Linearization and control of dual axis micromirror

LINEARIZATION AND CONTROL OF
DUAL-AXIS MICROMIRROR
ZHAO YI
NATIONAL UNIVERSITY OF SINGAPORE
2006
LINEARIZATION AND CONTROL OF
DUAL-AXIS MICROMIRROR
ZHAO YI
(M. Eng.)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF MECHANICAL ENGINEERING
NATIONAL UNIVERSITY OF SINGAPORE
2006
Acknowledgements
I would like to express my sincere gratitude to many people who have lent their
assistance throughout these years of research. This thesis would not have been
completed successfully without them.
First and foremost, I am deeply indebted to my advisors, Associate Professor Tay
Eng Hock and Associate Professor Chau Fook Siong, for their constant guidance
and valuable suggestions they gave me during the last three years. Associate
Professor Tay have helped me to gain insights into the world of MEMS. He provides
an environment, which allows creativity and free ideas, and instills in me the
confidence to carry out independent research. His undying enthusiasm propelled me
to explore new realms of research and be constantly in search of novel ideas. I would
also like to express my deep gratitude to Associate Professor Chau. I will always
remember his patience in imparting many tips and suggestions that are crucial in
overcoming the obstacles faced in this research. His constant encouragement and
ii
Acknowledgements iii
critical reviews prove vital to the success of this research.

Special thanks go out to Assistant Professor Zhou Guangya, for his timely and
helpful suggestions, especially at the beginning of this research.
I will like to show my sincere gratefulness to Professor Ben M. Chen for his
valuable suggestion for some of my papers and this thesis, particularly in the
control part. Sincere thanks also goes to Associate Professor Quan Chenggen for
his important comments during the qualification exam.
I would also like to thank the numerous anonymous referees who have reviewed
parts of this work prior to publication in journals and conference proceedings and
whose valuable comments have contributed to the clarification of many of the ideas
presented in this thesis.
I sincerely thank my parents for their support in my earlier years of study, my
wife for her love and encouragement and my one year old baby for the happiness
she brings to me.
My final thanks go to the National University of Singapore for awarding me
the research scholarship for my Ph.D. study here, and to the other staff in the
Department of Mechanical Engineering from whom I have learned much through
modules and seminars during these years.
Zhao Yi
Feb 2006
Summary
The electrostatically actuated dual-axis micromirror based on MEMS technology
has attracted much attention due to its promising applications. However, the
inherent nonlinearity of the electrostatic torques results in two problems. One is
the scanning distortion within the stable range. Another is the scanning instability,
known as the ”pull-in” problem, when the driving voltages go beyond certain
thresholds.
The objectives of this study are (1) to investigate the scanning nonlinearity
of a dual-axis micromirror and subsequently to propose methods to linearize the
distorted scanning field, and (2) to stabilize the device beyond the pull-in point
thereby extending its useful scanning range. Two linearization methods, i.e. Radial

basis function (RBF) neural network (NN) and Delaunay triangulation (DT) are
proposed to reduce or eliminate scanning nonlinearity, thus correcting the distorted
scanning field. A position feedback integral sliding mode control (ISMC) algorithm
iv
Summary v
is applied to stabilize the micromirror beyond its pull-in point.
Both static and dynamic p erformances are investigated experimentally. The
static tests show that the static scanning field of dual-axis micromirror is distorted.
And the distortion rates increase with the increment of tilt angles. For the
moderate 50 V bias voltage, the distortion rates observed are about 30%. On the
other hand, the dynamic testing shows that the system is severely under-damped,
which results in large percentage of overshoot (53% for x-axis and 90% for y-axis)
and long settling times (15 ms for x-axis and 24 ms for y-axis). The dynamic
testing also reveals that there exists quite significant cross-axis coupling.
RBF NN and DT methods are designed to linearize the distorted scanning field.
The nonlinearity mapping is firstly captured. Then an inverse mapping based
on RBF NN or DT is designed to counteract the nonlinearity. The results show
that both of the methods can capture the scanning nonlinearity very well and
produce linearized scanning field. The distortion rates are dramatically reduced.
The linearized scanning field demonstrates distortion rates of less than 5%.
In terms of closed-loop control, the PID method demonstrates the ability to
improve transient response, leading to short settling time (less than 5 ms for both
axes), and no overshoot. At the same time cross-axis coupling is eliminated. As
a nonlinear control method, the integral sliding mode control shows its ability
to stabilize the system beyond pull-in point. The system exhibits an extended
stable range, which is more than 30% larger than the system without applying the
method.
Summary vi
It is concluded that the proposed linearization and control techniques have
demonstrated their abilities to overcome the stated problems in dual-axis

