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dissertation entitled

CONTROL OF SINGLE-LINK FLEXIBLE MANIPULA’I‘ORS FABRICATED
FROM ADVANCED COMPOSITE IMNATES AND SMART MATERIALS
INCORPORATING ELECI'RO-RHEOWICAL FLUIDS

presented by

SEUNG-BOK CHOI

has been accepted towards fulfillment
of the requirements for

Ph-D. degree in M-E-

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CONTROL OF SINGLE-LINK FLEXIBLE HANIPUIATORS FABRICATED
I-‘ROH ADVANCED COHPOSITE IAHINATES AND SMART MATERIALS
INCORPORATING ELECTRO-RHEOLOGICAL FLUIDS

by

Seung-Bok Choi

A DISSERTATION

Submitted to
Hichigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHIIDSOPHY

Department of Hechanical Engineering

1990

boSdQQc

ABSTRACT

CONTROL OF SINGLE-LINK FLEXIBLE MANIPUIATORS FABRICATED
FROM ADVANC- COMPOSITE LAMINATES AND SMART MATERIALS
INCORPORATING ELECTRO-RHEOLOGICAL FLUIDS

3?
Seung-Bok Choi

Three different control schemes for a single-link flexible
manipulator fabricated from composite laminates or smart materials
incorporating electro-rheological fluids are developed in this study ; an
output feedback controller, a hybrid controller, and a nonlinear state
feedback controller.

The output feedback controller featuring two colocated sensors is
designed by utilizing the root-locus technique and the Lyapunov method
after a description of the dynamic modeling of the flexible manipulator.
It is shown in the analytical and experimental investigation that the arm
fabricated from composite laminates has superior system performances such
as faster settling-time, smaller input torque, smaller overshoot, and
superior stability characteristics relative to the arm fabricated from
aluminum.

The hybrid controller which consists of an output feedback
controller and a pseudo-state feedback controller is proposed for the

flexible manipulator fabricated from smart materials incorporating

electro-rheological fluids. A phenomenological equation ofrmnflon is
developed from experimental observations of the modal characteristics.
.And an empirical model of a distributed electro-rheological fluid
actuator which characterizes the pseudo-state feedback controller is
obtained as an explicit function of the electrical intensities. It is
shown in the experimental implementation that the robustness of the
output feedback controller subjected to a disturbance which has a forcing
frequency close to the clamped-mode natural frequency of the arm is
impressively enhanced by employing the proposed pseudo-state feedback
controller. I

The nonlinear state feedback controller, which is robust to system
parameter variations caused by uncertain parameters such as varying
'payload and mismatched lamina angle, is developed for the flexible
manipulator fabricated from composite laminates. The effects of each
uncertain parameter as well as the interaction of all parameters are
considered without assuming matching conditions. The properties of the
uniform and uniform ultimate boundedness of the solution of system state
equations are employed to formulate the controller. It is demonstrated
through the simulation that the proposed nonlinear feedback controller
has much better capability of arresting the unwanted vibratflxnn than the

linear controller in the presence of the uncertain parameters.

iii

This thesis is dedicated to the memory of
my father, Kyung-Hyun Choi (1909-1988) and
father-in-law, Bong-Nam Kim (1931-1989),
who passed away during the course of this
research program. Their immense love and

sacrifices remain forever.

iv

ACKNOWLEDGMENTS

I would like to express my deepest gratitude and appreciation to my
co-advisors, Dr. Brian S. Thompson and Dr. Mukesh V. Gandhi, for their
expert guidance, friendship and continuous encouragement throughout the
course of this research program, and for providing the unique research
environment. Sincere appreciation is extended to the other guidance
committee, Dr. Steven.Wfl Shaw and Dr. David H.Y. Yen, for their valuable
comments and suggestions. I also thank Dr. Clark Radcliffe for his
constructive comments on this research.

Particular gratitude goes to my lovely wife Yeon-Ook, my smart son
Min-Kyu and my pretty daughter Bun-Young for being a wonderful family
with patience and understanding.

I, of course, wish to express sincere gratitude to my mother, Bo-Bae
Cho, mother-in-law, Soon-Sun Jang, eldest brother, Seung-Bae and his
wife Soon-Lim Lee, elder brother, Seung-Keun and his wife Keun-Soon Kim,
younger sister, Seung-Yi and her husband Young-Hyun Kim, brother-in-law,
Ha Kim and his wife Soon-Lim Bee, and sister-in-law, Hyo-Sook Kim, for
their providing moral support and encouragement.

Special thanks go to my friends at Michigan State University for
making my graduate career a successful experience. Also, I would like to
thank all the students in Machinery Elastodynamics Laboratory and
Intelligent Materials and Structures Laboratory for their technical

ass is tance .

