conducted on two PKM-based meso-Milling Machine Tool (mMT) prototypes built in ... To this end, the modified IRC approach is shown via a .... 2.2.3 Structural Vibration Response of the Entire PKM-Based mMT . ..... Mode shapes of prototype III at configuation Home for (a) 1st mode at 85 Hz, ..... mass density of the beam ...
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
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
ii
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 configurationdependent 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.
iii
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.
iv
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
v
2.2
2.1.3
System Dynamic Modeling of the Overall PKM ........................................ 26
2.1.4
Admissible Shape Functions ....................................................................... 30
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 2.3
Structural Vibration Response of the Entire PKM-Based mMT ................ 39
Summary ............................................................................................................ 45
3 Dynamic Stiffness of Redundant PKM-Based Machine Tools ................................... 47 3.1
Dynamic Stiffness Definition ............................................................................. 48
3.2
Dynamic Stiffness Estimation ............................................................................ 50
3.3
3.4
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
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
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
vi
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 5
Summary ............................................................................................................ 86 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
5.3
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
Utilization of the IRC-Based Control Schemes in Quantitative Feedback Theory
(QFT)
5.4
............................................................................................................................ 94
5.3.1
Robust Stability........................................................................................... 95
5.3.2
Vibration Attenuation ................................................................................. 97
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 6
Summary .......................................................................................................... 110 Conclusions and Future Work ................................................................................. 112
vii
6.1.
Conclusions ...................................................................................................... 112
6.2.
Future Work ..................................................................................................... 115
References ....................................................................................................................... 119 Appendix A ..................................................................................................................... 138 Appendix B ..................................................................................................................... 139
viii
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
ix
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 Figure 2.15. 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 x
Figure 2.16. 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 . ....................................................................................................................... 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) configuration AA, (c) configuration BB, and (d) configuration CC......................... 57 Figure 3.10. FRFxz amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c) configuration BB, and (d) configuration CC......................... 57 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) configuration AA, (c) configuration BB, and (d) configuration CC......................... 59 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
xi
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) 2nd pair, and (c) 3rd 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) 2nd mode, and (c) 3rd 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 xii
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
xiii
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 mode modal residue of the plant transfer function minimum rth mode modal residue of the plant transfer function input matrix of the smart link in state-space representation coefficient of the in-plane shape function of the PKM link coefficient of the out-of-plane shape function of the PKM link 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 coefficient of the in-plane shape function of the PKM link coefficient of the out-of-plane shape function of the PKM link 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, xiv
moving platform, and spindle/tool ( ) ̂( )
transfer function of the compensator equivalent transfer function of the compensator constant feed-through term disturbance input signal coefficient of the in-plane shape function of the PKM link coefficient of the out-of-plane shape function of the PKM link transverse piezoelectric strain constant vertical position of the prismatic actuator column of prototype II vertical position of the linear prismatic joints for ith 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 xv
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
xvi
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 ith 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
xvii
PZT actuator coefficient for the jth 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 jth 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 ith 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
xviii
inertia matrix of the ith 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 xix
{O} ̅
inertial frame pole of the compensator
PM
phase margin reaction force vector acting on the ith link at reaction force vector acting on the ith 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 ith actuator position vector
( )
mth modal coordinate vector of modal coordinates for the ith 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 ith chain
xx
vector of the rigid-body motion coordinates of the entire n subchains 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 jth PZT actuator pair transformation matrix from the passive joint velocities to active joint velocities
xxi
(
)
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 jth 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 jth PZT sensor pair output controllability Grammian matrix state controllability Grammian matrix
(
)
local vector of the two elastic lateral displacements of the ith chain
xxii
state vector in state-space representation Cartesian position of the circular prismatic joints for ith chain Cartesian position of the spherical joint for ith 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
xxiii
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 kth 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
xxiv
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 resonanceshifted 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 mode natural frequency minimum rth mode natural frequency xxv
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
xxvi
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
xxvii
Chapter 1 Introduction This chapter provides the motivation of this thesis, followed by a review of the state-ofthe-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
1
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. 2
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 non3
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
4
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-topayload 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 PKMbased 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 configurationdependent variations within the workspace [31]. Knowledge of the configurationdependent structural dynamic characteristics of the PKM can provide an insight into 5
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]. 6
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 7
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].
8
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
9
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 configurationdependent 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
10
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 loopshaping, 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 11
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.
12
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 13
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 14
frequencies and response amplitudes of the structural dynamics of the PKM links experience configuration-dependent variation within the workspace. Such configurationdependent 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 activevibration-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 15
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.
16
Chapter 2 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].
17
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:
∫ (
where
,
, and
)
(2.1)
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
18
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:
∑∫ ( ̇
where
(2.2)
̇ )
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]:
∑∫ ( ̈
where
)
is the variation in the Cartesian coordinate of the position vector
(2.3)
. 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: ̇
̇
19
(2.4)
and, ̇ ̇ ̈ where
is the kinematic transformation matrix of the ith
] , and
[
(2.5) ̈
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 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 (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:
20
∑{
(
[
) (
)]
(
))
(
∫
(
(
]
)
]
) (2.7)
(
) (
∫
)]
(
[
(
(
(
)
]
))
)
∫
]
}
Virtual Work of External Forces The total virtual work done by external forces on the elastic links is given as: ( )
(
)
( )
(
)
( )
(
)
∑(
) ( )
where
[
(
]
)
( )
and
[
(
)
,
and
(
(2.8)
)
] are the two reaction forces
acting on the two end joints of the ith elastic link (i.e., and,
( )
and
), respectively,
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
measured in the same plane as the revolute joint angle is measured.
21
is
Boundary Conditions Substituting the results of Equations (2.3) and (2.7) along with Equation (2.8) into the extended Hamilton’s principle (Equation (2.1)), yields a set of equations of motions that represents the motion of active joints, links,
, passive joints,
, and elastic vibration of the
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,
(
and at
)
(
)
(
)
(2.10)
, (i.e. spherical joint) as follows:
(
)
(2.11)
and,
(
)
(
)
(
Similarly, the boundary conditions for out-of-plane vibration of the links,
)
(2.12)
, at,
are obtained as: ( and,
22
)
(2.13)
(
and at
)
(2.14)
, as follows:
(
(
)
)
(2.15)
and,
(
where
and
moment, and
(
)
)
(
)
(2.16)
are the in-plane and out-of-plane components of the bending and
force, respectively.
are the in-plane and out-of-plane components of the shear (.) 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:
23
(
)
∑
(
)
∑
( )
( )
( )
( )
(2.17)
( )
(2.18)
and,
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]: ( ) ̈ where
(
̇) ̇
( ) is the modal inertia matrix,
Coriolis and centrifugal effects, vector of modal gravity forces.
