with

respect to such variations. To address this issue, a QFT-based approach has been utilized.

It is shown that the proposed modified IRC scheme exhibits improved robustness

characteristics compared to the existing IRC schemes, while it can maintain its vibration

attenuation capability. The proposed IRC is implemented on the flexible linkage of PKM

to verify the methodology. The simplicity and performance of the proposed control

system makes it a practical approach for vibration suppression of the links of the PKM,

accommodating substantial configuration-dependent dynamic behavior.

1.5 Thesis Outline

This thesis presents the analysis of structural dynamics, dynamic stiffness, and active

vibration control of PKM with flexible links. The details involve the development of the

structural dynamic equations and link shape functions, development of FE-based models

for dynamic stiffness estimation and design improvements, conducting EMA, designing

and bonding PZT transducers to the PKM links, development and verification of the

electromechanical models of the PKM link with PZT transducers, investigation of the

variations of controllability of a proof-of-concept cantilever beam as a function of the tip

mass, development of the active-vibration-control system, design and synthesis of the

active-vibration-control scheme, and implementation of the control scheme in the active-

vibration-control system. The outline of the remainder of this thesis is as follows:

Chapter 2 presents the proposed method for structural dynamic modeling of the PKM

with flexible links and the accuracy of the PKM link shape functions. Chapter 3 presents

an FE-based modeling methodology to estimate the dynamic stiffness of the redundant

PKM-based machine tools at the TCP. The FE-based results are verified by EMA for

multiple configurations of the PKM. Chapter 4 presents the development and verification

of the electromechanical models of the PKM link with PZT transducers followed by the

16

controllability analysis of the smart link and its variations as a function of the tip mass.

Chapter 5 presents the design, synthesis and implementation of a new robust control

scheme for active vibration suppression of the PKM links. Finally, Chapter 6 summarizes

the findings of the thesis and offers concluding remarks as well as recommendations for

future work.

17

2 Chapter

Vibration Modeling of PKMs with Flexible Links:

Admissible Shape Functions

This chapter investigates the accuracy of various admissible shape functions for structural

vibration modeling of flexible intermediate links of Parallel Kinematic Mechanisms

(PKMs) as a function of the ratio of the effective mass of the moving platform with a

payload to the mass of the intermediate link (defined as mass ratio). The results are

applicable to any PKM architecture with intermediate links connected through revolute

and/or spherical joints. The proposed methodology is applied to a 3-PPRS PKM-based

meso-Milling Machine Tool (mMT) as an example.

2.1 Dynamics of the PKM with Elastic Links

A general PKM consists of a fixed base platform and a moving platform, as shown in

Figure 2.1‏. A number of actuators are mounted on the base platform and connected to the

moving platform through intermediate links. A payload is generally mounted on the

moving platform. Depending on the application of the PKM, the payload can perform

various tasks. For instance, for PKM-based milling machine tools, the payload can be the

spindle/tool which is mounted on the moving platform. Throughout the rest of this

chapter, the spindle/tool is assumed to represent the payload, although the developed

methodology is identical for PKM payloads used in applications other than machining.

The intermediate links may exhibit unwanted vibrations, and hence yield a “flexible”

PKM. In the following, the extended Hamilton’s principle with spatial beams utilizing

the Euler-Bernoulli beam assumption is used to systematically generate the flexible links

dynamics equations and boundary conditions [91], [92].

18

Figure 2.1‏. Schematic of a general PKM with kinematic notations

2.1.1 Modeling of the Elastic Linkages

The extended Hamilton’s principle for the elastic linkages of PKMs is given by:

∫ ( )

( (2.1‏

where , , and denote the variations of the total kinetic energy, total

potential energy, and the total virtual external forces done on the elastic linkages,

respectively .

Kinetic Energy

To derive the kinetic energy of the elastic links, we first assume that they are detached

from the moving platform. The resulting mechanism is a set of n serial sub-chains plus

the moving platform and spindle/tool. The dynamics of the n serial sub-chains is first

obtained and is superimposed on the dynamics of the moving platform and spindle/tool.

Having the superimposed dynamics of the PKM structural components, and considering

19

the PKM kinematic constraints, the dynamics of the entire PKM structure can be

obtained.

Let us define ( ) and

( ) as the joint-space position vectors of the actuated joints,

and passive joints of the ith

sub-chain of a general PKM, respectively, as given in Figure

( ) Also, let us define .2.1‏ [ ] as the local vector of the two elastic

lateral displacements of the ith

flexible links at a point and time , where and

are the in-plane and out-of-plane components of the lateral elastic displacements

of the of the ith

link, respectively. The absolute Cartesian position of an arbitrary point

along the ith

elastic link of a general PKM at time is given by (

). The total

kinetic energy of the elastic links is, then, given by:

∑∫ ( )

( (2.2‏

where and L are the mass per unit length and the total length of the flexible links,

respectively. Using calculus of variations, the variation in kinetic energy of the links is

written as [93]:

∑∫ ( )

( (2.3‏

where is the variation in the Cartesian coordinate of the position vector . Using

forward kinematics relationships of each sub-chain, the Cartesian components of velocity

and acceleration of the ith

elastic link are related to joint space velocities by the following

kinematic transformations:

( (2.4‏

20

and,

( (2.5‏

where [

]

, and is the kinematic transformation matrix of the i

th

elastic sub-chain. Substituting Equation ( ) into ,(2.5‏ the variation in kinetic energy of ,(2.3‏

the links can be represented in terms of joint space and elastic variables.

Potential Energy

The total potential energy of the elastic links is given by:

∑(∫ ( ( )

)

∫ ( ( )

)

∫

)

( (2.6‏

where and are the area moments of inertia of the links with respect to axes normal

to in-plane and out-of-plane surfaces, E is the Young’s modulus of the linkage. Also,

is the vertical component of the position vector . The first two terms on the right hand

side of Equation ( represent the elastic potential energy while the last term on the (2.6‏

right hand side represents the gravitational potential energy. The variation in potential

energy of the links is given by:

21

∑{ (

) (

)]

[

( (

)) ]

]

∫

( (

) )

(

) (

)]

[

( (

)) ]

]

∫

( (

) ) ∫

}

( (2.7‏

Virtual Work of External Forces

The total virtual work done by external forces on the elastic links is given as:

∑(

( )

( ) ( )

( ) ( )

( )

( )

( ) ( )

( ) ( )

( ))

( (2.8‏

where [

] and

[

]

are the two reaction forces

acting on the two end joints of the ith

elastic link (i.e., and ), respectively,

and, ,

and

are the variations of the Cartesian components of vector at the

boundaries. Without loss of generality, we assume that the links are connected to revolute

joints at , and spherical joints at , respectively. Assume that is

measured in the same plane as the revolute joint angle is measured.

22

Boundary Conditions

Substituting the results of Equations ( ) and (2.3‏ ) along with Equation (2.7‏ into the (2.8‏

extended Hamilton’s principle (Equation ( yields a set of equations of motions that ,((2.1‏

represents the motion of active joints, , passive joints,

, and elastic vibration of the

links, of the ith

sub-chain. Also, from the extended Hamilton’s principle, the boundary

conditions for in-plane vibration of the links, , at (i.e. revolute joint) are

obtained as:

( ) ( (2.9‏

and,

( )

( )

( (2.10‏

and at , (i.e. spherical joint) as follows:

( )

( )

( (2.11‏

and,

( ) ( )

(

) ( (2.12‏

Similarly, the boundary conditions for out-of-plane vibration of the links, , at,

are obtained as:

( ) ( (2.13‏

and,

23

( )

( (2.14‏

and at , as follows:

( )

( )

( (2.15‏

and,

( )

( )

(

) ( (2.16‏

where and are the in-plane and out-of-plane components of the bending

moment, and and are the in-plane and out-of-plane components of the shear

force, respectively. (.) and (.) are functions of the reaction forces at spherical

joints of the ith

chain for in-plane and out-of-plane, respectively. Since the Cartesian

components of the reaction force vector, , in (.) and (.) vary as a function of the

mass of the moving platform and spindle/tool, the realistic boundary conditions and the

resulting mode shapes and natural frequencies of the PKM links are dependent on the

mass of the moving platform and spindle/tool. To complete the structural dynamic

modeling methodology, we assume that there exist admissible shape functions ( ) and

( ) that can approximate the realistic in-plane and out-of-plane mode shapes of the ith

PKM link, respectively. These admissible functions, although unknown at the moment,

can be used in the Assumed Mode Method (AMM) to express in-plane and out-of-plane

elastic displacements of the ith

link. Note that the accuracy of these various admissible

shape functions in the context of the full PKM structure will be investigated after the

procedure for structural dynamic modeling is complete. The AMM can be expressed by

the following:

24

( ) ∑ ( )( )

( )( ) ( (2.17‏

and,

( ) ∑ ( )( )

( )

( ) (2.18)

where ( )( ) is the m

th modal coordinate of the i

th link. Assuming a p-mode truncation

for the ith

link, the vector of modal coordinates for the ith

link is as follows:

[ ] (2.19)

Considering the vector of modal coordinates [

]

of the n sub-

chains of the PKM, in conjunction with the rigid-body motion coordinates of the entire n

sub-chains of the PKM, [

]

, the complete set of

generalized coordinates of the PKM structure is given by [ ] .

Substituting Equations (2.17) and (2.18) into the variational dynamic model (Equation

( and performing the simplifications and integrations over the length of the links ,((2.1‏

will result in the following general discretized dynamic model for the coupled rigid-body

motion and elastic vibration of the elastic links [91]:

( ) ( ) ( ) (2.20)

where ( ) is the modal inertia matrix, ( ) is the modal matrix representing

Coriolis and centrifugal effects, is the modal stiffness matrix, and ( ) is the

vector of modal gravity forces. is a function of the reaction forces, and

at the

distal ends of the links.

25

2.1.2 Dynamics of PKM Actuators, Moving Platform, and

Spindle/Tool

Let us define the vector ( ) to represent the Cartesian task-space position and

orientation (pose) of the platform and spindle center of mass with respect to an inertial

frame {O}. The total kinetic energy of the actuators, the moving platform, and the

spindle/tool are given as follows:

( ) ∑(

(

)

( ))

(2.21)

where , , and are the inertia matrices of the i

th sub-chain actuator, the

moving platform, and the spindle/tool, respectively. The total potential energy of the

actuators, the moving platform, and the spindle/tool is given as:

( ) ∑

( )

(2.22)

where is the mass of each actuator, is the vertical component of the i

th actuator

position vector, and are the masses of the moving platform and spindle/tool,

respectively, and is the vertical distance of the mass center of the moving platform

from the base platform [94], [95]. Given the expressions for kinetic and potential energies

of the actuators, moving platform and spindle/tool, the energy expressions can be

substituted into the Lagrange’s equations to derive the equations of motion for the above

mentioned components. The Lagrange’s equations for the rigid body motion generalized

coordinates of the PKM for the dynamics of actuators, moving platform and spindle/tool

are given as:

26

(

)

(2.23)

where the vector contains the external input forces on the actuators, the platform and

spindle/tool system, as well as the reaction forces at the joints. [ ] is the

vector consisting of all dependent rigid coordinates used in the formulations. The

dynamics of the actuators, and moving platform and spindle for all the sub-chains is then

expressed as:

( ) ( ) ( ) (2.24)

where ( ) is the inertia matrix, ( ) is the matrix representing Coriolis and

centrifugal effects, and ( ) is the vector of gravity forces. These dynamic matrices

and vectors represent the contribution of all moving components of the PKM excluding

the links. The expanded partitioned form of the above mentioned generic matrices/vector

is given in the Appendix A.

2.1.3 System Dynamic Modeling of the Overall PKM

To derive the dynamics of the entire PKM, the matrix expressions of the dynamic

equations for the flexible links (Equation (2.20)) is superimposed with the corresponding

matrix expressions of dynamics of actuators, moving platform/spindle (Equation (2.24)).

In superimposing the dynamic equations, the virtual works done by reaction forces on the

links and the moving platform are essentially the summation of the works done by equal

and opposite forces, and do not appear in the expression for generalized forces.

Depending on the linkage configuration PKMs, one can note a number of closed-loop

kinematic chains. From the geometry of the closed-loop chains, the kinematic constraint

equations associated with the PKM closed-loop chains are given as:

27

(2.25)

where l is the number of the closed kinematic chains. The superimposed dynamics of the

PKM with n elastic links is given as:

( ) ( ) ( ) (

)

(2.26)

where [ ] , [ ] , and [ ]

is the vector of

Lagrange multipliers. Equation (2.26) with the constraint Equation (2.25) form a set of

differential-algebraic-equations (DAE) that represent the dynamic and vibration of the

entire PKM. The resulting equations are DAEs of index-3 which represent differential

equations with respect to the generalized coordinates and algebraic equations with respect

to Lagrange multipliers. The DAE index is the number of differentiations needed to

convert a DAE system into an Ordinary Differential Equation (ODE). The higher the

differentiation index, the more difficult it is to solve the DAEs numerically [22]. To solve

the above DAEs, they can either be utilized in their original differential-algebraic form,

or the equations may be reduced to an unconstrained differential form [15]. Treating the

DAEs in their original form requires less algebraic manipulation than the second

approach. The resultant dynamics of the PKM involves many terms and thus is very

complex. The current available software packages can solve index-1 DAEs in their most

original form. However, such software packages have limited ability to solve index-3

DAEs and thus it is not numerically efficient to have the developed DAEs of the PKM

solved without transforming the original equations into appropriate formulations.

Therefore, the DAE model must be transformed into an appropriate formulation which is

efficient for numerical simulation. The independent coordinate formulation is used in this

thesis to reformulate the dynamic equations of motion previously established, namely

Equations (2.25) and (2.26).

