PIEZOELECTRIC TOOL

TURNING

, David Eddy - •'•:" -.

: B.A.Sc. (Mechanical Engineering) . : , . :

University of British Columbia •

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

' MASTER OF APPLIED SCIENCE .•'

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_ THE FACULTY OF GRADUATE STUDIES

MECHANICAL ENGINEERING

; Weaccept thisi thesis as conforming. ;.

: : to the required standard '

July 1999 •>/>'•':'.

- vV " copyright 1999, David Eddy In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer• ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed with• out my written permission.

DEPARTMENT OF MECHANICAL ENGINEERING The University of British Columbia 2324 Mail Mall, Vancouver, BC, V6T 1Z4, CANADA Abstract ii

Abstract

High precision machining has the potential of increasing the efficiency of machining opera• tions and lowering costs in the manufacturing of many machined parts. This thesis investigates the use of a piezoelectric tool/actuator designed for the precision turning of cylindrical shafts.

The piezo tool actuator has four fundamental components. The piezo stack element delivers a dynamic displacement range of 40pm when 0 to -1000V voltage is applied. The piezo element is placed in a mechanical housing with leaf springs, which amplifies the displacement range to 62pm. The piezo element is activated by an amplifier which delivers a maximum 150W and 0- 1000 V voltage. The digital control law runs on a PC/DSP which has 5KHz loop frequency. The tool position is measured using a laser displacement sensor with 0.1 pm accuracy, and the work- piece surface is monitored using another laser sensor which has 1 pm resolution. The piezo tool actuator is mounted on a standard CNC turning center turret, and is intended to be used in preci• sion positioning of cutting tool within one micrometer during finish turning of shafts.

The mathematical model of each component in the dynamic system is identified. The piezo• electric hysteresis is modeled using a time-delay model, and the actuator's transfer function is identified experimentally by analytically indicating the source of dynamics. Several digital con• trol methodologies are developed for the actuator based upon the model generated. Pole place• ment, zero phase error tracking, and state feedback control with a disturbance observation model have been developed and cutting tests performed to evaluate their suitability for precision turning.

The mathematical model and experimental trials indicate that one to two micrometer dis• placement can be best achieved using a feed-forward controller with state feedback disturbance observer and rejection. The ability of the system in tracking highly dynamic position commands is also demonstrated by turning elliptical shafts. The identified bandwidth of the actuator is found to be about 500Hz, but the displacement ranged rapidly drops to ~12um at the peak frequency. Abstract 111

Higher dynamic bandwidth and surface finish accuracy can be obtained if the mechanical system is more rigidly designed, and the amplifier has more power to drive the piezoelectric ele• ment at higher actuation frequencies. Table of Contents iv

Table of Contents

Abstract ii Table of Contents iv List of Figures viii List of Tables x Acknowledgements xi Nomenclature xii

1 Introduction 1 1.1. Motivation 1 1.2. Outline of Thesis Content 2

2 Literature Review 3 2.1. Base Technology 3 2.1.1. Piezoelectric Devices 3 2.1.2. Magnetostrictive Devices 5 2.1.3. Electrostrictive Devices 6 2.1.4. MagLev Devices 7 2.2. Precision Tools and Actuators .7 2.3. Tool Sensors 8 2.3.1 .Laser Interferometry 8 2.3.2. Capacitive Sensors 8 2.3.3. LVDT Sensors 9 2.3.4. Laser Triangulation 10 2.3.5. Linear Gauge Sensors 11 2.4. External Sensors 11 2.4.1 .Roughness Measurement 13 2.4.2. Diameter Measurement 13 2.4.2. lLaser Curtain Sensors 13 2.4.2.2Snap Gauges 13 2.4.3. Run-Out 14 2.4.4. CNC Improvements 14 2.4.4. ILinear Slides 14 2.4.4.2Spindle Measurement 16 Table of Contents v

2.5. Conclusions 16

3 Piezoelectric Tool Actuator System 17 3.1. The Actuator System Components 17 3.2. Mathematical Modeling of Actuator Components 23 3.3. Modelling of Piezoelectric Actuators 23 3.4. Mathematical Models of Piezoelectric Stack Actuators 24 3.4.1.Strain Relationship 25 3.4.2.Dielectric Relationship 26 3.5. Equivalent Circuit For a Piezoelectric Translator 26 3.6. Modelling of Piezoelectric Element 28 3.6.1 .Piezoelectric Parameters 29 3.6.2. Piezoelectric Translator Simulation 30 3.6.3. Hysteresis 32 3.7. Hysteresis Background 34 3.8. Methods for Hysteresis Compensation 35 3.8.1 .Charge Controlled Piezoelectric Actuators ...35 3.8.2. Method of Equivalent Damping 36 3.8.3. Hysteretic Restoring Force Model 38 3.8.4. Phase Lead Compensation 40 3.9. Other Piezoelectric Non-Linearities 40 3.9.1. Capacitance 40 3.9.2. Elastic Compliance 41 3.9.3. Damping Coefficient 41 3.9.4. Temperature Effects 41 3.9.5. Electrostrictive Effects 42 3.9.6. Drift 42 3.10. Structural Dynamic Model of the Actuator Assembly 42 3.10.1 .Example Model 43 3.11. Factors Affecting Actuator Model 46 3.11.1 .Mounting of Actuator 46 3.11.2. Tool Holder 46 3.11.3. DSP Card : 49 3.11.4. LNS Sensor 49 Table of Contents vi

3.12. Summary 49

4 Digital Control of the Actuator Position 50 4.1. Introduction 50 4.1.1 .Traditional PID Control 50 4.2. Pole Placement 52 4.2.1. Background Theory 52 4.2.2. Previous Research 52 4.2.3. Discrete Model of the Actuator 53 4.2.3. IModel Form 53 4.2.3.2Continuous and Discrete Models 56 4.2.3.3Model Estimation Method 1 56 4.2.3.4Model Estimation Method 2 58 4.2.4. Desired Closed Loop Response 59 4.2.5. Controller Parameters 60 4.2.6. Pole Placement Control Tests 61 4.3. State Feedback Controller for Piezo Actuator 63 4.3.1 .Regulator Problem 63 4.3.2.Disturbance Observation 65 4.3.3.State Feedback Servo Problem 69 4.4. Zero Phase Error Tracking Control 72 4.5. Controller Tests With Actuator 74 4.5.1. Model of the Actuator System Mounted on the Lathe 74 4.5.1.lPole Placement Controller 75 4.5.1.2Zero Phase Error Tracking Controller (ZPETC) 77 4.5.1.3State Feedback Servo Controller 80 4.5.2. Disturbance Model 85 4.6. Cutting Force model 86 4.6.1.Cutting Tests 87 4.7. Conclusions 90

5 Precision Turning Tests with Piezoelectric Actuator 91 5.1. Introduction 91 5.2. Eccentric turning 91 Table of Contents vii

5.3. Extensions for on-line Sensors 95 5.3.1.Disturbance Rejection 104 5.3.l.lSpeed Reduction 106 5.3.1.2Retract Tool/Actuator 107 5.4. Eccentricity elimination 108 5.5. Turning for Precision Diameters 109 5.6. Conclusions Ill

6 Conclusions and Future Work 112 6.1. Conclusions 112 6.2. Recommendations for Future Research 113 Bibliography 115 List of Figures viii

List of Figures

Figure 2.1 Bi-Morph Actuation 5 Figure 2.2 Stack Actuation 6 Figure 2.3 Laser Interferometer Parts[9] 9 Figure 2.4 LVDT ...10 Figure 2.5 LNS and LTS Sensors [2],[3] 11 Figure 2.6 Laser Hologauge 12 Figure 2.7 Laser Scan Micrometer [4] 14 Figure 2.8 Snap Gauge[4] 15 Figure 2.9 Run-out Measurement 15 Figure 3.1 Actuator Mounted in Lathe 18 Figure 3.2 Essential Blocks of Tool/Actuator System 18 Figure 3.3 P-242.30 Stack Actuator 19 Figure 3.4 Dynamic model of the mechanical system 20 Figure 3.5 Sensor Signal Conditioning Circuitry 22 Figure 3.6 Amplifier Output Signal Conditioning 23 Figure 3.7 Equivalent Circuit - Piezoelectric Actuator 26 Figure 3.8 Piezoelectric Translator with Mechanical Load 28 Figure 3.9 Driven Piezoelectric Element 29 Figure 3.10 Simulation of Driven System 31 Figure 3.11 Hysteresis in Driven System 32 Figure 3.12 Typical Hysteresis Measured From the Actuator 33 Figure 3.13 Simulation of Equivalent Damping Hysteresis Compensation ..37 Figure 3.14 Simulation of Equivalent Damping Hysteresis Modeling 37 Figure 3.15 Hysteretic Restoring Force Model 39 Figure 3.16 Input-Output Curves for Hysteretic Restoring Force Model 39 Figure 3.17 Block Diagram Description with Hysteresis 43 Figure 3.18 System Model and Actuator Response 45 Figure 3.19 Actuator Mount at NRC 47 Figure 3.20 Actuator Mounting on the UBC CNC lathe 47 Figure 3.21 Hammer Tests on Actuator 48 Figure 3.22 Accelerometer Position 1, Hammer Hit at Point 1. (UBC mount) 48 Figure 4.1 Traditional PID Loop 50 Figure 4.2 Traditional PID Step Response 51 Figure 4.3 Frequency Response - PID 51 Figure 4.4 Pole Placement Control 52 Figure 4.5 Combined Dynamics of Translator 54 Figure 4.6 Combined Dynamics with Delay 54 Figure 4.7 Delay Comparison in Plant Model 55 Figure 4.8 Model Identification - Method 1 57 Figure 4.9 Step Response of the Pole Placement Controller (Test Bench) .62 List of Figures ix

Figure 4.10 Frequency Response of Pole Placement Controller (Test Bench) 63 Figure 4.11 State Feedback Regulator 63 Figure 4.12 State Feedback Regulator with Full State Observer 65 Figure 4.13 Plant with Applied Disturbance and Noise 66 Figure 4.14 State Feedback Regulator with Disturbance feedback 68 Figure 4.15 State Feedback Servo Simulation 70 Figure 4.16 State Feedback Servo Block Diagram 71 Figure 4.17 Zero Phase Error Tracking Controller 72 Figure 4.18 Step Response of Pole Placement Controller and Pole Placement Controller with ZPETC (Test Bench) 73 Figure 4.19 Frequency Response of ZPETC (Test bench) 73 Figure 4.20 Simulated Step Response of Plant and Controlled Plant 77 Figure 4.21 Simulated Step Response of ZPETC 79 Figure 4.22 Simulated Step Response (ZPETC, Pole Placement and Plant) 80 Figure 4.23 Step response of the State Feedback Servo Controller Simulation 84 Figure 4.24 Comparison of Pole Placement and State Feedback controllers Step Response and Response to Step Disturbance 84 Figure 4.25 Actual and Estimated Disturbance input to Plant 85 Figure 4.26 Orthogonal Cutting 86 Figure 4.27 Cutting Test Data Using State Feedback Servo Control with Disturbance Observer 88 Figure 4.28 Cutting Test Data using Polynomial Pole Placement Controller 89 Figure 4.29 Cutting Test Data PJX> Controller 89 Figure 5.1 Profile of an Elliptical Cut 92 Figure 5.2 Command and Error During Elliptical Cut 93 Figure 5.3 Laser Sensor Mount for Workpiece Measurement 95 Figure 5.4 Cutting with the Laser Twin Sensor 97 Figure 5.5 Phase I Workpiece Sensor Approach Phase 98 Figure 5.6 Phase II Cutting Workpiece 99 Figure 5.7 Laser Twin sensor measuring diminishing diameters 101 Figure 5.8 Phase III Holding Position 102 Figure 5.9 Phase IV of Operation - Retraction from the cut 103 Figure 5.10 Tool Position during cutting 104 Figure 5.11 Workpiece Measurement - No Compensation 105 Figure 5.12 LTS Sensor with Spindle Speed Reduction before Dwell Section 106 Figure 5.13 LTS Sensor Feedback Using Micro Motion Method 107 Figure 5.14 Eccentricity After Standard Cut - LTS Sensor 108 Figure 5.15 Actuator Position For Precision Diameter Cut, measured with the LNS sensor 109 Figure 5.16 Measured Eccentricity After Cut, measured with LTS sensor .110 List of Tables x

List of Tables

Table 3.1: Piezoelectric Translator P-242.30 19 Table 3.2 : LNS 18/60 Sensor Specifications 21 Table 3.3 : DSP Specifications 22 Table 3.4 : Parameters of Tool Actuator 43 Table 3.5 : Identified Model Parameters 44 Table 4.1 : Model Identified Parameters (Test Bench Actuator) 58 Table 4.2 : Initial Cutting Tests Mild Steel 87 Table 5.1 : Actuation Speed For Elliptical Cuts 94 Table 5.2 : Cutting conditions 96 Table 5.3 : Relative position of Workpiece and Sensor 101 Acknowledgements xi

Acknowledgements

The work presented in this thesis would not be possible without the support of many people and organizations. I would like to acknowledge the financial support of International Submarine Engineering Ltd. and especially Jim McFarlane who's boundless enthusiasm kept me going, the Science Council of British Columbia's GREAT scholarship program, and the National Research Council of Canada's IMTI West lab for allowing me to perform a great deal of my research there. I would also like to give special recognition to Dr. Yusuf Altintas and the people in the Manufac• turing and Automation Laboratory. Dr. Altintas has been an ardent supporter and has always given me the encouragement that I need. A master's degree has been a goal of mine for several years. Getting to this point has been an effort for me as well as my family. Special appreciation of my parents Laurance and Doreen Eddy for teaching me that you can achieve just about anything with hard work and determination. I love you both dearly. To my sisters Laureen, Barbara, Jennifer and Donna you all have shaped my life and have given me so much that you don't recognize, but I do. Nomenclature xn

Nomenclature

A : the area of the positive electrode of a piezoelectric stack. A(z) : the estimated plan denominator.

A0(z) : the observer polynomial.

Ac(z) : the controlled plant denominator.

Am(z) : the desired model denominator. b : the width of cut. B(z) : the estimated plant numerator.

Bm(z) : the desired model numerator. B+(z) : the cancelable zeros of the plant numerator. B (z) : the non-cancelable zeros of the plant numerator.

Bm(z) : the normalized model numerator B" (Z) : the phase shifted non-cancellable zeros of the controlled plant for the ZPETC.

B"c(z) : the non-cancellable zeros of the controlled plant for the ZPETC.

a B c(z) : the cancellable zeros of the controlled plant for the ZPETC.

Bc(z) : the controlled plant numerator. C : the estimated plant's output transition matrix.

C0 : the nominal capacitance of the piezoelectric ceramic.

Cj, Cm : represents the stiffness of the actuator in an equivalent circuit representation.

Cd : the nonlinear capacitance of a piezoelectric. Cjj : the state/output transformation matrix for the feed forward state model. C,., : relates how the disturbance states effect the disturbance. CNC: Computer Numerical Control CSO: CEI Des Senseurs Optique, Grenoble France : the piezoelectric stain constant for an electric field applied in direction i and strain pro• duced in direction j. DJJ : the input/output transformation matrix for the feed forward state model. 2

DP : electric flux density in a piezoelectric element (C/m ), Nomenclature xiii

DSP: Digital Signal Processor

Ep : electric field (V/m), EUVL: Extreme Ultraviolet Lithography

Ft : the tangential force in cutting. Fj : the feed force in cutting. g: Grams mass g(x, x) : the non-hysteretic component of Q{x, x). h : the chip thickness. Hz: Hertz (Frequency)

Hm(z) : the desired closed loop model of the actuator system for control. IMM: Intelligent Machining Module k : stiffness of the actuator, K : the state observer gain. KHz: Kilo Hertz (frequency) Kj-: the feed force cutting coefficient.

Kt : the tangential force cutting coefficient.

Kse : the tangential force edge forces.

Kje : the feed force edge forces.

Kpid : the proportional gain for the PID controller

Kw : the disturbance state observer gain. / : the unstrained length of the piezoelectric, L : the state feedback gain.

L,, Lm : represents the mass of the actuator in a equivalent circuit representation.

Lw : the disturbance feedback gain. LNS: Laser Nano Sensor LTS: Laser Twin Sensor LVDT: Linear Variable Differential Transformer m : the mass or dimension in Metres. MAL: Manufacturing & Automation Laboratory, UBC Nomenclature xiv

n : a parameter that controls the smoothness of the hysteretic restoring force model. N : the spindle RPM ORTS: Open Real Time Operating System P(z) : the plant model ^T^T A(z) Ph: Lead PC: Personal Computer PIM: Plug In software Module Qj : the free charge on a given positive electrode of a stack actuator. Q(x, x) : the hysteretic restoring force.

Rv Rm : the damping in the actuator in an equivalent circuit representation.

Rp : the portion of the R polynomial for the pole placement controller without the integral term. R(z) : the denominator for the feedback and feed forward transformations for the pole place• ment controller. s : elastic compliance of a piezoelectric element with a closed electric circuit,

Sp : strain in a piezoelectric element S(z) : the numerator of the feedback transformation for the pole placement controller. t : the thickness of a piezoelectric layer. 2

Tp : applied Stress on a piezoelectric element (A//m ), Ti: Titanium

TD : the time constant for the derivative portion of the PID controller. Tj : the time constant for the integral portion of the PID controller. T(z) : the numerator of the feed forward transformation for the pole placement controller. U : Electric Field Applied to Piezoelectric (Volts)

uamp '• Command voltage to Piezoelectric Power Amplifier

uc(k) : the controller input to the plant w^-+ Ujb.

udsp: Command output voltage from DSP

Uj^ik) : the controlled input to the plant model from the feedback controller.

Uj-j(k) : the feed forward part of the plant input. Nomenclature xv

V: Volts (Electric Potential) W: Watts (power) w(k) : the disturbance model state vector. w(k) : the difference between the actual disturbance state and the estimated. w(k) : the estimated disturbance state vector. x(k) : the state vector for the plant. x(k) : the estimated state vector for the plant. x(k) : the difference between the estimated state and actual state vector.

