Dissertation 5 : Zamberi Jamaludin

eri

DISTURBANCE COMPENSATION FOR MACHINE TOOLS WITH LINEAR MOTOR DRIVES

Promotoren: Prof. Dr. ir. H. Van Brussel Prof. Dr. ir. J. Swevers

Proefschrift voorgedragen tot het bekomen van de graad van Doctor in de Ingenieurswetenschappen door Zamberi JAMALUDIN

2008D09

September 2008

DISTURBANCE COMPENSATION FOR MACHINE TOOLS WITH LINEAR MOTOR DRIVES

Jury: Prof. Dr. ir. A. Haegemans, voorzitter Prof. Dr. ir. H. Van Brussel, promotor Prof. Dr. ir. J. Swevers, promotor Prof. Dr. ir. J. De Schutter Prof. Dr. ir. H. Ramon Prof. Dr. ir. P. Sas Prof. Dr. ir. G. Pritschow (University of Stuttgart)

2008D09

September 2008

Proefschrift voorgedragen tot het bekomen van de graad van Doctor in de Ingenieurswetenschappen door Zamberi JAMALUDIN

© Katholieke Universiteit Leuven Faculteit Ingenieurswetenschappen Arenbergkasteel, B-3001 Heverlee (Leuven), Belgium Alle rechten voorbehouden. Niets uit deze uitgave mag worden verveelvoudigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaandelijke schriftelijke toestemming van de uitgever. All rights reserved. No part of this publication may be reproduced in any form, by print, photoprint, microfilm or any other means without written permission from the publisher.

D/2008/7515/85 ISBN 978-90-5682-975-9 U.D.C. 681.5

My Parents My Wife

The beginning of knowledge is the intention, then listening, then understanding, then action, then preservation, and then spreading it.

Acknowledgements This journey begins in 2003. The common question at that time is “Why Belgium?” This is apparent especially because of the weak link in the educational relationship between Malaysia and Belgium. A nice and encouraging introduction by Dr. Indra Tanaya (a graduate of K.U.Leuven and who was once my colleague in Malacca) about PMA, its strong lists of academicians and researchers with dedicated research activities and the beautiful and peaceful city of Leuven have motivate me enough to begin this new chapter of my life here in PMA, Leuven. I am grateful that I have made this important choice because after these five wonderful and challenging years, both PMA and Leuven have never failed to contribute positively to my academic and personal development. There are two very important persons that have been very influential to my academic development here in Leuven. First and foremost, I would like to express my deepest gratitude to Prof. Van Brussel for accepting me into the PMA community and for the trust that he has put on me in realising this work. It is a great honour and pleasure to be able to work with him, a person of great mechatronics background, experience, success, ideas, and stature. It is certainly be a great challenge to me to emulate your success and to be as good as you are. Your kind advice, time, attention, and dedication towards realising this work are greatly appreciated. Finally, I would like to extend my deepest appreciation for your kind attentions on the well-being of myself (visiting me in the hospital on just my third week in Belgium) and my wife and your warm hospitality towards my parents. Secondly, I would like to express my special gratitude to Prof. Swevers – the control specialist, for nurturing me into the challenging world of control. Your knowledge in advanced control theory certainly amazed me and I wish that there is somebody of your calibre in Malacca (there is none at the moment) so that I can keep learning from the best mind in this field. Your guidance, attention, and time are greatly appreciated. I would like to thank you, your wife and family for being such a great and wonderful host during your yearly summer bbq sessions. I would like to extend my deepest gratitude to the thesis jury committee members; Prof. Dr ir. Herman Ramon, Prof. Dr ir. Joris De Schutter, Prof. Dr ir. Paul Sas, and Prof Dr ir. Gunther Pritshow of the Stuttgart University. Thank you for your valuable time, suggestions, comments, and critical i

Acknowledgements

thoughts that have enable me to improve the quality of this manuscript. I would also like to express my thanks to Prof. Dr Ann Haegemans for being the chairperson. I admired the great knowledge and experience of Prof. Vanherck, a person whom I regard as a walking technical library. Your help, assistance, teachings, and explanation on many technical issues have helped me tremendously throughout my work in PMA. Thank you for your dreams (I will always remember you saying this when there are problems with the xy table and the Ferraris sensor). I would like to extend my special thanks to the administrators of PMA - Luc Haine, Lieve Notre, Karin Dewit, Carine Coosemans, and Ann Letelier for their support, assistance and warm hearts. I am also grateful to all the technical personnel at the workshop, the electronic and the IT department for making my life easier at the workshop. My special thanks to Dirk Bastiaensen, Eddy Smets, Paul Van Cauwenbergh, Luc De Simpelaere, Bertram Van Soom, Jean Pierre Merkcx, Ronny Moreas, and Jan Thielemans. My life here at PMA is made easier by Dr T. Tjahjowidodo who has given me a nice introduction to life at PMA, its surrounding and its research environment. I still remember the confusions and funs that we had during my early days in PMA trying to understand each other in Bahasa Melayu (some similar Malaysian and Indonesian words can have totally different meanings). I would also like to express my appreciation for your time and assistance in explaining the world of friction to me. The nice explanations have certainly helped in my work on friction compensation. It is a great pleasure to be in the environment of the control group people and of my office colleagues – Bram, Christophe, David Vaes, Bart Paijmans, Kris Smolders, Goele, Jan De Caigny, Dimitri, Lieboud, Diederick, Bert Stalleart, Myriam, Leopoldo, Marnix and Maarten. Thank you for the company, assistance, and fruitful discussions. I would like to express my special thanks to Goele Pipeleers for assisting me in the design of the repetitive controller. The mechatronics group meetings have exposed me to many wonderful people. It is a pleasure to be in the company of these distinguished colleagues of various backgrounds and origins - Prof. Farid Al-Bender, Wim Symens, Brecht, Hsiao-Wei.Tang, Emmanuel Vander Poorten, Wim van de ii

Acknowledgements

Vijver, Tri, Gorka, Mohamed El-Said (also Neny and Zaharah for your kindness and warm hearts), Maira, Thierry, Kris, Pauwel, Bert, and Agusmian. Life in Belgium for a Malaysian can be a lonely experience (there are not many of us here). However, since the last two years, the students community has grown with the arrival of other Malaysian students and families (although there are all not in Leuven). I would like to extend my sincere appreciation to: Zaini and Ros, Saifullah, Yeong and family, Helmi and Natrah, Am and Nurul, Dr Razak and family, abang Zam, kak Zu and family, and Kelvin and family. Thank you for the friendship and hospitalities. I would also like to thank Akmal and Zura for their warm heart and hospitality towards me and my wife and for all the trips that you have taken us to. Finally, to the Embassy of Malaysia in Brussels, thank you for all the nice receptions and your kind treatment of myself and my wife for all the five years that we are in Belgium. I am also grateful to have known many kind and warm hearted Indonesian friends and families especially Pak Oemar, Mbak Leen and Esti, Gandjar and family, Tegoeh, Ira and family, Singgih, Uly and the kids, Arie and Sarah, and Freddy, Novita and family. This work is made possible with the financial supports of the Ministry of Higher Education Malaysia (SLAB) and the University Technical of Malaysia-Malacca (UTeM). These financial supports are greatly appreciated and indebted. I also wish to express my sincere gratitude towards the International Student Office of K.U.Leuven for the financial support that I have received during the final three months of my stay in Leuven. This financial support has enabled me to complete my study here uninterruptedly and successfully. Finally and most importantly, I wish to express my deepest gratitude to both my parents, Hj. Jamaludin and Hjh Khatijah Bee, my brother Saifullah, and the families in Kedah and Singapore for their prayers, loves, cares, and support. I am greatly indebted to them and may God repay all their deeds and sacrifices. To my wife Zila, there are no words to describe how grateful I am for your company during both the easy and the difficult times of this journey. I salute your patient, dedication, love, and care that have brought happiness and serenity to our life. Zamberi Jamaludin Leuven, Sept. 2008. iii

iv

Abstract Machining processes have evolved significantly over time in order to adapt to the increasing demand for speed, accuracy, and efficiency. This evolution or paradigm shift has created new and highly critical challenges. This thesis aims at addressing some of these issues, namely the compensation of the effect of friction and cutting forces on the accuracy of the machining process. Issues regarding friction effects in machining process have been studied extensively in the past and various techniques and ideas have been proposed and validated. Simple linear feedback control techniques such as PI, PID, or cascade control alone are insufficient to compensate the nonlinear friction behaviour. In this thesis compensating elements are proposed, developed and validated that can be added to existing feedback controllers in order to improve accuracy. Friction-model-based and friction-model-free approaches are considered to supplement the cascade P/PI position controller. The compensation performance is measured based on the magnitude of the “quadrant glitch” – a product of highly nonlinear complex friction behaviour near zero velocity or motion reversal identified by the appearance of “spikes” at each quadrant of a circle. In this thesis, the recently developed Generalized Maxwell-slip (GMS) friction model is used as a feedforward element in combination with the well-known and widely applied inversemodel-based disturbance observer for friction compensation on a linear drive based xy feed table of a high-speed milling machine. This combined approach almost completely compensates all friction effect. Besides friction forces, the effect of cutting forces on machining accuracy is significant. Several techniques described in literature are studied and their applicability to compensate cutting forces in machining process is evaluated. First, the application of the inverse-model-based disturbance observer is further extended to cutting force compensation. However, its performance is critically influenced by the limited bandwidth of a low pass filter often referred to as the Q-filter that is necessary to preserve the system stability. Second, cutting forces are estimated from the force balance acting on the drive using a Ferraris relative acceleration sensor measurement. The bandwidth is again restricted by a stability preserving low-pass filter, similar as for the inverse-model based disturbance observer. v

Abstract

Finally, a method that is renown for its excellent compensation of periodic disturbance signals is applied, namely, the repetitive controller (RC). A repetitive controller is developed for the considered linear drive based xy feed table. To validate the performance of this RC, an actual cutting process is performed on the test setup. It is shown that the developed RC is able to compensate almost completely the tracking errors introduced by the cutting forces. The repetitive controller, when combined with the previous friction compensation elements such as the GMS friction model feedfoward and the disturbance observer, almost completely removed the cutting forces during an actual cutting process. This thesis has successfully demonstrated that the tracking performance of a machine tool can be increased significantly by adding dedicated compensation elements to the simple and widely used cascade P/PI position controller. However, further studies are desired to include adaptive measures in both friction and cutting forces compensation using the advanced GMS friction model and the RC. This will ensure a robust friction compensation approach to changing friction behaviour and characteristics over time due to the influence of lubrication, heating and etc. An adaptive RC will compensate against changes in the cutting conditions, for example, changes in the spindle speed, tools diameter, tracking speed, and etc.

vi

Beknopte Samenvatting De technologische evolutie van conventionele materiaalbewerkingsmachines wordt gedreven door de vraag naar steeds hogere snelheden, nauwkeurigheden en efficiëntie. Deze evolutie creëert nieuwe uitdagingen. Deze thesis richt zich op enkele van deze uitdagingen, namelijk de actieve compensatie van de effecten van wrijving en snijkrachten op de nauwkeurigheid van het bewerkingsproces. Het effect van wrijving op de nauwkeurigheid van bewerkingsprocessen werd in het verleden reeds uitgebreid onderzocht. Verschillende technieken om het effect van wrijving te compenseren werden ontwikkeld en experimenteel gevalideerd. Eenvoudige lineaire regelaars op basis van terugkoppeling, zoals de PI-, PID-, en de cascaderegelaars voldoen niet om het effect van het niet-lineaire wrijvingsgedrag te compenseren. Deze thesis stelt een aantal wrijvingscompensatietechnieken voor die kunnen toegevoegd worden aan bestaande (lineaire) sturingen. Zowel technieken gebaseerd op een wrijvingsmodel als technieken die geen wrijvingsmodel vereisen, worden beschouwd. De performantie van deze technieken wordt gemeten aan de hand van de grootte van de “quadrant glitch”, het resultaat van het sterk niet-lineair gedrag van wrijving dat optreedt bij omkering van de bewegingrichting en bij zeer lage bewegingssnelheden, en dat zich uit als spikes in de volgfout. De compensatietechnieken die werden onderzocht worden ook experimenteel gevalideerd op een met lineaire motoren aangedreven XY voedingstafel van een hoge-snelheidsfreesmachine. De combinatie van voorwaartse koppeling aan de hand van het recent ontwikkelde “Generalized Maxwell-slip (GMS)” wrijvingsmodel en de gekende en vaak toegepaste storingsschatter op basis van een invers systeemmodel resulteert in de grootste verbetering van de volgnauwkeurigheid: wrijvingseffecten bij elke kwadrant van een circulaire baan worden bijna volledig gecompenseerd. Naast wrijving kunnen ook de snijkrachten een belangrijke invloed hebben op de bewerkingsnauwkeurigheid van de machine. Verschillende compensatietechnieken beschreven in de literatuur worden bestudeerd en hun toepasbaarheid om snijkrachten te compenseren wordt onderzocht en vii

Beknopte Samenvetting

experimenteel gevalideerd. Eerst wordt de storingsschatter op basis van een invers systeemmodel verder uitgebreid voor snijkrachtcompensatie. De performantie van deze storingsschatter wordt kritisch beïnvloed door de beperkte bandbreedte van de laagdoorlaatfilter (Q-filter) die aanwezig is in deze schatter om de systeemstabiliteit te vrijwaren. Daarnaast wordt een snijkrachtschatter bestudeerd die gebruik maakt van metingen van de relatieve versnelling van de lineaire motor op basis van een Ferraris sensor. Ook voor deze schatter gelden dezelfde beperkingen met betrekking tot bandbreedte en performantie. Tenslotte wordt de repetitieve regelaar bestudeerd. Deze regelaar richt zich uitsluitend op periodische storingen waarvan de periode gekend is of kan geschat worden. Deze regelaar maakt gebruik van niet-causale filters waardoor een aanzienlijk hogere bandbreedte kan bereikt worden. Een repetitieve regelaar wordt ontworpen en toegepast op de beschouwde testopstelling. Om de performantie van de ontwikkelde repetitieve regelaar te valideren wordt een freesbewerking uitgevoerd op de testopstelling. De testen tonen aan dat de ontworpen repetitieve regelaar in staat is om volgfout ten gevolge van het freesproces bijna volledig te elimineren. Deze repetitieve regelaar wordt toegepast in combinatie met de ontwikkelde wrijvingscompensatie op basis van het GMS model en de storingsonderdrukker zodat ook wrijving tijdens deze bewerking gecompenseerd wordt. De overblijvende volgfout is verwaarloosbaar. Deze thesis heeft met succes aangetoond dat de volgnauwkeurigheid van een werktuigmachine significant kan verbeterd worden door gerichte compensatietechnieken toe te voegen aan de eenvoudige klassieke cascade P/PI positiesturing van de machine. Echter, verder onderzoek is vereist om deze compensatie adaptief te maken om zo efficiënt te kunnen inspelen op veranderingen in de tijd van de wrijvingskarakteristieken, die beïnvloed worden door smering, temperatuur, en slijtage, en van de snijkarakteristieken, die o.a. beïnvloed worden door de spilsnelheid en de freesdiameter.

viii

Symbols and Abbreviations Symbols Control: d(t) ~ d (t ) ep(t) ev(t) f γp,∆ γnp kf kp ki kv

n(t) u(t) vref z(t) zref(t) ωo ζ F G Ĝ Gm Gn Gm’ M N(s) S T T0 Td Vest(s)

disturbance force signal estimated disturbance force

position tracking error signal velocity tracking error signal frequency robust periodic performance index Non-periodic performance index motor force constant proportional gain controller in velocity loop integral gain controller in velocity loop velocity gain factor, or the Proportional gain controller in cascade position loop noise control command signal desired reference velocity Output position signal Desired reference position signal Undamped natural frequency damping ratio Force acting on linear motor considered system FRF of the considered system system model transfer function nominal model system transfer function system model with GMS friction term mass of the sliding table notch filter sensitivity function complementary sensitivity function sampling period time delay estimated velocity signal

[N] [N]

[µm] [µm/s] [Hz] [N/V] [volt·s/µm] [volt/µm] [1/s] [volt] [µm/s] [µm] [µm] [rad/s] [N]

[kg]

[s] [s] [µm/s] ix

Symbols & Abbreviations

Friction:

δ σ αi ki s(v) Ff Fc Fs Vs Wi

Stribeck function shape factor viscous friction force elementary normalized friction force elementary spring constant of the Maxwell slip blocks element Stribeck function total friction force Coulomb friction force static friction force Stribeck velocity maximum elementary friction force

Abbreviations rms rpm DOB FRF FF GMS H∞ I/O PI PID RC RHP SISO SMC

x

root mean square revolution per minute disturbance observer frequency response function feedforward Generalized Maxwell-slip H-infinity input-output proportional plus integrator proportional plus integrator plus differentiator repetitive controller right half plane single input single output sliding mode control

[N·s] [N] [N/m] [N] [N] [N] [µm/s] [N]

Table of Contents Acknowledgements

i

Abstract

v

Beknopte Samenvetting

vii

Symbols and Abbreviations

ix

Table of Contents

xi

List of Figures

xv

List of Tables 1

2

xxiii

Introduction

1

1.1

Motivation ……………………………………………...

1

1.2

State of the Art on Motion Control …………………….

3

1.2.1

Mechanical Drive Systems …………………….

3

1.2.2

Disturbance Forces and Compensation Methods

6

1.3

Scope, Objective and Approaches ………….………….

9

1.4

Contributions …………………………………………...

10

1.5

Outlines ………………………………………………...

11

Classical Motion Control

13

2.1

Introduction …………………………………………….

13

2.2

High-Speed XY Milling Machine ……………………...

14

2.3

System Identification …………………………………..

17

2.4

Cascade Control Structure and Analysis ……………..

20

2.4.1

Cascade Controller Structure and Configuration

20

2.4.2

Analysis of the Closed Loops Behaviour with Proportional (P) Velocity Loop ………………..

22

xi

Table of Contents

2.4.3

2.5

2.6 2.6 3

25

Design and Validation of Cascade P/PI Controller Based on Measured FRFs ……………………….….….

27

2.5.1

Design and Analysis of the Velocity Loop...…..

27

2.5.2

Design and Analysis of the Position Loop ….…

34

2.5.3

Tracking Performance Numerical Validation …

39

2.5.4

Cascade P/PI with Feedforward …………...…..

40

Correlation between Velocity and Position Closed Loop Characteristics ………………………………………….

41

Summary ……………………………………………….

44

Disturbance Forces in Servo Drives System

45

3.1

Introduction ………………………………………….…

45

3.2

Friction Characterization and Model Structures ……….

45

3.3

The Dahl Model ………………………………………..

46

3.4

The LuGre Model ………………………………………

47

3.5

Static Friction Model ……………………………….….

48

3.2.1

Model Structure ………………………….….....

48

3.2.2

Identification of Static Friction Model ………...

49

Generalized Maxwell-slip Model (GMS) ……………...

50

3.2.1

Model Structure ………………………………..

50

3.2.2

Identification of GMS friction model ………….

52

Cutting Forces ………………………………………….

56

3.7.1

Cutting Force Measurements and Analysis …....

56

3.7.2

Artificial Cutting Force ……………….……….

60

Summary ……………………………………………….

62

3.6

3.7

3.8

xii

Analysis of the Closed Loops Behaviour with Proportional (P) and Integrator (I) Velocity Loop …………………….……………………...

Table of Contents

4

Friction Forces Compensation

63

4.1

Introduction …………………………………………….

63

4.2

Friction Model-Based Feedforward ………………...….

64

4.2.1

4.3

4.4 5

System Transfer Function with a GMS Friction Term for Friction Simulation ………………

64

4.2.2

Friction Feedforward: Numerical Results ……

67

4.2.3

Friction Feedforward: Experimental Results

69

Inverse-Model Based Disturbance Observer ……...…...

73

4.3.1

Q-filter Design and Stability Analysis ……...…

75

4.3.2

Loops Characteristic with Disturbance Observer

78

4.3.3

Numerical Validations…………………….........

87

4.3.4

Experimental Validations………...…………….

90

Summary ……………………………………………….

94

Cutting Force Compensation

97

5.1

Introduction ………………………………………….…

97

5.2

Inverse-Model-Based Disturbance Observer …………..

98

5.2.1

Numerical Validations…...……………………..

98

5.2.2

Experimental Validations...…….……...……….

100

Explicit Estimation of Cutting Force using Ferraris Relative Acceleration Sensor Measurements …………..

103

5.3.1

The Ferraris Principle ……………….…………

104

5.3.2

Cutting Force Estimator ……………………….

105

5.3.3

Experimental Validations...…………………….

106

Repetitive Controller …………………………………...

112

5.4.1

Design Structure of a Repetitive Controller……

112

5.4.2

Experimental Validations...…………………….

116

5.3

5.4

xiii

Table of Contents

5.5 5.6

Cutting Force Compensation during Actual Cutting Process ………………………………………………....

120

Summary ……………………….………………………

123

6

Conclusions and Future Studies

125

7

Bibliography

129

8

Curriculum Vitae

135

9

List of Publications

137

10

Appendices

139

A

System Dynamic Analysis

139

B

Calibration of Ferraris Sensor

140

C

Cutting Force Estimation from Ferraris Acceleration Sensor Measurement

143

General State Force Observer Design

147

D

xiv

List of Figures Chapter 1 1.1

Quadrant glitches in circular test ………………………...

2

1.2

Electromechanical ball-screw drive structure……………

4

1.3

Structure of an iron-core linear motor……………………

5

1.4

Schematic diagram for friction compensation techniques.

6

A linear-driven xy feed table of a high-speed milling machine ………………………………………………….

14

2.2

Schematic diagram of the xy table ….…………………..

15

2.3

Motion controller structure of a xy feed table with three linear drives for high speed milling application………….

17

2.4

FRFs measurement of the x and y axes…………………..

18

2.5

X-axis: FRF measurement and proposed model…………

19

2.6

Y-axis: FRF measurement and proposed model…………

19

2.7

General scheme of a cascade control structure…………..

20

2.8

Schematic diagram of the open loop system for force constant estimation……………………………………….

21

An ideal cascade control structure for a linear motor position control…………………………………………...

22

Frequency domain scheme of the cascade P/PI controller for control of a linear motor drive………………………..

28

2.11

Bode diagram of Vest(s)…………………………………..

28

2.12

X-axis: Bode diagram of a notch filter N(s) for frequency between 320 Hz and 340 Hz……………………………..

29

X-axis: Theoretical bode of the velocity open loop transfer function based on measured FRF of the system...

31

Chapter 2 2.1

2.9 2.10

2.13

xv

List of Figures

2.14

Y-axis: X-axis: Theoretical bode of the velocity open loop transfer function based on measured FRF of the system…………………………………………………….

31

Nyquist plots of the x-axis (left) and the y-axis (right) velocity open loop transfer functions based on measured FRFs of the system……………………………………….

32

Sensitivity function of x (left) and y (right) axes for the velocity loop……………………………………………...

33

Velocity closed loop transfer function for (a) x-axis and (b) y-axis ………………………………………………...

34

Position open loop transfer function for (a) x-axis and (b) y-axis based on measured FRF of the system ………...…

36

Nyquist plots of the x-axis (left) and the y-axis (right) of the position open loop transfer functions………………...

37

Sensitivity function of x (left) and y (right) axes for the position loop……………………………………………...

37

2.21

Position closed loop transfer functions for x and y axes…

38

2.22

Measured dynamic stiffness of x (left) and y (right) axes using cascade P/P and cascade P/PI controller…………..

39

Simulated tracking error of the x-axis (left) and the yaxis (right) for sinusoidal reference signal of an amplitude of 1mm and frequency of 10Hz………………

39

Position error transfer functions of the x-axis (left) and the y-axis (right)………………………………………….

40

Cascade P/PI with feedforward for improved tracking performance………………………………………………

40

Simulated tracking error of the x-axis (left) and the yaxis (right) for sinusoidal reference signal of amplitude 1mm and frequency of 10Hz……………………………..

41

2.27

Schematic diagram of a velocity closed loop…………….

42

2.28

Y-axis: Simulated speed closed loop based on measured system FRF ………............................................................

43

2.15

2.16 2.17 2.18 2.19 2.20

2.23

2.24 2.25 2.26

xvi

List of Figures

2.29

Measured FRF of the y-axis position closed loop with kv =150 s-1………………………………………………...

44

3.1

Pre-sliding and sliding friction regimes………………….

46

3.2

Friction components in static friction model (3.5)……….

48

3.3

Friction force from control command signals at constant velocity motion of 2.0 mm/s (y-axis)…………………….

49

Friction force-velocity mapping and the manually fitted static friction model using equation (3.5) …………….....

50

3.5

N-elementary configuration of the Maxwell-slip structure

51

3.6

Y-axis: Friction force and position for sinusoidal reference signal of 0.1Hz and amplitudes of (a) 15µm and (b) 450µm…………………………………………....

53

Y-axis: (a) virgin curve and (b) virgin curve with selected knots and slopes………………………………...

53

3.8

Y-axis GMS model parameters identification……………

54

3.9

X-axis: Friction force and position for sinusoidal reference signal of 0.1Hz and amplitudes of (a) 15µm and (b) 450µm……………………………………………

55

X-axis: (a) virgin curve (b) virgin curve with selected knots and slopes (right)…………………………………..

55

3.11

X-axis: GMS model parameters identification…………..

56

3.12

Cutting force in x and y direction for cutting depth 1mm..

57

3.13

Cutting force in x and y direction for cutting depth of 3mm ……………………………………………………..

58

Cutting force measurements on a work-piece using Kistler dynamometer force sensor along the x-axis……...

58

Spectral of cutting force along x and y direction, for cutting depth of 1mm…………………………………….

59

Spectral of cutting force along x and y direction, for cutting depth of 3mm…………………………………….

59

Chapter 3

3.4

3.7

3.10

3.14 3.15 3.16

xvii

List of Figures

3.17

X-axis: Comparison between the measured cutting force and the synthesized cutting force for 1mm depth of cut…

62

Friction compensation scheme using friction modelbased feedforward (kf is the force constant)……………...

64

System transfer function with a GMS friction model to characterize the complex friction behaviour……………..

65

System model parameters adjustment for system structure that includes the GMS friction term: (a) using the original model parameters and (b) using updated model parameters………………………………………...

66

Y-axis: Simulated position, velocity and tracking error for three different cases of friction compensation techniques (a) no friction feedforward, (b) Stribeck friction model feedforward, (c) GMS model feedforward.

67

Y-axis: Simulated tracking error using GMS model feedforward for (a) system model with delay and (b) with the delay removed.…………….…………………………

68

Y-axis: Simulated tracking error using (a) modified GMS model feedforward and (b) matched GMS model for system both without delay………………………………..

69

Y-axis: Measured position and tracking error for (a) no friction feedforward, (b) static friction model feedforward, (c) GMS feedforward …...…………………

70

X-axis: Measured quadrant glitches for maximum tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward…………...

71

Y-axis: Measured quadrant glitches for maximum tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward…………...

71

Measured contour and radial tracking error at tangential tracking velocity of 100mm/s: (a) no friction feedforward, (b) static friction model feedforward, (c) GMS model feedforward…………………………………

72

Chapter 4 4.1 4.2 4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

xviii

List of Figures

4.11

4.12 4.13 4.14 4.15 4.16 4.17

4.18

4.19

4.20 4.21

4.22

4.23

Measured contour and radial tracking errors for tangential tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward……

73

Block diagram of a system with an inverse-model- based disturbance observer……………………………………...

74

Equivalent block diagram of a system with an inverse model-based disturbance observer……………………….

74

Bandwidth limitation of the filter Q for (a) x-axis and (b) y-axis……………………………………………………..

77

Cascade P/PI position control with inverse model-based disturbance observer …......................................................

78

Cascade P/PI position control with equivalent block diagram of the inverse model-based disturbance observer

78

Velocity open loop transfer function for (a) x-axis and (b) y-axis for system with and without a disturbance observer…………………………………………………..

80

Velocity closed loop transfer functions for (a) x-axis and (b) y-axis for system with and without a disturbance observer…………………………………………………..

81

Effect of the disturbance observer on the sensitivity function of the velocity loop for x-axis (left) and y-axis (right) …………………………………………….………

82

Nyquist plots of the velocity loops with and without the disturbance observer……………………………………...

82

Position open loop transfer function for (a) x-axis and (b) y-axis for system with and without a disturbance observer…………………………………………………..

84

Position closed loop transfer functions for (a) x-axis and (b) y-axis for system with and without a disturbance observer…………………………………………………..

85

Effect of the disturbance observer on the sensitivity function of the position loop for the (a) x-axis and (b) yaxis …..…………………………………………………..

86

xix

List of Figures

4.24 4.25

4.26 4.27 4.28

4.29 4.30 4.31 4.32

Nyquist plots of the position loops with and without the disturbance observer……………………………………...

87

Simulated control scheme for friction compensation using friction model feedforward and a disturbance observer…………………………………………………..

88

Y-axis: Simulated position and tracking error for configuration (a), (b) and (c)……………………………..

89

X-axis: Measured position and tracking errors for configuration (a), (b) and (c)……………………………..

90

X-axis: Measured position and tracking errors for slower reference trajectory of 10mm/s for configuration (a) and (b)………………………………………………………...

91

Y-axis: Measured position and tracking errors for configuration (a), (b) and (c)……………………………..