Finally, I wish to gratefully acknowledge the financial support of
this research program by the Defence Advanced Research Projects Agency,
under contract DAAL03-87-K-0018, the U.S. Army Research Office, under
contract DAAL03-K-0022, and also the State of Michigan, Department of

Commerce, Research Excellence and Economic Development Fund.

vi

LIST OF

LIST OF

TABLE OF CONTENTS

TABLES ...........................................

FIGURES ..........................................

CHAPTER I. INTRODUCTION .................................

1.1

Motivation .......................................

1.2.1 Dynamic Modeling .............................
1.2.2 Controller Design ............................

1.2.3 Related Works on Modeling and Active Control

of Large Flexible Structures .................

1.2.4 Related Works on Smart Materials and Structures

1.3
1.4

CHAPTER

2.1
2.2
2.3
2.4
2.5

CHAPTER

Thesis Organization ..I ............................
Principal Contributions ..........................

II. MODELING OF A SINGLE-LINK FLEXIBLE MANIPULATOR
FABRICATED FROM ADVANCED COMPOSITE LAMINATES
Introduction .....................................
Derivation of Equations of Motion ................
Finite Element Formulation .......................
State Space Representation of the Model ..........
Summary of the Chapter ...........................

III. EXPERIMENTAL APPARATUS AND IDENTIFICATION OF
SYSTEM MODEL PARAMETERS ....................
Introduction .....................................

Experimental Apparatus ...........................

3.2.1 Flexible Arm .................................

.2.2 DC Motor and Servo-Drive

3.2.1.1 Aluminum Arm .............................
3.2.1.2 Composite Arm ............................

vii

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.........

000000000

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.........

000000000

19
26
36
38

41
41
42
50
S3
S6

57
57
57
59
59
61
66

3.2.3 Servo-Positioning Assembly ............................ 67

3.2.4 Sensors ............................................... 68
3.2.4.1 Angular Position Sensor ........................... 68
3.2.4.2 Angular Velocity Sensor ........................... 68
3.2.4.3 Tip Deflection Sensor ............................. 69

3.2.5 Microprocessor ........................................ 70

3.3 Identification of System Model Parameters ................. 71
3.3.1 Predicted ............................................. 72
3.3.2 Measured .............................................. 81

3.4 Summary of the Chapter .................................... 87

CHAPTER IV. OUTPUT FEEDBACK CONTROL ASSOCIATED WITH TWO

COLOCATED OUTPUT SENSORS ............................. 88

.1 Introduction .............................................. 88
.2 Controller Design ......................................... 89
.3 Implementation Results and Discussions .................... 99
4.3.1 Step Responses without Disturbances ................... 101
4.3.2 Step Responses with Disturbances ...................... 117
4.3.3 Instability of the System ............................. 128
4.4 Summary of the Chapter .................................... 136

CHAPTER V. HYBRID CONTROL FEATURING A DISTRIBUTED ELECTRO-

RHEOLOGICAL FLUID ACTUATOR ............................ 138

5.1 Introduction .............................................. 138
5.2 Background on Electra-Rheological Fluids .................. 139
5.3 Phenomenological Equation of Motion ....................... 144
5.4 Controller Formulation .................................... 151
5.5 Experimental Implementation ............................... 162
5.5.1 Apparatus and Instrumentation ......................... 162
5.5.2 Results and Discussions ............................... 170
5.6 Summary of the Chapter .................................... 181

(HflAPTER VI. NONLINEAR STATE FEEDBACK CONTROL ROBUST TO

SYSTEM PARAMETER VARIATIONS .......................... 183
6.1 Introduction .............................................. 183
6.2 Problem Formulation ....................................... 185

viii

6.3 Controller Design ......................................... 190

6.4 Illustrative Example ...................................... 209
6.5 Summary of the Chapter .................................... 222
CHAPTER VII. CONCLUSIONS AND RECOMMENDATIONS ..................... 224
7.1 Conclusions of the Thesis ................................. 224
7.2 Recommendations for Future Work ........................... 227
APPENDIX .......................................................... 231
List of Publications ...................................... 231
BIBLIOGRAPHY ...................................................... 236

ix

LIST OF TABLES

Page

Table 3.1 Material and Geometrical Properties of the

Aluminum Arm .......................................... 60
Table 3.2 Material and Geometrical Properties of the

Composite Arm ......................................... 64
Table 3.3 Comparison of the System Model Parameters of the

Aluminum and Composite Arm without Payload ............ 73
Table 3.4 Comparison of the System Model Parameters of the

Aluminum and Composite Arm with Payload of 0.061 kg .... 73
Table 3.5 Comparison of the Calculated and Measured Natural

