Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Zarif Mansour, Sepehr
Doctor of Philosophy - PhD
University of British Columbia
Piezoelectric actuators are commonly used in micro-positioning applications. However, they exhibit an inherent nonlinear relationship between their driving voltage and displacement, which is called hysteresis. To compensate for hysteresis, conventional control approaches use dedicated sensors that are often bulky and expensive. The main objective of this thesis is developing techniques to minimize the need for direct use of external sensors in micro-positioning work spaces. These techniques are categorized into two groups: 1- integrating actuators with dedicated sensors employed outside the actuators' work-space and 2- self-sensing, which is using an actuator simultaneously as a sensor. The first group consists of two techniques. The first technique moves the sensors from within a micro-positioning work-space to a location outside the work-space. The technique is experimentally validated, where about 1% average absolute errors are measured for both displacement and force. The second technique constructs strain gauges from top and bottom electrodes of a piezoelectric bender and uses them to measure the bender's tip displacement. The validation experiments show about 1% measurement error. The second group consists of three techniques. The first technique uses voltage and charge measurements of a piezoelectric actuator to estimate its displacement and force. The validation experiments show about 4% average absolute errors measured for both displacement and forces. The second technique refines the first technique by using a nonlinear capacitance and a novel exponential hysteresis operator. The experimental validations show about 2% and 3% average absolute errors for displacement and force respectively. The third technique is based on impedance measurement of a piezoelectric actuator. The presented technique measures the impedance at a mechanical resonance frequency and maps it to the displacement and force of the actuator. The experimental validation show about 3.7% and 2.4% average absolute errors for displacement force respectively. The presented techniques in this study will help future researchers design simple and economic micro-positioners with sensor-free work-spaces. The first group of the presented techniques is useful, when the design goal is to make compact micro-positioners with optimized work-spaces. The second group is useful, when dedicated sensors are unavailable due to work-space or financial restrictions.
Sensor-actuator integration andself-sensing techniques for spaceconstrained piezoelectric drivenmicro-positionersbySepehr Zarif MansourB.A.Sc., University of Tehran, Iran, 2013M.A.Sc., The University of British Columbia, Canada, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe College of Graduate Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)September 2019c© Sepehr Zarif Mansour, 2019The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a thesis/dissertation en-titled:Sensor-actuator integration and self-sensing techniques for space con-strained piezoelectric driven micro-positionerssubmitted by Sepehr Zarif Mansour in partial fulfilment of the require-ments of the degree ofDoctor of PhilosophySupervisor Dr. Rudolf Seethaler, School of Engineering, UBC OkanaganSupervisory Committee Member Dr. Abbas Milani, School of Engineering, UBCOkanaganSupervisory Committee Member Dr. Homayoun Najjaran, School of Engineering,UBC OkanaganUniversity Examiner Dr. Thomas Johnson, School of Engineering, UBC OkanaganExternal Examiner Dr. Ridha Ben Mrad, University of TorontoiiAbstractPiezoelectric actuators are commonly used in micro-positioning appli-cations. However, they exhibit an inherent nonlinear relationship betweentheir driving voltage and displacement, which is called hysteresis. To com-pensate for hysteresis, conventional control approaches use dedicated sensorsthat are often bulky and expensive. The main objective of this thesis is de-veloping techniques to minimize the need for direct use of external sensorsin micro-positioning work spaces. These techniques are categorized into twogroups: 1- integrating actuators with dedicated sensors employed outsidethe actuators’ work-space and 2- self-sensing, which is using an actuatorsimultaneously as a sensor.The first group consists of two techniques. The first technique moves thesensors from within a micro-positioning work-space to a location outside thework-space. The technique is experimentally validated, where about 1% av-erage absolute errors are measured for both displacement and force. Thesecond technique constructs strain gauges from top and bottom electrodesof a piezoelectric bender and uses them to measure the bender’s tip displace-ment. The validation experiments show about 1% measurement error.The second group consists of three techniques. The first technique usesvoltage and charge measurements of a piezoelectric actuator to estimateits displacement and force. The validation experiments show about 4%average absolute errors measured for both displacement and forces. Thesecond technique refines the first technique by using a nonlinear capacitanceand a novel exponential hysteresis operator. The experimental validationsshow about 2% and 3% average absolute errors for displacement and forcerespectively. The third technique is based on impedance measurement ofa piezoelectric actuator. The presented technique measures the impedanceat a mechanical resonance frequency and maps it to the displacement andforce of the actuator. The experimental validation show about 3.7% and2.4% average absolute errors for displacement force respectively.The presented techniques in this study will help future researchers designsimple and economic micro-positioners with sensor-free work-spaces. Thefirst group of the presented techniques is useful, when the design goal is toiiiAbstractmake compact micro-positioners with optimized work-spaces. The secondgroup is useful, when dedicated sensors are unavailable due to work-spaceor financial restrictions.ivLay SummaryMotion control in micrometer scales is a challenging task due to the re-quired stringent accuracies. Conventional sensors that can help to achievesuch accuracies are often bulky and expensive. The main goal of this study isto minimize the direct use of sensors in piezoelectric micro-positioning work-spaces. Piezoelectric actuators are popular devices in micro-positioning ap-plications, since they are very fast, very stiff and they can achieve nanometeraccuracies required.This thesis presents two groups of techniques aimed at reducing theimpact of conventional sensors. The first group moves the sensors fromwithin the micro positioning work space to a location outside this workspace. Thus, an indirect measurement of the actuator’s motion is providedand the micro-positioning work-space is freed from external sensors. Thesecond group uses a piezoelectric actuator simultaneously as a sensor andcompletely eliminates the need for sensors.vPrefaceA version of Figure 2.9 has been published. c© [ 2017], IEEE. Reprinted,with permission, from Sepehr Zarif Mansour and Rudolf J Seethaler. Si-multaneous displacement and force estimation of piezoelectric stack actu-ators using charge and voltage measurements. IEEE/ASME Transactionson Mechatronics, 22(6):2619-2624, 2017. I wrote the manuscript which wasfurther edited by Dr. Seethaler.A version of Chapter 3, has been published. c© [ 2017], IEEE. Reprinted,with permission, from Sepehr Zarif Mansour and Rudolf Seethaler. Designof a test setup to independently vary displacement and force of piezoelectricstack actuators. In 2017 IEEE SENSORS, pages 1-3. IEEE, 2017. I wrotethe manuscript which was further edited by Dr. Seethaler.A version of Chapter 3, (Sections 3.1, 3.2, and 3.3), has been published.c© [ 2017], IEEE. Reprinted with permission, from Sepehr Zarif Mansourand Rudolf J Seethaler. Simultaneous displacement and force estimationof piezoelectric stack actuators using charge and voltage measurements.IEEE/ASME Transactions on Mechatronics, 22(6):2619-2624, 2017. I wrotethe manuscript which was further edited by Dr. Seethaler.A version of Chapter 4, (Section 4.1), has been published. c© [ 2017],IEEE. Reprinted with permission, from Sepehr Zarif Mansour and RudolfJ Seethaler. Simultaneous displacement and force estimation of piezoelec-tric stack actuators using charge and voltage measurements. IEEE/ASMETransactions on Mechatronics, 22(6):2619-2624, 2017. I wrote the manuscriptwhich was further edited by Dr. Seethaler.A version of Chapter 5 has been published. c© [ 2018], IEEE. Reprintedwith permission, from Sepehr Zarif Mansour, Rudolf J Seethaler, Yik RTeo, Yuen Kuan Yong, and Andrew J Fleming. Piezoelectric bimorph ac-tuator with integrated strain sensing electrodes. IEEE Sensors Journal,18(14):5812-5817, 2018. I wrote the manuscript which was further edited byDr. Fleming, Dr. Seethaler, Dr. Teo, and Dr. Yong.A version of Chapter 6 has been published. c© [ 2017], IEEE. Reprintedwith permission, from Sepehr Zarif Mansour and Rudolf J Seethaler. Si-multaneous displacement and force estimation of piezoelectric stack actu-viPrefaceators using charge and voltage measurements. IEEE/ASME Transactionson Mechatronics, 22(6):2619-2624, 2017. I wrote the manuscript which wasfurther edited by Dr. Seethaler.A version of Chapter 7 has been published. c© [ 2019], IEEE. Reprintedwith permission, from Sepehr Zarif Mansour and Rudolf Seethaler. Dis-placement and force self-sensing technique for piezoelectric actuators usinga nonlinear constitutive model. IEEE Transactions on Industrial Electron-ics, 2019. I wrote the manuscript which was further edited by Dr. Seethaler.A version of Chapter 7, (Subsection 7.2.4), has been published. c© [2018], IEEE. Reprinted with permission, from Sepehr Zarif Mansour, RudolfSeethaler, and Andrew Fleming. A simple asymmetric hysteresis model fordisplacementforce control of piezoelectric actuators. In 2018 IEEE/ASMEInternational Conference on Advanced Intelligent Mechatronics (AIM), pages87-90. IEEE, 2018. I wrote the manuscript which was further edited by Dr.Seethaler.A version of Chapter 8 has been published. Reprinted with permission,from Sepehr Zarif Mansour and Rudolf Seethaler. Simultaneous quasi-staticdisplacement and force self-sensing of piezoelectric actuators by detectingimpedance. Sensors and Actuators A: Physical, 274:272-277, 2018. I wrotethe manuscript which was further edited by Dr. Seethaler.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . xxAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . .xxiiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Contributions of the proposed Research . . . . . . . . . . . . 3Chapter 2: Literature review . . . . . . . . . . . . . . . . . . . . 62.1 Piezoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Synthetic Piezoelectric Ceramics . . . . . . . . . . . . . . . . 72.3 Piezoelectric actuators . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Piezoelectric bender actuators . . . . . . . . . . . . . . 92.3.2 Piezoelectric stack actuators . . . . . . . . . . . . . . 102.3.3 Piezoelectric tube actuators . . . . . . . . . . . . . . . 112.4 Piezoelectric actuator modeling challenges . . . . . . . . . . 112.4.1 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 122.4.2 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . 13viiiTABLE OF CONTENTS2.4.3 Vibration . . . . . . . . . . . . . . . . . . . . . . . . . 142.4.4 Temperature change . . . . . . . . . . . . . . . . . . . 152.5 Modeling piezoelectric actuators . . . . . . . . . . . . . . . . 152.5.1 Piezoelectric standard constitutive relations . . . . . . 152.5.2 Linear electromechanical model of piezoelectric actu-ators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5.3 Non-linear electromechanical model of piezoelectric ac-tuators . . . . . . . . . . . . . . . . . . . . . . . . . . 182.6 Control techniques for piezoelectric actuators . . . . . . . . . 192.6.1 Feed-forward control . . . . . . . . . . . . . . . . . . . 192.6.2 Feed-back control . . . . . . . . . . . . . . . . . . . . . 252.6.3 Integrated sensors and actuators . . . . . . . . . . . . 262.6.4 Self-sensing . . . . . . . . . . . . . . . . . . . . . . . . 282.7 A comparison between the reviewed control structures . . . . 47Chapter 3: The design of a 2 DoF test setup . . . . . . . . . . 483.1 Setup design . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.2 Frequency response analysis . . . . . . . . . . . . . . . . . . . 523.3 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 Operational region . . . . . . . . . . . . . . . . . . . . . . . . 563.5 Sanity check . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Chapter 4: Integrating position and force sensors in spaceconstrained piezoelectric driven micro-positioners 594.1 Validation of the hysteresis voltage and charge relationship . 594.2 The proposed sensor-actuator integration technique . . . . . . 614.3 The experimental setup . . . . . . . . . . . . . . . . . . . . . 634.3.1 Mapping between driving voltages, force, and displace-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . 65Chapter 5: Piezoelectric bimorph actuator with integratedstrain sensing electrodes . . . . . . . . . . . . . . . 695.1 Tip displacement and surface-Strain relationship in benders . 695.2 Surface resistance change measurement . . . . . . . . . . . . . 715.3 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Validation of the proposed method . . . . . . . . . . . . . . . 755.5 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 775.5.1 Damping-integral controller . . . . . . . . . . . . . . . 775.6 Performance of the closed-loop system . . . . . . . . . . . . . 79ixTABLE OF CONTENTSChapter 6: A new self-sensing technique using charge andvoltage measurements . . . . . . . . . . . . . . . . . 816.1 Parameter identification . . . . . . . . . . . . . . . . . . . . . 816.2 Experimental validation of the self-sensing technique . . . . . 84Chapter 7: A modified electromechanical model for piezo-electric actuators . . . . . . . . . . . . . . . . . . . . 887.1 The effect of applied force and voltage on capacitance . . . . 887.1.1 The experimental setup . . . . . . . . . . . . . . . . . 887.1.2 The relationship between capacitance, voltage, and force 897.2 A new self-sensing technique for piezoelectric actuators . . . . 917.2.1 A new electro-mechanical model for piezoelectric ac-tuators . . . . . . . . . . . . . . . . . . . . . . . . . . 917.2.2 Displacement and force self-sensing . . . . . . . . . . . 927.2.3 A new parameter identification method . . . . . . . . 927.2.4 A simple asymmetric hysteresis model for displacement-force control of piezoelectric actuators . . . . . . . . . 957.3 Sensor-less force and displacement controller using the newself-sensing technique . . . . . . . . . . . . . . . . . . . . . . . 97Chapter 8: Self-sensing technique using impedance measure-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . 1018.1 The effect of force on the impedance of a piezoelectric actuator1018.2 Real time impedance measurement method . . . . . . . . . . 1048.3 Force and displacement detection . . . . . . . . . . . . . . . . 1068.4 Experimental validation . . . . . . . . . . . . . . . . . . . . . 1088.5 Controller design . . . . . . . . . . . . . . . . . . . . . . . . . 109Chapter 9: Conclusions and future works . . . . . . . . . . . . 1149.1 Research contributions and Conclusions . . . . . . . . . . . . 1149.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121xList of TablesTable 3.1 Reference values for PESA-1 and PESA-2 characteris-tic parameters. [Phy05] . . . . . . . . . . . . . . . . . 53Table 4.1 Displacement and force estimation average absoluteerror percentages . . . . . . . . . . . . . . . . . . . . . 67Table 5.1 Parameters of the damping-integral controller. . . . . . 78Table 6.1 Reference and identified characteristic parameters ofPESA-1. . . . . . . . . . . . . . . . . . . . . . . . . . . 83Table 7.1 Model parameters comparison. . . . . . . . . . . . . . 95Table 7.2 Identified parameters for the presented hysteresis model. 98Table 7.3 Performance of the proposed technique in comparisonwith some reported techniques in the literature. . . . . 100Table 9.1 A qualitative comparison of the presented techniquesin this thesis. . . . . . . . . . . . . . . . . . . . . . . . 118xiList of FiguresFigure 1.1 A one degree of freedom micro-positioning stage. Usedwith permission from [FL14]. . . . . . . . . . . . . . . 2Figure 2.1 A model for The lattice structure of quartz crystal(SiO2) proposed by Lord Kelvin in 1893 to explainthe piezoelectricity effect. (a) In an equilibrium state,centers of gravities of positive and negative ions coin-cide. (b) At the presence of mechanical stresses, cen-ters of gravities of positive and negative ions separate,which leads to generating an electrical voltage acrossthe surface of the crystal. (c) At the presence of anelectrical voltage, centers of gravity separate result-ing in mechanical strain. (Adopted with permissionfrom [FL14]) . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.2 A qualitative illustration of the poling process of syn-thetic piezoelectric ceramics. (Adopted with permis-sion from [FL14].) . . . . . . . . . . . . . . . . . . . . 8Figure 2.3 Two different types of bender actuators. In this study,a bimorph bender has been employed. . . . . . . . . . 9Figure 2.4 Two different electrical configurations for piezoelec-tric benders [RF14] c© [ 2014], IEEE. Used with per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.5 A piezoelectric stack actuator. (Adopted with per-mission from [FL14].) . . . . . . . . . . . . . . . . . . 10Figure 2.6 A piezoelectric tube actuator. (Adopted with permis-sion from [FL14].) . . . . . . . . . . . . . . . . . . . . 11Figure 2.7 A typical Hysteresis nonlinearity. . . . . . . . . . . . 12Figure 2.8 A qualitative representation of the creep nonlinearityin piezoelectric actuators. (Adopted with permissionfrom [FL14].) . . . . . . . . . . . . . . . . . . . . . . . 13xiiLIST OF FIGURESFigure 2.9 Frequency response of displacement over driving volt-age for a 2 DoF micro-positioning stage using stackactuators. . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.10 Electromechanical models for piezoelectric actuators[GC97]. Adopted with permission. . . . . . . . . . . . 17Figure 2.11 A feed-forward displacement control structure. . . . . 20Figure 2.12 Maxwell hysteresis model [CKC97] c© [ 1997], IEEE.Adopted with permission. . . . . . . . . . . . . . . . . 22Figure 2.13 Preisach hysteresis model. (Adopted with permissionfrom [GJ95].) . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.14 Prandtle-Ishlinskii hysteresis model [GZS13] c© [ 2014],IEEE. Adopted with permission. . . . . . . . . . . . . 24Figure 2.15 A feed-back control structure. . . . . . . . . . . . . . 25Figure 2.16 (a) A tube using four electrodes [BRFM07]. (b) Anew electrode pattern. (Reproduced from [MY08],with the permission of AIP Publishing. doi : 10.10631.2952506) . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.17 Integrating two identical piezoelectric transducers pro-posed in [PKS18] c© [ 2018], IEEE. Adopted withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.18 Self-Sensing techniques based on the piezoelectric ac-tuator’s electrical measurements. . . . . . . . . . . . . 29Figure 2.19 Bridge circuit employed in [PGCL06] for displace-ment self-sensing. When the hysteresis effect is neg-ligible the output voltage of the circuit is propor-tional to the displacement of the transducer, Vs(s) ∝VP (s) ∝ X(s) [PGCL06] c© [ 2008], IEEE. Adoptedwith permission. . . . . . . . . . . . . . . . . . . . . . 30Figure 2.20 Implementation of a passive shunt damping structureto damp disturbance vibrations [MFB03] c© IOPPublishing. Adopted with permission. . . . . . . . . . 31Figure 2.21 Passive shunt damping interpreted as a feed-back con-trol structure [EF10] c© [ 2010], IEEE. Adopted withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 2.22 The schematics of a conventional charge amplifier usedfor displacement self-sensing purposes [FM06] c© [2006], IEEE. Adopted with permission. . . . . . . . . 34xiiiLIST OF FIGURESFigure 2.23 A modified charge amplifier with improved low fre-quency performance.(Reproduced from [FM05], withthe permission of AIP Publishing. doi : 10.10631.1938952) . . . . . . . . . . . . . . . . . . . . . . . . 35Figure 2.24 A charge amplifier for displacement self-sensing. (Re-produced from [IRLC09b], with the permission of AIPPublishing. doi : 10.1063 1.3142486) . . . . . . . . . . 36Figure 2.25 The permittivity-displacement correlation under zeroexternal force [KKHM08]. (Adopted with permissionform Taylor & Francis.) . . . . . . . . . . . . . . . . . 37Figure 2.26 The displacement self-sensing techniques using per-mittivity measurements proposed by Morita et al.[KKHM08, IM10, IM11, SM15]. (Adopted with per-mission form Taylor & Francis and Sensors and Ac-tuators A: Physical ) . . . . . . . . . . . . . . . . . . 38Figure 2.27 The capacitance measurement method proposed byIslam et al. [ISM11a, ISM11b] c© [ 2011], IEEE. Adoptedwith permission. . . . . . . . . . . . . . . . . . . . . . 39Figure 2.28 The scheme of the complete low and high frequencydisplacement x˜ and low frequency force F˜lf estima-tion method [RIK+10] c© [ 2010], IEEE. Adoptedwith permission. . . . . . . . . . . . . . . . . . . . . . 41Figure 2.29 The scheme of the complete low and high frequencydisplacement x˜ and force F˜ estimation method [Rak13]c© [ 2013], IEEE. Adopted with permission. . . . . . . 42Figure 2.30 The schematic of a low frequency displacement andforce self-sensing method using Goldfarb et al.’s non-linear electromechanical model in (2.12 to 2.15) [BQN08b,KSUO12, MU14] c©[ 2008], IEEE. . . . . . . . . . . . 43Figure 2.31 The schematic of the displacement self-sensing tech-nique using charge and capacitance measurements [IS14,IS12a] c© [ 2014], IEEE. . . . . . . . . . . . . . . . . . 45Figure 2.32 The self-sensing technique using capacitance and chargemeasurements [LALR18] c© [ 2018], IEEE. . . . . . . 46Figure 3.1 Examples of setups in the literature. (a) A test setupthat operates under constant force [IS14] c©[ 2014],IEEE. (b) A one degree of freedom positioning stage[EVGP14]. (Adopted with permission from [FL14].) . 49xivLIST OF FIGURESFigure 3.2 The proposed setup to create independently varyingdisplacement and forces. Used with permission. . . . 50Figure 3.3 The test setup. Used with permission. . . . . . . . . . 51Figure 3.4 The preloading system. . . . . . . . . . . . . . . . . . 52Figure 3.5 Fitted frequency response from V1 to x1m and F1m.Used with permission. . . . . . . . . . . . . . . . . . . 54Figure 3.6 Feed-forward and feed-back control modes of the testsetup. Used with permission. . . . . . . . . . . . . . . 54Figure 3.7 A feed-back controller designed for the setup. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 55Figure 3.8 Decoupled displacement and force control structure.Used with permission. . . . . . . . . . . . . . . . . . . 55Figure 3.9 Feasible displacement-force region for the test setup.Used with permission. . . . . . . . . . . . . . . . . . . 56Figure 3.10 (a) Constant force trajectories at −100 N, −200 N,and −300 N. (b) Measured charge as a function ofdisplacement for the forces in (a). Used with permis-sion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Figure 3.11 (a) Constant displacement trajectories at−4 µm, 0 µm,and 4 µm. (b) Charge as a function of force for thedisplacements in (a). Used with permission. . . . . . 58Figure 4.1 The reference force and displacements values employedin validation of hysteresis voltage and charge relation-ship. In the constant force experiment shown in blue,the force is kept constant at -100 N and displace-ment varies from -3 µm to 5 µm and back to -3 µm.In the constant displacement experiment shown ingreen, the displacement is kept at constant at 0 µmwhile the force changes from 0 N to -350 N and backto 0 N. . . . . . . . . . . . . . . . . . . . . . . . . . . 60Figure 4.2 (a) Driving voltage vs. charge profiles for constantforce and constant displacement experiments. (b)Corresponding hysteresis voltages vs. charge profiles.Used with permission. . . . . . . . . . . . . . . . . . . 61Figure 4.3 The proposed displacement and force measurementtechnique. . . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.4 The experiential setup. . . . . . . . . . . . . . . . . . 64Figure 4.5 The schematic cross section of the sensing unit in Fig-ure 4.4 [Isl13a]. . . . . . . . . . . . . . . . . . . . . . . 65xvLIST OF FIGURESFigure 4.6 The measured F1m-x1m reference trajectory gener-ated with the setup outlined in Section 4.3. . . . . . . 66Figure 4.7 The experimental results of traversing the referencetrajectory at four different speeds. (a) The estimateddisplacement x˜1 vs. the measured displacement x1m.(b) The estimated force F˜1 vs. the measured force F1m. 67Figure 5.1 Electromechanical model of the bender for nonzerodriving voltage and zero external force. Used withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 70Figure 5.2 The electrical connection of the strain gauges (a) andthe equivalent circuit (b). Used with permission. . . . 72Figure 5.3 Experimental setup. Used with permission. . . . . . . 73Figure 5.4 Two electrical configurations for driving the bender.Used with permission. . . . . . . . . . . . . . . . . . . 74Figure 5.5 Block diagram of the piezoelectric bender system withintegrated strain sensors. The tip displacement of thebender is also measured using the laser (ylaser) forsensor calibration and comparison purposes. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 75Figure 5.6 The tip displacement versus driving voltage, whenthe actuator is driven by a triangular voltage withincreasing amplitude. Used with permission. . . . . . 76Figure 5.7 The recorded time signals (a) and difference betweenthe reference ylaser and proposed sensor y˜ (b). In (a),the peak-to-peak values are provided for the actua-tor voltage Vu, the reference sensor ylaser, and strainsensor y˜. Used with permission. . . . . . . . . . . . . 76Figure 5.8 The normalized open-loop frequency response in blue,measured from u to y˜, in mm/V. The closed-loop fre-quency response with damping-integral control, mea-sured from r to y˜, in mm/V. Normalizing the open-loop frequency response is achieved by multiplyingthe measured data by a gain of 7.9. Used with per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . 77Figure 5.9 Block diagram of the closed-loop system with thedamping-integral controller. The strain sensor out-put is the measurement in the feed-back loop. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 78xviLIST OF FIGURESFigure 5.10 Reference and measured closed-loop step response ofthe integrated bender using a damping-integral con-troller. Used with permission. . . . . . . . . . . . . . 79Figure 6.1 Reference and measured force-displacement profilesused for parameter identification. The measured pro-file is created using the feed-back control mode of thethe test setup described in Chapter 3. Used with per-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . 82Figure 6.2 A qualitative representation of the parameter iden-tification criteria used in [BQN08a]: the descendingbranch of the VH − q hysteresis loop must have zeroslope at the end of the branch. Used with permission. 83Figure 6.3 Measured and modeled hysteresis loops. Dashed bluelines represent the measured hysteresis loops using aCP value of 3.1 µF, which violates the hysteresis mod-eling criteria in Figure 6.2. The solid blue loops aremeasured with the CP value of 3.5 muF. Prandtle-Ishlinskii hysteresis operators have been used to gen-erate the green hysteresis fits. Used with permission. 84Figure 6.4 Red diamonds represent the sequence of the desiredpoints to follow validation force vs. displacement pro-file. Solid blue line shows force vs. displacement mea-surements. Used with permission. . . . . . . . . . . . 85Figure 6.5 Measured and estimated displacements and the esti-mation error. Used with permission. . . . . . . . . . . 86Figure 6.6 Measured and estimated forces and the estimationerror. Used with permission. . . . . . . . . . . . . . . 87Figure 7.1 The test setup employed to study the effect of voltageand force on the capacitance of a piezoelectric stackactuator. . . . . . . . . . . . . . . . . . . . . . . . . . 89Figure 7.2 The schematic representation of implementation ofthe setup. Used with permission. . . . . . . . . . . . . 90Figure 7.3 The experiments conducted to study the effect of ap-plied force and voltages on the capacitance of piezo-electric actuators. The capacitance measurements arecarried out at 30 Hz. Used with permission. . . . . . 91xviiLIST OF FIGURESFigure 7.4 The schematic of the new displacement-force self-sensingtechnique. The proposed techniques utilizes a voltagedependent capacitance and requires a single parame-ter identification experiment to identify n, k, CP , Cv,and H(q). Used with permission. . . . . . . . . . . . 93Figure 7.5 The experimental measurements collected for param-eter identification. Used with permission. . . . . . . . 94Figure 7.6 (a) The measured hysteresis voltage VH1, (shown inblue), using the identified parameters in Table 7.1and the fitted hysteresis voltage V˜H1 in red versusthe measured charge q1. (b) A qualitative represen-tation of the hysteresis model parameters. Used withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 96Figure 7.7 A sensor-less displacement-force controller using thenew self-sensing technique. Used with permission. . . 98Figure 7.8 Reference Fref vs. xref and measured F1m vs. x1mpaths shown in blue and red respectively. Used withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 99Figure 7.9 (a) Reference xref and measured x1m displacementsplotted against time. (b) Reference Fref and mea-sured F1m forces plotted against time. The absolutevalue of the control system errors, |esx| and |esF | areplotted in green in (a) and (b) respectively. Used withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 100Figure 8.1 The setup used for studying the effect of externalloads on the impedance of a piezoelectric actuator.Used with permission. . . . . . . . . . . . . . . . . . . 102Figure 8.2 Frequency sweep results for impedance measurementof PSA-I under F1m=0 N, -166 N, and -361 N ex-ternal force conditions. (a) magnitude. (b) phase.(c) Capacitance. (d) Resistance. Used with permission.103Figure 8.3 The setup used for the proposed impedance basedforce and displacement self-sensing technique. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 104Figure 8.4 Measured displacement-force map created for con-struction of the proposed self-sensing technique. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 105xviiiLIST OF FIGURESFigure 8.5 Measurements used to construct the self-sensing tech-nique. (a) Measured displacement vs. detected re-sistance. (b) Measured displacement vs. detectedcapacitance. (c) Measured force vs. detected resis-tance. (e) Measured force vs. detected capacitance.Used with permission. . . . . . . . . . . . . . . . . . . 107Figure 8.6 (a) Measured displacement vs. detected capacitanceand resistance profile is shown in red, together withthe fitted surface using (8.7). (b) Measured force vs.detected capacitance and resistance profile is shownin red, together with the fitted surface using (8.8).Used with permission. . . . . . . . . . . . . . . . . . . 108Figure 8.7 Real-time implementation of the impedance-changebased displacement and force self-sensing method. Usedwith permission. . . . . . . . . . . . . . . . . . . . . . 109Figure 8.8 Measured displacement-force map created for valida-tion of the proposed self-sensing technique describedin (8.7) and (8.8). Used with permission. . . . . . . . 110Figure 8.9 Measured and estimated displacement. Used withpermission. . . . . . . . . . . . . . . . . . . . . . . . . 111Figure 8.10 Measured and estimated force. Used with permission. 112Figure 8.11 A sensor less controller using impedance measure-ment self-sensing technique. Used with permission. . 113Figure 8.12 Experimental results for validating the control struc-ture. (a) The reference xref , the predicted x˜1 and themeasured x1m displacement profiles together with theabsolute values of the displacement actuating signal|Eax| and the overall error of the structure |Esx|. (b)The reference Fref , the predicted F˜1 and the mea-sured F1m force profiles together with the absolutevalue of the force actuating signal |EaF | and the over-all error of the structure |EsF |. Used with permission. 113xixGlossary of Notationα Linear model piezoelectric constant.An Nominal area.b Damping coefficient.C Capacitance.CP Clamped capacitance.d Piezoelectric material constant.D Electric displacement.eF Force error.eF Displacement error.E Electric field.F Force.F˜ Estimated force.Fs Measured force.Fref Reference force.G Transfer function.H(q) Hysteresis model.I Current.Ir Ripple current.T Permettivity at constant stress .i, j Lattice dimensions.xxGlossary of Notation~i, ~j, ~k Cartesian coordinates.m Actuator Mass.M Moment.L Inductance.ln Nominal length.γab Preisach hysteresis model operator.Γ Hysteretic voltage-displacement relation.m Actuator Mass.n Nonlinear piezoelectric coefficient.N Number of piezoelectric layers.P Polarization.q Charge.R Resistance.S Mechanical strain.sE Elastic compliance at constant electric field.τ Charge constant.T Mechanical stress.V Driving voltage voltage.VDC High DC voltage.VH Hysteresis voltage.Vr Ripple voltage.Vt Internal linear voltage.w Disturbance.wr Frequency.xxiGlossary of Notationx Displacement.x˜ Estimated displacement.xs Measured displacement.xref Reference displacement.x˙ Velocity.x¨ Acceleration.Y Electrical admittance.Z Impedance.xxiiAcknowledgmentsFirst and for most, I would like to thank Dr. Rudolf Seethaler for hisguidance and support during my two years of master’s and four years of PhDstudies at UBC. Dr. Seethaler made this six years a memorable period of mylife that I am and will be proud of. During my studies under his supervision,I felt supported, trusted, and valued, I learned how to work professionallyand patiently in an organized manner, I learned to pay attention to details, Ilearned how to present a scientific topic, and I learned how to write scientificpapers. In addition, I enjoyed the fact that he was always willing to listen tomy needs and opinions and was willing to discuss them patiently, he wantedme to learn and grow and never hesitated to encourage me in taking bigsteps in my academic carrier, and most importantly, his supervision styleencouraged me to think out of the box and dream big.I would like to I thank my committee members, Dr. Abbas S. Milani,Dr. Homayoun Najjaran, and Dr. Yang Cao for their valuable guidance inmy research. I also would like to thank Dr. Wilson Eberle for providingaccess to his laboratory facilities. I also acknowledge the support of mygreat laboratory group members, Mr. Brad Reinholz, Mr. Severin Nowak,Mr. Max Arzberger, Dr. Mohammad Nouroz Islam, Mr. Raed Mohsen, Mr.Yangi Yu, Mr. Toby Lindsay, and Mr. Peyman Amiri.I would like to thank Dr. Andrew Fleming for his guidance during myfour months of internship in his research laboratory at the University ofNewcastle, Australia. He taught me how to write scientific manuscript inan organized manner and he encouraged me to learn about new topics. Ialso acknowledge the support of Dr. Fleming’s research group members Dr.Yuen K. Yong, Dr. Yik R. Teo, Dr. Arnfinn A. Eielsen, Mr. Ben S. Routley,Mr Meysam Omidbeike, Dr. Steven I. Moore, Dr. Omid T. Ghalehbeygi,Dr. Michael G. Ruppert, Mr. Digvijay Raghuvanshi, and Dr. Mohd N.Islam.I would like to thank Dr. Micky Rakotondrabe and his PhD student,Mr. Joel Bafumba Liseli from Universite de Franche-Comte, France for theirsupport in my studies.I would like to acknowledge the funding support from the Natural Sci-xxiiiAcknowledgmentsences and Engineering Research Council (NSERC) of Canada which pro-vided the necessary financial support throughout my studies.xxivDedicationTo my parentsShiva HakimiMajid Zarif Mansourand my sisterRomina Zarif MansourxxvChapter 1IntroductionThis chapter provides with the motivation and the contributions of thisthesis.1.1 MotivationDuring the last meeting of the American Physical society in the 1950s,physicist Richard Feynman gave a talk entitled “There’s a plenty of roomat the bottom” [Fey92]. In his talk, he outlined an inspiring vision forfuture technologies by describing a process in which researchers are able tomanipulate individual atoms and molecules. The world waited over a decadeto observe the realization of this prophecy with the birth of nanotechnologyin 1981, when the first scanning tunneling microscope (STM) was developed.STMs are a type of scanning probe microscope that use a sharp tip as ascanning probe to interact with the surface of a sample material on theatomic level and provide a topographical surface image of it.In the past three decades, nanotechnology has experienced a tremen-dous growth in various areas [HHM95]. Different types of scanning prob mi-croscopes have been introduced, such as atomic force microscopes (AFMs)[BQG86, SS08], which helped researchers to produce clearer atomic level im-ages of the surface of different samples. In addition to surface imaging, thesedevices have been employed to “feel” the samples by measuring their me-chanical properties such as stiffness [FL14]. Also, AFMs have been employedto manipulate the molecular structure of different materials and develop ma-terials with new physical or chemical characteristics [FL14]. Nanotechnologyhas been used to increase a material’s conductivity, chemical reactivity, ormechanical strength [FL14]. Other applications of nanotechnology includeatomic or close to atomic manufacturing [LCWW05, FZG+19], ink jet print-ers [CYAC16], micro-grippers [YX17, WX15], adaptive optics [KKSK07],precise medical devices [MU14], and fuel injectors [NBMS16].A common feature among the applications listed above is the positioningstage, which is employed to move a probe or manipulate a sample objectat micro-scales. The literature shows a large verity of designs for these11.1. MotivationMovingplatformPiezoelectricactuatorFrame and flexuresFigure 1.1: A one degree of freedom micro-positioning stage. Used withpermission from [FL14].stages. An example of these designs is depicted in Figure 1.1. In thisconfiguration, a platform is moved via a linear actuator and is suspendedat the center of a solid frame via flexures, which provide preloading forceon the actuator [FL14] and prevent bending and torsion of the actuator.piezoelectric actuators have been widely employed in the positioning stages,due to their practically infinite resolution, large bandwidth, and high stiff-ness [Isl13b]. When an electrical voltage is applied to them, they elongateand move the platform. Reducing the applied voltage contracts the actu-ator, which retrieves the platform towards its initial position. However, amajor disadvantage of piezoelectric actuators is the inherent nonlinear re-lationship between the applied voltage and the resulting elongation of theactuator. This nonlinear relationship makes the motion control of the plat-form a challenging task. The conventional approach to tackle this problemis to use external displacement sensors and force sensors. These externalsensors add more complexity to the design of the positioning stage. In ad-dition, sensors with adequate resolution and bandwidths are expensive andbulky to be mounted in the micro-positioning working spaces. For example,hundreds of deformable mirrors are installed in an adaptive optics system,which makes it infeasible to monitor every single mirror with a separateexternal sensor.The current study proposes new techniques to minimize the implemen-tation of external sensors in piezoelectric actuator driven positioning stages.The proposed techniques are categorized into two main approaches. Thefirst approach integrates the actuators with sensors employed outside the21.2. Contributions of the proposed Researchmicro-positioning working space and provides indirect measurements of dis-placement and/or force. The second approach uses a piezoelectric actuatorsimultaneously as a sensor and completely eliminates the use of externalsensors.1.2 Contributions of the proposed ResearchThe main goal of this thesis is to minimize the use of sensors in piezoelec-tric actuator-driven micro-positioning work-spaces. To satisfy this goal, twogroups of techniques are presented: 1. integrating piezoelectric actuatorswith dedicated sensors outside their work-spaces and 2. introducing novelself-sensing techniques for piezoelectric actuators. This thesis is constructedwithin 9 chapters.Chapter 1 provides with an introduction for the presented thesis.Chapter 2 describes the background of the presented research. It explainsfundamental topics such as: piezoelectricity, different types of piezoelectricactuators, their modeling challenges, their existing electromechanical mod-els, and their existing displacement and force control techniques. For thefirst time in the literature, this chapter provides a detailed review of self-sensing techniques for piezoelectric actuators and categorizes them basedon the measurements of voltage, charge, and impedance. After reading thischapter, the reader will have the required knowledge to follow the contentin the later chapters. In addition, the researchers in the field of self-sensingwill have access to a comprehensive review of the existing techniques in theliterature.Chapter 3 describes the design of a 2 DoF setup that generates inde-pendently varying displacement and forces for piezoelectric stack actuators.The setup is employed in the experiments carried out in Chapters 4, 6, 7,and 8. First, an electromechanical model of the setup is derived. Then,the actual prototyped setup is described and a frequency response analy-sis of it is provided. Two control modes, voltage feed-forward control anddisplacement-force feed-back control, are designed and validated using ref-erence sensors. Feasible displacement-force operating regions are designedfor each of the control modes and experimentally validated.Chapter 4 describes a novel sensor-actuator integrating technique. Theproposed integrating technique employs a piezoelectric actuating transducerin a sensor-free micro-positioning work-space. The important novel featureof this arrangement is that the actuating transducer is electrically connectedin series to an equivalent sensing transducer that is clamped outside the31.2. Contributions of the proposed Researchwork-space between a solid frame and a dedicated force sensor. Exploitingthat hysteresis is purely a function of charge, a fact that is validated ex-perimentally for the first time as part of this work, this arrangement canestimate displacement and force of the actuating transducer from the forcemeasurement of the sensing transducer and voltage difference measurementof the two transducers. In contrary to other self-sensing techniques, there isno need for elaborate hysteresis models or complicated charge measurementcircuitry. The proposed technique is useful in applications where dedicatedsensors are not cost prohibitive, however the work-space limitations preventstheir implementation.Chapter 5 describes an elegant new sensor-actuator integration techniquefor piezoelectric benders, where two strain gauges are constructed from thetop and bottom electrodes and employed to measure the bender’s tip dis-placement under zero force condition. Chapter 5 first describes a linearrelationship between the surface strain of a bender and its tip displacementfor zero external forces. Then, it proposes a half-bridge circuit to measurethe bender’s surface strain and tip displacement. The proposed integratingtechnique is implemented in a displacement control structure on an off-theshelf piezoelectric bender. The proposed technique is not affected by creepand it exhibits no low frequency drifts. Therefore, it can be implementedon scanning probes.Chapter 6 describes a self-sensing technique that uses charge and voltagemeasurements of a piezoelectric actuator to estimate its displacement andforce. Compared to similar work in the literature, the contributions in thischapter include a novel, more accurate, and more efficient model parameteridentification technique, as well as the application of a more widely acceptedhysteresis operator. The performance of the presented self-sensing techniqueis experimentally validated for the setup described in Chapter 3.Chapter 7 further refines the self-sensing technique introduced in Chap-ter 6. This work is motivated by an experimental study that indicatesa linear dependency between applied voltage and capacitance of piezoelec-tric transducers. Consequently, Goldfarb et al.’s standard electromechanicalmodel is refined to include this effect. An elegant parameter identificationmethod is then presented that requires only a single hysteresis free exper-iment. In addition, the presented identification method employs a novelhysteresis operator that can model asymmetric hysteresis loops and doesnot require to be inverted for real-time implementations. The presentedoperator consists of two exponential functions which are described by onlyfour parameters. Another advantage of the presented operator over existinghysteresis operators is that it requires measurements of current rather than41.2. Contributions of the proposed Researchcharge. All of these novelties are added to the self-sensing technique fromChapter 6, which is then inserted into a control structure to regulate thedisplacement and force of the setup described in Chapter 3.Chapter 8 describes a novel self-sensing technique that uses impedancemeasurements to estimate the displacement and force of a piezoelectric ac-tuator. This work is motivated by the experimental results introduced inChapter 7 that indicated a linear relationship between a piezoelectric trans-ducer’s applied voltage and its capacitance. The main novelty of Chapter 8is the additional experiments that indicate that external forces result ina shift in the location of the resonance peaks. Unlike the similar worksin the literature that use permittivity to estimate the displacement underconstant forces, the presented self-sensing technique measures the actua-tor’s impedance close to a resonance peak and maps these measurements tothe actuator’s displacement and the force. The technique is experimentallyvalidated in a sensor-less displacement and force control structure that isimplemented on the setup described in Chapter 3.The techniques described in this thesis are experimentally validated atquasi-static frequencies. Therefore, they are applicable for applications suchas micro-grippers. The self-sensing techniques described in Chapters 6, 7,and Chapter 8 are preferable over the integrating techniques presented inChapters 4 and 5 in application where, the implementation of sensors isimpossible because of financial or work-space restrictions. The presentedtechnique in Chapter 8 is preferable over the techniques in Chapters 6 and 7at slow applications, where charge measurement is challenging.Finally, Chapter 9 provides conclusion for the presented thesis. A quali-tative comparison table is provided that compares the techniques describedin the thesis. In addition, Chapter 9 suggest the future studies that canbe conducted based on the proposed research. The result of the proposedresearch leads to the design of more compact and simpler positioners. Theremainder of this chapter provides a literature review of the fundamentalconcepts necessary to understand the behavior of piezoelectric actuators andthe existing displacement and force control techniques for them. Finally, thecontributions of this research are explained.5Chapter 2Literature reviewThis chapter provides with a literature review2.1 PiezoelectricityOne year after Thomas Edison’s invention of electric light bulbs lit upthe world1, French brothers Pierre and Jacques Currie discovered that crys-tals, such as quartz, Rochelle salt, and sodium chlorate, generate electricalcharges when subjected to mechanical stresses. In 1881, physicist Lippmannpredicted a converse effect for these crystals, in which mechanical strain isproduced when the crystals are subjected to electrical voltages [FL14]. TheCurie brothers confirmed this prediction in the same year [Mas81]. Thenewly discovered effect was named the piezoelectric effect. This name is de-rived from the Greek word piezein which means to press or squeeze [Ike96].In 1893, Lord Kelvin proposed a model for the lattice structure of quartzcrystals (SiO2) to explain the newly discovered effect. Figure 2.1 illustratesthis model for a unit cell of quartz. Positive Silicon ions are shown in bluecircles and negative oxygen ions are shown in red circles. This unit cell isunder an equilibrium condition, where centers of gravity of positive and neg-ative ions coincide. Figure 2.1 (b) shows that at the presence of mechanicalstress, the lattice structure deforms and the centers of gravities separate.As a result, a dipole moment is created which induces an electric voltage.This voltage can be measured across the surface of the crystal [FL14]. Theinverse piezoelectric effect is illustrated in Figure 2.1 (c), where an externalvoltage source is applied to the unit cell. The positive and negative ions areattracted to the opposite sign electrodes of the voltage source, resulting ina relative motion of centers of gravity. This motion leads to the physicaldeformation of the crystal and generates mechanical strain in it. The piezo-electric effect occurs only in the crystals with no center of symmetry in theirlattice structure. 