92

Measured contours and radial errors for configuration (a), (b) and (c)……………………………………………

93

Circular tracking tests at tangential tracking velocity of 10mm/s…………………………………………………...

93

Position and radial tracking error for different friction compensation approaches………………………………...

94

MATLAB/Simulink diagram of a cascade P/PI controller and a disturbance observer with a sinusoidal based disturbance input signal…………………………………..

98

Y-axis: The effect of a disturbance observer on tracking errors for disturbance input frequencies of (i) 1 Hz, (ii) 58 Hz, and (iii) 100 Hz…………………………………...

99

Y-axis: Position errors for system with and without a disturbance observer……………………………………...

100

Cutting force compensation using inverse-model-based disturbance observer……………………………………...

101

Chapter 5 5.1

5.2

5.3 5.4

xx

List of Figures

5.5

Y-axis: Measured position errors with inverse modelbased disturbance observer for synthesized cutting forces disturbance with different harmonics component………..

101

Y-axis: Spectral analysis of the position errors without the disturbance observer (top) and with the disturbance observer (bottom) for disturbance force signals with different harmonic contents………………………………

102

Measured contours and radial errors for circular tests with (right) and without (left) the inverse model based disturbance observer ……………………………………..

103

Schematic diagram of the Ferraris principle and the actual sensor used in measurement………………………

104

Schematic diagram of a cascade P/PI position control with cutting force estimator (without feedback)…………

107

5.10

Reference and estimated cutting forces………………….

107

5.11

Schematic diagram of a cascade P/PI position control with cutting force estimator (with feedback)…………….

108

Position errors for different cutting force harmonic components using estimator……………………………..

108

Spectral analysis of the position errors for different cutting force harmonic components using estimator…….

109

Tracking errors for different cutting force harmonic contents for cases with and without the estimator……….

110

Spectral analysis of the tracking errors for different cutting force harmonic components for cases with and without the estimator……………………………………..

110

Cutting Forces Estimation Analysis (without reference trajectory)………………………………………………...

111

Standard RC as an add-on module to a closed loop control scheme ……………………………..……………

113

Standard memory loop with periodic signal generator with period T0[s] …………………………….…………...

113

Discrete time implementation of a typical RC…………...

114

5.6

5.7

5.8 5.9

5.12 5.13 5.14 5.15

5.16 5.17 5.18 5.19

xxi

List of Figures

Schematic diagram of (a) nth order RC, (b) 2nd order RC ……………………………………………………..

115

Y-axis: Bode plots of (a) the loop gain, (b) sensitivity function and (c) complementary sensitivity function of the second order RC ……………………………………..

116

A schematic diagram of a cascade P/PI controller with a RC module and friction compensation elements ...……..

117

Y-axis: Measured position tracking errors with and without the RC for different harmonic component of the cutting forces……………………………………………..

118

X-axis: Measured position tracking errors with and without the RC for different harmonic components of the cutting forces. …………………………………….……...

118

Circle tests: Measured position and radial tracking errors for cases (i), (ii), and (iii) ………………..………………

119

5.26

Cutting force compensation during actual cutting process

121

5.27

Measured position tracking errors and spectral analyses for control configurations (i), (ii), and (iii) ...……………

122

Contour tracking measurement and radial tracking errors for control configurations (i), (ii), and (iii) ……………

122

5.20 5.21

5.22 5.23

5.24

5.25

5.28

xxii

List of Tables Chapter 2 2.1

System model parameters for x and y axes. …………….

19

2.2

Velocity loop PI controller parameters of x and y axes …

30

2.3

Gain margin and phase margin of x and y axes velocity open loop…………………………………………………

32

2.4

Bandwidths of the velocity loop…………………………

33

2.5

Gain margin and phase margin of x and y axes position open loop…………………………………………………

35

2.6

Bandwidths of the position loop…………………………

37

3.1

Static friction model parameters for x and y axes………..

49

3.2

GMS slip-blocks model parameters for the y-axis……….

54

3.3

GMS slip-blocks model parameters for the x-axis……….

55

3.4

Cutting process parameters………………………………

57

3.5

X-axis: Harmonic contents of the measured cutting force for 1mm depth of cut…………………………………….

60

Y-axis: Harmonic contents of the measured cutting force for 1mm depth of cut…………………………………….

61

4.1

Original and revised system transfer function parameters.

65

4.2

Simulated friction compensation performance for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s…………..

68

4.3a

Y-axis: Original GMS slip-blocks model parameters……

69

4.3b

Y-axis: Adjusted GMS slip-blocks model parameters…...

69

Chapter 3

3.6

Chapter 4

xxiii

List of Tables

4.4

Measured friction compensation performance for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s…………..

70

4.5

Characteristics of the low pass filter Q…………………...

77

4.6

Gain margin and phase margin of x and y axes velocity open loop…………………………………………………

80

4.7

Gain margin and phase margin of x and y axes position open loop…………………………………………………

83

4.8

Bandwidth of x and y axes position loops……………….

86

4.9

Simulated friction compensation performance for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s…………..

89

Measured friction compensation performance for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s…………..

92

Magnitude of the quadrant glitches for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s………………...

95

rms of position errors for different cutting force harmonics content using inverse-model-based disturbance observer …………………………………….

102

rms of position errors for different cutting force harmonics rms of position errors for different cutting force harmonics content using the explicit estimation of cutting force technique…………………………………...

109

rms of tracking errors for different cutting force harmonics content using the explicit estimation of cutting force technique…………………………………………...

111

5.4

rms of the radial tracking errors for cases (i), (ii),and (iii)

119

5.5

Cutting test characteristics used in cutting force compensation……………………………………………..

120

rms of the radial tracking errors for cases (i), (ii), and (iii) during actual cutting process

123

4.10

4.11

Chapter 5 5.1

5.2

5.3

5.6 xxiv

Chapter 1

Introduction

1.1 Motivation The constant demand for higher speed and accuracy in machine tools stimulates the development of machine tool technology and design methodologies. An integrated part of this technology and design is the machine tool controller. A coordinated and concurrent development of the different technology fields and a good knowledge and understanding of the factors that contribute to the machine speed and accuracy are essential. One of the factors that contribute to the accuracy of a machine tool is the tracking performance of its drive system, which is critically influenced by the following factors: •

The Mechanical structure can limit the system tracking performance. Mechanical resonances influence the dynamic and frequency response function (FRF) of a system. The resonances can be excited during motion and can limit the potential bandwidth and reduce the stability margins of the control system. The excitation of the system resonances is often accompanied by mechanical vibration of the moving structure which can then influence the tracking accuracy. This vibration can be damped with a good control design but this will result in a more complex control structure. The influence of mechanical structural vibration and mechanical resonance on positioning and tracking performances can be limited by a well-balanced and integrated mechanical design and good control design strategies – a mechatronics approach. 1

Introduction

•

Friction forces, a nonlinear phenomenon that originates from the motor and the support bearings of electromechanical servo drive systems. Friction is an undesired nonlinear phenomenon in electromechanical servo-drives systems. Friction compensation is therefore a prerequisite for accurate motion control applications. A distinct phenomenon that is caused by friction is “spikes” at quadrant locations during circular motion (see fig. 1.1). These are widely known as “quadrant glitches”. Quadrant glitches in circular motion [1] are the product of complex nonlinear behaviour of friction at motion reversal or near zero velocity on each axis of a motion system. Circle, 15mm radius; Controller: Cascade 15

position y [mm]

10 5 0 -5 -10 -15 -15

-10

-5

0

radial error x 90 5 10 15

position x [mm]

Fig. 1.1 Quadrant glitches in circular test

•

Cutting forces, in addition to friction force, act as an input disturbance on the motion control system during cutting processes. The cutting force disturbance signal, which is periodic, reduces the tracking performance of a system.

•

The work-piece mass can vary from a fraction of a table mass to several times the table mass. Robust controllers are needed to make the machine tool performing well for all these masses. This issue is not considered in this thesis, but has been treated extensively in [2].

Active and efficient compensation of both friction and cutting forces are required to achieve accurate tracking and positioning. This thesis focuses on the development and experimental validation of control techniques to 2

Introduction

actively compensate the influence of friction at motion reversal and cutting forces on the tracking error of drive systems. Both friction feedforward compensation based on the advanced nonlinear Generalized Maxwell Slip (GMS) friction model and various simple and advanced linear control techniques are added to a conventional machine tool controller. Compensation techniques that are highly practical and simple in their application are desired.

1.2 State of the Art on Motion Control Mechanical drive systems have evolved to meet the present demands for high speed (i.e. shorter transient response time) and high accuracy applications. This evolution however has created a new challenge to the control community with regard to the complexity in effectively rejecting disturbance forces and achieving the best possible tracking accuracy. This section presents a discussion of the paradigm shift in mechatronic drive system technology and a literature review on disturbance forces and their compensation techniques.

1.2.1 Mechanical Drive Systems Mechanical drives system technology has seen a shift from electromechanical drive systems to direct drive systems. The emergence of direct drive systems over conventional electromechanical drive systems has provided the industry with their speed and high tracking performance requirements while eliminating some disadvantages of electromechanical drives systems, as discussed in the following paragraph. Pritschow [3] discusses the principle differences between linear motors and the more conventional and still widely used ball-screw drives, and explains that a shift to linear motors is required to further increase the productivity level of machine tools. Fig 1.2 shows a schematic diagram of the conventional ball-screw drive system. The main characteristic of a conventional ball-screw drives structure is the transmission mechanism that converts rotary motion of the motor to linear motion. The transmission mechanism includes the gearing elements and the lead-screw. The leadscrew element contributes negatively to the drives’ performance. The pitch tolerances of the lead-screw generate transmission errors that reduce the tracking accuracy. Tracking accuracy is also compromised by the backlash 3

Introduction

effects that originate from the dead zones and the large friction forces that are generated in a high stiffness electromechanical ball-screw and bearing structure. The large friction force results in limit cycles and reduced tracking accuracy. High-gain feedback controllers reduce this effect but their bandwidth is often limited by the resonance mode of the structure. In addition, position, velocity, and acceleration are limited by the mechanical characteristics of the lead-screw (e.g. stiffness) itself. Also, the lead-screw adds large additional inertias to the system and reduces the first natural frequency of the system. This reduces the bandwidth of the system and represents a critical limitation to the system tracking accuracy. motor milling table Position / Velocity Controller

motor

ball-screw encoder position measurement

Fig. 1.2. Electromechanical ball-screw drive structure

Direct drives techniques with linear motors have been introduced as an enhancement over the ball-screw drives system. Direct drives system has no mechanical transmission between motor and load. This important characteristic removes friction and backlash that are previously present in the transmission mechanisms (gear boxes, harmonic drives, rack-and-pinion systems, …), such that the total system friction is reduced to the (low) friction in the guideways. Also, without transmission errors, better tracking performance can then be achieved. In addition, the absence of mechanical transmission elements enables higher velocities and accelerations. The first natural frequency normally associated with the ball-screw drives is removed thus extending the bandwidth of the system. The structure of an iron-core linear motor is shown in fig. 1.3. Linear motors are a special class of synchronous brushless servo motor. The structure consists mainly of lamination stacks, coils, and magnets. Electrical energy is converted to linear mechanical energy as a result of the electromagnetic interaction between a coil assembly (primary part) and a permanent magnet assembly (secondary part). 4

Introduction

Fig. 1.3. Structure of an iron-core linear motor (reproduced from the website of ETEL, www.etel.ch)

Direct drives have several disadvantages. First, because of the absent of the transmission mechanism, cogging forces, external disturbances and load variations have a direct influence on the positioning and tracking accuracy. As a consequence, the design of controllers that minimize these effects becomes more challenging. However, the potential to achieve controllers with better performance for direct drive systems is higher because of their simpler mechanical structure and consequently more favourable dynamics. Second, direct drives have a cost disadvantage compared to the conventional electromechanical drive. Pritschow [3] states that for equivalent power rating, the overall size of the linear drive is larger than the electromechanical drive. The larger size is caused by the unused velocity potential and the proportional relationship between the drive rated force and the surface area of the moving drive parts. The additional cost is required for cooling the high level of heat generation (in iron-core linear drive). Linear motors applications are justified for applications where positioning or tracking performance requirements out-weight the total initial cost. An example of such application is high-speed machining. Efficient control strategies are critical in machine tool direct drive applications as both friction force and cutting force are acting directly on the motor and could reduce the overall tracking performance. Successful compensation of these disturbance forces is essential for accurate and effective motion control. The next section investigates various disturbance compensation techniques and approaches that have been proposed and applied in literature. 5

Introduction

1.2.2 Disturbance Forces and Compensation Methods Two different types of disturbance forces that can critically affect tracking performance and work-piece finishing quality in a linear motor based highspeed milling cutting process are friction and cutting forces. Hard non-linear friction behaviour can only be partly compensated using linear feedback control strategies such as PID, cascade P/PI or statefeedback control. The performance of these control techniques are restricted by their limited bandwidth. More advanced techniques must be incorporated in order to achieve sufficiently high path and tracking accuracy. Various different approaches for friction compensation have been identified in the literature. In general, these techniques can be summarized as indicated in fig. 1.4. Friction Compensation Techniques Friction-Model Based

feedback

feedforward

Friction-Model Free

nonlinear

linear

Fig. 1.4. Schematic diagram for friction compensation techniques

Friction is compensated using friction-model based and friction-model free approaches. Friction-model based compensation is either implemented in a feedback or feedforward configuration. The difference between both configurations is further discussed below. Friction-model free techniques can be nonlinear or linear. Various model and non-model based friction compensation techniques have been proposed and discussed in the literature. Armstrong et al. [4] give an overview of the existing friction models and compensation approaches. The recently developed friction models such as the LuGre model, the Leuven model, and the Generalized Maxwell-slip model are obviously not mentioned in this survey. In friction-model based compensation, the accuracy of the model determines directly the efficiency of the friction compensation. The accuracy of the model depends on the selected model structure and accuracy of the model parameters. Several simple and more advanced friction models have been proposed in literature. Friction is categorized according to its pre-sliding and 6

Introduction

sliding regimes, and the most simple friction models consider the sliding regime only. These models are a static map between friction force and velocity, e.g. viscous, Coulomb and Stribeck effect friction models. A first attempt in describing the more complex friction behaviour in pre-sliding regime was accomplished by Dahl [5]. The Dahl model was applied extensively for systems with ball-bearing friction. In 1995, Canudas et al. [6] have proposed a new improved friction model, the LuGre model, for control of systems with friction. The model captures most of the observed frictional behaviours that include Coulomb friction, Stribeck effect, and hysteresis. The LuGre friction model is widely applied and accepted for its simplicity and relatively good performance. However, the LuGre friction model fails to describe the hysteresis non-local memory behaviour of friction force in presliding regime. Swevers et al. [7] have improved the LuGre model yielding the Leuven integrated friction model, which is further modified by Lampaert et al. [8]. Recently, Al-Bender et al. [9] developed the so-called Generalized Maxwell-slip (GMS) friction model and illustrate the superiority of the model with respect to simulation of friction behaviour in both the pre-sliding and sliding regimes. The main disadvantage of the GMS model is its complexity and large number of parameters, which complicates its application in control. In a feedback configuration, the measured position signal is used to generate a friction compensation force based on the available friction model. In a feedforward configuration, the reference position is used instead of the measured position signal. The feedback approach is generally designed on the basis of the Coulomb model and the Maxwell-slip model [10]. A careful design strategy is necessary to avoid instability problems. In this thesis, feedforward friction compensation is applied: friction forces predicted by the applied friction model are added to the control command signal to compensate friction forces in the system. The effectiveness of this approach, however, depends on the accuracy of the friction models. Changes in the system environment and structure (lubrication, wear, etc.) could change the friction behaviour and re-identification of the friction model will be necessary to avoid deterioration in the friction compensation performance. Besides model based friction compensation, various friction-model free compensation methods have been suggested and discussed in the literature. These include linear and non-linear control strategies. In a linear control approach, Tung, Anwar, and Tomizuka [11] have demonstrated the effectiveness of a repetitive controller, first introduced by Hara et al. [12], in improving tracking performance and quadrant glitches compensation. 7

Introduction

In addition, Lampaert et al. [13] have proposed a control approach that includes a disturbance observer based on a Kalman filter using a second order random walk model. Since friction is a highly nonlinear phenomenon it can be expected that nonlinear control approaches are more appropriate. Several nonlinear control strategies have been adapted for friction compensation. Tjahjowidodo et al. [14] have shown that a Maxwell-slip-model-based nonlinear gain scheduling controller yields fast response and low steady-state error for friction compensation in electro-mechanical systems. Sliding mode control (SMC) [15, 16] is an example of an important robust control design approach for linear and nonlinear systems. Two main features of SMC are the finite reaching time and the complete disturbance rejection of matched uncertainties. In SMC applications, the system states approach the switching line (or surface) and slide along this line to reach the final states. During sliding, the control behaviour is independent of the system dynamics and hence is independent of the influence of the acting disturbance forces. Altintas [17] has described the SMC design for high-speed feed drives. A tracking performance comparison between SMC and the classical cascade controller is discussed in [18]. SMC however, is widely known for its “chattering” problem [19, 20], a high-frequency switching of the SMC that is a result of the imperfect switching mechanism and discontinuous control signal around the switching surface. Chattering is highly undesirable since it involves large control activity and possible excitation of the resonance frequencies of the system. Many different techniques have been proposed in literature to eliminate chattering in SMC related applications [21, 22]. Besides friction force compensation, attenuation of other external disturbance forces, such as the cutting force that acts on a motion control system, is of equal importance. Various compensation methods for different applications have been proposed in literature. A repetitive controller is ideal for periodic reference command signals and disturbance inputs. The repetitive controller is only active in small frequency ranges around the harmonics and system phase lag is compensated for in its design [23]. The performance of a repetitive controller is highly influenced by the accuracy of the period of the harmonics signal. Disturbances can also be compensated using a robust control approach such as the H-infinity control [24] and SMC. Van Brussel and Van den Braembussche have investigated the robustness of both H∞ and SMC in linear motor systems [25].

8

Introduction

Ohnishi et al. present an inverse model-based disturbance observer [26, 27] control structure that can be applied for any type of input disturbance. It adds a high gain loop to an existing control configuration, thereby improving the disturbance rejection with its bandwidth, which depends on the accuracy of the available system model. The tuning of this bandwidth is a trade-off between performance and stability margin. Kempf et al. [28] have added the disturbance observer into an existing control system structure and have shown enhanced disturbance attenuation performance of a track following optical disk drive system against shock and vibration influence. The disturbance attenuation property is removed from the design specifications of the feedback controller resulting in simplified control design and greater stability. This approach can be applied to any disturbance and is effective up to a specified bandwidth which depends on the accuracy of the available system model. An explicit estimation of the cutting force has been attempted by Pritschow et al. [29]. The estimation is based on the balance of forces acting on the system and uses a relative acceleration sensor measurement. A reduced order state observer estimates the velocity and the cutting force. However, to the best of our knowledge, no report on the application of this estimator to effectively compensate the influence of the cutting force is available.

1.3 Scope, Objective, and Approaches Effective compensation of friction forces and cutting forces are a prerequisite for accurate tracking performance in machine tool drive applications. Classical motion controllers (discussed in chapter 2) are not able to fully reduce the effect of these disturbances sufficiently. Various other techniques have been reported as discussed in the previous section. However, very limited knowledge on the application of some of these techniques for friction and cutting forces compensation in high-speed milling machine direct drives is available. The objective of this research can be summarized as follow: “Improve the positioning and tracking accuracy of motion systems controlled by the classical cascade P/PI feedback control structure by adding simple but effective friction and cutting force compensation mechanisms.” 9

Introduction

These compensation mechanisms are developed for a linear drive based xytable of a high-speed milling machine. The modelling and experimental identification of this system, the design of the cascade P/PI feedback controller and various compensation mechanisms and their simulation-based and experimental validation are discussed in detail in this thesis. Friction forces are compensated using both friction-model based feedforward and the inverse-model-based disturbance observer. These techniques have been selected because of their simplicity. Feedforward is preferred over feedback in order to avoid stability issues. Two friction models are compared: a static friction model and the recent and more advanced Generalised Maxwell-slip model. The inverse model based disturbance observer [26, 27] is selected because of its simple structure and design methodology that accounts for model uncertainty. To compensate cutting forces, several techniques are implemented and compared. These include: the inverse model based disturbance observer, a repetitive controller, a force disturbance observer, and Pritschow’s explicit estimation of cutting forces using the Ferraris relative acceleration sensor measurement.

1.4 Contributions The thesis attempts to enrich the understanding and knowledge regarding accurate motion control of machine tool feed drive systems. The following contributions are presented: i.

Development of simple motion control structure based on the classical cascade P/PI controller combined with modules to improve the friction and cutting forces compensation.

ii.

Application of a simple identification procedure for the Generalized Maxwell-slip (GMS) friction model that produces superior friction compensation performance, especially at lower tracking velocity

iii.

Application of an inverse model based disturbance observer in combination with feedforward of the GMS friction model for efficient friction compensation.

iv.

Comparison of the tracking performance of different cutting force compensation techniques during actual circular cutting tests.

10

Introduction

1.5 Outline This thesis focuses on friction and cutting forces compensation of a linearmotor based xy feed drive system. The above mentioned friction feedforward, inverse model based disturbance observer, and repetitive controller are add-on devices, implying that they are added to an existing feedback control system, which is a classical cascade P/PI controller. The structure, design procedures, and analysis of a classical cascade P/PI position controller that is widely applied in many mechatronic systems are discussed first in Chapter 2. The minimization of the effect of disturbance forces on the system position and tracking performance requires precise knowledge and complete understanding of the characteristics of these disturbance forces. Chapter 3 discusses the modelling and identification of friction, and the spectral analysis of cutting force measurements of an actual milling cutting process. Chapter 4 discusses the implementation of the friction force compensation based on the feedforward of friction models. Simulations and experimental validations of friction compensation performances using a simple static friction model and the more advanced and recent GMS friction model are performed and compared. An inverse model-based disturbance observer module is introduced in combination with feedforward of the friction models and its influence on the magnitude of the quadrant glitches from circular tests are studied and analysed. Chapter 5 describes and discusses compensation of the cutting force using various selected control techniques. The experimental validation of the different compensation techniques are performed first using artificial cutting force synthesized from actual milling cutting force measurements and second during actual circular cutting processes. The practical implementation of friction and cutting forces compensation during these cutting tests and the tracking performances of the different techniques are discussed and compared. Finally, Chapter 6 presents the conclusions regarding the design of the cascade P/PI controller, the performance of friction and cutting forces compensation and some recommendations of future work.

11

12

Chapter 2

Classical Motion Control 2.1 Introduction Motion tracking controllers are designed with the objective of achieving maximal tracking accuracy and robustness against disturbances and plant uncertainties. The size of the tracking errors and actuator input signals are important indicators to validate a control system design. A good control design ensures that these indicators remain below some pre-specified conditions and hence exploiting the system to its fullest potential. Feedback control strategy is the basic principle of many control systems. The desired reference signals are compared with the actual output of the system and corrective measures are implemented to compensate these errors. Feedback control linearizes nonlinear elements and partly compensates the effects of disturbances and system variations. Due to its simplicity, PI and PID control form 90% of the practical control applications. Feedforward control strategies are normally introduced to compliment feedback control [30]. A combined feedforward and feedback control strategy improves tracking performance especially in systems with pre-knowledge of the inputs reference and disturbance signals. A prominent and classical tracking controller that exists in the majority of servo motion control systems is the classical cascade controller. Cascade control is widely applied for its simple structure and transparent design. In addition, the loop controllers are based on the familiar classical proportional P, and integrator I controllers. Various modifications to this controller have been suggested in literature with the aim of improving its tracking performance. For example, Doenitz [31] analysed the tracking error of a combination of a cascade controller with a disturbance observer. In addition, 13

Classical Motion Control

Boucher et al. [32] have presented a generalized predictive cascade control for control of machine tool drives. This chapter discusses the analysis and design of a classical cascade P/PI controller for position control of a linear drive based xy feed table. First, a detailed description of the experimental setup, that is, a high-speed xy milling machine is presented.

2.2 High-Speed XY Milling Machine The test setup that is considered for the experimental validation is a lineardrive based xy feed table of a high-speed milling machine (see fig. 2.1). This xy table is the result of a European project named MOTION (Modular Technologies for Intelligent Motion Unit with linear motors and axis control) that was implemented from 1996-2000. The partners included Philips CFT, Fidia S.p.A, SMT, Schneeberger, ETEL, Royal Institute of Technology (KTH), and K.U.Leuven (Div. PMA).

y-axis x-axis

Linear

Linear Motor Linear Motor

Motor

Fig. 2.1. A linear-driven xy feed table of a high-speed milling machine

14

Classical Motion Control

The axes are driven by three ETEL’s LMC 22-070-3TA linear motors. The upper stage y-axis is driven by a single ETEL iron-core linear motor while the bottom stage x-axis is driven by two ETEL iron-core linear motors. ETEL patented anti-cogging design ensures very low cogging and force ripple effects. The motions are driven in current mode using ETEL DSB2-S amplifiers. The schematic representation of the two axes is shown in fig.2.2. Encoder Y

Motor Y

Motor X1 Guideways

Milling Table

Encoder X Motor X2

y-axis x-axis Guideways

Fig. 2.2. Schematic diagram of the xy table

Both axes are equipped with a 0.25µm resolution Heidenhain LF481C linear encoder, mounted in the middle of the stages. The encoders have a signal period of 4µm and generate two sinusoidal signals of 1 Vpp (peak to peak) and a phase difference of π/2. A third index signal provides the motor absolute position. The signals are linked to the ETEL DSB2-S amplifiers for the stages position measurement. A dSPACE DS1103 controller board interfaced to the ETEL DSB2-S amplifiers receives an interpolated TTL analog position signal. The resolution of the encoder signal is determined by the interpolation setting factor of the DSB2-S amplifier. For an interpolation factor of 4, the encoder resolution is RES =

encoder period 4 μm = = 0.25μm . 4 ⋅ interpolation factor 4 ⋅ 4

(2.1)

15

Classical Motion Control

In the absence of a velocity sensor, the velocity signals are calculated by numerical differentiation of the position signals in combination with a firstorder low-pass Butterworth filter with a bandwidth of 150Hz. This filter is added to attenuate amplified measurement noise associated with the derivative action (see section 2.5). The stages are supported by 4 Schneeberger MRD35 preloaded roller carriages on 2 rails that are fixed on the middle part. For the y-axis, the stator of the linear motor is fixed to the middle part and the motor’s forcer is connected to the table. The total mass of the y-axis is 170kg. The x-axis of the table is supported in the middle by 6 MRD35 carriages on 3 rails that are fixed to the base. The stators of the two linear motors are fixed to the base while the forces are connected to the middle part. The total mass for the xaxis that includes the y-axis, the middle part, and the stators of the linear motors is 340kg. “Rubber wipers” protect the rolling elements in the linear bearing from any harsh environment. A u-shaped shield covers each linear motor, protecting the motors magnets from the cutting chips. The shields move in a resin profile for improved encapsulation. Fig. 2.3 shows a schematic diagram of the motion controller structure. The controller of each axis is implemented on a dSPACE 1103 DSP controller board, using the ControlDesk software from dSPACE to link the host computer to the ETEL drives. An additional DS1102 controller board provides the communications between the DSP controller board and the ETEL drive amplifiers using digital I/O interface. The digital I/O interface provides basic communications between the host computer and the linear drives. These actions include the drives power on/off, drive enable/disable commands, table absolute positioning, and homing procedures. The dSPACE 1103 controller board is responsible for position control of the drives. The controller settings and tracking commands are uploaded to the drives from the host computer via the dSPACE and tracking performances are monitored from the measured encoder signals.

16

Classical Motion Control

Fig. 2.3. Motion controller structure of a xy feed table with three linear drives for high speed milling application

The system identification is described in the next section.

2.3 System Identification A linear time-invariant model of the dynamic behaviour of each axis is obtained from frequency response function (FRF) measurements of the system. The dynamic coupling between both axes is negligible and the system dynamics can be described by two single-input single-output (SISO) models. In order to measure the two SISO FRFs of the system, the system is excited with band-limited white noise signals. The output encoder measurements and the excitation signal, that is the signal that is sent to the ETEL drive amplifiers, are recorded. The sampling frequency is 2000Hz and the total duration of the measurement is 5 min. A Hanning window is applied. The number of samples per window is 2048. This yields a sampling resolution of 1Hz. The SISO FRFs of the system are estimated using the H1 estimator [33]. 17

Classical Motion Control

Fig. 2.4 shows the measured FRFs for the x and the y axes. Bode Diagram

G =out. pos. / in. voltage [µm]/[volt]

6

10

x-axis

4

10

y-axis

2

10

0

10

-2

10

phase [degrees]

0 -90 -180 -270 -360

1

2

10

10 Frequency [Hz]

Fig. 2.4. FRFs measurement of the x and y axes

The FRF of each axis contains an anti-resonance and resonance combination near 45Hz. This is due to the relative motion between the base of the machine and the ground. A study of this is presented in detailed in Appendix A. The study shows a left shift in the anti-resonance and resonance frequencies as the tightness of the bolts that hold the base to the ground are loosened. Further theoretical explanation of this phenomenon is presented in detail in [34]. Parametric models are fitted on the FRFs measurement using the nonlinear least squares frequency domain identification method [33], yielding the following second order model with time delay: G m (s ) =

Z (s ) B = ⋅ e − sTd , U (s ) s (s + A)

(2.2)

Equation (2.2) relates the linear dynamic relation between input voltage u and table position z[m], with z = x and z = y for the x and y axes respectively. A summary of the estimated model parameters, A, B, and Td, for each axis is given in Table 2.1.