28

In order to derive a closed-form dynamic model which is expressed in terms of active

joint coordinates only, Equation (2.26) can be partitioned with respect to the vector of

active rigid/modal coordinates [ ] and the vector of passive/task space

coordinates[ ] . The dynamic equation for the active coordinates is given by:

( ) [ ] (2.27)

and for the passive coordinates by:

( ) [ ] (2.28)

Details of Equations (2.27) and (2.28) are given in Appendix B. The vector

[ ]

represents the external actuator forces and the vector

represents all external forces other than actuator forces. Elimination of Lagrange

multipliers from Equations (2.27) and (2.28) results in the following dynamic equation:

( )

( ) (2.29)

An expression for the passive coordinates, in terms of active independent coordinates,

can now be obtained via the kinematic analysis of the PKM:

(2.30)

where is the transformation matrix relating the passive joint velocities to active joint

velocities. Time differentiation of the inverse kinematic relationships of the PKM yields

the inverse Jacobian, of the PKM to be defined as:

(2.31)

Evaluating the time derivative of Equations (2.30) and (2.31), the acceleration vector of

dependent coordinates can be expressed in terms of independent coordinates as:

29

(

)

[

] (

) (

) (2.32)

Substituting Equation (2.32) into Equation (2.29), the equation of motion with

independent coordinates, in closed form is given as:

(

) (

) (2.33)

where

( ) [

], (2.34)

and,

, (2.35)

and,

( ) (

)(

) (2.36)

Equation (2.33) represents the explicit closed-form structural dynamics of a general PKM

in terms of active joints and modal coordinates. Using the developed model, the TCP

deviation due to linkage vibration of generic PKMs can be determined.

30

To summarize, using the adopted approach in this work, the Lagrange multipliers and

acceleration terms of passive coordinates ( and ) are eliminated using kinematics

relationships of the PKM, i.e. Equations (2.30) and (2.31). Such elimination leads to the

reduced order Equation (2.33). Solution of the dynamics equations is carried out as

follows. Using forward kinematics relationships, the constraint equations are solved for

passive coordinates in position and velocity at each time step and fed back to the dynamic

model to generate the active coordinates for the next time step. Given that the forward

kinematics is solved at the position level with this approach, no time integration of

constraint equations is involved and, thus, common numerical issues such as numerical

drift are avoided in this approach [22].

2.1.4 Admissible Shape Functions

To avoid the complexities associated with solving the exact frequency equation for the

entire PKM with flexible links, classical admissible shape functions that merely satisfy

the geometrical boundary conditions (i.e. Equations ( ) and ,(2.13) ,(2.9‏ and not ((2.14‏

necessarily the dynamic boundary conditions (i.e. Equations ( ) ,(2.10‏ ) ,(2.11‏ ,(2.15) ,(2.12‏

and (2.16)), may be used. The classical admissible functions to be considered are

“pinned-free”, “pinned-pinned”, and “pinned-fixed” for in-plane and “fixed-free”, “fixed-

pinned” and “fixed-fixed” for out-of-plane.

The use of classical admissible shape functions as mentioned above leads to a frequency

equation that is independent of the platform and spindle/tool mass which might result in

inaccurate mode shapes and natural frequencies. Thus, to incorporate the platform and

spindle/tool mass dependency on the natural frequencies and mode shapes, while

avoiding the complexities of solving the exact frequency equations, we propose to

consider “pinned-mass” and “fixed-mass” shape functions for in-plane and out-of-plane

motions, respectively, and check their accuracy for various ratios of the moving platform

and spindle/tool mass to link mass. Also, we assume that the mass attached to each

flexible link, is equal to the total mass of the moving platform and spindle/tool divided by

31

the number of the PKM links, i.e. n, that is, we divide the platform and spindle/tool into n

equal mass segments. We assume that the shape functions for in-plane and out-of-plane

motions can be expressed as:

( ) ( ) ( ) ( ) ( )

and

(2.37)

( ) ( ) ( ) ( )

( )

(2.38)

respectively, where and are the eigenvalue solutions associated with the in-plane

and out-of-plane natural frequencies of the link, , and , as:

√

(2.39)

and,

√

(2.40)

where is the mass of the link. Assuming harmonic motion and applying the boundary

conditions on the platform end joint (i.e. where the platform and spindle/tool mass is

assumed to be attached to the link) for the “pinned-mass” shape function leads to the

following frequency equation from which natural frequencies and mode shapes are

calculated:

32

[

( ) ( ) ( ) ( )

( )

( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )

]

(2.41)

Similarly, for the “fixed-mass” shape function, we get:

[

( ) ( ) ( ) ( )

( )

( ) ( )

( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )]

(2.42)

33

where is the ratio of the effective mass of the moving platform and spindle to the mass

of the link, i.e. ⁄ , called “mass ratio”, where ( ) ⁄ , with

( + ) being the total mass of the moving platform and spindle/tool. and

are the in-plane and out-of-plane components of the mass moment of inertia of the

effective portion of the platform and spindle/tool. The solution of Equations (2.41) and

(2.42) can then be obtained numerically for different values of the mass ratio.

2.2 Numerical Simulations

Numerical simulations are performed to examine the accuracy of the proposed “pinned-

mass” and “fixed-mass” admissible shape functions along with the classical shape

functions for the flexible links of the PKM for a range of mass ratios. Once the most

accurate set of shape functions have been obtained for a given mass ratio, they are used in

the dynamic model of the PKM to predict the structural vibration response at the tooltip

as shown in Figure 2.1‏. Numerical simulations which model a 3-PPRS PKM-based meso-

Milling Machine Tool (mMT), developed in our laboratory as an example architecture,

are carried out. Figure 2.2‏ shows the mechanical structure of the mMT, and Figure 2.3‏

provides its schematic representation.

Figure 2.2‏. Mechanical structure of the example

PKM-based mMT

Figure 2.3‏. Schematic of the PKM-based mMT

34

2.2.1 Architecture of the PKM-Based mMT

As noted in Figure 2.3‏, the PKM-based mMT consists of a circular base platform of

radius on which three circular prismatic joints (i=1, 2, 3) are mounted at points

. Three vertical columns are mounted to the circular prismatic joints. The vertical

prismatic joints (i=1, 2, 3) are situated on these three columns, respectively, at points

. The moving platform is connected to the three columns through three flexible

linkages of length . The linkages are connected to the three columns through revolute

joints . These linkages are connected to the moving platform through spherical joints at

points . The prismatic joints and are actuated joints and the revolute joints as

well as the spherical joint at are passive joints. The moving platform is approximated

with a cylindrical disk having a radius of , and the length between the center of the

moving platform and the tooltip is denoted as . A stationary coordinate reference

frame { } is defined at the centre of the circular base platform of the system. A moving

reference frame { } is defined at the tooltip.

The in-plane displacement component of the ith

elastic linkage is defined as shown in

Figure 2.4‏ as the lateral displacement of the linkage in the plane formed by the linkage

and the vertical column attached to it, denoted by ( ) . The out-of-plane

component is normal to the in-plane displacement and is given by ( ) (Figure

shows the reaction forces at the spherical joints of the moving platform 2.6‏ Figure .(2.5‏

applied to one of the linkages of the mMT.

The non-homogeneous boundary conditions of the PKM ith

linkage for in-plane motion

are obtained as:

( )

(2.43)

and for out-of-plane motion, the non-homogeneous boundary conditions are given as:

35

( )

(2.44)

Figure 2.4‏. Elastic displacement component of the

linkage for in-plane

Figure 2.5‏. Elastic displacement component of the

linkage for out-of-plane

Figure 2.6‏. Reaction forces at the spherical joints of the moving platform

As noted, Equations (2.43) and (2.44) contain the reaction forces that are dependent on

the mass of the platform and spindle as well as the joint space configuration of the PKM.

The natural frequencies associated with each admissible function are obtained using the

36

dimensions of the structural components given in Table 2.1‏ with physical parameters of

the PKM are given in Table 2.2.

Table 2.1‏. Dimensions of structural components

Dimension Value (m)

Linkage inner diameter 0.016

Linkage outer diameter 0.012

Length of the linkage 0.230

Radius of base 0.15

Radius of platform 0.0225

Thickness of platform 0.0225

Tool length 0.015

Table 2.2‏. Physical parameters of the PKM structure

Physical parameter Value

Elastic Modulus 205 GPa

Density 7850 Kg/m3

Circular actuators mass

each 0.328 Kg

Vertical actuators/joint

housing mass each 0.545 Kg

Vertical columns mass

each 0.976 Kg

Platform and spindle

mass 0.158 Kg

37

2.2.2 The Accuracy of Admissible Shape Functions as a Function

of Mass Ratio of the Platform/Spindle to Those of the Links

The focus of the simulations presented here is to examine the accuracy of the proposed

admissible shape functions to approximate the PKM link mode shapes for a wide range

of platform and spindle mass to link mass ratios. Herein, the accuracy of a given shape

function at a mass ratio is defined as the percentage error between the resulting natural

frequencies corresponding to that shape function, to those of the realistic

mode shapes, , of the PKM, which is expressed as:

(2.45)

The smaller the error, the more accurate a shape function is to the realistic PKM mode

shape. For each mass ratio, the eigenvalue problem associated with in-plane and out-of-

plane motion is solved for each shape function, and the natural frequencies for the first

two vibration modes of the link along each direction are calculated. The natural

frequencies associated with each shape function are then compared with the modal

analysis results obtained from the Finite Element Analysis (FEA) software package,

ANSYS, with an aim to compare the accuracy of each admissible shape function for a

given mass ratio. Moreover, comparison of the results of the mode frequencies obtained

from the proposed shape functions with those of the classical shape functions can

demonstrate how much improvement is achieved via the use of the proposed shape

functions.

Figure 2.7‏ shows the values of the natural frequencies of the first out-of-plane mode

obtained from the first mode of fixed-mass and first mode of fixed-free shape functions

compared with FEA versus mass ratio. It is noted that the natural frequencies obtained

from fixed-mass shape function yields close results to those from the fixed-free shape

function when the mass ratio is very small (i.e. ). This result is expected as the

38

links will behave dynamically close to the “free” boundary condition at the distal joint

when the platform and spindle mass is small compared to that of the link.

For , both fixed-free and fixed-mass shape functions predict the realistic mode

shape with an error of 15.6% (Equation (2.45)) compared with the result obtained with

FEA. However, as the mass ratio increases from , the natural frequencies

associated with fixed-mass shape function tends to give more accurate results than those

of the fixed-free shape function. It is noted that the use of first mode fixed-pinned, and

fixed-fixed shape functions yield the natural frequencies of 1170.3 Hz, and 1698.8 Hz

which are substantially far from first out-of-plane mode frequencies obtained from FEA

and thus are not given in Figure 2.7‏. Thus, the fixed-mass shape function is found to be

best mode shape approximation for the first out-of-plane mode.

Figure 2.7‏. Out-of-plane natural frequencies of the

PKM links for the first mode

Figure 2.8‏. Out-of-plane natural frequencies of the

PKM links for the second mode

The second out-of-plane mode frequencies versus mass ratio is given in Figure 2.8‏. Here,

in addition to the second fixed-mass and second fixed-free shape functions, the first mode

fixed-pinned and first mode fixed-fixed shape functions are considered for analysis, since

it is expected that distal joint may act like a “pinned” or “fixed” connection for the

second mode for large mass ratios. It is noted that for mass ratios of , the

second fixed-mass shape function can better approximate the second out-of-plane mode

shape than other shape functions with an error of 15.05%. However, it is seen that as the

mass ratio increases , the first mode fixed-pinned shape function gives closer

approximates of the second out-of-plane mode than other shape functions leading to a

0

100

200

300

0.001 0.01 0.1 1 10 100

Fir

st o

ut-

of-

pla

ne

mo

de

(Hz)

mass ratio (r)

First fixed-mass

First fixed-free

FEA200

700

1200

1700

2200

2700

3200

0.001 0.01 0.1 1 10 100

Sec

on

d o

ut-

of-

pla

ne

mod

e

(Hz)

mass ratio (r)

Second fixed-mass

First fixed-pinned

Second fixed-free

First fixed-fixed

FEA

39

maximum percentage error of 5.69% for first fixed-pinned. Thus, the bottom end joint

acts similar to a “pinned” connection for the second out-of-plane mode for .

Similar analysis was conducted for the first two in-plane modes of the links. Figure 2.9‏

shows natural frequencies of the first in-plane mode. It is noted that the first mode

pinned-pinned shape function can better approximate the first in-plane modes than other

shape functions for the whole range of mass ratio.

Figure 2.9‏. In-plane natural frequencies of the

PKM links for the first mode

Figure 2.10‏. In-plane natural frequencies of the PKM

links for the second mode

The second in-plane mode frequencies are given in Figure 2.10. Similar to the case for

the first in-plane mode, it is noted that the second pinned-pinned shape function gives a

better approximation of the natural frequencies than other shape functions for the whole

range of mass ratio.

2.2.3 Structural Vibration Response of the Entire PKM-Based

mMT

Simulations of the structural vibration of the entire PKM-based mMT were performed

using the parameters given in Table 2.1‏ and Table 2.2. The purpose of the simulations

was to examine the effect of using various shape functions on the time response of the

tooltip for a given mass ratio. Assuming the moving platform to be a rigid body, the time

response of the tooltip is a combination of contributions from the displacements due to

the in-plane and out-of-plane modes at the distal end of the flexible links of the PKM. To

0

400

800

1200

1600

2000

2400

0.001 0.01 0.1 1 10 100

Fir

st i

n-p

lan

e m

od

e (H

z)

mass ratio (r)

First pinned-massFirst pinned-pinnedFirst pinned-freeFirst pinned-fixedFEA

1000

2000

3000

4000

5000

6000

7000

0.001 0.01 0.1 1 10 100

Sec

on

d in

-pla

ne

mo

de

(Hz)

mass ratio (r)

Second pinned-mass

Second pinned-pinned

Second pinned-free

Second pinned-fixed

FEA

40

examine these contributions, the simulations were carried out in two sets. In the first set

of simulations, the effects of using various out-of-plane shape functions on the tooltip

response was examined with the in-plane shape functions unchanged. In the second set of

simulations, the effects of in-plane shape functions were considered, assuming that the

out-of-plane shape functions were unchanged. Both sets of simulations were carried out

for several mass ratios to examine the effects of the platform and spindle mass on the

elastic response at the tooltip.