Xjj(k) : the state vector for the feed forward state space system.

yd(k) : the desired output position for the Zero Phase Error Tracking Controller (ZPETC) *

yd(k) : the phase shifted output position for the ZPETC.

ydsp: Sensor Voltage converted into DSP voltage range

aw ysensor '• ^ voltage out of a sensor

Y3 : Young's modulus in the axial direction of the stack actuator. z(x) : the non-linear hysteretic restoring force of Q(x, x) a : the scale factor for the hysteretic restoring force model. P : a parameter for the non-linear hysteretic restoring force model. A/ : the change in length of the piezoelectric,

e3 : the electrical permittivity of a piezoelectric element in the axial (3) direction T e : the electrical permittivity (Stress(T) held to zero). e(k) : the difference between the actual plant output and the estimated plant output. Y : a parameter for the non-linear hysteretic restoring force model. K : the non-linearity constant for piezoelectric capacitance, and X : a scaling factor from the plant transfer function to the model transfer function. nF: Nano-Farad (electrical Capacitance) [im: Micrometers

(tin : the natural frequency of a system. C, : the damping factor of a system, v : the disturbance input to the plant Nomenclature xvi

T : the estimated plant's state input matrix TJ-J- : the input transition matrix for the feed forward state model.

TM : the desired input vector for the controlled system. *F : the electro-mechanical coupling factor which is the relation between force and current through the actuator <& : the estimated plant's state transition matrix. : the state transition matrix for the feed forward state model.

<3>m : the desired state transition matrix for the controlled system.

<&xw : defines how the disturbance effects the states of the process. : the disturbance model's state transition matrix. W 0 C: Temperature in Degrees Celsius Chapter 1

Introduction

1.1. Motivation Precision turning of shafts with dimensional accuracy and high quality surface finish started to gain importance with the advent of ceramic tools. The dimensional accuracy and surface qual• ity depends on the material properties of the shaft and cutting tool, rigidity of the shaft and cutting tool, positioning accuracy of the feed drives and the tool geometry. Conventional CNC lathes employ ball screw feed drives with rotary or linear encoders as position feedback sensors. In gen• eral, due to backlash, friction in the guideways, thermal expansion, position dependent static deformation and positioning loop errors, the conventional machines can not provide dimensional machining accuracy under 10 pm. The goal of this thesis is to develop an "add-on" tool position• ing system, which can deliver motion with sub-micron accuracy for ultra-precision turning of shafts on conventional CNC lathes. A sponsoring company (Pratt & Whitney Canada) raised one of the key applications. Gas tur• bine shafts are first turned on conventional CNC lathes, and the bearing mounting surfaces are later turned again on an ultra-precision CNC turning center. The process requires two set-ups, and the ultra-precision turning center is very costly. A piezo tool actuator was previously developed for active chatter vibration control and precision turning at UBC Manufacturing Automation Lab• oratory. The tool actuator can be mounted on a standard turret of conventional CNC lathes. This thesis investigates the ultra-precision positioning of the piezo tool actuator for precision finish turning of bearing surfaces on the shaft. The successful development of such a system would eliminate the secondary machine and set up, thus saving significant time and capital investment in turning shafts.

1 Chapter 1. Introduction 2

1.2. Outline of Thesis Content

The thesis first surveys actuators and precision turning techniques proposed in the literature in chapter 2. The use of piezoelectric elements as opposed to magnetostrictive and other actuator mechanisms is critically reviewed. It is explained that the actuator must have high bandwidth, rigid structure and positioning range of at least thirty micrometers for ultra-precision turning of shafts. The sensors, which are capable of measuring displacements under one micrometer, are investigated along with their integration to standard CNC lathes. Chapter 3 presents a description of the piezoelectric actuator used in this thesis. The actuator consists of a piezoelectric stack element with one end connected to a mechanical leaf spring sys• tem that amplifies the motion. The cutting tool is mounted on the actuator. The mathematical model of piezoelectric elements is presented based on their physical properties. The source of hysteresis and non-linearity in piezoelectric actuators are investigated. The mathematical model of the mechanical system which houses the piezo element and cutting tool is also presented. The transfer function of overall physical system, where the input is voltage command to piezo actuator and the output is the cutting tool displacement, is developed. The hysteresis is approximated by a time delay in the transfer function. In Chapter 4, the digital control of a piezoelectric actuator for precision turning is developed. Several control laws, including PID, pole placement, zero - phase error tracking controller and feed-forward state controller with disturbance compensation, are applied to the position control of piezo tool actuator. The proposed control algorithms are tested on the CNC lathe, and the results are presented in Chapter 5. The positioning accuracy of ±lpm during turning cylindrical disks was achieved. The bandwidth of the actuator was demonstrated in following an elliptical path in turning a disk. The thesis is concluded with a summary of contributions and further research required in achieving an industrially acceptable ultra-precision turning actuator. Chapter 2

Literature Review

Much Work has been done on the use of high precision actuators to improve the accuracy in turning operations. The base technologies used for fine motion actuation are surveyed first, fol• lowed by applications in precision motion that have been developed previously. In addition, an investigation to the application of precision actuators for turning and the control of precision devices are reviewed. 2.1. Base Technology In the area of precision motion actuators, there are many types of actuation technologies that compete with each other. These technologies include:

•Piezoelectric, • Magnetostricti ve, •Electrostrictive, and •MagLev Technologies. These above technologies have been used to develop high precision motion actuators for a variety of applications including fast tool servos, x-y stages for photolithography, printer heads and computer hard disks [16],[21],[25],[34],[37],[45]. 2.1.1. Piezoelectric Devices The piezoelectric effect was first discovered in the late 19th century by the Currie brothers. They found that an electric potential was created when a pressure was applied to certain types of crystals. This was dubbed piezoelectricity.

Piezoelectric actuators have been developed that exploit the features of the crystal. Because of their nature, piezoelectric devices can be used as either a sensor, actuator, or both. In this thesis, the use of a piezoelectric device as an actuator is investigated.

3 Chapter 2. Literature Review 4

The strengths of Piezoelectricity include:

• non-magnetic •high efficiency • good temperature stability • large force generation • high stiffness • good linearity •compact size • low power consumption, and • fast response time Piezoelectric devices have been used in many types of actuators for control of structures. Straub et. al. [44] and Chen et. al. [57] investigated using piezoelectric devices embedded in rotor blades for helicopters to effect the shape of the rotor blade. Dasgupta et. al. [58] and Bailey et. al. [126] investigated using piezoelectric elements to actively damp vibrations in cantilevered beams. Agrawal et. al. [75] and Charon et. al. [88] investigated using piezoelectric devices for vibration suppression in flexible space structures. The use of piezoelectric devices as actuation points for smart structures is one application of piezoelectric technology. This type of actuation typically relies on a bi-morph actuation of a piezoelectric (See Figure 2.1). This type of actuation is typical when embedding a piezoelectric device within a structure for the actuation of the structure and allowing the bimorph actuation to bend the structure. Another method of piezoelectric actuation is the use of multi-layer(stack) piezoelectric ele• ments that use the longitudinal expansion of the crystal under an applied electric field. Takahasi [129] performed an analysis of the two actuation schemes. The stacked or multi-layered actuators provide higher forces with higher natural frequencies than bimorph actuation. The bimorph, how• ever, offers a larger actuation range. Chapter 2. Literature Review 5

Figure 2.1 Bi-Morph Actuation

For the precision turning application, a stacked actuation scheme (Figure 2.2) would be the most appropriate. Piezoelectric actuators have been used to directly produce fine motion for sev• eral applications. Li et. al. [37] presented a piezoelectric fine positioning system for compensation of spindle errors in precision diamond turning. Moriwaki et. al. [39] used a piezoelectric stack actuator to reduce chatter vibrations in turning by using elliptical cutting. Youden [125] describes the special challenges in the design of ultra-precise machine tools. Patterson et. al. [128] describe a design of a fast tool servo used to increase the resolution and accuracy of high precision machine tools. Week et. al. [1] and Martinez et. al. [18] investigate using a piezoelectric actuator for vibration compensation in the machining process. 2.1.2. Magnetostrictive Devices Magnetostriction is a phenomenon exhibited by a very few materials whereby a mechanical expansion of the device is achieved when a magnetic field is introduced. A very good source for information on Magnetostriction comes from the book "Magnetostriction Theory and Applica• tions of Magnetoelasticity" [78]. Magnetostrictive devices have been used for vibration control in Chapter 2. Literature Review 6 the turning process. Sturos et. al. [45] describe a magnetostrictive tool actuator for the reduction of vibration in turning. Rubio et. al. [22] compared of control strategies for magnetostrictive and piezoelectric actuators.

<

Figure 2.2 Stack Actuation

Magnetostriction offers many of the desirable characteristics of piezoelectric devices, but requires a constant magnetic field in order for the actuator to be positioned. In order to produce this magnetic field, a coil is typically used to pass a current through and hence create the magnetic field. This generates significant heat. The temperature increase has an adverse affect on the sys• tems ability to control the precision. This seems to be the major drawback of Magnetostrictive actuation.

2.1.3. Electrostrictive Devices

Electrostrictive devices behave similarly to piezoelectric devices in the sense that expansion occurs upon an applied voltage to the crystal. Unlike piezoelectricity, strain in a electrostrictive ceramic is a quadratic function with respect to the applied voltage. Jones et. al. [77] perform an excellent technology trade-off between electrostrictive and piezoelectric devices. Although elec- Chapter 2. Literature Review 7 trostrictive devices offer a larger expansion per applied voltage than piezoelectric devices, they are more non-linear, and more temperature sensitive. 2.1.4. MagLev Devices

MagLev technology uses a coil suspended in a strong magnetic field. As electric current is introduced into the coil, the coil will impart a force into the environment. Gutierrez et. al. [34] investigated a magnetic servo levitation for a smart tool actuator to be used in the turning process. The actuator would be used to compensate for errors in the lathe system. Wronosky et. al. [24] proposed MagLev technology to build a fine positioning system for an Extreme Ultraviolet Lithography (EUVL) research tool.

In order to generate a magnetic field in the activation coil of these devices, a large amount of current is sometimes needed. The amount of heat generated by MagLev devices can be consider• able. The heat generation would have an adverse effect on the thermal stability of a precision actuator system.

This section has weighed the strengths and weaknesses of fine positioning actuation devices against each other. The objective of this evaluation is not necessarily to perform a rigorous analy• sis of these different technologies, but instead to show how piezoelectric devices compare to other devices and to show that piezoelectric devices are a more than suitable choice for precision tool actuation.

2.2. Precision Tools and Actuators

Many applications in precision engineering require the fine positioning of a device. This fine positioning is sometimes in the range of nano-meters in manufacturing. In the previous sections, several technologies have been introduced that can be used to actuate mechanical devices in very small and controlled motions. Chapter 2. Literature Review 8

2.3. Tool Sensors

In addition to the actuation elements, the sensors used to monitor the states of the actuator are of critical importance. When working with sub-nanometer motion, very precise sensors are required for the use of position feedback. The types of sensor used include:

•Laser Interferometers, • Capacitance Sensors, •LVDT Sensors, •Laser Triangulation, and •Linear gauge sensors. 2.3.1. Laser Interferometry Laser interferometry senses the position of a reflective target with respect to a fixed laser source. Laser interferometers consist of five basic parts. The Laser source, beam splitter, reflec• tive surface, diffraction grating, and two photo detectors. The laser source is split with one beam passing through the diffraction grating and one going to the reflective surface. The reflective sur• face sends the second beam to the diffraction grating. Two photo detectors compare the two beams to determine the position of the reflective surface. Figure 2.3 illustrates the components of a laser interferometry system. These devices are highly accurate to sub-nanometer precision. They require, however, a large amount of space and are not suitable for use in a precision tool actuator. A CSO laser interferom• eter [9] was used to calibrate sensors in the system and is a good tool for measuring sensor perfor• mance. 2.3.2. Capacitive Sensors Capacitive sensors consist of two metallic plates separated by an air gap. The air gap acts as the dielectric in the capacitive sensor. For air to be used as a dielectric, the air gap must be very small (< 100 pm). These sensors are only used to measure very small motions and are suitable for small motion actuators. The main drawbacks of these types of sensors is the fact they must be an integral part of the design, and there are few commercial vendors of such devices. In addition, Chapter 2. Literature Review 9 because these devices typically use air as the dielectric medium, they are susceptible to environ• mental contaminants unless special precautions are used to shield them. Because of the drawbacks and custom nature of capacitive sensors, they were not used in this research.

Laser diqd**-

Pfioiodetectors , ' '

Figure 2.3 Laser Interferometer Parts[9]

2.3.3. LVDT Sensors Linear Variable Differential Transformers represent a group of linear feedback devices that can be very accurate. These devices consist of three coils and a moving iron core. One coil is energized and two are used for sensing. As the iron core moves through the device, more or less of the current is transferred to the sensing coils (See Figure 2.4). LVDT sensors come in a variety of sizes and can be sealed for harsh environments like a machine tool. Chapter 2. Literature Review 10

Red Primary Coil Blue White

Secondary Coil 1

Yellow

Secondary Coil 2

Green Iron Core

Figure 2.4 LVDT

LVDT sensors work by driving the primary coil of the device with a base signal (typ. lOKHz) and sensing the relative signal transferred to the two secondary coils to get a measure of the position of the iron core. LVDT sensors were evaluated for this work, but they have poor high frequency (f> 100Hz) response due to heavy filtering of the carrier frequency.

2.3.4. Laser Triangulation

Laser triangulation uses a laser or a group of lasers, a reflecting surface and a photo detector.

As the laser source moves with respect to the mirror, the photo detector measures the position of the reflected beam. These types of sensors come in a range of sizes and shapes. The laser and photo detector are set up for specific sensing ranges. Examples of these types of sensors are the

LNS, and LTS sensors from Dynavision [2][3] (Figure 2.5). Accuracies in the sub-micron area can be achieved with these types of sensors. Chapter 2. Literature Review 11

LNS Sensor LTS Sensor

Figure 2.5 LNS and LTS Sensors [2],[3]

2.3.5. Linear Gauge Sensors

Linear gauge sensors (Figure 2.6) are high precision linear encoders. Encoder technology has progressed to the state that very high resolution encoders can be purchased. This type of feedback technology can be packaged in a sensor small enough to be incorporated in to a precision actuator.

The laser hologauge from Keyence is one of these examples. Unfortunately this type of sensor is prohibitively expensive.

2.4. External Sensors

One of the most important aspects of precision machining or the adaptation of existing machine tools to perform precision operations is the addition of sensors to measure the dimen• sions of the work-piece during the process. The process of in-situ measurement is an essential part of precision machining. A number of sensors can be evaluated for the purpose of-line measure• ment. Chapter 2. Literature Review 12

These sensors include:

• Proximity Sensors, •Laser range sensors, • Acoustic devices, and •Laser Curtain Measurement systems • Snap gauges The process involved in in-situ measurement requires one to precisely measure a machining parameter on-line and use this information for corrections to the cutting process. As in any preci• sion application, the error budget must be minimal in order to reduce the possibility of unwanted errors in the process. Laser Hologage SERIES 542 Resolution up to 0.00001mm

Figure 2.6 Laser Hologauge Chapter 2. Literature Review 13

2.4.1. Roughness Measurement

Zhang et. al. [85] showed how 3 and 4-point on-line roundness measurement can be achieved with proximity sensors and compared to 3-point and 4-point methods. They showed that the four point (sensor) method gives higher accuracy in roundness and spindle error measurements. The exact angle between the sensor is required for an accurate measurement. Yan et. al. [26] investi• gate measuring surface roughness during finish turning operations, and Kato et. al. [99] develop an in-situ measuring system for grinding operations.

2.4.2. Diameter Measurement Diameter measurement of the part is a major concern in precision machining. As a sample application of high precision turning of shafts, the diameter is a primary tolerance concern in the system. On the shop floor, a snap gauge is typically used to determine the diameter of a part after successive grinding operations. 2.4.2.1 Laser Curtain Sensors Laser curtain sensors can offer a very sophisticated real time measurement of part diameter down to nanometer level resolution (See Figure 2.7). Although these sensors are extremely costly, they offer the best accuracy for diameter measurement. However, the mounting of these expen• sive sensors within a CNC machine tool would be cumbersome.

2.4.2.2 Snap Gauges Snap gauges (See Figure 2.8) are used in industry quite regularly to measure the diameter of a part that has been machined. They are used as a final quality control check of a part after it has come out of the final grinding phase. This technology can be used as an off-line check of part diameter for precision machining. Chapter 2. Literature Review 14

544-112-1

Figure 2.7 Laser Scan Micrometer [4]

2.4.3. Run-Out

The run out of a shaft is a measure of the proximity sensors mounted at different parts of the lathe. Proximity and ultrasonic sensors have been used in the past to determine run out. Proximity sensors offer an inexpensive method for performing on-line measurements. Figure 2.9 shows how a proximity sensor can be used to measure run-out.

2.4.4. CNC Improvements

2.4.4.1 Linear Slides

Improvements to the CNC lathe's positioning sensors can also be achieved. By mounting higher precision sensors on the slides of the machine and incorporating this data to the machine tool controller. Chapter 2. Literature Review 15

Figure 2.9 Run-out Measurement Chapter 2. Literature Review 16

2.4.4.2 Spindle Measurement

Velocity and position feedback from the spindle could be used to support precision turning activities. Operations that require accurate knowledge of speed and position such as non-circular turning operations would require accurate spindle measurement. 2.5. Conclusions This chapter has given a basic introduction to the present work in the field of precision turn• ing with piezoelectric electric actuators. It gives the base technology for the precision tool actua• tor and gives the background references for this thesis. Chapter 3

Piezoelectric Tool Actuator System

This chapter outlines the Piezoelectric Actuator for Precision Turning. First, the actuator is introduced followed by a detailed analysis of each element in the actuator. 3.1. The Actuator System Components A photograph and a block diagram of the actuator mounted on the turret of standard CNC lathe are shown in Figures 3.1 and 3.2. The actuator was designed and built under a previous grad• uate student's work[6]. The actuator consists of piezoelectric stack actuator housed in a mechani• cal flexure. One end of the actuator is held by a rigid wall, while the other end is attached to a mechanical flexure which amplifies the stack's displacement by approximately 50%. The piezo• electric stack is powered by an amplifier which has 0-1000V voltage capacity and 150W of power. The amplifier receives +- 5V command voltage from the PC which is equipped with a Spectrum DSP controlled by ORTS/IMM open CNC platform developed in the UBC-MAL labo• ratory [15]. The actual displacement of the tool is measured by an optical LNS laser sensor [3], which has a resolution of 0.1 pm, absolute range of 60 pm with 50 KHz bandwidth. In addition, there is a load cell mounted co-linearly with the piezoelectric element to measure the expansion of the element and strain gauges mounted on the flexure structure to measure static cutting forces.