(
̇ ) is the modal matrix representing
is the modal stiffness matrix, and is a function of the reaction forces,
distal ends of the links. 24
(2.20)
( )
( ) is the and
at the
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:
̇
where
,
) ̇
(
∑( ( ̇ )
( ̇ ))
(2.21)
are the inertia matrices of the ith sub-chain actuator, the
, and
moving platform, and the spindle/tool, respectively. The total potential energy of the actuators, the moving platform, and the spindle/tool is given as:
(
where
)
is the mass of each actuator,
∑
( )
(2.22)
is the vertical component of the ith 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:
25
(
where the vector
(2.23)
)
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: ( where
(
) ̈
(
) is the inertia matrix,
centrifugal effects, and
(
̇ ) ̇
(
)
(2.24)
̇ ) is the matrix representing Coriolis 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:
26
(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:
(
where
[
) ̈
(
] ,
̇ ) ̇
[
(
] , and
)
(
[
)
(2.26)
] 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).
27
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 [ coordinates[ (
] and the vector of passive/task space
] . 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: ̇ where
(2.30) ̇
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: 28
̈ ̈ ̈ ̈ (
̈ ]( ̈ )
[ ̈
(
̇ ̇ ) ̇ ̇
(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.
29
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 (2.9), (2.13), and (2.14)) and not necessarily the dynamic boundary conditions (i.e. Equations (2.10), (2.11), (2.12), (2.15), 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”, “fixedpinned” 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
30
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: ( )
(
)
(
)
(
)
(
)
(2.37)
and ( )
(
)
( (
respectively, where
and
)
(
)
(2.38)
)
are the eigenvalue solutions associated with the in-plane
and out-of-plane natural frequencies of the link,
, and
, as:
√
(2.39)
√
(2.40)
and,
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:
31
(
)
(
[
)
( (
(
) (
)
(
)
( )
(
) )
(
(
)
)
(
(
)
(
)
(
)
)
) (
)
]
(2.41)
Similarly, for the “fixed-mass” shape function, we get:
( (
[
(
)
)
( (
)
) )
(
(
( )
(
) (
( )
(
(
)
)
) )
)
(
(
)
) (
)]
(2.42)
32
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 “pinnedmass” 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 mesoMilling 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
33
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 points
. These linkages are connected to the moving platform through spherical joints at . The prismatic joints
well as the spherical joint at
and
are actuated joints and the revolute joints
as
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
(
component is normal to the in-plane displacement and is given by
) . The out-of-plane (
) (Figure
2.5). Figure 2.6 shows the reaction forces at the spherical joints of the moving platform 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:
34
(
)
(2.44)
Figure 2.5. Elastic displacement component of the linkage for out-of-plane
Figure 2.4. Elastic displacement component of the linkage for in-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
35
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
36
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, mode shapes,
to those of the realistic
, 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-ofplane 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.
37
). This result is expected as the
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
300
Second out-of-plane mode (Hz)
First out-of-plane mode (Hz)
best mode shape approximation for the first out-of-plane mode.
200
100
0 0.001
First fixed-mass First fixed-free FEA 0.01
0.1 1 mass ratio (r)
10
100
3200
Second fixed-mass First fixed-pinned Second fixed-free First fixed-fixed FEA
2700 2200 1700 1200 700 200 0.001
0.01
0.1 1 mass ratio (r)
10
100
Figure 2.8. Out-of-plane natural frequencies of the PKM links for the second mode
Figure 2.7. Out-of-plane natural frequencies of the PKM links for the first 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 38
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.
First pinned-mass First pinned-pinned First pinned-free First pinned-fixed FEA
2000 1600
Second in-plane mode (Hz)
First in-plane mode (Hz)
2400
1200 800 400 0 0.001
0.01
0.1 1 mass ratio (r)
10
Figure 2.9. In-plane natural frequencies of the PKM links for the first mode
100
7000
Second pinned-mass Second pinned-pinned Second pinned-free Second pinned-fixed FEA
6000 5000 4000 3000 2000 1000 0.001
0.01
0.1 1 mass ratio (r)
10
100
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
39
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 ratioerror 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-ofplane
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 pinnedpinned
14.1%
Second pinnedpinned
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 “pinnedmass” 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. 40
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
2nd out-of-plane
Set 1(a)
1/300
1st fixed-free
2nd fixed-mass
Reference for set 1(a)
1/300
1st fixed-mass
2nd fixed-mass
Set 1(b)
2/3
1st fixed-free
2nd fixed-mass
Reference for set 1(b)
2/3
1st fixed-mass
2nd fixed-mass
Set 1(c)
150/3
1st fixed-mass
1st fixed-pinned
Reference for set 1(c)
150/3
1st fixed-mass
2nd fixed-mass
The MATLAB solver utilized was ode15s for stiff systems. The mechanism is initially positioned
at
the
following ] and
[ of
[
]
configuration: . An impulse force
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
41
function, accurately predicts the out-of-plane natural frequency for low mass ratios,
Tooltip response (µm)
simulation with this mode shape tends to under-predict the response amplitude. 1st fixed-mass for r=1/300
2
1st fixed-free for r=1/300
1 0 -1 -2 0
0.01
0.02
0.03
0.04
0.05 Time (s)
0.06
0.07
0.08
0.09
0.1
Figure 2.11. Tooltip time response for “1st fixed-mass” and “1st 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 2.11) to
⁄
(Figure
⁄ (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 fixedpinned” 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 2.5. Since the FEA frequencies, as shown in Figure 2.9 and Figure 2.10 are close to the “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. 42
Tooltip response (µm)
1st fixed-mass for r=2/3 1st fixed-free for r=2/3
16 8 0 -8 0
0.02
0.04
Time (s)
0.06
0.08
0.1
Tooltip response (µm)
Figure 2.12. Tooltip time response for “1st fixed-mass” and “1st fixed-free” shape functions for the first out⁄ of-plane mode at 400
1st fixed-pinned for r=150/3
300
2nd fixed-mass for r=150/3
200 100 0 0
0.02
0.04 Time (s)
0.06
0.08
0.1
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
2nd in-plane
Set 1(a)
1/300
1st pinned-mass
2nd pinned-mass
Reference for set 1(a)
1/300
1st pinned-pinned
2nd pinned-pinned
Set 1(b)
2/3
1st pinned-mass
2nd pinned-mass
Reference for set 1(b)
2/3
1st pinned-pinned
2nd pinned-pinned
Set 1(c)
150/3
1st pinned-mass
2nd pinned-mass
Reference for set 1(c)
150/3
1st pinned-pinned
2nd pinned-pinned
43
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 “pinnedpinned” shape functions leads to the same small contribution to the overall tooltip response, the “pinned-pinned” shape functions can more accurately predict the natural
Tooltip response (µm)
frequencies due to the in-plane modes. pinned-mass for r= 0.01/3 pinned-pinned for r=0.01/3
2 0
-2 0
0.02
0.04
Time (s)
0.06
0.08
0.1
Tooltip response (µm)
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 . 20
pinned-mass for r=2/3 pinned-pinned for r=2/3
10 0 -10 0
0.01
0.02
0.03
0.04
0.05 Time (s)
0.06
0.07
0.08
0.09
0.1
Figure 2.15. 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
Tooltip response (µm)
400
pinned-mass for r=150/3 pinned-pinned for r=150/3
200 0 0
0.02
0.04
Time (s)
0.06
0.08
0.1
Figure 2.16. 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 .
