Structural Vibration Modeling and Dynamic Stiffness - Mahmoodi Masih- 2014 - PhD Thesis

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 ( 2.45) .................................................. 40

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 1st fixed-free shape

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

Figure 2.12. Tooltip time response for 1st fixed-mass and 1st 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 rth


minimum rth


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 Youngs 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 segment of the smart link

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 ith 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

segment of the smart link

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

rth

mode pole of the plant

resonance frequency of the rth

mode of the smart link

maximum rth


minimum rth


xxvi

rth

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 Kanes

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, Lagranges 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 Hamiltons 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 Hamiltons 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

2.1. Also, let us define ( ) [ ] 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 ( 2.5), into ( 2.3), the variation in kinetic energy of

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 Youngs modulus of the linkage. Also,

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

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

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 ( 2.3) and ( 2.7) along with Equation ( 2.8) into the

extended Hamiltons principle (Equation ( 2.1)), yields a set of equations of motions that

represents the motion of active joints, , passive joints,

, and elastic vibration of the

links, of the ith

sub-chain. Also, from the extended Hamiltons 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 mth modal coordinate of the ith 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

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

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 Lagranges equations to derive the equations of motion for the above

mentioned components. The Lagranges 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

Documents

Structural Vibration Modeling and Dynamic Stiffness - Mahmoodi Masih- 2014 - PhD Thesis