Table 2.3 summarizes the shape functions with the closest mode frequencies to the FEA

results, as a function of the link to platform mass ratio. The recommended set of shape

functions can predict the realistic structural vibration behavior of the PKM links within

15.2% error for the whole range of mass ratios.

Table 2.3‏. Summary of the recommended shape functions for the PKM links with respect to the mass ratio- error defined by Equation (2.45)

Type of motion Recommended shape function

and maximum percentage

error for ⁄

Recommended shape function

and maximum percentage error

for ⁄

First out-of-plane First fixed-mass 14.9% First fixed-mass 2.56%

Second out-of-

plane

Second fixed-mass 15.05% First fixed-pinned 5.21%

First in-plane First pinned-pinned 11.6% First pinned-pinned 8.97%

Second in-plane Second pinned-

pinned

14.1% Second pinned-

pinned

15.2%

The shape functions with the closest natural frequencies to the FEA results for a given

mass ratio were selected for comparison with the presented “fixed-mass” and “pinned-

mass” shape functions in each simulation set. Table 2.4 shows the shape functions used

for comparison of the first simulation set for each mass ratio.

41

As shown in Table 2.4‏, the first and second modes of “fixed-mass” shape functions are

used as a reference for comparison of out-of-plane modes throughout the first simulation

set.

Table 2.4‏. Shape functions used for comparison in the simulation set 1.

Mass ratio 1st out-of-plane 2

nd out-of-plane

Set 1(a) 1/300 1st fixed-free 2

nd fixed-mass

Reference for set 1(a) 1/300 1st fixed-mass 2

nd fixed-mass

Set 1(b) 2/3 1st fixed-free 2

nd fixed-mass

Reference for set 1(b) 2/3 1st fixed-mass 2

nd fixed-mass

Set 1(c) 150/3 1st fixed-mass 1

st fixed-pinned

Reference for set 1(c) 150/3 1st fixed-mass 2

nd fixed-mass

The MATLAB solver utilized was ode15s for stiff systems. The mechanism is initially

positioned at the following configuration:

[ ] and

. An impulse force

of [ ] was applied at the tooltip at to excite the vibration

modes of the linkages.

Figure 2.11‏ corresponds to simulation set 1(a) which shows the elastic response of the

tooltip for mass ratio of ⁄ . The two responses are noted to have approximately

the same frequency, as predicted by Figure 2.7‏ for ⁄ However, the presence of

the inertia force, due to the end-mass in the “fixed-mass” shape function, leads to a

greater distal end displacement of the links than seen with the “fixed-free” shape function.

This leads to tooltip response amplitude of the “fixed-mass” shape function which is

greater than that of the “fixed-fee” shape function. Thus, while the “fixed-free” shape

42

function, accurately predicts the out-of-plane natural frequency for low mass ratios,

simulation with this mode shape tends to under-predict the response amplitude.


st fixed-free” shape functions for the first out-

of-plane mode at ⁄

Figure 2.12‏ is related to simulation set 1(b) and shows the elastic response of the tooltip

for mass ratio of ⁄ . It is noted that the difference in response amplitudes and

frequencies is more significant as the mass ratio increases from of ⁄ (Figure

⁄ to (2.11‏ (Figure 2.12‏). Simulation set 1(c) compares the effects of two shape

functions as the 2nd

out-of-plane mode in the tooltip response with the tooltip response

shown in Figure 2.13‏. It is noted that unlike the previous cases, the use of the “1st fixed-

pinned” shape function for high mass ratios does not lead to a noticeable difference

compared with use of the “2nd

fixed-mass” shape function.

Note that the use of “fixed-mass” shape functions, which accounts for the dynamic

effects of the platform and spindle, the general trend from Figure 2.11‏ to Figure 2.13‏

demonstrates the expected trend of a decrease in natural frequency with a corresponding

increase in the response amplitude, as the mass ratio increases from ⁄ to

⁄ .

The shape functions used for comparison in the second simulation set are given in Table

are close to the 2.10‏ and Figure 2.9‏ Since the FEA frequencies, as shown in Figure .2.5‏

“pinned-pinned” shape functions for both in-plane modes, they are used as a reference for

comparison with “pinned-mass” shape functions as given in Table 2.5‏.

-2

-1

0

1

2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

To

olt

ip r

esp

on

se (

µm

)

Time (s)

1st fixed-mass for r=1/300

1st fixed-free for r=1/300

43


st fixed-free” shape functions for the first out-

of-plane mode at ⁄

Figure 2.13‏. Tooltip time response for “2nd

fixed-mass” and “1st fixed-pinned” shape functions for the

second out-of-plane mode at ⁄ .

Table 2.5‏. Shape functions used for comparison in the simulation set 2.

Mass

ratio

1st in-plane 2

nd in-plane

Set 1(a) 1/300 1st pinned-mass 2

nd pinned-mass

Reference for set 1(a) 1/300 1st pinned-pinned 2

nd pinned-pinned

Set 1(b) 2/3 1st pinned-mass 2

nd pinned-mass

Reference for set 1(b) 2/3 1st pinned-pinned 2

nd pinned-pinned

Set 1(c) 150/3 1st pinned-mass 2

nd pinned-mass

Reference for set 1(c) 150/3 1st pinned-pinned 2

nd pinned-pinned

-8

0

8

16

0 0.02 0.04 0.06 0.08 0.1

To

olt

ip r

esp

on

se (

µm

)

Time (s)

1st fixed-mass for r=2/3

1st fixed-free for r=2/3

0

100

200

300

400

0 0.02 0.04 0.06 0.08 0.1To

olt

ip r

esp

on

se (

µm

)

Time (s)

1st fixed-pinned for r=150/3

2nd fixed-mass for r=150/3

44

Figure 2.14‏, Figure 2.15‏, and Figure 2.16‏ show the time response at the tooltip for mass

ratios of ⁄ , ⁄ , and ⁄ , respectively. It is noted that the use of

“pinned-pinned” and “pinned-mass” shape functions leads to negligible difference in the

tooltip response amplitude. In contrast, the use of these shape functions led to significant

differences in the natural frequencies of the response specially for very high and very low

mass ratios (see Figure 2.9‏ and Figure 2.10‏). The reason for such small difference is the

negligible contribution of the in-plane modes due to the assumption of a “pinned” joint at

the distal end of the flexible links. Thus, although the use of “pinned-mass” and “pinned-

pinned” shape functions leads to the same small contribution to the overall tooltip

response, the “pinned-pinned” shape functions can more accurately predict the natural

frequencies due to the in-plane modes.

Figure 2.14‏. Tooltip time response for “1st and 2nd pinned-mass” and “1st and 2nd pinned-pinned” shape

functions for the first and second in-plane modes at ⁄ .


functions for the first and second in-plane modes at .

-2

0

2

0 0.02 0.04 0.06 0.08 0.1To

olt

ip r

esp

on

se

(µm

)

Time (s)

pinned-mass for r= 0.01/3

pinned-pinned for r=0.01/3

-10

0

10

20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

To

olt

ip r

esp

on

se

(µm

)

Time (s)

pinned-mass for r=2/3

pinned-pinned for r=2/3

45


functions for the first and second in-plane modes at .

2.3 Summary

In this chapter, the accuracy of admissible shape functions used to predict the structural

vibration modes of Parallel Kinematic Mechanisms (PKMs) with flexible intermediate

links was investigated as a function of the ratio of the effective mass of the platform and

spindle to the mass of the flexible links (i.e. mass ratio). The modes of each admissible

shape function were calculated and compared to the modal analysis results of the PKM

from Finite Element Analysis (FEA) with respect to the mass ratio. The shape functions

with closest natural frequencies to the FEA results were selected for comparison with the

proposed “fixed-mass” shape functions for out-of-plane modes, and “pinned-pinned”

shape functions for in-plane modes in the vibration modeling methodology developed in

this chapter to predict the tooltip response.

As a result of the use of “fixed-mass” shape functions, the expected dependency of the

natural frequencies and response amplitudes of the whole PKM structure to the mass ratio

is taken into account. Comparison of the tooltip time responses shows that the use of

“fixed-mass” and “pinned-pinned” shape functions can accurately predict the out-of-

plane and in-plane vibration modes of the PKM with flexible links over a large range of

mass ratios. Furthermore, the in-plane modes are seen to have negligible contribution to

the overall response of the tooltip. Given the mass ratio, the results of this analysis can be

used as a guide to the selection of the most accurate shape function to represent the

realistic behavior of the structural vibration of a generic PKM with revolute and/or

spherical joints. Unlike FEA-based modal analysis, the presented method provides a

0

200

400

0 0.02 0.04 0.06 0.08 0.1To

olt

ip r

esp

on

se

(µm

)

Time (s)

pinned-mass for r=150/3

pinned-pinned for r=150/3

46

time-efficient solution for accurate prediction of the structural vibration response of the

PKM. The approach to model boundary conditions for PKMs leads to a better

approximation to the realistic dynamic behavior compared with other boundary

conditions. The resultant dynamic model, with more accurate structural vibration

modeling, can then be used for control system synthesis to design controllers for both

rigid body motion and suppression of the unwanted flexible linkage structural vibrations.

47

3 Chapter

Dynamic Stiffness of Redundant PKM-Based Machine

Tools

This chapter provides a methodology for estimation of the dynamics stiffness of

redundant PKMs within the workspace. The dynamic stiffness is extremely important is

machine tool design as it is directly related to the operational accuracy of the machine.

The cutting forces resulting from the interaction of the tool and the workpiece are

typically transferred to the machine tool structure. If the cutting force frequency is close

to one of the resonance frequencies of the machine tool, excessive structural vibration

will occur leading to process instability (i.e. chatter), or even damage to the machine tool

[2]. Therefore, the dynamic stiffness must be accurately predicted.

The dynamic stiffness of PKMs is typically known to exhibit configuration-dependent

behaviour within the workspace. Furthermore, as 6-dof PKMs are redundant for 5-axis

CNC machining, a given pose of the moving platform corresponds to infinitely many

joint-space configurations. Therefore, the model must be able to capture the variations of

the configuration-dependent dynamic stiffness both within the workspace for different

moving platform poses, and for a given pose of the moving platform.

In general, the directional displacement of the TCP at one of its resonance frequency

modes is the resultant contribution from its structural components such as links, and

columns, and the contributions from the clearance/preload of the joints, bearings, and

actuators [96]. The methodology and results of this chapter provides the basis for a fast

and accurate tool for on-line estimation of the dynamic stiffness for any PKM

configuration which could be later used in an optimization algorithm to select the

configuration of the redundant PKM with the highest dynamic stiffness. In addition, the

presented model can also be used for comparative analysis of dynamic stiffness among

various PKM-based machine tool designs.

48

3.1 Dynamic Stiffness Definition

Figure 3.1‏ shows the schematics of a generic PKM. As illustrated, the PKM undergoes an

elastic displacement of ( ) at the TCP when it is subjected to a dynamic loading ( ) at

the same point for the given configuration.

Figure 3.1‏. Schematic of a generic PKM

Now, considering the PKM as a general spatial structure, its directional dynamic stiffness

at the TCP can be represented via the Cartesian Frequency Response Function (FRF)

matrix with respect to a Cartesian frame which is expressed as [97]:

( ) ( )

( ) [

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

] ( (3.1‏

where i is the imaginary operator. ( ), and ( ) are the frequency spectrums (i.e. the

fast Fourier transforms) of the displacement and force vectors, ( ) , and ( ) ,

respectively. Therefore, the element ( ) in Equation ( can be obtained by (3.1‏

dividing the frequency spectrum of the displacement amplitude of the TCP along axis u,

by the frequency spectrum of the applied force to the TCP along axis v. represents

the direct-axes FRF component when u and v-axes are the same and it indicates cross-

axes FRF terms when u and v are different axes. Assuming the first -resonance modes

49

encompass the frequency range of interest used in the analysis, the FRF matrix element

( ) can be represented by [98]:

( ) ∑[ ] [ ]

( (3.2‏

where [ ] and [ ] are the eigenvectors of the entire PKM structure at the TCP, along

u, and v-axes, respectively. represents the mode of the vibration, is the damping

ratio; is the natural frequency for mode k. To obtain the minimum directional

dynamic stiffness for a given PKM configuration, one needs to obtain the peak amplitude

from each element of the FRF matrix. As an alternative to Equations ( ) and (3.1‏ the ,(3.2‏

dynamic stiffness matrix, , of a PKM for a given configuration can be defined as [99]:

( ) ( )

( ) √( ) ( ) ( (3.3‏

where , , and are the structural mass, equivalent damping, and static stiffness

matrices of the PKM, respectively. In Equation ( denotes the frequency of the ,(3.3‏

external force applied at the TCP. Assuming the PKMs as lightly damped structures, it is

noted from Equation ( that when the frequency of the applied force is close to one of (3.3‏

the structural resonance frequencies of the PKM, the term ( ) on the right hand

side of Equation ( becomes approximately zero leading to the minimum values for (3.3‏

dynamic stiffness. Therefore, it would be reasonable to consider the minimum dynamic

stiffness at the TCP of the PKM as the salient feature of the PKM structural dynamic

behaviour. In this thesis, dynamic stiffness is obtained using the FE software package,

ANSYS.

As a result of changes in the PKM joint-space configuration, the FRF peak amplitudes

experience configuration-dependent variations. As an example, if the PKM shown in

Figure 3.1‏ moves from an arbitrary configuration AA to another configuration BB, the

peak amplitude FRF could change from to leading to a change in the minimum

50

dynamic stiffness (Figure 3.2‏).

Figure 3.2‏. FRF amplitudes of a PKM for two example configurations

3.2 Dynamic Stiffness Estimation

The proposed FE-based methodology to calculate the dynamic stiffness is applied and

experimentally verified on two prototype PKM-based meso Milling Machine Tools that

were built at the Computer Integrated Manufacturing Laboratory (CIMLab) at the

University of Toronto. These PKM prototypes are both of 3×PPRS topology, where “P”,

“R”, and “S” denote prismatic, revolute, and spherical joints, respectively.