18

Classical Motion Control

Table 2.1. System model parameters for x and y axes A 28.57 [s-1] 20.00 [s-1]

x-axis y-axis

B 4.526 [m/volt.s2] 8.916 [m/volt.s2]

Td 0.00065s 0.00065s

Fig. 2.5 and fig. 2.6 show the frequency responses of the obtained models and corresponding measured FRFs for the x and y axes respectively. G= out. pos / in. volt [µm]/[volt]

Bode Diagram 10

4

FRF model

10 10 10

2

0

-2

Phase (deg)

-90 -135 -180 -225 -270 -315 -360 10

1

Frequency (Hz)

10

2

Fig. 2.5. X-axis: FRF measurement and proposed model G= out. pos / in. volt [µm]/[volt]

Bode Diagram 10

4

FRF model

10 10 10

2

0

-2

-90 Phase (deg)

-135 -180 -225 -270 -315 -360 10

1

Frequency (Hz)

10

2

Fig. 2.6. Y-axis: FRF measurement and proposed model 19

Classical Motion Control

The parametric model (2.2) estimates the models as a mass line with damping characteristic at lower frequency range. These models do not include the resonance and anti-resonance peaks at frequencies beyond 40Hz as shown in fig. 2.5 (e.g. the anti-resonance frequency at 47Hz for the x-axis, and at 44 Hz for the y-axis). The next section discusses the control analysis and design of a cascade controller that is applied for position control of the xy table. Analysis of the cascade controller based on an ideal configuration of the system is first considered.

2.4 Cascade Control Structure & Analysis 2.4.1 Cascade Controller Structure & Configuration A cascade control structure is used in many mechatronics applications. The control structure consists of a velocity loop and a position loop. The position loop is built around the velocity loop. Fig. 2.7 shows a general scheme of a cascade control structure. A disturbance force d(t) enters at the input of the system.

Fig. 2.7. General scheme of a cascade control structure

Tracking performance of a cascade controller is analysed based on the closed loop bandwidths, damping ratio, and dynamic stiffness. The compliance function is the transfer function from input disturbance d to the measured output position z. The dynamic stiffness is the inverse of this transfer function. Dynamic stiffness is an important characteristic of a tracking controller because it indicates the controller disturbance rejection capacity.

20

Classical Motion Control

This section analyses the characteristics of the cascade P/PI controller by first considering an ideal theoretical design configuration. The following idealized configuration is considered: ¾

the system behaves as an ideal rigid body system without friction, velocity v and position signals z are assumed to be available, measurement noise n(t) is only present on the velocity signal, not on the position signal, disturbances d(t) enter at the input of the system, that is, these disturbances are forces.

¾ ¾ ¾

The analysis (in section 2.4.2 and 2.4.3) is based on a model that is a further simplification of model (2.2), that is, the time delay Td and real pole s = - A are removed. The resulting model is a double integrator, or mass-line model, and is presented in fig. 2.8. M represents the mass of the considered system. kf [N/volt] represents in this model the motor constant relating the motor force F to the input voltage to the motor drive amplifier. Figure 2.7 indicates that this simplified model equals: Z (s)

U (s)

=

G s2

=

kf = G ⋅M

kf Ms 2

(2.3a)

(2.3b)

Based on the estimate of the mass M of each axis and the gain G [m/volts.s2] derived from the model (2.2) and Table 2.1, the motor constant of the x and y axes are estimated at 1450[N/volt] and 1470 [N/volt] respectively.

Fig. 2.8. Schematic diagram of the open loop system for force constant estimation

21

Classical Motion Control

Fig. 2.9 shows an ideal cascade control configuration.

Fig. 2.9. An ideal cascade control structure for a linear motor position control

The tracking error ep(t) is the difference between the reference position zref(t) and the measured position z(t). Structural resonances are not considered here. In practical applications, they are often present, and compensated in the control structure by adding notch filters [35]. The velocity loop controller is usually either a proportional (P) controller or a proportional and integral (PI) controller. Both a P and a PI velocity loop controllers are considered in the following theoretical analysis. The position loop controller is usually a proportional controller. First, the analysis of a system with a proportional velocity controller and a proportional position controller is considered.

2.4.2 Analysis of the Closed Loops Behaviour with Proportional (P) Velocity Control A velocity controller with a proportional gain kp and a position controller with a proportional gain kv is considered. The system dynamic open loop is: Z (s)

U (s)

=

kf Ms 2

(2.4)

The transfer functions relating input reference signal zref (t), velocity noise signal n(t), and input force disturbance d(t) to output position signal z(t) are:

22

Classical Motion Control

kv k p k f

Z (s) =

2

Ms + k p k f s + kv k p k f + +

⋅ Z ref ( s )

k pk f 2

Ms + k p k f s + kv k p k f 1 Ms 2 + k p k f s + kv k p k f

⋅ N (s)

(2.5)

⋅ D(s)

The dynamic stiffness of the controller, which is the inverse of

Z (s)

D(s)

in (2.5),

is D(s) Z (s)

= Ms 2 + k p k f s + kv k p k f .

(2.6)

The position error ep(t) can be expressed as a function of the reference signal zref (t), velocity noise signal n(t), and the input disturbance d(t), E p ( s ) = Z ref ( s ) − Z ( s ) Ep (s) = -

Ms 2 + k p k f s Ms 2 + k p k f s + kv k p k f k pk f Ms 2 + k p k f s + kv k p k f 1 2

Ms + k p k f s + kv k p k f

⋅ Z ref ( s )

(2.7)

⋅ N (s) ⋅ D(s).

The undamped natural frequency ω0 and the damping ratio ς of the system are obtained from the transfer function

ω0 =

ς =

1 2

kv k p k f M

k pk f Mk v

.

Z (s ) in Z ref (s )

(2.5),

(2.8)

(2.9)

23

Classical Motion Control

The tracking performance of the cascade P/P controller is analysed as follows: For a reference signal with a constant velocity, Z ref (s ) =

v s2

,

(2.10)

the steady-state tracking error obtained using the final-value theorem to equation (2.7) (ignoring noise signal n(t) and input disturbance d(t)) equals: ep (∞) =

v kv

(2.11)

The steady-state error is inversely proportional to the proportional gain kv and is directly proportional to the constant tracking velocity v. Next, the positioning performance is analysed for a step reference input signal zref(t)=k and for a static disturbance signal d(t) = k. The steady state position error for a step reference input signal and for a static disturbance signal, obtained by applying the final value theorem to equation (2.7) are: e pz _ step = 0

ep

d _ step

=−

(2.12a) k . kv k p k f

(2.12b)

The steady-state error is proportional to the static disturbance force k (equation 2.12b) and will always be a non-zero value. This also means that, in the absence of a disturbance signal, the steady-state error is zero (equation 2.12a).

24

Classical Motion Control

2.4.3 Analysis of the Closed Loops Behaviour with PI Velocity Control A velocity loop with proportional-integral (PI) controller and a position controller with a proportional gain kv is considered. The velocity loop PI controller has the following structure, PI = k p +

ki s

(2.13)

With reference to previous fig. 2.9, with velocity controller (2.13) and a proportional position controller kv, the transfer functions relating input reference signal zref (t), noise signal n(t), and the input force disturbance f(t) to the output position signal z(t) are: Z (s) =

kv k p k f s + kv ki k f

(

3

)

Ms + k p k f s 2 + ki k f + kv k p k f s + kv ki k f

+ +

k p k f s + ki k f 3

2

(

⋅ Z ref ( s )

)

Ms + k p k f s + ki k f + kv k p k f s + kv ki k f 3

2

s

(

)

Ms + k p k f s + ki k f + kv k p k f s + kv ki k f

⋅ N (s)

(2.14)

⋅ D(s)

The system dynamic stiffness, which is the inverse of Z(s)/D(s) in (2.14) equals: D(s) Z (s)

=

(

(

)

1 Ms 3 + k p k f s 2 + ki k f + kv k p k f s + kv ki k f s

)

(2.15)

The integrator in (2.15) gives an infinite static stiffness for the system at low frequency range (see fig. 2.22). The relationship between the position error ep(t) and the reference signal zref (t), the velocity noise signal n(t), and the input force disturbance d(t) are:

25

Classical Motion Control

E p ( s ) = Z ref ( s ) − Z ( s ) Ms 3 + k p k f s 2 + ki k f s

Ep (s) =

(

)

Ms 3 + k p k f s 2 + ki k f + kv k p k f s + kv ki k f k p k f s + ki k f

-

3

2

(

)

Ms + k p k f s + ki k f + kv k p k f s + kv ki k f

-

(

s

)

Ms 3 + k p k f s 2 + ki k f + kv k p k f s + kv ki k f

⋅ Z ref ( s ) ⋅ N (s)

(2.16)

⋅ D(s).

The undamped natural frequency ω0, and the damping ratio ς cannot be obtained directly from (2.14) because the system has 3 poles. An approximation of the transfer function is desired. J. Wang [36] approximated the transfer function Z(s)/Zref(s) in (2.14) as a second order model and suggested the following approximate relationships relating the controller parameters kv, kp, ki, with the damping ratio ς and the undamped natural frequency ω0: ω0 =

ς=

kv k p k f

1 2

(2.17)

M

k pk f Mk v

− 0.05 ≈

1 2

kf kp Mk v

(2.18)

The tracking performance of the cascade P/PI controller is analysed as follows: For a reference signal with a constant velocity, Z ref ( s ) =

v s2

,

(2.19)

the steady-state tracking error (with no disturbance and noise signals) obtained by applying the final-value theorem to equation 2.16 equals: ep (∞) =

26

v . kv

(2.20)

Classical Motion Control

Similar to the previous cascade P/P analysis, the steady-state error for constant velocity reference tracking is inversely proportional to the proportional gain kv and is directly proportional to the constant tracking velocity v. Next, the positioning performance is analysed for a step reference input signal zref(t) = k and for a static disturbance signal d(t) = k. The steady-state position error for the respective cases, obtained by applying the final value theorem to equation 2.16 equals: e pz _ step = 0

(2.21a)

e pd _ step = 0.

(2.21b)

Unlike the cascade P/P analysis, the integrator in the velocity controller eliminates position error for a step reference input signal and compensates the static disturbance force. Next, the design for the considered cascade P/PI controller is discussed. The design is based on the measured FRFs of the system shown in fig. 2.4.

2.5 Design and Validation of Cascade P/PI Controller Based on Measured FRFs The parameters of the cascade P/PI controllers are selected and tuned based on gain margin and phase margin considerations of the open loop transfer functions, using measured FRFs of the system Ĝ (fig. 2.4). First, the PI velocity loop controller parameters are designed and the loop characteristics are analysed based on the control scheme shown in Fig. 2.10.

2.5.1 Design and Analysis of the Velocity Loop The velocity loop of the cascade P/PI controller scheme includes the PI velocity controller (equation 2.13), a notch filter N(s), and the transfer function Vest(s) of the operator that estimates a velocity signal from the measured position encoder signal.

27

Classical Motion Control

Fig. 2.10. Frequency domain scheme of the cascade P/PI controller for control of a linear motor drive

This transfer function (2.22) is a combination of a continuous-time derivative, s, and a first order low-pass Butterworth filter with a cut-off frequency of 150 Hz for both axes. The filter is added to attenuate amplified measurement noise associated with the derivative action. Vest ( s ) =

942.5s s + 942.5

(2.22)

Fig. 2.11 shows the Bode diagram of Vest(s). Magnitude (dB)

60 40 20 0

Phase (deg)

-20 90 45 0 -45

low pass filter Vest(s)

-90 0 10

10

1

10

2

Fig. 2.11. Bode diagram of Vest(s)

28

10

3

Classical Motion Control

High frequency structural resonances can reduce the performance and stability of a system. Often, notch filters are added to reduce the influence of these resonances on the closed loop behaviour. With reference to the FRFs measurement shown in fig. 2.4, notable high structural resonance frequencies are observed between 320 Hz and 340 Hz and between 270Hz and 320 Hz for the x and the y axes respectively. Equation (2.23) describes a notch filter transfer function for the x-axis, based on a Butterworth band-stop filter. Fig. 2.12 shows the Bode diagram of (2.23). N (s ) =

s 2 + 4.3e6

(2.23)

s 2 + 125.7 s + 4.3e6

Bode Diagram

Magnitude (dB)

0

-50

-100

Phase (deg)

-150 450 405 360 315 270 250

300

350

400

Frequency (Hz)

Fig. 2.12. X-axis: Bode diagram of a notch filter N(s) for frequency between 320 Hz and 340 Hz

The transfer function of the notch filter selected for the y-axis equals: N (s ) =

s 2 + 3.41e6 s 2 + 314.2s + 3.41e6

(2.24)

Next, the velocity open and closed loop transfer functions are analysed based on the cascade control scheme in fig. 2.10. The velocity open and closed loop transfer functions are:

29

Classical Motion Control

v ol =

Z& est (s ) = PI ⋅ Gˆ ⋅ N ⋅ V est E v (s )

(2.25)

ˆ ⋅ N ⋅V Z& ( s ) PI ⋅ G est vcl = est = ˆ ⋅ N ⋅V U p ( s ) 1 + PI ⋅ G est

(2.26)

The compliance function and the sensitivity function of the velocity loop are: Gˆ / k f Z (s ) = D(s ) 1 + PI ⋅ Gˆ ⋅ N ⋅ Vest

(2.27)

1 ˆ 1 + PI ⋅ G ⋅ N ⋅ Vest

(2.28)

S v (s ) =

The velocity loop PI (equation (2.13)) controller parameters, kp and ki are selected based on gain margin and phase margin consideration of the velocity open loop transfer function. A reasonable value of gain margin and phase margin ensures stability and good transient response characteristics [32]. A minimum gain margin of 5dB and a minimum phase margin of 50 degree are desired. The velocity loop PI controller parameters of each axis are listed in Table 2.2. Table 2.2. Velocity loop PI controller parameters of x and y axes x-axis y-axis

kp [volt·s ] 0.00006 0.00003

ki [volt·s2] 0.00045 0.00025

Fig. 2.13 and fig. 2.14 show the velocity open loop transfer functions (equation (2.25)) for the x and y axes respectively, based on the PI controller’s parameters in Table 2.2. The influence of the structural resonance frequencies near 320 Hz is removed as indicated by the circle in the magnitude plots of fig. 2.13 and 2.14. Table 2.3 summarizes the gain margins and the phase margin of the x and y axes. These values ensure good transient response and stability margin for each axis velocity loops.

30

Classical Motion Control

Bode Diagram 40

Phase (deg)

Magnitude (dB)

without notch filter 20

with notch filter

0 -20 -40 0 -45 -90 -180 -270 -360 10

1

frequency [Hz]

10

2

Fig. 2.13. X-axis: Theoretical bode plot of the velocity open loop transfer function based on measured FRF of the system

Bode Diagram

40 Magnitude (dB)

without notch filter 20

with notch filter

0 -20

Phase (deg)

-40 0 -90 -180 -270 -360 10

1

10

2

Frequency (Hz)

Fig. 2.14. Y-axis: Theoretical bode plot of the velocity open loop transfer function based on measured FRF of the system

31

Classical Motion Control

Table 2.3. Gain margin and phase margin of x and y axes velocity open loop Gain Margin

Phase Margin

x-axis

12.332 dB (at 315.08 Hz)

74.514 deg (at 65.012 Hz)

y-axis

10.311 dB (at 307.63 Hz)

73.059 deg (at 34.922 Hz)

The magnitudes of the gain margin and the phase margin for the x and y axes velocity loops are conservative to ensure a good loop stability margin. The stability of both axes velocity loops is analysed from the Nyquist plots of the velocity open loop transfer function, that is, the point [-1,0] is not encircled. The Nyquist plots, each drawn with a unit circle, are shown in fig. 2.15. The positive influence of the notch filter in increasing the stability margin of the velocity loop is emphasized. Nyquist Plot (x-axis)

Nyquist Plot (y-axis)

2

2

1.5 1 Imaginary Axis

Imaginary Axis

1 0.5 0 -0.5 -1

-1

-1.5 -2 -1

0

w ith notch filter no notch filter

-0.5

0 0.5 real axis

1

-2 -1

w ith notch filter no notch filter

-0.5

0 0.5 real Axis

1

Fig. 2.15. Nyquist plots of the x-axis (left) and the y-axis (right) velocity open loop transfer functions based on measured FRFs of the system

The bandwidth of the velocity loop is determined from the magnitude plot of the sensitivity function Sv(s), that is the transfer function (2.28), shown in fig. 2.16. The bandwidth is defined as the frequency of which the magnitude of the sensitivity function crosses the -3dB line from below. Table 2.4 summarizes the values of the velocity loop bandwidth for both axes. Fig 2.16 also emphasizes the influence for the removal of the structural resonance frequency near 320 Hz, as indicated by the two circles.

32

Classical Motion Control

Bode Magnitude Diagram y-axis

2

Sv=pos err. / disturbance force [µm] / [N]

Sv=pos err. / disturbance force [µm] / [N]

Bode Magnitude Diagram x-axis

1

w ith notch filter no notch filter

0.1 0 10

1

10 10 Frequency [Hz]

2

1

w ith notch filter no notch filter

0.1 0 10

2

1

10 10 Frequency [Hz]

2

Fig. 2.16. Sensitivity function of x (left) and y (right) axes for the velocity loop Table 2.4. Bandwidths of the velocity loop bandwidth

x-axis

y-axis

28.9Hz

28.8Hz

This definition of a system bandwidth gives a more accurate representation of the system performance compared to the traditional definition based on the complimentary sensitivity function (equation 2.26) as indicated in [34]. Bode diagrams of the velocity closed loop transfer function of each axis based on (2.26) are shown in fig. 2.17. Bode Diagram

Magnitude (dB)

10 0 -10 -20 -30

without notch filter with notch filter

-40 90 Phase (deg)

0 -90 -180 -270 -360 10

1

10

2

Frequency (Hz)

(a) 33

Classical Motion Control

Bode Diagram

10 Magnitude (dB)

0 -10 -20 -30

without notch filter with notch filter

-40 90 Phase (deg)

0 -90 -180 -270 -360 10

1

10

2

Frequency (Hz)

(b) Fig. 2.17. Velocity closed loop transfer function for (a) xaxis and (b) y-axis

The positive effect of the notch filters is illustrated by the removal of the peak near 320Hz, as indicated by the two circles. This positive effect is specifically important because it removes the possibility of deterioration in performance at high frequency.

2.5.2 Design and Analysis of the Position Loop The second step of the design procedure includes the design and analysis of the position loop that is built around the velocity loop. The position open loop and closed loop characteristics are examined for a proportional gain controller kv equals: P = kv = 150 s −1

(2.29)

Based on fig. 2.10, the position open and closed loop transfer function are: posol =

34

Z (s)

Ep (s)

=

ˆ P ⋅ PI ⋅ G ˆ ⋅ N ⋅V 1 + PI ⋅ G

est

(2.30)

Classical Motion Control

poscl =

Z (s)

Z ref ( s )

=

ˆ P ⋅ PI ⋅ G . ˆ ˆ 1 + PI ⋅ G ⋅ N ⋅ Vest + P ⋅ PI ⋅ G

(2.31)

The compliance function, the sensitivity function, and the position steadystate error transfer function equals: Gˆ / k f Z (s ) = D(s ) 1 + PI ⋅ Gˆ ⋅ N ⋅ Vest + P ⋅ PI ⋅ Gˆ S p (s ) =

(2.32)

1

(2.33)

1 + PI ⋅ Gˆ ⋅ N ⋅ Vest + P ⋅ PI ⋅ Gˆ

Ep (s) =

ˆ ⋅ N ⋅V 1 + PI ⋅ G est Z s ˆ ˆ ref ( ) 1 + PI ⋅ G ⋅ N ⋅ V + P ⋅ PI ⋅ G est

-

ˆ /k G f ˆ ⋅ N ⋅ V + P ⋅ PI ⋅ G ˆ 1 + PI ⋅ G est

(2.34)

D(s).

The gain margin and the phase margin of the x and the y axes are identified from the position open loop transfer functions shown in fig. 2.18. These values are summarized in Table 2.5. Table 2.5. Gain margin and phase margin of x and y axes position open loop Gain Margin

Phase Margin

x-axis

15.1 dB (at 100 Hz)

68 deg (at 21.4 Hz)

y-axis

16.49 dB (at 100.62 Hz)

67.18 deg (at 22.47 Hz)

The gain and phase margins are conservative. A higher value of kv increases the sensitivity peak (refer to fig. 2.20) near and above the 6dB line (or the absolute magnitude value of 2), that is the normal design criteria for the sensitivity peak value. Higher sensitivity peak value deteriorates the system performance at high frequencies.

35

Classical Motion Control

x-axis

Phase (deg)

Magnitude (dB)

40 20 0 -20 -40 -60 -90 -180 -270 -360 10

1

Frequency [Hz]

10

2

(a) y-axis

Phase (deg)

Magnitude (dB)

40 20 0 -20 -40 -60 -90 -180 -270 -360 10

1

Frequency [Hz]

10

2

(b) Fig. 2.18. Position open loop transfer function for (a) x-axis and (b) y-axis based on measured FRF of the system

The stability of the x-axis and the y-axis position loops is analysed from the Nyquist plots of the position open loop transfer function, that is, the point [-1,0] is not encircled. Fig. 2.19 shows the Nyquist plots, each drawn with a unit circle. In both cases, the [-1, 0] point is not encircled, thus ensuring system stability.

36

Classical Motion Control

Nyquist Plot (y-axis) 2

1

1 Imaginary Axis

Imaginary Axis

Nyquist Plot (x-axis) 2

0

-1

-2 -1

-0.5

0

0.5

0

-1

-2 -1

1

Real Axis

-0.5

0

0.5

1

Real Axis

Fig. 2.19. Nyquist plots of the x-axis (left) and the y-axis (right) of the position open loop transfer functions

Next, the bandwidth of the position loop is identified from the -3dB crossing (from below) on the magnitude plot of the sensitivity function Sp(s), equation (2.33). Fig. 2.20 shows the magnitude plots of the sensitivity functions for the x and y axes. The bandwidths of both axes are summarized in Table 2.6. Bode Magnitude Diagram y-axis

0

10

w ith notch filter no notch filter

-1

10

1

10

Sp=pos err. / disturbance force [µm] / [N]

Sp=pos err. / disturbance force [µm] / [N]

Bode Magnitude Diagram x-axis

0

10

2

[Hz]

w ith notch filter no notch filter

-1

10

1

10

10

2

[Hz]

10

Fig. 2.20. Sensitivity function of x (left) and y (right) axes for the position loop Table 2.6. Bandwidths of the position loop bandwidth

x-axis

y-axis

26.6Hz

27.1Hz

The positive influence of the notch filters are emphasized from the removal of the structural resonance peak near 320 Hz, as indicated by the two circles. 37

Classical Motion Control

The peaks of the sensitivity magnitude curves lie below the conventional guideline of 6dB (or abs. value 2). This ensures good transient response characteristics. Also, the results of Table 2.6 confirm the fundamental characteristic of a cascade controller, that is, the bandwidths of the position loop is smaller than the bandwidth of the inner velocity loop. Next, the position closed loops of both axes are analysed and their Bode plots are shown in fig. 2.21. Bode Diagram

Magnitude (dB)

10 0

x-axis

-10

y-axis

-20 -30 -40 -50

Phase (deg)

-60 0 -90 -180 -270 -360 10

1

10

2

Frequency (Hz)

Fig. 2.21. Position closed loop transfer functions for x and y axes

In addition to the system bandwidth, dynamic stiffness gives a good measure of the controller disturbance rejection performance. Dynamic stiffness is the inverse of the transfer function from the input disturbance to the output position signals. In order to identify the system dynamic stiffness, a bandlimited white noise signal is introduced as a disturbance input to a system, and the position is measured. Based on these measurements, the dynamic stiffness is estimated using the H1 estimator [33]. The dynamic stiffness FRF measurements for the x and the y axes using a cascade P/PI controller, in comparison to the cascade P/P controller are shown in fig. 2.22. These measurements confirm the previous assertion that the cascade P/PI control structure has an infinite stiffness at lower frequency (equation 2.15) as a result of the integrator in the velocity loop. The minimum stiffness of the cascade P/PI controller is 15N/µm and 8N/µm for the x and the y axis respectively. 38

Classical Motion Control

10

10

10

y-axis

d istu rb an ce fo rce / p o sitio n [N] / [µ m ]

dis turba nc e forc e / pos ition [N ] / [µ m ]

x-axis

3

cascade P/PI

2

1

10

10

10

cascade P/P

0

10 0 10

3

1

0

1

10 frequency [Hz]

10

10 0 10

2

cascade P/PI

2

cascade P/P 1

10 frequency [Hz]

10

2

Fig. 2.22. Measured dynamic stiffness of x (left) and y (right) axes using cascade P/P and cascade P/PI controller

2.5.3 Tracking Performance Numerical Validation In addition to the disturbance forces rejection property, a good tracking performance is of equal importance in control design. The tracking performance is evaluated based on position and tracking errors. The tracking performance of the cascade P/PI controller is analysed for a sinusoidal reference input signal. A control scheme similar to fig. 2.10 with the system equals the system model Gm(s) is applied. Fig. 2.23 shows for both x and y axes, the simulated tracking errors for a sinusoidal reference signal with amplitude of 1mm and a frequency of 10Hz. The amplitude of these tracking errors corresponds to the magnitude of the error transfer function Ep(s)/Zref(s) (see equation 2.34) shown in fig. 2.24 (indicated by block square). x-axis

0.2 0 -0.2

-0.44 -0.5

0

0.2

0.4 0.6 time [s]

y-axis

0.5 0.42 position error [µm]

position error [µm]

0.5 0.44

0.8

1

0.2 0 -0,2 0,42 -0.5

0

0.2

0.4 0.6 time [s]

0.8

1

Fig. 2.23. Simulated tracking error of the x-axis (left) and the y-axis (right) for sinusoidal reference signal of an amplitude of 1mm and frequency of 10Hz 39

Classical Motion Control

Bode Magnitude Diagram

10

10

10

0

System: errorX Frequency [Hz]: 10 Magnitude (abs): 0.441

-1

-2

10

0

10

1

[Hz]

10

2

Magnitude (abs)

Magnitude (abs)

Bode Magnitude Diagram

10

10

10

0

System: errorY Frequency [Hz]: 10 Magnitude (abs): 0.423

-1

-2

10

0

10

1

[Hz]

10

2

Fig. 2.24. Position error transfer functions for the x (left) and y axes (right)

Significant tracking errors are still present. Tracking performance can be significantly improved by adding feedforward. This is discussed in the following section.

2.5.4 Cascade P/PI with Feedforward Various feedforward techniques are used to improve tracking performance in motion control system. Velocity feedforward, applied at the velocity loop of the cascade P/PI controller structure is considered. The tracking performance is further improved with an additional feedforward, based on the inverse open loop model of the system G-1(s) and position reference. Fig. 2.25 shows the resulting cascade configuration.

Fig. 2.25. Cascade P/PI with feedforwards for improved tracking performance 40

Classical Motion Control

The tracking performance analysis in 2.5.3 is repeated for cascade P/PI control with feedforward. A control scheme similar to fig. 2.25 is applied. A system model Gm(s) and the inverse open loop model feedforward Gm-1(s) are considered for this numerical validation purposes. Fig. 2.26 compares the tracking errors between the initial cascade P/PI control structure and the cascade P/PI with feedforward. x-axis

y-axis 0.5

position error [µm]

position error [µm]

0.5

0.25

0

-0.25

-0.5

with feedforward 0

0.5

time [s]

1

0.25

0

-0,25

-0,5

with feedforward 0

0.5

1

time [s]

Fig. 2.26. Simulated tracking error of the x-axis (left) and the y-axis (right) for sinusoidal reference signal of an amplitude 1mm and frequency of 10Hz

With feedforward, a zero tracking error is obtained (as indicated above by the thicker line). A complete removal of the tracking errors is accomplished using feedforwards of the velocity signal and of Gm-1(s). In the next section, correlation between a second order model time domain performance specifications and its frequency domain characteristics are analysed.

2.6 Correlation between Velocity Position Closed Loops Characteristic

and

Consider for the y-axis, a cascade control with PI velocity controller and a P position controller. A speed closed loop transfer function relating the actual velocity v and the reference velocity vref can be described according to the following form [35]: 41

Classical Motion Control

v 1 = 1 vref . s +1 ωos

(2.35)

ωos is the velocity closed loop bandwidth. Fig. 2.27 shows the velocity loop with a PI controller and the system model that is based on equation (2.2). The system model parameters (B and A) and the PI controller parameters are defined in Table 2.1 and Table 2.2 respectively.

Fig. 2.27. Schematic diagram of a velocity closed loop

The velocity closed loop transfer function based on the schematic diagram shown above is: v vref .