The mathematical model of the actuator assembly is developed by considering each element.

Piezoelectric Element: It is well documented in literature [98], that piezoelectric devices are governed by a coupled set of equations that relate the strain on the material and the stored electri• cal charge to the applied stresses and voltages to the material.

17 Chapter 3. Piezoelectric Tool Actuator System

Figure 3.1 Actuator Mounted in Lathe

Laser disriaoerrert sensor ^ool holder GCuir/N) tod Tool notionm ReeDelemert — 3*€p(rVV) 1

rigd rrouting plate

Hgh voltage 5

RezoArrriifier A Cbrrputer Ki(V/ujr| Control System Input x(um)

Figure 3.2 Essential Blocks of Tool/Actuator System Chapter 3. Piezoelectric Tool Actuator System 19

One of the major non-linearities in piezoelectrically driven devices is the hysteresis between the applied voltage and expansion of the piezoelectric stack. The motion of the piezoelectric actu• ator is dependent on the previous motion history. A standard piezoelectric actuator may exhibit a

10-15% hysteresis loop, which represents a phase lag between the input and output.

The piezoelectric stack actuator used in the tool/actuator is a Physik Instrumente model num• ber P-242.30 [10]. The following table outlines the specifications for this actuator:

Figure 3.3 P-242.30 Stack Actuator

Table 3.1 : Piezoelectric Translator P-242.30 Attribute Value Expansion @1000V 40 \xm Stiffness 325 N/(\im) Electrical Capacitance 650 nF Resonant Frequency 4.7 kHz Temperature Expansion 0.65 (\im)/K Weight 125g

Flexure Structure: The cylindrical piezoelectric ceramic element is housed between a rigid block and a flexure. The tool holder part of the system is connected to the actuator base with two Chapter 3. Piezoelectric Tool Actuator System 20 parallel flexures. The piezoelectric ceramic element is mounted under the tool, it pushes the tool indirectly so that its stroke is amplified by the leverage of the flexure bars. The flexures provide a theoretical mechanical amplification of two times the motion delivered by the piezoelectric actua• tors. In addition, the flexure structure provides the returning force for the piezoelectric element. The mechanical flexure can be modeled by a single degree of freedom, whose spring constant is dominated by the two flexures and the mass of the upper carriage, which holds the tool. Including the piezoelectric stack, the combined model of the mechanical assembly is shown in Figure 3.4.

Tool Extension •

Tool Holder

Figure 3.4 Dynamic model of the mechanical system

The transfer function between the applied force (F) and the tool tip displacement (x) is expressed by a single degree of freedom system:

X(s) = 1 (3.1). F(s) ms2 + cs + k

Laser Sensor: A sensor is mounted on the output of the actuator to measure the motion of the tool with respect to the actuator mounting location. The sensor used is a Laser Nano Sensor pro• duced by DynaVision[3]. The sensor uses a laser projected onto a reflecting surface via a system Chapter 3. Piezoelectric Tool Actuator System 21 of two high-grade lenses. The laser source is a miniature laser diode unit. Using the same system of lenses in conjunction with a beam splitter, part of the scattered light spot is reflected on a photo detector in the form of two light beams. The beam splitter is a holographic optical grating. When the object to be measured is precisely at the focal point of the lenses, the spots of the light beam are projected on the centre of the photodetector. When the object is shifted in relation to the sen• sor, causing it to move out of the focal point, the light spots also move. The object's movement with respect to the focal point of the sensor is calculated in the electronics by summing the out• puts from the photo detectors. In order to perform accurate measurements, a highly reflective sur• face is required to reflect the laser light. The following table outlines the specifications of the sensor. Table 3.2 : LNS 18/60 Sensor Specifications Measuring Distance (focal point) 18mm Effective Range ±60p,m Resolution 0.\\\,m Measuring Frequency 50kHz Temperature Range 0-50°C Humidity max 90% Output: Distance (X) ±7.8V Intensity (I) max. 1.5 Vdc Enable (E) -0.7 Vdc ....In Range +4.7 Vdc ... Out of Range Load (X,I,E) R > 10K2 Sensitivity 130— typ. pm Object Surface roughness polished mirror Angle between Sensor and object perpendicular ± 1 ° Weight 100 g Power Supply ±15Vdc±5% Power Consumption max. ± 200mA Laser Power 5mW (780nm) Chapter 3. Piezoelectric Tool Actuator System 22

Computer Controller: The computer control system for the tool actuator consists of a stan• dard X86 Computer with a Digital Signal Processor (DSP) card for running all of the advanced control and monitoring functions of the system. Table 3.3 outlines the specifications for the DSP.

Table 3.3 : DSP Specifications Processor Texas Instruments C31 Bus Format ISA Supplier Spectrum Signal Processing Analog Input 4 Channels 16 bit 3V Analog Output 4 Channels 16 bit 3V

Signal Conditioning: In order to support the computer control of the piezoelectric actuator, the signals input to and output from the DSP must be conditioned to the appropriate voltage lev• els. An ideal signal conditioning network will provide a constant gain of the signals over a wide frequency bandwidth. The output of the DSP card is ±3 V, and the input to the Power amplifier is ±5 V. The command voltage from the DSP card is amplified to the appropriate level by the signal conditioning. In addition to the command output to the power amplifier, the sensor feedback from the LNS sensor is conditioned to bring its values from the ±4V levels to ±3V. This is achieved with the following circuit shown in Figure 3.5.

Figure 3.5 Sensor Signal Conditioning Circuitry

The sensor signal conditioning has a bandwidth of over 10 kHz and has the following gain: Chapter 3. Piezoelectric Tool Actuator System 23

y.sensor (3.2) '•'P Rl+R2

The output signal conditioning circuit is shown in Figure 3.6.

u amp uldsp

Op Amp

Figure 3.6 Amplifier Output Signal Conditioning

The output gain of the output signal conditioning is as follows:

U u amp dsp (3.3)

Piezoelectric Amplifier: The piezoelectric amplifier is a model P-270 Power Amplifier from Physik Instrumente (PI) of Germany. This amplifier is specifically designed to drive Piezoelectric elements. The power amplifier provides the high voltage required to drive the piezoelectric stack actuator. The amplifier can supply 0-1000 Volts of output and is controlled by a ±5 Volt com• mand input. As will be seen in the following chapter, the instantaneous current that the amplifier can deliver is an important attribute in evaluating a power amplifier for piezoelectric devices. As an electrical circuit, a piezoelectric element behaves as capacitor. At higher frequencies, the ele• ments draw higher currents. If the power amplifier cannot supply enough current at frequency bandwidth required, it will begin to attenuate the output voltage. 3.2. Mathematical Modeling of Actuator Components 3.3. Modelling of Piezoelectric Actuators Piezoelectricity is the phenomenon in which an strain can be induced in an object by the application of a voltage. In addition, an electric potential is created when an external load is Chapter 3. Piezoelectric Tool Actuator System 24 applied to the crystal structure. First a mathematical model for a piezoelectric translator is out• lined. Then the non-linearities in the physical parameters describing the system are investigated. A detailed simulation of a translator with non-linear elements is presented and compared to a lin• ear model. Both driven and un-driven system are evaluated.

Piezoelectric devices are described by a set of coupled electro-mechanical equations. This makes it difficult to produce a single differential equation that describes the system satisfactorily. A piezoelectric element is a motion translator used for fine motion positioning. Typical motion translators have a range of less than 100 mm. Piezoelectric materials are generally made of a Pb-Zn-Ti ceramic which is very hard and brittle. Because of this, piezoelectric translators can only be used in compression. Although a piezoelectric translator will not pull with significant force, it the piezoelectric material will return to its zero voltage position. A set of differential equations that adequately describes the motion of a piezoelectric transla• tor is presented in this section. Initially a linear model is developed based upon the basic constitu• tive relationships of piezoelectric materials. The non-linearities in piezoelectric actuators are added in the following sections

3.4. Mathematical Models of Piezoelectric Stack Actuators It is well documented in literature [98] [133], that piezoelectric devices are governed by a coupled set of equations that relate the strain on the material and the stored electrical charge to the applied stresses and voltages to the material. The following two equations can be used:

(3.4)

D »P = d33Tp + E3Ep (3.5)

where:

Sp = strain, 2

Dp = electric flux density (C/m ), 2 T = applied Stress (N/m ), Chapter 3. Piezoelectric Tool Actuator System 25

Ep = electric field (V/m), E s = elastic compliance with a closed electric circuit,

e3 = is the electrical permittivity, and

d33 = piezoelectric strain constant (C/N). The following sections examine this relationship in more detail. 3.4.1. Strain Relationship The basic relationship for piezoelectric actuation relates the expansion of the ceramic to the applied electric field.

SP = J = diJEP 0-6)

Where:

Ep is the electric field, A/ is the change in length of the piezoelectric, / is the un-strained length of the piezoelectric,

Sp is the strain produced in the translator, and

dtj is the piezoelectric strain constant for an electric field applied in direction i and strain pro• duced in direction j.

In the stack actuators currently being used, expansion is in the same direction that the field is

applied. This is commonly denoted as d33 .

Of course strain is also produced by any external forces applied to the piezoelectric structure. So the equation for strain becomes:

E Sp = s Tp + d33Ep (3.7)

where:

Tp is the applied stress, and

s is the compliance of the piezoelectric with short circuit (E = 0). Chapter 3. Piezoelectric Tool Actuator System 26

3.4.2. Dielectric Relationship

A second relationship can be developed which relates the charge in a piezoelectric ceramic to applied voltage. Like a capacitor, the following relation holds (under no applied forces):

Dn = E En (3.8) p p Where: 2 T

Dp is the charge density C/m , e is the electrical permittivity (Stress(rp) held to zero), and

Ep is the electric field. Because this is a piezoelectric device, charge is also produced by an applied force to the translator. In this case, the equation for charge density becomes:

Dp = d33Tp + E3Ep (3.9)

3.5. Equivalent Circuit For a Piezoelectric Translator The response of a piezoelectric translator can be represented by an equivalent circuit consist• ing of an minimum of three elements. Figure 3.7 shows the equivalent circuit for a piezoelectric translator.[81][98]

LI Cl Rl

CO

Figure 3.7 Equivalent Circuit - Piezoelectric Actuator

Where:

CQ is the capacitance of the piezoelectric ceramic. Chapter 3. Piezoelectric Tool Actuator System 27

L represents the mass of the translator. 1/Cj represents the stiffness of the translator.

is the damping in the translator.

The values for Lx and Cx can be given by[81][98]:

L m ,2' (3.10)

c (3.11)

where: m is the mass of the piezoelectric, k is stiffness of the translator, and *F is the electro• mechanical coupling factor which is the relation between force and current through the translator ¥ is given by [98][81]:

(3.12)

In this circuit analogy, the piezoelectric device has two parallel branches. The lower branch of the circuit represents capacitor proper of the piezoelectric ceramic which is modeled by a

capacitor C0. If desired, a small leakage resistor RQ can be inserted in series with the capacitor for modeling and simulation purposes. The topmost branch of the circuit models the mechanical aspects of the piezoelectric actuator.

The inductor Lx represents the motion of the mass of the piezoelectric. 1/Cj represents the stiff• ness of the translator and R^ represents the mechanical damping in the translator. The voltage

drop across Cl in the simulation is directly related to the force corresponding to the stiffness term. The position of the translator corresponds to the charge q. The circuit analogy presented above only models the motion of the piezoelectric stack actua• tor. If the actuator was moving a mechanical system (e.g. the flexure structure), the model of the Chapter 3. Piezoelectric Tool Actuator System 28 mechanical system could be inserted in series with the topmost branch to simulate the effect of the external load(s). The equivalent circuit would have the circuit diagram depicted in Figure 3.8:

LI Cl Rl ZJVIech

CO

Figure 3.8 Piezoelectric Translator with Mechanical Load

If the translator is not moving any load, the mechanical connection would be a short circuit as shown in Figure 3.7 If the translator was blocked, so that no motion was allowed in the translator, the equivalent circuit would have an open circuit on the topmost branch. This would be equivalent to an infinite mechanical impedance connected to the system (extremely large mass). The equivalent circuit

would just be a capacitor with the capacitance C0. 3.6. Modelling of Piezoelectric Element

The analysis in the previous sections ignores the driving voltage of a supply. Figure 3.9 shows the schematic of a driven system

The following set of equations describe the dynamics of the driven system:

d JId (3.13)

2

1 dqx d q, i dq. (3.14)

l0 ~ ld + ll (3.15) Chapter 3. Piezoelectric Tool Actuator System 29

Figure 3.9 Driven Piezoelectric Element

As with the un-driven model, Lm represents the motion of the mass off the piezoelectric, Cm

represents the stiffness of the piezoelectric, and Rm represents the mechanical damping of the

device. The voltage drop across Cm is directly proportional to the force corresponding to the stiff•

ness term. The position of the translator is proportional to the charge qx. 3.6.1. Piezoelectric Parameters

The piezoelectric translator currently used in the tool actuator is a Physic Instrument systems model PI-242.30. Table 3.1 on page 19 outlines the characteristics of the translator [10]. Chapter 3. Piezoelectric Tool Actuator System 30

From this information, the following parameters can be derived:

C0 = 0.65 p,F , 40um .„ . „-8m d 40X1 0 » = T0W> = V (3.16)

33 £

for open circuit operation:

area „

.•.stiffness it = ^ (3-17)

Then the values of Cra and Lm

C = ^- = 4i* = f4.0xl0_8^2x325— = 0.52uF m k id \ VJ \xm

, m ... M

Lm = -;mAm = - (3ig)

im 0725*^05 = 2 w5mH '4.oxio-s^Y325^y 3.6.2. Piezoelectric Translator Simulation The driven piezoelectric actuator system is shown using a Simulink™ block diagram (Figure 3.10) The piezoelectric device simulation was connected to a sinusoid driving voltage. Figure 3.11 shows the output position vs. applied voltage for several driving frequencies. Chapter 3. Piezoelectric Tool Actuator System 31

Translator SmJation

1/s- 1/CD CU3 •CD 1/Fb Integatcr CU1 1/TJO Sun 1/Fb

Sure Orrert Supplied SneV\£we SUTB Vb Constant 1/s- vur>—' lrtegato2 1/cm QJ2 1/Lm Sum

Applied Voltage Frn Translator Fbsitkn

1/A.

1/A

Wee A is proportioned to the Bectro-N/bcharical Coupling Facta

Figure 3.10 Simulation of Driven System

With the increasing driving frequency, the hysteresis in the positioning of the translator

increases. The hysteresis is caused by the mechanical damping in the actuator Rm which creates a slight phase lag between the applied voltage and the strain produced. This is energy loss due to a linear damping. It does not adequately model the hysteresis of a driven piezoelectric translator at low driving frequencies. The following sections review hysteresis and hysteresis compensation models. Chapter 3. Piezoelectric Tool Actuator System 32

Simulated Translator Postiion Vs Input Voltage 5 x in/ PI-242.30 (translator) linear model 21 1 1 1 1 1 1 1 r

_21 I I i i i i I I i I -500 -400 -300 -200 -100 0 100 200 300 400 500 Input Voltage (Volts)

Figure 3.11 Hysteresis in Driven System

3.6.3. Hysteresis

The expansion/voltage graph for a piezoelectric translator shows a hysteresis loop. When the voltage is cycled between two voltages, the structure contracts differently to form a hysteresis loop. This phenomenon is common for all piezo-translators and is approximately 15%.

Hysteresis is a well known phenomenon exhibited by piezoelectric devices. The quasi-static expansion of a piezoelectric actuator is not directly proportional to the applied electric field. How much an actuator expands at a given voltage depends on whether it was previously operated at a higher or lower voltage. The typical hysteresis curves can be seen in Figure 3.12. Chapter 3. Piezoelectric Tool Actuator System 33

5 x 10" Translator Position Vs Input Voltage PI-242.30 (translator)

-400 -300 -200 -100 0 100 200 300 400 500 Input Voltage (Volts)

Figure 3.12 Typical Hysteresis Measured From the Actuator

It should be noted that the relative magnitude of the hysteresis remains the same relative to the distance moved. The typical solution to the open loop positioning of piezoelectric actuators is to provide a position feedback sensor and a closed loop controller to compensate for this non-lin• earity and produce reliable output. By adequately modeling the hysteresis, gains in controller per• formance can be obtained by recognizing that the hysteresis loop exhibited by piezoelectric systems is a manifestation of a non-linear term in the model of the system. Chapter 3. Piezoelectric Tool Actuator System 34

3.7. Hysteresis Background

Hysteresis is the phenomenon that is typically associated with the quasi-static motion of an

actuator system. It is typified by a position dependence on the previous motion of the actuator. For

example, a command position approached from one direction will correspond to a different final

position of the actuator then if the position is approached from the other direction. Experiments

have shown that the shape of the hysteresis loop is not a function of the actuation frequency of the

input signal [6]. These tests were performed with the piezoelectric actuator outside of the flexure

structure.

If the transfer function of a simple second order approximation for a the piezoelectric actua•

tor is considered as:

Y(s) 8.456 x 105 (3.19) V(s) s2 + 611.75+ 1.658 x 107

where Y is the position of the actuator (pm) and V is the input voltage to the actuator.

Using Matlab™ to simulate this actuator model, one can see the effect of different actuation

frequencies in Figure 3.11.

As the excitation frequency increases, the hysteresis between the input and output also

increases. As the excitation frequency increases, the phase lag between the input and output

increases, thus causing a hysteresis loop. The area of the loop represents the energy dissipated

within the damping of the second order system. The standard second order system does not ade•

quately describe the response of a piezoelectric actuator. The hysteresis experienced in the

response of a piezoelectric actuator is constant over a large frequency range. If we consider the

hysteresis in the system as the amount of energy loss in the actuation range, the hysteresis in the

piezoelectric actuator can be considered to be a non-linear damping term in the system.

As the driving frequency increases, the phase lag between applied field and strain increases.