3.2.1 Architecture of the Prototype PKMs

The two prototype PKMs, herein called prototype II and prototype III, are shown in

Figure 3.3‏, and Figure 3.4‏, respectively, with their architecture given in Figure 3.5, and

Figure 3.6‏. According to Figure 3.3‏, prototype II consists of a circular base platform on

which an actuator column and two vertical posts are mounted. The actuator column

consists of two actuators which can move in vertical and horizontal directions. The two

posts are bolted to the base platform; however, the radial position of the posts can be

adjusted in order to obtain a specific configuration. The angular positions of the actuator

column, and the two posts are measured counter-clockwise with respect to the center of

each of the chain’s corresponding rail and are denoted as , , and , respectively as

shown in Figure 3.5‏. The vertical positions of the two posts can be adjusted through

bolted connections. The vertical positions of the actuator column and the two posts are

FR

F m

agn

itu

de

(m/N

)

Frequency (Hz)

Configuration BB

Configuration AA

51

measured from the base platform to the corresponding revolute joints for each chain and

are denoted as , , and , respectively.

The architecture of prototype III consists of a (fixed) base on which three identical

kinematic chains are mounted (Figure 3.6‏). Each chain comprises two actuators: the first

(actuated) prismatic joint moves along a curvilinear rail, and its angular position is

denoted by , ; the second (actuated) prismatic joint, mounted on top of the

first one, moves linearly in the radial direction, and its linear position is denoted by ; a

(passive) revolute joint is mounted on top of the second prismatic joint, which connects a

fixed-length link to the moving platform via a spherical joint. Further details on the

dimensions of the prototypes can be found in [100].

Considering the 6 dof 3×PPRS PKM prototype III to be utilized for 5-axis machining, the

PKM shows kinematic redundancy. Specifically, for a given tool pose within the

workspace, there are infinite PKM configurations that lead to same platform roll angle i.e.

the rotation about the tool axis. This redundant dof , i.e. the platform roll angle can be

used for optimizing the dynamic stiffness.

3.2.2 FE-based Calculation of the Dynamic Stiffness

The FE model of the prototype PKMs at a given configuration was generated using the

CAD model of the corresponding mechanism in the software package, ANSYS. The

Cartesian FRFs of the PKM at the TCP are calculated via harmonic analysis using FE.

For the harmonic analysis, a 1 N sinusoidal force was applied to the TCP of the moving

platform for every PKM configuration along the x-axis. The 1 N harmonic force

represents periodic loads created during the meso-milling operations for which the

cutting force magnitude are expected to fall within the range of 100 mN-1N [101]. The

frequency for the harmonic force is varied from [0-1000] Hz. The displacement of the

TCP was calculated along the Cartesian coordinates for the frequency interval [0-1000]

Hz. This analysis was repeated with a force of the same magnitude/frequency range

applied along the y, and z-axes as well.

52

Figure 3.3‏. Prototype II

Figure 3.4‏. Prototype III

Figure 3.5‏. Architecture of PKM prototype II

Figure 3.6‏. Architecture of PKM prototype III

The “element type” used in the FE analysis was a 4-noded Tetrahedron. A convergence

test was done on the FE model to obtain the optimal mesh size. The optimal mesh size

were obtained as 0.8 mm for critical areas of the PKM structures (such as contact

interfaces), and 3.5 mm for non-critical areas. The contact interfaces that were

incorporated between the structural components of PKM were the rolling interfaces and

the bolted interfaces. The rolling interfaces included the joint bearings for the revolute

and spherical joints, the curvilinear guide bearings, and the prismatic actuator bearings.

The bolted interfaces included connections of the upper actuator stage to the revolute

joint housing, and the connections of the spherical joint housing to the links. Accurate

calculation of the dynamic stiffness required the rolling interfaces and bolted interfaces to

53

be modeled in the FE environment; which is non-trivial due to the dependence of joint

characteristics such as contact surface conditions, friction, and damping [98]. For the FE

model, the rolling interfaces were modeled using sliding contacts for the joint bearings

for the rolling interfaces. The bolted interfaces were modeled using frictional contact

with the friction coefficient set as 0.2.

3.2.3 Experimental Verification of the FE-Based Model

Verification of the FE model was performed via Experimental Modal Analysis (EMA).

The procedure for EMA is based on impact testing of the PKM structures. The set-up of

the EMA is shown in Figure 3.7‏.

Figure 3.7‏. Set-up of the experimental modal analysis

A Kistler 9724A2000 impulse force hammer is used to hit the moving platform in a given

direction for each configuration and a Kistler 8632C50 accelerometer is used to measure

the directional acceleration of the moving platform. The impulse force hammer and

accelerometer signals pass through a Kistler 5134 DC current supply. The time-domain

outputs are acquired at a rate of 10 KHz for 8 seconds using an NI-USB6211 data

acquisition (DAQ) device. The FRF of the time-domain signals is constructed using

LabVIEW user-generated code for a frequency range of [0-800] Hz. Each experiment is

54

repeated five times and averaged in order to establish the repeatability of the results and

to reduce the noise.

It is known that the development of an accurate damping model in mechanisms is

challenging, and the determination of damping is usually done through experiments.

Damping in mechanisms mainly results from contacting surfaces at bolted joints and

sliding joints. This type of damping constitutes more than ~90% of the total damping in

machine tools, and is referred to as interfacial slip damping [102]. Another type of

damping, referred to as material damping, results from the damping inherent to the

material the machine tool is made from. The material damping only accounts for ~10%

of the total damping in machine tools [102].

The damping ratios of the joints were incorporated by updating the FE model with the

modal damping obtained from experiments. To this end, a multimode partial fraction

curve-fitting algorithm was used for modal parameter estimation of the FRFs obtained

from FE model [103]. The Cartesian FRFs of the FE model were captured for 4

configurations for prototype II, and 8 random configurations of the prototype III. These

configurations are listed in Table 3.1‏ and Table 3.2.

Table 3.1‏. Joint space configurations chosen for prototype II

Configu-

ration (mm) (mm) (mm) (

o) (

o) (

o)

Home 65 65 65 0 0 0

AA 65 65 65 +30

30 0

BB 90 90 90 +30 30 0

CC 90 90 90 +30 30 +30

55

Table 3.2‏. Joint space configurations chosen for prototype III

Configu-

ration (mm) (mm) (mm) (

o) (

o) (

o)

Home 0 0 0 0 0 0

AA +20 0 +20 15 0 +15

BB +10 0 20 0 15 15

CC 10 0 20 +15 15 +15

DD 0 +20 0 15 0 15

EE 10 20 0 15 0 15

FF 20 20 20 0 15 15

GG +20 +20 0 +15 15 +15

3.3 Results and Discussions

3.3.1 Prototype II and Prototype III

Figure 3.8‏(a-d), Figure 3.9‏(a-d), and Figure 3.10‏(a-d) show the xx, xy, and xz-components

of FRFs of prototype II as an example for 4 of the random configurations, respectively. It

is noted that the FRFs obtained from the FE model exhibit reasonably similar behavior

with those of the experimental FRFs. Similar behavior is seen for other components of

the FRFs as well. Moreover, a strong dependence on the configuration is clearly seen in

the FRF peak amplitudes and corresponding frequencies.

56

Figure 3.8‏. FRFxx amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c)

configuration BB, and (d) configuration CC

Not surprisingly, the mode frequencies corresponding to the peak amplitude FRFs are the

same for a given configuration along various FRF components. These frequencies and the

corresponding mode shapes obtained from the FE model are listed in Table 3.3‏ and

Figure 3.11‏, respectively.

Table 3.3‏. Mode frequencies corresponding to the peal amplitude FRFs of prototype II

Configuration Mode frequency (Hz)

Home 104.9

AA 130.3

BB 100.1

CC 102.3

57

Figure 3.9‏. FRFxy amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c)


Figure 3.10‏. FRFxz amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c)


58

Figure 3.11‏.Mode shapes of prototype II at the dominant frequencies for (a) configuration Home, (b)

configuration AA, (c) configuration BB, and (d) configuration CC

From Figure 3.11‏(a-d), it is noted from that the bending vibration of the vertical posts and

that of the actuator column is responsible for the dominant modes of prototype II. The

results of such analysis assisted in the modification of the design of prototype II.

Specifically, it was noted that the elimination of the vertical column could result in

improved stiffness behavior [104]. An improved stiffness behavior was seen when the

vertical column was replaced with a horizontal one which lead to the design of prototype

III. Figure 3.12‏ shows the xx-components of the FRF magnitudes of prototype III for 4

configurations (out of 8 selected configurations) as an example.

(a) (b)

(c) (d)

59

Figure 3.12‏. FRFxx amplitudes of prototype III for (a) configuration Home, (b) configuration AA, (c)


The xx and zz components of FRF amplitudes of all eight configurations are given in

Figure 3.13‏ and Figure 3.14‏ for the FE model as an example. The two mode shapes of

prototype III for home configuration are also given in Figure 3.15‏.

Figure 3.13‏. FRFxx amplitudes of prototype III for 8 random configurations

60

Figure 3.14‏. FRFzz amplitudes of prototype III for 8 random configurations

Figure 3.15‏. Mode shapes of prototype III at configuration Home for (a) 1st mode at 85 Hz, and (b) 2

nd

mode at 157 Hz

The incorporation of the bolted interfaces in the developed FE model required the CAD

model of the PKM to include detailed geometrical features such as holes of small

diameters, leading to a computationally intensive calculations (~8h on Intel®

i7-2.80 GHz

with 12 GB RAM on 64 bit Windows 7). In order to reduce the computational time, a

simplified FE model was created with detailed CAD geometrical features of the bolted

interfaces being suppressed. The rest of the assumptions used to create the simplified FE

model were identical to those of the original model.

Due to the geometrical simplifications, the FE model was not able to predict the absolute

FRF amplitudes as the full-order model for each configuration, however, it was noted that

61

the simplified FE model was able to capture the relative dynamic stiffness behavior of

the original FE model. Since the ultimate objective of this analysis is to predict the

dynamic stiffness of the PKMs to optimize the configuration for maximized stiffness, it

would be sufficient to develop a model that can follow the same relative trend as for the

original FE model, even though the FE model is unable to predict the absolute stiffness

values.

Figure 3.16‏ shows a relative comparison of the FRF peak amplitudes of the simplified FE

model with those of the original model for the 8 random configurations (Table 3.2‏).

It is noted that the simplified FE model is able to capture the relative dynamic stiffness

behavior of the PKM. Therefore, the methodology utilized to develop the simplified FE

model can be used for comparative analysis and design purposes.

Figure 3.16‏. Variation of FRF peak amplitudes for 8 configurations using (a) original, and (b) simplified

FE model

62

3.3.2 Comparative Analysis of PKM Architectures

In addition to the optimization of the PKM configuration for maximized stiffness, the

developed methodology for obtaining the dynamic stiffness was used in comparative

analysis of various PKM architectures. Specifically, the proposed 3×PPRS PKM concept

(based on which Prototype III was built) was compared with similar three known 6-dof

PKM architectures which were capable of achieving a platform tilt angle of 90º. These

PKMs were the Eclipse PKM [38], the Alizade mechanism [105], and the Glozman

mechanism [106]. All of the compared mechanisms are redundant for 5-axis machining.

The CAD models of these PKMs are shown in Figure [99] 3.17‏.

Figure 3.17‏. Compared 6-dof PKMs (a) the Eclipse PKM, (b) the Alizade PKM, (c) the Glozman PKM,

and (d) the proposed PKM

63

Figure 3.18‏ shows the Cartesian xx, yy, and zz components of the FRFs for the compared

PKMs fore home configuration. It is noted that the proposed PKM has the highest

dynamic stiffness along the x and y axes, and the Eclipse and Alizade mechanisms have

higher dynamic stiffness along the z-axis. Also, it is noted that the dynamic stiffness of

each PKM is decreased along the axis, on which the first links act as cantilever beams.

For the Alizade mechanism, the chains are constructed from one prismatic kinematic

coupling that connects the base and the platform. Hence, it does not include a link that

acts as a cantilever beam, and it is stiffer along the z-axis.

Figure 3.18‏. FRF for all PKMs along the (a) xx, (b) yy, and, (c) zz directions

64

In addition to the comparative analysis of the above mentioned PKMs, the developed FE-

based methodology in this thesis was used for comparative analysis of a new redundant

Pentapod Parallel Kinematic Machine with further details given in [107], [108].

3.3.3 Redundancy

Considering 6-dof PKMs for 5-axis CNC machining, the roll angle of the platform (i.e.

the angle along the tool axis) can be regarded as redundant for machining. Therefore, for

a given (i.e. fixed) pose of the moving platform, there exists infinitely many distinct roll

angles, which correspond to infinitely many distinct joint-space configurations of the

PKM. Therefore, the roll angle of the platform can be potentially used for optimization of

the PKM configuration for a given tool pose.

In addition to the configuration-dependent behavior of the dynamic stiffness within the

workspace, it was noted that the model must be able capture the variation of the dynamic

stiffness of redundant PKMs for a given (i.e. fixed) pose of the moving platform. To this

end, three random distinct joint-space configurations were chosen for a given pose of the

moving platform for the proposed PKM architecture (Figure 3.17‏(d)). These three

configurations are given in Figure 3.19‏.

(a) (b)

65

Figure 3.19‏. Three redundant configurations for a given platform pose.

Figure 3.20‏ shows the FRFxx of the three redundant configurations at the TCP. It is noted

from that the peak amplitude and the resonance frequency of the FRFs undergoes

variations for these configurations, confirming that the model is able to capture the

kinematic redundancy of the PKM [109].

Figure 3.20‏. FRFxx of three redundant configurations for a given platform pose.