micromirrors.
Contents
Acknowledgements ii
Summary iv
Table of Content x
Nomenclature xi
List of Tables xv
List of Figures xvi
1 Introduction 1
1.1 Overview of Dual-Axis Micromirror . . . . . . . . . . . . . . . . . . 2
1.2 Operation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
vii
Contents viii
1.3.1 Scanning Field Distortion . . . . . . . . . . . . . . . . . . . 5
1.3.2 Pull-In Instability . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Purpose and Scope of Thesis . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2 Literature Review 12
2.1 Linearization Approaches . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.1 Differential Driving Scheme . . . . . . . . . . . . . . . . . . 13
2.1.2 Voltage Compensation . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 Radial Basis Function (RBF) NN . . . . . . . . . . . . . . . 16
2.1.4 Delaunay Triangulation (DT) . . . . . . . . . . . . . . . . . 17
2.2 Stable Range Extension Techniques . . . . . . . . . . . . . . . . . . 19
2.2.1 Geometry Modification . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Capacitor Feedback . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.3 Position Feedback Control . . . . . . . . . . . . . . . . . . . 24
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3 Theory 29
3.1 System Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2 Open-Loop Linearization . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1 RBF Neural Network Design . . . . . . . . . . . . . . . . . . 40
3.2.2 Delaunay Triangulation . . . . . . . . . . . . . . . . . . . . 43
3.3 Closed-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Contents ix
3.3.1 PID Control Design Within Stable Range . . . . . . . . . . 50
3.3.2 ISMC Design Beyond Stable Range . . . . . . . . . . . . . . 53
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 Experimentation 56
4.1 Fabrication Process PolyMUMPs . . . . . . . . . . . . . . . . . . . 56
4.2 Device Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5 Results and Discussion 67
5.1 Micromirror System Characterization . . . . . . . . . . . . . . . . . 67
5.1.1 System Simulation . . . . . . . . . . . . . . . . . . . . . . . 67
5.1.2 Experimental Testing . . . . . . . . . . . . . . . . . . . . . . 70
5.1.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.2 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2.1 Linearization Results of RBF Neural Networks . . . . . . . . 76
5.2.2 Linearization Results of Delaunay Triangulation . . . . . . . 82
5.2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Closed-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.3.1 PID Control Results . . . . . . . . . . . . . . . . . . . . . . 91
5.3.2 Stabilization Beyond Pull-In Point by ISMC . . . . . . . . . 99
5.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Contents x

6 Conclusion and Recommendations 104
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . 105
Bibliography 108
Appendices 123
A List of Publication 123
B RBF NN Code in C Language 125
C Micromorror Modelling in Matlab 128
D PID Control in Matlab 133
E Graphic Programme in LabVIEW 137
Nomenclature

0
Permittivity of free space
ω
n
Undamped natural frequency
φ Tilt angle about y-axis
τ
x
,τ
y
Damping coefficients of air
θ Tilt angle about x-axis
ζ Damping ratio
e Error between reference input and actual output
E
i
Range of the ith electrode
f

s
Sampling rate
g Gap between the mirror plate and the electrodes
xi
Contents xii
I
x
,I
y
Moments of inertia
K
x
, K
y
Stiffnesses of the torsional beams
s Switching function
T
s
Sampling interval
T
E
x
, T
E
y
Electrostatic torques
u Control output
V
i
Voltage applied to the ith electrode (i = 1 to 4)

V
x
,V
y
Driving control voltage
V
b
Bias voltage applied to the mirror plate
V
cx
,V
cy
Compensated driving control voltage
x
P SD tar
,y
P SD tar
Target laser sp ot position on PSD
x
P SD
, y
P SD
Lase spot position on PSD
A/D Analog-to-digital converter
AHDL Analog hardware description language
D/A Digital-to-analog converter
DAQ Data acquisition
DOF Degree-of-freedom
DT Delaunay triangulation
Contents xiii