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-ofplane 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 45
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.
46
Chapter 3 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. 47
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]:
(
)
( (
) )
[
where i is the imaginary operator. (
( ( (
) ) )
( ( (
), and (
) ) )
( ( (
) )] )
) are the frequency spectrums (i.e. the
fast Fourier transforms) of the displacement and force vectors, respectively. Therefore, the element
(
(3.1)
( ) , and
( ),
) in Equation (3.1) can be obtained by
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 crossaxes FRF terms when u and v are different axes. Assuming the first 48
-resonance modes
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. ratio;
represents the mode of the vibration,
is the damping
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 (3.1) and (3.2), the dynamic stiffness matrix,
( ) where
,
, and
, of a PKM for a given configuration can be defined as [99]:
( ) ( )
)
√(
(
)
(3.3)
are the structural mass, equivalent damping, and static stiffness
matrices of the PKM, respectively. In Equation (3.3),
denotes the frequency of the
external force applied at the TCP. Assuming the PKMs as lightly damped structures, it is noted from Equation (3.3) that when the frequency of the applied force is close to one of the structural resonance frequencies of the PKM, the term (
) on the right hand
side of Equation (3.3) becomes approximately zero leading to the minimum values for 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 49
leading to a change in the minimum
dynamic stiffness (Figure 3.2).
FRF magnitude (m/N)
Configuration BB Configuration AA
Frequency (Hz) 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
50
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. 51
Figure 3.3. Prototype II Figure 3.4. Prototype III
Figure 3.6. Architecture of PKM prototype III
Figure 3.5. Architecture of PKM prototype II
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 52
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 53
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
Configuration
(o)
(o)
(o)
(mm)
(mm)
(mm)
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
54
Table 3.2. Joint space configurations chosen for prototype III
Configuration
(o)
(o)
(o)
(mm)
(mm)
(mm)
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.
55
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
56
Figure 3.9. FRFxy amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c) configuration BB, and (d) configuration CC
Figure 3.10. FRFxz amplitudes of prototype II for (a) configuration Home, (b) configuration AA, (c) configuration BB, and (d) configuration CC
57
(a)
(b)
(c)
(d)
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.
58
Figure 3.12. FRFxx amplitudes of prototype III for (a) configuration Home, (b) configuration AA, (c) configuration BB, and (d) configuration CC
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
59
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) 2nd 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 60
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
61
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 3.17 [ 99].
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 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
63
In addition to the comparative analysis of the above mentioned PKMs, the developed FEbased 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)
64
(c) 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.
65
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 FEmodels 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 FEbased modeling methodology was utilized in comparative dynamic stiffness analysis of new PKM architectures for 5-axis machining.
66
Chapter 4 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
67
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, it is subjected to a transverse mechanical displacement at point The thicknesses of the beam and each transducer are denoted as
, when
in Figure 4.1. 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].
68
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.
69
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]:
( where
(
( )
)
(
( )
)
( ) is the variable flexural rigidity,
)
(4.1)
( ) 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 (4.1), the eigenvalue problem associated with the ith beam segment is given as:
(
)
( )
; where
(4.2)
( ) ,
( ) is the mode shape function of the ith segment. The general solution for the
mode shapes for the ith segment is given as:
( ) where transducers.
(
(
,
)
)
. and ,
, and
(
)
(
)
(
)
(4.3)
is the natural frequency of the combined beam and PZT 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]: 70
( )
( )
( )
(
)
(
)
( )
( )
( )
(
)
(
)
( )
(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:
[ ( The values of
)]
(4.5)
satisfying Equation (4.5) constitute the natural frequencies of the
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:
∑∫
(
( )
( ))
The final normalized mode shapes of the system for the rth mode are given as:
71
(4.6)
( )
( )
( )
( )
( )
( )
( )
{
(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]:
( )
where
(
)
(
(
( )
)
)
∑
( )
( ) is the input voltage to each actuator in the jth pair and
( )
(4.9)
( ) is the second
spatial derivative of the distribution function of the input voltage over the jth PZT actuator 72
pair. For the configuration as given by Figure 4.1, the distribution function
( ) is
given as:
( ) where
[ (
)
(
(4.10)
)]
( ) is the Heaviside function. Equation (4.10) shows that the voltage input to the
jth PZT actuator has a uniform profile over the PZT actuator length and is zero elsewhere. The coefficient
is defined as follows [112]:
( where
(4.11)
)
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:
( )
where
(
[
)
]
(4.12)
, is the position of the center point of the jth 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:
(
where
is the capacitance of the PZT sensor.
73
)
(4.13)
4.1.4
System Modeling of the Combined Beam and PZT Transducers
Substituting Equation (4.8) into (4.9) and utilizing the orthonormality of the mode shapes, 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: ̇
where
[
{
]
}
(4.16)
,
̅(
,
̅(
)
[
( )(
)
( )(
of
modal
] ̅(
̅(
)
)
] . Also,
the input voltage to the actuators and vector
coordinates.
)
[ [
where
̅(
)
[
[ ( )
( )
Defining 74
)
the
,
[ ̇]
]
( ) ( )
,
( )]
( ) ( )
( ) sensor
( )] voltage
is is the as
[
( ) ( )
̅
] ̅
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, output voltage vector,
, and the sensor
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]:
( ) Due to the presence of
(4.18)
∫
in Equation (4.18), the state controllability is dependent on the
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 75
controllable. The corresponding performance index defined for controllability is defined as:
(∑
√∏
) (
where
(4.19) )
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:
(
)
∫
Similar to the state controllability, the performance index can be expressed as:
76
(4.21)
(∑
(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 77
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 ( )
78
.
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 70
Beam Young’s modulus (
)
PZT Young’s modulus (
)
Beam density(
62 2700
⁄
7800
⁄
)
PZT transducer density(
)
PZT transducer strain constant (
4.3.1
Unit
)
⁄
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, 2nd, and 3rd PZT transducer pairs as an example. It is noted from Figure 4.4 that the natural frequencies of the stepped beam
79
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.