(c)

66

3.4 Summary

An FE-based methodology was proposed in this chapter to estimate the dynamic stiffness

of redundant PKMs at the TCP. The FE-model was developed via a harmonic analysis of

the PKM structure in ANSYS for a given PKM architecture. The FE-model was verified

through experimental modal analysis of two PKM-based meso-Milling Machine Tool

prototypes built in the CIMLab at the University of Toronto. It was shown that the

dynamic stiffness of the PKMs undergo strong configuration-dependent behaviour in

terms of amplitude and mode frequency both within the workspace and for a given

platform pose due to kinematic redundancy. The methodology utilized to develop the FE-

models can provide a basis for optimization of the redundant 6-dof PKM configuration,

to achieve the highest stiffness along the tool path for 5-axis machining. Also, the FE-

based modeling methodology was utilized in comparative dynamic stiffness analysis of

new PKM architectures for 5-axis machining.

67

4 Chapter

Electromechanical Modeling and Controllability of

PZT Transducers for PKM Links

This chapter provides the methodology for electromechanical modeling of a set of bender

piezoelectric (PZT) transducers to suppress the unwanted transverse vibrations of PKM

links. Development of an accurate electromechanical model of the PZT-actuated (i.e.

smart) PKM links enables successful synthesis and implementation of the vibration

control algorithm in the closed-loop system. To this end, the “stepped beam model” is

adopted in this thesis which takes into account the added mass and stiffness of the PZT

transducers to those of the PKM link. The resonance frequencies and mode shapes (and

spatial derivatives) of the smart PKM link obtained from the “stepped beam model” are

compared to the commonly used “uniform beam model” which neglects the mechanical

effects of the PZT transducers.

In addition to the methodology presented for electromechanical modeling of the smart

PKM link, the variations of the controllability of the PKM flexible links, from a set of

PZT actuator pairs, is investigated as a function of the platform mass. It is known that

effective vibration control of the smart structures for a number of modes can be achieved

through proper placement of the PZT transducers. To this end, various optimization

algorithms have been employed in the literature to achieve maximized controllability.

Herein, a simplified methodology is proposed to obtain the desired controllability for a

proof-of-concept cantilever beam for a set of PZT actuators by adjusting the tip mass.

Given the mode shapes of the PKM links in general are dependent on the platform mass,

the methodology proposed for the controllability analysis of the cantilever beam is

directly applicable to predict the controllability of the PKM links. Specifically, the

methodology can be used in the design of the platform and its mass so as to adjust the

68

controllability of the PKM with flexible links to a desired value. In addition, the results

of this chapter can be used to gain an estimation of the relative control input required for

each PZT actuator pair.

4.1 Electromechanical Modeling

4.1.1 Stepped Beam Model

Let us consider a uniform flexible beam with p identical PZT transducer pairs. For the

sake of modeling simplicity, we assume that the PZT transducers are perfectly bonded on

the top and bottom surfaces, as shown in Figure 4.1‏. Herein, we consider each PZT

transducer to comprise a PZT actuator and a PZT sensor, where the latter is positioned at

the center of the transducer through an electrode isolation process from the PZT actuator.

The PZT transducers enable sensing and actuation of the transverse vibration of the link.

The jth

PZT actuator generates a bending moment, , when a voltage, is applied

across the actuator electrodes. Similarly, the jth

PZT sensor generates a voltage, , when

it is subjected to a transverse mechanical displacement at point in Figure 4.1‏.

The thicknesses of the beam and each transducer are denoted as and , respectively.

The PZT transducers are bonded to the beam such that the direction of polarization for

each PZT actuator pair is the same, i.e., the combined beam and PZT actuators operate in

a bimorph configuration with parallel operation. The bimorph configuration refers to the

beam and PZT transducer structural arrangement where two identical PZT transducers

are mounted on the top and bottom of the host structure (e.g. the beam). For the same

motion, the parallel operation chosen here requires half the voltage required for the series

operation, where the polarization direction of the two PZT actuators are opposite to each

other [110].

69

Figure 4.1‏. Schematic of the beam and the PZT actuator pairs

The “stepped beam model” adopted here takes into account the effects of the added mass

and stiffness of the PZT transducer pairs to those of the beam by adopting a

discontinuous Euler-Bernoulli beam with N jump discontinuities as shown in Figure 4.2‏.

According to this figure, the beam is partitioned into segments ( ), where

the mass per length and the flexural rigidity of the ith

segment are denoted as and

( ) , respectively. The positions of the discontinuities of the ith

segment with respect to

the beam origin O to the are denoted as and and the width of the beam and the PZT

transducer is denoted as b. In order to obtain the relationship between the input voltage to

the PZT actuators and the output voltage from the PZT sensors, the transverse vibration

behavior of the combined beam and PZT transducers must be known first.

Figure 4.2‏. Euler-Bernoulli beam model for N jump discontinuities.

70

To this end, the governing equations of the transverse vibration of the combined beam

and PZT transducers, with arbitrary boundary conditions are given as follows [111]:

( ( )

( )

) ( )

( )

( (4.1‏

where ( ) is the variable flexural rigidity, ( ) is the variable mass per unit length of

the combined beam and PZT transducers, and ( ) is its transverse displacement.

Assuming the solution is separable in time and space and applying the harmonic time

solution into Equation ( the eigenvalue problem associated with the i ,(4.1‏th

beam segment

is given as:

( )

( )

( ) ( (4.2‏

; ,

where ( ) is the mode shape function of the ith

segment. The general solution for the

mode shapes for the ith

segment is given as:

( ) ( ) ( ) ( ) ( ) ( (4.3‏

where

( ) . and is the natural frequency of the combined beam and PZT

transducers. , , , and are mode shape coefficients that are determined by

applying the arbitrary boundary conditions at and along with the

continuity conditions on the ith

segment. For the first and last segments, the boundary

conditions are applied on one end of these segments and the continuity conditions are

applied at the other end. For all other segments, the continuity conditions are applied on

both ends of the segment. The continuity conditions are applied for the displacement,

slope, bending moment, and shear force at the points of discontinuity and are given by

[111]:

71

( ) ( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( (4.4‏

In order to obtain the mode shape coefficients for each segment, the characteristic matrix

of the system, ( ) is formed by applying the continuity conditions along with the

boundary conditions on each segment. The characteristic matrix of the system is a

matrix with being its only variable [111]. In order to determine a non-trivial

solution for the mode shape coefficients, the frequency equation is formed by setting the

determinant of ( ) equal to zero, as:

[ ( )] ( (4.5‏

The values of satisfying Equation ( constitute the natural frequencies of the (4.5‏

combined beam and PZT transducers. The mode shape coefficients associated with each

natural frequency are normalized so as to satisfy the following orthonormality condition

for the rth

mode shape:

∑∫ ( ( )

( ))

( (4.6‏

The final normalized mode shapes of the system for the rth

mode are given as:

72

( )( )

{

( )( )

( )

( )

( )

( )

( (4.7‏

The mode shapes obtained are further used in the development of an input-output

relationship between the PZT actuator and PZT sensor voltages as follows. Using these

normalized mode shapes, the response of the system can be given as:

( ) ∑ ( )( ) ( )( )

( (4.8‏

Before proceeding with the system dynamic model, the constitutive equations for bender

PZT actuators in bimorph configuration, for parallel operation are given in Section 4.1.2.

4.1.2 PZT Actuator Constitutive Equations

Consider the jth

PZT transducer pair that is perfectly bonded to the surfaces of a beam in

bimorph configuration, (Figure 4.1‏). The arbitrary jth

PZT transducer pair consists of two

identical PZT transducers with the PZT actuators that constitute the majority of the

transducer area. The constitutive relationship between the input voltage to each actuator

pair and the resulting transverse displacement of the compound beam and PZT pair,

neglecting the viscous and structural damping effects, is given as [62]:

( ) ( )

( ( )

( )

) ∑

( ) ( )

( (4.9‏

where ( ) is the input voltage to each actuator in the j

th pair and

( ) is the second

spatial derivative of the distribution function of the input voltage over the jth

PZT actuator

73

pair. For the configuration as given by Figure 4.1‏, the distribution function ( ) is

given as:

( ) [ ( ) ( )] ( (4.10‏

where ( ) is the Heaviside function. Equation ( shows that the voltage input to the (4.10‏

jth

PZT actuator has a uniform profile over the PZT actuator length and is zero elsewhere.

The coefficient is defined as follows [112]:

( ) ( (4.11‏

where is the Young’s modulus of the PZT actuator material, and is the transverse

piezoelectric strain constant.

4.1.3 PZT Sensor Constitutive Equations

Considering the jth

PZT sensor pair perfectly bonded to a beam, the voltage that is

generated across the sensor electrodes is approximated as:

( )

[ ( )

] ( (4.12‏

where

, is the position of the center point of the j

th actuator i.e. the location

of the sensor. It is assumed that the actuator constitutes the majority of the PZT

transducer. The coefficient is given as:

( )

( (4.13‏

where is the capacitance of the PZT sensor.

74

4.1.4 System Modeling of the Combined Beam and PZT

Transducers

Substituting Equation ( ) into (4.8‏ ,and utilizing the orthonormality of the mode shapes (4.9‏

the mass-normalized electromechanical equations of the cantilever beam with a tip mass

are expressed as:

( )( ) ( )( )

( )( )

∑ [ ( )( )

( )( )

]

( )

( (4.14‏

( )

∑

( )( )

( )( )

( ) ( (4.15‏

where, , and are the rth

mode damping ratio and resonance frequency. The

electromechanical equations in state-space form are expressed as:

, ( (4.16‏

where [

{ }

]

,

[

[

( )

( )

( )

( )

]

]

, [ ]

,

where ( )

[ ( )( )

( )( )

] . Also, [

( ) ( )

( )] is

the input voltage to the actuators and [ ( )( ) ( )( ) ( )( )] is the

vector of modal coordinates. Defining the sensor voltage as

75

[ ( )

( ) ( )]

, matrix is given as

[

( )

( )

( )

( )

]

where ( )

[ ( )( )

]. It is noted that matrix

contains terms that depend on the slope (i.e. first spatial derivative) of the mode shapes at

the distal ends of the PZT actuators. This matrix will be of particular importance in the

subsequent controllability analysis as will be discussed in Section ‎4.2.

Having all of the dynamic matrices between the PZT actuator and sensor voltages, the

transfer function matrix between the actuator input voltage vector, , and the sensor

output voltage vector, is obtained as:

( )

( )

( ) ( ) ( (4.17‏

4.2 Controllability

As noted, the control influence matrix , is a function of the slope of the mode shapes at

the two distal ends of the PZT actuator. The standard measure of controllability adopted

herein is based on the eigenvalues of the Grammian matrix [72], [113]. The eigenvalues

of the output controllability matrix represent the ability of a particular PZT actuator pair

to control the transverse vibration modes of the smart link within a frequency range of

interest. The state controllability Grammian matrix can be expressed as [72], [114]:

( ) ∫

( (4.18‏

Due to the presence of in Equation ( the state controllability is dependent on the ,(4.18‏

location of the PZT actuators. Specifically, matrix is a function of the spatial

derivatives of mode shapes as mentioned in Section 4.1.4 The eigenvalues of

are a measure of the control energy that is required to bring all the states (i.e. all modes)

of the system to a desired value. The higher the eigenvalues of , the less control

energy is required to bring all the states to desired values, namely, the system is more

76

controllable. The corresponding performance index defined for controllability is defined

as:

(∑

)

(

√∏

)

( (4.19‏

where is the eigenvalue of the state controllability Grammian matrix. To obtain

the “output controllability”, an output vector based on the actual elastic displacement of

the beam at a point of interest, ( ) is defined. Each element of the output vector

represents the contribution of a particular mode to the elastic displacement. The output

vector is defined as follows:

[ ( )( ) ( )( ) ( )( ) ( )( )]

{ ( )( ) ( )( ) ( )( ) ( )( )}

⏟

[ ( )( ) ( )( ) ( )( ) ( )( )]

( (4.20‏

The above equation can be regarded as a transformation from modal variables to physical

output variables. Matrix is also the transformation matrix. The output controllability

Grammian matrix at is then expressed as:

( ) ∫

( (4.21‏

Similar to the state controllability, the performance index can be expressed as:

77

(∑

)

(

√∏

)

( (4.22‏

where is the eigenvalue of the output controllability Grammian matrix.

To apply the controllability measures on the smart link, it is assumed that the beam with

p simultaneous input voltages from the p PZT actuators is equivalent to the superposition

of one PZT actuator attached to beam at a time. Mathematically, the superposition can be

expressed as:

[

( )

( )

( )

( )

( )

( )

( )

( )

( )

]

[

( )

( )

( )

]

⏟

[

( )

( )

( )

]

⏟

( (4.23‏

The output controllability of the beam with a set of p PZT actuators for the simultaneous

suppression of the first n modes, can be obtained by calculating the output controllability

of each of the individual PZT actuators, and superimposing the controllability results of

the individual PZT actuators

4.3 Numerical Simulations and Experimental Validation

The simulations and experiments are performed on a proof-of-concept “clamped-mass”

smart link which is made of Aluminum with a tip mass attached to its free end as shown

in Figure 4.3‏. The tip mass used in the modeling represents an equivalent mass of the

78

moving platform of the PKM (see Chapter 1). Three PZT transducer pairs are bonded on

the aluminum beam in a bimorph configuration. The dimensions of the aluminum beam

and each PZT transducer is given in Table 4.1‏. The electrode configuration for each PZT

transducer is designed as follows: each PZT transducer sheet has an

electrode isolated region such that a area at the middle of the transducer is

electrically isolated from the rest of the transducer. The area is used as the

sensor and the rest of the PZT is utilized as the actuator in the experiments as shown in

Figure 4.3‏. Without loss of generality, we assume that the center-point of the three PZT

transducer pairs are located at ⁄ ,

⁄ , and ⁄ .

Figure 4.3‏. PZT transducer configuration of the smart link

Table 4.1‏. Dimensions of the beam and PZT transducer.

Dimension (in millimeter) Value

Beam length ( )

PZT actuator length ( )

Beam and PZT transducer width ( )

79

Beam thickness ( )

PZT actuator thickness ( )

The PZT transducers are made of 5H4E material from Piezo Systems Inc. with the

properties given in Table 4.2‏. The tip mass is 0.0132 kg.

Table 4.2‏. Materials of the beam and PZT transducer.