B + s A = B 1 + PI ⋅ s+A PI ⋅

(2.36)

A comparison between (2.35) and (2.36) is performed in the next paragraph. 1 PI ⋅ B 1 = ≈ 1 1 + + ⋅ s ( A PI B ) s +1 s +1 ωos PI ⋅ B ωos = PI ⋅ B ≈ 267.5 rad / s = 42.6 Hz

(2.37)

(2.38)

A bode plot of the velocity closed loop transfer function based on the control scheme in fig. 2.10 and transfer function 2.26 is shown in fig. 2.28. The velocity closed loop shown in fig. 2.28 has a bandwidth of 41 Hz. This value is closely estimated in (2.38).

42

Classical Motion Control

Bode Diagram

Magnitude (dB)

0 -20

System: spdtf_cl Frequency (Hz): 40.9 Magnitude (dB): -3.03

-40 -60 -80

Phase (deg)

0 -180 -360 -540 -720 0 10

1

2

10

10 Frequency (Hz)

Fig. 2.28. Y-axis: Simulated speed closed loop based on measured system FRF

Next, the position closed loop bandwidth ωop is estimated based on the following relationship: ζ=

1 ωos 2 kv

(2.39)

ωop = kv ωos

For kv = 150s-1 and ωos = 267.5rad/s, ωop = 200 rad / s ≈ 31 Hz ζ = 0.67

(2.40)

Fig 2.29 shows the measured FRF of the position closed loop obtained using band-limited white noise as the reference input. The FRF relates the reference position to actual position. The measured bandwidth of the system is identified at 28 Hz. This is nearly identical to the estimated bandwidth of 31Hz obtained in (2.40). A definite correlation between theoretical estimation and actual measurement result concerning the velocity and position closed loop characteristic has been established.

43

Classical Motion Control

Bode Diagram 20

Magnitude (dB)

0 System : frdFRF Frequency (Hz): 28 Magnitude (dB): -3

-20 -40 -60 -80 -100

Phase (deg)

0 -90 -180 -270 -360 10

0

10

1

10

2

Frequency (Hz)

Fig. 2.29. Measured FRF of the y-axis position close loop with kv =150 s-1

2.7 Summary This chapter discusses classical feedback control of motion systems. A cascade controller structure with a PI velocity loop controller and a P position loop controller is considered. Theoretical analysis of an ideal system using cascade P/P and cascade P/PI is presented. A cascade P/PI is then designed for the linear drive-based xy feed table considered in this thesis based on measured FRFs of the system. The cascade P/PI controller parameters are tuned according to gain margin and phase margin considerations of the model open loop transfer function. Velocity feedforward and feedforward of the inverse open loop model of the system G -1(s) are added to improve the tracking performance. A known correlation between velocity and position closed loop characteristics is successfully established. This correlation is confirmed using an actual experimental measurement of the position closed loop.

44

Chapter 3

Disturbance Forces in Servo Drives System 3.1 Introduction Disturbance forces directly affect the positioning and tracking accuracy of linear drive systems because of the lack of transmission. Two important disturbance forces are the highly nonlinear friction forces and cutting forces that occur during milling operation. Effective reduction of the effects of these disturbances is highly desirable. This requires a deep and detailed knowledge of the characteristics of these disturbances. This chapter focuses on the understanding, characterization and modelling of friction behaviour and cutting forces. Two different friction models are considered and the identification of the parameters of these models is discussed. A frequency domain analysis of the spectrum of cutting forces based on a sequence of cutting force measurements on a milling machine is provided. First, the friction force characterization and model structures are discussed and explained in the next section.

3.2 Friction Structures

Characterization

and

Model

Friction, by definition, refers to the resistance to motion during sliding of two opposing objects against one another. Friction is present in various applications and influences many aspects of our daily life. It brings benefit in various forms. Braking systems and clutches are two examples of friction effects that benefit us. However, friction is highly undesirable in most 45

Disturbance Forces in Servo Drives System

industrial and mechanical applications. Uncompensated friction force increases energy consumption and reduces efficiency of these processes. In a positioning system, at near zero velocity and during motion reversal, the highly complex non-linear friction behaviour introduces a unique form of contour tracking error, namely, the quadrant glitch error (fig. 1.1). Friction is categorized according to its pre-sliding and sliding regimes (see fig. 3.1). In pre-sliding regime, friction force is predominantly dependent on displacement. In sliding regime, the friction force is predominantly dependent on the sliding velocity. The most simple friction models consider the friction sliding regime only. These models are a static map between friction force and velocity, e.g. viscous, Coulomb and Stribeck friction models (see section 3.3).

Fig. 3.1. Pre-sliding and sliding friction regimes (reproduced from [14])

A first attempt in describing the more complex friction behaviour in presliding regime was accomplished by Dahl [5]. Since then, several other advanced and complex friction models have been proposed. This includes the LuGre friction model [6], the Leuven model [7, 8], and the Generalized Maxwell-slip (GMS) model [9]. The basic structures of the Dahl and the LuGre friction models are summarized in the following sections.

3.2.1 The Dahl Model The Dahl model is used extensively for simulating friction behaviour in ballbearing system. The model is an extension to the classic Coulomb friction, with smooth transitions around the critical zero velocity regions. A generalized first order differential equation of the position that is a function of the sign of the velocity v, approximates the hysteresis at presliding regime. The function is describes below: 46

Disturbance Forces in Servo Drives System

dF f dx

= σo 1 −

Ff Fs

sgn ( v )

δd

Ff ⎛ ⎞ sgn ⎜⎜1 − sgn ( v ) ⎟⎟ . Fs ⎝ ⎠

(3.1)

Ff and Fs are the total friction force and the static friction force respectively, σo is the initial stiffness of the contact at velocity reversal, and δd determines the shape of the hysteresis. The LuGre friction model is an extension and an improvement over the Dahl model. The model combines the pre-sliding friction behaviour of the Dahl model with the steady-state friction characteristic of the sliding regime.

3.2.2 The LuGre Model The model is based on the concept of averaging deformation of the contact asperities. The friction force is defined as: F f = σo z + σ1

dz + σ 2v. dt

(3.2)

σo, σ1, and σ2 are the asperity stiffness, micro-viscous friction coefficient, and viscous friction coefficient respectively. The state variable z represents the average deflection of the asperities and v is the velocity. dz v = v − σo z. dt s (v)

(3.3)

s(v) is the Stribeck curve, that is, a decreasing function for increasing velocity with upper limit and lower limit bounds corresponding to the static friction force Fs and Coulomb friction force Fc respectively (see section 3.3.1). Vs is the Stribeck velocity and δ is the Stribeck shape factor. ⎛ v − ⎜ Vs s ( v ) = sgn ( v ) ⎜ Fc + ( Fs − Fc ) e ⎜ ⎜ ⎝

δ

⎞ ⎟ ⎟. ⎟ ⎟ ⎠

(3.4)

The LuGre model fails to characterize the non-local memory behaviour of the presliding regime. Further improvement to this model is accomplished by the Leuven model that incorporates a hysteresis function with non local memory behaviour in presliding regime using a Maxwell-slip (see section 47

Disturbance Forces in Servo Drives System

3.3.3) approximation. The Leuven model is unattractive due to its difficult implementation. A recent improvement of the Leuven model is the GMS friction model [9]. This model is discussed in detailed in section 3.3.2. This thesis considers the widely known static friction model and the recent and more advanced GMS friction model for friction compensation. First, the static friction model is discussed in the following section.

3.3 Static Friction Model 3.3.1 Model Structure Static friction models describe the steady-state friction behaviour in sliding regime and hence are dependent on the sliding velocity v. The well-known static friction model considers Coulomb, viscous, and Stribeck friction. This is described in equation (3.5). Fc, Fs, and σ represent the Coulomb, static and viscous friction forces respectively. The Stribeck effect represents a decreasing effect of friction forces with increasing velocity. The Stribeck friction model parameters are the Stribeck velocity Vs and the Stribeck shape factor δ. ⎧ ⎛ v ⎪ F f ( v ) = ⎨ Fc + ( Fs -Fc ) ⋅ exp ⎜ − ⎜ Vs ⎪⎩ ⎝

δ

⎫ ⎞ ⎟ + σ ⋅ v ⎪⎬ ⋅ sign ( v ) . ⎟ ⎪⎭ ⎠

(3.5)

Fig. 3.2 illustrates the different friction components in the static friction model.

Fig. 3.2. Different friction components in static friction model (3.5).

The next section discusses the estimation of the model parameters for the considered linear drive xy feed table. 48

Disturbance Forces in Servo Drives System

3.3.2 Identification of the Static Friction Model At constant velocity, the motor force equals the friction force and is represented by the force control command signal. An illustration of a control command signal measurement during a constant velocity motion is shown in fig. 3.3.

control command signal [N]

160 140

this average is equivalent to the friction force

120 100 80 60 40 20 0 -20

0

5

10

15

20

25 30 time [s]

35

40

45

50

Fig. 3.3. Friction force from control command signals at constant velocity motion of 2.0 mm/s (y-axis)

Constant velocity is enforced using a manually tuned PID controller with a constant velocity reference signal (a ramp signal). The control command signal measurements for constant reference velocities of 0.010, 0.040, 0.080, 0.2, 0.4, 0.5, 1, 2, 4, 5, 8, 10, 14, 16, 20, 25, and 30 mm/s were averaged and recorded. Fig. 3.4 shows these averaged measured friction forces and the manually fitted friction-velocity map of the static friction force model (3.5). The identified parameters of the static friction model for both the x and y axes are tabulated in Table 3.1. Table 3.1. Static friction model parameters for x and y axes Fc

Fs

1/Vs

σ

δ

x-axis

122 N

177 N

0.00045 s/µm

0.00095Ns/µm

1

y-axis

105 N

165 N

0.001s/µm

0.00004Ns/µm

1

49

Disturbance Forces in Servo Drives System

Y-Axis

X-Axis

200

200 static friction,Fs

150

Coulomb friction,Fc

100

50 0 -50 -100

50 0 -50 -100

measured model

-150 -200 -40

static friction,Fs Coulomb friction,Fc

friction force [N]

friction force [N]

100

150

-20

0 20 velocity [mm/s]

40

measured model

-150 -200 -40

-20

0 velocity [mm/s]

20

40

Fig. 3.4. Friction force-velocity mapping and the manually fitted static friction model using equation (3.5).

The friction forces that act on the bottom x-axis are considerably larger than the y-axis friction forces. This is because the larger mass x-axis has more influence on the bearings than the top stage y-axis (the x-axis is supported by 6 Schneeberger MRD 35 carriages on 3 rails compared to the y-axis that is supported by 4 Schneeberger MRD 35 carriages on 2 rails). The static friction model fails to describe friction behaviour in pre-sliding regime. Therefore, a more advanced friction model, that is, the recently developed Generalized Maxwell-slip (GMS) model is considered.

3.4 Generalized Maxwell-slip Model (GMS) 3.4.1 Model Structure The GMS friction model is a relatively new model that has great potential for efficient friction compensation applications. Its main advantage is that it contains a hysteresis function with non-local memory behaviour to describe the pre-sliding regime. The GMS friction model incorporates the following components: (i) the Stribeck curve for constant velocity, (ii) hysteresis function with non-local memory for the pre-sliding regime, and (iii) frictional memory for the sliding regime. The structure of this model is similar to the Maxwell-slip structure [37, 38], that is, it consists of a parallel connection of N different elementary slip-blocks and springs (see fig. 3.5).

50

Disturbance Forces in Servo Drives System

Fig. 3.5. N-elementary configuration of the Maxwell-slip structure

Each block represents a generalized asperity of the contact surface that can either stick or slip and each element i has an elementary stiffness ki, a state variable zi that describes the element position, a maximum elementary Coulomb force Wi and a friction output Fi. A new state equation that characterizes sliding dynamics of each elementary slip-block replaces the original Coulomb law in the Maxwell-slip friction model structure. The dynamic behaviour of an elementary model is captured and described mathematically based on its current condition. At the time of sticking, which occurs during motion reversal and as velocity approaches zero, the dynamic behaviour of an elementary slip-block is described by a spring model with stiffness ki: dFi = ki v. . dt

(3.6)

On the other hand, slipping occurs if the elementary friction force Fi equals a maximum value Wi = αis(v), αi is the normalized sustainable maximum friction force of each element during sticking and s(v) is the Stribeck curve. During slipping, the state equation describing the dynamic behaviour of the elementary slip-block is represented as ⎛ dFi F ⎞ = sign ( v ) ⋅ C ⋅ ⎜ αi − i ⎟ . ⎜ dt s ( v ) ⎟⎠ ⎝

(3.7)

The constant parameter C (equals to 1/Vs in 3.5) indicates the rate at which the friction force follows the Stribeck effect in sliding. The total friction force is the summation of the output of all elementary state models and a viscous term σ (if viscous friction is present). 51

Disturbance Forces in Servo Drives System

N

F f ( v ) = ∑ Fi ( v ) + σ ⋅ v ( t ). i =1

(3.8)

A more detail description of the GMS friction model can be found in [10, 14]. The following section discusses the identification of the GMS model parameters.

3.4.2 Identification of GMS Model For the considered test setup, a GMS model with 4 elementary slip-blocks is selected, yielding a total of 13 model parameters: two parameters (αi‘s and ki’s) from each of the four elements and another five parameters from the state equations (3.5) in sliding regime. Increasing the number of elementary slip-block has minimal effect on the compensation performance of the GMS model [14]. The identification of the parameters of the state equation (3.5) is discussed in previous section 3.3.2. These values, for both the x and y axes, are presented in Table 3.1. In pre-sliding regime, friction is dominated by the displacement and behaves as a hysteretic function of displacement with non-local memory behaviour. This behaviour is characterized by the so-called virgin curve [39]. The virgin curve is derived from a sinusoidal excitation of the system. The frequency and amplitude of the sinusoidal input signal are selected to minimize inertia effect and to remain in the pre-sliding regime. Virgin curves for both axes are then constructed based on these measurements, and their parameters are identified. The identification procedures for the construction of the y-axis virgin curve are explained in the following paragraph. The virgin curve is constructed using two measurements obtained with different excitation amplitude. The measurement obtained with the small excitation amplitude (fig. 3.6(a)) provides detailed information about the friction-displacement behaviour in pre-sliding regime away from breakaway, while the measurement obtained with the larger excitation amplitude (fig. 3.6(b)) provides information close to breakaway or in the transition to sliding. Fig. 3.6(a) and 3.6(b) clearly indicate which part of the measurement is selected to compose the virgin curve shown in fig. 3.7(a). The selected parts are combined, reduced by a factor of 2 (the combined parts constitute a double-stretched version of the virgin curve), and shifted to the origin (0μm, 0N) to generate the virgin curve. 52

Disturbance Forces in Servo Drives System

100

80

80

A

60

friction force,[N]

friction force,[N]

B

60

40 20 0 -20 -40

A

40 20 0 -20 -40 -60

-60

-80

-80 -20

-10

0 position,[µm]

10

-100 -600

20

-400

-200 0 200 position,[µm]

(a)

400

600

(b)

90

90

α4

80

80

α3

70

70

friction force,[N]

friction force,[N]

Fig. 3.6. Y-axis: measured friction force and position for sinusoidal reference signal of 0.1Hz and amplitudes of (a) 15µm and (b) 450µm

60 50 40 30

60

40

10

10 10

20 30 40 position,[µm]

50

60

Ka

α1

30 20

0

α2

50

20

0 -10

Kc Kb

0 -20

Ko

0

20 position,[µm]

40

60

(a) (b) Fig. 3.7. Y-axis: (a) virgin curve and (b) virgin curve with selected knots and slopes

The four αi’s and ki’s slip-block parameters are estimated by approximating this virgin curve with a piecewise linear function. The knots and slopes of this piecewise linear function determine the αi’s and ki’s parameters. First, the knots of the piecewise linear curve are selected manually. Fig 3.7b shows the selected knots αi’s and the corresponding slopes Ki’s. Equation (3.9) summarizes the relationship between [K0, Ka, Kb, Kc], [α1, α2, α3, α4], and [k1, k2, k3, k4] based on fig. 3.8 and the theory of superposition.

53

Disturbance Forces in Servo Drives System

friction force,[N]

100 80

Kc

Kb 60

Ka

40

Ko 20

virgin curve 0

0

10

20

30

40

50

friction force,[N]

position,[µm] 50

α1

40

k1

30 20

k2

0

α2 α3 α4

k3

10

k4 0

10

20

30

40

50

position,[µm]

Fig. 3.8. Y-axis GMS model parameters identification α1 + α 2 + α3 + α 4 = ∑ Wi k1 + k2 + k3 + k4 = K 0 k2 + k3 + k4 = K a

(3.9)

k3 + k4 = Kb k4 = K c .

ΣWi is the friction force at the moment of breakaway. First, the values of [K0, Ka, Kb, Kc] are obtained from the measured virgin curve in fig. 3.7b for the manually selected knots [α1, α2, α3, α4]. Finally, based on these values and using (3.9), the values for [k1, k2, k3, k4] are obtained. Table 3.2 summarizes the identified GMS model parameters of the y-axis. Table 3.2. GMS slip-blocks model parameters for the y-axis αi [N] ki [N/μm]

α1 = 50

α2 = 20

α3= 15

α4 = 3.4

k1 = 99.94

k2 = 1.364

k3 = 1.081

k4 = 0.119

This technique is repeated for the GMS friction identification of the x-axis. The virgin curve is first constructed (see fig. 3.9 and 3.10) based on sinusoidal excitation at amplitudes of 15µm and 450µm, and a frequency of 0.1Hz. 54

Disturbance Forces in Servo Drives System

200

200

100

friction force [N]

100

friction force [N]

B

150

A

150

50 0 -50

50 0 -50

-100

-100

-150

-150

-200 -15

-10

-5

0

5

10

A

-200 -400

15

-200

position [µm]

0

200

400

position [µm]

160

160

140

140

120

120

friction force [N]

friction force [N]

(a) (b) Fig. 3.9. X-axis: Measured friction force and position for sinusoidal reference signal of 0.1Hz and amplitudes of (a) 15µm and (b) 450µm

100 80 60

60

20

20 10

20

30

40

50

α2 Ka

80

40

0

Kc Kb

100

40

0

α4 α3

0 -20

α1

-10

Ko

0

10

20

30

40

50

position [µm]

position [µm]

(a) (b) Fig. 3.10. X-axis: (a) virgin curve (b) virgin curve with selected knots and slopes (right)

The four αi’s and ki’s slip-block parameters are obtained based on fig. 3.10, fig. 3.11, and equation (3.9). The identified GMS model parameters for the x-axis are summarized in Table 3.3. Table 3.3. GMS slip-blocks model parameters for the x-axis αi [N] ki [N/μm]

α1 = 65

α2 = 55

α3= 26

α4 = 6.5

k1 = 61.80

k2 = 9.852

k3 = 4.490

k4 = 0.325

55

friction force, [N]

Disturbance Forces in Servo Drives System

150

Kc Kb

100

Ka 50

Ko 0

0

5

10

15

20

25

30

35

40

position [µm] friction force[N]

80

α1 α2

60

k1 40

α3

k2 20 0

k3 0

5

10

k4 15

20

α4 25

30

35

40

position [µm]

Fig. 3.11. X-axis: GMS model parameters identification

3.5 Cutting Forces Cutting forces act directly on the linear motor. Uncompensated cutting forces generate position and contour tracking errors. Knowledge of the cutting force characteristics provides valuable information on possible approaches and strategies for its effective compensation. For this purpose, a series of milling cutting processes were executed to measure and analyse the resulting cutting forces. These cutting processes were performed on an actual industrial milling machine and not on the experimental test setup itself. Details of these experiments are discussed in the following section.

3.5.1 Cutting Force Measurements and Analysis Cutting force measurements were performed using a Kistler 3-component dynamometer (type 9265B) force measurement unit, yielding cutting forces measurements in two horizontal directions, Fx and Fy. The experimental conditions of the milling process are summarized in Table 3.4.

56

Disturbance Forces in Servo Drives System

Table 3.4. Cutting process parameters Parameters

Value

Work piece material

Aluminium

Spindle speed

1360 rpm

10mm

Table feed rate

320mm/min

Cutter diameter

Parameters

3

No. of edges

1mm and 3mm

Depth of cut Sampling frequency

0.1 mm per tooth

Feed per tooth

Value

3000Hz

The cutting force measurements Fx and Fy for cutting depth of 1mm and 3mm are illustrated in fig. 3.12 and fig. 3.13 respectively. The cutting processes were performed along the x-axis of the milling table (see fig. 3.14). x-axis 200

start of cutting

150

150

100

100

50

50

force [N]

force [N]

200

y-axis

0

0

-50

-50

-100

-100

-150

-150 0

2

time [s]

4

6

0

2

4

6

time [s]

Fig. 3.12. Cutting force in x and y direction for cutting depth of 1mm

57

Disturbance Forces in Servo Drives System

x-axis

y-axis 400

start of cutting

300

300

200

200 force [N]

force [N]

400

100

100

0

0

-100

-100

-200

0

2

4 time [s]

6

-200

0

2

4

6

time [s]

Fig. 3.13. Cutting force in x and y direction for cutting depth of 3mm

cutting tool workpiece Kiestler measurement unit

Fig. 3.14. Cutting force measurements on a workpiece using Kistler dynamometer force sensor along the x-axis

Spectral analysis of the measured cutting forces for 1mm and 3mm depth of cut is shown in fig. 3.15 and fig. 3.16 respectively.

58

Disturbance Forces in Servo Drives System

y-axis

120

120

100

100

80

80

force [N]

force [N]

x-axis

60

60

40

40

20

20

0

0

100 200 frequency [Hz]

0

300

0

100 200 frequency [Hz]

300

Fig. 3.15. Spectral of cutting force along x and y direction, for cutting depth of 1mm y-axis

120

120

100

100

80

80

force [N]

force [N]

x-axis

60

60

40

40

20

20

0

0

100 200 frequency [Hz]

300

0

0

100 200 frequency [Hz]

300

Fig. 3.16. Spectral of cutting force along x and y direction, for cutting depth of 3mm

The fundamental frequency is 23.3Hz, that is, the frequency of the spindle rotation. Table 3.5 and Table 3.6 summarized the first 15 harmonic components of the measured cutting forces for 1mm depth of cut for x and y axes.

59

Disturbance Forces in Servo Drives System

3.5.2 Artificial Cutting Force An artificial cutting force, based on the first fifteen harmonic components of the measured cutting force, is synthesized for simulation and experimental validation purposes. The synthesized artificial cutting force is generated according the following equation (z = x or y axis): 15

(

)

F%z ( t ) = ∑ Ai ⋅ cos ( ωi t ) − Bi sin ( ωi t ) . i =1

(3.10)

Ai and Bi are the amplitude and the phase of the ith harmonic component of the measured cutting forces. These values are represented in Table3.5 and 3.6. Table 3.5. X-axis: Harmonic contents of the measured cutting force for 1mm depth of cut Harmonics

Freq. (Hz)

Ai

Bi

1

23.29

-2.147e+004

6.615e+004

2

46.59

1.45e+004

7.434e+004

3

69.88

1.544e+005

-1.353e+005

4

93.18

-2.491e+004

-4.705e+004

5

116.47

9.209e+003

1.860e+004

6

139.76

-7.812e+004

-1.512e+004

7

163.06

9.260e+002

1.460e+004

8

186.35

-1.589e+004

5.733e+002

9

209.65

3.384e+004

7.720e+003

10

232.94

8.626e+003

-1.410e+003

11

256.24

1.836e+003

1.618e+003

12

279.53

6.953e+003

9.00e+002

13

302.83

-4.101e+003

-1.294e+003

14

326.12

1.305e+004

2.9406e+002

15

349.41

8.234e+003

3.321e+003

60

Disturbance Forces in Servo Drives System

Table 3.6. Y-axis: Harmonic contents of the measured cutting force for 1mm depth of cut Harmonics

Freq. (Hz)

Ai

Bi

1

23.29

-6.792e+004

-1.263e+005

2

46.59

4.497e+003

-7.883e+004

3

69.88

4.833e+004

1.453e+005

4

93.18

4.546e+004

2.417e+004

5

116.47

-1.020e+004

1.563e+004

6

139.76

9.306e+004

2.657e+004

7

163.06

-4.213e+003

-1.180e+004

8

186.35

1.061e+004

3.850e+003

9

209.65

-6.261e+004

-1.425e+004

10

232.94

-4.878e+003

6.895e+003

11

256.24

-6.341e+003

-2.983e+003

12

279.53

1.574e+004

1.280e+004

13

302.83

-5.417e+003

4.901e+003

14

326.12

-3.133e+003

2.225e+003

15

349.41

-8.592e+003

-2.971e+003

A comparison between the measured cutting force and the synthesized artificial cutting force is shown in fig. 3.17. The synthesized cutting force signal closely resembles the actual cutting force.

61

Disturbance Forces in Servo Drives System

100 actual cutting force synthesized cutting force error

80 60

Force [N]

40 20 0 -20 -40 -60 -80 -100

0

0.02

0.04

0.06

0.08

0.1 time [s]

0.12

0.14

0.16

0.18

0.2

Fig. 3.17. X-axis: Comparison between the measured cutting force and the synthesized cutting force for 1mm depth of cut

3.6 Summary This chapter discusses the characteristics of the disturbance forces considered in the thesis; namely, the friction and the cutting forces. Two friction models, the widely known static friction model, and the recent more advanced GMS friction model are discussed. Unlike static friction model, the GMS friction model describes friction behaviour in both the sliding and the pre-sliding regimes. The GMS friction model is more complex and more difficult to identify. The identification of the two friction models for the two axes of the experimental test setup considered in this thesis is presented in detail. The characteristics of cutting forces based on cutting force measurements performed during an actual milling cutting process is presented. The spectral analysis of the cutting force reveals its harmonics component. Artificial cutting forces are synthesized from these measured cutting forces for simulation and experimental validation of cutting forces compensation performance of different techniques discussed in Chapter 5.

62

Chapter 4

Friction Force Compensation Design 4.1 Introduction Friction forces reduce the positioning and tracking accuracy in mechanical systems. Various friction compensation techniques have been described and presented in literature and have been highlighted in section 1.2.2. This chapter focuses on the application of two compensation techniques on the considered test setup: friction model feedforward and a disturbance observer that does not rely on a friction model. Feedforward friction compensation is validated for two different friction models: a static friction model (section 3.3) and a Generalized Maxwell-slip model (section 3.4). The disturbance observer is based on the inverse-model based disturbance observer design by Ohnishi [26]. The design and the analysis of this disturbance observer for the considered system are presented. The model applied in the disturbance observer is an approximate second order linear model of the considered system (equation 2.2). Finally, both techniques are combined and the friction compensation performances are compared [40]. Both numerical and experimental validations are performed. The magnitude of the quadrant glitch that appears near zero velocity or during motion reversal becomes the basis of performance comparison.

63

Friction Force Compensation Design

4.2 Friction Model-Based Feedforward The performance of feedforward friction compensation based on a static and a GMS friction model is evaluated numerically and experimentally. The general control scheme is shown in fig. 4.1 and is based on the control structure presented in section 2.5.4 (fig. 2.25). The control scheme of each individual axis consists of a cascade P/PI position controller that includes a proportional (P) position and a PI velocity feedback controllers, a notch filter N(s), a first order velocity estimator Vest(s) (2.22), a velocity feedforward, and a feedforward Gm-1 based on the inverse system model Gm (equation 2.2) and reference position. The friction model feedforward is inserted at the input of the plant.

Fig. 4.1. Friction compensation scheme using friction modelbased feedforward (kf is the force constant).

The following section discusses the numerical validation of friction feedforward compensation using two different friction models.

4.2.1 System Transfer Function with a GMS Friction Term for Friction Simulation The friction compensation performance is validated numerically in MATLAB/ Simulink. The system model Gm’ (see figure 4.1) consists of a linear second order model with delay (equation 2.2) in combination with the GMS friction model. Fig. 4.2 shows this system model in detail.

64

Friction Force Compensation Design

Fig. 4.2. System transfer function with a GMS friction model to characterize the complex friction behaviour.

The second order model parameters (parameters A and B in equation (2.2) and Table 2.1) are adapted manually such that the FRF of this system model Gm’ corresponds to that of the original linear model Gm (equation 2.2 with parameters in Table 2.1). Extending this original linear model with the GMS model yields a FRF that differs significantly from the FRF of the original linear model only. This is shown in figure 4.3a for the y-axis. The FRF of the original linear model is obtained by evaluating equation 2.2 with parameters in Table 2.1 in the frequency domain (replacing s by jω). The FRF of the linear model in combination with the GMS model is obtained by exciting this combined model with a band-limited white noise and estimating the FRF based on input and simulated output using the H1 [33] estimator. Fig. 4.3a clearly shows a difference in the phase diagram between both FRF's at low frequencies. In order to improve the correspondence between both FRF's, the A and B parameters of the linear model (2.2) were adapted manually. Better correspondence is obtained with the A' and B' parameter values shown in Table 4.1. This improved correspondence is clearly visible in fig. 4.3b. This procedure is applied to the models of both the x and y axes. To conclude, the system model used for numerical analysis in this section is model equation (2.2) with the A' and B' parameters of Table 4.1 in combination with the GMS friction model (equation 3.6, 3.7, 3.8) and GMS parameter values of Tables 3.2 and 3.3. Table 4.1. Original and revised system transfer function parameters A [s-1]

A’ [s-1]

B [m/volt.s2]

B’ [m/volt.s2]

x-axis

28.57

2.5

4.526

4.526

y-axis

20

0.095

8.916

8.916 65

Friction Force Compensation Design

Bode Diagram

Magnitude (dB)

150 100 50 0 -50

original linear plant tf. based on measured FRF original linear plant tf. linear model+ GMS model

Phase (deg)

-100 0 -90 -180 -270 -360 -450 -540 10

1

Frequency (Hz)

10

2

(a) Bode Diagram Magnitude (dB)

150 100 50 0 -50

original linear plant tf. based on measured FRF revised linear plant tf + GMS model

Phase (deg)

-100 0 -100 -200 -300 -400 -500 10

1

Frequency (Hz)

10

2

(b) Fig. 4.3. System model parameters adjustment for system structure that includes the GMS friction term: (a) using the original model parameters and (b) using updated model parameters.