(a)
(b)
(c) Figure 4.4. FRFs of the PZT transducer pair obtained from experiments, uniform model, and stepped beam mode for (a) 1st pair, (b) 2nd pair, and (c) 3rd pair
80
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)
81
(c)
Figure 4.5. First three mode shapes of the beam with PZT transducer pairs: (a) 1st mode, (b) 2nd mode, and (c) 3rd mode
(a)
(b)
82
(c) Figure 4.6. First three modal strain distributions along the beam with PZT transducer pairs: (a) 1 st 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 optimizationbased 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).
83
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.
(a)
(b)
84
(c) Figure 4.7. Variation of the mode shapes as a function of the tip mass for (a) 1 st mode, (b) 2nd 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 2nd 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)
85
(b) 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, 86
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.
87
Chapter 5 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 88
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]:
̈ where
,
, and
(5.1) ̇
are the modal mass, modal damping, and modal stiffness
matrices of the PKM links, respectively.
and
are the
coordinates and the PZT actuator voltages vectors, respectively,
and is the
modal 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 (5.1)). Furthermore, the variable tip
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 (4.17) can be written as:
89
(5.2) ( )
where
∑
is the modal residue of the transfer function, and can be
expressed as:
( )
(
)
( )
[
(
)
( )
(
For (nearly) collocated PZT actuators and sensors, we must have
)
]
(5.3) .
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.
90
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 rth zero and
(
margin of
) is the rth pole, if
. “A negative integral controller in negative feedback, which adds a
constant phase lead of between
(
and
, will yield a loop transfer function whose phase response lies ; that is, the closed-loop system has a highly desirable phase
,” [87].
(a)
(b) Figure 5.1. ( a) IRC scheme proposed in [87], and (b) its equivalent representation.
91
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].
92
Application of the unity feedback with the constant compensator gain on the plant transfer function given by Equation (5.2) results in a stable equivalent plant transfer 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 (5.6) that the natural frequencies of the equivalent plant transfer function are increased to
5.2.3
̂
√
, increasing the system bandwidth.
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
93
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:
( )
( ) ̂ ( )
∑
Comparing Equations (5.6) and (5.7), it is noted that for ̂ function of the proposed IRC scheme, shifted IRC,
(5.7)
̂
, the equivalent transfer
( ), is smaller than those of the resonance-
( ), 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
94
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:
95
(
where
(
( )
)
in terms of
( )
)
(
|
) (
)
(5.8)
|
for the three control systems. The stability margins are expressed
as [121]:
(5.9) ( For the standard IRC scheme (
[
)
]
), the loop gain is given by
Similarly, for resonance-shifted IRC ( the loop gains are expressed by
(
(
)
̂(
) (
), and the proposed IRC schemes ( ̂(
)
)
(
), 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.
96
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
97
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 (5.2)) were 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
41.8%
14.1%
26.7%
65.8%
71.3%
75.5%
Percentage variation
(
)
98
To compare the (i) robust stability, and (ii) vibration attenuation of the proposed IRCbased 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 ensured that the condition
{
( )
̅
, and
( ) } 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 3rd modes due to the placement of the 1st 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 5.6(a-c) shows the closed-loop system obtained from simulations and experiments for the 99
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.
(a)
(b)
(c) 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 100
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 3rd 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.
101
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 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 103
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 resonanceshifted 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 configurationdependent structural vibration of PKM links is satisfied.
Figure 5.9. QFT robust stability of the compared control schemes.
104
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 5.11. The active vibration control system utilized the LabVIEW Real-Time Module [123], [58].The diagram of the control system is shown in Figure 5.12.
105
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 postprocessed 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 realtime code. The output command signal from the controller was amplified using the SS08 power amplifier from SensorTech and applied to the PZT actuator.
106
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
107
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, 2nd, and 3rd set of modes occurs at the frequency ranges of [153-229] Hz, [368-465] Hz, and [12961342] 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 2nd 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.
108
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 for home configuration as an example. The results of Figure 5.14 and 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.
109
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 configurationdependent 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
110
scheme, compared to the existing robust controllers, makes it a viable solution for vibration suppression of the configuration-dependent links of the PKMs.
111
Chapter 6 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.
112
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.
113
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-ofconcept 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 highfidelity 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
114
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 configurationdependent 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 configurationdependent 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
115
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
116
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.
117
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.
118
References
[1]
X. Zhang, J. Mills and W. Cleghorn, "Dynamic Modeling and Experimental Validation of a 3-PRR PArallel Manipulator with Flexible Links," Journal of Intelligent Robotic Systems, vol. 50, pp. 323-340, 2007.
[2]
Y. Altintas, Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, New York: Cambridge University Press, 2000.
[3]
B. Chung, S. Smith and J. Tlusty, "Active Damping of Structural Modes in HighSpeed Machine Tools," Journal of Vibration and Control, vol. 3, no. 3, pp. 279295, 1997.
[4]
P. Mukherjee, B. Dasgupta and A. Malik, "Dynamic Stability Index and Vibration Analysis of a Flexible Stewart Platform," Journal of Sound and Vibration, vol. 307, pp. 495-512, 2007.
[5]
M. Mahboubkhah, M. J. Nategh and S. Esmaeilzadeh Khadem, "A Comprehensive Study on the Free Viration of Machine Tool's Hexapod Table," International Journal of Manufacturing Technology, vol. 40, pp. 1239-1251, 2009.
[6]
J. Chen and W. Hsu, "Dynamic and Compliant Charactristics of a CartesianGuided Tripod Machine," ASME Journal of Manufacturing Science and Engineering, vol. 128, pp. 494-502, 2006.
119
[7]
Z. Zhou, J. Xi and C. K. Mechefske, "Modeling of a Fully Flexible 3PRS Manipulator for Vibration Analysis," Transactions of the ASME journal of Mechanical Design, vol. 128, pp. 403-412, 2006.
[8]
L. Shanzeng, Z. Zhencai, Z. Bin and C. L., "Dynamics of a 3-DOF Spatial Parallel Manipulator with Flexible Links," in IEEE International Conference on Mechanic Automation and Control Engineering (MACE), Wuhan, China, 2010.
[9]
X. Wang and J. Mills, "A FEM Model for Active Vibration Control of Flexible Linkages," in Proc. of the IEEE International Conference on Robotics and Automation (ICRA), New Orleans, LA, USA, 2004.
[10]
A. Gasparetto, "On the Modeling of Flexible-Link Planar Mechanisms: Experimental Validation of an Accurate Dynamic Model," ASME Journal of Dynamic Systems, Measurement, and Control, vol. 126, no. 2, pp. 365-375, 2004.
[11]
R. Katz and Z. Li, "Kinematic and Dynamic Synthesis of a Parallel Kinematic High Speed Drilling Machine," International Journal of Machine Tools and Manufacture, vol. 44, no. 12-13, pp. 1381-1389, 2004.