Material property Value Unit

Beam Young’s modulus ( ) 70

PZT Young’s modulus ( ) 62

Beam density( ) 2700 ⁄

PZT transducer density( ) 7800 ⁄

PZT transducer strain constant ( ) ⁄

4.3.1 Stepped Beam Model Verification

Experiments were conducted to verify the electromechanical model of the PZT

transducer pairs with the beam. A chirp signal (i.e. a sinusoidal input voltage with the

frequency that varies from zero to 1000 Hz with a constant rate) is applied on the PZT

actuator and the output voltage of the corresponding sensor is captured. Figure 4.4 shows

the FRF of the experiments compared with those of the “Stepped Beam Model” and

“Uniform Beam Model” model for the 1st, 2

nd, and 3

rd PZT transducer pairs as an

example. It is noted from Figure 4.4 that the natural frequencies of the stepped beam

80

model are closer to the experimental values than those of the uniform model. Therefore,

the stepped beam model provides a more realistic electromechanical behavior than the

uniform model. The improvement on the use of the stepped beam model is observed from

Figure 4.4.

Figure 4.4‏. FRFs of the PZT transducer pair obtained from experiments, uniform model, and stepped beam

mode for (a) 1st pair, (b) 2

nd pair, and (c) 3

rd pair

(a)

(b)

(c)

81

The mechanical damping ratio of the stepped beam model is identified graphically by

matching the peaks of the experimental data [115]. Figure 4.5‏ and Figure 4.6‏ show the

first three normalized mode shapes and normalized modal strain distributions along the

beam with PZT transducer pairs versus normalized link length, respectively. The modal

strains are obtained by twice differentiating the mode shapes with respect to the beam

length. The jumps in the strain values for the stepped beam model in Figure 4.6‏ result

from enforcing the shear force and bending moment balance conditions at the boundaries

of the PZT pairs. It is noted that the use of the uniform beam model tends to overestimate

the strain distribution of the link for those portions where PZT transducers are bonded.

(a)

(b)

82

Figure 4.5‏. First three mode shapes of the beam with PZT transducer pairs: (a) 1st

mode, (b) 2nd

mode, and

(c) 3rd

mode

(c)

(a)

(b)

83

Figure 4.6‏. First three modal strain distributions along the beam with PZT transducer pairs: (a) 1st mode, (b)

2nd

mode, and (c) 3rd

mode

4.3.2 Controllability Analysis as a Function of the Tip Mass

Assuming simultaneous control of the first three modes, the “state controllability” and

“output controllability” of the proof-of-concept cantilever beam with three PZT actuator

pairs were calculated for each individual PZT actuator. The tip mass were varied from 0

to 10X (10 times its actual value) in the simulations and the “state controllability” and

“output controllability” at each PZT actuator location was calculated for each tip mass.

It should be noted that the objective, herein, is not to conduct optimization-based

methods to determine location, and dimensions of the PZT transducers for maximized

controllability. The proposed method is just an alternative to the commonly used

optimization methodologies. Herein, we state that it is possible to achieve the desired

controllability, to some degree, by adjusting the moving platform mass of the PKM. The

advantage of the proposed method is its relative simplicity compared to optimization-

based methods.

The proposed method is not directly comparable to the optimization-based methods in the

literature, as the variables are different, (location/dimension of the PZT transducers in the

optimization-based method, and the moving platform mass in the proposed method).

(c)

84

The underlying idea of the proposed methodology is that by changing the tip mass, the

resulting mode shape (and its slope) would undergo variations. Therefore, it is possible to

achieve the desired controllability by obtaining a specific mode shape (and slope), which

indeed, corresponds to a specific tip mass. Figure 4.7‏ shows the variation of the mode

shapes as a function of the tip mass for the first three resonance modes of the smart

cantilever beam. The general trend of decrease in mode shape amplitudes (and slopes) is

observed from the graphs.

(b)

(a)

85

Figure 4.7‏. Variation of the mode shapes as a function of the tip mass for (a) 1st mode, (b) 2

nd mode, and (c)

3rd

mode

Figure 4.8‏ shows the state and output controllability of the three PZT pairs. The general

trend shows a decrease of the controllability for the 1st and 2

nd PZT actuators as the tip

mass increases. For the 3rd

PZT actuator, there is a noticeable increase from 0X to 1X.

Furthermore, it is seen that both state and output controllability results show almost the

same trend of variations although the results of the two controllability indices are

completely different.

(a)

(c)

86

Figure 4.8‏. Variation of the controllability indices of the individual PZT pairs based on (a) state

controllability (b) output controllability

4.4 Summary

In this chapter, a methodology based on the “stepped beam model” was proposed for

electromechanical modeling of a set of bender piezoelectric (PZT) transducers to

suppress the unwanted transverse vibrations of PKM links. The “stepped beam model”

was adopted herein which takes into account the added mass and stiffness of the PZT

transducers to those of the PKM link. The resonance frequencies and mode shapes (and

spatial derivatives) of the smart PKM link obtained from the “stepped beam model” were

compared to the commonly used “uniform beam model” which neglects the mechanical

effects of the PZT transducers.

The developed electromechanical model of the smart PKM link was utilized in a

simplified methodology to obtain the desired controllability for a proof-of-concept

cantilever beam for a set pf PZT actuators by adjusting the tip mass. Given the mode

shapes of the PKM links depend on the platform mass, the methodology proposed for the

controllability analysis of the cantilever beam is applicable to the PKM links. Specifically,

(b)

87

the methodology can be used in the design of the platform and its mass so as to adjust the

controllability of the PKM with flexible links to a desired value. In addition, the results

of this chapter can be used to gain an estimation of the relative control input required for

each PZT actuator pair.

88

5 Chapter

Design, Synthesis and Implementation of a Control

System for Active Vibration Suppression of PKMs with

Flexible Links

In this chapter, a new modified Integral Resonant Control scheme is proposed for

vibration suppression of the flexible links of Parallel Kinematic Mechanisms (PKMs).

Typically, the resonance frequencies and response amplitudes of the structural dynamics

of the PKM links experience configuration-dependent variation within the workspace.

Such configuration-dependent behavior of the PKM links requires a vibration controller

that is robust with respect to these variations. To address this issue, a Quantitative

Feedback Theory (QFT) approach is utilized herein. In this chapter, we provide both

simulation and experimental evidence of the performance of this approach. First, we

present results utilizing a simple cantilever beam, with a variable tip mass to change the

structural mode frequencies and response amplitudes, (called plant templates). The

proposed IRC scheme is synthesized with the plant templates within the QFT

environment to compare its (i) robust stability and (ii) vibration attenuation with the

existing IRC schemes. It is shown that the proposed modified IRC scheme exhibits

improved robustness characteristics compared to the existing IRC schemes, while it can

maintain its vibration attenuation capability. The proposed IRC is subsequently

implemented on a flexible linkage mounted in a PKM at four different configurations to

verify the methodology. The simplicity and performance of the proposed control system

makes it a practical approach for vibration suppression of the links of the PKM,

accommodating substantial configuration-dependent dynamic behavior [116].

5.1 System Model

To apply the active vibration control to the PKM links, it is assumed that multiple PZT

bending transducers are mounted on the surface of the flexible links of a PKM. The

89

eletromechanical equations of the PKM flexible links relate the input voltage to the PZT

bender actuators to the output voltage from the PZT bender sensors. We utilize existing

dynamic models of this structure, with appropriate citations of the literature. The

truncated -mode modal equations of the combined PKM links with PZT transducer(s)

pairs in its general form can be expressed as [58]:

( (5.1‏

where , , and are the modal mass, modal damping, and modal stiffness

matrices of the PKM links, respectively. and are the and modal

coordinates and the PZT actuator voltages vectors, respectively, is the matrix

containing actuator electromechanical coefficients as well as the mode shape derivatives.

Finally, is the vector that reflects (i) the modal forces resulting from the

inertial forces due to the coupling effect among the various PKM links and (ii) the modal

forces resulting from the motor dynamics of the PKM. Further details and explanation of

the coupling terms is given in [58]. Matrices and in their general form contain

nonlinear terms that are dependent on the joint-space configuration of the PKM. The

resulting response under this configuration-dependent dynamics would be variations in

the structural dynamic characteristics.

In order to illustrate the performance of the proposed control scheme, a cantilever beam

with variable tip mass is considered as a proof-of-concept. Such a choice of the cantilever

beam avoids the complications arising from the coupling effects between the PKM links

and, the motor/joint dynamics (i.e. in Equation ( Furthermore, the variable tip .((5.1‏

masses of the cantilever beam can represent the variable structural dynamics of the PKM

link. Subsequently, the approach is implemented on the flexible link of the PKM.

The transfer function of the cantilever beam with a variable tip mass, following Equation

( :can be written as (4.17‏

90

( ) ∑

( (5.2‏

where is the modal residue of the transfer function, and can be

expressed as:

( )( )

[

( )( )

( )( )

] ( (5.3‏

For (nearly) collocated PZT actuators and sensors, we must have .

Herein, to account for the variations of the structural dynamics of the PKM link, the tip

mass is treated as a variable. As a result of the changes in the tip mass, the natural

frequencies and modal residues of the transfer function vary within the range of

[

] and [

], respectively. As we shall see in Section 5.3, such

variations in the structural dynamic characteristics are treated as system uncertainties to

be accommodated by the controller design.

5.2 Controller Design

The proposed control scheme is a new modification of the Integral Resonant Control

(IRC) scheme, that was originally introduced in [87] and was later modified in [90]. The

proposed control scheme is implemented on a proof-of-concept cantilever beam with

variable tip mass. To account for the parameter uncertainty in the controller design, a set

of plants (i.e. plant template) are generated within the QFT design environment. The

modified IRC scheme is designed based on a nominal plant within the template and

synthesized with it to compare its (i) robust stability and (ii) vibration attenuation

characteristics with the existing IRC methods. In the following, we briefly review the

existing related IRC literature, to provide the basis for the modified IRC approach,

presented in this thesis.

91

5.2.1 Overview of the Standard Integral Resonant Control (IRC)

The design procedure for the standard IRC was originally provided in [87] and is briefly

reviewed in this Section. Figure 5.1‏(a) shows the block diagram of the IRC scheme

introduced in [87], where ( ) is the compensator transfer function, ( ) is the plant

transfer function, and ( ), , and ( ) are the reference input, disturbance input, and

plant output signals for the closed-loop system, respectively. It is known that the phase

response of flexible collocated systems lies between and and it exhibits a pole-

zero alternating pattern in the frequency domain [87], [117]. It was shown in [87] that by

adding a constant term, (called feed-through) to ( ), a zero less than the first natural

frequency of the plant is added. Furthermore, the modified plant, ( ), shows zero-pole

alternating pattern of [89]:

( (5.4‏

where ( ) is the r

th zero and

( ) is the rth

pole, if

( ) . “A negative integral controller in negative feedback, which adds a

constant phase lead of , will yield a loop transfer function whose phase response lies

between and ; that is, the closed-loop system has a highly desirable phase

margin of ,” [87].

Figure 5.1‏. (a) IRC scheme proposed in [87], and (b) its equivalent representation.

(a) (b)

92

To avoid high controller voltages at low frequencies, and to facilitate the stability

analysis, the above IRC control scheme was rearranged in an equivalent form as shown in

Figure 5.1‏(b), where ( ) is the input to the plant [118]. In the equivalent form, ( ) is

obtained in its general form as [119]:

( )

( )

( ) ( (5.5‏

Therefore, if an integral compensator ( )

is used, the equivalent compensator can

be rearranged as ( )

5.2.2 Resonance-Shifted IRC

The resonance-shifted IRC was introduced in [90] to order to improve the bandwidth of

the standard IRC scheme. To assist the reader with the IRC scheme presented in this

chapter, the resonance-shifted IRC is briefly reviewed. The resonance-shifting IRC closes

a unity feedback loop with a constant gain compensator, ( ), as given in Figure

.5.2‏

Figure 5.2‏. Resonance-shifted IRC scheme in [90].

93

Application of the unity feedback with the constant compensator gain on the plant

transfer function given by Equation ( results in a stable equivalent plant transfer (5.2‏

function from ( ) to ( ), which is expressed as:

( ) ( )

( ) ∑

( (5.6‏

We assume that the modes are well-spaced, and therefore the mode-coupling is neglected

here. It is noted from Equation ( that the natural frequencies of the equivalent plant (5.6‏

transfer function are increased to √

, increasing the system bandwidth.

5.2.3 Proposed Modified IRC

The modified IRC scheme presented herein is obtained by removing the compensator

gain from the feed-forward path of the resonance-shifted IRC and placing it in the

feedback loop (Figure 5.3‏). The equivalent representation of the block diagram of the

proposed control system is given in Figure 5.4‏.

Figure 5.3‏. Proposed modified IRC scheme

94

Figure 5.4‏. Equivalent representation of the proposed modified IRC scheme

Similar to the resonance-shifted IRC, the equivalent transfer function of the plant for the

proposed resonance-shifting IRC, from ( ) to ( ) is expressed as:

( ) ( )

( ) ∑

( (5.7‏

Comparing Equations ( ) and (5.6‏ it is noted that for ,(5.7‏ , the equivalent transfer

function of the proposed IRC scheme, ( ), is smaller than those of the resonance-

shifted IRC, ( ), and the standard IRC, ( ). ( ( ) ( ), and ( ) ( )). As

we shall see in the Section 5.3.1, the reduced equivalent transfer function of the proposed

IRC scheme leads to improved robust stability compared to the other two control

schemes.

5.3 Utilization of the IRC-Based Control Schemes in

Quantitative Feedback Theory (QFT)

The QFT was originally introduced in [120] as a robust control methodology that aims to

attain the desired performances for the closed-loop system under the existence of plant

uncertainty and plant disturbances. Herein, an overview of the existing QFT method in

95

the literature is briefly provided to facilitate subsequent analyses with further details

given in [86].