FPGA Field programable gate array
HLS Hidden layer size
IC Integrated circuit
ISMC Integral sliding mode control
LPCVD Low pressure chemical vapor deposition
LSE Least squares error
MEMS Micro-Electro-Mechanical-System
MOEMS Micro-opto-electromechanical system
MSE Mean squared error
NN Neural networks
PID Proportional-Integral-Derivative
Poly-MUMPs Polysilicon Multi-User MEMS Processes
PSD Position sensitive detector
PSG Phosphosilicate glass
RBF Radial basis function
RT Real-time
SEM Scanning electron microscopy
SMC Sliding mode control
Contents xiv
TITO Two-input two-output
List of Tables
3.1 Optical system parameters . . . . . . . . . . . . . . . . . . . . . . . 35
4.1 MUMPs layers names, thicknesses and lithography level [105]. . . . 57
4.2 Material properties of PolyMUMPs . . . . . . . . . . . . . . . . . . 58
4.3 Designed structure dimensions (µm) . . . . . . . . . . . . . . . . . . 59
5.1 Damping ratios and natural frequencies identified. . . . . . . . . . . 74
5.2 PID control parameters. . . . . . . . . . . . . . . . . . . . . . . . . 94
xv
List of Figures
1.1 3D model of MEMS dual-axis micromirror with electrostatic actuation. 3

1.2 Schematic of one-DOF electrostatic actuation model. . . . . . . . . 5
1.3 Normalized displacement and driving voltage. . . . . . . . . . . . . 6
1.4 Typical distorted scanning field of dual-axis micromirror. . . . . . . 6
1.5 Pull-in problem of parallel plate model . . . . . . . . . . . . . . . . 9
2.1 Differential driving scheme vs. non-differential driving scheme.
(a) Simulated scanning field obtained from non-differential scheme.
(b)Simulated canning field from differential scheme [19]. . . . . . . . 14
2.2 A schematic of Delaunay triangulation. (a) 2D scattered points
input. (b) Delaunay 2D mesh generated. . . . . . . . . . . . . . . . 18
2.3 The idea of leverage bending [48]. . . . . . . . . . . . . . . . . . . . 19
xvi
List of Figures xvii
2.4 The idea of sidewall-electrodes [50]. (a) Cross-section schematic of
the mirror cavity structure with sidewall electrodes. (b) Top-view
schematic of the mirror bottom electrodes plane layout. . . . . . . . 20
2.5 A schematic of voltage-controlled capacitor feedback [51]. . . . . . . 21
2.6 A schematic of charge-controlled capacitor feedback [52]. . . . . . . 22
2.7 Feedback control algorithm. (a) Electrostatic torque. (b) Actuation
voltage [61]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Phase portraits for control input U = +V and -V for the electrostatic
microactuator [63]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 The mirror plate with a tilt angle. . . . . . . . . . . . . . . . . . . . 31
3.2 Electrostatic torques with respect to tilt angles for fixed voltages
V
x
= V
y
= 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Schematic of optical system. (a) Optical path. (b) Reflecting plane.
(c) Coordinate of laser spot on PSD plane. . . . . . . . . . . . . . . 34

3.4 Mapping from tilt angles to positions on PSD. (a) x
P SD
with respect
to angles θ and φ. (b) y
P SD
with respect to angles θ and φ. . . . . . 37
3.5 Schematic of the linearization. . . . . . . . . . . . . . . . . . . . . . 38
3.6 Radial basis function neural network . . . . . . . . . . . . . . . . . 40
3.7 Delaunay mesh (solid lines) is obtained by drawing connecting lines
between nodes perpendicular to the edges of the Voronoi diagram
(dashed lines). When four nodes are co-cyclic, quadrangular
elements are produced but they can be correctly divided into two
triangles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
List of Figures xviii
3.8 Procedure of Delaunay triangulation. (a) Input 2D scattered points
for triangulation. (b) Initialize the boundary edges. (c) Generate
initial triangles. (d) Remove the triangle, which encompasses the
new inserted point, and its neighbors. (e) Update the mesh after
inserting a new point. (f) Obtain the final Delaunay mesh. . . . . . 44
3.9 Analytical result: a typical sampling. (a) The command voltage
pairs applied. (b) The sampled positions on the PSD. . . . . . . . . 47
3.10 Analytical result: Delaunay mesh. The shaded area is the target
linear scanning field. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.11 Barycentric coordinates of a point P in triangle P
1
P
2
P
3
. . . . . . 49