66

Friction Force Compensation Design

4.2.2 Friction Feedforward: Numerical Results Feedforward friction compensation performance for both axes is validated numerically using MATLAB/Simulink, based on the control scheme shown in fig. 4.1. The reference trajectory is a sinusoidal signal which corresponds to a frequency of 0.53 Hz with an amplitude of 30mm and a maximum tracking velocity of 100mm/s. Friction model feedforward using the static friction model (see equation 3.5 and Table 3.1) and the GMS friction model (see equation 3.6-3.8 and Table 3.2) are applied. The position signal and the tracking errors for the y-axis are shown in fig. 4.4. Significant quadrant glitch magnitudes are observed for the case where no friction compensation is applied (indicated by the circle in fig. 4.4a). No Friction Model FF.

0 0.5 5 x 10

0 -1 0.5

1

0 -20 0.5

1 time [s]

-2

1.5

0 0.5 5 x 10

1

0 -1 0.5

1

10 0 -10 -20

0

0.5

1 time [s]

1.5

GMS Friction Model FF.

0 -2 0 0.5 5 x 10

1

1.5

0

0.5

1

1.5

0

0.5

1 0 -1

1.5

20

x 10

2

-4

1.5

1

0 tracking error,[µm]

20

0

0

1.5

40

-40

2

-4

1.5

1

0 tracking error,[µm]

1

4

4 position [µm]

-2

x 10

velocity [µm/s]

position [µm]

0

Static Friction Model FF.

4

4

velocity [µm/s]

position [µm]

2

-4

velocity [µm/s]

x 10

tracking error,[µm]

4

4

20 10 0 -10 -20

1 1.5 time [s]

(a) (b) (c) Fig. 4.4. Y-axis: Simulated position, velocity and tracking error for three different cases of friction compensation techniques (a) no friction feedforward, (b) Stribeck friction model feedforward, (c) GMS model feedforward.

The results show a significant reduction of the quadrant glitch magnitude using the static and GMS friction model feedforward. The GMS friction model feedforward almost completely compensates the hard non-linear friction behaviour. A complete compensation of friction using the GMS friction model feedforward is not possible as a result of the delay term in the model structure. Fig. 4.5 shows that complete friction compensation is obtained for system where the delay term is removed. 67

3

3

2

2 tracking error [µm]

tracking error [µm]

Friction Force Compensation Design

1 0 -1 -2 -3

1 0 -1 -2

0

0.5

1 tim e [s]

1.5

-3

0

0.5

1 tim e [s]

1.5

(a) (b) Fig. 4.5. Y-axis: Simulated tracking error using GMS model feedforward for (a) system model with delay and (b) with the delay removed.

Similar simulations are performed for the x-axis. Table 4.2 summarizes the results of friction compensation for both axes based on the magnitude of the quadrant glitches (shown in circle). The magnitude of the quadrant glitches are reduced from a maximum of 31 micrometer (no friction compensation) to near zero (GMS model feedforward). Table 4.2. Simulated friction compensation performance for a reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s Friction Compensation Methods

Quadrant Glitch [µm] x-axis

y-axis

No friction feedforward

15

31

Static friction feedforward

5

8.3

GMS friction feedforward

4

1.2

In feedforward friction compensation, the performance depends on the accuracy of the friction model. The performance robustness against friction model parameters variation is analysed next. A simulation using a feedforward of an adjusted GMS friction model parameters is performed (system has no delay). Fig. 4.6 shows the position error in comparison to the result of fig. 4.5(b). Table 4.3a and 4.3b list the original GMS friction model parameters and the adjusted model parameters respectively.

68

4

4

2

2

position error [µm]

position error [µm]

Friction Force Compensation Design

0

-2

-4

0

-4 0

0.5

1 time [s]

1.5

2

0

0.5

1 time [s]

1.5

2

(a) (b) Fig. 4.6. Y-axis: Simulated tracking error using feedforward of (a) adjusted GMS model and (b) matched GMS model, for system both without delay. Table 4.3a. Y-axis: Original GMS slip-blocks model parameters αi [N] ki [N/μm]

α1 = 50

α2 = 20

α3 =15

α4 = 3.4

k1 = 99.94

k2 = 1.364

k3 = 1.081

k4 = 0.119

Table 4.3b. Y-axis: Adjusted GMS slip-blocks model parameters αi [N] ki [N/μm]

α1 = 57

α2 = 26.5

α3 = 15

α4 = 3.4

k1 =110.0

k2 =2.0

k3 = 1.5

k4 = 0.119

The experimental validation of the friction model-based feedforward compensation is discussed in the next section.

4.2.3 Friction Feedforward: Experimental Results Experimental validation of the feedforward friction model compensation is performed on both axes individually. The reference is again a sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s. Friction model feedforward, using the static friction model (see equation 3.5 and Table 3.1) and the GMS (see equation 3.6-3.8 and Table 3.2) friction model are applied. The measured position and the position tracking errors are recorded. Fig. 4.7 shows for the y-axis, a significant improvement in tracking accuracy when applying the static friction model and GMS friction model feedforward. Table 4.4 lists a summary of the friction compensation results for both axes. 69

Friction Force Compensation Design

4

No Friction Feedforward

4

x 10

-3 0.5

0

1 1.5 time,[s]

0.5

1 1.5 time,[s]

20 0 -20 -40 0

0.5

1 1.5 time,[s]

(a)

10 0 -10 -20

0

0.5

1 1.5 time,[s]

0

0.5

1 1.5 time,[s]

20 tracking error,[µm]

tracking error,[µm]

20

40

0

-3 0

60

GMS Friction Model

3

-3 0

tracking error,[µm]

x 10

3

0

-60

4

Static Model FF.

position,[µm]

3 position,[µm]

position,[µm]

x 10

0

0.5

1 1.5 time,[s]

(b)

10 0 -10 -20

(c)

Fig. 4.7. Y-axis: Measured position and tracking error for (a) no friction feedforward, (b) static friction model feedforward, (c) GMS feedforward. Table 4.4. Measured friction compensation performance for a reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s Friction Compensation Methods

Quadrant Glitch [µm] x-axis

y-axis

No friction feedforward

15

35

Static friction feedforward

5

13

GMS friction feedforward

4

10

The GMS friction model feedforward yields the least quadrant glitch magnitudes on both x and y axes. A 71% reduction in the magnitudes of the quadrant glitches are observed with GMS friction model feedforward (compared to the uncompensated case). The improvement obtained using the GMS friction model over static friction model is small for this reference trajectory which has a maximum tracking velocity of 100mm/s. The use of a complex GMS friction model cannot be motivated for this relatively fast motion. However, by further reducing the tracking velocity to 10mm/s, the pre-sliding regime becomes more dominant and the benefit of using the GMS friction model is more pronounced. This is shown explicitly in fig. 4.8 and fig. 4.9 for the x and y-axis respectively. 70

Friction Force Compensation Design

4

4

0 -2

GMS friction Model FF.

0 -2

0

5

10 time [s]

-4

15

20

20

10

10

0 -10 -20

x 10

2 position

2

-4

glitch [µm]

4

Static friction FF.

x 10

glitch

position [µm]

4

0

5

10 time [s]

15

0

5

10 time [s]

15

0 -10

0

5

10 time [s]

-20

15

(a)

(b)

Fig. 4.8. X-axis: Measured quadrant glitches for maximum tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward.

4

4

Static Model FF.

x 10

x 10 3 position,[µm]

position,[µm]

3

0

-3 5

10 time [s]

15

20

0

5

10 time [s]

15

20

0

5

10 time [s]

15

20

20 tracking error,[µm]

20 tracking error,[µm]

0

-3 0

10 0 -10 -20

GMS Friction Model FF.

0

5

10 time [s]

15

20

10 0 -10 -20

(a) (b) Fig. 4.9. Y-axis: Measured quadrant glitches for maximum tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward. 71

Friction Force Compensation Design

Next, circular tests are performed, where friction is compensated with static and GMS friction models feedforward, simultaneously on the x and y axes. These results are shown in fig. 4.10 for a tangential tracking velocity of 100mm/s. The tracking errors are amplified by a factor of 400 for graphical presentation purposes. static friction model ff.

no friction feedforward

0 -20

-40

0 -20

10 0

0

90 180 270 angle [degree]

(a)

360

20 10 0 -10 0

0 -20

-40

-20 0 20 40 position X [mm]

30 radial error [µm]

20

20

-40

-20 0 20 40 position X [mm]

30 radial error [µm]

radial error [µm]

30

-10

20

-40 -40

-20 0 20 40 position X [mm]

40 position Y [mm]

position Y [mm]

position Y [mm]

20

-40

GMS friction model ff.

40

40

90 180 270 angle [degree]

(b)

360

20 10 0 -10 0

90 180 270 angle [degree]

360

(c)

Fig. 4.10. Measured contour and radial tracking error at tangential tracking velocity of 100mm/s: (a) no friction feedforward, (b) static friction model feedforward, (c) GMS model feedforward.

Previous results have shown that the advantage of using the GMS friction model is not clearly demonstrated at higher tracking velocity. The circular tests are repeated at a lower tangential tracking velocity of 10mm/s. The measured contour and the radial errors for the two friction models are shown in fig. 4.11.

72

Friction Force Compensation Design

static model (error x400)

GMS model (error x400) 30 position Y [mm]

position Y [mm]

30 15 0 -15

-30

-15

-30

-15 0 15 30 position X [mm]

20

-15 0 15 position X [mm]

30

20 radial error [µm]

radial error [µm]

0

-30

-30

10 0 -10 -20 0

15

90

180 270 angle [degree]

360

10 0 -10 -20 0

90

180 270 angle [degree]

360

(a) (b) Fig. 4.11. Measured contour and radial tracking errors for tangential tracking velocity of 10mm/s: (a) static friction model feedforward and (b) GMS model feedforward.

The experimental results presented thus far have identified the advantage of the GMS friction model over the more simple static friction model for friction compensation especially at lower tracking velocity, where the presliding regime becomes more dominant. The results are not perfect because of the imperfection of the GMS friction model. The next section considers a friction model-free approach for friction compensation, namely, the inverse-model-based disturbance observer [26, 27].

4.3 Inverse- Model- Based Disturbance Observer Inverse-model-based disturbance observer has been studied and applied in various high accuracy positioning systems; namely the hard disk drive and high-speed servo systems [41, 42]. In this thesis, the disturbance observer is applied as an add-on module to the cascade P/PI controller for friction force and cutting force compensation. Its structure and design is based on the work of Ohnishi [26] that has been further refined by Umeno and Hori [43]. The 73

Friction Force Compensation Design

disturbance observer estimates disturbance forces that can be either real disturbances acting on the system, for example cutting forces if present, or forces representing the linear model and friction model inaccuracies. The disturbance force estimate equals the difference between the control command signal and the input obtained by filtering the system output signal z with the inverse of a selected plant model Gn(s). Fig. 4.12 shows a schematic diagram of the system G(s) and the inverse-model-based disturbance observer. A low pass filter, known as the Q-filter [28], is added to preserve stability. Fig. 4.13 represents an equivalent block diagram to the control scheme shown in fig. 4.12.

Fig 4.12. Block diagram of a system with an inverse-model-based disturbance observer.

Fig 4.13. Equivalent block diagram of a system with an inverse model-based disturbance observer.

The disturbance observer design freedom is restricted to the selection of the inverse model Gn-1 and the characteristics of the filter Q. For the considered application, the inverse model Gn-1 is the inverse of the nominal second order plant model which corresponds to model (2.2) without delay and parameter values presented in Table 2.1. The delay is omitted from model (2.2) because this would yield a non-causal inverse. The selection of the filter Q is discussed in the following section 4.3.1. 74

Friction Force Compensation Design

4.3.1 Q-filter Design and Stability Analysis Consider the equivalent control scheme shown in fig. 4.13. The following transfer functions can be derived: GUpi − z ( s ) =

Gd − z ( s ) =

Gn − z ( s ) =

Z (s)

U pi ( s )

Z (s)

D(s) Z (s)

N (s)

=

=

=

G ⋅ Gn ; Gn + Q ( G − Gn )

G ⋅ Gn (1 − Q )

(4.1)

;

(4.2)

G ⋅Q . Gn + Q ( G − Gn )

(4.3)

Gn + Q ( G − Gn )

The above transfer functions show that if Q → 1, the followings are true: GUpi − z ( s ) = Gn ; Gd − z ( s ) = 0.

(4.4)

Secondly, if Q → 0, then, Gn ( s ) = 0.

(4.5)

Equation (4.4) shows that as Q → 1 at low frequency, the closed loop behaves as the nominal plant Gn(s), and the effect of the disturbance force on the output position z is zero. In addition, as Q → 0 at higher frequency, the effect of noise on the output position z becomes zero (equation 4.5). These analyses confirm the necessity of the transfer function Q being a low pass filter. Q is also necessary in order to realize G-1(s). Ryoo, Doh, and Chung [44] have suggested the following structure for the low pass filter Q: m

Q(s) =

i ∑ qi s

i =0

( s + ωc ) n

(4.6)

where; qi’s are the filter numerator coefficients, ωc is the cut-off frequency and m and n are the order of the numerator and the order of the denominator respectively, with n > m. The difference in n-m is used to make Q/Gn proper. 75

Friction Force Compensation Design

According to Kwon [45], the coefficients qi’s can be determined from binomial coefficients of the following form: qi =

n!

⋅ ωcn −i

( ( n − i )!i! )

(4.7)

Kempf and Kobayashi [24] have illustrated the significance of ωc on the performance of the disturbance observer. A higher cut-off frequency is desired for better disturbance attenuation. However, the bandwidth of the filter Q is limited by the un-modelled dynamics of the system. These unmodelled dynamics are expressed as a multiplicative perturbation Δ(f):

(

ˆ ( f ) = G ( f ) 1+ Δ( f ) G n Δ( f ) =

)

(4.8)

ˆ ( f )−G ( f ) G n

(4.9)

Gn ( f )

Ĝ(f) represents the real system dynamics. Since the real system dynamics are nonlinear and the expression is only valid for linear systems, the measured FRF of the system, presented in fig. 2.4 for the x and y axes respectively, are used for G(f) in equation (4.9). In the absence of any un-modelled dynamics, the disturbance observer loop, described by the equivalent diagram shown in fig. 4.13, has an open loop transfer function equal to: Gol ( s ) =

D% ( s )

U PI ( s )

=

Q(s)

1 − Q(s)

..

(4.10)

The complementary sensitivity transfer function T(s) of the disturbance observer loop is Q(s). Based on H∞ theory, it can be stated that the robust stability of the disturbance observer loop is guaranteed only if, T ( jω ) ⋅ Δ ( jω )

∞

≤1

(4.11)

or if, Q(s) ≤

76

1 Δ(s)

(4.12)

Friction Force Compensation Design

This means that the magnitude of the filter Q has to lie below the |1/Δ| line. The |1/Δ| line is constructed based on equation (4.9). Fig. 4.14 shows the |1/Δ| curves for both axes and the designed Q filters of which the parameters are presented in Table 4.5. The bandwidth of the filter Q is 60 Hz and 90 Hz for the x and y axes respectively. Figure 4.14 illustrates that further increasing the bandwidth of the filter Q reduces the stability margins and increases the possibility of the filter Q line intersecting the |1/Δ| curve. A reasonable margin is required to ensure stability during actual practical implementation of the disturbance observer on the considered test setup. Bode Magnitude Diagram (x-axis)

Bode Magnitude Diagram (y-axis)

40

30

1/∆

20

1/∆

20

Magnitude (dB)

Magnitude (dB)

10 0 Q(s) -20

0 Q(s) -10 -20 -30

-40 -40 -60 1 10

2

10 Frequency [Hz]

-50 1 10

2

10 Frequency [Hz]

(a)

(b)

Fig. 4.14. Bandwidth limitation of the filter Q for (a) x-axis and (b) y-axis. Table 4.5. Characteristics of the low pass filter Q m

n

wc

Q

x-axis

0

2

60Hz

Q(s ) =

y-axis

0

2

90Hz

Q(s ) =

1.421e005 2

s + 754 s + 1.421e005 3.198e005 2

s + 1131s + 3.198e005

The order of the filter numerator m and denominator n are selected such that the transfer function Q(s)/Gn(s) is proper that is n-m ≤ 2 (since the relative degree of the nominal plant model Gn(s) is two).

77

Friction Force Compensation Design

4.3.2 Loops Characteristic with Disturbance Observer Adding this disturbance observer changes the characteristic of the velocity loop and the position loop. This section analyses these loops. The loops transfer functions are analysed to determine the influence of the added disturbance observer on the loops characteristics (i.e. gain margin, phase margin, and stability). Fig. 4.15 shows the schematic diagram of the control scheme consisting of the cascade P/PI controller and the disturbance observer.

Fig. 4.15. Cascade P/PI position control with an inverse model-based disturbance observer.

In order to calculate these loops’ characteristics, G(s) is replaced by the measured system FRFs (Ĝ) shown in figure 2.4, and the different transfer functions in this control scheme are evaluated at the corresponding frequencies, that is s is replaced by jω. Fig. 4.16 shows an equivalent control scheme that is based on fig. 4.13 and 4.15.

Fig. 4.16. Cascade P/PI position control with equivalent block diagram of the inverse model-based disturbance observer.

78

Friction Force Compensation Design

First, the velocity loop with the added disturbance observer is considered. The velocity open loop, closed loop, and sensitivity transfer functions equal: Vdob _ ol =

Vdob _ cl =

Z& ( s )

=

Ev ( s ) Z& ( s )

U p (s)

Sv _ dob ( s ) =

=

ˆ ⋅ G ⋅V ( s ) ⋅ N ( s ) PI ⋅ G n est , ˆ −G +G Q G

(4.13)

ˆ ⋅ G ⋅V ( s ) ⋅ N ( s ) PI ⋅ G n est , ˆ ˆ ⋅ G ⋅V ( s ) ⋅ N ( s) Gn + Q G − Gn + PI ⋅ G n est

(4.14)

(

n

)

(

n

)

1 . ˆ ˆ ⋅ G ⋅V ( s ) ⋅ N ( s ) Gn + Q G − Gn + PI ⋅ G n est

(

(4.15)

)

Fig. 4.17 and 4.18 illustrate the influence of the disturbance observer on the velocity open loop and closed loop characteristics for the x and y axes respectively. x-axis Magnitude (dB)

40 20 0 -20 -40

Phase (deg)

-60 90 0 -90 -180 -270 -360

no observer with observer 1

10

2

Frequency (Hz)

10

(a)

79

Friction Force Compensation Design

y-axis Magnitude (dB)

40 20 0 -20 -40

Phase (deg)

-60 90 0 -90 -180 -270

no observer with observer

-360

1

10

2

Frequency (Hz)

10

(b) Fig. 4.17. Velocity open loop transfer function for (a) x-axis and (b) y-axis for system with and without a disturbance observer.

Table 4.6 summarizes the gain margin and phase margin for the system with and without the disturbance observer, based on fig. 4.17. Table 4.6. Gain margin and phase margin of x and y axes velocity open loop Gain Margin

Phase Margin

no observer

with observer

no observer

with observer

x-axis

12.332 dB

11.02 dB

74.514 deg

67.345 deg

y-axis

14.622 dB

12.938 dB

71.888 deg

65.11 deg

The inverse model-based disturbance observer reduces the gain margin and the phase margin of the cascade P/PI controller. The gain margin and the phase margin of both axes were reduced by nearly 1.5dB and 7 degrees respectively. For both axes, the gain margin and the phase margin remain at acceptable values and ensure good transient response and stability margin.

80

Friction Force Compensation Design

x-axis Magnitude (dB)

0 -10 -20 -30 -40 -50

Phase (deg)

0 -90 -180 -270

no observer with observer

-360

1

10

2

Frequency (Hz)

10

(a) y-axis

Magnitude (dB)

0 -10 -20 -30 -40 -50

Phase (deg)

0 -90 -180 -270 -360

no observer with observer 1

10

2

Frequency (Hz)

10

(b) Fig. 4.18. Velocity closed loop transfer functions for (a) x-axis and (b) y-axis for system with and without a disturbance observer.

Fig. 4.18 shows the bandwidth improvement for velocity loops of both axes. A nearly 5 Hz increase in bandwidth is recorded for each of the axes. Similar conclusion follows from the velocity loops sensitivity curves (equation 4.15, shown in fig. 4.19). Here, an average bandwidth improvement of 2Hz is observed. The arrows clearly indicate the reduction in the sensitivity magnitude at lower frequency and the corresponding increases in the magnitude at the higher frequency- the waterbed effect [46]. 81

Friction Force Compensation Design

Bode Magnitude Diagram (x-axis)

Bode Magnitude Diagram (y-axis)

5

5 Sspd

Sspd

0 Freq(Hz): 28.9

0 Freq (Hz): 28.9

-5

Magnitude (dB)

Magnitude (dB)

Mag(dB): -3.01

Sspd_dob Freq(Hz): 29.9 Mag(dB): -3.01

-10

-15

Mag(dB): -3.02

-5

Sspd_dob Freq (Hz): 32 Mag(dB): -3.02

-10

-15 no dist. observer w ith dist. observer

-20 1 10

no dist. observer w ith dist. observer

-20 1 10

2

10 Frequency [Hz]

10 Frequency [Hz]

2

Fig. 4.19. Effect of the disturbance observer on the velocity loop sensitivity transfer functions for x-axis (left) and y-axis (right)

Next, the velocity loop stability margin with the disturbance observer is identified. The Nyquist plot of equation (4.13) (see fig. 4.20) indicates the velocity loop stability and robustness. Nyquist Diagram (y-axis) 3

2

2

1

1

Imaginary Axis

Imaginary Axis

Nyquist Diagram (x-axis) 3

0 -1 -2

0 -1 -2

no observer with observer

-3 -1.5

-1

-0.5

0 0.5 Real Axis

1

1.5

-3 -1.5

no observer with observer -1

-0.5 0 0.5 Real Axis

1

1.5

Fig. 4.20. Nyquist plots of the velocity loops with and without the disturbance observer.

This analysis is further extended to include the position loop. The position open loop and closed loop transfer functions (based on control scheme shown in fig. 4.16) with the disturbance observer equals: L pos =

82

Z (s)

Ep (s)

=

P ⋅ PI ⋅ GGn , Q ( G − Gn ) + Gn + PI ⋅ GGn ⋅ Vest ( s ) ⋅ N ( s )

(4.16)

Friction Force Compensation Design

T pos =

Z (s)

Z ref ( s )

=

P ⋅ PI ⋅ G ⋅ Gn . Q ( G − Gn ) + Gn + PI ⋅ GGn ⋅ Vest ( s ) ⋅ N ( s ) + P ⋅ PI ⋅ G ⋅ Gn

(4.17)

The compliance transfer function is Z (s)

D(s)

=

G (1 − Q ) ⎛ G ⎞ − 1⎟ + P ⋅ PI ⋅ G + GPI ⋅ Vest ( s ) ⋅ N ( s ) + 1 Q⎜ G ⎝ n ⎠

(4.18)

.

The sensitivity function is

(1 − Q )

S p _ DOB ( s ) =

⎛ G ⎞ − 1⎟ + P ⋅ PI ⋅ G + G ⋅ PI ⋅ Vest ( s ) ⋅ N ( s ) + 1 Q⎜ G ⎝ n ⎠

(4.19)

.

The tracking error is ⎛ ⎞ ⎛ G ⎞ − ⎜1 + Q ⎜ − 1⎟ + PI ⋅ G ⋅ Vest ( s ) ⋅ N ( s ) ⎟ ⎜ ⎟ G ⎝ n ⎠ ⎝ ⎠ E p _ DOB ( s ) = Z ref ( s ) ⎛ G ⎞ 1+ Q⎜ − 1⎟ + PI ⋅ G ⋅ Vest ( s ) ⋅ N ( s ) + P ⋅ PI ⋅ G ⎝ Gn ⎠ _

G (1 − Q )

⎛ G ⎞ 1+ Q⎜ − 1⎟ + PI ⋅ G ⋅ Vest ( s ) ⋅ N ( s ) + P ⋅ PI ⋅ G G ⎝ n ⎠

(4.20)

D(s).

Fig. 4.21 and 4.22 show the Bode plots of the position open loop and closed loop transfer functions based on the measured system FRFs (fig. 2.4) respectively. The plots compare the loop characteristics with and without the disturbance observer. Table 4.7 summarizes the gain margin and the phase margin of the position open loop transfer function (4.16). These values ensure good transient response characteristic and stability margin of the position loop. Table 4.7. Gain margin and phase margin of x and y axes position open loop Gain Margin

Phase Margin

no observer

with observer

no observer

with observer

x-axis

15.071 dB

13.621 dB

67.959 deg

69.561 deg

y-axis

16.331 dB

13.773 dB

67.295 deg

70.113 deg 83

Friction Force Compensation Design

The gain margin and the phase margin of both axes were reduced by about 1.5-2 dB and 3 degrees respectively. As with velocity loop, the resulting gain and phase margins are appropriate to ensure good system transient response and stability margin. x-axis

Magnitude (dB)

20 0 -20 -40 -60

Phase (deg)

-90

-180

-270

-360

no observer with observer 1

10

Frequency (Hz)

(a)

20 0 -20 -40 -60

1

10

84

2

10

Friction Force Compensation Design

Next, the position closed loop transfer functions of both axes are analysed. Fig. 4.22 shows the Bode diagram of the position closed loop transfer functions. x-axis Magnitude (dB)

0

-20

-40

-60 Phase (deg)

0 -90 -180 -270

no observer wth observer

-360

1

10

2

Frequency (Hz)

10

(a) y-axis

Magnitude (dB)

0

-20

-40

-60 Phase (deg)

0 -90 -180 -270 no observer with observer

-360

1

10

2

Frequency (Hz)

10

(b) Fig. 4.22. Position closed loop transfer functions for (a) x-axis and (b) y-axis for system with and without a disturbance observer.

The disturbance observer increases the bandwidth of both axes position loops by nearly 5Hz. The updated bandwidths of the x-axis and the y-axis are 40.6Hz and 43.2Hz respectively. 85

Friction Force Compensation Design

A comparison in the sensitivity magnitude Bode plots based on equation (4.19) and (2.33) for a system with and without the disturbance observer respectively is shown in fig. 4.23 below. Bode Magnitude Diagram (y-axis) 10

0

0

-10

-10 Magnitude (dB)

Magnitude (dB)

Bode Magnitude Diagram (x-axis) 10

-20 -30 -40 -50

-30 -40 -50

-60 -70

-20

no observer with observer 1

10

2

-60 -70

10 Frequency (Hz)

no observer with observer 1

10

(a)

2

10 Frequency (Hz)

(b)

Fig. 4.23. Effect of the disturbance observer on the sensitivity function of the position loop for the (a) x-axis and (b) y-axis

Table 4.8 summarizes the bandwidths of the position loop based on the sensitivity curve shown in fig. 4.23. Moderate improvements in the bandwidths of both axes are observed. Table 4.8. Bandwidth of x and y axes position loops Bandwidth no observer

with observer

x-axis

26.6 Hz

29.2 Hz

y-axis

27.2 Hz

34 Hz

The disturbance observer improves tracking and disturbance rejection performances in the lower frequency range (below the system bandwidth), as seen by shifted sensitivity curves (illustrated by the arrows). This, however, increases the magnitude of the sensitivity function at higher frequency range. This phenomenon is generally known as the “waterbed effect” [46].

86

Friction Force Compensation Design

Finally, the stability of each axis position loop is analysed. Fig. 4.24 shows the Nyquist plots of each axis position open loop transfer function Lpos(s) that confirm the loop stability and robustness. x-axis

1.5

1.5

1

1

0.5 0 -0.5

0.5 0 -0.5

-1

-1

-1.5

-1.5

-2 -1.5

no observer with observer -1

-0.5 0 Real Axis

y-axis

2

Imaginary Axis

Imaginary Axis

2

0.5

1

-2 -1.5

no observer with observer -1

-0.5 0 Real Axis

0.5

1

Fig. 4.24. Nyquist plots of the position loops with and without the disturbance observer.

Based on the presented Bode and Nyquist plots of the open-loop transfer functions and on the comparison of the Q filter frequency characteristic and the inverse of the amplitude of the multiplicative uncertainty (fig. 4.14), it can be concluded that there is still room to increase the bandwidth of the disturbance observer and closed loop transfer function. However, implementing this higher bandwidth inverse model-based disturbance observer in the experimental test setup resulted in actuator saturation and thus limits the performance potential of the disturbance observer. Next, the friction compensation performance of the inverse-model-based disturbance observer is analysed numerically using MATLAB/Simulink.