[12]
X. Wang and J. Mills, "Dynamic Modeling of a Flexible-link Planar Parallel Robot Platform Using a Substructuring Approach," Mechanism and Machine Theory, vol. 41, no. 6, pp. 671-687, 2005.
[13]
Y. Yun and Y. Li, "Modeling and Control Analysis of a 3-PUPU Dual Compliant Parallel Manipulator for Micro Positioning and Active Vibration Isolation," ASME Journal of Dynamic Systems, Measurement and Control, vol. 134, no. 2,
120
pp. 021001-1-9, 2012.
[14]
K. Stachera and W. Schumacher, "Derivation and Calculation of the Dynamics of Elastic Parallel Manipulators," in Automation and Robotics, I-Tech Education and Publishing, 2008.
[15]
X. Zhang, J. Mills and W. Cleghorn, "Coupling Characteristics of Rigid Body Motion and Elastic Deformation of a 3-PRR Parallel Manipulator with Flexible Links," Multibody System Dynamics, vol. 21, pp. 167-192, 2009.
[16]
X. Zhang, J. Mills and W. Cleghorn, "Investigation of Axial Forces on Dynamic Properties of a Flexible 3-PRR Planar Parallel Manipulator Moving With High Speed," Robotica, vol. 28, pp. 607-619, 2010.
[17]
G. Cai, J. Hong and S. Yang, "Dynamic Analysis of a Flexible Hub-Beam System with Tip Mass," Mechanics Research Communications, vol. 32, pp. 173-190, 2005.
[18]
S. Esmaeilzadeh Khadem and A. Pirmohammadi, "Analytical Development of Dynamic Equations of Motion for a Three-Dimensional Flexible Link Manipulator With Revolute and Prismatic Joints," IEEE Transactions on Systems, Man, and Cybernetics. Part B Cybernetics, vol. 33, pp. 237-249, 2003.
[19]
M. Ansari, E. Esmaeilzadeh and N. Jalili, "Exact Frequency Analysis of a Rotating Cantilever Beam With Tip Mass Subjected to Torsional-Bending Vibrations," ASME Journal of Vibration and Acoustics, vol. 133, p. CID: 041003, 2011.
121
[20]
H. Gokdag and O. Kopmaz, "Coupled Bending and Torsional Vibration of a Beam with In-span and Tip Attachments," Journal of Sound and Vibration, vol. 287, pp. 591-610, 2005.
[21]
J. Li and H. Hua, "The Effects of Shear Deformation on the Free Vibration of Elastic Beams With General Boundary Conditions," Proceedings of the Institute of Mechanical Engineering, Part C: Journal of Mechanical Engineering Science, vol. 224, pp. 71-84, 2009.
[22]
J. De Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems-The Real-Time Challenge, New York: Springer-Verlag, 1994.
[23]
S. Dwivedy and P. Eberhard, "Dynamic Analysis of Flexible Manipulators, a Literature Review," Mechanism and Machine Theory, vol. 41, pp. 749-777, 2006.
[24]
R. Milford and S. Asokanthan, "Configuration Dependent Eigenfrequencies for a Two-Link Flexible Manipulator: Experimental Verification," Journal of Sound and Vibration, vol. 222, no. 2, pp. 191-207, 1999.
[25]
K. Morris and K. Taylor, "Variational Calculus Approach to the Modelling of Flexible Manipulators," Society for Industrial and Applied Mechanics (SIAM), vol. 38, no. 2, pp. 294-305, 1996.
[26]
E. Barbieri and U. Ozguner, "Unconstrained and Constrained Mode Expansion for a Flexible Slewing Link," in American Control Conference, Atlanta, GA, USA, 1988.
122
[27]
L. Meirovitch, Principles and Techniques of Vibrations, Prentice-Hall Inc., 1997.
[28]
L. Lopez de Lacalle and A. Lamikiz, Mchine Tools for High Performance Machining, London: Springer-Verlag, 2009.
[29]
G. Wiens and D. Hardage, "Structural Dynamics and System Identification of Parallel Kinematic Machines," in Proc. of IDETC/CIE ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Philadelphia, 2006.
[30]
K. Cheng, Machining Dynamics: Fundamentals, Applications and Practices, London: Springer, 2009.
[31]
B. Thomas, C. Helene, B. Belhassen-Chedli and R. Pascal, "Dynamic Analysis of the Tripteor X7: Model and Experiments," in Proceesing of IDMME-Virtual Concept, Bordeaux, France, 2010.
[32]
M. Law, Y. Altintas and A. Srikantha Phani, "Rapid Evaluation and Optimization of Machine Tools with Position-Dependent Stability," International Journal of Machine Tools and Manufacture, vol. 68, pp. 81-90, 2013.
[33]
C. Henninger and P. Eberhard, "Computation of Stability Diagrams for Milling Processes with Parallel Kinematic Machine Tools," Proc. of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, vol. 223, pp. 117-129, 2009.
123
[34]
X. Wang and J. Mills, "Dual-Modal Control of Configuration-Dependent Linkage Vibration in a Smart Parallel Manipulator," in Proceedings of the IEEE International Conference on Robotics and Automation, Orlando, Florida, 2006.
[35]
A. Cammarata, "On the Stiffness Analysis and Elastodynamics of Parallel Kinematic Machines," in Serial and Parallel Robot Manipulators - Kinematics, Dynamics, Control and Optimization, InTech, 2012.
[36]
Z. Zhou, C. Mechefske and F. Xi, "Nonstationary Vibration of a Fully Flexible Parallel Kinematic Machine," Transactions of the ASME Journal of Vibration and Acoustics, vol. 129, pp. 623-630, 2007.
[37]
J. Corral, C. Pinto, M. Urizar and V. Petuya, "Structural Dynamic Analysis of Low-Mobility Parallel Manipulators," Mechanism and Machine Science, vol. 5, pp. 387-394, 2010.
[38]
J. Kim, F. Park, S. Ryu, J. Kim, J. Hwang, C. Park and C. Iurascu, "Design and Analysis of a Redundantly Actuated Parallel Mechanism for Rapid Machining," IEEE Transactions on Robotics and Automation, vol. 17, no. 4, pp. 423-434, 2001.
[39]
A. Moosavian and F. Xi, "Design and Analysis of Reconfigurable Parallel Robotics with Enhanced Stiffness," Mechanism and Machine Theory, vol. 77, pp. 92-110, 2014.
[40]
D. Chakarov, "Study of the Antagonistic Stiffness of Parallel Manipulators with Actuation Redundancy," Mechanism and Machine Theory, vol. 39, no. 6, pp.
124
583-601, 2004.