This is the expected response of this type of damping. Using this as a model however does not Chapter 3. Piezoelectric Tool Actuator System 35 accurately model the hysteresis in the system because at low driving frequencies, there is very lit• tle phase lag in the modeled system.

In contrast, the actual input/output graphs of the piezoelectric actuator at the same actuation frequencies are shown in Figure 3.12

Modeling the actuator with a fixed linear damping term may adequately describe the motion of the actuator at one frequency, but at higher driving frequencies, the actuator will exhibit less phase lag than modeled. Conversely, at lower driving frequencies, the actuator will exhibit a larger phase lag than modeled. This is apparent changing of the damping of the system at different driving frequencies is main hysteresis problem encountered in piezoelectric actuators. The fol• lowing sections outlines possible methods to compensate for hysteretic damping in piezoelectric actuators. 3.8. Methods for Hysteresis Compensation 3.8.1. Charge Controlled Piezoelectric Actuators The method that Janocha et. al.[16] and Main et. al.[38] chose to use charge control of a piezoelectric actuator to eliminate the hysteresis in the open loop control of piezoelectric actua• tors. Using the basic constitutive relationship between strain, stress, electric field, and dielectric displacement it was proven that the expansion of the actuator was not a linear function of applied voltage, but linear with respect to the free charge present on the piezoelectric. It is shown that there is a linear relationship between the charge and the strain in the piezoelectric ceramic over a wide range of operation. Insight into the constitutive relationship and the piezoelectric effect is gained by reviewing this derivation as presented by Main e.t al.[38] They reformulated the main constitutive relationship in terms of the applied stress and the electric displacement. The formula• tion has the following form:

td™ Qf Al = nt\ T + (3,20) Y T P TA V J E 3 E 3 33 Chapter 3. Piezoelectric Tool Actuator System 36

where Al is the expansion, n is the number of piezoelectric layers, t is the thickness of each layer,

Y3 is the Young's modulus in the axial direction,

J33 is the piezoelectric expansion coefficient in the axial direction, T

e3 is the electrical permittivity in the axial direction with the applied stress held constant,

Tp is the applied axial stress, A is the area of the positive electrode, and Qj is the free charge on a given positive electrode. The major drawback with this method for hysteresis compensation is inherent in the control of the charge. Janocha et. al.[16] used a current feedback monitor to observe the charge on the piezoelectric element and was successful for a small f< 150Hz control bandwidth. 3.8.2. Method of Equivalent Damping The concept of structural or hysteretic damping, is worthwhile investigating. Structural damping is caused by internal friction in the material as relative slipping or sliding of internal planes occurs during deformation [134]. The area enclosed by the hysteresis loop represents the energy loss per loading cycle. Most models assume that this energy loss is proportional to the stiffness of the material and the square of the displacement amplitude, but is independent of fre• quency. From this assumption, an equivalent viscous damping coefficient is derived. For har• monic motion, the equivalent damping coefficient would be: (3.21)

where h is a material constant, and 00 is the frequency.

The simulation of this non-linear damping can be seen in the Figure 3.13 Chapter 3. Piezoelectric Tool Actuator System

Translator Simulation With Non-Linear Damping

Sine Wave

Constant

Figure 3.13 Simulation of Equivalent Damping Hysteresis Compensation

Simulated Translator Postiion Vs Input Voltage

x 10 PI-242.30 (translator) non-linear damping model

-500 -400 -300 -200 -100 0 100 200 300 400 500 Input Voltage (Volts)

Figure 3.14 Simulation of Equivalent Damping Hysteresis Modeling Chapter 3. Piezoelectric Tool Actuator System 38

This simulation was driven with the same frequencies that were used in the linear system analysis given in Figure 3.10 and Figure 3.11. Figure 3.14 shows the results at 10,100, 500 and

900 Hz respectively.

3.8.3. Hysteretic Restoring Force Model

It should be noted that hysteresis is a normal phenomenon in highly strained materials (piezoelectric or not), and a wealth of insight into hysteretic system can be gained from the inves• tigation of highly strained materials. A method proposed by Wen[132] uses an added term in the system of equations that describe the motion of a structure. Wen states that the force in a hyster• etic system is described by Equation (3.22).

Q(x,x) = g(x,x) + z(x) (3.22)

Where g(x, x) is the non-hysteretic component and z(x) is a function of the time history of x. Wen described z by the following non-linear differential equation:

_1 z(x) = -Y|jc||zr -px|zr + ax (3.23) Wen has shown that a hysteretic relationship exists between z and x and that if used as a restoring force in the model of the actuator can effectively estimate the hysteresis in piezoelectric devices. The scale and shape of the hysteresis loop are governed by the parameters a, y, and (3 and the smoothness of the curve is controlled by n. These parameters are typically found using a non-linear identification techniques[137]. A simulation of this model (See Figure 3.15 and Figure 3.16) was performed with representative hysteretic parameters. Chapter 3. Piezoelectric Tool Actuator System 39

Scope1

Figure 3.15 Hysteretic Restoring Force Model Intput-Ouptut Hysteresis for Hysteretic Restoring Force Model 30

Alpha = 0.9 Beta = 0.5 Gamma = 0.51 n = 1

-30 -500 -400 -300 -200 -100 0 100 200 300 400 500 Input Voltage (V)

Figure 3.16 Input-Output Curves for Hysteretic Restoring Force Model Chapter 3. Piezoelectric Tool Actuator System 40

3.8.4. Phase Lead Compensation

Duong and Garcia [31] introduced a method for hysteresis compensation that attempted to adjust for the phase lag inherent in piezoelectric driven systems. They demonstrated that by using a phase lead compensator, a simple and effective means of hysteresis compensation was achieved in the open-loop case. One of the drawbacks of this method was that the bandwidth achievable with this method was limited to a range of (100-400 Hz)[31], which is well within the actuation frequencies of the tool actuator. The flexure structure has a natural frequency of 600 Hz and the dynamics will be dominated by the flexure structure near these frequencies. In addition, because a piezoelectric actuator behaves largely as a capacitive load on the power amplifier, the maximum actuation frequency is greatly reduced to below 500Hz for full range motion of the actuator. Therefore, a simple method of modeling hysteresis is to introduce a time delay in the model of the system, which is adopted in this work. This model considers only a fixed time delay for the hysteresis, but there is evidence that the time delay varies over various driving frequencies. For example, in quasi-static operation of the piezoelectric device, when a voltage is applied to a piezo-translator and held at that voltage, the translator slowly "drifts" over time. The translator drifts in the direction of the previously applied motion. This drift could be considered as a delay in the motion of the actuator, or a manifestation of hysteresis at low frequencies. This low frequency phenomenon can be easily compensated for by the controller. Due to its simplicity in feedback control, a fixed time delay is used in this work which gives the best fit for actuation frequencies

(100-500Hz). 3.9. Other Piezoelectric Non-Linearities 3.9.1. Capacitance The capacitance of a piezoelectric ceramic increases by 30% as the applied voltage increases to maximum[10]. This variation in capacitance can have a large effect on the required current to drive the actuator. In order to model the non-linearity, the following equation was used for the Capacitance: Chapter 3. Piezoelectric Tool Actuator System 41

Cd = C0(1+K£/) (3.24)

where:

U = the applied voltage,

K = non-linearity constant, and

Cd = non-linear capacitance of the device.

This non-linearity does not effect the output performance of the translator, but puts extra demands on the driving amplifier for the translator. A piezoelectric device puts a largely capaci• tive load on the drive amplifier for the system. As the driving frequency increases, the amount of current to drive the translator increases.

3.9.2. Elastic Compliance

The elastic compliance of the material does change nominally over the operating range of the crystal[52]. The elastic compliance will, however, change as the strain on the crystal is increased.

3.9.3. Damping Coefficient

A linear model for the mechanical damping in the actuators may not be appropriate. As with many mechanical structures that move very small distances, stiction may exist for small motions.

This may cause a hysteresis like effect in the positioning of the actuator.

3.9.4. Temperature Effects

Most piezoelectric ceramics that are produced for actuators are quite stable with respect to temperature. If the temperature of the ceramic approaches the Curie temperature (~300°C), the ceramic will de-polarize and become ineffective. Typical ambient temperatures for the actuators in this application will be well below 300°C, however, heat is generated by the energy losses in the ceramic if the actuator is operated closer to its resonance. If the polarization is lost, the piezo• electric effect will diminish. In fact, a piezoelectric actuator operated near its Curie temperature will become more ineffective with time. Actuators are periodically re-poled to ensure proper polarization of the crystal structures. The resonant frequency of piezoelectric actuators is typically Chapter 3. Piezoelectric Tool Actuator System 42

» 10 times the nominal driving frequency. Therefore, heat generation in the piezo is generally neglected.

The temperature expansion of this translator is stated to be 0.65 \im/K. This corresponds to 1.6% of the full range per degree Celsius. It is assumed that the lab environment is kept a constant ambient temperature during cutting operations, however, this should not be neglected if an accu• rate model is desired. This can be compensated for by using the laser feedback sensor on the actu• ator. 3.9.5. Electrostrictive Effects All piezoelectric devices show some electrostrictive effects. Electrostiction is expansion of a crystal relative to the square of the applied voltage. Expansion will occur whether the applied voltage is positive or negative as a function of the square of the applied voltage. In the derivation of complete equations that describe the motion of a piezoelectric element, a factor as a function of voltage squared would be expected. The strength of the electrostrictive effect in most piezoelec- trics is negligible[52]. 3.9.6. Drift When a voltage is applied to a piezotranslator and held at that voltage, the translator will slowly "drift" over time. The translator will drift in the direction of the applied motion. For exam• ple, if the voltage applied to the actuator goes from 0 V to 500V and is held there. The actuator will continue to expand over time. This expansion rate decays over time. 3.10. Structural Dynamic Model of the Actuator Assembly The parameters of both piezo stack and mechanical flexure system are summarized in Table 3.4 on page 43. From Figure 3.4 on page 20, it can be seen that the dynamics of the transla• tor (piezoelectric element) and the tool holder (flexure structure) act in parallel. Because the dynamics of the piezoelectric translator are so high, the dynamics of the combined system is very similar to that of the flexure structure alone. The Mass M in Figure 3.4 is the combined equiva• lent mass of the piezoelectric translator and the flexure structure. In this case, the flexure struc- Chapter 3. Piezoelectric Tool Actuator System 43 ture's dynamics can be modeled as an impedance in the upper most branch of the simulation circuit Figure 3.8 Table 3.4 : Parameters of Tool Actuator Flexure Structure Natural Frequency 610 Hz Damping Factor -5% Stiffness 0.11a pm/N Piezo Element Natural Frequency 4.7 KHz Stiffness 325 N/um

The open loop block diagram of the amplifier, piezo stack with hysteresis model and mechan• ical flexure is shown in Figure 3.17 Here, both the amplifier and piezo stack are treated as gains, since they have a flat bandwidth well above the first natural frequency of the flexure (600Hz). The open loop transfer function of the whole assembly can be expressed in both continuous (s) and discrete (z) time domain as described in the following sections.

[V] [V] [m] Process Input Process Output (input voltage) j Gamp Delay ^Tool Displacement) Amplifie Piezo- Hysterisis Element Translator / Tool

Figure 3.17 Block Diagram Description with Hysteresis

This produces the following discrete transfer function form using a a zero order hold approx• imation [103].

Y(z) -d biZ + bo = (3.25) U 2 ^ z + axz + a0

3.10.1. Example Model Model identification was performed on a test bench at the National Research Council Labora• tory. Model identification involved exciting the actuator system with a band limited white noise Chapter 3. Piezoelectric Tool Actuator System 44 signal and recording the response. Standard, system identification, routines were used to deter• mine the parameters of the system.

Using a white noise signal generator in conjunction with a step function, a voltage command signal was generated for the power amplifier. This data was later post processed to determine the model of the system. The generated model was then evaluated based upon the expected natural frequency. The data collection was performed at a rate of 5KHz. Ten data sets were collected, ana• lyzed and post-processed. Using a standard least squared based estimations method, the parame• ters of the system model were determined. One of the most crucial determinations was the number of delay steps present in the system. The following parameters were determined. Table 3.5 : Identified Model Parameters d 4 (0.8ms) bl 0.01791 bO 0.01147 al -1.262 aO 0.9174

This leads to the following Transfer function:

Y(z) _ -4 0.0179U +0.01147 (3.26) Z 2 U(z) ~ z -1.262z +0.9174

The continuous time conversion of this model using a zero order hold approximation[103] is:

,5 Y(s) 17.75s+ 8.148x10 -o.ooos* ——- = e (3.27) U(s) / +431.1s + 1.818xl07

This transfer function has two poles and one zero. The zero is very far to the left of the poles

(i.e. very stable), and for preliminary analysis it can be ignored. The natural frequency of this

model, damping factor and DC gain all must correspond to what was determined theoretically for

the system. These values are:

(On = 618Hz, C = 0051, DCGain = 0.0448 (3.28) Chapter 3. Piezoelectric Tool Actuator System 45

The dimension of the transfer function is \x.m/V where the voltage is the amplifier voltage to the piezoelectric element (0-1000V). The actuator has motion range of approximately 60 p.m. It would be expected that the DC Gain would be ~0.06. The attenuation of the expected DC Gain could be due to poor low frequency characteristics of the CSO Laser Interferometer[9]. The dynamics of the system however seem to be well modeled. Figure 3.18 shows the frequency response of the system modeled and measured in addition to a sample output of the actuator and model in the time domain. The attenuation of the expected DC Gain could also be due to the poor high frequency performance of the power amplifier[6]. Because the piezoelectric element behaves as a largely capacitive load, the maximum amplitude of the actuation is attenuated due to current limitations in the power amplifier.

This model of the actuator was developed with no tool mounted in the tool holder. As will be seen in Chapter 4, the transfer function will change to reflect the extra mass of the cutting tool.

Time Domain Response of Bench Mouted Tool Actuator Vs Model Response Recpercy Rssponse of actuator Solid LNSi dashed Mxfel

Blue Dashed: Actuator Output Red: Model Output

0 200 403 600 800 1000 1200 1400 160C 1.21 1.22 1.23 1.24 1.26 1.26 1.27 1.28 Time (second) Freqjarcy (Hz)

Figure 3.18 System Model and Actuator Response Chapter 3. Piezoelectric Tool Actuator System 46

3.11. Factors Affecting Actuator Model 3.11.1. Mounting of Actuator

When the actuator was moved from NRC and mounted on the lathe at UBC, the controllers were no longer stable. There was a noticeable oscillation at -1100 Hz. The only change in the setup was the mounting of the actuator.

At NRC, the spine of the actuator (where it interfaces to the lathe's tool holder), was clamped with screws spaced ~ 1.0" on centre for the entire length of the actuator (See Figure 3.19). At UBC, a wedge system is used to clamp the actuator on the tool stock (See Figure 3.20). With this method, a section ~ 1/3 the length of the actuator is not clamped in the centre. This could lead to the actuator bowing out as the piezo-element expands. It would also show a second dynamic in the device. It was noticed that the static range of the actuator is smaller ~±25 \im at the UBC lab as opposed to ~±30 \x.m motion at the NRC lab. This suggests that some of the force supplied by the piezoelectric element is being used somewhere else; perhaps to bow the spine of the actuator. A basic modal analysis of the actuator was performed for each setup of the actuator. Several Hammer tests on the actuator were performed to attempt to identify the problems with the Control of the actuator when mounted on the lathe. The accelerometer was mounted on the actuator in three different locations as outlined in the figure below. Impact tests were performed in four loca• tions in the actuator for each accelerometer position. The average stiffness of the actuator mounted on the lathe was found to be 0.11 \im/N as can be seen from Figure 3.22. 3.11.2. Tool Holder The tool/actuator allows for different cutting tools to be mounted in the actuator. Different tools will have different mass properties and significantly affect the equivalent mass of the tool/ actuator model. Three different tools were used in the experiments, and a different model natural frequency was found for each one. Chapter 3. Piezoelectric Tool Actuator System 47

Actuator Back View of Actuator Holder

Piezoelectric Actuator

Figure 3.19 Actuator Mount at NRC

Lathe Tool Stock

Piezoelectric Undamped Actuator Space

Figure 3.20 Actuator Mounting on the UBC CNC lathe Chapter 3. Piezoelectric Tool Actuator System 48

I Point 4

Figure 3.21 Hammer Tests on Actuator

10"7 Frequency Response from Hammer Tests. (Displacement)

200 400 600 800 1000 1200 1400 1600 1800 2000

100

200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency (Hz)

Figure 3.22 Accelerometer Position 1, Hammer Hit at Point 1. (UBC mount) Chapter 3. Piezoelectric Tool Actuator System 49

3.11.3. DSP Card The DSP card at the NRC lab uses the Texas Instruments C32 processor while the UBC lab uses a card with the older/slower C31 card. The control code ran on the C32 card with a processor usage of -25%. At the lab on the C31 card, the same code at the same control frequency ran with a processor usage of -65%. One thing that this will cause is an increase in the computational delay of the computer. This should be compensated in the control laws. Essentiality the question is: Is the control output assumed to come out instantaneously or at the end of the control cycle? The later would introduce yet another step delay in the system model.

3.11.4. LNS Sensor The LNS sensor [3] was calibrated at NRC against the laser interferometer. However, as men• tioned earlier, the static range of the actuator was smaller at UBC than at NRC. This could be due to a sensor problem, possibly different gains as a result of different ambient conditions. The chance of this being the source of the problem at UBC is small. All literature points out that the LNS sensor is a very stable device and should not be effected by small changes in temperature and ambient lighting. 3.12. Summary

This chapter has presented the basic mathematical relationships for piezoelectric devices. Each component of the tool/actuator has been evaluated and a model for the piezoelectric actuator developed. This model will be used to develop a control strategy for the tool/actuator and perform cutting experiments in the following chapters. Chapter 4 Digital Control of the Actuator Position

4.1. Introduction Traditionally, control for piezoelectric actuators is typically limited to basic PI or PID con• trollers. These controllers provide a basic control of the device but do not lead to more advanced forms of control. In Chapter 3, a model of the tool actuator was developed. This chapter presents the advanced control algorithms developed for the actuator. These algorithms are proven in a cut• ting test example with the tool actuator mounted in a CNC lathe. 4.1.1. Traditional PID Control

The closed loop control diagram and PID control law are shown in the Figure 4.1.