4.3.3 Numerical Validations The friction compensation performance of the inverse-model-based disturbance observer is validated numerically with MATLAB/Simulink, using the simulation model presented in section 4.2.1. The trajectory is a sinusoidal signal with an amplitude of 30 mm, a frequency of 0.53Hz, and a maximum tracking velocity of 100 mm/s. The simulated control scheme is shown in fig. 4.25. 87

Friction Force Compensation Design

Fig. 4.25. Simulated control scheme for friction compensation using friction model feedforward and a disturbance observer.

The tracking and friction compensation performances are analysed using 3 different compensation configurations: a) An inverse-model-based disturbance observer only b) An inverse-model-based disturbance observer with a static friction model feedforward c) An inverse-model-based disturbance observer with a GMS model feedforward Fig. 4.26 shows the simulated results for the y-axis for these different compensation configurations. Similar simulations are performed for the xaxis. The results are tabulated in Table 4.9.

88

Friction Force Compensation Design

x 10

0 -2 -3

10 0 -10 0

0.5

3 2 0 -2 -3

1 1.5 time [s]

0

tracking error [µm]

0.5

1 1.5 time [s]

Observer + Static Model

4

x 10 position [µm]

position [µm]

position [µm]

3 2

0 tracking error [µm]

4

Observer Only

0.5

10 0 -10 0

0.5

1 1.5 time [s]

Observer + GMS Model

3 2 0 -2 -3

1 1.5 time [s] tracking error [µm]

4

x 10

0

0.5

1 time [s]

0

0.5 1 time [s]

1.5

10 0 -10 1.5

(a) (b) (c) Fig. 4.26. Y-axis: Simulated position and tracking error for configuration (a), (b) and (c). Table 4.9. Simulated friction compensation performance for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s Friction Compensation Methods

Quadrant Glitch [µm] x-axis

y-axis

Disturbance observer only

9

8.5

Disturbance observer and static friction feedforward

5.5

5

Disturbance observer and GMS friction feedforward

1

1

A combination of GMS friction model feedforward and the inverse modelbased disturbance observer yields a near zero quadrant glitch magnitude. As discussed previously in section 4.2.2, the system delay has prevented a complete removal of the quadrant glitch, as expected in an ideal case. Next, experimental validations are performed on the test setup. The results are analysed and discussed in the next section.

89

Friction Force Compensation Design

4.3.4 Experimental Validations Fig. 4.27 shows the measurement results on the real test setup (x-axis) obtained by applying the inverse model-based disturbance observer with static and GMS friction model feedforward for a similar sinusoidal reference trajectory as applied in the previous section. Observer Only

4

x 10

4

x 10

-3 1 time [s]

1.5

0

tracking error,[µm]

15 7 0 -7 -15 0

0.5

1 time [s]

1.5

0

-3 0.5

1 time [s]

1.5

20

20

15

15 tracking error,[µm]

0.5

20 tracking error,[µm]

0

-3 0

3 0 -3

-15 -20

Observer + GMS Model FF.

3 position,[µm]

0

-20

4

x 10

3 position,[µm]

position,[µm]

3

Observer + Static Model FF.

0

0.5

1 time [s]

1.5

0

0.5

1 time [s]

1.5

0

0.5

1 time [s]

1.5

3 0 -3

-15 -20

(a) (b) (c) Fig. 4.27. X-axis: Measured position and tracking errors for configuration (a), (b) and (c).

The combination of the inverse model-based disturbance observer and the GMS friction model feedforward yield the most significant reduction of the magnitude of the quadrant glitches (as concluded previously using the simulated results). The results however do not clearly indicate the difference in friction compensation performance between conditions (b) and (c). The advantage of the GMS friction model is not clearly demonstrated. Similar experiments for configuration (b) and (c) are repeated with a reduced maximum tracking velocity of 10mm/s instead of 100mm/s. Fig. 4.28 shows the result for this slower reference trajectory.

90

Friction Force Compensation Design

4

x 10

Observer + Static Friction FF.

4

4

2

position,[µm]

position,[µm]

4

0 -2 -4

0

5

10 time [s]

10 5 0 -5 -10 -20

0 -2

0

5

10 time [s]

15

0

5

10 time [s]

15

20 tracking error,[µm]

tracking error,[µm]

20

Observer + GMS Friction FF.

2

-4

15

x 10

0

5

10 time [s]

15

10 5 0 -5 -10 -20

(a) (b) Fig. 4.28. X-axis: Measured position and tracking errors for slower reference trajectory of 10mm/s for configuration (a) and (b).

This figure clearly demonstrates the advantage of using a GMS friction model at lower tracking velocity. At a lower tracking velocity, the presliding regime is more dominant and the benefit of using the GMS friction model that includes hysteresis with non-local memory is more pronounced. A similar experimental validation was repeated for the y-axis. The results are shown in fig. 4.29. Similar to the x-axis, the combination of the disturbance observer and the GMS friction model feedforward yields the smallest quadrant glitch magnitude. A quadrant glitch magnitude of about 3µm was recorded. The tracking error that remains besides the quadrant glitches is due to the imperfection of the inverse model feedforward which result from modelling errors. The measurement results for both the axes are summarized in Table 4.10.

91

Friction Force Compensation Design

4

x 10

Observer + 4 x 10 Static Friction FF.

Observer Only

0

-3 0.5

1 time [s]

1.5

0

-3 0

0.5

1 time [s]

1.5

10 0 -10

0

0.5

1 time [s]

1.5

10 5 0 -5 -10 -20

0

0.5

1 time [s]

1.5

0

0.5

1 time [s]

1.5

20 tracking error [µm]

20 tracking error [µm]

20 tracking error [µm]

0

-3 0

-20

3 position [µm]

3 position [µm]

position [µm]

3

Observer + 4 x 10 GMS Friction FF.

0

0.5

1 time [s]

10 3 0 -3 -10 -20

1.5

(a) (b) (c) Fig. 4.29. Y-axis: Measured position and tracking errors for configuration (a), (b) and (c).

Table 4.10. Measured friction compensation performance for a reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s Friction Compensation Methods

Axial Quadrant Glitch [µm] x-axis

y-axis

Disturbance observer only

7

11

Disturbance observer and static friction feedforward

3

5

Disturbance observer and GMS friction feedforward

2.5

3

Next, circular tests using the three different compensation configurations are performed on the xy table when both axes are controlled simultaneously. Fig. 4.30 shows the contour measurements and the radial tracking errors.

92

Friction Force Compensation Design

observer only (error x400)

observer + static model (error x400)

0

-30

0

-30 -30

0 30 position X [mm]

0

-30 -30

0 30 position X [mm]

-30

10

10

5

5

5

0 -5 -10 0

rmsX=1.13um rmsY=1.02um

90 180 270 angle [degree]

360

radial error [um]

10 radial error [um]

radial error [um]

30 position Y [mm]

30 position Y [mm]

position Y [mm]

30

observer + GMS model (error x400)

0 -5 -10 0

rmsX=0.666um rmsY=0.675um

90 180 270 angle [degree]

360

0 30 position X [mm]

0 -5 -10 0

rmsX=0.84um rmsY=0.62um

90 180 270 angle [degree]

360

(a) (b) (c) Fig. 4.30. Measured contours and radial errors for configuration (a), (b) and (c).

Observer + Static Friction Model

Observer + GMS Friction Model

30

30

position, y-axis [mm]

position, y-axis [mm]

Circular tests at tangential tracking velocity of 10mm/s are performed to illustrate more clearly the advantage of the GMS model over the static friction model. The results are shown in fig. 4.31.

20 10 0 -10 -20

20

radial tracking error [µm]

radial tracking error [µm]

-30 error: x400 -20 0 20 position, x-axis [mm]

8 0 -8

-20 0

90 180 270 angle [ degree]

360

20 10 0 -10 -20 -30 error: x400 -20 0 20 position, x-axis [mm] 20

8 0 -8

-20 0

90 180 270 angle [degree ]

360

Fig. 4.31. Circular tracking tests at tangential tracking velocity of 10mm/s. 93

Friction Force Compensation Design

4.4 Summary This chapter discusses friction compensation on a direct driven linear motors XY feed table using friction model based and friction model-free approaches. Static and GMS friction model feedforward, and an inversemodel-based disturbance observer are compared and combined. The friction compensation performances are compared based on the magnitude of the quadrant glitches. Quadrant glitches, which are caused by the complex nonlinear behaviour of friction at velocity reversal, can be compensated effectively using a combination of the GMS friction model feedforward and an inverse-model based disturbance observer. The benefits of using an advanced friction model like the Generalized Maxwell-slip (GMS) friction model are especially clear at slow motions where the pre-sliding friction is dominant. Fig. 4.32 and Table 4.11 summarize the friction compensation performances for all six different compensation configurations tested on the experimental test setup using axial and circular trajectory with an amplitude of 30mm, a frequency of 0.53Hz, and a maximum tracking velocity of 100mm/s. static model feedforward

position Y [mm]

no compensation

30

static model + observer

30

30

0

0

0

0

0

0

-30

-30

-30

-30

-30

-30

-30

0

error x400

30

position X [mm] 30

-30

0

error x400

30

position X [mm]

-30

0

30

GMS model + observer

30

error x400

radial error [µm]

disturbance observer only

GMS model feedforward

30

error x400

error x400

30

position X [mm]

-30

0

-30

30

position X [mm]

0

error x400

30

position X [mm]

-30

15

15

15

15

15

10

10

10

10

10

15

5

5

5

5

5

10

0

0

0

0

0

-5

-5

-5

-5

-5

-10

-10

-10

-10

-10

-15 0

-15

25 20

5

0

0 -5 0

-15 90 180 270 360 0

angle [degree]

90 180 270 360

angle [degree]

90 180 270 360

angle [degree]

-15 0 90 180 270 360 0

angle [degree]

90 180 270 360

angle [degree]

-15 0

90 180 270 360

angle [degree]

Fig. 4.32. Position and radial tracking error for different friction compensation approaches.

94

30

position X [mm]

Friction Force Compensation Design

Table 4.11. Measured magnitude of the quadrant glitches for reference sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s. x-axis

y-axis

Quadrant Glitch (µm)

Quadrant Glitch (µm)

No friction feedforward

15

35

25

Static friction feedforward

5

13

7

GMS friction feedforward

4

10

6

Disturbance observer only

7

11

7

Disturbance observer + static friction model feedforward

3

5

5

Disturbance observer + GMS friction model feedforward

2.5

3

3

Friction Compensation Methods

Radial Error (µm)

95

96

Chapter 5

Cutting Force Compensation Design 5.1 Introduction Friction force compensation alone is insufficient to obtain high tracking accuracy in high-speed linear drive based milling processes. Cutting forces generated by the cutting tool and work-piece interaction act directly on the linear motors and influence positioning and tracking accuracy. Cutting force compensation is essential and is traditionally accomplished with high gain feedback control yielding high dynamic stiffness. High dynamic stiffness means high disturbance rejection capacity. High controller gain however reduces the stability margins. This chapter analyses several disturbance compensation methods that have been proposed in literature for various applications but limited knowledge is available with regards to their application and performance for compensation of cutting forces. The different compensation methods will supplement the existing cascade P/PI position controller in a modular based approach. Three compensation methods are considered, namely; (i) an inverse-model-based disturbance observer (section 5.2), (ii) an explicit estimation of cutting force using relative acceleration measurement (section 5.3), and (iii) a repetitive controller (section 5.4). First, the performance of the inverse-model-based disturbance observer is analysed for cutting force compensation, an extension from its previous friction force compensation.

97

Cutting Force Compensation Design

5.2 Inverse-Model-Based Disturbance Observer The structure, design and analysis of the inverse-model-based disturbance observer [26] for friction compensation have been discussed extensively in the previous chapter (section 4.3). Here, the application of this disturbance observer to compensate cutting forces is analysed. Cutting forces are composed of several harmonic components as shown previously in fig. 3.13 and 3.14. In section 4.3 it is shown that the performance of the inverse- model-based disturbance observer depends on the bandwidth of the low pass filter Q. The number of cutting force harmonic components that are compensated is therefore limited by the bandwidth of the filter Q. The disturbance observer cutting force compensation performance is first validated numerically using MATLAB/Simulink.

5.2.1 Numerical Validations Fig. 5.1 shows a MATLAB/Simulink diagram of the control structure, including the cascade P/PI position controller and the inverse-model-based disturbance observer. In this analysis, only the cutting force compensation is considered (no friction force compensation) and no reference tracking is applied. First, a simulated disturbance force d(t) is applied. Its amplitude is 1 volt which corresponds to a force of 1450 N. The performance of the disturbance observer is compared based on the position error ep(t), for three different frequencies: (i) 1 Hz, (ii) 58 Hz, and (iii) 100 Hz.

Fig. 5.1. MATLAB/Simulink diagram of a cascade P/PI controller and a disturbance observer with a sinusoidal based disturbance input signal

98

Cutting Force Compensation Design

Fig. 5.2 shows the position tracking error of the system, before and after the disturbance observer is activated. 1 Hz

150

2.97 µm

-50 -100

observer on 0

2

4 time [s]

6

100

position error [µm]

50

position error [µm]

position error [µm]

100

0

100Hz

150

109µm

100

-150

58Hz

150

133.9µm

50 0 -50 -100

50

56.1 µm

34.91µm

0 -50 -100 observer on

-150 0

observer on 0.1

(i)

0.2 0.3 time [s]

0.4

0.5

-150 0

(ii)

0.1

0.2 0.3 time [s]

0.4

0.5

(iii)

Fig. 5.2. Y-axis: The effect of a disturbance observer on tracking errors for disturbance input frequencies of (i) 1 Hz, (ii) 58 Hz, and (iii) 100 Hz

The position tracking error of the system without a disturbance observer is described in equation (2.34). In the absence of a reference input signal, the tracking error from the input disturbance signal is simply, E p ( s )noDOB = −

Gm D(s) 1 + Gm ⋅ P ⋅ PI + Gm ⋅ PI ⋅ Vest ⋅ N

.

(5.1)

E p ( s )noDOB = −Gm ( s ) ⋅ S ( s ) ⋅ D ( s ) .

S(s) is the sensitivity function. Equation (4.20) describes the position tracking error of the system with a disturbance observer. In the absence of a reference input signal, the tracking error equals: E p ( s ) DOB = − E p ( s ) DOB

Gm (1 − Q )

⎛G ⎞ 1 + Q ⎜ m − 1⎟ + PI ⋅ Gm ⋅ Vest ⋅ N + P ⋅ PI ⋅ Gm G ⎝ n ⎠ = − Gm ⋅ S ( s ) ⋅ D ( s ) .

D(s)

(5.2)

Fig. 5.3 shows the Bode magnitude plot of the position error transfer functions (5.1) and (5.2). Indicated on the two curves are the theoretical positions tracking errors for frequencies of (i) 1Hz, (ii) 58Hz, and (iii) 100Hz. 99

Cutting Force Compensation Design

Bode Magnitude Diagram

3

10

Freq (Hz): 58 Mag (abs): 109

Freq (Hz): 1 Mag(abs): 133

2

Freq (Hz): 58.2 Mag (abs): 112

Magnitude (abs)

10

Freq (Hz): 100 Mag (abs): 57.8 Freq (Hz): 100 Mag (abs): 36.4

1

10

Freq (Hz): 1 Mag(abs): 2.96

0

10

-1

10

no observer w ith observer -2

10

-1

10

0

10

1

10

2

10

3

10

Frequency (Hz)

Fig 5.3. Y-axis: Position errors for system with and without a disturbance observer

Fig. 5.3 shows an almost exact match of position errors for each of the three different frequencies, for cases with and without the disturbance observer, when compared to the simulated position errors shown in fig. 5.2. The system is able to partly compensate the purely sinusoidal signal until a frequency of nearly 60Hz. The position error is amplified beyond this frequency, as shown in fig. 5.2 (iii) and fig. 5.3.

5.2.2 Experimental Validations Experimental validation is performed on the test setup using a synthesized cutting force with different harmonic contents. Fig. 5.4 shows the control scheme that includes the cascade P/PI controller, GMS friction model feedforward, the inverse-model reference feedforward, and the inversemodel-based disturbance observer. First, a synthesized cutting force described in section 3.4.2 is applied at the input of the system with no reference trajectory motion. The position errors for a synthesized cutting force with 1, 2, 3, and 15 harmonics content is shown in fig. 5.5 (the fundamental frequency is 23.3Hz, see Table 3.5).

100

Cutting Force Compensation Design

Fig. 5.4. Cutting force disturbance compensation using inversemodel-based disturbance observer

30

10 0 -10 -20

0 -10 -20

0.5

1 time [s]

1.5

-30 0

3 harmonics

30

20

10

observer on

-30 0

30

20 position error [µm]

position error [µm]

20

2 harmonics

10 0

-10

observer on

-20

-30 0.5 time1[s] 1.5 0

15 harmonics

20 position error [µm]

1 harmonic

position error [µm]

30

10 0

-10

observer on

0.5

1 time [s]

-20 -30 1.5 0

observer on

0.5

1 1.5 time [s]

Fig. 5.5. Y-axis: Measured position errors with inverse model-based disturbance observer for synthesized cutting forces disturbance with different harmonics component

The mean value of the position errors for input disturbance signals with two or more harmonic components are not zero due to the differences in the amplitudes and phases of the various harmonics. The position errors are reduced by applying the inverse model-based disturbance observer even if the disturbance input signal contains the full fifteen harmonics (the third harmonic is already beyond the Q-filter bandwidth). This is possible because of the much smaller amplitudes of the high frequencies harmonic components. A detailed analysis of the frequency content and the root mean square (rms) values of the position errors are shown in fig. 5.6 and Table 5.1 respectively.

101

Cutting Force Compensation Design

spectrum of pos. error (no observer)

1 harmonic

3 harmonics

15 harmonics

20

20

20

15

15

15

15

10

10

10

10

5

5

5

5

0

spectrum of pos. error (with obse)rver

2 harmonics

20

0

50

100 150 200

0

0

50

100 150 200

0 0

50

100 150 200

0 0

20

20

20

20

15

15

15

15

10

10

10

10

5

5

5

5

0

0

0

0

50 100 150 200 freq. [Hz]

0

50 100 150 200 freq. [Hz]

0

50 100 150 200 freq. [Hz]

0

0

50

100 150 200

50 100 150 200 freq. [Hz]

Fig. 5.6. Y-axis: Spectral analysis of the position errors without the disturbance observer (top) and with the disturbance observer (bottom) for disturbance force signals with different harmonic content.

The spectral analyses show a significant amplitude reduction (nearly 60%) only for the first harmonic (23.3Hz). This is consistent with the fact that the performance of the disturbance observer is limited by the bandwidth of the filter Q. Table 5.1. rms of position errors for different cutting force harmonics content using the explicit estimation of cutting force technique 1 harmonic

2 harmonics

3 harmonics

15 harmonics

no feedback

9.62µm

10.83 µm

13.99 µm

14.67 µm

with compensation

5.63 um

6.34 µm

6.98 µm

11.11 µm

% reduction

41.47%

41.45%

50.00%

24.26%

Next, circular tests are performed to measure both the x-axis and the y-axis cutting force compensation performance using the inverse-model-based disturbance observer. A synthesized cutting force with 15 harmonics is applied at both inputs of the system. The control scheme includes also the GMS friction model feedforward and the inverse model reference feedforward. Fig. 5.7 compares the circular test results with and without the disturbance observer. 102

Cutting Force Compensation Design

No Cutting Force Compensation

Inverse-Model-Based Disturbance Observer 30

10 0 -10 -20 -30 -40

radial error [µm]

position, y-axis [mm]

20

error: x200

-20 0 20 position, x-axis [mm]

20 10 0 -10 -20 -30 -40

40

30

30

20

20 radial error [µm]

position, y-axis [mm]

30

10 0 -10 -20 -30 0

error: x200

-20 0 20 position, x-axis [mm]

40

10 0 -10 -20

90

180 270 angle [degree]

360

-30 0

90

180 270 angle [degree]

360

Fig. 5.7. Measured contours and radial errors of circular tests with (right) and without (left) the inverse model based disturbance observer

The results show a reduction in the magnitude of the radial tracking errors for the system with disturbance observer. The radial tracking error is reduced from a maximum of 20µm to about 10 µm. A complete compensation of the cutting force is not possible due to the limited bandwidth of the filter Q. In the next section, an approach to explicitly estimate the cutting forces from relative acceleration measurements is described.

5.3 Explicit Estimation of Cutting Force via Ferraris Sensor Measurements Pritschow [29] has presented an approach to reconstruct process forces in direct drives using relative acceleration measurements. However, there is limited information on the application of this estimator to effectively compensate the influence of cutting forces. This section analyses the proposed estimation method on the experimental test setup.

103

Cutting Force Compensation Design

5.3.1 The Ferraris Principle The relative acceleration signal, which is the main element of this approach, is captured by a sensor that is known as a Ferraris sensor, named after the Ferraris principle [47]. Fig. 5.8 shows a schematic diagram of the Ferraris sensor.

*

copper strip Ferraris sensor Fig. 5.8. Schematic diagram of the Ferraris principle (* republished from [36] ) and the actual sensor used in measurement

The sensor has three main elements, namely: • • •

104

A block consisting of the lamination, permanent magnet, and the windings. A non-magnetic eddy current strip that is either Aluminium or copper (in this application a copper strip is used) An amplifier for the sensor signal

Cutting Force Compensation Design

The eddy current strip that lies within the slot of the block is attached to the moving object while the other parts are fixed to the frame. A relative motion between the eddy current strip and the block generates eddy current in the eddy current strip due to the permanent magnet that induces a voltage. The resulting eddy current generates magnetic fields and the strength of this field changes as the velocity of motion changes. This results in a voltage being induced in the winding. This voltage is proportional to the change of velocity, that is, the acceleration of the motion. The amplifier then amplifies the signal. A Ferraris sensor is attractive in mechatronic application because it provides an alternative method for generating a velocity signal from the acceleration signal [48]. Noise amplification and quantization of velocity signals that are generated by numerically differentiating position encoder signals can be avoided. J. Wang [36] has described the effect of eddy current strip characteristics, placement, and structure on the quality of the sensor signals. The disadvantage of this sensor is that, it has to be calibrated on-site. This is because the installation of the sensor part can affect the sensor sensitivity. The calibration of the sensor is based on the comparison between the position encoder measurement and the position signal that is generated from the Ferraris sensor measurements. The calibration of the Ferraris sensor sensitivity and offset value are described in Appendix B.

5.3.2 Cutting Force Estimator The Pritschow’s cutting force estimator [29] is designed based on a reduced order observer [35] that estimates state variables that cannot be measured directly. The observer is designed by considering the following model equation that describes the balance between forces that acts on a direct drive system (here, the &x& term refers to absolute acceleration). && x (t ) =

kf M

u (t ) +

1 1 d (t ) − Ff (v) M M

(5.3)

x is the position, d is the disturbance force, kf is the motor force constant, M is the mass, Ff is the friction force, and u is the voltage to the drive amplifier. The state space representations of the system and the measurement equation are: 105

Cutting Force Compensation Design

⎡ x& ⎤ ⎢ ⎥ ⎢ v& ⎥ = ⎢ d& ⎥ ⎣ ⎦

⎡0 1 ⎢ ⎢0 0 ⎢ ⎣0 0

0 ⎤ ⎡x⎤ ⎥ 1 ⎥ ⎢v ⎥ + M ⎢ ⎥ ⎥ 0 ⎦ ⎢⎣ d ⎥⎦

⎡ x ⎤ ⎡1 0 y=⎢ ⎥=⎢ 0 0 ⎣ a ⎦ ⎢⎣

⎡ 0 ⎢ ⎢kf ⎢M ⎢ ⎢⎣ 0

x 0 ⎤ ⎢⎡ ⎥⎤ ⎡ 0 ⎥ ⎢v⎥ + ⎢ k f 1 ⎥ ⎢ M ⎦ ⎢ d ⎥ ⎣⎢ M ⎣ ⎦

⎤ ⎥ ⎥ ⎡ u ⎤ ⎥ ⎢⎢ F f ( v ) ⎥⎥ ⎦ ⎥ ⎣ ⎦⎥

(5.4)

0 ⎤ ⎥ ⎡ u ⎤ ⎥ 1 ⎢ − ⎥ ⎣⎢ F f ( v ) ⎦⎥ M ⎦⎥

(5.5)

0 1 − M 0

The detailed derivation of the reduced order observer for the estimation of position x, velocity v, and the disturbance forces d is presented in Appendix C. In fact, the disturbance force d can be estimated directly from (5.3) and is described as: kf ⎛ ⎞ 1 dˆ = M ⎜⎜ a − u+ F f ( v ) ⎟⎟ M M ⎝ ⎠

(5.6)

a is the Ferraris relative acceleration signal measurement. Next, the estimator is designed based on (5.6) to compensate cutting forces and its experimental validation is performed on the considered test setup.

5.3.3 Experimental Validations The first part of the experimental validation analysed the quality of the estimated disturbance forces only, without feedback. Fig. 5.9 shows the schematic diagram of the control structure that includes the cascade P/PI position controller and the estimator. The estimator is similar to equation (5.6) without the friction term. The effect of friction is not considered at this moment. The disturbance force is the only input to the system. The estimator is experimentally validated on the y-axis where the Ferraris sensor is attached.

106

Cutting Force Compensation Design

Fig. 5.9. Schematic diagram of a cascade P/PI position control with cutting force estimator (without feedback)

The inputs to the estimator are the control command signal u and the calibrated relative acceleration signal a. A synthesized cutting force signal with 1, 3, and 15 harmonic contents are introduced at the input of the system G(s). The axial position errors ep(t) are recorded. Fig. 5.10 compares the reference cutting force d and the estimated cutting force dˆ . The estimated cutting force is delayed due to the delay in the drive. 3 harmonics

cutting force [N]

1 harmonic

15 harmonics

600

600

600

400

400

400

200

200

200

0

0

0

-200

-200

-200

-400

-400

-400

reference estimated

-600 0

0.05

0.1 time [s]

0.15

-600 0.2

-600 0

0.05

0.1 0.15 time [s]

0.2

0

0.05

0.1 0.15 time [s]

0.2

Fig 5.10. Reference and estimated cutting forces

In the second part of the experimental validation, a feedback of the estimated cutting force is applied. Fig. 5.11 shows the applied control scheme. A low pass filter Q similar to equation (4.7) and desc

107

Cutting Force Compensation Design

Fig. 5.11. Schematic diagram of a cascade P/PI position control with cutting force estimator (with feedback)

Synthesized cutting force signals with 1, 2, 3, and 15 harmonic contents are applied at the input of the system G(s). The axial position errors ep(t) are recorded. Fig. 5.12 illustrates the synthesized cutting force and compares the position tracking errors with and without the estimator feedback. The spectral analysis of the position errors is shown in fig. 5.13.