3.12 Phase Portrait of SMC . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.13 Schematic of ISMC . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1 Cross-Sectional View of PolyMUMPs [105]. . . . . . . . . . . . . . . 56
4.2 Top view of the dual-axis micromirror. . . . . . . . . . . . . . . . . 59
4.3 Cross-section view of the dual-axis micromirror. . . . . . . . . . . 60
4.4 Mask design of mirror. . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5 Support layout. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Wire-bonded dual-axis micromirror ready for testing. . . . . . . . . 62
4.7 SEM view of the dual-axis micromirror. . . . . . . . . . . . . . . . . 62
4.8 Schematic of system testing set-up. . . . . . . . . . . . . . . . . . . 63
4.9 Schematic for differential voltage op eration. . . . . . . . . . . . . . 64
4.10 Picture of the experimental set-up. . . . . . . . . . . . . . . . . . . 66
4.11 A close view of the optical detecting system. . . . . . . . . . . . . 66
List of Figures xix
5.1 Schematic of simulation in Matlab. . . . . . . . . . . . . . . . . . . 68
5.2 Static scanning range. (a) Driving voltage pairs. (b) Static p ositions
on PSD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3 Stable scanning range. (a) Stable driving voltages range. (b) Stable
position range on PSD. . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.4 Experimental results of static scanning performance. . . . . . . . . 72
5.5 System damping ratios and natural frequencies identification from
experiment. (a) Normalized step response of x-axis. (b) Normalized
step response of y-axis. ∆θ and ∆φ are the increments of tilt angles,
the steady states of which are denoted as Θ and Φ respectively. . . 73
5.6 Simulation layout in Saber
TM
with analog hardware description
language. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.7 Simulated result of original distorted scanning field. . . . . . . . . . 76
5.8 Simulated results of training RBF NN. (a) Selecting the hidden layer

size. (b) Expected output and simulated output for the validation
data set (31 pairs of data). . . . . . . . . . . . . . . . . . . . . . . . 78
5.9 Simulated result of linearized scanning field by RBF NN. . . . . . . 79
5.10 Program in RT embedded micro-controller PXI 8175. . . . . . . . . 80
5.11 Experimental results of scanning field (V
b
= 50 V). (a) Scanning
field without RBF NN. (b) Scanning field with RBF NN. . . . . . . 81
5.12 DT simulated sampled voltage pairs. . . . . . . . . . . . . . . . . . 82
5.13 DT simulated scanning points on PSD. . . . . . . . . . . . . . . . . 83
List of Figures xx
5.14 DT simulated result: Delaunay mesh generated and the target area
(shaded rectangle) specified. . . . . . . . . . . . . . . . . . . . . . . 83
5.15 Simulated linearization results: input voltage pairs. . . . . . . . . . 85
5.16 Simulated linearization results: compensated voltage pairs. . . . . . 85
5.17 Simulated linearization results: linearized scanning field. . . . . . . 86
5.18 Experimental result: original scanning test. (a) The command input
voltage pairs. (b) The corresponding scanning field. The distortion
rates, 33%, 26%, 34% and 30%, for the left, bottom, right and top
boundary are observed respectively. . . . . . . . . . . . . . . . . . . 87
5.19 Experimental result: sampled points and Delaunay mesh. (a) The
command input voltage pairs for sampling. (b) The sampled points
on the PSD and the Delaunay mesh. The shaded rectangle is the
target linear scanning field. . . . . . . . . . . . . . . . . . . . . . . . 88
5.20 Experimental result of linearization. (a) Command input voltage
pairs. (b) Compensated voltage pairs. (c) Linearized scanning field. 89
5.21 Schematic of PID simulation in Matlab
TM
. . . . . . . . . . . . . . . 91
5.22 Simulation result: closed-loop static positioning performance. The

references of x
P SD
and y
P SD
are changing from -4 mm to 4 mm with
a step of 1 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.23 Simulation result: open-loop transient response. (a) Normalized
transient response of x-axis. (b)Normalized transient response of
y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.24 Simulation result: closed-loop transient response. . . . . . . . . . . 93
5.25 Simulation result: Control Outputs. . . . . . . . . . . . . . . . . . . 94
List of Figures xxi
5.26 Schematic of programming executed in the embedded controller PXI
8175. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.27 Experimental result: closed-loop performance of static scanning. . . 96
5.28 Experimental result: open-loop transient resp onse. (a) Step for
x-axis. (b) Step for y-axis. . . . . . . . . . . . . . . . . . . . . . . . 97
5.29 Experimental result: closed-loop transient response. (a) Step for
x-axis. (b) Step for y-axis. . . . . . . . . . . . . . . . . . . . . . . . 98
5.30 Simulation results in Matlab
TM
: transient responses for the set point
(x
P SD
= 4, y
P SD
= 3.5), which is beyond the pull-in point. . . . . . 99
5.31 Simulated result of control outputs for both axes. (a) Controlled
voltage V
x