Figure 4.1 Traditional PID Loop

The values of the constants, Kpid, Tt and TD are the proportional, integral, and derivative constants respectively. They are chosen initially based on a simulated system response. Upon

implementation, they are optimized to produce the best actual response. The constant TQ is the sample interval. Figures 4.2 and 4.3 show the response of the PID controller This controller was developed and tested by Steeves and Yu [6].

50 Chapter 4. Digital Control of the Actuator Position

Closed Loop Step Response 1.2 1

Io i 0.8 in 0) = 0.6 n > - Process Input U Jjj|jj T; 0.4 - Reference Input R #tll 0.2 Process Output Y (tool displacement)

0.01 0.02 0.03 0.04 0.05 Time [seconds]

Figure 4.2 Traditional PID Step Response

There is a -10% of step size overshoot with a settling time of -0.004 seconds.

Frequency Response for Closed Loop System

10 100 Frequency [Hz]

Figure 4.3 Frequency Response - PID

The frequency response shows a -3dB bandwidth at -220 Hz with a phase margin of degrees. During operation of this controller significant phase lag was noted. Chapter 4. Digital Control of the Actuator Position 52

4.2. Pole Placement 4.2.1. Background Theory

Pole placement control is based on the input-output model of a given system. Control input to the plant is generated by feed forward and feedback quantities. These values are generated by two distinct transformations, one on the feedback value and one on the reference set point. The block diagram of the controller can be seen in Figure 4.4

T(z)

uc = command S(z) y = measured output R(z) u = Ujjj + Ujy = input to plant H(z) = plant model

Figure 4.4 Pole Placement Control

The pole placement controller is governed by the polynomials T, S, and R. These polynomi• als are chosen such that the closed loop response of the system exhibits the dynamics that are cho• sen by the control system designer. In order to accomplish this, the dynamics of the plant must be cancelled by the combination of these polynomials. In order to cancel the dynamics of the plant, the poles of the plant must be known. In addition, the poles of the desired response must be included in the closed loop transfer function. The method of pole placement control is to cancel the poles of the plant and place new ones that represent the response of the closed loop controller.

In order to implement pole placement control properly, an accurate model of the plant to be controlled is essential. 4.2.2. Previous Research

Pole placement controller's have been attempted before in the control of the piezoelectric actuator. The past attempts [6], however, have been unsuccessful in producing a stable controller. Pole placement control requires the accurate model of the plant to be controlled. The dynamics of Chapter 4. Digital Control of the Actuator Position 53 the plant were determined in the previous studies, but the time delay of the plant was assumed to be zero. Initial investigation into the response of the actuator showed that there was significant delay present in the actuator (See Section 3.10.). This delay is mainly due to the charge time of the piezoelectric capacitor and lags due to hysteretic effects in the actuator. It is a well known fact that pole placement controllers are sensitive to the time delay present in the plant it must be included in the model of the plant in order for pole placement control to be successful. 4.2.3. Discrete Model of the Actuator 4.2.3.1 Model Form A thorough analysis of the system components to determine the order of the system and the dominant dynamics must be performed. The system is currently configured to be a single input single output system (SISO). The input to the plant is the command input to the piezoelectric amplifier and the output of the plant is the position feedback from the position sensor. Figure 3.4 on page 20 shows the graphical representation of the system. The actuator contains a piezoelectric element moving a flexure structure. The dynamics of the system is obtained by a combination of these two dynamic systems. Previous tests on the flex• ure structure by itself have given its dynamic properties. In addition, data sheets for the piezoelec• tric element give its dynamic properties. Table 4.2 on page 46 outlines the properties of both elements. From Figure 3.4, it can be seen that the dynamics of the translator (piezoelectric element) and the tool holder (flexure structure) act in parallel. Because the dynamics of the piezoelectric trans• lator are -7.8 times higher than the flexure structure, it is expected that the dynamics of the com• bined system is very similar to that of the flexure structure alone. Figure 4.5 illustrates the combined dynamics of the tool actuator. Chapter 4. Digital Control of the Actuator Position 54

[V] [V] [N] [m] 1 Process Input U Process Output Y

(Input Voltage) MeffS + CeffS+ Keff (Tool Displacement)

Amplifier Piezo-Element Translator / Tool Holder

Figure 4.5 Combined Dynamics of Translator

This is an initial estimate of a model of the piezoelectric tool actuator. It is a purely linear rep• resentation of the system to be controlled. One of the major non-linearities in piezoelectrically driven devices is the hysteresis in the expansion/voltage graph of the device. Hysteresis is the phenomenon that is typically associated with the quasi-static motion of an actuator system. It is typified by a position dependence on the previous motion of the actuator. The hysteresis loop rep• resents a phase lag between the input and output. As outlined in Section 3.8.4., the method used for modeling hysteresis in a system is to introduce a time delay in the model of the system. The model block diagram is shown in Figure 4.6.

[V] M [N] I , I | [m] 1 Process Input U Process Output Y Qirnp Delay (Input Voltage) MeffS'+QffS+Keff (Tool Displacement) Amplifier Piezo-Element Hysterisis Translator / Tool Holder

Figure 4.6 Combined Dynamics with Delay

This hysteresis model considers only a fixed time delay for the hysteresis, but there is evi• dence that the time delay varies over various driving frequencies. For example, in quasi-static operation of the piezoelectric device, a phenomenon known as drift occurs. When a voltage is applied to a piezotranslator and held at that voltage, the translator will slowly "drift" over time.

The translator will drift in the direction of the previously applied motion. This drift could be con- Chapter 4. Digital Control of the Actuator Position 55 sidered a delay in the motion of the actuator, or a manifestation of hysteresis at low frequencies. For the continued analysis in this thesis, a fixed time delay will be used to model hysteresis.

Using several sample data sets that represent actuator being excited over a 0-500 Hz fre• quency range, the number of delay steps to be used in the model was determined. By minimizing the error variance for the model over a series of data sets, the delay in the plant that compensates for the inherent phase delays of the piezoelectric tool actuator is determine. Figure 4.7 shows on of the sample data runs. Each run in the test showed a consistent pattern.

Delay Comparison of different number of delays Sample set #1

4 5 6 7 8 10 Delay Steps in Plant

Figure 4.7 Delay Comparison in Plant Model Chapter 4. Digital Control of the Actuator Position 56

4.2.3.2 Continuous and Discrete Models The continuous time input-output model of the actuator system can be defined with the fol• lowing transfer function:

2 ( Georzr.,; \ Y(S) = sTd\ (4.1) U(s)

where:

G is the DC Gain, con is the Natural Frequency, C, is the Damping Factor, and Td is the delay time. A zero hold equivalent discrete time model of the system can be determined using the fol• lowing discrete time input-output transfer function [103]:

m = Z-*J^>_ (4.2)

U 2 ^ z + alz + a0

Where:

, 2 f^n n\ 2

b0 = a +ccl — Y-3J a0 = a

bx = l-a^ + -^YJ «i = -2a|3 (4.3) a = e P = cos(co/i)

Y = sin(co/z) co = 0)n7l - C d = delay steps

4.2.3.3 Model Estimation Method 1 Steeves and Yu[6] performed a model estimation of the tool actuator. This model was deter• mined by exciting the actuator with a sine wave excitation at frequencies from 10 to 1000 Hz and amplitudes from 0.8-74% of full scale. The motion of the actuator was recorded. The points were then plotted on a bode diagram and a curve fit to this data to determine the transfer function. Figure 4.8 shows the test results. Chapter 4. Digital Control of the Actuator Position 57

Frequency Response for Entire Assembly 50

45 + 7.40 V p-p In m 2.30 V p-p In 40 A 0.70 V p-p In 35 • 0.08 V p-p In 30 S imu lation

25 +

•o 8- 20 - % a 15 - 10 -- I §|—fit ^ -M: 5 0 0 400 600 800 1000 Frequency [Hz]

400 600 1000 Frequency [Hz]

Figure 4.8 Model Identification - Method 1

From this figure, the following transfer function was determined:

Y(s) _ C\ (4.4) U(s) s2 + CV + Cn

Where:

Cj = 57.420 C2 = 301.15 (4.5)

and C3 = 1.5983 x 10

This method of model determination is not adequate, because it does not determine the time delay caused by the hysteresis in the piezoelectric element. Because the hysteresis is so large (10- 15%), it will have a profound effect on the control. Controllers based on a model of the actuator Chapter 4. Digital Control of the Actuator Position 58

(without delay) do not lead to stable control of the actuator. This was shown in preliminary tests and well documented in literature [6],[103]. 4.2.3.4 Model Estimation Method 2 In the second method of model identification, involved exciting the actuator system with a band limited white noise signal and recording the response. Standard, discrete time system identi• fication, algorithms were used to determine the parameters of the system. As discussed above the form of the model to be defined is:

Y(z) _.d blZ + b0 = Z —;

U(z) z +a}z + a0

Using a white noise generated step function PIM [15], a voltage command signal was gener• ated for the power amplifier. This data was later post processed to determine the model of the sys• tem. The generated model was then evaluated based upon the expected natural frequency. The data collection was performed at a rate of 5KHz. The CSO laser interferometer was used at a posi• tion feedback device. The data was collected and post-processed using a least squares based sys• tem identification routines. One of the most crucial determinations was the number of delay steps present in the system. The following parameters were determined with the tool actuator mounted at the NRC test bench. Table 4.1 : Model Identified Parameters (Test Bench Actuator) Parameter Value d 5 0.01768 0.005469

a\ -1.36

a0 0.9024

A delay step size of 5 was identified on the NRC test bench with the CSO laser interferometer as the position feedback response. In the UBC lab, it was determined that 4 steps was a better esti• mate (see Section 4.2.3.1). It is expected that the natural frequency of the actuator be smaller now Chapter 4. Digital Control of the Actuator Position 59 than without a tool mounted in the actuator. The natural frequency from Section 3.10. is 61SHz compared to the 616Hz natural frequency determined with the tool mounted in the tool holder.

This leads to the following Transfer function:

Y(z) _ _-50.01768z +0.005469 (4 6) U^ z2-1.36z +0.9024

The continuous time conversion of this model using a zero order hold approximation[103] is:

Y(s) _ 34.4s + 6.404 x IQ5 -o.ooi* . . u(s> s + 513.4s + 1.499x10

This transfer function has two poles and one zero. The zero is very far to right of the poles

(sz = -18616 ) (i.e. very stable), and for preliminary analysis can be ignored. The poles of the

system are Sj 2 = -255.32+"j'3860. The natural frequency of this model, damping factor and DC gain all must correspond to what was determined theoretically for the system. These values are:

(0„ = 616//z C = 0.066 DCGain = 0.04272 (4.8)

The dimension of the transfer function is um/V. Where the voltage is the amplifier volts to the piezoelectric element (0-1000V), which leads to a 42.72 pm maximum displacement. 4.2.4. Desired Closed Loop Response

The pole placement controller is designed to achieve a desired closed loop response of the system. The choice of the response depends upon the sampling rate of the control system. Because the discrete model of the system was developed at a sampling rate of 5 KHz, the control will also be done at this rate. The natural frequency and damping of the closed loop response can be chosen to have approximately 2-4 sample intervals in the rise time. This will give a good con• trol that is not too close to deadbeat. The rise time is given by:

= ro* =ra^2 (4-9) Chapter 4. Digital Control of the Actuator Position 60

If the desired closed loop response is given by:

0)n = 500Hz (4.10) C = 1.0 Then:

tr = 0.000255 (4.11)

This value is very close to the limit for control. Faster poles would translate to a straight deadbeat controller. 4.2.5. Controller Parameters The controller parameters are determine by deriving the coefficients of the R, S, and T poly• nomials (Figure 4.4). In an attempt for conciseness, only a brief overview of the controller deriva• tion is given here. The plant model is given by (Derived during experiments on a bench testbed Section 4.2.3.4):

(4.12)

The process poles will be cancelled by the controller and the new closed loop poles will be

used based upon a natural frequency con = 500Hz and a damping factor C, = 1.0. The open loop zero will not be cancelled because it is close to being unstable. The sample rate is 5 KHz, and the desired closed loop response is:

0.010674Z + 0.003306 (4.13) z7- 1.764z6 + 0.7778z5

The first step is to determine the orders of the polynomials. The control will be designed with an integrator term so that low frequency servo characteristics are good. In addition, the observer polynomial will be set to deadbeat. The polynomial orders are as follows (based on causality con• ditions):

order(R) = 7, order(T) = l,order{AQ) = 7 (4.14)

where ;A0= observer polynomial Chapter 4. Digital Control of the Actuator Position 61

In order to solve of the parameters of these polynomials the following equations have to be solved simultaneously. The observer polynomial is chosen for the fastest response in this case.

Therefore Ao is:

1 A0 = z (4.15)

Then the following equation is solved:

AR + BS = A0Am (4.16)

• Am, desired response denominator, •A, plant denominator, and • B, plant numerator. The last polynomial to determine is found using:

T = B'mA0 (4.17)

where B'm is the normalized model numerator.

The solution to these polynomials is straightforward, although a little tedious for the higher orders that are present here. A Matlab™ program for automating this procedure was developed. In order to solve them via the program, a Sylvester matrix[103] was generated which solves for all the parameters simultaneously. The polynomials for this model are:

R(z) = z1 - 0.4033z6 - 0.6734z5 - 0.552U4 - 0.1436z3 + 0.3029z2 + 0.3810z + 0.0884 S{z) = 9.0782z7 + 6.0264z6 - 14.5035z5 (4-18) T(z) = 0.6011z7

These three polynomials were used to create custom control PIM for IMM[15]. 4.2.6. Pole Placement Control Tests

The step response of the pole placement controller can be seen in Figure 4.9. Chapter 4. Digital Control of the Actuator Position 62

Step response of Pole Placement Control using Laser Interferometer 15i 1 1 1 1 1 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Time (s)

Figure 4.9 Step Response of the Pole Placement Controller (Test Bench)

The step response shows a very good response to a step input of 20 um (-33% of the range of motion). This is a very large step and a good test for the control algorithm. The overshoot is -10% of the step size, and settling time is -0.005 seconds. This test shows that the basic model of the actuator is a good estimate for the actuator. The step response follows the design parameters for the controller.

It is also useful to evaluate the controller in the frequency domain. The frequency response shows the performance of the actuator system over the desired frequency range. A frequency response test was performed over a 0-1000 Hz frequency range.

The frequency response of the pole placement controller can be seen in the Figure 4.10. Chapter 4. Digital Control of the Actuator Position 63

Frequency Response of Pole Placement Control Using Laser 5 +3dB 0 -3dB -5 dB -10

-15

—I I I l_L -20 _i i • • i i _i i i • • ' 1 10 10 10

10 10 10 Frequeny (Hz)

Figure 4.10 Frequency Response of Pole Placement Controller (Test Bench)

The frequency response graph shows a -3dB bandwidth of ~140Hz at a phase margin of -50 degrees.

4.3. State Feedback Controller for Piezo Actuator 4.3.1. Regulator Problem In introducing State Feedback control, it is useful to investigate the State Feedback Regulator problem. The state feedback regulator has a block diagram shown in Figure 4.11.

Figure 4.11 State Feedback Regulator Chapter 4. Digital Control of the Actuator Position 64

where x represents the states of the plant. The gain vector L can be chosen using pole place• ment techniques similar to the polynomial pole placement technique. The input to the plant is given by:

u = -Lx (4.19)

The plant can be described by:

X(k+i) = ®x(k) + ru(k) y(k) = Cx(k) With the regulator feedback, the system has the following form:

x(k+l) = (

The system response is now governed by the matrix (3> - TL). The gain vector L is chosen by choosing the poles of the characteristic equation to have the desired response of the system The characteristic equation is given by:

|z/-0 + rL| (4.22).

A suitable state feedback gain vector L can be determined by solving this characteristic equation to get the desired response for the system. In order to implement state feedback control a valid state vector must be produced. All of the states of a system will have to be observed or mea• sured. Consider the plant a described by Eq. (4.20), the state x(k) must be approximated by an esti• mated state x(k). The estimated state is given by:

x(k+ 1) = ®x(k) + Tu(k) (4.23)

Matrices O and T are identical between the two formulations. This estimate of the state will

only be true if the initial conditions are the same between the estimate and the actual plant states.

Estimation can be improved by introducing feedback using the measured output(s) as follows: Chapter 4. Digital Control of the Actuator Position 65

jc(fc+l|Jfc) =

To get the gain vector K is obtained by introducing a variable representing the observer error:

~x(k) = *(*)-*(*) (4.25)

ThenfromEq. (4.20) and Eq. (4.24):

~x(k+l\k) = ®x(k\k-l)-K(y(k)-Cx(k\k-l)) y(k) = Cx{k) (4.26) ~x(k+l\k) = (®-KC)x(k\k-l)

Therefore, K can be chosen in the same manner as the state feedback problem. This presents the basis of state feedback control for the state regulator problem. To extend the state feedback control methodology to the servo control problem is a fairly straightforward extension.

Figure 4.12 shows the block diagram for the state feedback regulator with full state observer.

u ^ y — -L Plant meas

X Observer

Figure 4.12 State Feedback Regulator with Full State Observer

4.3.2. Disturbance Observation

On of the advantages of the state feedback formulation is that it allows for the easy imple• mentation of an observer to estimate the disturbances in the system. In the current application, the disturbance can be the cutting forces. The cutting forces act externally on the actuator and pro• duce a compression of the piezoelectric stack. This will lead to position errors while cutting and increase the error. The polynomial pole placement controller and, to some extents, the PID con• troller described previously eliminated this by simple integration. This method, however, is exten- Chapter 4. Digital Control of the Actuator Position 66 sible to many different disturbance models and is not limited to using an integral feedback for the disturbance elimination.

Under pole placement control, the closed loop system is forced to have a response that is designed into the system by the design engineer. Assuming a good knowledge of the model and the physical limits of the components in the system are not exceeded, one can design a controller that has the performance characteristics required. This was demonstrated in the previous sections. In order to compensate for un-modelled or unknown disturbances in the system, an observer can be used to determine the these disturbances and compensate appropriately for them. As with all good control system design, a model of the disturbance is preferred for defining an appropriate disturbance observer. The current disturbance observer is designed using a constant cutting force model. The expected cutting forces for a precision turning application can be estimated by analyz• ing the cutting conditions. Time varying disturbances can be modelled by using a different distur• bance model form.