0.2 time [s] observer on

20 0 -20 0

0.2

time [s]

0.4

0 40

0.4

0.2 time [s] observer on

20 0 -20 -40

15 harmonics

0

0.2

time [s]

0.4

600 400 200 0 -200 -400 -600 0

40

0.4

0.2 time [s] observer on

20 0 -20 -40

0

0.2

time [s]

0.4

0 40

position error [um]

40

-40

3 harmonics 600 400 200 0 -200 -400 -600

position error [um]

0

position error [um]

2 harmonics 600 400 200 0 -200 -400 -600

position error [um]

synthesized cutting force [N]

1 harmonic 600 400 200 0 -200 -400 -600

0.4

0.2 time [s]

observer on

20 0 -20 -40

0

0.2

time [s]

Fig. 5.12. Position errors for different cutting force harmonic components using estimator

108

0.4

0.4

2 harmonics

0

100 freq. [Hz]

0

100 200 freq. [Hz]

15

10 5

0

100 freq. [Hz]

5 0

10

0

0

100 freq. [Hz]

0

5 0

10

0

0

100 200 freq. [Hz]

0

100 200 freq. [Hz]

15

10

100 200 freq. [Hz]

20

200

15

10

200

position error [um]

0

200

15

0

10

position error [um]

0

15 harmonics

20 position error [um]

position error [um]

10

3 harmonics

20

position error [um]

1 harmonic 20

position error [um]

position error [um] with compensation

position error [um] without compensation

Cutting Force Compensation Design

0

100 freq. [Hz]

10

200

5 0

Fig. 5.13. Spectral analysis of the position errors for different cutting force harmonic components using an estimator

Table 5.1 summarizes the rms of the position errors for cases with and without the estimator compensation. The results show limited performance of the estimator. An average of 33% reduction in the magnitude of the position errors is achieved. The limited bandwidth of the filter Q reduces the overall compensation performance (similar argument as previously discussed in section 4.3). The estimator has an almost equal percentage reduction in the position errors for all the different cutting force harmonic contents (see Table 5.2). However, the magnitude of the rms values are lower compared to the results obtained using the inverse-model based disturbance observer shown in Table 5.1. Table 5.2. rms of position errors for different cutting force harmonics content using the explicit estimation of cutting force technique 1 harmonic

2 harmonics

3 harmonics

15 harmonics

no feedback

10.76µm

12.04 µm

18.27 µm

18.53 µm

with compensation

7.16 um

8.15 µm

11.57 µm

11.89 µm

% reduction

33.46%

32.37%

36.67%

29.77%

109

Cutting Force Compensation Design

Finally, the cutting force is compensated during a reference tracking of a sinusoidal signal with an amplitude of 30mm and a maximum tracking velocity of 100mm/s. Fig. 5.14 shows the tracking errors with and without the estimator feedback. The spectral analysis of these tracking errors is shown in fig. 5.15. 3 harmonics

0

0.5

1

1.5

40 20 0 -20 -40

0

0.5

1

1.5

40 20 0 -20 -40

0

0.5

1 time [s]

1.5

0 -50

0

0.5

1

1.5

40 20 0 -20 -40

0

0.5

1

1.5

40 20 0 -20 -40

15 harmonics position [mm]

-50

50

0

0.5

1 time [s]

1.5

position error [um] position error [um]

position [mm]

0

position error [um] position error [um]

position error [um] (with estimator)

position error [um] (no estimator)

position [mm]

1 harmonic 50

50 0 -50

0

0.5

1

1.5

0

0.5

1

1.5

0

0.5

40 20 0 -20 -40 40 20 0 -20 -40

1 time [s]

1.5

Fig. 5.14. Tracking errors for different cutting force harmonic contents for cases with and without the estimator 3 harmonics

10 5 0

50 100 150 freq. [Hz]

200

20

10 5 0

50 100 150 200 freq. [Hz]

20

15 10 5 0

15

0

15 harmonics

0

50 100 150 freq. [Hz]

200

15 10 5 0

20 15 10 5 0

0

50

100 150 freq. [Hz]

200

0

50

100 150 freq. [Hz]

200

20 position error [um]

0

20

position error [um]

position error [um]

15

position error [um]

position error [um] (with estimator)

position error [um] (no estimator)

1 harmonic 20

0

50 100 150 200 freq. [Hz]

15 10 5 0

Fig. 5.15. Spectral analysis of the tracking errors for different cutting force harmonic components for cases with and without the estimator 110

Cutting Force Compensation Design

Table 5.3 summarizes the rms of the tracking errors. Table 5.3. rms of tracking errors for different cutting force harmonics content using the explicit estimation of cutting force technique 1 harmonic

3 harmonics

15 harmonics

8.96µm

10.06 µm

10.11 µm

compensation

5.67 um µm

6.60 µm

6.73 µm

% reduction

36.72%

34.39%

33.43%

no compensation

The estimator is able to partly compensate the cutting forces that act on the system during tracking. An average of 33% reduction in the magnitude of the position errors is observed. Fig. 5.16 shows a comparison between the reference cutting forces and the estimated feedback cutting force. The results illustrate the poor performance of the estimator. This is a stark contrast to the estimator performance when the estimated cutting force is not fed back into the system as previously shown in fig. 5.11. The performance is degraded during feedback due to the limited bandwidth of the filter Q. The delay that has already influenced the estimated cutting force further reduces the estimator performance.

cutting force [N]

1 harmonic

3 harmonics

600

600

600

400

400

400

200

200

0

0

15 harmonics

200 0 -200

-200

-200 reference

-400

-400

-400

-600

estimated -600 0.2

0.25

0.3

time [s]

0.35

0.4

-600 0.2

0.25

0.3

time [s]

0.35

0.4

-800 0.2

0.25

0.3

0.35

0.4

time [s]

Fig 5.16. Cutting Forces Estimation Analysis (without reference trajectory)

A more efficient and complete compensation of the cutting force is desirable. An approach that incorporates the characteristic of the cutting force signals in its design structure is presented next. This approach is based on the design of a general order state observer. 111

Cutting Force Compensation Design

Cutting forces consist mainly of harmonic components with a fundamental frequency equal to the frequency of rotation of the tool (as indicated in Table 3.5 and 3.6). It is expected that by taking this property into account, improved cutting force rejection can be obtained. Two approaches are considered: a disturbance force observer based on a state estimator that includes the knowledge of the harmonic contents of the cutting force and a repetitive controller. The first approach has been designed and validated numerically and experimentally on the considered test setup. Preliminary results indicate that the method has good potential in compensating cutting forces. However, due to practical limitations (drive saturation etc.) of the test setup, compensation of cutting forces beyond the first three harmonic contents was not possible. The design, analysis, and preliminary results of this technique are presented in Appendix D. The second approach that is using the repetitive controller is discussed in the following section.

5.4 Repetitive Controller Repetitive control (RC) is an appropriate control strategy if the dominant disturbances or the tasks to be performed are periodic. For example, in the track-following servo system of optical disk drives, peristaltic pumps used in medical devices, active noise control, robotized laparoscopic surgery, or tracking of periodic trajectories as, for instance, in robots performing repetitive tasks and during non-circular machining. RC is also appropriate to compensate the effect of the cutting forces since these forces are mainly periodic.

5.4.1 Design Structure of a Repetitive Controller Including a repetitive controller (RC) in the feedback loop improves the attenuation of periodic disturbance inputs. RC is based on the internal model principle [49], which states that, if a disturbance signal can be regarded as the output of an autonomous system, including this system in a stable feedback loop guarantees asymptotically perfect rejection. A memory loop is the most frequently used generator of periodic signals. Fig. 5.17 shows a RC (indicated in blue) as an add-on module in a closed loop of a general control scheme for a plant with a periodic disturbance input d of known period time T0[s]. 112

Cutting Force Compensation Design

Fig. 5.17. Standard RC as an add-on module to a closed loop control scheme

The memory loop, a basic element of a RC, is a delay line of T0 [s], in a positive feedback loop. T0 equals the period time of the disturbance signal d (T0 = 2π/ωo, where ωo is the period frequency). Figure 5.18 shows the memory loop diagram. ep(t) is the input to the closed loop and can include both periodic and non-periodic components.

Fig. 5.18. Standard memory loop with periodic signal generator with period T0[s].

In most practical implementations of a RC, this delay is implemented in discrete-time for a selected sampling period Ts. The sampling period is chosen such that T0 = N·Ts, where N ∈ . If the ratio of T0 /Ts is not exactly an integer number, it is rounded to the lower integer value. The transfer function from ep to ep’ equals: ep ' ( s) ep ( s)

=

e −T0 s 1 − e −T0 s

(5.7)

For frequency ω = kωo (k an integer and ωo=2π/T0), the magnitude of the denominator of equation (5.7) becomes zero and the transfer function is infinite. This means that an infinite loop gain at all the harmonics of the disturbance signal is realised using this memory loop. High loop gains, however, can cause instability. Stability analysis in RC design is thoroughly 113

Cutting Force Compensation Design

discussed in [50, 51]. Generally, two discrete time filters are added: L(z) and Q(z). The filters L(z) and Q(z) are necessary to preserve stability, modulus margin and high-frequency rolls-off of the original feedback controller [52]. L(z) and Q(z) are usually designed as follows [50, 51]:

•

L(z) is set equal to the inverse of the complementary sensitivity function of the system, augmented with a low pass filter with a cutoff frequency above or equal to that of the Q filter. L(z) is implemented as a FIR filter.

•

Q(z) is a low-pass linear-phase FIR filter with unity dc-gain and cutoff frequency ωQ.

The RC performance is limited by the cut-off frequency ωQ. The RC is not active beyond the pass band of the filter Q(z) (that is when Q(ω) ≈ 0). It is important to note that due to the presence of the delay z-N, L(z) and Q(z) filters can be designed as non-causal filters. For the implementation of these filters, the non-causal part is absorbed into the delay z-N. Fig. 5.19 shows the discrete-time implementation of a typical RC with the delay line z-N and the filters L(z) and Q(z).

Fig. 5.19. Discrete time implementation of a typical RC.

This typical RC is often referred to as a first order RC, a special form of a so called high order RC where the delay z-N is replaced by a polynomial of the delay: n

∑ wi z −iN

(5.8)

i =1

n equals the order of the RC. High order RC gives the designer more degrees of freedom (wi parameter) to achieve a better robustness against uncertainties of the period T0 of the disturbance signal. For the considered application, i.e. the cutting force compensation, a second order RC (n = 2) is designed. Fig. 5.20 shows the schematic diagrams of an nth order RC and a second order RC. RC‘s with an order higher than two were not considered because of processor hardware limitations. 114

Cutting Force Compensation Design

Fig. 5.20. Schematic diagram of (a) nth order RC, (b) 2nd order RC

The two parameters w1 and w2 of the second order RC were designed using the method developed by G.Pipeleers et. al. [52, 53] as a trade-off between periodic performance considering a relative uncertainty of 1% on the period of the disturbance and non-periodic performance degradation, that is the amplification of non-periodic disturbances such as measurement noise that are present between the harmonics of the cutting force. The periodic and non-periodic performances of a RC are quantified by the robust periodic and non-periodic performance indices respectively [51]. The robust periodic performance index γp,∆ [51] is defined as the smallest reduction of the sensitivity function over all considered harmonics of the disturbance and over all potential frequencies within the specified (in this case 1%) relative uncertainty interval. Attenuation of the periodic disturbances corresponds to γp,∆ < 1, where γp,∆ = 0 indicates perfect rejection. The non-periodic performance index γnp [51] is defined as the highest amplification of the sensitivity function over all frequencies within the bandwidth of the RC. The RC is designed to have a bandwidth of 200 Hz (ωQ =200Hz), such that the first eight harmonics (the fundamental frequency is 23.3 Hz, for a spindle speed of 1360rpm) of the synthesised cutting force are compensated. This bandwidth is chosen due to the model accuracy limitation beyond a frequency of 200Hz. This limitation will have a minimal effect on the controller compensation performance because the influence of the 8th or higher harmonic components of the cutting forces is not significant (Table 3.5 and 3.6). The designed RC has a periodic performance index γp,∆ of 0.063 and a non-periodic performance index γnp of 6dB. 115

Cutting Force Compensation Design

Fig. 5.21 shows the Bode diagrams of the loop gain, sensitivity function, and the complementary sensitivity function of the y-axis based on the designed second order RC and a general control scheme that is the cascade P/PI controller (designed in section 2.5). The high loop gain magnitudes at the harmonic frequencies (fig. 5.21(a)) are translated to the reduction in the magnitude of the sensitivity function at the corresponding frequencies. Fig. 5.21(b) compares the sensitivity curves between the cascade P/PI control scheme and the control scheme with the RC added. The non-periodic performance degradation is identified by the arrows A, and is smaller than 6dB as predicted by γnp. Bode Plot of Loop Gain 80

Loop Gain [dB]

60 40 20 0 -20 -40 -60 0 10

1

2

10

10 frequency (Hz)

(a) Complementary Sensitivity Function

Sensitivity: S(z) 10

10

0 A

0

-20

-10 Magnitude [dB]

Magnitude (dB)

-10 S-RC

-30 S-Cascade -40

-20 -30 -40

-50

-50

-60

-60

-70 0 10

1

2

10 10 Frequency (Hz)

(b)

-70 0 10

1

2

10 10 Frequency (Hz)

(c)

Fig. 5.21. Y-axis: Bode plots of (a) the loop gain, (b) sensitivity function and (c) complementary sensitivity function with second order RC

116

Cutting Force Compensation Design

5.4.2 Experimental Validations The repetitive controller is implemented on the test setup to compensate the synthesized cutting forces applied at the input of the system. Fig. 5.22 shows a schematic diagram of the overall control structure. The RC (the blue line) is structured as an add-on module to the existing cascade P/PI position controller. A GMS friction model feedforward (green line) and an inversemodel-based disturbance observer (red line) compensate mainly the friction forces.

Fig. 5.22. A schematic diagram of a cascade P/PI controller with a RC module and friction compensation elements.

First, the cutting forces compensation is validated on each individual axis during reference tracking. The reference trajectory is a sinusoidal signal with amplitude of 30mm and a maximum tracking velocity of 100mm/s. Synthesized cutting forces with 1, 2, 3 and 15 harmonic contents are applied at the input of the system. Fig. 5.23 and fig. 5.24 show the position tracking errors for cases with and without the repetitive controller along the x-axis and the y-axis respectively. The RC is able to compensate these repetitive disturbances effectively up to the 8th harmonic. The influence of the remaining harmonics (9-15) on the tracking error is negligible. The experimental validation is extended to include simultaneous compensation of cutting forces in both axes. Three different control configurations are compared, namely: i. no friction and no cutting force compensation ii. inverse-model-based disturbance observer with GMS friction model feedforward iii. repetitive controller with the inverse-model-based disturbance observer and with the GMS friction model feedforward 117

Cutting Force Compensation Design

2 harmonics

3 harmonics

15 harmonics

400

400

400

200

200

200

200

0

0

0

0

-200

-200

-200

-200

-400 0

position error [µm] with RC

position error [µm] no RC

cutting force [N]

1 harmonic 400

0.1

-400 0.2 0

0.1

-400 0.2 0

0.1

-400 0.2 0

20

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-20 2 0

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2

-20 0

0.1

0.2

1

2

1 time [s]

2

Fig. 5.23. Y-axis: Measured position tracking errors with and without the RC for different harmonic component of the cutting forces.

2 harmonic

3 harmonic

15 harmonics

400

400

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position error [µm] with RC

position error [µm] no RC

cutting force [N]

1 harmonic 400

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0.2

1

2

1 time [s]

2

Fig. 5.24. X-axis: Measured position tracking errors with and without the RC for different harmonic components of the cutting forces. 118

Cutting Force Compensation Design

Fig. 5.25 shows for each case the reference contour, the measured contour, and the radial tracking errors. The measured contours have been amplified by a factor of 200 for display purposes. The root mean square (rms) values of the radial tracking errors for the three cases are summarized in Table 5.4.

position y [mm]

0 -20

0 -20

-20 0 20 position x [mm]

-40 -40

40

-20 0 20 position x [mm]

10 0 -10 -20

0 -20

180

270

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360

angle [degree]

0

measured pos. reference pos. error = x200

-20 0 20 position x [mm]

40

90 180 270 angle [degree]

360

20

-20 90

20

-40 -40

40

20 radial error [µm]

radial error [µm]

20

0

RC + Inv. Mod. Based Dist. Observer + GMS FF. 40

error = x200

error = x200

-40 -40

20

radial error [µm]

position y [mm]

20

position y [mm]

Inverse-Model-Based Disturbance Observer + GMS FF. 40

No Friction + Cutting Force Compensation 40

10 0 -10 -20

90 180 270 angle [degree]

360

0

(i) (ii) (iii) Fig. 5.25. Circle tests: Measured position and radial tracking errors for cases (i), (ii), and (iii) Table 5.4. rms of the radial tracking errors for cases (i), (ii), and (iii) rms

case (i)

case (ii)

case (iii)

7.68µm

4.43 µm

1.09 µm

The repetitive controller yields the lowest rms of the measured radial tracking errors. A reduction of 85% is obtained compared to case (i) where no friction and cutting forces compensation are applied. These results show the ability of the repetitive controller to effectively eliminate the first 8 harmonics of the cutting force. Higher harmonics have limited influence on the tracking error. The 8th harmonic (186 Hz) lies well beyond the bandwidth of the position controller (45Hz) and of the inverse model disturbance observer (60Hz). This is possible since the repetitive controller is only active in small frequency ranges around these harmonics and the system phase lag is compensated for in its design [21]. 119

Cutting Force Compensation Design

The disturbance observer performance is limited by its inherent phase lag. A higher bandwidth is possible with a more accurate model, but will always be lower than the bandwidth of the repetitive controller due to uncompensated system delay. In the next section, actual cutting operations are performed on the considered test setup. Because of the superior performance of the RC, only this technique is implemented and validated.

5.5 Cutting Forces Compensation during an Actual Cutting Process Actual cutting tests are performed on the test setup. The cutting test characteristics are summarized in Table 5.5. Table 5.5. Cutting test characteristics used in cutting force compensation Cutting conditions Work piece material Cutter diameter No. of edges

Descriptions Aluminium 16mm 4

Feed per tooth

0.1mm/tooth

Spindle speed

1360rpm

Axial depth of cut

4mm

Radial depth of cut

1mm

Sampling frequency

2000Hz

The following control configurations are considered for the evaluation of the cutting force compensation: (i)

no friction and no cutting force compensation

(ii) inverse-model-based disturbance observer + GMS friction model feedforward (iii) inverse-model-based disturbance observer and GMS friction model feedforward with cutting force compensation using a second order repetitive controller 120

Cutting Force Compensation Design

The cutting tests are performed on the considered test setup for a circle with a radius of 25mm and at a tangential tracking velocity of 5mm/s (see fig. 5.26). The slower tracking velocity will emphasize the advantage of the GMS friction model in compensating the highly nonlinear friction behaviour during velocity removal.

Fig. 5.26. Cutting force compensation during actual cutting process

Fig. 5.27 compares the table axial tracking errors and their spectral contents for the three control configurations. A gradual improvement in tracking performance is observed from case (i) to case (iii). The spectral analysis results shown in fig. 5.27 indicate an almost complete removal of the cutting force harmonics using the repetitive controller. However, due to structural vibration during the cutting process, the magnitudes of some non-harmonic frequencies are amplified, especially between the 250 Hz and 300 Hz frequency range (indicated by the circle A) and also at the lower frequency range between 1Hz and 20 Hz (indicated by the circle B). These effects are clearly identified by the circles A and B on the spectral analysis results in fig 5.27. Fig. 5.28 compares the reference contour and the measured contour of each of the three cases. The tracking performance is analysed from the magnitude of the radial tracking errors. The control configuration using the repetitive controller yields the least radial tracking error and the most accurate contour. This is clearly indicated by the rms measurements of the radial tracking errors for the three cases shown in Table 5.6.

121

Cutting Force Compensation Design

Position Error (X-Axis) [µm]

No Friction and No Cutting Force Compensation

Position Error (Y-Axis) [µm]

GMS + Inv. Model Observer + Repetitive Controller

20

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GMS + Inv. Model Observer

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0 0

100 200 [Hz]

300

0 0

100 200 [Hz]

(i)

0 0

300

(ii)

(iii)

Fig. 5.27. Measured position tracking errors and spectral analyses for control configurations (i), (ii), and (iii). No Friction + No Cutting Forces Comp.

GMS + Inv. Model Observer

0 -20

-20

20 0 -20 error x 200

error x 200

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40

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position X [mm]

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(i)

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30

position X [mm]

radial error [µm]

position Y [mm]

20

GMS +Inv. Model Observer + RC

40

40 position Y [mm]

position Y [mm]

40

10 0 -10 -20

90

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angle [degrees]

(ii)

360

-30 0

90

180

270

360

angle [degrees]

(iii)

Fig. 5.28. Contour tracking measurement and radial tracking errors for control configurations (i), (ii), and (iii). 122

Cutting Force Compensation Design

Table 5.6. rms of the radial tracking errors for cases (i), (ii), and (iii) during actual cutting process radial error

case (i)

case (ii)

case (iii)

rms

3.04 µm

1.72 µm

0.86 µm

5.6 Summary This chapter discusses the compensation of the cutting forces. Several techniques that are available in literature but have not been applied specifically for cutting forces compensation in milling machine have been discussed and presented. These techniques include the inverse-model-based disturbance observer, the explicit estimation and compensation of cutting forces using the Ferraris sensor relative acceleration measurements, and finally the repetitive controller. The compensation performances are first validated experimentally using synthesized cutting forces that are applied at the input of the system. The performance of the inverse-model-based disturbance observer and the second method that is based on the Ferraris sensor measurement are restricted by the bandwidth limitation of the low pass filter. The best performance is observed with the repetitive controller. An almost complete removal of the cutting forces harmonics is achieved. Finally, actual cutting process was performed on the test setup. It is shown that the developed second order repetitive controller is able to almost completely compensate the cutting forces.

123

124

Chapter 6

Conclusions & Future Study This thesis investigates friction and cutting forces compensation in a linear drive xy feed table based on several techniques that have been discussed in literature. A simple and highly practical cascade control configuration is selected as the primary controller. The design of a cascade P/PI controller is first discussed. Careful attention is given during the design steps to minimize the influence of the system natural resonance frequencies on system stability and tracking behavior of the drives. A PI velocity controller and a P position controller for each axis are chosen. The system bandwidth of each axis is limited by the first anti-resonance and resonance frequency at 40 Hz and 45 Hz for x and y axes respectively. These anti-resonance and resonance frequencies have been shown to be caused by the relative motion between the base of the machine and the floor. During the design analysis, notch filters are added to suppress the high resonance frequencies. The advantages of adding the notch filters have been clearly demonstrated in the design analysis discussed in Chapter 2. Velocity feedforward and an inverse model position feedforward compliment the cascade P/PI structure in order to eliminate the reference tracking errors. Compensation of disturbance forces acting on the linear drives is critical for high tracking performance. The non-linear friction behaviour is compensated using both friction model-based and friction model-free approaches. The friction compensation performances of these methods are compared based on the magnitude of the quadrant glitches – that are “spikes” at each quadrant of a circle, resulting from the high and complex nonlinear friction behaviour during velocity reversal. The combination of friction feedforward based on the recently developed Generalized Maxwell-slip (GMS) friction model and the inverse-model-based disturbance observer yields the smallest quadrant glitch magnitudes. The magnitude of the quadrant glitches were reduced from a maximum of 27 µm to only 3µm. This thesis has demonstrated the 125

Conclusions and Future Study

effectiveness of the GMS friction model especially at lower tracking velocity. This can be explained from the model’s ability to accurately describe the friction pre-sliding regime that is dominant at slower motion. In addition, the accuracy of the friction model especially in describing the presliding behaviour of the friction regime determines the degree of performance. This thesis does not consider an adaptive friction model that will be necessary to improve the robustness of this approach. Further study on the application of such technique is desired. On the other hand, the performance of the inverse-model-based disturbance observer is limited by the bandwidth of the applied low pass filter Q. The filter Q is necessary for system stability. A low pass filter Q with a bandwidth of 60 Hz and 90 Hz are designed for the x and y axes. These bandwidths are selected to ensure a reasonable stability margin and are restricted by the unmodeled system dynamics. The inverse-model-based disturbance observer is able to compliment the GMS friction model feedforward approach because it estimates the disturbance forces for example cutting forces if present, or forces representing the linear model and friction model inaccuracies that cannot be explained by the system output and the available system model. Several different techniques are studied for cutting force compensation. The application of the inverse-model-based disturbance observer is further extended to include cutting forces compensation. As discussed earlier, the performance is again limited by the bandwidth of the low pass filter Q. Next, a method that is based on explicit estimation of the cutting forces from relative acceleration measurement using the Ferraris sensor is applied. The compensation performance is negatively influenced by the delay in the estimator caused by the delay in the drive. In addition, this thesis has shown that the application of a low pass filter in the feedback loop degrades the compensation performance. A successful compensation of the synthesized cutting forces is observed using a repetitive controller. The spectral analysis of the tracking errors shows complete removal of the cutting force up to its eighth harmonic. However, due to the waterbed effect, an improvement in the performance at the harmonic frequencies has yielded the degradation of performance in other frequencies.

126

Conclusions and Future Study

A study on other compensation techniques is desirable. This could include the application of the nonlinear sliding mode control (SMC) or the robust H∞ approach. The application of a sliding mode controller is attractive because SMC is widely known for its robustness against matched disturbances, which means that the disturbance forces are completely rejected once sliding has occurred. An actual cutting process was performed on the test setup. The performance of the inverse-model-based disturbance observer and the repetitive controller are compared. Analysis of the drive position tracking errors indicates a successful compensation of the cutting forces using the repetitive controller. However, the roundness measurements of the finished product (a circular shape work-piece) indicate some structural problem of the test setup. An additional study is necessary to clearly identify the cause of this deficiency. In addition, a surface roughness analysis on the finished products (circles) is desired in order to establish the effect of the high frequencies component of the cutting forces on surface finish quality. In the test setup, the tracking performance is highly influenced by the high friction and actuator saturation. It is believed that the full potential of these techniques has not yet been fully explored. Also, a future study on the application of adaptive repetitive controller is desired to compensate changing frequencies of the cutter drive because of the changes in the setpoint reference and also due to the cutting tools and work-piece interactions.

127

128

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134

Curriculum Vitae Name Date of Birth Place of Birth Nationality Home Address

: : : : :

Telephone No. Position Office Address

: : :

Office Tel. No. Email Address

: :

Zamberi Jamaludin 4th September 1972 Malacca, Malaysia Malaysian 22 Jalan Burung Pikau, Taman Bukit Maluri, 52100 Kepong, Kuala Lumpur, Malaysia. (60) 12-3911912 Lecturer Faculty of Manufacturing Engineering, Department of Robotics and Automation, P.O.Box 1200, 75450 Ayer Keroh, Malacca, Malaysia (60) 6-233 2421 [email protected]

Academic Backgrounds 1. Bachelor of Engineering (Chemical Engineering) Lakehead University (Department of Chemical Engineering), 955 Oliver Road, Thunder Bay, Ontario, Canada. 1993-1997. Thesis: Study on Optimal Parameters for Pulp and Paper Sludge Dewatering Process Using Polymer Based Material. Supervisor: Prof. Dr. Robert Rosehart 2. Master Engineering (Manufacturing Eng.) National University of Malaysia (U.K.M) Bangi, Selangor, Malaysia. 1999-2001. Thesis: Analysis for Optimal Parameters For Filling Process In Metal Injection Molding Utilising the Moldflow Flow Simulation Software. Supervisor: Associate Prof. Dr. Norhamidi Muhammad 135

Curriculum Vitae

Working Experiences 1.

Japan-Malaysia Technical Institute, Vocational Training Officer, Manpower Dept., Ministry of Human Resources. 01/08/1997 – 09/06/2002

2.

Malaysia-Malacca Technical University (UTeM) Lecturer, Ministry of Higher Education. 10/06/2002 - present

Awards 1.

Malaysian Federal Government Scholarship High School and Bachelor Studies in Canada 1990-1997.

2.

Japan Int. Cooperation Agency (JICA) Scholarship Malaysian-Japanese Counterpart Training in Japan 27/10/1997 – 08/02/1998.

3.

Japan Int. Cooperation Agency (JICA) Scholarship

Seminar on Enhancing Vocational Training, Japan 23/10/2000 – 08/12/2000. 4.

136

Malaysian Ministry of Higher Education Scholarship PhD Studies at K.U.Leuven, Belgium 29 Sept. 2003 – 28 Sept. 2008.

List of Publications Journals 1. Z. Jamaludin, H. Van Brussel, J. Swevers, “Friction Compensation of a XY Feed Table using Friction Model-Based Feedforward and an Inverse-Model-Based Disturbance Observer”, Submitted for publication in IEEE Trans. on Industrial Electronics- Advances in Motion Control . 2008. 2. Z. Jamaludin, H. Van Brussel (1), G. Pipeleers, J. Swevers, “Accurate Motion Control of XY High-Speed Linear Drives using Friction Model Feedforward and Cutting Forces Estimation”, Annals of the CIRP-Manufacturing Technology, vol. 57/1, pp 403-406, 2008. 3. Z. Jamaludin, H. Van Brussel, J. Swevers, “Classical Cascade and Sliding Mode Control Tracking Performances for a X-Y Feed Table of a High-Speed Machine Tools”, Int. J. Precision Technology, vol.1/1, pp. 65-74, 2007.

Proceedings 1. Z. Jamaludin, H. Van Brussel, J. Swevers, "Design of a Disturbance Observer and Model-Based Friction Feedforward to Compensate Quadrant Glitches", The 9th International Conference on Motion and Vibration Control - MOVIC, Munich, Sept. 15-18, accepted, 2008. 2. Z. Jamaludin, H. Van Brussel, J. Swevers, “Quadrant Glitch Compensation using Friction Model-based Feedforward and an Inverse-Model-Based Disturbance Observer”, Proc. of the 10th International Workshop on Advanced Motion Control, Trento, Italy, pp. 212-217, 2008. 3. Z. Jamaludin, H. Van Brussel, J. Swevers, “Tracking Performances of Cascade and Sliding Mode Controllers with Application to a XY Milling Table”, Proc. on Int. Conference on Noise and Vibration Engineering, Leuven, Belgium, Sept. 18-20, 2006. 137

List of Publications

4. Z. Jamaludin, H. Van Brussel, J. Swevers, “Comparison of Classical Cascade and Sliding Mode Control of a X-Y Feed Table of a HighSpeed Machine Tools”, Proc. of Int. Conf. on Manufacturing Science and Technology-ICOMAST, August 28-30, Malacca, Malaysia, 2006. 5. Z. Jamaludin, H. Van Brussel, J. Swevers, “Comparison of Classical Cascade and Sliding Mode Control for a X-Y Feed Table of a HighSpeed Machine Tools”, Book of Abstracts - 25th Benelux Meeting on System and Control, March 15-18, Heeze, The Netherlands, 2006.

138

Appendix A Machine Dynamics Analysis The machine conditions are adjusted and the frequency response function relating the input voltage to the drive amplifier to output position is measured. Fig. A1 shows the FRF measurement for two different machine conditions: The bolts that hold the machine to the ground are removed (indicated by the solid line). The bolts remained in place (indicated by the dash line)

(i) (ii)

Bode Diagram of TF from Voltage to Drive Amplifier to Encoder (micrometer) 120 no mat, unbolted no mat, bolted

Magnitude (dB)

100

A

80 60 40

A

20 0 -20 0

Phase (deg)

-45 -90

B

-135

B

-180 -225 -270 -315 0 10

10

1

10

2

Frequency (Hz)

Fig. A1. Y-axis: FRF measurement for different machine conditions (bolted and unbolted)

The anti-resonance and resonance frequencies shifted from 45Hz to 12Hz with the removal of the bolts (indicated by the circle A and B). *Two bolts are located along the y-direction and 1 bolt each along the x-direction of the machine 139

Appendix B Ferraris Acceleration Sensor Calibration This section describes the procedures and the results of the identification of the Ferraris relative acceleration sensor. The sensor gain sensitivity and offset values are determined. The Ferraris sensor is graphically shown in fig. B.1.