[41]
J. Kotlaraski, B. Heimann and T. Ortmaier, "Influence of Kinematic Redundancy on the Singularity-Free Workspace of Parallel Kinematic Machines," Frontiers of Mechanical Engineering, vol. 7, no. 2, pp. 120-134, 2012.
[42]
J.-P. Merlet, "Redundant Parallel Manipulators," Laboratory Robotics and Automation , vol. 8, no. 1, pp. 17-24, 1996.
[43]
D. Zlatanov, R. Fenton and B. Benhabib, "Analysis of the Instantaneous Kinematics and Singular Configurations of Hybrid-Chain Manipulators," in Proceedings of the ASME 33rd Biennial Mechanisms Conference, Minneapolis, NM, DE-70, 1994.
[44]
D. Zlatanov, R. Fenton and B. Benhabib, "Identification and Classification of the Singular Configurations of Mechanisms," Mechanism and Machine Theory, vol. 33, no. 6, pp. 743-760, 1998.
[45]
C. Pinto, J. Corral, S. Herrero and B. Sandru, "Vibratory Dynamic Behaviour of Parallel Manipulators in their Workspace," in 13th World Congress in Mechanism and Machine Science, Guanajuato, Mexico, 2011.
[46]
A. Ghazavi, F. Gardanine and N. Chalhout, "Dynamic Analysis of a Composite Material Flexible Robot Arm," Computers and Structures, vol. 49, pp. 315-325, 1993.
[47]
C. Sung and B. Thompson, "Material Selection: An Important Parameter in the Design of High-Speed Linkages," Mechanism and Machine Theory, vol. 19, pp. 125
389-396, 1984.
[48]
X. Zhang, J. Mills and W. Cleghorn, "Flexible Linkage Strcutural Vibration Control on a 3-PRR Parallel Manipulator: Experimental Results," Proceedings of the Institution of Mechanical Engineering, Part I: Journal of Systems and Control Engineering, vol. 223, pp. 71-84, 2009.
[49]
X. Zhang, Dynamic Modeling and Active Vibration Control of a Planar 3-PRR Parallel Manipulator With Three Flexible Links, Toronto: Ph.D. Thesis, University of Toronto, 2009.
[50]
L. Kunquan and W. Rui, "Active Vibration Isolation of 6-RSS Parallel Mechanism Using Integrated Force Feedback Controller," in IEEE Third International Conference on Measuring Technology and Mechatronics Automation, Shanghai, China , 2011.
[51]
A. Ast and P. Eberhad, "Active Vibration Control for a Machine Tool with Parallel Kinematics and Adaptronic Actuators," Journal of Computational and Nonlinear Dynamics, vol. 4, p. CID: 031004, 2009.
[52]
A. Ast, S. Braun, P. Eberhad and U. Heisel, "An Adaptronic Approach to Active Vibration Control of Machine Tools with Parallel Kinematics," Production Engineering, vol. 3, pp. 207-215, 2008.
[53]
Y. Yun and Y. Li, "Active Vibration Control Based on a 3-DOF Dual Compliant Parallel Robot Using LQR Algorithm," in IEEE/RSJ International Conference on Intelligent Robots and Sysyems, 2009.
126
[54]
S. Algermissen, M. Rose, R. Keimer and E. Breitback, "High-Speed Parallel Robots with Integrated Vibration-Suppression for Handling and Assembly," Proc. of the SPIE, Smart Structures and Materials, 2004.
[55]
M. Kermani, R. Patel and M. Moallem, "Multi-Directional Stabilization of A Large-Scale Robotic Manupulator," in IEEE International Conference on Robotics and Automation, Orlando, Florida, 2006.
[56]
M. R. Kermani, R. V. Patel and M. Moallem, "Multimode Control of a LargeScale Robotic Manipulator," IEEE Transactions on Robotics, vol. 23, no. 6, 2007.
[57]
R. Neugebauer, V. Wittstock, A. Bucht and A. Illgen, "Active Component and Control Design for Torsional Mode Vibration Reduction for a Parallel Kinematic Machine Tool Structure," Proc. of SPIE, Industrial and Commercial Applications of Smart Structures Technologies, vol. 6930, p. 69300F, 2008.
[58]
X. Zhang, J. Mills and W. Cleghorn, "Experimental Implementation on vibration Mode Control of a Moving 3-PRR Flexible Parallel Manipulator with Multiple PZT Transducers," Journal of Vibration and Control, vol. 16, no. 13, pp. 20352054, 2010.
[59]
X. Zhang, J. Mills and W. Cleghorn, "Vibration Control of Elastodynamic Respons of a 3-PRR Flexible Parallel Manipulator Using PZT Transducers," Robotica, vol. 26, no. 5, pp. 655-665, 2008.
[60]
J. Niu, A. Y. T. Leung and P. Q. Ge, "An Active Vibration Control Model for Coupled Flexible Systems," Journal of Mechanical Engineering Science, vol.
127
222, pp. 2087-2098, 2008.
[61]
J. J. Liu and B. Liaw, "Effiiciency of Active Control of Beam Vibration Using PZT Patches," in Proc. of the SEM X International Congress and Exposition on Experimental and Applied Mechanics, Costa Mesa, CA, USA, 2004.
[62]
N. Jalili, Piezoelectric-Based Vibration Control, From Macro to Micro/Nano Scale Systems, New York: Springer, 2010.
[63]
A. Salehi-Khojin, S. Bashash and N. Jalili, "Modeling and Experimental Vibration Analysis of Nanomechanical Cantilever Active Probes," Journal of Micromechanics and Microengineering, vol. 18, p. CID: 085008 , 2008.
[64]
N. Maxwell and S. Asokanthan, "Modal Characteristics of a Flexible Beam With Multiple Distributed Actuators," Journal of Sound and Vibration, vol. 269, pp. 19-31, 2004.
[65]
D. Sun, J. Mills, J. Shan and S. Tso, "A PZT Actuator Control of a Single-Link Flexible Manipulator Based on Linear Velocity Feedback and Actuator Placement," Mechatronics, vol. 14, pp. 381-401, 2004.
[66]
E. Crawley and J. de Luis, "Use of Piezoelectric Actuators as Elements of Intelligent Structures," AIAA Journal, vol. 25, no. 10, pp. 1373-1385, 1987.
[67]
Q. Wang and C. Wang, "A Controllability Index for Optimal Design of Piezoelectric Actuators in Vibration Control of Beam Structures," Journal of Sound and Vibration, vol. 242, no. 3, pp. 507-518, 2001.
128
[68]
Q. Wang and C. Wang, "Optimal Placement and Size of Piezoelectric Patches on Beams From the Controllability Perspective," Smart Materials and Structures, vol. 9, pp. 558-567, 2000.