. (b) Controlled voltage V
y
. . . . . . . . . . . . . . . . . . 100
5.32 Experimental result of transient responses for the reference point (4,
3.5), which is beyond the pull-in point (2.6, 1.5). . . . . . . . . . . . 101
5.33 Experimental result of control outputs for both axes. (a) Controlled
voltage V
x
. (b) Controlled voltage V
y
. . . . . . . . . . . . . . . . . . 102
Chapter 1
Introduction
In the early of 1990s, microelectromechanical systems (MEMS) emerged with the
aid of the development of integrated circuit (IC) fabrication process, and numerous
novel devices have been reported in diverse areas of engineering and science. The
term MEMS refers to a collection of microsensors and actuators which can sense
its environment and have the ability to react to changes in that environment with
the use of a microcircuit control. MEMS technology based system is faster, more
reliable, cheaper and capable of incorporating more complex functions.
So far, remarkable research progress has been achieved under the support of
governments as well as industries. In addition to the commercialization of
some less-integrated MEMS devices, such as microaccelerometers, inkjet printer
heads, micromirrors for projection, etc, the concepts and feasibility of more
complex MEMS devices have been proposed and demonstrated for a wide range
of applications, such as microfluids, aerospace, biomedicine, chemical analysis,
wireless communication, data storage, display and optics. During the last two
decades, all kinds of MEMS have been reported, such as optical MEMS [1][2],
1
1.1 Overview of Dual-Axis Micromirror 2

bioMEMS [3][4], power MEMS [5][6], microfluidics [7], miniaturized total analysis
systems (µTAS) [8] and RF MEMS [9][10].
Optical MEMS, also called as micro-opto-electromechanical system (MOEMS), has
attracted a great deal of research interest since their potential application market
is enormous. The dual-axis micromirror is one of the most important devices in
the MOEMS area. However, there are still some basic problems unsolved due to
its complexity of the structure and difficulty in control.
1.1 Overview of Dual-Axis Micromirror
The dual-axis micromirror has attracted much attention because of its promising
applications, such as free-space fiber optic switch [11]-[13], miniaturized pro jection
display [14]-[16] and endoscopic optical coherence tomography [17]. This device is
generally a double-gimballed structure and has 3-DOF (degrees of freedom), two
rotation motions (around x -axis and y-axis) and one translation motion (along
-z-axis), as shown in Fig. 1.1.
Both the bulk micromachining process [18] and the surface micromachining process
[19] have been reported for fabricating the dual-axis micromirror. The bulk
micromachining process has the ability to fabricate a thick and flat mirror plate.
On the other hand, the surface micromachining can fabricate very complex
multi-layer structures and is compatible with standard IC fabrication pro cesses.
One disadvantage of surface micromachining is that it may result in a curved
mirror plate, which is undesirable in optical applications.
All kinds of dual-axis micromirrors have been fabricated by increasing numbers
of researchers around the world. Actuation schemes, such as thermal [20]-[22],
1.2 Operation Principle 3
Inner frame
Mirror plate
Support
plate
Sliding plate
Bond pad

Substrate
Outer frame
Torsional beam
Electrodes
y
x
o
z
Figure 1.1: 3D model of MEMS dual-axis micromirror with electrostatic
actuation.
piezoelectric [23]-[27], electromagnetic [28]-[32], magnetostrictive [33]-[35] and
electrostatic have been reported. Among them, electrostatic actuation is a popular
actuation scheme for the dual-axis micromirror, since it has the merits of low power
consumption, simple driving electronics and ease of fabrication and integration.
However, electrostatic actuation suffers from nonlinearity, which will be discussed
later.
1.2 Operation Principle
A 3D view of the surface micromachined dual-axis micromirror is shown in Fig. 1.1.
The mirror plate, coated with a gold layer for reflectivity enhancement, is used
both as top electrode and as effective reflective surface. The mirror plate is
suspended with two torsional beams inside a gimbal, which is then suspended by
the outside torsional beams perpendicular to the direction of the inside torsional