Fig B.1: Ferraris relative acceleration sensor

Procedures ¾ The milling table was excited with sinusoidal signals at selected frequencies between 0.2-100Hz. ¾ Large gain was applied at lower frequencies (until 25Hz for displacement of ~10mm) and the gain was reduced for higher frequencies excitation to avoid saturation effect. ¾ The encoder measurement and the Ferraris sensor signals were recorded using the dSPACE. Gain Sensitivity and Zero Offset Determination

The calibrated relative acceleration signal, a(t) is defined as follow: a (t ) = m(t ) ⋅ G + Z

140

(B.1)

Appendix B

m(t) is the Ferraris measurement signal, G(t) is the sensor sensitivity, and z(t) is the offset. Velocity signal v(t) can derived from the integration of (1), v(t ) = G ⋅ ∫ m(t ) dt ⋅ + Zt + v0

(B.2)

the initial velocity. Finally, position, p(t) is obtained from integration of the velocity function: v0 is

z p(t ) = G ⋅ ∫ ∫ m(t ) dτdt ⋅ + t 2 + v0 t + p0 2

(B.3)

is the initial position.The sensor sensitivity, G and offset Z are then obtained by comparing equation (B.3) to the actual position measurement recorded by the encoder, e (t). p0

In matrix form:

⎡ e ( t ) = ⎢ ∫ ∫ m dτdt ⎢⎣

2

t 2

⎡G ⎤ ⎤⎢ Z ⎥ t 1⎥ ⎢ ⎥ ⎥⎦ ⎢ v0 ⎥ ⎢ ⎥ ⎣ p0 ⎦

(B.4)

At t=kT, T being the sampling time, ∫ ∫ m dτdt = M

⎡ e (t ) = ⎢M ⎣⎢

2

k T 2

2

⎡G ⎤ ⎤⎢ Z ⎥ kT 1⎥ ⎢ ⎥ ⎦⎥ ⎢⎢ v0 ⎥⎥ ⎣ p0 ⎦

(B.5)

A least square method is then applied to solve for G and Z. Results:

The Ferraris sensor sensitivity, G and the zero offset Z as a function of frequency are illustrated below.

141

Appendix B

3.7

x 10

7

Ferraris sensor sensitivity 2

3.6

x 10

4

Ferraris sensor zero offset

1

3.5 zero offset,

sensitivity,

0 3.4 3.3

-1

-2 3.2 -3

3.1 3

0

20

40 60 frequency,[Hz]

80

-4

100

0

20

40 60 frequency,[Hz]

80

100

Fig B.2. Gain sensitivity and offset value of Ferraris sensor as a function of frequency

The sensitivity and the zero offset values are rather uniform until 40Hz before fluctuating drastically. The mean sensitivity and zero offset values within this frequency range are G = 3.298e7

µm volt ⋅ s

; Z = -8580

2

µm

(B.6)

s2

The following figure illustrates the validity of the gain sensitivity and offset in (B.6). The actual encoder measurement was compared to the position signal generated from the double integration of the calibrated Ferraris signal. A close comparison was observed. position,[micrometer]

x 10

4

1

0

0

4

2Hz

10Hz

2000 1000 0 -1000

-1 1

2 time,[s]

3

4 20Hz

500 300 0 -300 -500

-2000

-1 0

position,[micrometer]

x 10

0.4Hz

1

0

0.2

0.4 0.6 time,[s]

0.8

1

40Hz

200

0

0.1

0.2 0.3 time,[s]

0.4

100

100

50

0

0

-100

-50

-200

-100

encoder pos. from Ferr. sensor

0

0.02

0.04 0.06 time,[s]

0.08

0.1

0

0.02

0.04 0.06 time,[s]

0.08

0.1

0

0.02

0.04 0.06 time,[s]

0.08

Fig. B3. Comparison between encoder signal and position signals acquired from the calibrated Ferraris relative acceleration signal 142

0.5

50Hz

0.1

Appendix C Cutting Force Estimation from Acceleration Sensor Measurement

Ferraris

The state space representations of the system, including the measurement equation are: 0 ⎤ ⎡ x⎤ ⎡ 0 ⎡ x& ⎤ ⎡0 1 ⎢ v& ⎥ = ⎢0 0 1 / M ⎥ ⎢ v ⎥ + ⎢ k f ⎥⎢ ⎥ ⎢M ⎢ ⎥ ⎢ ⎢⎣d& ⎥⎦ ⎢⎣0 0 0 ⎥⎦ ⎢⎣d ⎥⎦ ⎢⎢ 0 ⎣

0 ⎤ u ⎤ 1 ⎥⎡ − ⎥⎢ ⎥ ( ) F v M ⎥ ⎣ friction ⎦ 0 ⎥⎦

(C.1) ⎡ x⎤ 0 ⎤⎢ ⎥ ⎡ 0 ⎡ x ⎤ ⎡1 0 y=⎢ ⎥=⎢ ⎥⎢v ⎥ + ⎢ k f ⎣a ⎦ ⎣0 0 1 / M ⎦ ⎢d ⎥ ⎢⎣ M ⎣ ⎦

0 ⎤⎡ u ⎤ 1 ⎥⎢ − ⎥ ⎣ F friction (v )⎥⎦ M⎦

(C.2)

Next, a reduced order observer is designed. Consider a general state equation of the form: x& = Ax + Bu y = Cx, C = ⎣⎡ E 0 ⎦⎤

(C.3)

E is a unit matrix. A state of the system that contains a combination of the measured variable y and the non-measured state variable w are: ⎡ y⎤ x=⎢ ⎥ ⎣⎢ w⎦⎥

(C.4)

The system equation then becomes,

143

Appendix C

⎡ y& ⎤ ⎡ A11 ⎢ ⎥=⎢ ⎣⎢ w& ⎦⎥ ⎣ A21

A12 ⎤ ⎡ y ⎤ ⎡ B1 ⎤ ⎥ ⎢ ⎥ + ⎢ ⎥u A22 ⎦ ⎣⎢ w⎦⎥ ⎣ B2 ⎦

(C.5)

or, y& = A11 y + A12 w + B1u

(C.6a)

w& = A21 y + A22 w + B2u

(C.6b)

w is the only unknown in (C.6a). Therefore, a measuring equation for w becomes: y = y& − A11 y − B1u = A12 w

(C.7)

The observer equation for w then equals: ˆ& = ( A22 − LA12 ) w ˆ + A21 y + B2u + Ly , w & ˆ = ( A22 − LA12 ) w ˆ + A21 y + B2u + L ( y& − A11 y − B1u ) w

(C.8)

Equation (C.8) is modified to avoid using the derivative of y. Let: ˆ − Ly z=w ∴ z& = ( A22 − LA12 ) z + ( A22 − LA12 ) Ly + ( A21 − LA11 ) y + ( B2 − LB1 ) u

(C.9)

The matrix L is the observer gain. A transformation matrix T is required since the measured acceleration signal is not a state variable. Pritschow [29] has suggested a transformation matrix T that is equal to: ⎡1 0 ⎢ T = ⎢0 0 ⎢⎣0 M

0⎤ ⎥ 1⎥ 0 ⎥⎦

(C.10)

The state equation for the transformed system equals (with x’=Tx): ⎡ ⎢ 0 ⎡0 0 1 ⎤ ⎢ ⎢ ⎥ & = 0 0 0 x' + ⎢ 0 x' ⎢ ⎥ ⎢ ⎢⎣0 1 0 ⎥⎦ ⎢kf ⎢⎣ M

144

⎤ 0 ⎥ ⎥ ⎡ u ⎤ 0 ⎥ ⎢ ⎥ F v ⎥ ⎣⎢ friction ( ) ⎦⎥ 1 − ⎥ M ⎥⎦

(C.11a)

Appendix D

⎡ 0 ⎡ x ⎤ ⎡1 0 0 ⎤ y=⎢ ⎥=⎢ x'+ ⎢ k f ⎥ ⎢ ⎣ a ⎦ ⎣0 1 0 ⎦ ⎣M

0 ⎤⎡ u ⎤ 1⎥ − ⎥ ⎢⎣ F friction (v )⎥⎦ M⎦

(C.11b)

By definition, x ⎡ ⎤ ⎡ y '⎤ ⎡1 0 0⎤ k ⎢ ⎥ y = ⎢ 1 ⎥ = y − Du = ⎢ x ' 1 = f ⎥ u+ F friction (v )⎥ ⎢a − ⎣0 1 0 ⎦ ⎣ y 2 '⎦ M M ⎣ ⎦

(C.12a)

and, ⎡ x1 ' ⎤ ⎢ ⎥ x' = ⎢ x2 ' ⎥ ⎢⎣ x3 ' ⎥⎦

(C.12b)

The matrices A11, A12, A21, A22, B1, B2 (based on (B.11a) and the observer gain L for the designed of the reduced order observer of the transformed system are: ⎡1⎤ ⎡0 0 ⎤ A11 = ⎢ ⎥; A12 = ⎢ ⎥; A21 = [0 1]; A22 = [0]; 0 0 ⎣0 ⎦ ⎣ ⎦ ⎡kf ⎡0 0 ⎤ 1 ⎤ − ⎥; B1 = ⎢ ⎥; B2 = ⎢ 0 0 M M ⎣ ⎦ ⎣ ⎦ L = [l1 0]

(C.13)

Substituting (C.13) into the reduced order observer of equation (C.9) yields,

[

]

⎡ y '⎤ ⎡ k f zˆ& = −l1 &z& + − l1 2 1 ⎢ 1 ⎥ + ⎢ ⎣ y 2 '⎦ ⎣⎢ M

−

u ⎤ 1 ⎤⎡ ⎥⎢ ⎥ F v ( ) M ⎦⎥ ⎣ friction ⎦

(C.14a)

⎡y '⎤ ˆ = ˆz + ⎣⎡l1 0⎦⎤ ⎢ 1 ⎥ = ˆz + l1 y1 ' w ⎣ y2 ' ⎦

(C.14b)

ˆ ∴ ˆx3 ' = ˆz + l1x y1' = x, ˆx3 = w,

(C.14c)

⎡ y⎤

Based on x = ⎢ ⎥ , ⎢⎣ w⎥⎦

145

Appendix C

⎡ ˆx1 ⎤ ⎡ y1' ⎤ ⎡ 0 ⎤ ⎡ 1 0⎤ ⎡ y1' ⎤ ⎢ˆ ⎥ ⎢ ⎥ ⎢ ⎥ ˆ ⎢ ⎥ ⎡ y1 ' ⎤ ⎢ ⎥ 0 0 1 x y ' z = = + = ⎢ ⎥ 2 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ y ' ⎢ y2 ' ⎥ 2 ⎣ ⎦ ⎢⎣ ˆx3 ⎥⎦ ⎢⎣ w ⎥⎦ ⎢⎣1 ⎥⎦ ⎢⎣l1 0 ⎥⎦ ⎢⎣ ˆz + l1 y1 ' ⎥⎦

(C.15)

Equations (C.14a) and (C.15) can be simplified as follows: ˆz& = A' ˆz + B1' y' + B2 ' u ˆx' = D1' ˆz + D2 ' y'

(C.16)

Next, re-transformation into original physical states are accomplished, using the transformation matrix T and by replacing y’ with y: ˆz& = A' ˆz + B1 ' ( y − Du ) + B2 ' u

(C.17)

ˆx = T ( D1' ˆz + D2 ' y' )

or,

[

]

x ⎤ ⎡k ⎡ kf ⎥+⎢ f 1 z&ˆ = −l1&z& + − l12 1 ⎢ a − u + F v ( ) ⎢ friction ⎥ ⎣ M M M ⎦ ⎣

⎡1 0 xˆ = ⎢⎢0 0 ⎢⎣0 M

−

u ⎤ 1 ⎤⎡ ⎥ ⎥ ⎢F M ⎦ ⎣ friction (v )⎦

⎡ ⎤ ⎢ ⎥ x x 0⎤ ⎡ 1 ' ⎤ ⎢ ⎥ ⎡ xˆ ⎤ ⎥⎢x ⎥ = ⎢ ⎥ = ⎢ vˆ ⎥ zˆ + l1 x 1⎥ ⎢ 2 '⎥ ⎥ ⎢ ⎥ ⎢ kf ⎞⎥ ⎢⎣dˆ ⎥⎦ 1 0⎥⎦ ⎢⎣ x3 '⎥⎦ ⎢ ⎛⎜ M a− u+ F friction (v )⎟ ⎟⎥ ⎢ ⎜ M M ⎠⎦ ⎝ ⎣

(C.18a)

(C.18b)

Fig. C.1 shows the structure of this reduced order observer based on (C.16)

Fig. C.1. Schematic diagram of the cutting forces estimator 146

Appendix D General State Force Observer Cutting forces contain harmonic components (as shown in Table 3.5 and 3.6) 3.4). A cutting force state observer estimates and compensates the cutting forces based on these harmonic frequencies. The fundamental frequency equals the frequency of the spindle speed (1360Hz).

Structure of a Disturbance Force Observer Consider a sinusoidal disturbance force input d (t), its first derivative d& ( t ) , and the second derivative d&&( t ) : d ( t ) = q1 = A' sin ( ωt + θ ) d& ( t ) = q&1 = q2 = A' ω cos ( ωt + θ )

(D.1)

d&& ( t ) = q&2 = − A' ω sin ( ωt + θ ) = −ω q1 2

2

A’ is the amplitude of the sinusoidal signal, t is the time, ω and θ are the signal frequency and offset respectively. Equation D.1 shows that the second derivative of the reference disturbance force is directly proportional to the reference signal itself. A force disturbance observer can then be designed according to this observation. The following is the state space representation of a general order state observer: ˆx& = Fx + Gu + L ( y − ˆy ) y = Hx

(D.2)

x is the state variable matrix, y is the output, and L is the observer gain matrix. Equation (D.2) is extended based on the system transfer function (2.2) for cutting force compensation of a single harmonic content.

147

Appendix D

⎡ ˆx& ⎤ ⎡0 1 0 ⎢ ⎥ ⎢ & ˆ − A B 0 ⎢v ⎥ ⎢ ⎢ & ⎥ = ⎢0 0 0 ⎢ qˆ 1 ⎥ ⎢ ⎢ && ⎥ ⎢0 0 −ω2 ⎣ qˆ 2 ⎦ ⎣

0⎤ ⎥ 0⎥ 1⎥ ⎥ 0 ⎥⎦

⎡ ˆx ⎤ ⎡ 0 ⎤ ⎡ l1 ⎤ ⎢ vˆ ⎥ ⎢ ⎥ ⎢l ⎥ ⎢ ⎥ + ⎢ B ⎥ [u ] + ⎢ 2 ⎥ ⎢ qˆ 1 ⎥ ⎢ 0 ⎥ ⎢ l3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ qˆ 2 ⎦ ⎣ 0 ⎦ ⎣l4 ⎦

( y − ˆy )

⎡ ˆx ⎤ ⎢ vˆ ⎥ y = [1 0 0 0] ⎢ ⎥ ⎢ qˆ 1 ⎥ ⎢ ⎥ ⎣ qˆ 2 ⎦

(D.3a)

(D.3b)

The output y is equal to z in (2.2), where with z=x and z=y for the x and y axes respectively. The term A and B in matrix F and G are the system model parameters (Table 2.1). Fig. D.1 shows a schematics diagram of the system and the observer, with disturbance force input signal d, and a disturbance force.

Fig. D.1. Schematic diagram of a force observer structure 148

Appendix D

The fundamental equation (D.3) can be further extended to include other cutting forces harmonic contents. The following is an example of a disturbance force observer structure for compensation of cutting forces with three harmonics components: d1 ( t ) = q1 = A1 ' sin ( ω1t + θ1 ) d&1 ( t ) = q&1 = q2 = A1 ' ω1 cos ( ω1t + θ1 ) d&&1 ( t ) = q&2 =

− A1 ' ω12

sin ( ω1t + θ1 ) =

(D.4a)

−ω12 q1

d 2 ( t ) = q3 = A2 ' sin ( ω2 t + θ 2 ) d&2 ( t ) = q&3 = q4 = A2 ' ω2 cos ( ω2 t + θ 2 )

(D.4b)

d&&2 ( t ) = q&4 = − A2 ' ω22 sin ( ω2 t + θ 2 ) = −ω2 2 q3 d3 ( t ) = q5 = A3 ' sin ( ω3t + θ3 ) d&3 ( t ) = q&5 = q6 = A3 ' ω3 cos ( ω3t + θ3 ) d&&3 ( t ) = q&6 =

− A3 ' ω32

(D.4c)

sin ( ω3t + θ3 ) = −ω3 q5 2

The total forces equal: dtotal = d1 + d 2 + d3 or

(D.5)

dtotal = q1 + q3 + q5 .

The structure of this observer is an extension to the fundamental structure shown in fig. D.1. The disturbance observer design is based on the selection of the design parameters, that is, the observer gains l1, l2, l3, l4. Large LHP poles results in faster response but with reduced stability margin while small LHP poles have slower response but better stability margin.

149

Appendix D

Design of a Disturbance Force Observer The disturbance force observer is designed as an add-on module to the primary cascade P/PI position controller, discussed in Chapter 2. The schematic diagram of a control scheme that includes the cascade P/PI position controller and the disturbance force observer is shown in Fig. D.2.

Fig. D.2. A schematic diagram of a cascade P/PI controller with a general state force observer.

For analyses purposes, a single harmonic cutting force is considered. The disturbance observer state space model equals: ˆx& = Fxˆ + Gu + L ( y − ˆy ) ; but ˆy = Hxˆ ˆx& = Fxˆ + Gu + Ly − LHxˆ

(D.6)

With, 0 ⎡0 1 ⎢ 0 − A B ⎡⎣F ⎤⎦ = ⎢⎢ 0 0 0 ⎢ ⎢⎣0 0 −ω2

0⎤ ⎡0⎤ ⎥ ⎢ ⎥ 0⎥ B = ⎢ ⎥ ; ⎡⎣ H ⎤⎦ = ⎡⎣1 0 0 0⎤⎦ ; ; G ⎡ ⎤ ⎣ ⎦ 1⎥ ⎢0⎥ ⎥ ⎢ ⎥ 0 ⎥⎦ ⎣⎢ 0 ⎦⎥

⎡ ˆx ⎤ ⎢ˆ ⎥ v dˆ = Hxˆ = ⎡⎣0 0 1 0⎤⎦ ⎢ ⎥ ⎢ qˆ1 ⎥ ⎢ ⎥ ⎢⎣ qˆ 2 ⎥⎦

150

(D.7a)

(D.7b)

Appendix D

Taking the Laplace Transform, sXˆ ( s ) = FXˆ ( s ) + GU ( s ) + LY ( s ) − LHXˆ ( s )

(D.8)

However, U ( s ) = U PI ( s ) − HXˆ ( s )

(

(D.9)

)

U ( s ) = P ⋅ PI Z ref − Y − PI ⋅ Vest ⋅ Y − HXˆ

Substituting (D.9) into (D.8),

(

(

)

)

sXˆ ( s ) = FXˆ ( s ) + G P ⋅ PI Z ref − Y − PI ⋅ Vest ⋅ Y − HXˆ + LY ( s ) − LHXˆ ( s ) (D.10) Xˆ ( s ) =

( L − G ⋅ P ⋅ PI − G ⋅ PI ⋅Vest ) G ⋅ P ⋅ PI Z ref ( s ) + Y (s) sI − F + GH + LH sI − F + GH + LH

(D.11)

Also, (let the system be Gsys) Y ( s ) = Gsys (U + D )

(

(

(D.12)

)

Y ( s ) = Gsys P ⋅ PI Z ref − Y − PI ⋅ Vest ⋅ Y − HXˆ + D

)

(D.13)

Substituting (D.11) into (D.13) and with further simplification, ⎛ Gsys H ⋅ GP ⋅ PI ⎞ ⎜ Gsys PPI ⋅ − ⎟ ⎜ sI − F + GH + LH ⎟⎠ ⎝ Y (s) = Z (s) L − GPPI − G ⋅ PI ⋅ Vest ref 1 + Gsys PPI + Gsys PI ⋅ Vest + Gsys H sI − F + GH + LH Gp D(s) + L − G ⋅ P ⋅ PI − G ⋅ PI ⋅ Vest 1 + Gsys PPI + Gsys PI ⋅ Vest + Gsys H sI − F + GH + LH

(D.14)

The sensitivity function, S(s) equals: S (s) =

1 1 + Gsys P ⋅ PI + Gsys PI ⋅ Vest + Gsys H

L − G ⋅ P ⋅ PI − G ⋅ PI ⋅ Vest sI − F + GH + LH

(D.15)

151

Appendix D

Fig. D.3(a) and (b) show sensitivity functions based on equation (D.15) for a force observer that is designed for a single harmonic (46Hz) and a multiple harmonics (46Hz and 69Hz) component. Large gain is applied at each specific frequency. Bode Diagram

Magnitude (dB)

100 0 -100 -200

Freq(Hz): 46 Mag (dB): -307

-300 180

Phase (deg)

135 90 45 0 -45 0 10

1

2

10

10

Frequency (Hz)

(a) Bode Diagram

Magnitude (dB)

100 0 -100 -200 -300

System: frdy_d Frequency (Hz): 46 Magnitude (dB): -346

System: frdy_d Frequency (Hz): 69 Magnitude (dB): -310

180

Phase (deg)

90 0 -90 -180 -270 -360 -450 0 10

1

2

10

10

Frequency (Hz)

(b) Fig. D.3. Y-axis: Sensitivity function for (a) one harmonic (46Hz) and (b) two harmonics (46Hz and 69Hz) 152

Appendix D

Next, the transfer function (D.14) and (D.15) are validated numerically using MATLAB/Simulink based on the control scheme in fig. D.2. Fig. D.4(a) shows the simulated position error for sinusoidal disturbance signal at 23Hz. Fig. D.4(b) shows the estimated position error obtained using the sensitivity function (D.15). 1000

position error, [µm]

782

0

-782 -1000

2

2.02

2.04 2.06 tim e,[s]

2.08

2.1

(a) Bode Magnitude Diagram

5

10

0

Position error (abs) [µm]

10

Freq (Hz): 23 Mag (abs): 783

-5

10

-10

10

-15

10

-20

10

0

10

1

2

10

10

Frequency (Hz)

(b) Fig. D.4. Y-axis: (a) Simulated position error and (b) estimated position error for force observer designed for 46Hz and 69Hz harmonics

An exact match between the simulated position error and the estimated position error are observed.. This confirms the transfer functions (D.14) and (D.15). 153

Appendix D

A reduction in position error at a harmonic frequency, however, inflates the position errors of neighbouring frequencies in the opposite direction. This effect is widely known as the waterbed effect [46]. Fig. D.5 illustrates this waterbed effect. Bode Magnitude Diagram

0

Position error (abs) [µm]

10

-5

10

-10

10

-15

10

1 harm onic 2 harm onics

-20

10

1

10

2

Frequency (Hz)

10

Fig. D.5. Waterbed effect observed from additional harmonic component

The effectiveness of the force observer depends on the accuracy of the predicted frequencies used in the observer model. Fig. D.6 shows the force observer performance when a significant mismatch exists between the designed frequencies and the actual frequencies. The figure shows the effect of the disturbance force observer on the position error of the system. The input disturbance force contains harmonics at 23.3Hz and 40Hz. However, the observer is designed for disturbance harmonics at 23.3Hz and 46.6Hz.

154

Appendix D

15

position error [µm]

10

Force observer is activated

5

0

-5

-10

-15

0

0.2

0.4

0.6 0.8 time [s]

1

1.2

Fig. D.6. Position error for cutting force signal with 23.3Hz and 46.6Hz harmonic components associated with force disturbance observer with unmatched harmonic (23.3Hz and 40Hz)

Fig. D.7 shows a spectral analysis of the position error. With Force Observer 10

8

8 position error [µm]

position error [µm]

Without Force Observer 10

6

4

2

0

6

4

2

0

20

40 60 frequency [Hz]

80

0

0

20

40 60 frequency [Hz]

80

Fig. D.7. Spectral analysis of position error for force observer with unmatched harmonics

The spectral analysis shows total compensation of the matched harmonic alone. The mismatched harmonic remained uncompensated. This illustrates the non-robustness property of the disturbance force observer.

155

Appendix D

Experimental Validation The disturbance force observer is validated experimentally on the test setup. Synthesized disturbance forces with 1, 2, and 3 harmonics content are applied at the input of the system. Fig. D.8 illustrates the control scheme. The control scheme includes the inverse-model-based disturbance observer that is added to compensate friction during reference tracking.

Fig. D.8. Schematic diagram of a control scheme with a general state force disturbance observer and friction compensation

The performance is validated for two different cases, namely, (i) without reference tracking and (ii) with reference tracking. Fig. D.9 shows for case (i), the position errors of the y-axis when synthesized disturbance forces with 1, 2, and 3 harmonics are inserted at the input of the system. 500

500

3 harmonics [23.25Hz 46.6Hz, 69.9Hz] 500

200

200

200

0

0

0

-200

-200

-200

20

-500 0.1

0.2

0.3

observer activated

10 0 -10 -20 0

0.2 0.4 time [s]

0.4

0

20

-500 0.1

0.2

0.3

observer activated

10 0 -10 -20 0

0.2 0.4 time [s]

0.4

0

position error [µm]

-500 0

position error [µm]

2 harmonics [23.25Hz 46.6Hz]

position error [µm]

synthesized cutting force [N]

1 harmonic [23.25Hz]

20

0.1

0.2

0.3

observer activated

10 0 -10 -20 0

0.2 0.4 time [s]

Fig. D.9. Y-axis: Measured position errors for 1, 2, and 3 harmonic synthesized cutting force. 156

0.4

Appendix D

The above results indicate a complete removal of the cutting forces influence on the position errors. Fig. D.10 shows the compensation results during reference tracking. 4

1 harmonic reference [µm]

x 10

2 0 -2 0.5

1 time [s]

20

1.5

1.5

position error [µm] no observer

-4 0

1.5

position error [µm] with observer

position error [µm] with observer

position error [µm] no observer

reference [µm]

4

4

10 0 -10 -20 0

rms =3.1µm 0.5

1 time [s]

10 0 -10 0

rms = 0.66µm 0.5

1 time [s]

4

3 harmonics

x 10

2 0 -2 -4 0

0.5

1 time [s]

20

1.5

10 0 -10 rms = 7.12µm -20 0 0.5

1

1.5

10 0 -10 0

rms = 3.90µm 0.5

1 time [s]

1.5

Fig. D.10. Y-axis: Measured tracking errors for synthesized cutting force with 1 (left) and 3 (right) harmonics.

Fig. D.11 shows the spectral content of the tracking errors. 1 harmonic

3 harmonics 10 position error [µm] no observer

position error [µm] no observer

10 8 6 4 2 0

0

20

40 60 frequency [hz]

80

4 2 0

20

40 60 frequency [hz]

80

100

0

20

40 60 frequency [hz]

80

100

10 position error [µm] with observer

position error [µm] with observer

6

0

100

10 8 6 4 2 0

8

0

20

40 60 frequency [hz]

80

100

8 6 4 2 0

Fig. D.11. Spectral analysis of the tracking error for synthesized cutting force with 1 harmonic (left) and 3 harmonics (right). 157

Appendix D

Tracking error is completely eliminated for disturbance force with one harmonics content. In case (ii), the harmonics components are removed from the tracking errors, as indicated by the spectral analysis result in shown in fig. D.11. However, the lower frequency components of the signal is amplified. This is shown by the non-zero tracking error in fig. D.11 (indicated by the circle). This amplification is described as a waterbed effect. Next, circular tests are performed using a synthesized cutting force with 1 harmonic component. Fig. D.12 compares between the reference and the actual contours, and the radial tracking errors between two cases, (i) without a disturbance force observer, (ii) with a disturbance force observer. The actual contours have been amplified by a factor of 200 for graphical display purposes. The results show that the synthesized cutting force that contributes to the radial tracking errors is completely compensated by the disturbance force observer. During reference tracking, the experimental validations for both the radial and circular tracking tests are restricted to compensation of cutting force of lower harmonic components. Compensation of synthesized cutting forces with 3 or more harmonics components is not possible due to the physical limitation of the drives. This is explained in fig. D.13. With Observer ( 1 harmonic) position, y-axis [mm]

position, y-axis [mm]

No Observer (1 harmonic) 20

0

-20

20

0

-20 error: x200

error: x200 -20 0 20 position, x-axis [mm]

40

15

15

10

10 radial error [µm]

radial error [µm]

-40

5 0 -5 -10 -15

0

100 200 angle [degree]

300

-40

-20 0 20 position, x-axis [mm]

0

100 200 angle [degree]

40

5 0 -5 -10 -15

300

Fig. D.12. Circular test results for synthesized cutting force with 1 harmonic component for system without (left) and with (right) the force observer. 158

Appendix D

200 no observer

100

with observer

position error [µm]

0 -100 -200 -300 -400 -500 -600 -700 -800

0

1

2

3

4

5 time [s]

6

7

8

9

10

Fig. D.13. Position errors signal when the disturbance force observer is activated

This large transient response saturates the motor and result in stability. As a result of this limitation, a complete cutting force compensation analysis using synthesized cutting forces with 15 harmonics components is not possible. This preliminary results, however, indicates the effectiveness of this method for compensation of cutting forces of known frequency contents.

159