20 out of 32 classes of crystals are piezoelectric [FL14].1The first electric light bulb was invented in 1879.62.2. Synthetic Piezoelectric CeramicsSiSi SiOOO+++- --SiSi SiOOO+---+ +SiSi SiOOO+---+ +- - - - - - - - - -+ + + + + + + + + +(a)(b) (c)Figure 2.1: A model for The lattice structure of quartz crystal (SiO2)proposed by Lord Kelvin in 1893 to explain the piezoelectricity effect. (a)In an equilibrium state, centers of gravities of positive and negative ionscoincide. (b) At the presence of mechanical stresses, centers of gravities ofpositive and negative ions separate, which leads to generating an electricalvoltage across the surface of the crystal. (c) At the presence of an electricalvoltage, centers of gravity separate resulting in mechanical strain. (Adoptedwith permission from [FL14])2.2 Synthetic Piezoelectric CeramicsThe early studies on piezoelectricity were limited to the materials thatnaturally exhibit this effect. In the 1940s, a new class of synthetic piezo-electric ceramics were developed that exhibit an enhanced piezoelectricityeffect, meaning that their response to an applied voltage or stress is morepronounced in comparison to natural piezoelectric crystals. The manufac-turing process of the synthetic ceramics consists of a number of steps. First,raw materials such as lead, zirconium, titanium, etc. are combined througha set of milling processes to make a powder. The powder enters the poly-crystalline phase through a heat treatment process. It then undergoes anadditional milling process and is combined with a binder. The resultingpowder is pressed into molds with desired shapes and dimensions, where itunder goes a sintering process. The developed ceramic is then polished and72.3. Piezoelectric actuators(a) Before poling (b) Applying DC voltage at the Curie tempature (c) After poling +-DomainFigure 2.2: A qualitative illustration of the poling process of syntheticpiezoelectric ceramics. (Adopted with permission from [FL14].)electrodes are attached to it [FL14]. At this stage, the produced ceramicconsists of microscopic regions called domains that represent a group ofunit cells with similar dipole orientations. Figure 2.2(a) illustrates the ran-dom orientation of these domains. Therefore, at the presence of mechanicalstresses or electrical voltages the domains will reorient in random directionsand no net piezoelectric effect will be measured. In order to maximize thepiezoelectric effect, domains should have the same direction. This can beachieved through a process called poling, which consists of three steps. Atfirst, the ceramic is heated to reach its Curie temperature2. Then a high DCelectric field is applied to the ceramic as illustrated in Figure 2.2(b). Thisforces the domains to align with respect to the applied electric field. Whilethe electric field is applied, the ceramic goes under a cooling down process.Consequently, the majority of the dipoles stay in the desired alignment asdepicted in Figure2.2(c). In addition, this process permanently changes thedimensions of the ceramic [FL14].2.3 Piezoelectric actuatorsThe synthetic piezoelectric ceramics have been widely employed to de-velop micro-positioning actuators, due to their enhanced piezoelectricity ef-2Curie temperature is defined for certain materials. Above this temperature, they loosetheir permanent magnetic properties.82.3. Piezoelectric actuatorsElectrodeReinforcement Piezo plate(a) A unimoph piezoelectric ben-der.ElectrodeReinforcement ElectrodePiezo plates(b) A bimorph piezoelectric ben-der.Figure 2.3: Two different types of bender actuators. In this study, a bimorphbender has been employed.Electric voltageTip displacement(a) Series electrical configuration.Electric voltageTip displacement(b) Parallel electrical configuration.Figure 2.4: Two different electrical configurations for piezoelectric ben-ders [RF14] c© [ 2014], IEEE. Used with permission.fect [Isl13b]. These actuators are manufactured in different shapes and forms[ZHB+01]. The remainder of the current section describes three commonlyused types of piezoelectric actuators: 1. piezoelectric benders, 2. piezoelec-tric stack actuators, and 3. piezoelectric tubes.2.3.1 Piezoelectric bender actuatorsPiezoelectric benders are employed in micro-positioning applications withlarge ranges of motion and small forces. An example of such an applicationis the scanning probe of an AFM [FL14]. Piezoelectric unimorphs and bi-morphs are the two main types of benders [RF14]. Figure 2.3(a) illustratesthat a unimorph bender consists of a thin piezoelectric ceramic plate bondedto a non-piezoelectric elastic reinforcement layer such as aluminum or brass.92.3. Piezoelectric actuatorsFigure 2.3(b) depicts a bimorph bender, which consists of two piezoelectricceramic plates bonded together. The reinforcement is provided via a thirdelastic plate, which is sandwiched between the piezoelectric plates.In order to drive a bimorph bender, it is cantilevered and an electricfield is applied to it, so that one piezoelectric plate expands while the otherone contracts and consequently, net forces and displacements are generatedat the tip of the bender. The electric field can be applied in two differentconfigurations [RF14, Saw36]. Figure 2.4(a) depicts the series configuration,in which the piezoelectric plates are electrically connected in series and theirpoling directions are opposite. Figure 2.4(b) shows the parallel configura-tion, in which the piezoelectric plates are electrically connected in paralleland their poling directions are aligned. Although the two configurationachieve the same theoretical tip displacement and forces, from a safety per-spective, the parallel configuration is preferable, since both top and bottomplates can be grounded [RF14].2.3.2 Piezoelectric stack actuatorsPiezoelectric stack actuators are employed in applications with largeforces and small motion ranges. As an example, they have been used in highpressure fuel injectors [NBMS16]. Figure 2.5 shows a piezoelectric stack ac-tuator, which is made of hundreds of thin piezoelectric ceramic plates, (eachone is about 100 µm thick), that are stacked on top of each other. Anytwo adjacent plates have opposite poling directions and they are electricallyElectricvoltageTip displacementFigure 2.5: A piezoelectric stack actuator. (Adopted with permissionfrom [FL14].)102.4. Piezoelectric actuator modeling challengesElectricvoltageMovement in directionFigure 2.6: A piezoelectric tube actuator. (Adopted with permissionfrom [FL14].)connected in parallel [FL14]. As a result, all plates extend and contract inthe same direction and a net displacement is generated in the stack actuator,which is about 0.2% of its nominal length [AI02].2.3.3 Piezoelectric tube actuatorsPiezoelectric tube actuators are made of cylindrical shape synthetic piezo-electric materials that are poled radially. They generate motion in two orthree dimensions [FL14]. Figure 2.6 depicts the electrode configuration ofa tube actuator that is employed to generate motion in ~i and ~j directions.For clarity reasons, only the electrical connections for the motion in the ~idirection is depicted in Figure 2.6.In conclusion, this section reviewed three conventional types of piezoelec-tric actuators. Benders and stack actuators are employed for large motionrange/low force or low motion range/large force applications respectively.Tube actuators have the advantage of generating motion in two directions.2.4 Piezoelectric actuator modeling challengesPiezoelectric actuators are popular tools in micro-positioning devices,due to their quick response to both electrical and mechanical signals, largebandwidth, and high stiffnesses. However, their performance can suffer due112.4. Piezoelectric actuator modeling challenges0 5 10Time [s]-1000100Driving Voltage [V]-100 0 100Driving Voltage [V]-101Tip Displacement [mm](a) (b)Figure 2.7: A typical Hysteresis nonlinearity.to hysteresis, creep, vibration, and temperature change. This section dis-cusses these effects [CC15].2.4.1 HysteresisHysteresis is conventionally described as a nonlinear relationship betweenthe applied voltage to a piezoelectric actuator and its resulting displacement.Figure 2.7 illustrates a typical hysteresis nonlinearity measured for a piezo-electric bender actuator at a fixed quasi-static frequency and under zeroexternal forces. Figure 2.7 (a) shows a tooth shape driving voltage that isapplied to the bender. Figure 2.7 (b) depicts the resulting nonlinear rela-tionship between the applied voltage and the bender’s tip displacement. Thepresence of external forces will vary the shape of the hysteresis profile. Theshape can also change under different frequencies, indicating that hysteresisis a rate-dependent phenomenon. Figure 2.7 (b) consists of four hystereticloops which correspond to four cycles of the applied driving voltage shown inFigure 2.7 (a) [GZS+16, Vis13, LSGQ13]. In addition, Figure 2.7 (b) showsthat the ascending branches of the loops line up. This is due to the memoryeffect inherent of hysteresis. As a result of the memory effect, the instan-taneous displacement values are not only a function of their correspondingvoltage inputs, but also a function of the previous voltage inputs togetherwith the previous displacement values [KJ01].The main cause of hysteresis nonlinearity is believed to be the interac-122.4. Piezoelectric actuator modeling challengesTip displacementTimeDesiredActualCreepTransient response (Couple of seconds )Figure 2.8: A qualitative representation of the creep nonlinearity in piezo-electric actuators. (Adopted with permission from [FL14].)tions in the piezoelectric ceramics’ molecular structure [FL14]. The piezo-electric ceramics consist of microscopic polarized regions called domains.The imaginary boundaries between these domains are called domain walls.Under an applied electrical or mechanical field, the domains expand or con-tract and orient themselves with the field. As a result, the domain wallsinteract with each other. A significant form of this interaction is attributedto friction between the walls, which results in irreversible energy losses andconsequently, the presence of hysteresis [JA86]. Subsection 2.6.1 provides adiscussion of hysteresis compensation methods.2.4.2 CreepCreep is referred to the drift in a piezoelectric actuator’s position undera static voltage [CC15]. Figure 2.8 shows a qualitative representation ofcreep. A step driving voltage is applied to a piezoelectric actuator to achievea desired position. Due to the large bandwidth of the actuator, it shows aquick transient response. However, the actuator’s position slowly creepsto a new actual position. The time constant for creep is typically a fewminutes [FL14].The reason for creep is believed to be due to the slow polarization in-crease in the domains after applying a constant voltage. This was experimen-tally confirmed in a microscopic level experiment [TPGT02], where it wasshown that the domain sizes creep logarithmically under constant applied132.4. Piezoelectric actuator modeling challengesFigure 2.9: Frequency response of displacement over driving voltage for a2 DoF micro-positioning stage using stack actuators.voltages [TPGT02]. A number of studies in the literature are dedicated tocreep compensation. The experiments in this study are conducted at quasi-static frequencies that would not be exposed to creep. For more informationon the creep modeling, the reader is referred to [ERYT02, VCIM17, SS02].2.4.3 VibrationPiezoelectric actuator-driven Micro-positioners exhibit lightly dampedresonance peaks in their frequency responses [EVGP14]. For example, Fig-ure 2.9 illustrates the modeled frequency response of a micro-gripper, wherea major resonance peak can be observed at 4 kHz.Voltage inputs with high frequency content can excite the frequency reso-nances. To avoid the positioning errors due to excitation, micro-positioners’operational frequency ranges are typically limited to 1/10 or 1/100 of theirmajor resonance frequency. However, this limits the performance of highspeed devices such as AFMs. Using stiffer piezoelectric actuators with largerbandwidths leads to increased operational speeds [SAD+07]. However, stifferpiezoelectric actuators have limited displacement strokes [FL14]. Alterna-tively, the resonances can be actively damped using appropriate controlstrategies, which are discussed in Section 2.6.142.5. Modeling piezoelectric actuators2.4.4 Temperature changePiezoelectric actuators are highly sensitive to their ambient tempera-ture changes. An experimental study in [FL14] indicated that the displace-ment/voltage sensitivity of a tube actuator increased by 10% over 25◦Cincrease in its ambient temperature. Such a high variation in sensitivityhinders the performance of piezoelectric actuators.The effect of temperature change on a piezoelectric actuator is explainedvia its intrinsic, (referring to the individual piezoelectric domain behavior),and extrinsic, (referring to the piezoelectric domain wall activity) responses.In order to separate the two responses, parameters such as capacitance aremeasured at temperatures close to the absolute zero temperature. In the-ory, at this temperature, the domain walls are frozen (which eliminatesthe extrinsic contribution) and only the intrinsic contribution can be ob-served [Hal01]. A mathematical description of the intrinsic contribution isgenerated and employed to derive the extrinsic contribution at higher tem-perature [Hal01]. The reported results indicate that the extrinsic contribu-tions account for the majority of the parameter changes at temperaturesclose to room temperature and low electric fields [Hal01]. Although thesestudies provide an insight to the physics of piezoelectric ceramics, they areunable to suggest practical solutions to compensate for the temperatureuncertainties. In control applications, temperature uncertainties are com-pensated for with sufficiently large gain margins. [FL14].2.5 Modeling piezoelectric actuatorsThe previous sections have introduced the electromechanical behaviorof piezoelectric actuators on a microscopic level. The current section intro-duces three macroscopic mathematical models that describe the behavior ofpiezoelectric actuators.2.5.1 Piezoelectric standard constitutive relationsIn 1987, the standards committee of the IEEE Ultrasonics, Ferroelectric,and Frequency Control Society published a set of linear and static descrip-tions of piezoelectricity [SCotIUS87]. A well known set of these equations isreprinted here:Si = sEijTj + dkiEk, (2.1)Dm = dmjTj + TmkEk, (2.2)152.5. Modeling piezoelectric actuatorswhere, S, E, T , and D represent strain (the length change divided by thematerial’s nominal length), electric field (the applied electrical voltage di-vided by the material’s nominal length), stress (the applied force dividedby the material’s nominal cross sectional area), and electric displacementrespectively (the electrical charge divided by the material’s nominal crosssectional area). The matrices sE , d, and T represent the elastic complianceat constant electric field, the piezoelectric material constant, and the per-mittivity at constant stress. The indexes i, j = 1, 2, ..., 6 and m, k, l = 1, 2,3 refer to the dimensions of the lattice structure of piezoelectric materials.In (2.1) and (2.2), the strain and the electric displacement generated ina piezoelectric ceramic are linearly related to the stress and the electricalfield applied to it. These equations can be employed for different typesof actuators by selecting the appropriate index values that correspond tothe actuators’ polarization and movement directions. However, (2.1) and(2.2) fail to explain the dynamic behavior and hysteresis nonlinearity inpiezoelectric actuators [ADKB00, QEF+09].2.5.2 Linear electromechanical model of piezoelectricactuatorsThe IEEE model in (2.1) and (2.2) is limited to static operations andsmall voltages. In the 1990s, a large number of studies focused on developingelectromechanical models that could describe the dynamics and nonlinearbehavior of piezoelectric actuators [GC97].A popular linear electromechanical model is depicted in Figure 2.10(a),which is constructed based on the IEEE standard relations (2.1) and (2.2).The model consists of an electrical domain and a mechanical domain that arelinked via a piezoelectric constant α. In the mechanical domain, the actuatoris modeled as a mass-spring-damper system. In the electrical domain, theactuator is modeled as a clamped capacitor CP in series with an internalvoltage source VP . The voltage source magnitude is proportional to theactuator’s displacement x via VP = −αx. Applying a driving voltage Vgenerates an internal force FP in the mechanical domain, which is modeledas FP = αCPV . This force together with an external force F act on theactuator. The mathematical representation of this model ismx¨+ bx˙+ kx = F + αCPV, (2.3)∫Idt = q = αCPx+ CPV, (2.4)162.5. Modeling piezoelectric actuatorsMechanical Domain Electrical Domainlinks the two domains:links the two domains:(a) A linear electromechanical model(b) A nonlinear electromechanical model +Figure 2.10: Electromechanical models for piezoelectric actuators [GC97].Adopted with permission.where, x˙, and x¨ are the velocity and the acceleration of the actuator andm, b, and k, are mass, damping coefficient, and the spring stiffness of theactuator respectively. q is the charge passing through the actuator.The transformation from the IEEE standard model in (2.1) and (2.2)with the static description, x˙ = x¨ = 0, to the model in (2.3 and 2.4) requiresreplacing the states in (2.1) and (2.2) with172.5. Modeling piezoelectric actuatorsSi =xNln, (2.5)Tj =FAn, (2.6)Dm =qAn, (2.7)Ek =Vln, (2.8)where, ln and An are the nominal length and the nominal cross sectionalarea of the actuator respectively and N is the number of piezoelectric layersthat compose the actuator. Similarly, material properties in (2.1) and (2.2)are required to be replaced withsEij =AnNlnk, (2.9)dki = dmj =αCPk, (2.10)Tmk =lnCPNAn+lnα2C2PNAnk. (2.11)The model in (2.3) and (2.4) only includes 2nd order mechanical dynamics,which is sufficient for many applications. Higher orders for multiple modesand anti-resonances can easily be added.2.5.3 Non-linear electromechanical model of piezoelectricactuatorsThe linear model depicted in Figure 2.10(a), is incapable of modeling hys-teresis. In 1997, Goldfarb and Celonovic presented a new electromechanicalmodel that includes both dynamic behavior and hysteresis nonlinearity inpiezoelectric actuators [GC97]. A graphical representation of this model isdepicted in Figure 2.10(b). Similar to the linear model in 2.10(a), the pre-sented model consists of an electrical domain and a mechanical domain thatare linked via a piezoelectric coupling parameter n. The differences betweenthe two models are in the electrical domain, where 1. the internal voltage182.6. Control techniques for piezoelectric actuatorssource is replaced with an internal current source i in parallel with the ac-tuator’s clamped capacitance CP and 2. the hysteresis effect is modeled asthe voltage VH across a dipole H in series with CP .In this model, the internal current is proportional to the actuator’s move-ment via i = nx˙. In addition, the linear voltage across the capacitance Vtis proportional to the internal mechanical force through FP = nVt. Themathematical description of this model is given by,mx¨+ bx˙+ kx = F + nVt, (2.12)q = nx+ CPVt, (2.13)V = Vt + VH , (2.14)VH = H(q), (2.15)Equation (2.14) indicates that the total driving voltage of the actuator Vis the sum of the linear voltage Vt and the nonlinear hysteresis voltage VH .Equation (2.15) suggests that the hysteresis voltage is purely a function ofthe charge q passing through the actuator. In the absence of hysteresis non-linearity, the models in Figures 2.10(a) and (b) are equivalent [DIG92], andtheir coupling parameters are related via n = αCP . In order to implementthis model in a control structure, a hysteresis model H is required. Subsec-tion 2.6.1 discusses some of the available hysteresis models in the literature.2.6 Control techniques for piezoelectric actuatorsIt was described in Section 2.4 that hysteresis, creep, temperature change,and vibration hinder the performance of piezoelectric actuator-driven micro-positioning systems. However, there is a wide range of control techniquesavailable for piezoelectric actuators in the literature. This section, catego-rizes these techniques into four groups of: 1. Feed-forward, 2. Feed-back,3. Integrated sensor and actuator, and 4. Self-sensing techniques. The re-mainder of the current section discusses these group of techniques in detail.2.6.1 Feed-forward controlFeed-forward is a conventional technique in control theory that relieson a well-described model of the system to be controlled. It has been em-ployed in the literature to compensate for creep, vibrations, and hysteresisof piezoelectric actuator-driven systems [GZS+16]. Since the main focus192.6. Control techniques for piezoelectric actuatorsFigure 2.11: A feed-forward displacement control structure.of this study is on short time quasi-static applications, it only overviewsfeed-forward techniques for hysteresis compensation.Subsection 2.4.1 introduced hysteresis as an inherent nonlinear relation-ship between the driving voltage of a piezoelectric actuator and its resultingdisplacement at zero external forces. Goldfarb et al.’s model can describethis observation. Under zero external forces and at quasi-static applications,velocity and acceleration are negligible in comparison to displacement and(2.12) is reduced tokx = αVP . (2.16)Substituting VP = kx/α from (2.16) into (2.13) indicates that the chargeand displacement are linearly related:q = (α+kα)x. (2.17)Using this equation together with (2.14), (2.15), and (2.16) yieldsV =kαx+H((α+kα)x) = Γ(x), (2.18)in which, Γ represents a hysteretic voltage-displacemnt relationship. Thisconfirms that the Goldfarb et al.’s model is consistent with the hysteresisdescription in Subsection 2.4.1.Feed-forward displacement control structures apply the reference inputto a model of Γ(x) to calculate the required driving voltage of the actuatoras depicted in Figure 2.11. This structure is designed to follow a referencedisplacement xref under zero external forces at quasi-static frequencies. Theconventional approach to frame the feed-forward structure first constructsa model for the piezoelectric actuator as L(V ) = x. Then the model isinversed to L−1 = Γ and used as the feed-forward block.Models for L are divided into two groups of physical and phenomeno-logical models [GZS+16]. The physical models use domain wall theories on202.6. Control techniques for piezoelectric actuatorsa microscopic scale to model hysteresis [SO00]. They lead to a better un-derstanding of the physics of piezoelectric ceramics. However, the real-timeimplementation of physical models is not computationally efficient.Phenomenological models only consider the shape of the hysteretic voltage-displacement profile and fit mathematical operators to it. Recently, a com-prehensive review of phenomenological hysteresis models was conductedin [GZ19]. The remainder of this subsection reviews three widely used phe-nomenological hysteresis models.Maxwell hysteresis modelThe Maxwell model was originally proposed to describe the hystereticrelationship between force and displacement in a massless slider-spring sys-tem that is depicted in Figure 2.12(a) [CKC97]. In this system, FN is thenormal force from the ground to a mass-less slider, F is the external forceapplied to the system, x is the displacement of the slider, and kb is the stiff-ness of the spring. In addition, f is the friction force on the system, whichcan be calculated as f = µFN , where µ is the friction coefficient betweenthe ground and the slider. The force equilibrium for the slider yieldsF ={kb(x− xb), kb|x− xb| < f,fsign(x˙), else,, (2.19)where, xb is the initial position of the slider. Figure 2.12(b) depicts a graph-ical representation of (2.19). This model can be refined by employing nnumber of slider-spring elements, which is depicted in Figure 2.12(c). As aresult of this modification, (2.19) is generalized toFi ={kbi(xi − xbi), kbi|xi − xbi| < fi,fisign(x˙i), else,, (2.20)F =n∑i=1Fi. (2.21)A qualitative representation of the refined F -x profile is depicted in Fig-ure 2.12(d). This behavior of the slider-spring system, is similar to hys-teresis in piezoelectric ceramics, where each domain can be modeled as asingle slider-spring, and the domain interactions are modeled as the fric-tion. Goldfarb et al. employed this model to describe the hysteresis opera-tor VH = H(q) in their model in (2.15), where F is replaced by VH , and xis replaced by q.212.6. Control techniques for piezoelectric actuators(a) A mass-less slider spring system(b) displacement-force relationshipof the system in (a)(c) Modified Maxwell system (d) Displacement-force relationshipof the modified Maxwell systemFigure 2.12: Maxwell hysteresis model [CKC97] c© [ 1997], IEEE. Adoptedwith permission.The Maxwell model has been used to model Γ in many zero force ap-plications. However, since this model is based on Coulomb friction, it isrestricted to actuators with rate-independent hysteresis.Preisach hysteresis modelThe Preisach model uses an elementary operator called the Relay op-erator [GJ95], which is depicted in Figure 2.13 and it is mathematicallydescribed asγab(u(t)) =1, u(t) > a,u(t− 1), b < u(t) < a,0, u(t) < b,(2.22)222.6. Control techniques for piezoelectric actuators(a) A Relay operator+(b) Preisach hysteresis modelFigure 2.13: Preisach hysteresis model. (Adopted with permissionfrom [GJ95].)where, u(t) is the input, and a and b are the thresholds of the elementaryoperator. The hysteresis model L can be framed using n number of Relayoperators, which is shown in Figure 2.13(b), and mathematically describedasx = L(V ) =n∑j=1wtjγajbj (V ), (2.23)where, wj are the weights given to each Relay operators. Since this modeluses a double integral, inverting it to obtain Γ = L−1 is relatively compli-cated [GJ95].Prandtle-Ishlinskii hysteresis modelThis model is a simplified version of the Preisach model [GZS13]. It doesnot require double integration and its parameters can be determined througha simple numerical process. The elementary operators employed in Prandtle-Ishlinskii model are called Play operators. A graphical representation of thisoperator is depicted in Figure 2.14(a) and its mathematical representationis given asy(u(t)i) = max{u(t)i − r,min{u(t)i + r, y(u(t)i−1)}}, (2.24)where, r is called the threshold parameter. An initial value of y(u(t)0)corresponding to the initial value of u(t)0 is required to build the operators.232.6. Control techniques for piezoelectric actuators(a) A Play operator+(b) Prandtle-Ishlinskii hysteresis modelFigure 2.14: Prandtle-Ishlinskii hysteresis model [GZS13] c© [ 2014], IEEE.Adopted with permission.Figure 2.14 indicates that hysteresis can be modeled by employing n numbersof weighted Play operators asx(t) = L(V ) =n∑j=1wtjyj(V (ti)), (2.25)where each operator is defined byyj(V (t)i) = max{V (t)i − rj ,min{V (t)i + rj , yj(V (t)i−1)}} (2.26)and wt represents the weights. Fitting algorithms are available to determineoptimum weights. In addition, a number of procedures have been introducedto define the threshold values. A simple and effective procedure is to dividethe voltage range into n equal segments. Then, r would correspond to themaximum voltage at each segment [KJ01].The advantage of the Prandtle-Ishlinskii hysteresis model over previouslydiscussed models is its ease of inversion to determine Γ = L−1. For detailson inverting procedure, the reader is referred to [KJ01].In summary, feed-forward control structures for hysteresis compensationemploy phenomenological hysteresis operators to model a piezoelectric ac-tuator as L(V ) = x under zero external force conditions and at quasi-staticfrequencies. They invert the model to Γ = L−1 and use it as a feed-forward242.6. Control techniques for piezoelectric actuatorsorController unitorPiezoelectricactuatorFeedbacksensororFigure 2.15: A feed-back control structure.controller. Since they do not require any feed-back sensors, they are ex-cellent for micro-positioning applications with space limitations. However,they are not robust against un-modeled disturbances such as external forcesor changes in the environmental condition such as temperature.2.6.2 Feed-back controlConventional feed-back control structures use dedicated sensors to mon-itor the performance of piezoelectric actuators. A block diagram of a feed-back control structure is depicted in Figure 2.15. A large variety of sensorscan be employed in the control structure’s feed-back loop. Laser triangula-tion sensors, laser vibrometers, and strain gauges are widely used examplesof displacement sensors. For force sensing, strain gauges and piezoelectricload-cells are popular.Ensuring the stability of the structure is the main challenge of feed-back structures. In most industrial applications, hysteresis nonlinearity istreated as a disturbance input d to the system (See Figure 2.15) [GZS+16].The dedicated sensors in the feed-back loop provide powerful disturbancerejections and excellent reference signal tracking as well as stability androbustness against un-modeled dynamics [GZS+16]. Due to the efficientsteady-state and quasi-static performance of PID control units, they havebeen used in a large number of commercial micro-positioning stages. How-ever, PID controllers suffer from phase lags at higher frequencies [GZS+16].Some studies have addressed this by introducing complex controllers such asH-infinity [ZZW+19], positive position feed-back (PFF) [FC90], integral res-onant control (IRC) [AFM07], and integral force feed-back (IFF) [PDM92].In conclusion, feed-back control offers excellent performances in a largenumber of motion control applications specifically at macro scales. How-252.6. Control techniques for piezoelectric actuators(a) (b)+-Figure 2.16: (a) A tube using four electrodes [BRFM07]. (b) A new