Utilizing a frequency-domain approach, the QFT method takes into account the

parameter uncertainty by systematically generating the set of all possible plants (called

the plant template) that can be achieved using the parameter ranges given in the problem

[86]. The plant template contains a number of possible plants for a given frequency range

of interest. The plant template is represented in the Nichols chart, where each plant at a

given frequency can be presented by a point in the Nichols chart. A different plant at the

same frequency may be represented by a different point than the previous plant in the

Nichols chart. Therefore, all possible plants at a given frequency would constitute a set of

points in the Nichols chart. Same would apply to other frequencies. A nominal plant (i.e.

with specific parameters) is chosen for the entire frequency range of interest for

subsequent analyses. Once the plant template and the nominal plant are obtained, a set of

bounds must be defined to ensure that all possible plants in the template can meet the

requirements. For vibration control structures undergoing parameter uncertainty, these

requirements are the (i) robust stability and (ii) vibration attenuation (represented via

disturbance rejection) which are further discussed here.

5.3.1 Robust Stability

The stability margin is represented via gain margin (GM) and phase margins (PM) or the

correlated contour (called U-contour), as discussed in detail in [83]. The specified

gain and phase margins of every plant within the plant template must be sufficient to

ensure robust stability against parameter variation. The U-contour is represented in the

Nichols chart. “To guarantee a sufficient phase margin, the loop gain (denoted as ( ))

must not enter the U-contour in the Nichols chart at any of the given frequencies,” [121].

The U-contour for a unity-feedback system is defined as:

96

( ) |

( )

( )| (5.8)

where ( ) ( )

( ) for the three control systems. The stability margins are expressed

in terms of as [121]:

(

) [ ]

(5.9)

For the standard IRC scheme ( ), the loop gain is given by ( ) ( ) ( ).

Similarly, for resonance-shifted IRC ( ), and the proposed IRC schemes ( ),

the loop gains are expressed by ( ) ( ) ( ), and ( ) ( ) ( ),

respectively. It was noted in Section 5.2.3 that the equivalent transfer function for the

proposed IRC was smaller compared to those of the resonance-shifted IRC and the

standard IRC. Therefore, it is concluded that:

| ( )

( )| |

( )

( )| |

( )

( )| |

( )

( )| (5.10)

The above inequalities imply that the closed-loop magnitude of the proposed IRC scheme

is smaller than those of the standard IRC and the resonance-shifted IRC. Namely, smaller

values of can be set for the proposed IRC scheme compared to the other two control

schemes which leads to larger gain margin, and phase margins. Therefore, the magnitude

of the FRF, ( ), is considered as the index of robust stability for the closed-loop

system.

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5.3.2 Vibration Attenuation

The vibration attenuation is represented via the input disturbance of the control system in

the presence of disturbances at the input of the plant. To satisfy the disturbance rejection

requirement, the FRF from the plant disturbance to its output must be less than or equal

to the required value over a frequency band of interest. In other words, we must have:

| ( )

( )| ( ) { } (5.11)

5.4 Results and Discussions

The simulation and experimental results of the proposed modified IRC method is

presented and compared with the standard IRC and resonance-shifted IRC schemes. As a

first step, the results are given for the proof-of-concept flexible beam with variable tip

mass, followed by comparative analysis of the (i) robust stability and (ii) vibration

attenuation based on the QFT method. Following this, the proposed modified IRC

scheme is implemented on a PKM prototype with flexible links at multiple configurations.

5.4.1 Proof-of-Concept

The plant used as the proof-of-concept is a cantilever beam with three pairs of (nearly)

collocated actuators and sensors and a tip mass (Figure 4.3‏). The dimensions and

properties of the aluminum beam and each PZT transducer used in the simulations is

given in Table 4.1‏and Table 4.2‏.

Herein, the control design is presented for the 1st PZT transducer pair only. To represent

the variable structural dynamics in the plant, the tip mass was varied from its nominal

value of 1X (i.e. 13.2 grams) to 4 times its nominal value, or 4X (i.e. 52.8 grams) by

manually adding additional masses to the tip of the beam. Figure 5.5‏ shows the

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experimental FRFs of the beam when the tip mass was increased from 1X to 4X. As

expected, the resonance frequencies of the system with the additional mass (4X) are

reduced compared to those of the nominal mass.

Figure 5.5‏. Open-loop FRFs for variable tip mass.

As a result of changing the tip mass, the first three resonance modes and their

corresponding modal residues of the open loop transfer function (Equation ( were ((5.2‏

calculated to vary within the ranges, as given in Table 5.1‏.

Table 5.1‏. Variation ranges for the beam resonance frequencies and modal residues.

Tip mass ( ) ( ) ( )

1X (nominal) 222.63 1787.1 5188 2206 13645 996162

4X 129.46 1533.5 3803 754.7 3916.2 243929

Percentage

variation

(

)

41.8% 14.1% 26.7% 65.8% 71.3% 75.5%

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To compare the (i) robust stability, and (ii) vibration attenuation of the proposed IRC-

based controller, with those of the standard IRC and resonance-shifted IRC schemes, the

following procedure was followed. The nominal plant transfer function was selected to be

. A set of performance specifications with respect to robust stability and disturbance

attenuation was defined for the closed-loop system, for the frequency band of 0-1000 Hz.

A standard IRC compensator, ( )

, was synthesized with the nominal plant

using MATLAB Control System ToolboxTM

. The controller gain , and pole

were tuned to satisfy the constraint on the performance specifications using numerical

optimization of the toolbox. The feed-through term was calculated from

, and

ensured that the condition { ( ) ( ) } is satisfied for all possible

plants. If the condition was not satisfied, the numerical optimization was repeated to

obtain a different value of the gain and the pole. For the nominal system at hand, the

controller parameters were calculated as and . For resonance-

shifted IRC and proposed IRC schemes, the standard IRC was synthesized using the

tuned parameters with the equivalent plants ( ), and ( ), respectively. It should be

noted that the feed-forward gain for the resonance-shifted IRC scheme and feedback gain

for the proposed IRC scheme must be selected so as to ensure closed-loop stability. This

can be checked via the root-locus of open-loop system. From the root-locus plot of the

resonance-shifted IRC scheme, the range of the compensator gain to achieve stability is

obtained as . The gain value of was chosen for subsequent analysis.

Herein, the objective of the control system design was focused on suppressing the 1st and

3rd

modes of the cantilever beam. The 2nd

mode exhibited relatively lower controllability

index compared to the 1st and 3

rd modes due to the placement of the 1

st PZT actuator pair

along the beam, and hence vibration suppression of this mode was not pursued in the

analysis.

The simulation results of the closed-loop system were verified with experiments. Figure

shows the closed-loop system obtained from simulations and experiments for the (a-c)5.6‏

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three control schemes tested for 1X tip mass, as an example. Good agreement is observed

between the simulation and experimental results for the three control schemes.

Figure 5.6‏. closed-loop FRFs of the proof-of-concept for 1X for (a) strandard IRC, (b) resonance-shifted

IRC, and (c) proposed modified IRC schemes

Figure 5.7‏(a-c) show the experimental FRF magnitudes of the closed-loop system using

the standard IRC, resonance-shifted IRC, and the proposed IRC schemes, respectively, all

(a)

(b)

(c)

101

with tip masses of 1X and 4X. It is noted that the standard IRC is able to attenuate the

first resonance modes for 1X and 4X by at least 10 dB. However, the standard IRC

provided less attenuation for the 3rd

resonance mode, due to the limited controller

bandwidth. Using the resonance-shifted IRC scheme (Figure 5.7‏(b)), it was noted that the

attenuation for the 3rd

modes was improved, while the attenuation of the 1st modes was

comparable to those of the standard IRC scheme.

For the proposed IRC scheme (Figure 5.7‏(c)), the attenuation of the 1st and 3

rd modes was

noted to be approximately similar to that of the resonance-shifted IRC scheme. To further

compare the robustness of the proposed IRC with the resonance-shifted IRC, the QFT

analysis is conducted using the QFT Toolbox in MATLAB [122]. In the following, we

utilize the approach taken in [121] to analyze the robustness and vibration attenuation of

the proposed IRC controller. As the first step, the plant template is created given the

parameter variation range in Table 5.1‏.

102

Figure 5.7‏. FRF magnitudes of the proof-of-concept for open-loop and with (a) standard IRC, (b)

resonance-shifted IRC, and (c) proposed IRC.

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Figure 5.8‏ shows the plant template for the proposed IRC scheme for a frequency range

of [ ] as an example. This frequency range encompasses the variation in

frequency of the first resonance mode (Table 5.1‏). It should be noted that the frequency

vector for the QFT design environment must have sufficient resolution to capture all

plant variations within the band of interest. Herein, for the sake of clarity, the plant

template is only shown for a limited set of frequency points.

Once the plant template is created, the (i) robust stability and (ii) vibration attenuation

requirements are defined. At every frequency point, the robust stability and the vibration

attenuation requirements set bounds upon the closed-loop FRF magnitude of the system,

given by Equation (5.8), and Equation (5.11), respectively. To satisfy both requirements

simultaneously, the union of the bounds, called the U-contours for each requirement is

obtained from the Nichols chart [121].

Figure 5.8‏. Plant template in the QFT design environment.

The next step is the synthesis of the controller with the plant template. The synthesis is

performed with the Nichols chart with all the U-contours for the frequency band of

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interest. To compare the robust stability and disturbance attenuation capabilities of the

control schemes, the three IRC-based control algorithms previously designed are utilized

for synthesis with the plant template. Figure 5.9‏ and Figure 5.10‏ compare the robust

stability and disturbance attenuation of the closed-loop systems with the resonance-

shifted IRC and the proposed IRC schemes under the worst case scenarios of the plant

parameter variation. The worst case scenario corresponds to the maximum FRF

magnitude of the plant open-loop among possible open-loop plants within the template, at

a specified frequency.

It is noted from Figure 5.9‏ and Figure 5.10‏ that the utilization of the proposed IRC leads

to a closed-loop response with less sensitivity, and improved robustness to parameter

variations than that of the resonance-shifted IRC for almost the entire frequency range.

Furthermore, the proposed IRC is able to maintain its disturbance attenuation capability

as shown in Figure 5.10‏. Considering the above, conclusions are based on the application

of the proposed IRC scheme on the proof-of-concept cantilever beam with variable tip

mass, our premise of utilizing the proposed IRC scheme to suppress the configuration-

dependent structural vibration of PKM links is satisfied.

Figure 5.9‏. QFT robust stability of the compared control schemes.

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Figure 5.10‏. QFT disturbance attenuation of the compared control schemes.

5.4.2 Application of the Proposed IRC-Scheme to Vibration

Suppression of the PKM with Flexible Links

The PKM utilized herein is Prototype III (see Chapter 3) with one of its links is made to

be flexible with three pairs of PZT transducers attached on its surface (Figure 5.11‏). The

flexible modes of this flexible linkage have substantially lower resonant frequencies than

the other two linkages; hence, the modal coupling due to the presence of other links is

avoided as much as possible.

It should be noted that the other two existing IRC schemes were already compared with

the proposed IRC scheme in sub-section 5.4.1. Specifically, it was shown that the

standard IRC scheme has limited capability of suppressing the 3rd mode due to its

limited bandwidth. Also, the resonance-shifting IRC scheme was shown in the robustness

analysis to exhibit lower robust stability than the proposed IRC scheme. Therefore, the

existing IRC schemes were excluded from the closed-loop analysis of the PKM with

flexible links. The proposed IRC scheme was implemented on the PKM given in Figure

,The active vibration control system utilized the LabVIEW Real-Time Module [123] .5.11‏

[58].The diagram of the control system is shown in Figure 5.12‏.

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Figure 5.11‏. PZT transducers bonded on flexible link of a PKM.

The sensor signals were acquired and filtered using an NI SCXI 1531 signal conditioning

unit, with a 4-pole low-pass Bessel filter of 2.5 KHz cut-off frequency. For the control

processing unit, we used a desktop-PC with Intel E8400 Core 2 Duo processor with 3 GB

of memory, running the LabVIEW Real-Time Operating System (RTOS), as the Target

PC. The sampling frequency was 4 KHz. A swept sine (chirp) signal of 3V (peak-to-peak)

was applied over a frequency band of 0-1000 Hz to the 1st PZT actuator as the

disturbance input and the sensor signal from the 1st PZT sensor was captured and post-

processed using a Host PC as the user interface. The transfer function of the controller

obtained from the simulations was discretized and implemented in the LabVIEW real-

time code. The output command signal from the controller was amplified using the SS08

power amplifier from SensorTech and applied to the PZT actuator.

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Figure 5.12‏. Diagram of the active vibration control system.

To examine the performance of the controller under variable structural dynamics

behavior of the PKM, four different joint-space configurations were chosen as an

example, as given in Table 5.2‏.

Table 5.2‏. Four configurations selected for vibration control experiments.

Configuration

name (mm) (mm) (mm) (degree) (degree) (degree)

Home 0 0 0 0 0 0

AA 0 0 -20 +15 -15 +15

BB +20 0 -20 +15 -15 0

CC -20 0 -20 -15 +15 -15

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For each configuration, the open-loop transfer function of the flexible link of the PKM

was measured. Figure 5.13‏ shows the open-loop FRFs of the PKM link for four

configurations. It is noted that the modes undergo variations in terms of both resonance

frequencies and amplitudes.

Specifically, three set of modes are noted in the open-loop response. The 1st, 2

nd, and 3

rd

set of modes occurs at the frequency ranges of [153-229] Hz, [368-465] Hz, and [1296-

1342] Hz, respectively.

Figure 5.14‏(a-d) shows the FRFs of the PKM links with and without the control applied

for each configuration. It is noted that the 1st and 2

nd set of modes, herein called low

frequency modes, do not undergo significant changes with configuration, and likely arise

due to joint clearances.

Figure 5.13‏. Open-loop FRF pf the PKM link for four example configurations.

109

Figure 5.14‏. FRF of the flexible PKM link with and without controller for (a) configuration AA, (b)

configuration BB, (c) configuration CC, and (d) configuration Home.