[69]
M. Kermani, M. Moallem and R. Patel, "Parameter Selection and Control Design for Vibration Suppression Using Piezoelectric Transducers," Control Engineering Practice, vol. 12, pp. 1005-1015, 2004.
[70]
Z. Qiu, X. Zhang, H. Wu and H. Zhang, "Optimal Placement and Active Vibration Control for Piezoelectric Smart Flexible Cantilever Plate," Journal of Sound and Vibration, vol. 301, pp. 521-543, 2007.
[71]
A. Jha and D. Inman, "Optimal Sizes and Placements of Piezoelectric Actuators and Sensors for an Inflated Torus," Journal of Intelligent Material Systems and Structures, vol. 14, pp. 563-576, 2003.
[72]
B. Dunn and E. Garcia, "Optimal Placement of a Proof Mass Actuator for Active Strcutural Acoustic Control," Mechanics of Structures and Machines: An International Journal, vol. 27, pp. 23-25, 2007.
[73]
V. Gupta, M. Sharma and N. Thakur, "Optimization Criteria for Optimal Placement of Piezoelectric Sensors and Actuators on a Smart Structure: A Technical Review," Journal of Intelligent Material Systems and Structures, vol. 21, pp. 1227-1242, 2010.
[74]
Z. Luo, "Direct Strain Feedback Control of Flexible Robot Arms: New Theoretical and Experimental Results," IEEE Transactions on Automatic Control,
129
vol. 38, no. 11, pp. 1610-1622, 1993.
[75]
J. Fanson and T. Caughey, "Positive Position Feedback Control for Large Space Structures," American Institutue of Aeronautics and Astronautics, vol. 28, no. 4, pp. 717-724, 1990.
[76]
L. Meirovitch, Dynamics and control of structures, Canada: John Wiley & Sons, Inc., 1990.
[77]
G. Natal, A. Chemori and F. Pierrot, "Nonlinear Control of Parallel Manipulators For Very High Accelerations Without Velocity Measurement: Stability Analysis and Experiments on Par2 Parallel Manipulator," pp. 1-28, 2014.
[78]
X. Zhang, J. Mills and W. Cleghorn, "Multi-mode Vibration Control and Position Error Analysis of Parallel Manipulator with Multiple Flexible Links," Transactions of the Canadian Society for Mechanical Engineering, vol. 34, no. 2, pp. 197-213, 2010.
[79]
X. Wang, J. Mills and S. Guo, "Experimental Identification and Active Control of Configuration-Dependent Linkage Vibration in a Planar Parallel Robot," IEEE Transtactions on Control Systems Technology, vol. 17, no. 4, pp. 960-969, 2009.
[80]
S. Algermissen, R. Keimer, M. Rose, M. Straubel, M. Sinapius and H. Monner, "Smart-Structures Technology for Parallel Robots," Journal of Intelligent Robotics Systems, vol. 63, pp. 547-574, 2011.
[81]
S. Karande, P. Nataraj and M. Deshpande, "Control of Parallel Flexible Five Bar Manipulator Using QFT," in IEEE International Conference on Industrial 130
Technology (ICIT), Gippsland, Australia, 2009.
[82]
S.-B. Choi, "Vibration Control of a Smart Beam Structure Subjected to Actuator Uncertainty: Experimental Verification," Acta Mechanica, vol. 181, pp. 19-30, 2006.
[83]
S. Choi, S. Cho, H. Shin and H. Kim, "Quantitative Feedback Theory Control of a Single-Link Flexible Manipulator Featuring Piezoelectric Actuator and Sensor," Smart Materials and Structures, vol. 8, pp. 338-349, 1999.
[84]
S. Choi, M. Seong and S. Ha, "Accurate Position Control of a Flexible Arm Using Piezoactuator Associated With a Hysteresis Compensator," Smart Materials and Structures, vol. 22, p. CID: 045009, 2013.
[85]
M. Kerr, S. Jayasuriya and S. Asokanthan, "QFT Based Robust Control of a Single-Link Flexible Manipulator," Journal of Vibration and Control, vol. 13, no. 1, pp. 3-27, 2007.
[86]
C. Houpis, S. Rasmussen and M. Garcia-Sanz, Quantitative Feedback Theory: Fundamentals and Applications, Boca Raton: CRC/Taylor & Francis, 2006.
[87]
S. S. Aphale, Andrew J. Fleming and S. R. Moheimani, "Integral Resonant Control of Collocated Smart Structures," Smart Materials and Structures, vol. 16, pp. 439-446, 2007.
[88]
E. Pereira, S. Aphale, V. Feliu and S. Moheimani, "Integral Resonant Control for Vibration Damping and Precise Tip-Positioning of a Single-Link Flexible Manipulator," IEEE/ASME Transactions on Mechatronics, vol. 16, no. 2, pp. 131
232-240, 2011.
[89]
B. Bhikkaji, S. Moheimani and I. R. Petersen, "A Negative Imaginary Approach to Modeling and Control of a Collocated Structure," IEEE/ASME Transactions on Mechatronics, vol. 17, no. 4, pp. 717-727, 2012.
[90]
M. Namavar, A. J. Fleming and S. Aphale, "Resonance-Shifting Integral Resonant Control Scheme for Increasing the Positioning Bandwidth of Nanopositioners," in European Control Conference (ECC), Zurich, Switzerland, 2013.
[91]
M. Mahmoodi, J. Mills and B. Benhabib, "Structural Vibration Modeling of a Novel Parallel Mechanism-Based Reconfigurable meso-Milling Machine Tool (RmMT)," in 1st International Conference on Virtual Machining Process Technology, Montreal, Canada, 2012.
[92]
M. Mahmoodi, J. Mills and B. Benhabib, "Vibration Modeling of Parallel Kinematic Mechanisms (PKMs) With Flexible Links: Admissible Shape Functions," Under review in: Transactions of the Canadian Society for Mechanical Engineering, 2014.
[93]
L. Meirovitch, Fundamentals of vibrations, McGraw-Hill, 2000.
[94]
M. Mahmoodi, Y. Le, J. Mills and B. Benhabib, "An Active Dynamic Model for a Parallel-Mechanism-Based meso-Milling Machine Tool," in 23rd Canadian Congress of Applied Mechanics (CANCAM), Vancouver, Canada, 2011.
132
[95]
A. Le, J. Mills and B. Benhabib, "Dynamic Modeling and Control Design for A Parallel-Mechanism-Based meso-Milling Machine Tool," Robotica, vol. 32, no. 4, pp. 515-532, 2014.
[96]
L. Mi, G. Yin, M. Sun and X. Wang, "Effects of Preloads on Joints on Dynamic Stiffness of a Whole Machine Tool Structure," Journal of Mechanical Science and Technology, vol. 26, pp. 495-508, 2012.