electrodepattern. (Reproduced from [MY08], with the permission of AIP Publishing.doi : 10.1063 1.2952506)ever, the often bulky and expensive sensors employed in these structureare difficult to apply to micro-positioning applications. Subsection 2.6.3reviews control techniques that integrates piezoelectric actuators with sen-sors to eliminate or minimize the direct implementation of external sensors.Subsection 2.6.4 provides a detailed review of control techniques that takeadvantage of the reversible characteristic of piezoelectric materials that al-lows using a piezoelectric actuator simultaneously as a sensor.2.6.3 Integrated sensors and actuatorsThe main problem faces by the feed-back structures in Subsection 2.6.2,is cost and space constraints of micro-positioning applications. In order tosave space, a group of studies have suggested to mount strain gauges to thesurface of the actuator [FL10]. The benefit of this technique is its smallform factor and low cost. The strain gauge was employed in the feed-backloop of a displacement controller [FL10].Another group of studies split an actuator into actuation and sensingregions. Benders and tube piezoelectric actuators are typically used for thistechnique. This technique uses voltage measurements to estimate the dis-placement of the actuator under zero force conditions. While the electrodesof the actuating region are employed to drive the actuator, the reading ofthe induced voltage on the sensing region electrodes are used to estimate thedisplacement. This technique uses bridge circuits to measure the internal262.6. Control techniques for piezoelectric actuators+ -Ps PaA smallmicro-positioning work-space- +Capacitive sensorController unitFeedbacksensorPaPs(a) Integrating two identical trasducers(b) The feedback control structure for (a)Figure 2.17: Integrating two identical piezoelectric transducers proposedin [PKS18] c© [ 2018], IEEE. Adopted with permission.voltage VP of the sensing region, which is in a linear relationship with thedisplacement VP = −αx (See Subsection 2.5.2).Figure 2.16(a) shows the electrode geometry presented in [BRFM07] usedfor simultaneous 2D actuation and sensing of tube actuators. A 2D motionis generated by applying actuating voltages Vx+ and Vy+ to actuating elec-trodes X+el and Y+el respectively. The generated 2D motion at the tip of thetube induces sensing voltages Vx− and Vy− on sensing electrodes X−el andY −el respectively [BRFM07, BRM07]. Since the system contains two inputs,it is modeled as a MIMO system which accounts for cross coupling. Thisapproach has two drawbacks: 1- Using one electrode for actuation in eachdirection reduces the actuation range by half. 2- The actuator is drivenasymmetrically [BRFM07].The electrode pattern shown in Figure 2.16(b) address the aforemen-tioned challenges [MY08]. The outer electrode is divided into 12 equal272.6. Control techniques for piezoelectric actuatorselectrodes of 30◦ angular sweep each. To drive the actuator in the ~i di-rection, two opposite driving voltages with the same magnitude are appliedto two electrodes on opposite sides. The central electrodes are then usedto measure the induced voltage as a result of the tube’s deformation. Thesame approach is used on the rest of the electrodes to drive the tube inthe ~j direction. For clarity, only the ~i direction electrodes are labeled inFigure 2.16(b). This electrode geometry provides symmetric actuation andimproves the operational range. It was used widely in early atomic forcemicroscopy [YAM10, BYMS12]. Further developments have mainly focusedon controller design for reducing vibrations due to cross coupling betweenthe axes of an AFM [TZD05, YZL09, BYMM13]. Integrating actuating andsensing electrodes is a popular technique for low amplitude applications,where hysteresis effects are small. For applications with larger displace-ment amplitudes, the hysteresis effect introduces significant errors on thistechnique.Recently, a new displacement measurement method was introduced thatintegrates two identical transducers [PKS18]. Figure 2.17(a) depicts the im-plemented configuration. While the actuating transducer Pa performs themicro-positioning task in a limited work-space, the sensing transducer Psis employed to measure the actuator’s displacement by mimicking its per-formance outside the work-space. The two transducers are electrically con-nected in parallel. When an identical driving voltage V is applied to thetwo identical transducers, both transducers move with the same displace-ment xa = xs. The displacement of the sensing transducer xs is measuredwith a corresponding sensor and employed in a feed-back loop, (see Fig-ure 2.17(b)), to control the position of the actuating transducer. In order toachieve the full stroke of the actuator, the required power for the combinedsensing and actuation system is doubled. In addition, the setup is not robustagainst environmental mismatches between the transducers and it requiresconstant external forces.In conclusion, Integrating sensors with actuators ensures robustness againstun-modeled dynamics. The main disadvantage of this technique over othercontrol techniques is that it reduces the operational range of the actuators.2.6.4 Self-sensingSelf-sensing employs an actuator simultaneously as a sensor. It is robustagainst disturbances and can be used in miniature micro-positioners withoutexternal sensors. Figure 2.18 depicts the self-sensing techniques for piezo-electric actuator reviewed in this subsection. These techniques use different282.6. Control techniques for piezoelectric actuatorsSelf-sensing techniques reviewed in this studyP.A.VAB:A:D:F:G:E:C:Figure 2.18: Self-Sensing techniques based on the piezoelectric actuator’selectrical measurements.measurement combinations of voltage V , charge q, or impedance Z = V/Ifor displacement and/or force self-sensing.A. Displacement estimation using voltageThe methods described herein employ the linear electromechanical modeldescribed in Subsection 2.5.2 to estimate displacement of a piezoelectricactuator from itsvoltage measurements. These methods are most efficientfor damping applications where low driving voltage amplitudes are used.Capacitive bridges: The first self-sensing method in the literature waspresented in 1991, where a bridge circuit was employed for vibration controlof a piezoelectric cantilever [DIG92]. The linear electromechanical model de-picted in Figure 2.10(a), suggested that the displacement of an actuator isproportional to the generated internal voltage, (x ∝ VP ). Therefore, by ex-tracting VP one can estimate displacement. Figure 2.19 shows the schematicsof a bridge circuit employed in [PGCL06]. The resistances R1 and R2 are292.6. Control techniques for piezoelectric actuators++Figure 2.19: Bridge circuit employed in [PGCL06] for displacement self-sensing. When the hysteresis effect is negligible the output voltage of the cir-cuit is proportional to the displacement of the transducer, Vs(s) ∝ VP (s) ∝X(s) [PGCL06] c© [ 2008], IEEE. Adopted with permission.employed to prevent DC charge drifts in the circuit. The capacitors are se-lected equal to the piezoelectric clamped capacitance (C1 = C2 = C3 = CP ).The output voltage Vs of the circuit is then proportional to VP and conse-quently proportional to the displacement of the actuator [PGCL06]:Vs ∝ VP ∝ x. (2.27)Although this technique is popular in the literature for vibration control[KS10, SJHM+04, HCLQ18, HWP+18, PHT+08, LLH03], it performs poorlyat low frequencies and constant driving voltages, due to internal chargeleakages of the employed amplifier [PGCL06]. Also, a close match betweenthe actuator capacitance and the circuit capacitors is difficult to achieve inpractice. In addition, neglecting hysteresis causes imbalances in the bridgecircuit. To address this problem, an auxiliary adaptive balancing circuit wasdesigned in [KS10, KS09], which adds more complexity to this technique.Shunt circuits: The electromechanical behavior of piezoelectric actua-tors contains lightly damped resonances [EF10], which limit their high fre-quency performance. To address this challenge, a passive control tech-nique was introduced in [HvF91], where an RL circuit was attached in302.6. Control techniques for piezoelectric actuatorsa) Equivalent circuit b) Feedback structure++++--+Figure 2.20: Implementation of a passive shunt damping structure to dampdisturbance vibrations [MFB03] c© IOP Publishing. Adopted with permis-sion.series with a piezoelectric cantilevered beam. By tuning the resistanceand inductance values, the beam vibrations due to disturbances were dissi-pated [HvF91]. Since then, a large number of studies have focused on passiveshunt damping [Car01, KPvdVG97, RGAC99, TW01, TRB01, Wu98]. In2003, a comprehensive review of these methods was provided in [Moh03].The initial methods were dedicated to one mode damping, (mainly the firstmode) [HCVF90, yW96]. More recent studies focus on multiple mode damp-ing [Wu98, BM02, Hol94]. A more recent review on shunt damping waspublished in [YWH+17]. This subsection reviews the key milestones of thistechnique that are applicable to self-sensing applications.A study in [MFB03] suggested that a passive shunt vibration dampingcontrol structure can be interpreted as a feed-back control structure. Thiscan be explained with Figure 2.20(a), where a piezoelectric transducer isattached to a flexible structure.The transducer is shunted to an electrical admittance Y (s). An inducedmechanical disturbance force w(s) on the structure will result in a voltagedifference V (s) on the transducer’s electrodes and eventually a current I(s)passing trough the admittance. The voltage drop across Y (s) can be tunedto maximize the extracted mechanical energy from the disturbance.The corresponding block diagram is shown in Figure 2.20(b). In the ab-sence of a driving voltage V (s), (closed circuit), the generated voltage VP (s)due to w(s) is VP (s) = GVPw(s)w(s) . In the absence of the disturbance,VP (s) is due to the driving voltage V (s), VP (s) = GVPV (s)V (s) [MFB03].312.6. Control techniques for piezoelectric actuatorsAssuming a linear systemVP (s) = GVPw(s)w(s) +GVPV (s)V (s). (2.28)The current passing through the admittance can then be expressed asI(s) = Y (s)V (s) = (VP (s)− V (s))CP s. (2.29)Using (2.28-2.29) a feed-back structure is framed and depicted in Figure2.20(b), where the transfer function between VP (s) and w(s) is given byTVPw =VP (s)w(s)=GVPw(s)1− GVP V1+Y (s)CP s. (2.30)The poles of GVPw and GVPV are associated to the base structure, so theyare the same [MFB03]. To damp the modes of the structure, the poles ofTVPw need to move towards the left in the s plane. This can be achievedwith a properly designed admittance Y (s) [MFB03]. This method has beenemployed in a large number of studies [FM03a, BFM03, FM03b, MFB04,FM06].The reviewed vibration damping methods above assume that zero refer-ence voltage is applied to the structure. The feed-back structure in [MFB03]is employed in the passive voltage control structure [EF10], and it is shownin Figure 2.21(a), where a simple RL circuit is attached in series with thetransducer. A simplified block diagram of the presented circuit is depictedin Figure 2.21(b), where G(s) represents the transducer and K(s) representsthe feed-back controller.Resistance and inductance values are selected in such a way that thel2 norm for GxVW (s) =X(s)VW (s)and consequently the energy of the systemis minimized. The obtained impedance Z(s) is implemented syntheticallyusing digital signal processors (DSP) [FMM14]. The challenges with this ap-proach are: 1- Hysteresis is neglected. 2- Determining the optimum dampingimpedance Z is complicated. Recent developments of this technique focuson implementation issues such as better tuning algorithms and adaptivityof the synthetic impedances [NVM16, PKDS07, MCCSR14].In conclusion, the main purpose of shunt circuits is damping the vibra-tion modes in piezoelectric actuator-driven systems. The advantage of shuntcircuits over similar methods such as notch filters [FTL15] is that they canbe interpreted as feed-back control structures, (see Figures 2.20 and 2.21).For more information on shunt circuits, the reader is referred to [YWH+17].322.6. Control techniques for piezoelectric actuatorsa) Equivalent circuit b) Feedback structure representation++Filter -+Figure 2.21: Passive shunt damping interpreted as a feed-back control struc-ture [EF10] c© [ 2010], IEEE. Adopted with permission.B. Displacement or Force Estimation Using ChargeCombining equations (2.3) and (2.4) provides a relationship betweencharge, displacement and force,q(s) = (mαs2 +bαs+kα+ αCP )X(s)− 1αF (s). (2.31)At frequencies well below the first mechanical resonance, displacement andforce are linearly related to the charge passing through the actuator,q(s) = (kα+ αCP )X(s)− 1αF (s). (2.32)This relationship can be derived from Goldfarb et al.’s equations as well. Byeliminating Vt from (2.12) and (2.13) at low frequencies, where displacementis dominant over velocity and accelerationq(s) = (n+CPkn)X(s)− CPnF (s). (2.33)A number of self-sensing techniques measure charge and map it to eitherdisplacement or force [IRLC09b].The simplest way to measure charge is integrating the voltage dropacross a shunt resistor in series with the actuator [WBBK05, GKR+08].To measure small charges, the unit resistance needs to be high leading toconsiderable ohmic losses. To minimize these losses without incurring ex-cessive noise, analog current integrator amplifiers [FUM98, CZDS06] can beemployed. Although, this approach has been successful in high frequency,332.6. Control techniques for piezoelectric actuators+Figure 2.22: The schematics of a conventional charge amplifier used fordisplacement self-sensing purposes [FM06] c© [ 2006], IEEE. Adopted withpermission.(below first resonance of the system), applications such as atomic force mi-croscopy [ASDR+03, RM13, AdRS+10, AMR+05, LIS99, RM16] and inkjetprinting [KK07], its low frequency performance is poor due to charge driftsin the integrator circuits. Some studies have also used numerical current in-tegration instead of analog current integrator amplifiers [IS14, IS12b, IS12a].However, the accuracy of this technique is limited to the resolution of thecurrent probes and there is no improvement in terms of integration driftcompared to analog current integrator amplifiers.Charge amplifiers are another implementation of charges based positionself sensing. Rather than inferring position from a measurement of charge,this approach directly controls charge. Figure 2.22 depicts the schematicsof a generic charge amplifier [FM06, FM03c]. Neglecting hysteresis, theactuator is modeled as a capacitance CL in series with a voltage source VP .A feed-back loop gainKc is employed to equate the voltage across the sensingcapacitance Cs with a reference voltage Vr. Thus a charge proportional to Vris supplied to the actuator,qL = VrCs. (2.34)For a load resistance, RL, the actual charge passing through the actuator, q342.6. Control techniques for piezoelectric actuators+Figure 2.23: A modified charge amplifier with improved low frequency per-formance.(Reproduced from [FM05], with the permission of AIP Publishing.doi : 10.1063 1.1938952)is governed by a high pass filterQ(s)Vr(s)= Cs(s)ss+ 1RLCL, (2.35)with a cutoff frequency of ωc =1RLCL. For a typical piezoelectric actuator,this cutoff frequency has a very low value of a few tens of Hertz [FM06],which is satisfactory for high frequency applications. However, many micro-positioning applications, such as atomic force microscopes, require betterlow frequency performance.The poor low frequency performance of generic charge amplifiers can beimproved with a secondary voltage feed-back loop. The second loop reducesthe charge drifts at low frequencies [FM04, FM06, YV05]. None of theabove methods can drive grounded loads [BRFM07]. Figure 2.23 shows aschematic charge amplifier that is able to drive grounded loads [FM05]. Inthis model, the circuit leakage is modeled as Rs. By letting RLCP = RsCs,the amplifier transfer function is Q(s)Vr(s) = Cs(s), which has no low frequencydynamics and has shown excellent performance at frequencies as low as afew mHz [FM05]. However, exactly matching RLCL with RsCs is impossiblefor all operating conditions.An alternative method to address the poor low frequency performance ofcharge amplifiers is presented in 2.24 [IRLC09b]. This study uses the model352.6. Control techniques for piezoelectric actuatorsFigure 2.24: A charge amplifier for displacement self-sensing. (Reproducedfrom [IRLC09b], with the permission of AIP Publishing. doi : 10.10631.3142486)described in Subsection 2.5.2. Assuming that hysteresis is absent in (2.13),Vt = Vx =1n(q − CPV ). (2.36)Figure 2.24 shows the circuit that is employed in [IRLC09b]. A leakingresistance RFP is modeled parallel to the internal current source i = nx˙.A capacitor Cr is used in parallel with the actuator with a value close tothe actuator’s capacitance Cr ∼= CP . The current passing through the ac-tuator is integrated via a capacitor C. The actuator’s low frequency freedisplacement (F = 0 N) is estimated asx˜free,lf =1n(CVo − CrV − 1RFP∫ t0V dt). (2.37)This method can only estimate the quasi-static displacement at zero forceconditions. Recently, the method was adopted for a 2 DOF bender thatconsists of four separate electrodes. Charge was modeled as the sum ofinduced charges resulting from strain in the two motion axes. A numericalprocedure is then framed to extract the displacements in different directionsfrom the four voltage readings together with the measured charge [IAA+17].In conclusion, charge based self-sensing techniques benefit from the linearrelationship between charge, displacement, and force. They require sophis-ticated charge measurement circuits and they are better suited for high fre-quency applications. Their low frequency performance is poor due to circuit362.6. Control techniques for piezoelectric actuatorsOperating rangeSlope(a) Polarization-Electric field (b) Permittivity-Electric field (c) Displacement-Electric field (d) Permittivity-DisplacementFigure 2.25: The permittivity-displacement correlation under zero externalforce [KKHM08]. (Adopted with permission form Taylor & Francis.)charge leakages. A number of methods were reviewed herein that improvethe quasi-static performance of this technique. However, its static perfor-mance still suffers from circuit charge leakages. In addition, the op-ampsare highly sensitive to temperature changes and disturbance noises.C. Displacement Estimation Using PermittivityThe techniques described in Subsection 2.6.4.B compensates for hys-teresis using a linear relationship between charge, displacement, and force.However, it suffers from a weak low frequency performance of charge de-tection circuits. Recently, an alternative technique has been proposed thatuses permittivity instead of charge to estimate displacement under constantforce conditions.Figure 2.25(a) shows a strong hysteretic relation between the electricfield E applied to a piezoelectric actuator and the developed polarization Pin it. The practical operating range for the actuator is shown as a red loopin the figure, which is limited to positive electric field values. Assuminga linear lossy behavior for piezoelectric actuators, the polarization and thedielectric displacement are nearly equal (P ∼= D) [HBOS98]. Therefore, theslope of the P -E profile represents the permittivity of the material: = 0P =DE∼= PE. (2.38)In (2.38), 0 is the vacuum permittivity and P is the actuator’s relative372.6. Control techniques for piezoelectric actuators+++++Lock-in amplifierLock-in amplifierDetecting voltage, A++ AADetecting voltage, Detecting voltage, Band pass filter(a) Using the detecting current magnitude(b) Using the imaginary part of the detecting current (c) Using equal and opposite detecting currentsDriving voltage, Driving voltage, Driving voltage, AFigure 2.26: The displacement self-sensing techniques using permittivitymeasurements proposed by Morita et al. [KKHM08, IM10, IM11, SM15].(Adopted with permission form Taylor & Francis and Sensors and ActuatorsA: Physical )permittivity. The butterfly curve depicted in Figure 2.25(b), qualitativelyrepresents changes in  as E varies. Figure 2.25(c) shows a similar re-versed butterfly curve between the x and E. Using this similarity, Morita etal. [KKHM08] hypothesized a linear correlation between  and x, which isdepicted in Figure 2.25(d). They experimentally validated their hypothesisemploying a real-time permittivity measurement method.Figure 2.26(a) shows a schematic description of the method in [KKHM08],which was employed on a biomorph bender actuator. A permittivity detect-ing voltage Vr with a fixed small amplitude and a fixed high frequency ωris added to the driving voltage of the actuator V and the resulting volt-age Vt = V + Vr is then applied to the actuator. The generated current Irdue to Vr is extracted from the total current It, using a band-pass filter.Impedance of the actuator can then be determined as Z = VrIr . For an ac-tuator modeled as a capacitor C and resistor R in parallel, the relationshipbetween Z, R, and C can be arranged as1Z=IrVr=1R+ jωrC, (2.39)where, j represents the imaginary unit. Assuming an infinite value for theresistance R of the actuator, and given the fact that Vr has a fixed amplitude,the magnitude of the detecting current is proportional to the permittivity (|Ir |∝ C ∝ ) [KKHM08]. The amplitude of Vr needs be small, so it doesnot affect the movement of the actuator. However, it needs to be large382.6. Control techniques for piezoelectric actuatorsDetecting voltage, ++Driving voltage, ABand Pass FilterFourier TransformsPSAVFigure 2.27: The capacitance measurement method proposed by Islam etal. [ISM11a, ISM11b] c© [ 2011], IEEE. Adopted with permission.enough to induce a measurable current Ir. In [KKHM08], the amplitude ofthe detecting voltage | Vr | is selected to be about 2% of the magnitude ofthe driving voltage | V |. In addition, the higher the frequency, the larger |Ir | gets. This is experimentally validated in [IM10], where substantialimprovement of the signal-to-noise ratio (S/N) was obtained by increasing ωrfrom 100 kHz to 500 kHz.Morita et al. modified their method assuming a finite value for R [IM11].Expanding (2.39) to1Z=Ir,realVr+ jIr,img.Vr=1R+ jωrC, (2.40)shows that the imaginary part of the detecting current Ir,img. is proportionalto the permittivity (| Ir,img. |∝ C ∝ ). The modified method is depicted inFigure 2.26(b), where a lock-in amplifier extracts Ir,img.. A positioning errorof less than 0.4µm over a movement range of 80µm is achieved in [IM11].The main challenge with this method is achieving a desirable S/N ra-tio. Morita et al. improved upon it on a positioning stage that consists oftwo stack actuators assembled in series [SM15]. The method is depicted inFigure 2.26(c). The two actuators are fed with equal and opposite voltagesresulting in equal and opposite currents. A differential current measurementtechnique is then employed which subtracts the currents from each other and392.6. Control techniques for piezoelectric actuatorsuses a band-pass filter to extract the imagery part of the current. Therefore,the available current for for estimation is doubled 2Ir,img..Islam et al. performed a numerical approach to measure the capaci-tance [ISM11a, IS13]. Their method is depicted in Figure 2.27. The detect-ing signals Vr and Ir are extracted from Vt and It respectively using an ana-log filter. Fourier transformations are carried out in a MATLAB/Simulinkenvironment, where Fourier transform coefficients for Vr, AV , BV as wellas coefficients for Ir, AI , BI are determined. These coefficients are thenemployed to determine resistance R and capacitance C. This method re-quires less electrical devices (e.g., lock-in amplifiers) at the expense of morecomputation. Islam et al. implemented their technique in a sensor-lessdisplacement control setup of a piezoelectric stack actuator [IS12b].Islam et al. extended their technique to an impedance measurementmethod [ISM11b, ISA15]. They experimentally showed that the resistance Ris in correlation with the temperature change of the actuator. In [ISA15],they proposed a second-order polynomial correlation between R and thetemperature change of a bender while C is being used to estimate x.In the last decade, this technique has attracted growing attention in theliterature [BUH+12, ARR+15]. It has demonstrated good low frequencyperformance, and requires relatively simple hardware. However, there aresome questions that need to be answered:− The linear correlation between x and  was experimentally observed.The reason for this correlation still needs to be explained. It is wellknown that the domain structure change affects the permittivity changeas well as the displacement of an actuator. Therefore, domain struc-tures need to be studied on a microscopic level to achieve a betterunderstanding of the experimental observations [IM11].