Disturbances are unknown inputs to the plant system that effect the states of the system and the control. By developing a model for a disturbance, it can be observed and compensated for in the control law. Information about the dynamics of the disturbance can be used to refine the dis• turbance observation model. Figure 4.13 shows a plant acted on by disturbances.

Figure 4.13 Plant with Applied Disturbance and Noise

In the above diagram the plant process is augmented by two signals input into the system.

The external disturbance v acts on the input to the plant and measurement noise e acts on the out• put of the plant. The disturbance can be modelled as a dynamic system based upon the expected Chapter 4. Digital Control of the Actuator Position 67 disturbances on the plant. In the case of the tool actuator, a step disturbance with noise is expected based upon basic cutting force equations for orthogonal turning. For other forms of cutting, more complex disturbance models can be introduced. For this exercise, the measurement error, e, is neglected. The disturbance can be modeled by:

w(k+l) = ww(lc) (4.27)

v(*) = Cww(k)

where w(k) is the state vector of the disturbance model.

The revised state equations for the combine plant/disturbance system are expressed by the following state equations: x(k+ 1) XIV X(k))+r\u(k) (4.28) w(k+ 1). 0 $ w(k)J VoJ w This formulation effectively augments the state vector with a state of the disturbance. The disturbance can have any order required to model the outside disturbance of the system. The

model <&xw defines how the disturbance effects the states of the process. The following discus•

sion describes how

JC(*+1) = <&x(k) + Tu'(k) (4.29)

where u\k) is the input to the plant after the disturbance and is given by:

u\k) = u(k) + v(k) (4.30)

jc(Jfc+l) = x(k) + T(Cww(k) + u(k)) (4.31)

*™ = rCw (4-32)

This leads to the combined state transformation equations described in Eq. (4.28).

v(k) represents the disturbance into the plant. Time varied disturbances can be modeled by

choosing appropriate models , Oxw, and Cw. Chapter 4. Digital Control of the Actuator Position 68

The input disturbance will effect the states of the plant and subsequently effect the output of the plant. The states of the disturbance can be used to develop a feedback regulator for the distur• bances. The feedback regulator for a plant with disturbances would be:

u(k) = - Lx(k) - Lww(k) (4.33)

The block diagram for feedback regulator with disturbance rejection can be seen in

Figure 4.14:

-L u(k) Process

x(k) v(k) disturbance

Figure 4.14 State Feedback Regulator with Disturbance feedback

From Eq. (4.28) and Eq. (4.33), the overall state transition equations for the closed loop state regulator with disturbance feedback is:

= (*-rL)x(k) + <&xw-rLw)w'k) (4.34)

w(k+l) = ®ww(k)

The state observer, including disturbances, can be formulated by extending the standard observer formulation. The following system can be defined:

x(k+ 1) X w(k + 1). 0 o w(k)J VOJ (4.35)

The complete state(jc, w) is not reachable, but the states are observable if the system () is observable[103]. The control law is formulated as:

u(k) = - Lx(k) - Lww(k) (4.36) Chapter 4. Digital Control of the Actuator Position 69

where x and w are the estimated states given by:

(K 0 O, + i J"W+^ lEW (4.37)

£(*) = y(k)-Cx(k)

To determine the disturbance observer gain, Eq. (4.35) is subtracted from Eq. (4.37) and the following substitutions are used to get.

w = w — w

x(k+l\k) x(k\k-l) K (4.38) K \(y(k)-Cx(k\k-l)) w{k+\\k\ 0 3>„ w(k\k-l\

The gain Kw can be determined from Eq. (4.38) using standard pole placement techniques. 4.3.3. State Feedback Servo Problem The state feedback regulation presented above drives the user defined states to zero with the response of the pole placement control law. In order to servo the controller to a specific state, additions to the controller must be put into place. Figure 4.15 shows a state feedback servo con• troller. This controller is a standard feed forward model based servo controller[103]. The equations that define this controller are as follows:

Desired model States : xm(k + 1) = ®mxm(k) + Tmuc(k)

Estimated State: x(k + 1) = <£>x(k) + <&xww(k) + Tu(k) + Ke(k) (4.39)

Disturbance Observer : w(k + 1) = <&ww(k) + Kwe(k) Error": E(fc) = y(k)-Cx(k)

with:

Ufb(k) = L(xm(k)-x(k))-Lww(k) (4.40) Chapter 4. Digital Control of the Actuator Position 70

where ufb(k) is the feedback term which drives the system states to the desired state xm(k).

State Servo Controller With Disturbance Observer

Gotol +3 r«H uc uff From Muxl [^.v] Sine Wave Feed Forward v(k) Generator -J*" Mux Sum

K |ant Plant i-HMux 2 yp ufb(k) [B] Model Sum1 Gain Sum2 L Si Froml Goto Gain wjiat Lw x_hat u e y_hat y

e(k) Observers 3r Mux EstY [Y,Y_hatrVY ha ] Mux

Figure 4.15 State Feedback Servo Simulation

In order to provide the feedforward term in the control law a suitable feed forward transfer function needs to be defined. The feed forward transformation is:

(4.41) H(z) =U]X

where X is a scaling factor. Hm(z) is the desired model transformation, and H(z) is the plant

transformation. The feedforward transformation assumes that both Hm and H share the same zeros. This allows the feed forward transformation to be expressed in terms of the denominators of the model and plant transfer functions. X is the scaling factor between the to allow for the DC transformation between the input and output of the plant transfer function. This transformation can be represented in State Space form by converting the transfer function into a standard canoni• cal form: Chapter 4. Digital Control of the Actuator Position 71

xff(k + 1) = %xff{k) + Tff uc(k) (4.42)

uff(k) = Cffxff(k) + Dffuc(k)

The following parameters are given in the design of the state feedback servo controller. Plant Model: 4>, T, C Disturbance Model: 3> , and

Desired Model: <3>m, Tm

From the plant model and the Desired model the feed forward model Ej-, Cjy, D^ can be determined. Model & FF Generator

plant & measurement

Observer

Figure 4.16 State Feedback Servo Block Diagram

The parameters that constitute the state feedback part of the controller are derived as described in the previous sections. This design of the state feedback servo problem allows de-cou•

pling of the design process. It should be noted that the two states xm(k) and x(k) need to be in the same domain. This means that the form of the state space equations needs to be equivalent between the model generated form and the state feedback representation. The desired model form

of the controller Om, Tm are derived from the desired model transfer function Hm. The state

space form for Om, Tm is in the same canonical form as the state observer. Chapter 4. Digital Control of the Actuator Position 72

4.4. Zero Phase Error Tracking Control

Tomizuka[123] first introduced zero phase error tracking control (ZPETC). This method of control attempts to compensate for the inherent phase lag in a control system by introducing a feed-forward controller in the control system. This controller uses the setpoint from several steps ahead to obtain the control input. The feed-forward controller cancels the closed loop poles and cancelable zeros. The block diagram of the controller is shown in Figure 4.17.

Feedforward Controller Controlled Plant

X y (k + d + s) [ Bc (z ) y*(k + d) d f"Bc{z- ) y{k) ,

1 • -1 S"(z )B"(1) Ac(z-') 1 I

Figure 4.17 Zero Phase Error Tracking Controller

One of the main drawbacks with Zero Phase Error Tracking Control is that it produces a heavy command action into the actuator. This can excite un-modelled or un controllable dynamics of the plant. Figure 4.18 shows an example step response of the actuator with ZPETC and the pole placement controller. An unwanted oscillation at 1.1 KHz is being exhibited by the actuator. This response is difficult to control because the bandwidth of the power amplifier - piezoelectric ele• ment pair is severely attenuated at that frequency. In order to compensate, an extension to the Zero Phase Error Tracking control algorithm has been proposed by [91] to filter the input to the plant thereby eliminating the higher order excitation. The extended bandwidth produced by the ZPETC is shown in Figure 4.19. In this frequency plot, the frequency response of the actuator system is clearly improved over the basic PID control• ler (Figure 4.3) and the pole placement controller (Figure 4.10).

The ZPETC algorithm only effects the input/output response of the controlled system. Because the derivation of the ZPETC algorithm is based solely on the commanded setpoint to the actuator system, the response of the ZPETC to external disturbances is exactly the same as the Chapter 4. Digital Control of the Actuator Position 73 underlying pole placement controller. This is a key factor when evaluating controllers during cut• ting operations. Because the ZPETC algorithm is purely a feed forward algorithm, the response of the ZPETC and the pole placement controller that it is derived from to disturbances (e.g. cutting forces) is exactly the same.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 Time (s)

Figure 4.18 Step Response of Pole Placement Controller and Pole Placement Controller with ZPETC (Test Bench)

102 Frequency (Hz)

Figure 4.19 Frequency Response of ZPETC (Test bench) Chapter 4. Digital Control of the Actuator Position 74

4.5. Controller Tests With Actuator 4.5.1. Model of the Actuator System Mounted on the Lathe

The actuator models used at UBC differ from the one implemented on the test bench. The model of the system being controlled is an inherent part of pole placement control. For the tests described in this section, a different cutting tool was used with the actuator than used in the iden• tification of Section 4.2.3.4. This effectively changed the mass of the motion flexure. The natural frequency of the actuator would decrease accordingly. In addition, the actuator was mounted on the UBC CNC lathe which has a different mounting method. These factors effect the dynamics of the system.

The model that was used for the basic of control of the actuator was a second order model, which adequately modeled the DC-gain of the system and the dynamics at 610 Hz. This was iden• tified by performing hammer tests on the actuator and determining the system parameters by per• forming a least squares based identification on input/output data. Extensive controller testing on the test bench proved that this model represents the actuator for control system development. When the actuator was installed on the CNC lathe at UBC, however, the model parameters were found to be changed. The model parameter estimates were determined separately with the actua• tor mounted on the machine. Based upon the methods discussed in Section 4.2.3.4, the following model of the actuator was used for the actuator cutting tests.

Transfer function for the actuator on the lathe:

B(z) _ -40.00288Z + 0.006975 H(z) (4.43) AW z2- 1.502z + 0.793

where the sampling time is 0.0002 s From this model, controller parameters were generated for the pole placement, ZPETC, and state feedback servo controller. Chapter 4. Digital Control of the Actuator Position 75

The desired closed loop model of the overall system is:

2

Hm{s) = ^ (4.44)

+2 * ODmS + CDm

where, com is the model natural frequency and L,m is the model's damping coefficient. These two design parameters are used to determine the closed loop response of the system. The transfer function is converted to a discrete time equivalent with enough input delays to satisfy causality constraints.

Q)B = 300Hz C = 0.8 0.05803? + 0.04744 (4-45)

mKz) z2_L442z + 0.5471

where the sampling time is 0.0002s The plant model's unstable zero (Eq. (4.43)) must be included in the model transfer function:

= ,-4 0-03082z +0.07465 z2- 1.442z +0.5471

This desired model transfer function is used in the pole placement, feed forward and state feedback control laws used in controlled the actuator. 4.5.1.1 Pole Placement Controller The pole placement controller was developed based upon the traditional techniques proposed by Astrom [103]. As with any model based control, the desired plant model is first defined. In this case the following causality conditions must be satisfied:

order(Am) - order(Bm) > order(A) - order(B) , and

order(R) = 6, order(T) = 6, order(AQ) = 6, order(S) = 6

where R, S, T, and A0 are control polynomials (Figure 4.4). A and B are the denominator

and numerator of the plant model. Bm and Am are the denominator and numerator of the desired model of controller. Chapter 4. Digital Control of the Actuator Position 76

The characteristic equation Eq. (4.18) must be solved to determine the Polynomials S and R. In order to allow for simple disturbance rejection, an integral term is added to the R polyno• mial

R = (z-l)Rp (4.47)

where Rp is the portion of the R polynomial without the integral term. The characteristic equation (Eq. (4.16)) is then:

A(z-l)Rp + B'S = A0Am (4.48)

where: A0 is the observer polynomial, Am is the model characteristic equation, B is the non-cancellable zeros of the plant model, and A is the plant's characteristic equation. In this case,

the order of A0Am is (6+6)=12. To solve the Diophantine equation a matrix method is used for the high order of the system. First define:

-1 n (z-l)A = A, = aiQ + a/jz + + ainz~

x n 4 49 B~ = b0 + hxz~ + ... + bnz~ ( - )

A(Am ~ Cp = 1+ C\Z~X + -•+ ck + l+ 1

where k is the order of the Rp polynomial and / is the order of the S polynomial. The characteristic equation is then:

AjRp + BS = Cp (4.50)

which can be solved using the Sylvester matrix [103] to get the polynomials R, &,S . The

T polynomial is determined using Eq. (4.17). Using the desired model transfer function Hm defined by Eq. (4.46), the polynomials are solved for. The polynomial are given in Eq. (4.51).

R(z~l) = 1 - 1.3816Z-1 + 0.3048z~2 - 0.024U-3 + 0.0214z-4 + 0.0468z-5 + 0.0328z Siz'1) = 1.578 + 3.2754Z-1 - 3.7246z~2 (4.51) T(z~l) = 10.7026 - 15.4288Z"1 + 5.855z-2 Chapter 4. Digital Control of the Actuator Position 11

The controller was simulated before being tested on the actuator. In Figure 4.20, the simu• lated step response of the uncontrolled plant is compared to the controlled plant. This confirmed the basic control law formulation for the pole placement controller and allows it to be tested with the actual actuator/tool combination in sections to follow.

Simulated Step Response of Plant and Controlled Plant 1.6 Green Dashed: Plant Model 1.4 Blue/Red: Controlled Plant (Pole Placement)

1.2

E '

CD "5 0.8 Q. E < 0.6

0.4

0.2

0.005 0.01 Time (sec)

Figure 4.20 Simulated Step Response of Plant and Controlled Plant

4.5.1.2 Zero Phase Error Tracking Controller (ZPETC)

The zero phase error tracking controller is an extension of the pole placement controller. This controller is a true feed forward implementation and will have no effect on the disturbance rejec- Chapter 4. Digital Control of the Actuator Position 78 tion of the controller. This formulation is presented as a extension to the pole polynomial pole placement control law, but can be extended for use in any controller formulation.

The controller cancels the phase shift created by non-cancellable zeros of the plant and assures zero phase shift for all the frequencies[123]. Two feed forward transfer functions are required to be derived for the zero phase error tracking controller. The two transfer functions are illustrated in Figure 4.17 on page 72. The controlled plant is defined by:

~~AJX^Y - "™(Z) (4'52)

where Hm(z) is the desired model transformation described by Eq. (4.46).

Bm(z) = 0.03082z + 0.07465 AJz) = z* - 1.442z + 0.5471 (4.53)

The first transfer function for the controller cancels the cancellable zeros and the controlled plant's characteristic equation.

—u— <4-54> a l u B m(z-)B m(l)

There are no cancellable zeros in the model therefore:

a 1 u :.B c(r ) = l B m(z) = 0.03082z + 0.07465 (4.55)

The first transfer function becomes:

A l 1 2 c(Z~) = 1 - 1.442Z- + 0.5471 z~

a u B c(z-i)B c(l) ~ 01055

The second transfer function compensates for the phase lag due to the non-cancellable zeros

K 0.07465 - 0.03082Z-1 (4.57) 5"(1) °-1055

The controlled plant transfer function used in the derivation of the ZPETC is assumed to be equivalent to the desired model form described in Eq. (4.46). The step response of the ZPETC Chapter 4. Digital Control of the Actuator Position 79 was simulated before being used on the actuator, is shown in Figure 4.21. The step response fig• ure shows a zero phase lag response to a step input.

Simulated Step Response of ZPETC Controller

Solid: Input to Controller Dashed: Simulated Plant Response

0.8

E 3 0.6 c o w o 0_ 0.4

0.2

6 8 10 12 Time (s) x 10

Figure 4.21 Simulated Step Response of ZPETC

A comparison of the ZPETC with the standard polynomial pole placement controller, and the plant model is showing in Figure 4.22. chapter 4. Digital Control of the Actuator Position 80

Step Response Green: Plant, Red: Pole Placement, Blue: ZPETC

0 0.005 0.01 Time (seconds)

Figure 4.22 Simulated Step Response (ZPETC, Pole Placement and Plant)

4.5.1.3 State Feedback Servo Controller

The state feedback servo controller is developed piece by piece as described in Section 4.3.

The first step is to represent the model of the actuator in a stare space representation.Comparison of State Feedback and Pole Placement Controllers

Several design parameters are required to derive the sate feedback servo controller. The first part of the design process is to design a state feedback controller in a traditional manner. The State

feedback gain L, state observer K, and disturbance observer gain Kw are determined using stan• dard Pole placement techniques described in Section 4.3. The plant model is given by Eq. (4.46). Chapter 4. Digital Control of the Actuator Position 81

This model can be transformed into a state space canonical form seen in Eq. (4.58).

1.5016 -0.7930 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 r = 0 0 0 10 0 0 0 (4.58) 0 0 0 10 0 0 0 0 0 0 10 0

C = [o 0 0 0 0.0029 0.0070]

The model reference for the servo controller to follow has the following characteristics:

co = 300Hz C = 0.8 (4.59)

Using Eq. (4.1), Eq. (4.2) and Eq. (4.3) it is possible to derive the characteristic equation for a model that has the dynamics described by Eq. (4.59). This dynamics produces the following characteristic equation:

z2- 1.4390z + 0.5442 (4.60)

This will give the following model form in a canonical form:

1.4390 -0.5442 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 q> = 0 1.0000 (4.61) m 0 0 1.0000 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0

In order to allow for a the gain of the model to be compensated for, a input vector is defined:

10.6772 0 0 (4.62) 0 0 0 Chapter 4. Digital Control of the Actuator Position 82

State feedback gain is derived by forcing the characteristic equation of (O-TL) (see Eq.

(4.21))to have the dynamics that have a natural frequency of ton = 500Hz, and a damping factor

£ = 0.8. The characteristic equation is developed based upon the method described in Section

4.2.3.2 and reiterated on the previous page. The gain T is then chosen by equating eigenvalues of

(0 - TL) to the desired characteristic equation The following gain is obtained.