[97]
A. Iglesias, J. Munoa and J. Ciurana, "Optimization of Face Milling Operations With Structural Chatter Using a Stability Model Based Process Planning Methodology," International Journal of advanced Manufacturing Technology, vol. 70, pp. 559-571, 2014.
[98]
M. Law, A. Srikantha Phani and Y. Altintas, "Position-Dependent Multibody Dynamic Modeling of Machine Tools Based on Improved Reduced Order Models," ASME Journal of Manufacturing Science and Engineering, vol. 135, p. CID: 021008, 2013.
[99]
H. Azulay, M. Mahmoodi, R. Zhao, J. Mills and B. Benhabib, "Comparative Analysis of A New 3×PPRS Parallel Kinematic Mechanism," Robotics and Computer-Integrated Manufacturing, vol. 30, no. 4, pp. 369-378, 2014.
[100] G. Zhao, "Design, Analysis, and Prototyping of A 3PPRS Parallel Kinematic Mechanism for meso-Milling," Master's Thesis, 2013.
[101] M. Volgar, X. Liu, S. Kapoor and R. DeVor, "Development of meso-Scale Machine Tool (mMT) Systems," Transactions of NAMRI/SME, vol. 30, pp. 653-
133
661, 2002.
[102] S. Lee, R. Mayor and J. Ni, "Dynamic Analysis of a Mesoscale Machine Tool," ASME Journal of Manufacturing Science and Engineering, vol. 128, pp. 194-203, 2006.
[103] M. Mahmoodi, J. Mills and B. Benhabib, "Configuration-Dependency of Structural Vibration Response Amplitudes in Parallel Kinematic Mechanisms," in 2nd International Conference on Virtual Manufacturing Process Technology, Hamilton, Canada, 2013.
[104] R. Zhao, H. Azulay, M. Mahmoodi, J. Mills and B. Benhabib, "Analysis of 6-dof 3×PPRS Parallel Kinematic Mechanisms for meso-Milling," in 2nd International Conference on Virtual Machining Process Technology (VMPT 2013), Hamilton, Canada, 2013.
[105] R. Alizade, N. Tagiyev and J. Duffy, "A Forward and Reverse Displacement Analysis of a 6-DOF In-Parallel Manipulator," Mechanism and Machine Theory, vol. 29, pp. 115-124, 1994.
[106] D. Glozman and M. Shoham, "Novel 6-DOF Parallel Manipulator With Large Workspace," Robotica, vol. 27, pp. 891-895, 2009.
[107] A. Alagheband, R. Zhao, M. Mahmoodi, J. Mills and B. Benhabib, "Analysis of a Kinematically-Redundant Pentapod for meso-Milling," in 3rd International Conference on Virtual Machining Process Technology, Calgary, Alberta, Canada, 2014.
134
[108] A. Alagheband, M. Mahmoodi, J. Mills and B. Benhabib, "Comparative Analysis of A Redundant Pentapod Parallel Kinematic Machine," Under review in the "ASME Journal of Mechanisms and Robotics", 2014.
[109] M. Luces, P. Boyraz, M. Mahmoodi, J. Mills and B. Benhabib, "Trajectory Planning for Redundant Parallel-Kinematic-Mechanisms," in 3rd International Conference on Virtual Machining Process Technology, Calgary, Alberta, Canada, 2014.
[110] P. Inc., Catalog #8, Woburn, MA, 2011.
[111] S. Bashash, A. Salehi-Khojin and N. Jalili, "Forced Vibration Analysis of Flexible Euler-Bernoulli Beams with Geometrical Discontinuities," in American Control Conference, Seattle, Washington, USA, 2008.
[112] J. Dosch, D. Inman and E. Garcia, "A Self-Sensing Piezoelectric Actuator for Collocated Control," Journal of Intelligent Material Systems and Structures, vol. 3, pp. 166-185, 1992.
[113] M. Mahmoodi, J. Mills and B. Benhabib, "Controllability of PiezoelectricallyActuated Links of Parallel Kinematic Mechanisms," in 3rd International Conference on Vitural Machining Process Technology, Calgary, Alberta, Canada, 2014.
[114] D. Inman, Vibration with Control, Chichester, West Sussex, England: John Wiley & Sons Ltd., 2006.
135
[115] A. Erturk and D. Inman, Piezoelectric Energy Harvesting, Chichester, West Sussex, United Kingdom: John Wiley & Sons, Ltd, 2011.
[116] M. Mahmoodi, J. Mills and B. Benhabib, "A Modified Integral Resonant Control Scheme for Vibration Suppression of Parallel Kinematic Mechanisms With Flexible Links," Submitted to "Smart Materials and Structures", 2014.
[117] A. Preumont, Vibration Control of Active Structures, Berlin: Springer Netherlands, 2011.
[118] E. Pereira, S. Moheimani and S. Aphale, "Analog Implementation of an Integral Resonant Control Scheme," Smart Materials and Structures, vol. 17, pp. 1-6, 2008.
[119] A. Al-Mamun, E. Keikha, C. Singh Bhatia and T. Heng Lee, "Integral Resonant Control for Suppression of Resonance in Piezoelectric Micro-Actuator Used in Precision Servomechanism," Mechatronics, vol. 23, pp. 1-9, 2013.
[120] I. Horowitz and M. Sidi, "Synthesis of Feedback Systems With Large Plant Ignorance For Prescribed Time Domain Tolerances," International Journal of Control, vol. 16, pp. 287-309, 1972.
[121] M.-S. Tsai, Y.-S. Sun and C.-H. Liu, "Robust Control of Novel Pendulum-Type Vibration Isolation System," Journal of Sound and Vibration, vol. 330, pp. 43844398, 2011.
136
[122] M. Garcia-Sanz, A. Mauch and C. Philippe, "The QFT Control Toolbox (QFTCT) for MATLAB," CWRU, UPNA and ESA-ESTEC, Version 4.20, October 2013.
[123] N. Instruments, "Getting Started with the LabVIEW Real-Time Module," 2012. [Online]. Available: http://digital.ni.com/manuals.nsf/websearch/8EB98552B3EC7474862579F80083 D16E. [Accessed 1 June 2014].
137
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:
(
)
[
138
]
(A.3)
Appendix B Partitioned Matrix and Vector Representations for Active and Passive Joints
Inertia matrix (active coordinates): (
)
[
Stiffness matrix (active coordinates): [
)
[
]
(B.3)
Jacobian for active coordinates: (
)
(B.4)
(
)
(B.5)
Jacobian for passive coordinates:
Inertial and gravity forces on active coordinates: [
(B.2)
]
Inertia matrix (passive coordinates):
(
(B.1)
]
(
]
̇ ) ̇
(
)
(B.6)
Inertial and gravity forces on passive coordinates: [
(
]
139
̇ ) ̇
(
)
(B.7)