However, note that the 3rd

set of mode amplitudes, arising from the bending vibration of

the links, are suppressed using the proposed IRC control scheme. The time-response of

the PKM link, when the mode corresponding to the link is suppressed, is shown in Figure

5.15‏ and Figure 5.14‏ for home configuration as an example. The results of Figure 5.15‏

show that the proposed IRC scheme is able to suppress the configuration-dependent

vibrations resulting from the links of the PKM with reasonable amount of suppression.

Specifically, the proposed controller is robust in the presence of variations of resonance

frequencies and mode amplitudes while the vibration attenuation capabilities are

maintained.

110

Figure 5.15‏. Time-response of the PKM link for configuration Home.

5.5 Summary

In this chapter, a new modified Integral Resonant Control scheme is presented and

implemented for vibration suppression of the flexible links of PKMs exhibiting

configuration-dependent resonance frequencies and mode amplitudes. The proposed IRC

scheme is compared with the existing IRC schemes in terms of its robust stability and

vibration attenuation under variations in the natural frequencies and mode amplitudes.

Using a Quantitative Feedback Theory method, it is demonstrated that the presented IRC

scheme has improved robustness over the existing IRC schemes while maintaining its

vibration attenuation capabilities.

The significance of the robust performance is that in addition to the configuration-

dependent structural dynamics of the PKMs, it is expected that in the typical use of the

PKM, the vibration frequencies, and mode amplitudes change due to unknown changes in

the physical parameters of the PKM, such as added masses/payloads to the moving

platform. The proposed modified IRC control methods exhibits improved robustness over

existing approaches, as outlined in this work. Hence this approach permits the good

attenuation of linkage vibration characteristics which change as a result of both dynamic

model uncertainty caused by unknown payloads, and PKM configuration dependent

behavior. Such improvement in the robust performance is very important, and is provided

by our approach. Moreover, the simplicity and performance of the proposed control

111

scheme, compared to the existing robust controllers, makes it a viable solution for

vibration suppression of the configuration-dependent links of the PKMs.

112

6 Chapter

Conclusions and Future Work

6.1. Conclusions

This thesis was focused on the structural dynamic modeling, dynamic stiffness analysis,

and development of an active vibration control system for PKMs with flexible links using

PZT transducers. The contributions achieved in this thesis are summarized as follows:

The complete coupled rigid-body and structural dynamic models of a PKM with

flexible links were developed using extended Hamilton’s principle, Lagrange’s

equations and Assumed Mode Method (AMM). Subsequently, to avoid the

complexities associated with analytical solution of the frequency equation for PKM

links, a set of admissible shape functions were proposed to be used in the AMM. The

proposed admissible shape functions reflected the effects of the mass of the adjacent

structural components (e.g. moving platform, payload) to those of the flexible links.

Specifically, a “mass ratio” was defined as the ratio of the effective mass of the

moving platform and payload (e.g. spindle/tool) to the mass of the flexible links. The

accuracy of the proposed admissible shape functions was examined by comparing the

natural frequencies calculated from the solution of the frequency equation of the

shape function, with the natural frequencies of the entire PKM obtained from FE

analysis as a function of the mass ratio. Finally, the most accurate shape function was

recommended for a given “mass ratio”. The methodology developed in this thesis led

to a more accurate and computationally-efficient structural dynamic model for the

generic PKMs with flexible links by incorporating shape functions that take into

account the mass/inertia effects of the adjacent structural components to the PKM

links. The developed model for the PKM with flexible links can be synthesized with

real-time model control design to suppress the unwanted vibrations of the PKM links.

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An FE-based methodology was developed to estimate the natural frequencies, mode

shapes, and dynamic stiffness of PKM-based machine tools. The developed FE model

utilized the CAD model of the PKM with detailed geometrical features, and was

obtained via a harmonic analysis in software package, ANSYS. Specifically, the

objective of the analysis was to predict the mode shapes and the directional elastic

displacement of the Tool Center Point (TCP) as a result of the excitation of the

structural resonance modes of the PKM-based machine tool due to the exertion of

cutting forces at the TCP. Specific attention in the analysis was on 6-dof PKM-based

machine tools that are kinematically redundant for 5-axis machining. It was shown

that the developed model was able to capture both the configuration-dependent

variations of the dynamic stiffness within the workspace, and the variations of the

dynamic stiffness for a given platform position and orientation due to the redundancy

of the machine tool. The developed FE model was validated via Experimental Modal

Analysis i.e. impact hammer testing of two prototype PKM-based meso-Milling

Machine Tools (mMT) designed and built in the CIMLab at the University of Toronto.

The FE simulations and experiments were performed for multiple joint-space

configurations of the prototypes. Strong configuration-dependent behaviour for the

PKM prototypes was observed in terms of resonance frequencies and TCP

displacement amplitudes, which were represented via Frequency Response Function

(FRF) curves in this analysis. Subsequently, a simplified, and hence more efficient FE

simulation model was also developed for relative estimation of the dynamic stiffness

of a generic PKM for multiple configurations. The developed FE models provided a

basis for comparative analysis of various and/or new PKM architectures for design

improvements from a stiffness point of view. For 6-dof PKMs performing 5-axis

machining, the FE model can potentially provide the input information required to

perform an on-line optimization of the tool path so as to achieve the PKM joint-space

configuration with the highest dynamic stiffness (among infinitely many redundant

joint-space configurations) along the tool path during on-line operation of the

machine tool.

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Piezoelectric (PZT) actuators and sensors were designed and bonded to the flexible

link of the PKM to suppress the unwanted vibration of the PKM resulting from the

links. An electromechanical modeling methodology was presented in this thesis to

obtain the relationship between the input voltage of the PZT actuators to the output

voltage from the PZT sensors. It was shown that the incorporation of the added mass

and stiffness properties of PZT transducers to those of the link in the

electromechanical model resulted in a more accurate prediction of the resonance

frequencies and mode shape (and mode shape slope) amplitudes of the smart link.

The presented electromechanical model was verified via experiments on a proof-of-

concept cantilever beam with three pair of PZT transducers. Since the resonance

frequencies and mode shape amplitudes of the smart link are directly utilized in the

controller design and synthesis, accurate prediction of these variables through a high-

fidelity electromechanical modeling approach is of crucial importance. The

developed electromechanical model was subsequently used in the controllability

analysis of the smart link for a set of resonance modes targeted for control. In this

regard, the available literature focused on the implementation of optimization

algorithms on smart cantilever beams to obtain the location (and dimension) of PZT

actuators for which maximized controllability is achieved. In this work, the location

and dimension of the PZT actuators along the link were fixed. Instead, it was shown

that it is possible to achieve a desired controllability by adjusting the mass of moving

platform of the PKM. The methodology was implemented on a proof-of-concept

cantilever beam with a tip mass, where it represented a portion of the moving

platform mass in a PKM. The methodology presented in this chapter provides a basis

for electromechanical modeling for subsequent controller design and synthesis of the

PKMs with flexible links. Also, the controllability analysis and the methodology

presented to adjust its value to the desired one can be utilized in the design of the

moving platform of PKMs with flexible links, for effective vibration suppression of a

set of modes targeted for control. Moreover, the controllability analysis can be used

115

to gain an estimation of the relative control voltages needed for each PZT actuator to

suppress a set of resonance modes of a PKM link.

An active vibration control methodology was designed and implemented on the

flexible links of the PKM to suppress the unwanted structural vibrations of the PKM.

It is known that the structural dynamics of the PKM links undergo configuration-

dependent variations within the workspace. Therefore, the controller must be robust

in the presence of such configuration-dependent variations. To address this issue,

various model-based robust control techniques methods have been proposed. In this

thesis, a new control scheme based on Integral Resonance Control (IRC) method was

proposed. Specifically, the proposed IRC method is this thesis was modified to

achieve improved robustness over the existing IRC schemes. To examine the

performance of the proposed control scheme, a proof-of-concept cantilever beam with

a variable tip mass was taken to represent the configuration-dependent structural

dynamics of PKM flexible links. The performance of the proposed modified IRC was

examined in terms of (i) robust stability and (ii) vibration attenuation capabilities

using the Quantitative Feedback Theory (QFT). Specifically, the configuration-

dependent dynamics of the proof-of-concept were represented via a number of points

in the Nichols chart for a given frequency to form the plant template. Subsequently,

the loop-shaping in the QFT environment was conducted using the designed modified

IRC method. The QFT analysis results showed that the modified IRC scheme exhibits

improved robustness over the existing IRC methods, making it a simple and viable

approach to suppress the configuration-dependent vibrations of the PKM links. Using

LabVIEW Real-Time module, the proposed IRC scheme was experimentally

implemented on the PKM flexible links on distinct configurations of the PKM.

6.2. Future Work

While this thesis addressed the research issues associated with the structural vibration,

dynamic stiffness, and active vibration control of PKMs, there are still a number of open

116

topics that can be potentially investigated as future research in this area. Below is a brief

discussion on the open research areas in this topic:

With respect to the dynamic stiffness estimation of PKM-based machine tools, it

is well known that the total displacement at the TCP is the resultant contribution

from the (i) structural components such as links, and (ii) the contacting interfaces

such as joint bearings, joint clearances, bolted connections, and actuators. This

thesis was mainly concerned with the structural dynamic modeling of PKMs as a

result of elasticity in the structural components. However, during the

experimental modal analysis of the PKM prototypes, it was noted that the joint

dynamics greatly affects the total stiffness of a PKM at the TCP. Particularly, it

was noted that the resonance modes, and dynamic stiffness of the PKM are

greatly reduced when joint dynamics are taken into account. While the FE model

developed in this thesis takes into account the contact interfaces by using an

equivalent coefficient of friction at the joints, the joint clearances, and joint

stiffness/damping were not incorporated in the analysis. Therefore, as a future

step in the refinement of the FE model, it would be beneficial to incorporate the

joint effects for a more accurate estimation of the dynamic stiffness. However, as

analytical identification of the joint dynamics is typically difficult in general, they

must be obtained through experiments. To this end, various methodologies based

on Component Mode Synthesis can be utilized to obtain the joint parameters via

experiments to be further utilized in the analytical or FE models of the PKM.

As mentioned in Chapter 3, the FE model provides a basis for subsequent path

planning of the tool path. Specifically, for 6-dof PKM based machine tools used

for 5-axis machining, the robot joint-space configuration can be optimized on-line

so as to achieve maximized dynamic stiffness along the tool path. To perform the

on-line optimization, the dynamic stiffness must be estimated via time-efficient

methods. One approach to obtain a time-efficient and yet reliable tool for

estimation of dynamic stiffness is to utilize the dynamic stiffness data obtained

117

from the FE models in training emulators such as Artificial Neural Networks

(ANN). The trained ANN can then be used in the on-line optimization procedure

to achieve the desired robot configuration with the maximized dynamic stiffness.

The methodology proposed to achieve the desired controllability in this thesis

(Chapter 4) was applied on a proof-of-concept cantilever beam with variable tip

mass. The next step would be to implement the controllability analysis on the

flexible PKM links, and examine the effects of the moving platform mass.

Moreover, the variation of the controllability as a function of the PKM joint-space

configuration of the PKM within the workspace can be another interesting topic

to investigate. This topic could be of particular interest in PKM-based machine

tools as 3-dimensional flexible mechanisms, since, one might be interested in

knowing how well, a set of PZT actuators can affect the modes in the Cartesian

direction on the moving platform.

Although the electromechanical model and the controllability analysis was

performed for all three PZT transducer pairs of the smart link, the implementation

of the closed-loop control scheme was only carried on the 1st PZT transducer pair

for vibration suppression. An immediate extension could be to implement the

control scheme on the other 2nd

and 3rd

PZT transducer pairs to further verify the

control methodology.

In this thesis, to demonstrate the active vibration control methodology, only one

of the PKM links was made to be flexible. In addition, the use of one flexible

smart link with the other two links being as rigid avoided the complications

resulting from the mode coupling from other linkages. This is because the modes

associated with the other two links as significantly higher than the flexible link.

Investigations and experiments with two and three flexible links may be carried

out in future work.

118

In this thesis, to facilitate the implementation of the control scheme, the PZT

transducers were designed so as to achieve a collocated sensor actuator

configuration. Generally, collocated configuration for sensors and actuators yield

minimum phase systems for which better closed-loop characteristics such as

robustness can be achieved. It is well known that the overall objective in vibration

control design of PKM-based machine tools is to reduce the vibration as the TCP.

In other words, regardless of the vibration amplitudes along the flexible links, it is

important to reduce the vibration transmitted to the TCP as much as possible.

Given this discussion, it would be beneficial to design and synthesize control

schemes that can reduce the vibration at the TCP, and not necessarily the link

itself. To this end, the open-loop transfer function from the PZT actuators on the

links to the sensing element on the moving platform (e.g. an accelerometer) must

be obtained. Unlike the collocated case, this transfer function will represent a

non-minimum phase system for which robustness analyses are not as

straightforward as they are for minimum phase systems. Therefore, the

development of vibration suppression controller for non-minimum phase system

that undergoes variations in structural dynamic properties could be an excellent

research area to be investigated.

119

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Appendix A

Partitioned Matrix and Vector Expressions For Structural

Components of the PKM Excluding the Links

Inertia matrix :

( ) [

] (A.1)

Coriolis/centrifugal matrix :

( ) [

] (A.2)

Gravity vector:

( ) [

] (A.3)

139

Appendix B

Partitioned Matrix and Vector Representations for Active and

Passive Joints

Inertia matrix (active coordinates):

( ) [

] (B.1)

Stiffness matrix (active coordinates):

[

] (B.2)

Inertia matrix (passive coordinates):

( ) [

]

(B.3)

Jacobian for active coordinates:

(

)

(B.4)

Jacobian for passive coordinates:

(

)

(B.5)

Inertial and gravity forces on active coordinates:

[ ]

( ) ( ) (B.6)

Inertial and gravity forces on passive coordinates:

[

]

( ) ( ) (B.7)

Documents

STRUCTURAL DYNAMIC MODELING, DYNAMIC STIFFNESS… · Structural Dynamic Modeling, Dynamic Stiffness, and Active Vibration Control of Parallel Kinematic Mechanisms with Flexible Linkages