L = [0.3687 -0.4180 -0.0030 0.0000 0.0000 0.0000] (4-63)

The state observer is derived by forcing the eigenvalues of (<&-KC) (see Eq. (4.26)) to

have the dynamics that have a natural frequency of ton = 2500Hz and a damping factor £ = 0.5 derived from Eq. (4.1), Eq. (4.2), and Eq. (4.3). The following gain is obtained. -65.8491 13.2143

K = 108.0652 187.9689 219.6588 178.8945

The disturbance observer gain is generated by the exact same methodology for finding the

above two gains. The gain Kw is determined be forcing the characteristic equation (see Eq.

(4.38)) of the augmented states to have dynamics corresponding to the a natural frequency

con = 350Hz and a damping factor £ = 0.7 . The following disturbance observer gain

Kw = 11.4890 (4.64)

The model feed-forward system defined by Eq. (4.41) is expressed in Eq. (4.65), (4.66),

(4.67), and (4.68). Chapter 4. Digital Control of the Actuator Position 83

0 0 0.0000 0.0000 0.0000 0.0000 1.000 0 0 0 0 -0.0000 0 1.0000 0 0 0 0 (4.65) 0 0 1.0000 0 0 -0.0000 0 0 0 1.0000 0 -0.5442 0 0 0 0 1.0000 1.4390

T (4.66) ff ~

Cff = [-0.6682 1.6946 2.8021 3.1100 2.9503 2.553o] (4.67)

Dff = 10.6772 (4.68)

Figure 4.23 shows the step response of the State Feedback Servo controller. This demon• strates that the basic formulation is correct for this plant model. In addition to the basic step response, a disturbance was input to the system half way through the simulation run. Figures 4.23 and 4.24 show the comparison of the State Feedback controller with the Pole placement control• ler. The response to disturbances is better controlled with the disturbance observer formulation. Figure 4.25 shows the actual and estimated disturbance input to the plant. The state feedback con• troller has the desired response to disturbance inputs. Chapter 4. Digital Control of the Actuator Position 84

Simulation of State Feedback and Pole Placement Controllers 14 Dashed: Setpoint Dotted: Pole Placement Control 12 Solid: State Feedback Control

10 [f 8 ? 5 6 3 O

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (seconds)

Figure 4.23 Step response of the State Feedback Servo Controller Simulation

Simulation of State Feedback and Pole Placement Controllers Simulation of State Feedback and Pole Placement Controllers

Dashed: Setpoint Dashed: Setpoint Dotted: Pole Placement Control Dotted: Pole Placement Control Solid: State Feedback Control Solid: State Feedback Control | \ /l \

i r i

5 9.5 Q. 3 i \1 i 1V

10 . i 1 S* l 8.5 ' \> 9.5

0.02 0.022 0.024 0.026 O.C 0.03 0.032 0.038 0.04 0.042 0.044 0.046 0.043 0.05 0.052 0.054 0.056 0.058 Time (seconds) Time (seconds)

Figure 4.24 Comparison of Pole Placement and State Feedback controllers Step Response and Response to Step Disturbance Chapter 4. Digital Control of the Actuator Position 85

Disturbance Input to Plant at 0.05s 60 I I I I I I I I I

Dashed: Actual Disturbance Value

50 Solid: Estimated Disturbance — / / 1 40

w o 30 i£ 20 10

fx I,

-10 i i i i i 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time (s)

Figure 4.25 Actual and Estimated Disturbance input to Plant

4.5.2. Disturbance Model The proposed actuator is intended for "precision finish turning of hardened shafts" on a CNC turning center. Here the piezoelectric tool holder actuator is used as a secondary precision posi• tioning device after the gross motion is completed by the conventional servos. The secondary pre• cision motion requires 60 \im position range, 0.1 pm position accuracy and 100 N force delivery for precision machining. The developed actuator delivers about 60 pm dynamic displacement at a lower frequency range (-100 Hz), and more than 20 pm at the highest operation frequency (400Hz) with 1000 N dynamic force delivery capacity. The control algorithms for precision turn• ing have been presented in the previous sections. Chapter 4. Digital Control of the Actuator Position 86

Figure 4.26 Orthogonal Cutting

4.6. Cutting Force model One of the biggest concerns with precision turning over the traditional grinding process is the surface finish that one can achieve with very small depths of cuts as compared to the grinding pro• cess. Momper[112] looked into this problem. The use of hardened steel such as P20 mold steel as a work-piece material is essential in achieving a good surface finish in cutting operations with small depths of cut. The fine precision turning operation is the final operation of the turned part. This operation is characterized by very small depths of cut and consequently small cutting forces. If we look at a

simple orthogonal cutting scenario (See Figure 4.26 where Fc = Ft), it has the following charac• teristics.

Ft = Ktbh + Kseb (4.69)

Ff=KfFt + Kfeb

The mechanistic cutting coefficients for P20 mould steel given in Eq. (4.70). Chapter 4. Digital Control of the Actuator Position 87

Kt = 1800^- Kse = 18 AN/mm mm (4.70)

Kf = 0.556 Kfe = 35JN/mm

4.6.1. Cutting Tests

Several cutting tests with the actuator were performed which included using various control methodologies. The objectives of this test were to test the performance of various controllers under servo control while cutting a disk. The actuator was mounted in the lathe and the electronics connected. Orthogonal turning was performed on a mild steel disk. A zero rake manually ground carbide tool was used for cutting. Three basic controllers were used in a set of three different cuts. An single order integral term was used in the pole placement controller and a basic distur• bance observer based on the cutting force model was used for the state feedback servo controller. The controllers used are described in Section 4.5.1. Three cutting tests were performed to test the controllers. The actuator was commanded to a setpoint of 0p,m, which means that the actuator is commanded to hold its position. The figures (4.27, 4.28, and 4.29) clearly show when the tool is engaged with the workpiece. During the cut• ting, the actuator is kept within ±0.2pm for all controllers. The time of cut is indicated on the fig• ures.

The pole placement and state feedback controllers were based upon the exact controllers defined in Section 4.5.1.1 and Section 4.5.1.3.

The following cutting parameters were used: Table 4.2 : Initial Cutting Tests Mild Steel Cutting Test 1 2 3 Controller Pole Placement PID State FB w/ Distur• bance Observer Cutting Speed 360 m/min 360 m/min 360 m/min Feed rate 50 ^ 50 ^ 50 rev rev rev Width of Cut 2.45 mm 2.45 mm 2.45 mm Chapter 4. Digital Control of the Actuator Position 88

Table 4.2 : Initial Cutting Tests Mild Steel Start Diameter 95.28 mm 90.28 mm 85.28 mm End Diameter 90.28 mm 85.28 mm 80.28 mm

The results show that the controllers are comparable. It can be seen (Figure 4.27) that the state feedback controller with disturbance observer works as intended by compensating for the cutting forces experienced by the tool/actuator.

Laser Sensor position Info Vs Time (State feedback with disturbance observer)

0.6 r—T 1 1 1 1 1 1 1 1 r

0.4

o Q_

-0.4 Start of Cut End of Cut

-0.6

5 6 10 Time (s)

Figure 4.27 Cutting Test Data Using State Feedback Servo Control with Disturbance Observer

The state feedback servo controller successfully controls the actuator and performs the required disturbance rejection to achieve an actuator position to within ±0.5 pm. The controller parameter developed in Section 4.5.1.3 on page 80 were used in the control algorithm for the actuator. These parameters were converted into a specialized Plug in Module for the IMM soft• ware package [15].

The pole placement controller parameters used were derived in Section 4.5.1.1 on page 75. The polynomials R, S, and T are given by Eq. (4.51). Chapter 4. Digital Control of the Actuator Position 89

Laser Sensor position Info Vs Time (Pole Placement Controller) 0.3 -

0.2 -

-0.4 -

2 4 6 8 10 12 14 Time (s)

Figure 4.28 Cutting Test Data using Polynomial Pole Placement Controller

Laser Sensor position Info Vs Time (PID Controller)

0.4 F—' ' ' ' ' • =

0.3 -

0.2 - I

-0.4 - I

2 4 6 8 10 12 Time (s)

Figure 4.29 Cutting Test Data PID Controller Chapter 4. Digital Control of the Actuator Position 90

The basic PID control law (Section 4.1.1.) was used with the following parameters:

K = 0.13 Tj = 0.001 TD = 0.00001 TQ = 0.0002s (4.71)

Each of the cutting tests demonstrate how the controllers are able to be used with the tool/ actuator during a cutting process. 4.7. Conclusions

Using a model of the piezoelectric actuator developed in previous chapters, advanced model based control algorithms were presented and tested. This model includes a method for hysteresis modeling. Controllers based on PID, pole placement and state feedback with disturbance observer were presented. Overall this chapter has shown that a valid model of the piezoelectric actuator can be devel• oped and a model based controller developed. Significant gains in the phase margin can be achieved with little compromise on bandwidth using a Zero Phase Error Tracking Control (ZPETC) methodology. The ZPETC is used to improve the input/output bandwidth of the actuator system and it has been shown by demonstration that this controller provide this. In order to pro• vide a mechanism to compensate for plant disturbances, a disturbance observer was created. This general disturbance observer can be modified by the control system designer to compensate for any disturbance that can be modeled by the state space representation given in Eq. (4.27). The feed forward term of the controller (ZPETC or other) and the disturbance observer can be devel• oped separately. The cutting tests were performed to test the validity of the disturbance observa• tion method. Chapter 5 Precision Turning Tests with Piezoelectric Actuator

5.1. Introduction In the previous chapters, the piezoelectric tool actuator has been introduced and modeled. The specific challenges faced by precision turning have been introduced and methods to mitigate these obstacles have been presented. Ways of overcoming the challenges of precision turning task on standard CNC machines have included: the use of high precision sensors for on-line and off• line measurement of critical dimensions, a high bandwidth control system for the tool actuator that used the cutting force model to model the external disturbances, a high performance data acquisition and control system for the implementation of the system. The following sections out• line the tests performed on the tool actuator on a lathe in the UBC Manufacturing Automation Laboratory. 5.2. Eccentric turning A fast tool/actuator can be used for performing turning operations that require a non-circular shape. The turning of elliptical or cam like profiles can be achieved with this type of actuator which has a high bandwidth. The main limitation with using this piezoelectric actuator is the range of motion of the actuator. Because the range of motion of the actuator is < 60 \xm, the mag• nitude of the eccentricity is limited. An experiment was performed to see if an elliptical cut could be produced on a disk and to what fidelity the cut could be made at. In order to perform an ellipti• cal cut, the tool/actuator must be actuated with a sinusoidal input to the tool. Two periods of the sine wave will be imparted on the workpiece during one revolution. In order to accomplish this task, the position and speed of the workpiece is required in order that the movement of the tool is in phase with the spindle motion. For these tests, only the open loop spindle speed was known.

91 Chapter 5. Precision Turning Tests with Piezoelectric Actuator 92

Any variance in the actual spindle speed from the commanded could result in machining errors. The frequency of the imparted sine wave produced by the actuator is related to the spindle speed

N(rev/min) by the following equation:

(5.1)

Because the waveform put onto the disk is directly related to the spindle speed, the knowl• edge of the spindle speed is essential. For these experiments, the spindle speed was assumed to remain at the setpoint value. No feedback of spindle speed was given to the tool/actuator control• ler. Of course, accumulated error in the estimate of the spindle speed will result in machining errors. In order to provide some form of compensation, the system should include feedback from the spindle for non-circular machining tasks. For example, if on-line spindle velocity and position sensors are available, proper eccentric turning could be performed with the actuator.

Plot of LNS feedback vs Time (Ellipitcal Cut)

Dashed: Command

180

Solid: Feedback

270 Figure 5.1 Profile of an Elliptical Cut Chapter 5. Precision Turning Tests with Piezoelectric Actuator 93

Actuator Command and Position Error vsTime (Elliptical Cut)

0 0.02 0.04 0.06 0.08 0.1 0.12 Time (s)

Figure 5.2 Command and Error During Elliptical Cut

Figure 5.1 and 5.2 show an example run for the cutting of an elliptical cut. The workpiece cut was a 12L14 steel disk with a width of cut of 2.4 mm. The tool in the tool holder was a carbide parting tool. Orthogonal cutting was performed. The actuator was commanded to generate an elliptical cut on the workpiece. The actuators start/stop position was -15 um. The tool was posi• tioned so that it was just touching the workpiece then commanded to move in two sine waves to impart an elliptical profile on the surface of the workpiece. The peak to peak amplitude of the cut was 30 um. The frequency of the motion is governed by the spindle speed. The spindle speed was set to 1280 rpm. Because two waveforms are needed to perform the elliptical cut, a actuation fre• quency of twice the spindle speed is required (41.66667 Hz). The basic PID control law (Section 4.1.1.) was used with the following parameters:

K = 0.13 T, = 0.001 TD = 0.00001 T0 = 0.0002s (5.2) Chapter 5. Precision Turning Tests with Piezoelectric Actuator 94

It is interesting to note the required actuator command frequencies for certain cutting condi• tions. Assuming a cutting speed of 360 m/min, the following spindle speeds and corresponding actuation frequencies of the tool actuator would be required. Table 5.1 : Actuation Speed For Elliptical Cuts Diameter Spindle Speed Elliptical Cut Freq. mm inches RPM Hz 114.3 4.5 1003 33.4 101.6 4.0 1128 37.6 88.9 3.5 1289 43.0 76.2 3.0 1504 50.1 63.5 2.5 1805 60.1 50.8 2.0 2256 75.2 38.1 1.5 3008 100.3 25.4 1.0 4511 150.4 12.7 0.5 9023 300.8 6.35 0.25 18046 601.5

The frequency required to perform an elliptical cut is proportional to the inverse of the diam• eter. In order to perform a non-circular cut on a shaft, a high bandwidth is required from the actu• ator. In addition, the phase shift from command position to feedback position becomes crucial. If, for instance, a 30 degree phase lag between input and output of the actuator could have an adverse effect on the resultant work piece. The use of on-line sensors are important for non-circular turning. In order to perform a proper elliptical cut, the exact spindle speed would be necessary for feedback to the controller. In addi• tion, the spindle position would be used to determine the relative position of the shaft with respect to the tool. In the tests presented here, the commanded spindle speed was used for the prediction of the eccentric cut with the actuator. In order to improve the process, the spindle position and speed would be required to be measured. The Hardinge lathe uses a position resolver on the spin• dle to measure both position and velocity. Interfaces to these devices can be done, but a serious retrofit of the machine will need to be performed. Another method would be to use a high preci- Chapter 5. Precision Turning Tests with Piezoelectric Actuator 95 sion velocity and position measurement scheme as an add on device to the lathe. This could be achieved with he use of a strober type device. Table 5.1 shows the actuation frequencies required for elliptical cuts. The frequencies show that a high bandwidth tool actuator such as the one presented here would be suitable. This type of actuation frequencies would not be suitable for standard CNC drives, which have a low band• width. The tool command, however, would be required to be in phase with the spindle rotation. The phase lag in the controlled plant should be minimized and well known. Because of this fact, the Zero Phase Error Tracking controller would be the most suitable for this application. This type of controller has been shown to effectively ensure zero phase lag tracking of the setpoint position up to a 600Hz actuation frequency (See Chapter 4). 5.3. Extensions for on-line Sensors On-line sensors can be used to determine the shape and machining errors in the part being turned. In these experiments a laser sensor was used to determine the shape of the piece being turned during the cutting process. A sensor mount (Figure 5.3) was designed to be used with the Hardinge lathe in conjunction with the precision actuator.

Actuator

Work Piece

LTS Sensor — 04,7240

Figure 5.3 Laser Sensor Mount for Workpiece Measurement Chapter 5. Precision Turning Tests with Piezoelectric Actuator 96

A large amount of on-line sensors should be used to assist in the precision machining task. These sensors should include spindle speed and position sensors combined with spindle shaft run• out sensors. The work piece should also be measured on line. For the purposes of initial tests, only one sensor was used to determine the position of the workpiece with respect to the tool turret. A data collection experiment was set up to measure a workpiece before and after a cut. A workpiece disk was put into the lathe and the tool actuator brought into position close to the workpiece and the workpiece rotated. The relative position of the workpiece with respect to the LTS sensor was measured. The work piece shows a noticeable eccentricity in the disc before it is cut. This eccen• tricity is well beyond the specifications for the lathe, and is assumed to be produced by the mount• ing of the steel disc. The Laser Twin Sensor was used in conjunction with a preliminary cutting test to evaluate the output data of the sensor. A cutting test was set up with a standard parting tool cutting a mild steel disk. Regular circular cutting was performed for the following tests. The fol• lowing cutting parameters were used.

Table 5.2 : Cutting conditions Work Piece Diameter 120.0 mm Cutting Speed 240 & 360 m/min Width of Cut 2 mm Feed Rate 50 [im/rev End Diameter 115 mm

The following process is envisioned for the actuator. CNC Program Outline

(1) Approach Work Piece (2) Set Constant cutting speed motion of 360 m/min. Um (3) perform a 5 mm cut at a cutting rate of 50 rev (4) Wait at the surface for 10 seconds (Dwell) (5) Retract from surface and return tool to home position Chapter 5. Precision Turning Tests with Piezoelectric Actuator 97

During step 4 of the process, the tool actuator can be used to perform precision cuts and elim• inate errors in the process. This was achieved by putting a dwell command in the NC program. Based on the number of on-line sensors available to the process, different operations can be per• formed. This time can be used to eliminate eccentricity of a part, or add a non-circular profile to the shaft being turing. Figure 5.4 shows a typical result from the Laser Twin Sensor.

The turning run can be broken into four distinct phases. The four phases of the cut as defined above are shown in Figure 5.4. The tool enters the cut at approximately 2.0 seconds into the run. Cutting finishes at 7 seconds and the tool "rests" on the surface of the workpiece for approxi• mately 10 seconds. The purpose of the dwell time in the process is to allow for future enhance• ments. The State feedback controller with disturbance observer was used in this tests. The parameters of the transfer function and controller are given in Section 4.5.1.3.

Laser Twin Sensor position Info Vs Time Workpiece Measurement

500

July 30, 1998 Tests State Feedback w/ disturbance Observer Cutting Speed 240 m/min

o Q.