− It was experimentally shown that higher detecting frequencies ωr re-sults in better S/N ratios. However, there is no clear guideline in theliterature to find the best detecting frequency for different applications.The techniques reviewed in Subsections 2.6.4A to 2.6.4C use a single electri-cal parameter measurement, (charge, voltage, or permittivity), to estimateeither displacement or force. However, monitoring both displacement andforce is of great importance in many micro-positioning applications. Theremainder of this chapter reviews the techniques for simultaneous displace-ment and force estimation.402.6. Control techniques for piezoelectric actuators-+ +-PZTCharge amplifier Static estimator(1.37) and (1.41)Dynamic estimatorFigure 2.28: The scheme of the complete low and high frequency displace-ment x˜ and low frequency force F˜lf estimation method [RIK+10] c© [ 2010],IEEE. Adopted with permission.D. Displacement and Force Estimation Using Voltage and ChargeThis technique is most efficient at high frequencies. However, due to thehigher importance of slow applications, most of the methods introduced forthis technique are designed for low frequencies.Rakotondrabe et al. added low frequency force detection to their chargebased low frequency free displacement self-sensing method, (reviewed in Sub-section 2.6.4B), by including voltage measurements [IRLC09a].First, the circuit described in Figure 2.24 is employed to estimate an ac-tuator’s low frequency free displacement xfree,lf from charge measurementsusing (2.37). Taking advantage of the linear relation between charge, dis-placement, and force, the same circuit is also employed to estimate a lowfrequency external force Flf in a similar fashion as xfree,lfF˜lf =1β(CVo − CrV − 1RFP∫ t0V dt− qH − qcr), (2.41)where, β is the force sensitivity coefficient, and qH and qcr are nonlinearcharges due to hysteresis and creep respectively. The total low frequencydisplacement is then estimated by super-positioning the displacement dueto the external force to the free displacement of the actuator asx˜lf = x˜free,lf − 1kF˜lf . (2.42)A detailed procedure was presented for identifying the model parameters β,k, qH , and qcr. A dynamic observer was added to this method to estimatelow and high frequency displacements and low frequency forces in [RIK+10].The extended method is depicted in Figure 2.28. The dynamic observer412.6. Control techniques for piezoelectric actuators-+ +-P. A.Charge amplifier(Figure 10) Low frequencyestimator(1.37) Dynamic displacement estimatorInverse dynamcis based uknown input observer [LP02]Figure 2.29: The scheme of the complete low and high frequency displace-ment x˜ and force F˜ estimation method [Rak13] c© [ 2013], IEEE. Adoptedwith permission.consists of three gains G1, G2, and G3 in an inverse multiplicative structureand it estimates the free high frequency displacement x˜free through realtime numerical processing [RIK+10, AR18]. The complete low and highfrequency displacement estimation in the presence of external forces is thendetermined via:x˜ = x˜free − 1kF˜lf . (2.43)The parameters for the observer are conveniently determined from the samemodel parameters already determined for (2.37) and (2.41) [RIK+10].Rakotondrabe et al. further extended their method to low and highfrequency displacement and force self-sensing [Rak13]. The extended tech-nique treats forces as unknown inputs to the system. The schematics of thistechnique is depicted in Figure 2.29. An inverse dynamic unknown-input-observer is employed to estimate force [LP02]. Validation experiments arecarried out by driving a bender actuator against a stationary force sensor.Therefor, the displacement and force are not completely independent fromeach other. The performance of the method needs to be tested using a 2 DoFsetup that is capable of creating simultaneously varying displacement andforces. Although this method provides a complete low and high frequencyestimation of both displacement and forces, its complexity both in termsof the employed circuitry and required computation makes it difficult to beimplemented in real-time control applications.422.6. Control techniques for piezoelectric actuatorsPEAALow frequency displacement and force self-sensingBased on the piezoelectric nonlinear model (1.12-1.15)VFigure 2.30: The schematic of a low frequency displacement and force self-sensing method using Goldfarb et al.’s nonlinear electromechanical model in(2.12 to 2.15) [BQN08b, KSUO12, MU14] c©[ 2008], IEEE.A number of studies rearrange Goldfarb et al.’s electromechanical modelin (2.12-2.15), and use it to estimate displacement and force of a piezo-electric actuator from its charge and voltage measurements at quasi-staticfrequencies. The rearranged form of Goldfarb et al.’s isx˜ =qn+CPH(q)n− CPVn, (2.44)F˜ =nV − nH(q)− kx˜. (2.45)The graphical representation of a self-sensing method that uses (2.44) and(2.45) is depicted in Figure 2.30. The physical implementation of thismethod requires the characteristic parameters of the actuator, Cp, n, and k,together with a hysteresis operator H.A group of studies implemented this self-sensing method to control thedisplacement and force of piezoelectric driven magnetic resonance imaging(MRI)-Compatible tweezers [KSUO12, MU14]. These studies do not con-duct a parameter identification task and rely on the parameters reportedin the data-sheet provided by the actuators’ manufacturer. A drawbackof this approach is that the characteristic parameters of an actuator canchange their values due to their variations in its operational environment.In addition, the parameters vary from one actuator to another.Badel et al. suggested using three sets of measurements to identify theparameters of a piezoelectric actuator [BQN08b]. 1. In the first experi-ment, a major V vs. q loop is generated at zero external force, which thenis employed to construct hysteresis function H between V and q. Badelet al. used their previously introduced hysteresis operator, which consists432.6. Control techniques for piezoelectric actuatorsof two hyperbola functions for ascending and descending branches of q-VHloops [BQN08a]. 2. The maximum displacement xmax is measured at max-imum driving voltage Vmax and zero force. 3. The maximum force Fmax ismeasured at the same maximum driving voltage Vmax, but at zero displace-ment. The stiffness is then determined ask =Fmaxxmax. (2.46)Using (2.12) at zero displacement, the value of n can be determined asn =FmaxVt,max=FmaxVmax − VH,max , (2.47)where, VH,max is the maximum hysteresis voltage measured during the firstset of measurements. Using (2.12) and (2.13) at zero force and quasi staticconditions a relationship between the charge and the linear voltage is deter-mined asq=(CP +n2k)Vt. (2.48)By fitting the first set of measurements to this relationship, (CP +n2k ) andeventually CP are determined.Badel et al. implemented the characterized model in Figure 2.30 in adisplacement and force control structure where, less than 2% error valueswere measured for both displacement and force. However, their identificationmethod contains some simplifying assumption that need to be noted. Forexample, in order to model the hysteresis in their first set of experiments, itis assumed that the slope of the descending branch in VH versus q approacheszero. Also, the effect of hysteresis is not considered in determining k.In conclusion, Subsection 2.6.4D reviewed the self-sensing technique thatuses charge and voltage measurements to estimate both displacement andforce. These techniques face two main challenges. Its low frequency perfor-mance is relatively weak due to different charge leakages, (More informationis provided in Subsection 2.6.4B). Also, modeling the hysteresis nonlinearityis required.E. Displacement Estimation Using Charge and PermittivityThe charge based technique, described in Subsection 2.6.4B, showed re-liable performance at high frequencies, but it suffered at low frequencies. Incontrast, the permittivity/capacitance based technique, described in Subsec-tion 2.6.4C, showed good performance at low frequencies [MS16]. In order442.6. Control techniques for piezoelectric actuatorsDriving voltage, Detecting voltage, ++++-+APSAVL.P.FMS16Figure 2.31: The schematic of the displacement self-sensing technique usingcharge and capacitance measurements [IS14, IS12a] c© [ 2014], IEEE.to benefit from the strengths of both techniques, Islam et.al fused them intoa hybrid displacement self-sensing technique [IS14, IS12a]. The target fre-quency range of their presented technique was [0 Hz to 100 Hz] and the goalwas to use charge based measurements at high frequencies and capacitancebased measurements to compensate for the charge drifts at low frequencies.A schematic of this technique is depicted in Figure 2.31.A low frequency position estimate, x˜C , is determined using the real-timecapacitance measurement, which is illustrated in Figure 2.27 and is detailedin Subsection 2.6.4C.The current due to the driving voltage I is obtained by low-pass filter-ing the total current It. The obtained current is then used to estimate acharge based displacement x˜q from (2.33) assuming zero external forces. Atlow frequencies, x˜q is prone to drift due to current Io representing chargeleakage and current measurement offsets. In order to address this issue, x˜Cis employed in a feed-back structure to identify Io and subtract it from I:Imod = I + Io = I + L(x˜c − x˜q). (2.49)Where the feed-back term L contains a proportional and an integral opera-tor,L = LP +LIs. (2.50)The feed-back operators are selected as LP = 2ωtran(n +kCPn ) and LI =452.6. Control techniques for piezoelectric actuatorsVANumerical processing using (1.51)Figure 2.32: The self-sensing technique using capacitance and charge mea-surements [LALR18] c© [ 2018], IEEE.ω2tran(n +kCPn ). Consequently, the transition ωtran between xq and xC isbased on the frequency range of the two position estimates. However, thepresented technique does not provide a clear instruction on how to select anappropriate ωtran.Liesli et al. [LALR18] employed permittivity measurements to modifythe charge based displacement self-sensing method that was described inFigure 2.24.The circuit in Figure 2.24 assumed a constant value for CP and em-ployed a capacitor Cr ≈ CP to compensate for the capacitive feed-throughcharge CPV in (2.13) and estimate the displacement from (2.36). However,as described in Figure 2.25(b), the actuator’s capacitance varies under dif-ferent input voltages. Liesli et al. suggested to replace the constant CP in(2.36) with a voltage dependent capacitance C∗r ,x˜free =1n(∫I − qDA − C∗rV ). (2.51)For this reason, an impedance analyzer is employed to create a map betweenimpedance Z, frequency f , and input voltage V under zero external forces.This map is then employed to generate the numerical parameter C∗r = γVat any frequency of interest.In conclusion, Subsection 2.6.4E reviewed the methods that use chargeand permittivity/capacitance to estimate the displacement of piezoelectricactuators. The advantage of these methods is that they compensate forhysteresis and they can be used across a large frequency range. However,the real time implementation of these methods is computationally expensive.462.7. A comparison between the reviewed control structuresF. Displacement and Force Estimation Using ImpedanceIn theory, using the complete phase and magnitude information of themeasured impedance can be used to find two detect two unknowns. Thistechnique is introduced in this study and explained in Chapter 8.G. Combining Voltage and Impedance MeasurementsTo best of author’ knowledge, there is no study in the literature thatcombines voltage and impedance measurements. Chapter 8 will show thatthe impedance itself can be used to estimate both displacement and forcevalues. Therefore, the effect of adding voltage measurements could provideimproved accuracy and help estimate a large frequency range. Since, thevoltage measurement is hysteretic, hysteresis models will be required, whichwill complicate the method.2.7 A comparison between the reviewed controlstructuresIn conclusion, this chapter reviewed the available motion control meth-ods for piezoelectric actuators in the literature. Feed-forward methods relyon an accurate models of the actuator to be controlled and do not requireexternal sensors. However, their performance is weak when subjected to un-modeled disturbances. Feed-back controllers employ external sensors in theirfeed-back loop, which give them robustness against disturbances. However,it makes their implementation a challenging task due to space limitationsin many micro-positioning applications. An alternative method to the twoconventional feed-forward and feed-back control methods is to integrate anactuator with a sensor outside the actuator’s work-space. A second alterna-tive is called self-sensing which uses an actuator simultaneously as a sensor.This thesis proposes novel actuator-sensor integrations as well as novel self-sensing methods.47Chapter 3The design of a 2 DoF testsetupThis chapter describes the design of an experimental test-bed that gen-erates independently varying displacement and forces for piezoelectric actu-ators. The setup is instrumental to validate the proposed displacement andforce control techniques in the later chapters of this thesis.3.1 Setup designThe majority of the setups to study piezoelectric actuators in the lit-erature either generate constant forces, or forces that follow the actuator’sdisplacement. Examples are depicted in Figure 3.1. In Figure 3.1(a), apiezoelectric stack actuator (PESA) is mounted between a force sensor anda needle [IS14]. In order to minimize the mounting misalignments and elim-inate the undesired moment loads on the actuator, ball seats have beenemployed. A spring with the stiffness of ks is employed to exert desiredpreloads on the actuator. To ensure that the preload remains relatively con-stant while the actuator is driven, the stiffness of the spring is selected to beconsiderably less than the actuator’s stiffness ks k [IS14]. Figure 3.1(b)depicts a typical single degree of freedom positioner, in which both displace-ment and force vary, increase or decrease, in the same direction [EVGP14].The setups in Figure 3.1 have been employed in a large number of studieson displacement control structures. However, the main goal of the currentthesis is to control displacement and force independent from each other.A 2 DoF setup is presented herein that consists of two equivalent PESAsassembled in series. The schematic of the proposed setup is depicted in Fig-ure 3.2(a). In this setup, PESA-1 is considered to be the main actuator to becontrolled and PESA-2 is employed to vary the external forces. Therefore,the setup benefits from the fast response time and high stiffness of piezo-electric actuators. Large external loads can be generated almost instantlyand independent from the displacement.483.1. Setup designForce s ensorPESA(a) (b)Force s ensorPESADisplacement DisplacementForce ForceMoving platformFigure 3.1: Examples of setups in the literature. (a) A test setup that oper-ates under constant force [IS14] c©[ 2014], IEEE. (b) A one degree of freedompositioning stage [EVGP14]. (Adopted with permission from [FL14].)The outer ends of the actuators are fixed using ball seats, which pre-vent misalignments and the actuators’ tips are pressed against a thin metalsheet in the middle of the setup. This configuration will serve three mainpurposes: 1. Piezoelectric stack actuators have to be preloaded to providedomain alignment and prevent structural damage during operation. 2. Theactuators are always in contact and experience the same displacement. 3.The tip displacement can be detected by measuring the displacement ofthe metal sheet. By applying driving voltages V1 and V2 to PESA-1 andPESA-2 respectively, the location of the metal sheet and consequently thedisplacement of the actuators, as well as the preload on the actuators canbe controlled independently. To visualize this, consider applying equal andpositive voltages to these equivalent actuators. As a result, the metal sheetwill not move, but the actuators experience a preload increase. If equal andopposite voltages are applied, one actuator will contract while the other onewill extend the same amount. Therefore, the location of the metal sheet will493.1. Setup designForce SensorLaser SensorPESA-1(a) Schematic of the proposed setup PESA-2Force SensorForce Sensor(b) Electromechanical model of the setup(c) Free body diagram of the setupPre-loadingassemblyFigure 3.2: The proposed setup to create independently varying displace-ment and forces. Used with permission.change and the preload will stay the same. Displacement of the metal sheetbetween the actuators is measured with a differential laser vibrometer. Aforce sensor is mounted between PESA-1 and the solid frame of the setup.Using (2.12), the mechanical model of the setup is depicted in Fig-ure 3.2(b). Since PESA-1 and PESA-2 are equivalent, their characteristicsparameters are assumed to be the same. In addition, in this configurationthe displacement and force of the actuators are the same. However, sincePESA-1 is the main actuator, the displacement and force are labeled as x1and F1 respectively. The model combines the lumped masses of the actuatorsinto a single mass m. Neglecting hysteresis, piezoelectric forces nV1 and nV2are exerted on PESA-1 and PESA-2 respectively. The free body diagramsof the force sensor and the mass m are depicted in Figure 3.2(c). Forceequilibrium equations yield0 =− nV1 + kx1 + bx˙1 − F1, (3.1)mx¨1 =n(V1 − V2)− 2kx1 − 2bx˙1. (3.2)503.1. Setup designVAVPESA-1PESA-2Rigid FrameLaser VibrometerForce SensorAmp-2Amp-1Metal SheetBall SeatPreload Assembly DS 1103data acquisitionboardFigure 3.3: The test setup. Used with permission.Equations (3.1) and (3.2) can be framed in a state space form as[x˙1x¨1]=[0 1−2km −2bm] [x1x˙1]+[0 0nm − nm] [V1V2], (3.3)[x1F1]=[1 0k b] [x1x˙1]+[0 0−n 0] [V1V2]. (3.4)Using either (3.1) and (3.2) or (3.3) and (3.4), the transfer function of thesetup can be framed in Laplace domain as[x1(s)F1(s)]= G(s)[V1(s)V2(s)], (3.5)where,G(s) =[G11(s) G12(s)G21(s) G22(s)]=[nms2+2bs+2k− nms2+2bs+2k−n(ms2+bs+k)ms2+2bs+2k−n(bs+k)ms2+2bs+2k]. (3.6)The transfer function in (3.5) confirms that by varying V1 and V2, theproposed setup can generate independently varying displacement and forces.The actual setup is depicted in Figure 3.3. The actuators PESA-1 andPESA-2 are mounted in series within a rigid frame using ball seats. Both513.2. Frequency response analysisactuators are PI-885.90 from Physik Instrumente with the dimensions of5 mm × 5 mm × 36 mm. The remaining actuator specifications are listed inTable 3.1. A 0.5 mm thick steel sheet, is sandwiched between the tips of theactuators. The setup’s frame consists of a preloading system that pressesthe actuators against the metal sheet and creates sufficient preloads, (seeTable 3.1), on them. Figure 3.4 depicts the preloading mechanism, whichconsists of a threaded hole machined in the setup’s frame. A bar is placedwithin the hole to push PESA-1. The bar rotation of the pushing bar isminimized with a key-seat and set screw.Driving signals u1 and u2 are created in a dSpace DS 1103 data acquisi-tion board and fed to amplifiers Amp-1, a PI-E617-00F high power amplifierfrom Physik Instrumente, and Amp-2, which is a PZD-700 linear amplifierfrom Trek, respectively. The amplifiers generate driving voltages V1 and V2for PESA-1 and PESA-2 respectively. A Tektronix-P2220 voltage probeand a Tektronix-TCPA300 current probe are employed to measure the volt-age and current of PESA-1. By integrating the current, charge q1 passingthrough PESA-1 is detected. A Polytec-HSV2002 differential high-speedlaser vibrometer with a resolution of 0.3 µm measures the tip displacementof PESA-1 x1m. Force F1m is measured by a PCB-208C2 force sensor witha resolution of 4 mN that is placed between the frame and PESA-1.3.2 Frequency response analysisFrequency domain least square fitting techniques and a second order ap-proximation of the piezoelectric actuators’ dynamics have been employedto determine the open loop frequency responses for PESA-1. Figure 3.5Rigid FramePESA-2 Threaded boltSet screwPushing barFigure 3.4: The preloading system.523.2. Frequency response analysisTable 3.1: Reference values for PESA-1 and PESA-2 characteristic param-eters. [Phy05]Parameter Data sheet valueDriving voltage range -20 to 120 VFree displacement 32 µm @ 100 VClamped force 950 N @ 120 VInitial preload 360 NCP @ 1 kHz 3.1 ± 20 % (µF )n 7.9 (N/V)*k 25 (106 N/m)*Data sheet value for α is determined by α = FmaxVmaxdepicts frequency responses from V1 to x1m and from V1 to F1m. The twofrequency responses exhibit slightly different resonance peak locations thatare attributed to measurement error. However, in this study, large displace-ments with the full voltage range are studied, where the operating frequencyrange of the setup is limited by the current saturation limit of the amplifiers.Assuming that the impedance of PESA-1 consists of a pure capacitance andusing the maximum driving voltage V1, the operating frequency range islimited tof =75 mA2pi(3.1 µF )(100 V )= 39 Hz, (3.7)where the current saturation of the PI-E617-00F amplifier is 75 mA and thecapacitance value of 3.1 µF is chosen from Table 3.1. Therefore, the ex-periments carried out in this study are limited to low frequency quasi-staticexperiments, which are well below the resonance frequencies measured fordisplacement or force. At low frequencies, the velocity and the accelerationare negligible compare to the displacement. Therefore, the transfer functionof the setup G(s), defined in (3.5) and (3.6), simplifies to[x1F1]= G′[V 1V 2], (3.8)where,G′ =[G′11 G′12G′21 G′22]=[n2k − n2k−n2 −n2]. (3.9)533.3. Controller designFigure 3.5: Fitted frequency response from V1 to x1m and F1m. Used withpermission.3.3 Controller designThe setup can be operated either in Voltage feedforward mode or in adisplacement and force feed-back control mode as shown in Figure 3.6. Inthe feed-forward control mode, reference voltages V1ref and V2ref are feddirectly to the actuators. These voltages need to be within the availablevoltage range of (0 V to 100 V) of the amplifiers. DS 1103Data acquisition board Control mode Feed-forwardFeed-backFigure 3.6: Feed-forward and feed-back control modes of the test setup.Used with permission.543.3. Controller design+- ++ +++++- ++Coupled systemForce SensorLaser SensorDecoupling termsDecoupled systemAmp-2Amp-1Figure 3.7: A feed-back controller designed for the setup. Used with per-mission.Alternatively, a MIMO feed-back controller was also designed for thesetup to follow reference x1ref and F1ref values. Figure 3.7 shows the feed-back control structure [VM02]. Integral actions for displacement Kix/s andforce Kif/s have been employed for tracking reference displacement andforces respectively, since they provide efficient tracking performances at lowfrequencies. Using decoupling constantsT12 = −G′12G′11= 1, (3.10)T12 = −G′21G′22= −1, (3.11)+- -+- + Force SensorLaser SensorDecoupled systemFigure 3.8: Decoupled displacement and force control structure. Used withpermission.553.4. Operational regionthe decoupled block-diagram in Figure 3.8 is obtained. For the quasi-staticexperiments conducted in this study, a 3-dB closed loop bandwidth fre-quency of ωc =120 Hz is selected for both displacement and force controllers.The required integral gains can then be calculated asKix = ωc(G′11 −G′12)−1 =kωcn, (3.12)Kif = ωc(G′22 +G′21)−1 = −ωcn. (3.13)3.4 Operational regionThe borders of the feasible displacement and force region for the setupare created using the extermum values of driving voltages. These bordersare experimentally generated and depicted in Figure 3.9. The gray arrow inthe figure shows the direction that the borders are created in. Figures 3.9(a)shows the feasible region bounded by the amplifier maximum voltages. Dueto hsytersis and differences between PESAs1 and PESA-2, this region isslightly asymmetrical. Figure 3.9(b)(b) shows the region adopted for feed-back control. It needs to be a slightly smaller region in order to allow for-20 -10 0 10 20-600-400-2000(a) Feed-Forward control mode-20 -10 0 10 20-600-400-2000(b) Feed-back control modeFigure 3.9: Feasible displacement-force region for the test setup. Used withpermission.563.5. Sanity checksufficient control authority, (that is limited by the available driving voltages),when getting close to the region boundaries.3.5 Sanity checkIt was discussed in Sub-subsection 2.6.4 that at low frequency appli-cations, the charge passing through a piezoelectric actuator is in a linearrelationship with its displacement and the external force applied to it. Thishysteresis free relationship is selected in this study as the criteria to carryout a sanity check of the designed setup.The setup is operated in feed-back control model. Constant displacementand constant force experiments are carried out within the feasible operatingregion of the setup. Figure 3.10(a) illustrates the constant force experiment.Forces of −100 N, −200 N, and −300 N are tracked across the feasible regionof the test setup. Charge q1 passing through PESA-1 is measured and plot-ted against xs in Figure 3.10(b), where The hysteresis between charge anddisplacement is almost insignificant. Figure 3.11(a) illustrates the constantdisplacement experiment. Displacements of −4 µm, 0 µm, and 4 µm aretracked across the feasible region of the test setup. In this experiment, thecharge q1 is plotted against the force in Figure 3.11(b) where, the hysteresisis also almost insignificant. Consequently, the setups performance conformswell to (2.32).In summary, this chapter described the design of 2 DoF setup that canindependently vary displacement and forces for a PESA. The setup consistsof two PESAs. PESA-1 is designated the main actuator and PESA-2 isemployed to vary external forces. Later in this thesis, the setup is employedin Chapter 4 to proposes a new sensor-actuator integration technique. Thesetup is also employed in Chapters 6, 7 and 8 to validate new self-sensingtechniques.573.5. Sanity check-15 -10 -5 0 5 10 15-600-500-400-300-200-1000100(a)-15 -10 -5 0 5 10 15-0.100.10.20.3(b)-300 N -200 N-100 NFigure 3.10: (a) Constant force trajectories at −100 N, −200 N, and −300 N.(b) Measured charge as a function of displacement for the forces in (a). Usedwith permission.-15 -10 -5 0 5 10 15-600-500-400-300-200-1000100-400 -300 -200 -100 0-0.0500.050.10.150.20.25(a) (b)Figure 3.11: (a) Constant displacement trajectories at −4 µm, 0 µm, and4 µm. (b) Charge as a function of force for the displacements in (a). Usedwith permission.58Chapter 4Integrating position andforce sensors in spaceconstrained piezoelectricdriven micro-positionersThis Chapter proposes a displacement and force measurement techniquefor micro-positioning applications with limited space for sensors. It pro-vides a technique to mount sensors outside the actuator’s work-space. Thismethod is builds on Goldfarb et al.’s assumption that hysteresis is purelya function of charge passing through a piezoelectric actuator. Thus, Sec-tion 4.1 first experimentally validates this assumption. The theory of theproposed technique is then detailed in section 4.2. Section 4.3 describes theexperimental setup that has been employed for the validation. The experi-mental results and discussions are provided in Section 4.4.4.1 Validation of the hysteresis voltage andcharge relationshipThe proposed displacement and force estimation technique in this chap-ter builds on Goldfrab et al.’s assumption that the hysteresis voltage of apiezoelectric actuator is solely a function of the charge passing through it.This assumption is experimentally validated with the setup described inChapter 3. Using the setup’s feed-back control mode, constant force andand constant displacement experiments are carried out within the feasibleregion of the setup. Figure 4.1 depicts the reference displacement and forcesemployed for these experiments. In the first experiment, the force is keptconstant at 100 N and the displacement varies from -3 µm to 5 µm and backto -3 µm. In the second experiment, the displacement is kept at constant at0 µm while the force varies from 0 N to -350 N and back to 0 N. These ranges594.1. Validation of the hysteresis voltage and charge relationship-15 -10 -5 0 5 10 15-600-500-400-300-200-1000100Figure 4.1: The reference force and displacements values employed in val-idation of hysteresis voltage and charge relationship. In the constant forceexperiment shown in blue, the force is kept constant at -100 N and dis-placement varies from -3 µm to 5 µm and back to -3 µm. In the constantdisplacement experiment shown in green, the displacement is kept at con-stant at 0 µm while the force changes from 0 N to -350 N and back to0 N.are selected through a try and error processes to achieve the same chargerange in both experiments. The hysteresis voltage of PESA-1 is measuredusing voltage, charge, and displacement measurements of PESA-1 viaVH1 = V1 − kn2 + nkCPq1 +nn2 + nkCPF1m, (4.1)which is derived using (2.12), (2.13), and (2.14) at low frequencies. The char-acteristic parameters, n, k, and CP are listed in Table 6.13. Figure 4.2(a)depicts driving voltages vs. charge for constant force and constant displace-ment. The charge range is the same for both cases. Figure 4.2(b). shows thecorresponding hysteresis voltages VH1 vs. charge q1. The match between thetwo profiles confirms that the relationship between charge and VH1 is purely3The characteristic parameters are selected through an identification procedure, whichis detailed in Chapter 6.604.2. The proposed sensor-actuator integration techniqueFigure 4.2: (a) Driving voltage vs. charge profiles for constant force andconstant displacement experiments. (b) Corresponding hysteresis voltagesvs. charge profiles. Used with permission.a function of charge for constant force and constant displacement scenarios.This is an important result for the proposed sensing method outlined in thenext section.4.2 The proposed sensor-actuator integrationtechniqueFigure 4.3 depicts the implementation of the proposed technique forpiezoelectric stack transducers. A piezoelectric stack actuator PESA-1 isemployed in a small micro-positioning work-space. PESA-1 is electricallyconnected in series to an identical sensing stack PESA-s, which is easilyaccessible outside the actuating work-space and is clamped between a forcesensor and a solid frame. A voltage source of VT is connected in series withthe stacks, which applies voltages of Vs and V1 on PESA-s and PESA-1respectively. For PESA-1, (2.12) and (2.13) yieldk1x1 = F1 + n1Vt1, (4.2)q1 = n1x1 + CP1Vt1 (4.3)and for PESA-s they yieldksxs = Fs + nsVts, (4.4)614.2. The proposed sensor-actuator integration technique+ + --Force SensorA smallmicro-positioning work-spacePESA-1PESA-sFigure 4.3: The proposed displacement and force measurement technique.qs = nsxs + CPsVts. (4.5)Although, it is shown in Figure 4.3 that the proposed technique completelyclamps PESA-s, in practice this is hard to achieve. Thus, (4.4) and (4.5) al-low for a finite clamping stiffness ks to improve the accuracy of the proposedtechnique.Since, PESA-1 and PESA-s are identical, their characteristic parametersare equivalent, n1 = ns = n, CP1 = CPs = CP , and k1 = ks = k. Furthermore, these transducers are electrically connected in series. Thus the samecharge passes through them, q1 = qs. Therefore, by equating (4.3) and (4.5)and using (2.14), x˜1 is estimated asx˜1 = xs +CPn((Vs − VHs)− (V1 − VH1)). (4.6)However, the results from the previous section indicate that the hysteresisvoltages from PESA-1 and PESA-2 must be the same, since the same chargepasses through them, VH1(q) = VHs(q). Hence, (4.6) is simplified tox˜1 = xs +CPn(Vs − V1). (4.7)The small movements of the clamped transducer x˜s can be estimated us-ing (4.4) asx˜s =1kFs +αk(Vs − VHs) ≈ 1kFs +αkVs, (4.8)624.3. The experimental setupwhere the hysteresis voltage VHs is neglected, since xs is much smallerthan xa. Substituting (4.8) into (4.7) yieldsx˜1 =1kFs +αkVs +CPα(Vs − V1). (4.9)The external force on PESA-1 can be obtained by substituting (4.7) into (4.2)and adding the updated equation to (4.4)F˜1 = Fs + (n+kCpn)(Vs − V1). (4.10)The proposed method only requires a dedicated force sensor and two voltagereadings to estimate the displacement and force of a piezoelectric actuatorin small-space micro-positioning applications.4.3 The experimental setupThis section describes the setup that has been employed to validatethe proposed sensor-actuator integration technique. Figure 4.4 depicts thesetup, which consists of a sensing unit and an actuating unit. Indeed, theactuating unit is the 2 DoF setup that is described in Chapter 3. The mainactuator is PESA-1 and PESA-2 generates external forces.Within the sensing unit, a piezoelectric transducer PESA-s is clampedbetween a solid frame and a PCB 260A12 force sensor with 1% nonlinearityover the range of 0kN-4.45kN. This force sensor measures Fs. Similar tothe actuating unit, PESA-s is also a PI-885.90 piezoelectric stack actuatorfrom Physik Instrumente and its data sheet values are listed in Table 3.1.Figure 4.5 depicts the schematic of the sensing unit. A threaded cap is usedto apply the required preload to the actuator (see Table 3.1). PESA-s iselectrically connected in series to PESA-1. The linear PZD-700 amplifierprovides a driving voltage V2 to PESA-2 and generates varying forces F1.The high power PI-E617-00F amplifier is connected in series with PESA-1and PESA-s and provides a total voltage of VT .VT = V1 + Vs. (4.11)Two Tektronix-P2220 voltage probes are employed to measure VT and V1.The following subsection derives a relationship between x1, F1, VT , and V2to run the experiments.634.3. The experimental setup+ -PI-E617-00FAmp.PZD-700Amp.HSV2002vibrometerPCB 260A12PCB-208C2stiffness Sensing Unit Actuating UnitPESA-2PESA-1PESA-sFigure 4.4: The experiential setup.4.3.1 Mapping between driving voltages, force, anddisplacementVoltage is applied across PESA-1 and PESA-s in a series arrangement.However, the voltages across the two actuators are not identical, since theyhave different clamping conditions. This subsection derives the mapping ofthe driving voltages VT and V2 to desired values of x1 and F1 for the setupdescribed in Figure 4.4. Using the hysteresis free constitutive equations forPESA-s yieldkxs = Fs + nVs, (4.12)qs = αxs + CPVs, (4.13), for PESA-1 they yieldkx1 = F1 + nV1, (4.14)q1 = αx1 + CPV1, (4.15), and for PESA-2, they yieldkx1 = −F1 + nV2, (4.16)644.4. Experimental resultsCap FramePESA-sPCB 260A12Figure 4.5: The schematic cross section of the sensing unit in Fig-ure 4.4 [Isl13a].q2 = nx1 + CPV2. (4.17)Since, PESA-1 and PESA-s are electrically connected in seriesVT = V1 + Vs. (4.18)Also, the relationship between Fs and xs is given byFs = Ksxs, (4.19)where, Ks is the stiffness of the frame in the sensing unit. For given referencevalues of x1 and F1, the required voltages VT and V2 are determined from(4.12-4.19) asVT = ((k −Ks)(kCP + n2)nCP (k −Ks) + n3 +kn)x1−(CP (k −Ks)nCP (k −Ks) + n3 +1n)F1,(4.20)V2 =knx1 +1nF1. (4.21)4.4 Experimental resultsUsing the setup described in Subsection 4.3 an independently varying F1-x1 reference trajectory is generated. Figure 4.6 depicts the corresponding654.4. Experimental results-6 -4 -2 0 2 4 6 8-400-350-300-250-200-150-100-50050222Figure 4.6: The measured F1m-x1m reference trajectory generated with thesetup outlined in Section 4.3.force and position values obtained with the reference position sensor for x1and the reference force gauge for F1. The gray arrows show the trajectory’straverse direction. The extermum position and force values at the corners arerestricted by the allowable positive V2, V1, and Vs voltages. The advantage ofusing this trajectory is that it covers all four possible increase/decrease com-binations of displacement and force. The reference trajectory is traversedat four different speeds and the proposed technique in (4.9) and (4.10) isemployed to estimate the displacement and force.Figure 4.7 shows the experimental results, where close agreement be-tween the estimated and the reference measurements can be observed. Theaverage absolute differences between the two measurements are calculatedover the full ranges of displacement and force and listed in Table 4.1. Dur-ing the slowest experiment, 2 s, a small amount of drift is observed in bothestimated displacement and force, which leads to average absolute errors ofµ(|ex|) =2.9% and µ(|eF |) =1.5% respectively. The reason for this measure-ment drift will be a part of future studies. The error values in the remainingexperiments are less than 1%. In conclusion, this chapter proposes a sim-664.4. Experimental results-10 -5 0 5 10-50510-100 0 100 200 300 400-10001002003004002 s1 s0.6 s0.2 s(a) Displacement (a) Force Figure 4.7: The experimental results of traversing the reference trajectory atfour different speeds. (a) The estimated displacement x˜1 vs. the measureddisplacement x1m. (b) The estimated force F˜1 vs. the measured force F1m.ple sensor and force integration technique for piezoelectric actuator drivenmicro-positioners. The study suggests to employ a piezoelectric positioningtransducer in a sensor-free work-space and electrically connect it to an iden-tical sensing transducer outside the work-space. The sensing transducer isclamped between a solid frame and a force sensor. The proposed techniqueuses voltage and force measurements of the sensing transducer to compen-sate for hysteresis nonlinearity and estimate both displacement and force ofthe working actuator. This technique is applicable to quasi-static frequen-cies. It does not require sophisticated computation or complex electrical cir-cuits. Therefore, it can easily be applied to real-time implementations. TheTable 4.1: Displacement and force estimation average absolute error per-centagesTraverse time µ(|ex|) [%] µ(|eF |) [%]2 s 2.9 1.51 s 1.0 0.10.6 s 0.9 0.60.2 s 0.9 0.2674.4. Experimental resultsproposed technique was experientially validated for simultaneously varyingdisplacement and force scenarios at different speeds. Less than 1% averageabsolute errors are calculated for both position and force estimations. How-ever, this value grows at very low speeds due to measurement drifts. Lowfrequency drift and the effect of temperature remain to be studied in thefuture.68Chapter 5Piezoelectric bimorphactuator with integratedstrain sensing electrodesChapter 4 introduced a sensor-actuator integration technique for stackactuators. The current chapter presents a novel sensor integration techniquefor benders. It integrates strain gauges into the top and bottom electrodesof a bimorph bender, (See Sub-subsection 2.3.1). This provides an indi-rect tip-displacement measurement method that can be employed in limitedmicro-positioning work-spaces. The proposed method uses the same prin-ciple as a resistive metal foil strain gauge: The electrical resistance changeis proportional to the surface-strain. In benders, surface strain is relatedto the tip displacement. Hence, the tip displacement of a bender can beestimated by measuring its surface resistance change.Section 5.1 derives the relationship between tip-displacement and surface-strain in benders. Section 5.2 describes the proposed surface strain measure-ment method using an electrical half-bridge circuit. The implementation ofthe proposed technique is described in Section 5.3. Validation experimentsare carried out in Section 5.4. A displacement controller is designed forthe setup in Section 5.5. The performance of this controller is validated inSection 5.6.5.1 Tip displacement and surface-Strainrelationship in bendersFigure 5.1 depicts an electromechanical model of a bimorph bender.It is clamped on its left end and no external forces are exerted on it. Adriving voltage V is applied in a parallel electrical configuration, (see Sub-subsection 2.3.1), and assumed to be effective across the thickness h of eachpiezoelectric plates. The bender has a length L. On its top and bottomelectrodes, ls is the length of the surface region that its strain is measured695.1. Tip displacement and surface-Strain relationship in bendersFigure 5.1: Electromechanical model of the bender for nonzero driving volt-age and zero external force. Used with permission.via integrated strain gauges and mapped to the tip-displacement. The twostrain sensing regions are separated from the high-voltage electrodes via agap t. Rearranging the constitutive equation in (2.1) yieldsTkl =dkijsEijklEk − 1sEijklSij . (5.1)This equation adapted to the bender depicted in Figure 5.1, using its elec-tromechanical orientationsT1 = d31YV3h+ Y S1, (5.2)where Y = 1sE12is the Young’s modulus of the bender and E3 is replacedwith V3h . For the sake of simplicity, in the remainder of this chapter thesubscripts are dropped from this equationT = d31YVh+ Y S. (5.3)Assuming that the bender is an Euler-Bernoulli beam with relatively smalltip displacements, S in (5.3) is replaced with S = −y d2ydx2, where y is thevertical distance from the neutral axis. The moment M due to the externalforces can then be calculated by taking the integral of Ty, over the cross-sectional area of the top half of the bender [RF15] that isM = d31Y V wh2− Y I d2ydx2. (5.4)705.2. Surface resistance change measurementIn (5.4),I =∫d2ydx2y2dy (5.5)is the second moment of area and w is the width of the bender. For zeroexternal forces, M(x) = 0, which simplifies (5.4) tod2ydx2= d31V wh2I. (5.6)This indicates that the bender is experiencing pure bending due to the ap-plied driving voltage. Solving for y, at x = L, the tip displacement is,ytip =d31V whL22I. (5.7)Using (5.6), the average surface strain due to electrical voltage S¯t along thelength of ls isS¯t =1ls∫ ls+ttS =−d31V wh22I. (5.8)Comparing 5.7 with 5.8, the tip displacement ytip and the average strain S¯tare proportional. The proportionality isytipS¯t= −L22h= Ce. (5.9)Section 5.2 describes an approach to measure the surface strain.5.2 Surface resistance change measurementThis work proposes to cut strain-gauges out of the electrodes of a benderand use them to infer tip displacement. Figure 5.2(a) shows two U-shapedstrain-gauges integrated into the top and bottom electrodes with electricalconnections Vt, Vc, and Vb. A conditioning circuit provides the top andbottom strain gauges with constant DC voltages of Vt and Vb. The equivalenthalf-bridge circuit is shown in Figure. 5.2(b), where the nominal resistanceof the top and bottom gauges are Rt and Rb respectively. When the bendermoves, the top and bottom gauges experience a change in their resistances∆Rt and ∆Rb, respectively. Assuming that the top and bottom sensorsexperience identical strains, S¯ = S¯t = −S¯b,S¯ ∝ ∆RtRt∝ ∆RbRb. (5.10)715.3. Experimental setup(b) Strain gauge circuit(a) Electrical connections of starin gaugesFigure 5.2: The electrical connection of the strain gauges (a) and the equiv-alent circuit (b). Used with permission.Using (5.9) and (5.10), the tip displacement can be estimated by measuringthe resistance change, which is proportional to Vc. The excitation voltagesVt and Vb are tuned so that the nominal value of Vc is zero. When this isachieved, VtVb = −RtRb. In the presence of strain, assuming that the straingauges are identical (Rt = Rb), the voltage Vc isVc ∝ ∆RtRt∝ ∆RbRb. (5.11)Using a gain of g, this voltage is amplified to Vg = gVc. Equations (5.9-5.11)give,Vg ∝ ytip. (5.12)The proportionality (5.12), is experimentally derived in the next section.The estimated tip-displacement y˜ is then given byy˜ = gsVg. (5.13)5.3 Experimental setupThe proposed sensor-actuator integration method is implemented on aY-poled T220-A4-503Y bimorph bender from Piezo Systems. The benderdimensions are 63.5×31.8×0.51 mm. It has a free displacement of ± 1 mmand a blocked force of ± 0.35 N. The internal construction is illustratedin Figure 5.3. The layers include a 0.1 mm thick brass reinforcement layershown in orange, two 0.2 mm thick piezoelectric plates shown in yellow, and725.3. Experimental setupResin MaskHNO3 (30%)1(a) Construction of strain gauges (b) Experimentation procedureBottomElectrodeTopElectrodeBrassLayerPiezoPlatePiezoPlateLaser SensorLinear AmplifierMatlab/SimulinkFigure 5.3: Experimental setup. Used with permission.two 5 µm layers of Nickel electrodes shown in blue. Figure 5.3(a) illustratesthe process used to create the integrated strain gauge. First, a 0.1 mm thickresin mask is printed on the electrode using a Form 2 Desktop 3D printerwith 5 µm resolution. Then, 30% nitric acid is used to etch the electrodearea not covered by the mask. The desired electrode feature is an electricallyisolated U-shaped strain gauge with an effective length of 2ls =40 mm anda width of 1 mm. The electrode feature is isolated from the high voltageelectrode with a gap of t =0.5 mm. This process is carried out on both topand bottom electrodes.Figure 5.3(b) shows the experimental setup. The bender is cantileveredwith a free length of 57 mm. It is driven in a parallel electrical configurationwhere the middle brass layer is grounded and ±90 V signal u is applied to theexternal electrodes by a PiezoDrive PD200 amplifier. An alternative config-uration is to ground the external electrodes and drive the middle brass layer.The two configurations and their equivalent electrical circuits are depictedin Figure 5.4. Configuration 1 is recommended by the manufacturer since itis convenient to have grounded external electrodes. However, the followingparagraph will show that this increases dynamic measurement errors.The electrical connections in Configuration 1 are depicted in Figure 5.4(a).The electrodes are grounded and V is applied to the middle layer. The top735.3. Experimental setup(a) Configuration 1 (b) Equivalent circuit (c) Simplified circuit (d) Configuration 2 (e) Equivalent circuit (f) Simplified circuit Figure 5.4: Two electrical configurations for driving the bender. Used withpermission.and bottom electrodes are electrically modeled as distributed resistances inseries, while piezoelectric layers are modeled as distributed capacitors. Theelectrical equivalent circuit is depicted in Figure 5.4(b). Assuming that theresistances are equal, the simplified circuit in Figure 5.4(c) is a high-pass fil-ter which allows direct feed-through from V to Vc at high frequencies, whichis highly undesirable.In Configuration 2, the equivalent circuit and simplified circuit is de-picted in Figures 5.4(d) and (e) respectively. In this configuration the directfeed-through is eliminated. In practice, there still exists a small capacitivecoupling between the driven surface electrode and strain sensor; however,this is small in the frequency range of interest. Therefore, Configuration 2is employed herein.The strain gauges are excited in a half-bridge configuration by applying745.4. Validation of the proposed methodFigure 5.5: Block diagram of the piezoelectric bender system with integratedstrain sensors. The tip displacement of the bender is also measured usingthe laser (ylaser) for sensor calibration and comparison purposes. Used withpermission.a DC voltage to the top Vt and bottom strain gauge Vb. By tuning either Vtor Vb, the voltage at the center of the half-bridge Vc can be set to zero. Vc isthen amplified by a gain g, which provides a voltage Vg that is proportionalto the tip displacement. The sensitivity gs is found experimentally usinga Di-Soric LAT61 laser displacement reference sensor. The estimated tipdisplacement is labeled y˜ in Figure 5.3(b).A block diagram of the setup is depicted in Figure 5.5 where, W repre-sents the linear amplifier dynamics, P represents the bender dynamics, Hlrepresents the laser dynamics, and Hsg represents the strain gauge dynamics.Therefore, the open-loop transfer function of the integrated bender isG(s) =y˜u= W (s)P (s)Hsg(s)gs. (5.14)5.4 Validation of the proposed methodA calibration experiment is first carried out, where the maximum avail-able driving voltage range V= +/-90 V is applied to the bender and the tipdisplacement is measured using the laser sensor. The sensitivity gain is thencalculated as Kg = ylaser/Vg = 0.9 mm/V.The performance of the proposed strain sensor is further evaluated byapplying a triangular driving voltage V to the actuator at 33%, 67%, and100% of the full-scale voltage (±90 V). The resulting tip displacements mea-sured by both the reference and strain sensor are plotted in Figure 5.6. Thevoltage-displacement hysteresis can be clearly observed in both signals.755.4. Validation of the proposed method-100 0 100-101-101Figure 5.6: The tip displacement versus driving voltage, when the actuatoris driven by a triangular voltage with increasing amplitude. Used withpermission.0 5 10Time [s](a) Time Signals180 V, 2 mm, 2 mm, -1 0 1-101(b) NonlinearityError (%)-101Figure 5.7: The recorded time signals (a) and difference between the refer-ence ylaser and proposed sensor y˜ (b). In (a), the peak-to-peak values areprovided for the actuator voltage Vu, the reference sensor ylaser, and strainsensor y˜. Used with permission.Time domain signals are plotted in Figure 5.7(a). A peak-to-peak drivingvoltage of 180 V results in a full-range of 2 mm of the displacement. Aclose agreement between the reference and strain sensor can be observed.In Figure 5.7(b), the output of the reference and strain sensor are plottedagainst each other. The difference between the two sensors is calculated765.5. Controller design-100-50050-720-3600360Magnitude (dB)Phase (deg)100 101 102 103Frequency (Hz)Figure 5.8: The normalized open-loop frequency response in blue, measuredfrom u to y˜, in mm/V. The closed-loop frequency response with damping-integral control, measured from r to y˜, in mm/V. Normalizing the open-loopfrequency response is achieved by multiplying the measured data by a gainof 7.9. Used with permission.as a percentage of full-range, based on a least-squares fit for the sensitivityand offset. The maximum difference is 1.1 % of the full-range displacement.A polynomial calibration function could be used to significantly reduce theresidual error.5.5 Controller designIn this section the performance of the proposed strain sensor is eval-uated in a closed-loop position control system. A damping-integral con-troller [EVGP14] is designed and implemented in the feed-back loop for thestrain sensor measurement.5.5.1 Damping-integral controllerThe normalized open-loop frequency response from u to y˜, measuredusing a Quattro Data Physics signal analyzer, is plotted in Fig 5.8. Thefirst resonance occurs at 57 Hz. A damping-integral controller is designed775.5. Controller designFigure 5.9: Block diagram of the closed-loop system with the damping-integral controller. The strain sensor output is the measurement in thefeed-back loop. Used with permission.Table 5.1: Parameters of the damping-integral controller.Parameter Valueωc 123 Hzki 406to damp this resonance. Figure 5.9 shows the block diagram of the closed-loop system with the damping-integral control strategy that includes anintegral action.Ci =kis, (5.15)a reconstruction filter Fr, and an anti-aliasing filter Fa. Both filters aresecond-order low-pass Butterworth filters [EVGP14].Fr = Fa =ωcs2 +√2ωcs+ ω2c. (5.16)The controller design criteria is to maximize the closed-loop bandwidth with-out destabilizing the closed-loop system [EVGP14]. The closed-loop transferfunction isT =CiFrFaG1 + CiFrFaG. (5.17)Employing the Nelder-Mead optimization algorithm [LRWW98], the cut-offfrequency of the filters ωc as well as the integral control gain ki are optimizedto fulfill the design criteria. The optimal parameters are summarized in Ta-ble 5.1. The resultant gain and phase margins are 6 dB and 61◦ respectively.785.6. Performance of the closed-loop system0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Time [s]-0.200.20.40.60.8Reference1Figure 5.10: Reference and measured closed-loop step response of the inte-grated bender using a damping-integral controller. Used with permission.5.6 Performance of the closed-loop systemThe measured closed-loop frequency response of the bender with a damping-integral control is plotted on top of the open-loop frequency response inFigure 5.8. It is observed that the first resonant peak is substantially sup-pressed by 33 dB and the resultant 3-dB closed-loop bandwidth of the systemis 60 Hz.The step response of the closed-loop system is plotted in Figure 5.10. Asettling time of 0.06 s is recorded with no noticeable overshoot.In conclusion this chapter presents a new technique for measuring the tipdisplacement of piezoelectric benders by integrating resistive strain gaugesinto their top and bottom electrodes. The change in surface resistance isshown to be proportional to the tip displacement in the absence of externalforces. The strain gauges are employed in a half-bridge configuration tomeasure the resistance change and tip displacement.The presented technique is experimentally compared to a reference sen-sor. The maximum difference between the two sensor outputs was 1.1%795.6. Performance of the closed-loop systemof full range. The performance of the presented technique for control ap-plications is experimentally demonstrated by employing the technique ina damping-integral control structure. The proposed sensing technique haslimitations for high frequency applications due to the capacitive couplingbetween the driven surface electrodes and the strain gauges, which is in thetens of kilohertz. Future work includes validating the high frequency per-formance of the proposed sensing technique. The sensor geometry will alsobe optimized for external force measurement.80Chapter 6A new self-sensing techniqueusing charge and voltagemeasurementsChapters 4 and 5 integrated piezoelectric actuators with dedicated sen-sors for space-limited micro positioning work-spaces. The current chapterintroduces a self-sensing technique based on Goldfarb et al.’s electromechan-ical model to completely eliminate external position and force sensors. Itimplements the self-sensing technique in Figure 2.30 to estimate the displace-ment and force of a piezoelectric stack actuator. The main advantage of theproposed method over the similar methods [BQN08b, KSUO12, MU14] isthat a new parameter identification method is presented herein that requiresonly a single experiment. Section 6.1 introduces the novel parameter iden-tification method and validation experiments are discussed in Section 6.2.6.1 Parameter identificationUsing the setup described in Chapter 3 in its feed-back control mode, aforce-displacement profile is generated that consists of four possible increasing-decreasing combinations of force and displacement. Figure 6.1 depicts thegenerated reference and measured profiles. The borders of these profiles aregenerated using the extermum values of the driving voltages at feed-backcontrol mode. The gray arrow indicates the traversing direction along theprofiles. In this experiment, PESA-1 