-1000

-1500

2 4 6 8 10 12 14 16 18 Time (s)

Figure 5.4 Cutting with the Laser Twin Sensor Chapter 5. Precision Turning Tests with Piezoelectric Actuator 98

Phase I - Approach: represents is the approach of the tool to the workpiece. The eccentricity in the mounting of the work piece is clearly shown in Figure 5.5. Each cycle represents the eccen• tricity on the shaft, and measured by the laser twin sensor mounted underneath the actuator assembly (See Figure 5.3). Laser Twin Sensor position Info Vs Time Workpiece Measurement

Time (s)

Figure 5.5 Phase I Workpiece Sensor Approach Phase

Phase II - Cutting: During this phase, the cutting of the work piece can be seen. The cutting

conditions are outlined in Figure 5.2. The approximate cutting forces given by (Eq. 4.69) and

(4.70) and using parameter from Table 5.2 are:

Ft = 216N Ff=\9\N (5.3)

The force acting on the tool in the direction of actuation is the feed force Fj-. The tangential

force Ft acts perpendicular to the tool and is assumed to have little effect on actuator position. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 99

The diameter of the work piece is being reduced. The relative distance from the workpiece sensor and the workpiece is increasing as shown in the following figure(Figure 5.6 on page 99). At the start of the cut, the peaks of the original eccentricity are eliminated first followed by the cutting down of the diameter. With the mounting of the workpiece sensor (LNS), the measured distance to the workpiece is affected by due to the sensor beam measuring a diminishing circle as illustrated in Figure 5.7 on page 101 Laser Twin Sensor position Info Vs Time Workpiece Measurement

600

400

200 July 30, 1998 Tests State Feedback w/ disturbance Observer Cutting Speed 240 m/min

-1000

2 3 4 5 6 7 Time (s)

Figure 5.6 Phase II Cutting Workpiece

In using the LTS sensor for cutting operations, it is common to get a "spike" in the data dur• ing a typical cutting operation. Figure 5.6 Note A, shows an example of such a phenomenon. The spike has a magnitude of almost 1 mm. It is assumed that these are caused by errant chips from the cutting process. These spikes are only seen during the cutting phase of the operation. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 100

Because of the physical size of the LTS sensor, the mounting of it was placed relative to the tool turret. At a work piece diameter of 120 mm (4.724") the LTS sensor is perpendicular to the work piece surface with the line of the laser passing through the centre of the work piece. As the work piece diameter decreases, the values from the LTS sensor need to be compensated. As the diameter decreases from 120 mm, the measurement of the eccentricity needs to compensated by the following equations: The relative position of the centre of the work piece with respect to the laser twin sensor in the axis of sensor is: (see Figure 5.7)

(5.4)

where xc, yc is the shift of the origin of the work piece from the baseline 120 mm diameter,

rrej- is the calibrated workpiece radius of 60 mm, and r is the new radius of the work piece. There are three components to the relative distance coming from the LTS sensor. First the diameter decreases which should effectively move the work piece away from the sensor head. Secondly, the relative position of the centre of the workpiece changes with respect to the sensor head. Finally, the relative position of the workpiece changes based on the curvature of the work- piece. The relative sensor change e it then defined by:

(5.5)

where: e is the relative distance from the laser sensor, and a is the angle from the true centre of the work piece and the sensor origin. In the case where the diameter of the workpiece is 120 mm, a = 0°. The sensitivity of the sensor will increase as the diameters is decreased. For these experiments, the smallest work piece diameter was 115 mm. Table 5.3 outlines the relative posi• tion of the center of the workpiece disk and the sensor origin. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 101

Table 5.3 : Relative position of Workpiece and Sensor Diameter a £ (mm) deg. (mm) (mm) 120 0 0 0 115 1.76 1.77 0.759 110 3.69 3.54 1.58 105 5.79 5.30 2.47

Figure 5.7 Laser Twin sensor measuring diminishing diameters Chapter 5. Precision Turning Tests with Piezoelectric Actuator 102

Phase III - Holding Position: After the CNC cut, the NC program was paused to measure the workpiece with the LTS sensor. At this point of the process, the precision actuator can be used to perform cutting operations. The zoom of this section is seen in Figure 5.8.

Laser Twin Sensor position Info Vs Time Workpiece Measurement

11 12 13 Time (s)

Figure 5.8 Phase in Holding Position

Phase IV - Depart: The depart part of the cut monitors the motion of the tool from the work- piece seen in the Figure 5.9. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 103

Laser Twin Sensor position Info Vs Time Workpiece Measurement

-800

-900

-1000

-1100

E

C -1200 g '5o5 •1300

-1400

-1500

-1600

16.5 17 17.5 18 18.5 19.5 Time (s)

Figure 5.9 Phase IV of Operation - Retraction from the cut

It is interesting to also look at the tool position from the tool actuator's position sensor (LNS). The Laser Nano Sensor is the tool actuator's position sensor and it gives an accurate representa• tion of tool position. For a typical cutting test, the following tool position (Figure 5.10) was logged: Chapter 5. Precision Turning Tests with Piezoelectric Actuator 104

Laser Sensor position Info Vs Time -18 F—i 1 1 1 1 1 1 1 1 1 r

-18.5 -

-19- I

-21.5 -

i i i i i i i i i i 1 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Time (s)

Figure 5.10 Tool Position during cutting.

The actuator is commanded to a -20pm displacement. Figure 5.10 shows the tool position during the cutting operation. The actuator is held in position to within 1pm during the cut. 5.3.1. Disturbance Rejection

Disturbances on the actuator must be compensated for by the controller for the actuator. The types of disturbances that are observed by the actuator are mainly due to external cutting forces from the cutting zone and fluctuations in cutting forces from the previous pass of the cutting tool. For example, when the tool approaches the workpiece for the first time, Phase I, the eccentricity from the mounting acts as a periodic disturbance on the actuator at the control frequency. As dis• cussed in Chapter 4, the disturbance rejection is best suited for lower frequency disturbances that are modeled by cutting forces (Eq. 5.3) during turning. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 105

The dwell period of the precision turning operation is a critical section of the process. In order to reduce the amount of disturbances on the actuator, the disturbance during the dwell should be known. Two methods to reduce the disturbances on the tool/actuator during the dwell period were experimented with based on initial tests with the actuator. Figure 5.11 shows the cut• ting procedure when no disturbance compensation was done.

Laser Twin Sensor position Info Vs Time fa a Cutting Run

1000 Workpiece Measurement For Full Cutting Run

500 July 22,1998 tests Pete Placement Controller Cutting Speed 360 nVrrin 0 .1 -500 1 Q. -1000

-1500

-2000 .0 2 4 6 8 10 12 14 16 18 Time(s)

Laser Twin Sensor position Into Vs Time tor a Cutting Run Laser Nano Sensor position Into Vs Time

•950 Y

1. -1000

| -1050

9 10 11 12 13 Time (s) Workpiece Measurement Actuator Position Blowup of the Dwell Time During the Dwell Time

Figure 5.11 Workpiece Measurement - No Compensation Chapter 5. Precision Turning Tests with Piezoelectric Actuator 106

These initial tests were performed with a basic pole placement controller (See parameters in Section 4.5.1.1). Continuing tests were performed with the state feedback controller with distur• bance model. Two additional actions were taken in addition to the cutting force disturbance model. These were: 1) To reduce the spindle speed after the cut. 2) To retract the actuator a small amount after the cutting phase. 5.3.1.1 Speed Reduction Reducing the spindle speed during the dwell phase of the process effectively reduces the fre• quency of any workpiece surface waves that may excite the actuator. Figure 5.12 shows the results of reducing the spindle speed by -50%. Laser Twin Sensor position Info Vs Time State Feedback Controller

200 h

oh

-200 July 28, 1998 Tests Cutting Speed 240 m/min -400 Spindle Speed Reduction to 300 RPM E

§ -600

-800

-1000

-1200

-1400

0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 5.12 LTS Sensor with Spindle Speed Reduction before Dwell Section Chapter 5. Precision Turning Tests with Piezoelectric Actuator 107

This experiment was performed using the state feedback controller with disturbance observer

(See parameters from Section 4.5.1.3). The cutting conditions were as follows: Depth of cut 50

\im per revolution, cutting speed 240m/min, and a width of cut of 2 mm. The reduction of the spindle speed shows a significant improvement in the process.

5.3.1.2 Retract Tool/Actuator

Another method to reduce the problem with actuator run away in the dwell time is to back the actuator or tool turret away from the work piece. In the following experiment, the tool turret was moved away from the work pieces by 25 \xm. Figure 5.13 shows the results of this method.

Laser Twin Sensor position Info Vs Time Workpiece Measurement

500 h

July 30, 1998 Tests State Feedback w/ disturbance Observer Cutting Speed 240 m/min

-1000

-1500

2 4 6 8 10 12 14 16 18 Time (s)

Figure 5.13 LTS Sensor Feedback Using Micro Motion Method Chapter 5. Precision Turning Tests with Piezoelectric Actuator 108

5.4. Eccentricity elimination One of the major errors that can be introduced in turning operations is due the eccentricity of the spindle motion and the workpiece. The integration of sensors which can measure the rotating shaft at several location can be used to determine the error and the tool/actuator can consequently be used to compensate for these errors. The laser twin sensor can be used after the cut to deter• mine the eccentricity and provide the commands to the micro-actuator to attempt to remove the eccentricity. As seen in Figure 5.5, the eccentricity of a part the workpiece is fairly large due in large part to the mounting of the workpiece disk in the lathe. Much of that eccentricity can be removed with the first cuts on the disk. Figure 5.14 shows the eccentricity left on the disk after a typical cutting run.

Laser Twin Sensor position Info Vs Time State Feedback Controller

Figure 5.14 Eccentricity After Standard Cut - LTS Sensor Chapter 5. Precision Turning Tests with Piezoelectric Actuator 109

The peak to peak eccentricity is approximately 20 p.m. The sources of the eccentricity error include any errors in the axis drives of the lathe. In order to eliminate this effect the actuator con• trol was used to cut a small depth of cut during the dwell portion of the cut. This should eliminate the eccentricity error introduced by the axis drives. Section 5.5. demonstrates this. 5.5. Turning for Precision Diameters

Several cutting tests were performed in which P20 mold steel disks were cut with the actua• tor. The process of integrating the high precision sensors with the advanced control to produce parts was investigated. The externally mounted Laser Twin Sensor was used to measure the part before and after the cuts. One of the main considerations in using the precision tool/actuator is the effect on surface finish with such small depth of cut in precision turning operations. During the "dwell" portion of the precision turning process, a precision diameter cut was performed. The actuator was commanded to move 20 \im and the results are outlined in Figure 5.15 and Figure 5.16. A standard 5° Positive rake carbide tool was used for the cutting. The work piece material was a P20 mould steel disk. Tool Feedback and Setpoint Vs Time

10 Feedback 8 6 Note A 4

E 2

I ° Setpoint s. -2 -4

-6h -8 -10

_i i i i i i i i i_ 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 Time (s)

Figure 5.15 Actuator Position For Precision Diameter Cut, measured with the LNS sensor Chapter 5. Precision Turning Tests with Piezoelectric Actuator 110

Figure 5.15 shows the command and feedback position of the actuator. The state feedback servo controller with disturbance observer was used in this control test. The controller parameters are given in Section 4.5.1.3 on page 80. The control input command to the actuator was per• formed via a user input button in the IMM software. Note A on Figure 5.15 shows a glitch in the input command. This is a possible error in the IMM software interface. The glitch was noticed again when the command was change to retract the actuator from the workpiece.

Laser Twin Sensor position Info Vs Time (Set 1)

P-P distance = 15.4 um

Time (s)

Figure 5.16 Measured Eccentricity After Cut, measured with LTS sensor

The measured eccentricity of 15 \im peak to peak is measured. This is comparable to the sur• face left by the drive axis cut. The error in motion of the actuator is less than 1 \im, therefore it is postulated that the addition eccentricity error is due to the spindle. Because the depths of cut are so small, a careful analysis of the cutting of small depths of cuts is require, but is beyond the scope of this thesis. Chapter 5. Precision Turning Tests with Piezoelectric Actuator 111

Figure 5.14 indicates that the piezoactuators is capable of achieving a very high positioning accuracy during cutting tests (within 2 pra). The large error showing in Figure 5.16 must be due to other factors. The error is due mainly to the LTS sensor. The LTS sensor requires a smooth sur• face to get correct reflective feedback of the laser sources. If the surface finish is not to a high quality, error in the LTS sensor data will occur. This fact is not discussed in the product manual, but off lathe tests in the MAL laboratory have shown that a polished surface is required to get the most accurate data. 5.6. Conclusions

The use of the piezoelectric tool translator in conjunction with on-line sensors has been shown to improve the accuracy of the machining process. In addition, by modeling the cutting forces and having an accurate model of the actuator, it has been shown that advanced control tech• niques can be used to enhance the precision turning operation. Care must be taken in the precision turning process to ensure that the actuator is not excited by surface waves left on the workpiece after the standard cut. By the introduction of more on-line sensors on the turning machine and careful process control, enhancements to the turning process can be achieved. Chapter 6

Conclusions and Future Work

6.1. Conclusions

Mathematical modeling, control and testing of a piezoelectric tool actuator for ultra precision turning of shafts have been presented.

The piezoelectric actuator system has the following sub-components: Piezoelectric element and its amplifier, mechanical housing used to amplify the piezoelectric motions, cutting tool, pre• cision displacement sensors integrated to the actuator and a digital control law running on a high performance Digital Signal Processing board housed on a PC bus. It is assumed that the conven• tional CNC lathe delivers the gross motion of the tool, while the final but small ultra-precision motion is delivered by the piezoelectric tool actuator in one direction for finish turning of shafts.

The mathematical model of the entire system was constructed both as an independent module and when mounted on the turret of the CNC lathe. The main non-linearity in the amplifier is the saturation when high frequency displacements may require current beyond the capacity of the amplifier. The non-linearity of piezoelectric element is mainly its hysteresis. The mathematical model of the hysteresis is developed based on the physical properties of the piezoelectric element.

The simulations are compared with the experimental observations, which validate the proposed mathematical model of the piezoelectric element hysteresis. It is argued that the full hysteresis model is difficult to integrate to the control model due to its position and velocity dependent vari• ation. Instead, the hysteresis is considered as a time delay in the transfer function of the overall system.

The dynamics of the overall actuator system is dominated by the structural mode of the mechanical housing. The tool carriage acts like a mass attached to the piezoelectric element, and

112 Chapter 6. Conclusions and Future Work 113 the damping is contributed by the materials and strain feedback within the amplifier. The first nat• ural mode of the overall system is about 610Hz, therefore the actuator can be said to have about 500Hz bandwidth. The maximum displacement of the tool is about 60 \im up to about 100Hz, but drops off approximately 30% at 500Hz due to saturation of the piezoelectric amplifier. This is mainly due to the fact that the piezoelectric element is largely a capacitive load on the amplifier unit.

A Zero Phase Error Tracking Controller (ZPETC) compensates the first natural mode of the actuator (610Hz) for non-circular machining. However, the ZPETC's performance is heavily dependent on the cutting force disturbances, which caused transient oscillations during actual machining. PID and Pole Placement controllers also provided sluggish response if tuned to have large damping, and they were not capable of compensating cutting force disturbances quickly. A feed forward state controller with cutting force disturbance observer-compensator is shown to deliver most accurate tool positioning for piezoelectric actuators. A tool accuracy of < 1 \im was achieved. Cutting tests with the controller were performed and the disturbance observer operated properly. This method for disturbance observation allows the control system designer to base the controller about any disturbance model and to compensate for such disturbances in the control law. An on-line sensor was used in conjunction with the tool actuator to measure the workpiece. The sensor was evaluated and used in a turing application. The Work piece measurement system employed a laser range sensor to measure the worpiece eccentricity. This range sensor was found to be very sensitive to surface finish of the workpiece. 6.2. Recommendations for Future Research The tool/actuator used in this work was initially developed for use in chatter vibration sup• pression. Although this actuator is suitable for proof of concept a more refined actuator with increased structural rigidityi s needed for future development. Chapter 6. Conclusions and Future Work 114

The actuator natural frequency (-610 Hz) is too low for cutting applications where this fre• quency can be persistently excited. A wider bandwidth for the piezoelectric element is required to actively damp out natural modes within lOOOHz-frequency band. In order to support this, a higher power current amplifier would be required. This would increase the linear bandwidth of the actu• ator system. The frequency bandwidth of the controller was increased to close to this frequency, but can not be increased because of limitations of the power amplifier.

A more detailed analysis of the cutting conditions for precision turning needs to be investi• gated. As the depths of curt are reduced, the edge forces on the cutting tool dominate the cutting forces. The mechanics of precision turning operations needs to be investigated further. Development of on-line sensors to enhance the precision turning operation is required. Development of spindle measurement sensors for eccentric turning applications and more robust workpiece measurement devices for precision turning are required. The current actuator design occupies two tool locations in the turret of the lathe. The method for mounting the actuator needs to be improved to provide a more stable mounting of the tool/ actuator. A new actuator that fits in a smaller space and has a more solid mounting would be ben• eficial for extending this work to industry. Bibliography

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[8] Diana Zankowsky. "High-Performance Tools Position Accurately", Laser Focus World, pp. 135-138, 1997. [9] "CSO HC-250 Interferometer Manual", CIE Des Senseurs Optique, Grenoble, France, 1997. [10] "Physik Instrumene (PI) Catalog", Physik Instrumente, Germany, 1997. [II] N. Aurelle, D. Guyomar, L. Eyraud. "Power Limitations in Acoustic Transducer Due to PZT Material", Paper Presented at the Third ICIM/ECSSM '96, pp. 592-597, 1996.

[12] S. Blochwitz, R. Habel, M. Diestelhorst, H. Beige. "Application of Nonlinear Dynamics in the Study of Ferroelectric Materials", Paper Presented at the Third ICIM/ECSSM 1996, pp. 586-591,1996.

[13] Daniel Enke. "Piezoelectric Microactuator Including Compliant Mechanism", Proceedings - ASME 1996 Annual Meeting, Vol. 14, pp. 306-309, 1996.

115 Bibliography 116

[14] David Henderson, Dana Jensen, and Peter Piccirilli. "Recent Advancements in Piezoelec• tric Stepping Motors", Proceedings - ASPE 1996 Annual Meeting, Vol. 14, pp. 318-321, 1996. [15] "IMM Version 1.2 Real-Time Signal Processing Software User's Guide, Script Language Guide, Programmer's Guide", Manufacturing Automation Laboratories Inc., 1996.

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