variables, x1m, F1m, V1, and q1 arerecorded.The hysteresis free charge, displacement, and force relationship, firstshown in (2.33), is then employed for parameter identification and reprintedhere for reference:q(s) = (n+CPkn)X(s)− CPnF (s). (6.1)816.1. Parameter identificationFigure 6.1: Reference and measured force-displacement profiles used forparameter identification. The measured profile is created using the feed-back control mode of the the test setup described in Chapter 3. Used withpermission.Measurements of x1m, F1m, and q1 are fitted to (6.1) providing coeffi-cients n + CP kn , andCPn . An R2 value of 99.3% is achieved in this fittingoperation.Another constraint is required to identify model parameters n, CP , and kfrom these two coefficients. This constraint is dictated via hysteresis voltagemodeling requirements: Most phenomenological hysteresis operators, (SeeSection 2.6.1), require the slope at the end of the descending branch of theVH − q hysteresis loop to asymptoticly approach zero slope. This criteriais qualitatively depicted in Figure 6.2. The hysteresis voltage VH1 can bedetermined using (4.1) reprinted here:VH1 = V1 − kn2 + nkCPq1 +nn2 + nkCPF1m, (6.2)which is also described with the characteristic parameters. By selectinga value for CP , the other two model parameters can be calculated from826.1. Parameter identificationZero SlopeFigure 6.2: A qualitative representation of the parameter identification cri-teria used in [BQN08a]: the descending branch of the VH − q hysteresis loopmust have zero slope at the end of the branch. Used with permission.Table 6.1: Reference and identified characteristic parameters of PESA-1.Parameter Data sheet value [Phy05] Identified valueCP 3.1 ± 20 % (µF ) 3.5 (µF )α 7.9 (N/V) 8.2 (N/V)k 25 (106 N/m) 22.4 (106 N/m)coefficients n+ CP kn , andCPn . VH1 − q1 loop is plotted and CP is optimizedto meet the criteria in Figure 6.2.Using the data sheet value of 3.1 µF for CP , Figure 6.3(a), shows thatthis modeling requirement is violated. Following [BQN08a], CP=3.5 µFis selected to ensure that the VH1 vs. q1 hysteresis loop, shown in solidblue in Figure 6.3(b), has zero slope at the end of its descending branch.This leads to the solid blue line in Figure 6.3(a), which can be fit withstandard hysteresis model operators. The identified parameters are reportedin Table 6.1.A Prandtl-Ishlinskii hysteresis model with 150 Play operators [KJ01] isemployed to fit the experimental measurements toq1 = L(VH1). (6.3)Figure 6.3(a) shows the modeled trace of q˜1 vs. VH . Finally, a simple nu-merical procedure, outlined in [KJ01], is carried out to obtain H = L−1.836.2. Experimental validation of the self-sensing techniqueFigure 6.3: Measured and modeled hysteresis loops. Dashed blue linesrepresent the measured hysteresis loops using a CP value of 3.1 µF, whichviolates the hysteresis modeling criteria in Figure 6.2. The solid blue loopsare measured with the CP value of 3.5 muF. Prandtle-Ishlinskii hysteresisoperators have been used to generate the green hysteresis fits. Used withpermission.Figure 6.3(b) shows the modeled trace of V˜H1 vs. q1. The identified param-eters n, CP , and k together with the calibrated hysteresis operator H areemployed in the self-sensing structure depicted in Figure 2.30.6.2 Experimental validation of the self-sensingtechniqueThis section implements the self-sensing technique using the setup de-scribed in Chapter 3. A force vs. displacement trajectory is created forthe validation of (2.44) and (2.45) and shown in Figure 6.4. Red numbereddiamonds represent the sequence of the desired motion endpoints to follow.Displacement and forces are within the ranges used in the identification ex-periment (See Figure 6.1). Small displacement fluctuations observed in theconstant displacement portion are close to the resolution of the displace-ment sensor (0.3 µm). Figures 6.5 and 6.6 show the estimation results.Figure 6.5 shows a good match between measured x1m and estimated x˜1displacements. Estimation error is also depicted in Figure 6.5. Following846.2. Experimental validation of the self-sensing technique-15 -10 -5 0 5 10 15-600-500-400-300-200-1000 1, 3, 94, 8265, 7Desired pointsMeasuredFigure 6.4: Red diamonds represent the sequence of the desired points tofollow validation force vs. displacement profile. Solid blue line shows forcevs. displacement measurements. Used with permission.the example in[MU14], an average value of the error of 0.5% with a standarddeviation of 2.7% is determined.Figure 6.6 also shows a good match between measured F1m and esti-mated F˜1 forces. An average force estimation error of −0.4% with a standarddeviation of 3.1% is determined.In summary, this chapter presents a self-sensing technique for piezoelec-tric actuators using Goldfarb et al.’s model. The presented self-sensing tech-nique benefits from a new parameter identification procedure that requiresa single identification experiment. Experimental validation of the techniqueis provided, in which strong performance in both displacement and forceestimations are observed.856.2. Experimental validation of the self-sensing technique0 0.05 0.1 0.15 0.2 0.25-20-1001020Displacement [m]-10-50510Error [%]Figure 6.5: Measured and estimated displacements and the estimation error.Used with permission.866.2. Experimental validation of the self-sensing technique0 0.05 0.1 0.15 0.2 0.25-600-500-400-300-200-1000100Force [N]-8-6-4-20246Error [%]Figure 6.6: Measured and estimated forces and the estimation error. Usedwith permission.87Chapter 7A modifiedelectromechanical model forpiezoelectric actuatorsSo far, this thesis proposed two novel sensor-actuator integration tech-niques in Chapters 4 and 5, together with an improved model identifica-tion procedure for a force and displacement self-sensing technique in Chap-ter 6. A common assumption among the introduced techniques is that thecapacitance of a piezoelectric actuator stays constant throughout a micro-positioning application.This chapter first investigates the constant capacitance assumption forstack actuators in Section 7.1 and experimentally shows that the capaci-tance is a function of the applied voltage to the actuator at low frequencies.Section 7.2 then applies this finding to a new charge and voltage based self-sensing technique. Section 7.3 implements the new self-sensing technique ina sensor-less control structure.7.1 The effect of applied force and voltage oncapacitanceIn the majority of piezoelectric driven micro-positioning systems, thedriving voltage and the external forces are considered to be the main inputson the system. This section studies their effect on a piezoelectric stackactuator’s capacitance at quasi-static frequencies.7.1.1 The experimental setupFigure 7.1 shows the experimental setup. It contains a DS 1103 dataacquisition board for providing a DC offset voltage, a PI-E617-00F Poweramplifier to feed the offset voltage to a DC bias protection circuit [Key]for the Agilent-E4980A LCR meter used to measure impedance. A PI-887.1. The effect of applied force and voltage on capacitancePI-E617-00FAmp.Agilent E4980ALCR meterPESA-sPCB 260A12DC biasScrew capDS 1103 data acquisition boardprotection circuitFigure 7.1: The test setup employed to study the effect of voltage and forceon the capacitance of a piezoelectric stack actuator.885.90 piezoelectric stack actuator from Physik Instrumente is held in asolid frame and connected to the LCR meter through the DC protectioncircuit. A functional diagram of the setup is shown in Figure 7.2. The forceof the pre-load spring acting on PESA-s is adjusted using a screw cap. Thestiffness of the pre-loading Ks is much smaller than the actuator’s stiffness(Ks k). Therefore, the applied external load stays essentially constantwhen the actuator’s displacement changes due to a DC offset voltage appliedto the actuator. The LCR meter uses a small detecting voltage Vr = 1 Vto measure the capacitance at a frequency of 30 Hz, which corresponds totypical quasi-static applications for this actuator.7.1.2 The relationship between capacitance, voltage, andforceIn the first experiment illustrated in Figure 7.3(a), VDC=0 V and avarying preload forces of Fex= 0 N to -240 N are applied to the actuator insteps of 60N. This experiment aims to study the effect of applied force onthe actuator’s capacitance while keeping the applied voltage constant. Tocompensate for creep, preload forces are held constant for five minutes ateach level before capacitance readings are recorded. In the whole range ofthe applied forces, less than 5% variation in capacitance is observed. In the897.1. The effect of applied force and voltage on capacitancePESA-sPCB 260A12PI-E617-00FAmp.Agilent-E4980ALCR meterAdjustable preload, (screw cap)SumingcircuitFigure 7.2: The schematic representation of implementation of the setup.Used with permission.second experiment illustrated in Figure 7.3(b), Fex=0 N and DC voltages,starting from VDC= 0 V increasing to VDC=100 V and decreasing back toVDC=0 V, are applied to the actuator in steps of 10 V. To mitigate creep,offset voltages are held constant for five minutes before recording capacitancevalues.A strong dependency between capacitance and the voltage applied to thepiezoelectric actuator is experimentally observed in Figure 7.3(b). This canbe explained with the electric field-polarization relationship shown in Fig-ure 2.25. In the operational region, the slope of the plot, which representspermittivity, decreases as the electric field increases. Since, the piezoelectricdomain structure plays a major role in the permittivity change [KKHM08],one would also expect to see a change in capacitance as a result of appliedforce. Little is shown in the literature about this relationship. However, theexperiments shown in Figure 7.3(a) indicate that the change in capacitanceas a result of applied force is negligible. More studies on the molecular struc-ture of the piezoelectric materials are required to support this experimentalobservation.907.2. A new self-sensing technique for piezoelectric actuators-250 -200 -150 -100 -50 02.62.833.23.43.60 50 1002.62.833.23.43.6(a) Capacitance under differnt forces measured @ 30 Hz(b) Capacitance under differnt voltages measured @ 30 HzFigure 7.3: The experiments conducted to study the effect of applied forceand voltages on the capacitance of piezoelectric actuators. The capacitancemeasurements are carried out at 30 Hz. Used with permission.7.2 A new self-sensing technique for piezoelectricactuatorsThis section modifies Goldfarb et al.’s electromechanical model, (2.12-2.15), with the findings from Subsection 7.1 and introduces a new self-sensingtechnique for piezoelectric stack actuators at quasi-static frequencies.7.2.1 A new electro-mechanical model for piezoelectricactuatorsThe experimental results in Figure 7.3 demonstrate that in contrast tothe conventional model for piezoelectric materials in (2.12-2.15), the actu-ator’s capacitance is strongly dependent on the applied voltage. However,the effect of external forces on the capacitance were found to be negligible.This study proposes to use a first order polynomial to model the observedcapacitance-voltage relationship,C∗ = CP + CvVt. (7.1)Utilizing the proposed fit for the experimental data in Figure 7.3(b), a CPvalue of 3.34 µF and a Cv value of -0.007 µF/V is obtained with a coefficientof determination of 99.2%.917.2. A new self-sensing technique for piezoelectric actuatorsFollowing these findings, a modification is applied to Goldfarb et al.’squasi-static model by replacing the constant capacitance CP in (2.13) witha voltage dependent capacitance C∗kx = F + nVt, (7.2)q = nx+ (CP +Cv2Vt)Vt. (7.3)V = Vt + VH , (7.4)VH = H(q). (7.5)7.2.2 Displacement and force self-sensingA new displacement-force self-sensing technique is proposed, which uses(7.2-7.5). A schematic diagram of this technique is depicted in Figure 7.4.The actuator undergoes a displacement x as a result of an applied volt-age V and an external Force F . The driving voltage V and the current Iare measured using respective probes. The charge q passing through theactuator is determined by integrating I. Measured values of voltage V andcharge q are then employed in a rearranged form of (7.2-7.5) to estimatethe displacement x˜, and the force F˜ . This algorithm is very similar to thework shown in Chapter 6, except that the capacitance term Cv is addedand a computationally efficient hysteresis model is employed. The imple-mentation of the proposed self-sensing technique, requires the identificationof n, k, CP , Cv, and a hysteresis model H(q). Subsection 7.2.3 proposes anefficient parameter identification method that uses a single experiment toidentify all parameters.7.2.3 A new parameter identification methodThe proposed identification method eliminates the non-measurable valueof Vt, by substituting Vt as a function of x and F from (7.2) into (7.3)q = a1x+ a2F + a3x2 + a4F 2 + a5xF, (7.6)where,a1 = n+ CPkn, (7.7)a2 = CP1n, (7.8)927.2. A new self-sensing technique for piezoelectric actuatorsAA new technique for estimating the displacement and the force from charge and voltage measurementsVPEAFigure 7.4: The schematic of the new displacement-force self-sensing tech-nique. The proposed techniques utilizes a voltage dependent capacitanceand requires a single parameter identification experiment to identify n, k,CP , Cv, and H(q). Used with permission.a3 = Cvk22n2, (7.9)a4 = Cv12n2, (7.10)a5 = Cvkn2. (7.11)In order to identify the coefficients a1−5, a fitting experiment is created us-ing the setup described in Chapter 3, in which displacement and force varyindependent from each other. The corresponding measurements for PESA-1are shown in Figure 7.5. All possible increase and decrease combinationsof displacement and force in their maximum ranges, (-12 µ to 12 µm) and(0 N to 550 N) respectively, are part of this profile. During the experi-ments, x1, F1, V1, and q1 are recorded. To minimize the effects of currentsensor offsets and noise, thirty experiments are conducted and the average ofthe 30 experiments is used for identification purposes. This is highlighted inFigure 7.5, where the original collected data is shown with blue crosses, andthe averaged values are shown in a solid red line. To ensure the same initialcondition for each experiment, the waiting time in-between the experimentsis the same, (a few seconds). In addition, all measurements are set to zeroat the beginning of each experiment.Fitting experimental data in Figure 7.5 to the model in (7.6) allows forthe identification of five fitting coefficients a1 to a5. These can then be937.2. A new self-sensing technique for piezoelectric actuators0 0.2 0.4 0.6-20-10010200 0.2 0.4 0.6-600-400-20002000 0.2 0.4 0.6-500501000 0.2 0.4 0.6-2000200400(a) Displacement(b) Force(c) Voltage(d) ChargeTime [s]Time [s]Figure 7.5: The experimental measurements collected for parameter identi-fication. Used with permission.transformed to the desired model parametersk =√a3a4, (7.12)n = a1 − a2k, (7.13)CP = a2n, (7.14)Cv = 2a4n2. (7.15)The model parameters obtained for the proposed electromechanical modelare compared to the data sheet values in Table 7.1. Stiffness, k, piezoelectriccoupling n, and the clamped capacitance at zero applied electric field CPare very similar.947.2. A new self-sensing technique for piezoelectric actuatorsTable 7.1: Model parameters comparison.k(kN/m)α(N/V)CP(µF )Cv(µF/V)Data sheet Values [Phy05] 25 7.9 3.1 -Extended Goldfarb withvoltage dependent Capacitance 23.1 8.2 3.37 -0.008Linear model between capacitanceand voltage deriven in Section 7.2.1 - - 3.34 -0.007The identified CP and Cv values are also very similar to the values ob-tained from the fit to (7.1) illustrated in Figure 7.3(b). This indicates thatthe proposed model is consistent with the measurements discussed in Sub-section 7.2.1.Equations (7.12-7.15) do not make use of fitting parameter a5, because a5is dependent on the other four fitting parameters, a1 to a4. However, a5 canbe used to validate the model consistency by comparing it toa5 = 2√a3a4. (7.16)In this experiment case, a5=0.003 C/Nm and 2√a3a4=0.0027 C/Nm. Thisis a good match indicating that the proposed extended Goldfarb model isconsistent with the fitted model parameters. In order to build the self-sensing structure in Figure 7.4, a hysteresis operator H is also required.Any conventional hysteresis operators can be employed as H, (See Sub-section 2.6.1). However, this chapter introduces a novel and very simplehysteresis operator, which is detailed in the next subsection.7.2.4 A simple asymmetric hysteresis model fordisplacement-force control of piezoelectric actuatorsConventional hysteresis operators are typically computationally expen-sive. For example, the Prandtl-Ishlinskii operator used in Chapter 6, em-ployed 150 Play operators. In this section, a new and computationally ef-ficient phenomenological model for rate independent hysteresis with bothmajor and minor loops is introduced that maps the hysteresis voltage ofPESA-1, VH1 to the charge q1 passing through it. the model can map asym-metric hysteresis loops and unlike most conventional hysteresis models, ut957.2. A new self-sensing technique for piezoelectric actuators-200 0 200 400-505101520[V](a) (b)Figure 7.6: (a) The measured hysteresis voltage VH1, (shown in blue), usingthe identified parameters in Table 7.1 and the fitted hysteresis voltage V˜H1in red versus the measured charge q1. (b) A qualitative representation ofthe hysteresis model parameters. Used with permission.does not require to be inverted for real-time implementations, (See Sub-section 2.6.1). The presented model consists of two exponential functions,which are described by only four parameters that are identified from theidentification experiment in Figure 7.5. Another advantage of the presentedmodel over the existing models is that it requires measurements of currentrather than charge.Using the identified parameters in Table 7.1 the measured hysteresisvoltage is calculated from the experimental data using (4.1). The profile ofmeasured VH1 vs. charge q1 is plotted in Figure 7.6(a) in blue.Figure 7.6(b) provides a qualitative representation of the model. It con-sists of two exponential functions for the ascending and descending branchesof the hysteresis loop. In Figure 7.6(b), VA, qA, and τA are asymptotic max-imum hysteresis voltage, zero crossing, and the charge constant for the as-cending branches respectively. VB, qB, and τB are the minimum asymptotichysteresis voltage, zero crossing, and the charge constant for the descendingbranches respectively. The mathematical representation of the model isVH =VA(1− e− q−qAτA ), if I1 ≥ 0VB(1− e−q−qBτB ). otherwise(7.17)The finite difference formulation of this model uses current instead of charge967.3. Sensor-less force and displacement controller using the new self-sensing techniquemeasurements, it eliminates the need for calculating qA and qB parameters,and it can model both major and minor hysteresis loops:V˜H1(i) ={V˜H1(i−1) + (VA − V˜H1(i−1))∆tτA I1(i), ifI1(i) ≥ 0V˜H1(i−1) + (VB − V˜H1(i−1))∆tτB I1(i), otherwise(7.18)where, ∆t is the sampling period. The model parameters are fitted usingleast square error minimization, where experimental data from ascendingbranches provide fitted values for VA and τAVAτA =∑∆VH1(i)∆tI1(i) −∑(∆VH1(i))2∑(∆tI1(i))2 −∑∆VH1(i)∆tI1(i)−1×∑VH1(i)∆VH1(i)∆tI1(i)∑VH1(i)(∆tI1(i))2 ,(7.19)and data from the descending branches provide fitted values for VB and τBVBτB =−∑∆VH1(i)∆tI1(i) −∑(∆VH1(i))2∑(∆tI1(i))2∑∆VH1(i)∆tI1(i)−1×−∑VH1(i)∆VH1(i)∆tI1(i)∑Vh1(i)(∆tI1(i))2 ,(7.20)where ∆VH1(i) = VH1(i) − VH1(i−1). The identified values are summarizedin Table 7.2. The fitting results V˜H1-q1 are depicted in Figure 7.6(a) inred together with the measured VH1-q1 hysteresis loop. Figure 7.6(a) showsthat the employed model fits the major and minor loops of the measuredhysteresis loop.7.3 Sensor-less force and displacement controllerusing the new self-sensing techniqueThe values n, k, CP , and Cv identified in Subsection 7.2.3, together withthe hysteresis operator characterized in Subsection 7.2.4, are employed inthe displacement-force self-sensing technique structure in Figure 7.4.This structure is implemented in a sensor-less controller for the 2 DoFsetup explained in Chapter 3. The control structure is depicted in Fig-ure 7.7. This control structure has already been described in Chapter 3,977.3. Sensor-less force and displacement controller using the new self-sensing techniqueTable 7.2: Identified parameters for the presented hysteresis model.Parameter Identified ValueVA 18.5 VτA 136 µCVB -1 VτB 113 µC+- -+- + Force SensorLaser SensorSelf-sensing technique in Figure 6.4Decoupled systemFigure 7.7: A sensor-less displacement-force controller using the new self-sensing technique. Used with permission.(see Section 3.3), and applied in the validation experiments in Chapter 6.The current study replaces the measured displacement x1m and force F1mwith their corresponding estimated values x˜1 and F˜1 respectively in thefeed-back loop of the controller. A simultaneously varying displacement-force test scenario is generated to evaluate the performance of the presentedself-sensing technique shown in Figure 7.4 and the control structure shownin Figure 7.7. Figure 7.8 shows the reference xref vs Fref profile in blue.The gray arrow shows the travel direction along the path.The actual displacement-force profile developed by the setup is measuredand shown in red in Figure 7.8, which follows the reference path. Displace-ment and forces are shown separately in Figure 7.9. Reference xref andmeasured x1m displacements are plotted against time in Figure 7.9(a). Theabsolute value of the difference between these two values |esx| =|xref −x1m|represents the displacement control system error. An average absolute error987.3. Sensor-less force and displacement controller using the new self-sensing technique-10 -5 0 5 10-500-450-400-350-300-250-200-150-100-50050Figure 7.8: Reference Fref vs. xref and measured F1m vs. x1m paths shownin blue and red respectively. Used with permission.value of µ(|esx|) = 2% is obtained.Similarly, Figure 7.9(b) depicts the reference Fref and measured F1mforces. An average absolute error value of µ(|esF |) = 3% is obtained for theforce control system error. The obtained errors for displacement and forceare similar indicating equal control performance.Table 7.3 compares the performance of the presented technique withsimilar displacement and force self-sensing techniques based on Goldfarb etal.’s model in Chapter 6 and other techniques in the literature. Comparedto the results in Chapter 6, the presented modification to the Goldfarb etal.’s model has led to slight improvement. However, this study has presentedother more valuable findings:A new electromechanical model is presented for piezoelectric actuatorsthat takes the voltage dependency of its actuator’s capacitance into account.The presented electromechanical model led to a new parameter identificationmethod that requires a single experiment. Therefore, the identified param-997.3. Sensor-less force and displacement controller using the new self-sensing technique0 0.05 0.1 0.15 0.2 0.25 0.3-10-8-6-4-20246810012345678910(a) (b)0 0.05 0.1 0.15 0.2 0.25 0.3-500-450-400-350-300-250-200-150-100-50050051015Figure 7.9: (a) Reference xref and measured x1m displacements plottedagainst time. (b) Reference Fref and measured F1m forces plotted againsttime. The absolute value of the control system errors, |esx| and |esF | areplotted in green in (a) and (b) respectively. Used with permission.Table 7.3: Performance of the proposed technique in comparison with somereported techniques in the literature.Self-sensingtechniqueµE ,Chapter 7µE ,Chapter 6µE ,[BQN08b]µE ,[MU14]µE ,[KSUO12]Displacement 2% 4% 2% 12% 11%Force 3% 4% 2% 12% 11%eters are more reliable compared to existing methods that require severalexperimental setups. Furthermore, employing this parameter identificationmethod, the measured hysteresis voltage-charge loop shown in Figure 7.6(a)confirms the assumption made in Figure 6.2 that the major hysteresis loopshould have zero slope at the end of its descending branch. This behavioralso allows for the use of an efficient exponential hysteresis operator thatfurther simplifies the presented model.100Chapter 8Self-sensing technique usingimpedance measurementsFor constant external forces, permittivity of a piezoelectric actuator islinearly related to its displacement. A number of studies have used the linearrelationship for displacement self-sensing. Chapter 7 studied the effect ofapplied force on the capacitance of a stack actuator at quasi-static frequen-cies, where no noticeable relationship between the force and the capacitancewas observed. However, this chapter hypothesizes that using the completeinformation provided by the impedance measurements, external force anddisplacement can simultaneously be detected.A detailed study on the effect of force on the impedance of a piezo-electric actuator is presented in this chapter. The current chapter uses thecomplete impedance information, to detect the external force as well as thedisplacement of an actuator. Section 8.1 describes the feasibility of the pro-posed technique. Section 8.2 describes a real time impedance measurementmethod. Section 8.3 describes the implementation of the proposed self-sensing technique. The proposed technique is experimentally validated inSection 8.4. Finally Section 8.5 describes the implantation of the proposedtechnique in a sensor-less control structure.8.1 The effect of force on the impedance of apiezoelectric actuatorThis section describes a frequency sweep impedance measurement of apiezoelectric actuator to gather a better understanding of the effect of exter-nal forces on the impedance of a piezoelectric actuator. The 2 DoF setup de-scribed in Chapter 3 is employed together with an Agilent-E4980A LCR me-ter for this purpose. The LCR meter performs a frequency sweep impedancemeasurement for PESA-1, while PESA-2 varies the external load on PESA-1. The schematics of this experiment is depicted in Figure 8.1.A small driving voltage of V 1=1 V is fed to PESA-1 from the LCR meter1018.1. The effect of force on the impedance of a piezoelectric actuatorForce SensorLCRMeter+-PESA-2 PESA-1Figure 8.1: The setup used for studying the effect of external loads on theimpedance of a piezoelectric actuator. Used with permission.for measurement purposes. A driving voltage of V 2 in the range of 0 to 100 Vis fed to PESA-2 to vary the external forces. The linear approximation of thedisplacement and force created by the setup for quasi-static applications wasgiven by (3.8). Since, the voltage from the LCR meter is small (V1 V2),force approximation in (3.8) could be simplified toF1 = −n2V2. (8.1)The impedance of PESA-1 is measured under different compressive forcesof Fs = 0 N, -166 N, and -361 N by exiting PESA-2 with V 2=0 V, 50 V,and 100 V respectively. It is observed that the theoretical force values us-ing (8.1), and the corresponding measured values at 50 V and 100 V, have19% and 9% discrepancy respectively. This is due to hysteresis which isneglected in (8.1). The Agilent-E4980A LCR meter performs a frequencysweep impedance measurement at each loading condition. To eliminate theeffect of creep, each frequency sweep is carried out approximately one minuteafter applying the preload voltage to PSA-2. The frequency range for thesweep is 60 kHz to 180 kHz, which is well above the first mechanical res-onance frequency of 4 kHz of the test setup. Thus, the displacement ofPESA-1 is not influenced by the impedance measurement [IM10].Magnitude and phase of zero force condition impedance are depictedin blue in Figures 8.2(a) and 8.2(b) respectively. The corresponding ca-pacitance and resistance values are depicted in Figures 8.2(c) and 8.2(d)respectively. Four major resonance peaks at frequencies of 60 kHz, 95 kHz,1028.1. The effect of force on the impedance of a piezoelectric actuator50 100 150 200051015Impedance 50 100 150 200-90-80-70-60-50-40Phase (deg)50 100 150 200Frequency (kHz)11.522.533.5Capacitance50 100 150 200Frequency (kHz)00.20.40.60.8Resistance (a) (b)(c) (d)= 0 N= N= N80 100 120 140 16000.51-166-361Figure 8.2: Frequency sweep results for impedance measurement of PSA-Iunder F1m=0 N, -166 N, and -361 N external force conditions. (a) magni-tude. (b) phase. (c) Capacitance. (d) Resistance. Used with permission.125 kHz, and 135 kHz are detected at zero force. The red and yellow profilesdepict the results for external forces of F1m= -166 N and F1m = -361 Nrespectively.No noticeable change in the impedance is observed outside the resonancefrequencies for the three levels of applied external load, which is consistentwith the studies in Chapter 7. Thus, a permittivity measurement betweenresonance peaks is not a good indicator of applied force.A clear shift in the location of the resonance peaks in Figure 8.2 is de-tected as a result of external forces. The reason for the observed behaviorof the impedance at the resonance frequencies is partly related to the ge-ometry change of the actuator at elevated pre-loads where the actuator’slength decreases and its cross sectional area increases. This in turn in-1038.2. Real time impedance measurement methodForce SensorLaser SensorAmp.-2DS 1103 Data Acquisition BoardAmp.-1 VABand Pass FilterPESA-2 PESA-1Figure 8.3: The setup used for the proposed impedance based force anddisplacement self-sensing technique. Used with permission.creases its stiffness and resonance frequency [IM11]. Therefore, in contrastto conventional permittivity change based displacement self-sensing tech-nique that uses frequencies far from the resonance frequencies, the proposedtechnique suggests measuring impedance at frequencies close to resonancepeaks to detect changes in external forces. Two measurements are requiredto simultaneously predict displacement and force. The proposed self-sensingtechnique measures capacitance and resistance of a piezoelectric actuator ata specific electrical resonance frequency, where both are affected by externalforces and the actuator displacement.8.2 Real time impedance measurement methodSimilar to permittivity detection methods in the literature, a ripple volt-age Vr is added to the driving voltage V1 of PESA-I in order to carry outa real-time impedance measurement. The frequency of the impedance mea-surement voltage needs to be: 1- Far from the first mechanical resonancefrequency of the setup (4 kHz). 2- Close to a higher order frequency todetect any impedance change as a result of applied external forces. In thisstudy, Vr has a frequency of 100 kHz which meets both criteria. The mag-1048.2. Real time impedance measurement method-20 -15 -10 -5 0 5 10 15 20-700-600-500-400-300-200-100021,910345627822Figure 8.4: Measured displacement-force map created for construction ofthe proposed self-sensing technique. Used with permission.nitude of Vr is selected to provide a a small, but measurable Current ripple.At a ripple frequency of ωr = 100 kHz, a voltage ripple magnitude of Vr =0.1 V provides a current ripple magnitude of Ir = 0.1 A.Figure 8.3 illustrates how the setup described in Chapter 3 is employedto frame the impedance measurement technique. The ripple voltage Vr isadded to the desired driving voltage V1 via a high power PI-E617-00F ampli-fier AMP.-I. The resulting voltage Vt = V1 +Vr is in the range of 0 to 100 V.It is fed to PESA-I. In order to detect the ripple voltage V r and ripplecurrent Ir, voltage and current of PSA-I are measured via a Tektronix-TCPA300 voltage probe and a Tektronix-TCPA300 current probe respec-tively and filtered at 100 kHz using a Krohn-Hite-3384 bandpass filter.In order to efficiently sample the measured signals, the frequency foldingprinciple around the Nyquist frequency is utilized for a sampling rate of45.455 kHz in a DS 1103 data accusation board. The folded 100 kHz ripplesignals is then folded to 9.09 kHz. Therefore the current and voltage signalsare sampled using N = 45.4559.09 = 5 samples. Using discrete Fourier transform1058.3. Force and displacement detectionin a Matlab/Simulink environment, the ripple voltage and current can bedetermined asF(Vr(t)) =N−1∑n=0Vr(t)ne− j2pinN = AVr + iBVr , (8.2)F(Ir(t)) =N−1∑n=0Vr(t)ne− j2pinN = AIr + iBIr . (8.3)The impedance can be measured asZ = R+ i12piωrC=F(Vr(t))F(Ir(t)) . (8.4)Therefore using the Fourier transform coefficients of Vr, AVr and BVr as wellas Fourier transform coefficients of Ir, AIr , capacitance and resistance valuesare obtains asC = − 12piωrAI r2 +BI r2AI rBV r −AV rBI r , (8.5)R =AI rAV r +BI rBV rAI r2 +BI r2. (8.6)A more detailed explanation of the impedance measurement procedure isfound in [ISM11a, MS16].8.3 Force and displacement detectionA single calibration experiment is required to generate reference mapsbetween impedance, displacement, and force. In later sections, these mapswill be employed in real time applications to detect displacement and forcefrom measurements of impedance.The displacement-force illustrated in Figure 8.4 is generated using thesetup described in Chapter 3 in feed forward control mode.Corners of the map are generated using the extermum voltage values.The gray arrows indicate the order of the traced path segments.Using the impedance measurement method described in Section 8.2, se-ries capacitance C, and resistance R of PESA-I are measured while tracingthe displacement-force path. Figure 8.5 shows the measurement results.Figures 8.5(a) and (b) show the relationship between the measured values1068.3. Force and displacement detectionof displacement x1m, capacitance C, and resistance R. To frame the pro-posed self-sensing method, displacement is modeled as a third order (usingt-test) least square fit of capacitance and resistancex˜1 = 3.1e3− 4.6e3C − 2.6e3R+ 0.8e3CR+ 2.5e3C2 + 3.1e3R2 − 0.5e3C3 − 2.0e3R3. (8.7)A Similar relationship between the measured force F1m, capacitance C, andresistance R is shown in Figures 8.5(c) and (d) respectively. Another third-0.2 0.4-20020(a)1.2 1.4 1.6-20020(b)0.2 0.4-600-400-2000(d)1.2 1.4 1.6-600-400-2000(e)Figure 8.5: Measurements used to construct the self-sensing technique. (a)Measured displacement vs. detected resistance. (b) Measured displacementvs. detected capacitance. (c) Measured force vs. detected resistance. (e)Measured force vs. detected capacitance. Used with permission.1078.4. Experimental validation-200120(a)1.5 0.5020(b)-6002-4001.5-2000.501Figure 8.6: (a) Measured displacement vs. detected capacitance and resis-tance profile is shown in red, together with the fitted surface using (8.7).(b) Measured force vs. detected capacitance and resistance profile is shownin red, together with the fitted surface using (8.8). Used with permission.order polynomial (using t-test) least square fit is framed for the force modelF˜1 = −0.1e5 + 0.2e5C + 1.2e5R− 1.6e5CR− 0.1e5C2 − 0.6e5R2 + 0.5e5C2R+ 0.5e5CR2. (8.8)Figure 8.6(a) shows the fit using (8.7). The confidence value for this fitis 0.989. The fit for force using (8.8) is shown in Figure 8.6(b) and has aconfidence value of 0.994.So far, this chapter proposed a impedance-change based self-sensing tech-nique utilizes a non-hysteretic relationship between displacement, force andimpedance of piezoelectric actuators that is experimentally validated. Fig-ure 8.7 depicts a graphical overview of the proposed technique.8.4 Experimental validationThe framed self-sensing technique in Figure 8.7 is experimentally val-idated with the force-displacement path shown in Figure 8.8. Using theimpedance measurement method described in Subsection 8.2, capacitanceand resistance values are obtained and substituted into (8.7) and (8.8) toestimate displacement and force. The performance of the proposed tech-nique is evaluated using mean value µE and standard deviation σE of theestimation errors. Figure 8.9 shows the measured displacement xs togetherwith the estimated displacement x˜. The displacement estimation has a mean1088.5. Controller design++Band Pass FilterFourier TransformsTo drive the actuatorTo detect the impedancePrecalibrated mapsPESAAVEq. (7.7) Eq. (7.8)Figure 8.7: Real-time implementation of the impedance-change based dis-placement and force self-sensing method. Used with permission.error of 1.8%, and a standard deviation of the error of 1.9%. Measured Fsand estimated F˜ force values are depicted in Figure 8.10. Force estimationhas a mean error of 1.7%, and a standard deviation of 1.8%.8.5 Controller designThe proposed force and displacement self-sensing technique introducedin the previous section is now integrated into a sensor-less displacementand force control structure. The controller is implemented on the setupdescribed in Chapter 3. The measured feed-back signals in the original feed-back mode of the setup are replaced with the estimated values using the1098.5. Controller design-20 -15 -10 -5 0 5 10 15 20-700-600-500-400-300-200-1000162534222Figure 8.8: Measured displacement-force map created for validation of theproposed self-sensing technique described in (8.7) and (8.8). Used withpermission.proposed technique. Figure 8.11 depicts the resulting sensor-less controlstructure.A simultaneously varying displacement-force test scenario is created tovalidate the performance of the control structure. To show results moreclearly, Figure 8.12 depicts the experimental forces and displacements sep-arately against time. The reference displacement xref , the estimated dis-placement x˜1, and the laser measured displacement x1m are plotted in Fig-ure 8.12(a). The figure shows good matching between, xref , and x˜1. Thedifference between these two values is the actuating signal of the displace-ment controller. The absolute value of this signal |Eax| =|xref − x˜1| isplotted in black. An average absolute error of µ(|Eax|) =2.3% is obtained.This value measures the performance of the displacement integral controller.Figure 8.12(a) also shows that the laser measured displacement profile x1mfollows the reference profile xref well. The absolute value of the differencebetween these two profiles |Esx| =|xref −x1m|, plotted in purple, represents1108.5. Controller design0 0.2 0.4 0.6-20-1001020-30-20-100102030Figure 8.9: Measured and estimated displacement. Used with permission.the overall displacement control system error. An average absolute error ofµ(|Esx|) =3.7% is obtained.Similarly, the reference force Fref , the estimated force F˜1 and the mea-sured force F1m, together with the absolute values of force actuating signal|EaF | =|Fref−F˜1| and overall force control system error |EsF | =|Fref−F1m|are plotted in Figure 8.12(b). Average absolute error values of µ(|EaF |) =2.1% and µ(|EsF |) = 2.4% are obtained. These are similar in magnitude tothe displacement errors.The overall performance of the proposed control structure is similar tothe best reported in the literature [BQN08b]. However, actuating errors aresmaller than control system errors. This indicates that the main source ofthe error is the impedance-change based displacement and force estimationmethod. Figures 8.12(a) and (b) show peak error values at the corners ofthe reference profiles, where the frequency content is higher than in other re-gions. This indicates that impedance-change based measurements are mostpromising for quasi static control conditions.In summary, this chapter evaluated the performance of a complete sensor-less displacement and force control structure using an impedance-change1118.5. Controller design-30-20-1001020300-200-400-6000 0.2 0.4 0.6Figure 8.10: Measured and estimated force. Used with permission.based self-sensing method. The main application of the proposed controlstructure is precise positioning, where both displacement and force need tobe controlled.The experimental results show average absolute values of µ(|Esx|) =3.7%,µ(|EsF |) =2.4% for overall control system displacement and force errors re-spectively. These values are similar to the best reported in the literature.The experimental results indicate that the main source of errors is the es-timation error at higher contouring frequencies, which were not taken intoaccount in the quasi-static fitting experiments. The end of the experimentcontains a short steady-state section, which shows that the controlled forceand displacement values do not drift. However, the effect of creep will bemore pronounced if the steady-state experiment runs for a longer time.1128.5. Controller design+- -+- + Force SensorLaser SensorSelf-sensing technique in Figure 7.10Decoupled systemFigure 8.11: A sensor less controller using impedance measurement self-sensing technique. Used with permission.Time [s] Time [s]Error [%]Error [%]0 0.5 1-15-10-5051015-500500 0.5 1-600-500-400-300-200-1000100-50050(b)(a)Figure 8.12: Experimental results for validating the control structure. (a)The reference xref , the predicted x˜1 and the measured x1m displacementprofiles together with the absolute values of the displacement actuating sig-nal |Eax| and the overall error of the structure |Esx|. (b) The reference Fref ,the predicted F˜1 and the measured F1m force profiles together with the ab-solute value of the force actuating signal |EaF | and the overall error of thestructure |EsF |. Used with permission.113Chapter 9Conclusions and future worksThe current chapter provides conclusions for the research carried out inthis thesis. Section 9.1 discusses the thesis contributions and Section 9.2describes the suggested future work for the research.9.1 Research contributions and ConclusionsThis thesis presents novel techniques to minimize the need for sensors inpiezoelectric actuator-driven micro-positioners. The proposed techniquesare categorized into two groups of sensor-actuator integration and self-sensing techniques.Chapter 2 described the background of the presented research, whichconsists of the definition of piezoelectricity, different piezoelectric actuatortypes, their implementation challenges, their electromechanical models, andthe existing control approaches for them. The main contribution of Chap-ter 1 was presenting a through review of self-sensing control techniques forpiezoelectric actuators. The reviewed techniques were categorized based onthe actuators’ charge, voltage, and permittivity measurements. The readerof this chapter will gain the required knowledge to follow the information inthe later chapters. In addition, this chapter provided the researchers in thefield of self-sensing with a comprehensive review of the existing techniquesin the literature.Chapter 3 provided a detailed description of a 2 DoF setup that gener-ates independently varying displacement and forces for a piezoelectric stackactuator. The setup was employed in the experiments in Chapters 4, 6,7, and 8. First, Chapter 3 described the electromechanical model of thesetup. Then, the actual prototyped setup was described and a frequency re-sponse analysis was provided. Two control modes, voltage feed-forward anddisplacement-force feed-back control, were designed and validated experi-mentally. The feasible displacement-force operation regions were designedand validated experimentally using reference sensors.Chapter 4 described a novel sensor-actuator integration technique. Thetechnique was built on Goldfarb et al.’s assumption that a piezoelectric ac-1149.1. Research contributions and Conclusionstuator’s hysteresis voltage is only a function of the charge passing throughit. For the first time in the literature, this assumption was experimentallyvalidated in this chapter. The proposed technique employed a piezoelectrictransducer as an actuator in a sensor-free work-space. The novelty of theproposed technique was that the actuating transducer was electrically con-nected in series to a sensing transducer, which was clamped between a solidframe and a force sensor outside the work-space. The two transducers wereequivalent. Therefore, since the same charge passed through them, theyhad the same hysteresis voltage. The proposed technique employs measure-ments of the voltage difference of the two transducers and the force sensorto compensate for the hysteresis voltage and to estimate the displacementand force of the actuating transducer. The main novelty of this chapter liesin the application of a sensing actuator outside the physical workspace usinga series electrical connection. Compared to the only other techniques thatuses a dedicated sensing actuator, the proposed technique better eliminateshysteresis and it can identify force in addition to displacement. The tech-nique does not require sophisticated electrical circuits. Less than 1% averageabsolute errors were measured for both displacement and force estimations.Chapter 5 described a sensor-actuator integration technique that wasdedicated to tip displacement measurement of piezoelectric benders. Thenovelty of the presented technique lied in the fact that two strain gaugeswere constructed from top and bottom electrodes of a bender and no ex-ternal sensors were added to the system. The presented technique is notaffected by creep. First, a linear relationship between the bender’s surfacestrain and its tip displacement was derived. Then, a surface strain measure-ment method was proposed that used the constructed strain gauges in anelectrical half-bridge configuration. Open and closed-loop performances ofthe presented technique were experimentally compared to a reference sensor.The maximum difference between the two tip displacement measurementswas recorded as 1.1% of the bender’s full tip displacement. However, theproposed technique had limitations for high frequency applications due tominimum phase zeros caused by capacitive feed-through from the drivingelectrodes. This limits the control bandwidth to quasi-static frequencies.Chapter 6 presented a self-sensing structure that used voltage and chargemeasurements of a piezoelectric actuator to estimate its displacement andforce. Two similar techniques had been previously reported in the litera-ture [MU14, BQN08b]. The contributions in this chapter included a novel,more accurate, and more efficient model parameter identification technique,as well as the application of a more widely accepted hysteresis operator. Thepresented technique was experimentally validated on the setup described in1159.1. Research contributions and ConclusionsChapter 3 for frequencies below the first mechanical resonance of the setup.About 4% average absolute error values were calculated for both estimateddisplacement and forces.Chapter 7 builds on and improves the self-sensing technique presentedin Chapter 6. The contributions of this chapter are based on a novel con-stitutive model that postulates that the piezoelectric capacitance is linearlyrelated to the applied voltage. This allows for an elegant new parameteridentification method that identifies the characteristic parameters of thepiezoelectric actuator through a single hysteresis free experiment. In addi-tion, a novel exponential hysteresis operator was proposed that can modelasymmetric hysteresis loops and does not need to be inverted for real-timeimplementations. Another advantage of the presented hysteresis model overthe existing models is that it requires measurements of current rather thancharge. The proposed self-sensing technique was implemented in a sensor-less control structure for the setup described in Chapter 3. Average absoluteerror values of about 2% and 3% had been measured for displacement andforce respectively. The presented technique exhibited marginal improve-ments over the technique in Chapter 6. However, the main contributionof the presented technique lied in the introduction of a more experimen-tally consistent electromechanical model of piezoelectric actuators and anexceedingly simple hysteresis operator.Chapter 8 described a novel impedance based displacement and forceself-sensing technique for piezoelectric actuators. Chapter 7 experimentallyshowed that a piezoelectric actuator’s capacitance is almost linearly relatedto its applied voltage. Chapter 8 for the first time studied the effect ofexternal forces on the impedance of piezoelectric actuators. No noticeablechange in the impedance was observed for frequencies between mechanicalresonances peaks. This is in agreement with the results reported in Chap-ter 7. However, at frequencies close to resonance peaks, a clear shift inthe resonance location was observed. The reason for this observation is be-lieved to be the dimensional change of the actuator in response to an appliedforce. Guided by these observations, a novel self-sensing technique was pro-posed for the first time that measures the impedance of an actuator at afrequency close to a resonance peak and maps it to the actuator’s displace-ment and force. The presented technique was implemented in a sensor-lessdisplacement and force control structure on the setup described in Chap-ter 3. The experimental results showed average absolute values of 3.7% and2.4% for overall control system displacement and force errors respectively.The performance of the presented technique was limited to low frequencyapplications.1169.1. Research contributions and ConclusionsIn summary, this thesis proposed techniques for adding position andforce feed-back to piezoelectric micro-positioning systems without infringingon their work-space. Two groups of solutions were offered: sensor-actuatorintegration and self-sensing techniques. Table 9.1 provides a qualitativecomparison between these techniques.1179.1. Research contributions and ConclusionsTable9.1:Aqualitativecomparisonofthepresentedtechniquesinthisthesis.ChapterTechnique:Sensor-actuatorintegration(SAI)Self-sensing(SS)Measureda(Estimated)AdvantageDisadvantageComplexityb3SAIVandF(xandF)Hysteresis-freeRequiresrelativelylargedrivingpowerLow4SAIV (x)NotaffectedbycreepSuffersfromcapacitancecouplingsathighfrequenciesModest5SSVandq(xandF)SingleparameteridentificationexperimentNumericallyheavyhysteresismodelhigh6SSVandq(xandF)SimplehysteresismodelNewphysicalmodelOnlyquasi-staticModest7SSZ(xandF)HysteresisfreeNofrequencyselectionguidelineHighaV=voltage,q=charge,Z=impedance,x=displacement,F=Force.bLow=passivecomponentsonly,Modest=analogueimplementationpossible,High=DSPrequired.1189.2. Future work9.2 Future workChapter 3 described a 2 DoF setup that consists of two stack actuatorsassembled in series. By varying the driving voltages of the actuators, in-dependently varying displacement and forces are generated. However, dueto the power limits of the available amplifiers, only quasi-static experimentscould be carried out with this setup. In the future, more powerful ampli-fiers will be employed to create higher frequency experiments. In addition,the environment temperature influences on the performance of piezoelectricactuators.Chapter 4 employed an actuating piezoelectric transducer in a sensor-free micro-positioning work-space and integrated it with a sensing trans-ducer that is clamped out-side the work-space between a force sensor anda solid frame. The displacement and force of the actuating piezoelectrictransducer are estimated using measurements of the force sensor and thevoltage difference of the two transducers. A future study could improvethe technique’s slow speed performance by compensating for measurementdrifts. In addition, future studies will improve on the achievable actuatingdisplacement and forces by introducing different boundary conditions for thesensing transducer.Chapter 5 integrated a bender with two strain gauges to measure thebender’s tip displacement at zero external forces. No guideline is presentedin this chapter for choosing a proper strain gauge length to achieve goodstrain sensitivity. The future work will optimize the gauge lengths, so thatthe tip displacement can be measured regardless of the presence of an ex-ternal force. In addition, the current technique suffers from capacitancecoupling between the gauges and the bender electrodes. The future workwill address this problem. The future work will also propose a technique forsimultaneously measuring force and tip displacement.Chapters 6 and 7 introduced two quasi-static self-sensing techniques thatuse measurements of charge and applied voltage to infer an actuator’s forceand displacement. In the future, dynamic performance of the techniques,temperature behavior, and creep effect on the techniques need to be studiedin detail.Chapter 8 maps the real-time impedance measurements close to a reso-nance peak to the displacement and the force of a piezoelectric actuator. Itis believed that the change in permittivity due to an applied voltage is theprominent effect that leads to the measured change in impedance. On theother hand, an external force changes the geometry of the actuator and con-sequently varies the resonance peak locations. The future work will study1199.2. Future workthese effects in more detail. In addition, the effect of creep and temperaturechange also need to be studied. In addition, the future work will benefitfrom machine learning algorithms to improve the presented techniques.120Bibliography[ADKB00] HJMTS Adriaens, Willem L De Koning, and Reinder Ban-ning. Modeling piezoelectric actuators. 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Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Sensor-actuator integration and self-sensing techniques for space constrained piezoelectric driven micro-positioners
Text