Frequency without Payload ............................. 85
Table 3.6 Comparison of the Calculated and Measured Natural

Frequency with Payload of 0.061 kg .................... 85
Table 4.1 Closed-Loop Poles with the Compensator Zero Zco- -2 .... 102
Table 4.2 Closed-Loop Poles with the Compensator Zero 200- -5 .... 107
Table 4.3 Closed-Loop Poles with the Compensator Zero Zco- -1 .... 111
Table 4.4 Closed-Loop Poles with Kp-0.406, Kv-0.254 for

the Aluminum Arm, and Kp- 0.403, Kv- 0.252 for

the Composite Arm ...................................... 114
Table 5.1 Coherence Factor of the First and Second Mode at

Different Voltage ..................................... 149
Table 5.2 Measured System Model Parameters of the Smart ER Arm

in the Absence of the Electric Field .................. 169
Table 6.1 Geometrical and Material Properties of the

Nominal Composite Arm ................................. 210
Table 6.2 System Model Parameters of the Nominal System ......... 212
Table 6.3 Comparison of the Tip Displacement Response (Settling

Time and Maximum Overshoot) ........................... 220

Figure

Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.10

.11

.12

.13

.14

.15

LIST OF FIGURES

Schematic Diagram of a Single-Link Flexible
Manipulator

Composite Laminate Configuration
Two-Node Beam Element
Photograph of Overall Robotic System
Schematic Diagram of Overall Robotic System
Photograph of the Experimental Aluminum Arm
Design Methodology of the Composite Arm
Curing Cycle for AS4/3501 Graphite-Epoxy
Photograph of the Experimental Composite Arm

Load-Deflection Curve for the Aluminum
and Composite Arm

Relationship between the Velocity and the Measured
Voltage of the Tachometer

Relationship between the Amplified Voltage and
the Tip (End-Point) Deflection

Frequency Variation with respect to the Ratio of the

Hub Moment of Inertia to the Beam Moment of Inertia ....

Frequency Variation with respect to the Ratio of the
Payload to the Beam Mass

First Three Pinned-Mode Shape for the Aluminum
and Composite Arm

Calculated Open-Loop Transfer Function for the Angular
Velocity without Payload

Calculated Open-Loop Transfer Function for the Angular
Velocity with Payload of 0.061 kg

Calculated and Measured Frequency Response of the
Aluminum Arm without Payload

xi

00000000000000000

OOOOOOOOOOOOOOOOOOOOOOO

51

58

58

59

62

63

65

66

69

7O

76

77

78

79

80

83

Figure 3.16 Calculated and Measured Frequency Response of the
Composite Arm without Payload ........................ 84

Figure 3.17 Calculated Open-Loop Transfer Function for the Angular
Velocity with é(s)-(s/(s/a+l))9(s) ................... 86

Figure 4.1 Block-Diagram of the Output Feedback
Control System ....................................... 89

Figure 4.2 Open-Loop Pole-Zero Location with the Compensator
Zero zco- -1 ........................................ 91

Figure 4.3 Root-Locus with the Compensator Zero Zoo. -1 ......... 92

Figure 4.4 Open-Loop Pole Zero Location with the Compensator

Zero 2 - -15 ....................................... 94

co
Figure 4.5 Root-Locus with the Compensator Zero ZCO- -15 ........ 95
Figure 4.6 Experimental Set-Up for Output Measurements ........... 99

Figure 4.7 Circuit Diagram of Two Channel Operational
Amplifier ............................................ 100

Figure 4.8 Measured Step Responses without Payload.
Kp-0.534, KVP0-257 for the Aluminum Arm, and
Kp-0.292, KV-0.l46 for the Composite Arm ............. 103

Figure 4.9 Simulated Step Responses without Payload.
Kp-0.534, Kv-0.267 for the Aluminum Arm, and

Kp-0.292, Kv-0.146 for the Composite Arm ............. 104

Figure 4.10 Simulated Open-Loop Step Responses of the Aluminum
Arm without Payload .................................. 105

Figure 4.11 Measured Step Responses without Payload. Kp-1.0

and Kv-0.2 for both Aluminum and Composite Arms ....... 108

Figure 4.12 Simulated Step Responses without Payload. Kp-1.0
and Kv-0.2 for both Aluminum and Composite Arms ....... 109

Figure 4.13 Measured Step Responses with Payload of 0.061 kg.

Kp-Kv70.48 for the Aluminum Arm, and Kp-Kv-0.24 for
the Composite Arm .................................... 112

xii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.14

.15

.16

.17

.18

.19

.20

.21

.22

.23

.24

Simulated Step Responses with Payload of 0.061 kg.
Kp-Kv-0.48 for the Aluminum Arm, and Kp-Kv=0.24 for

the Composite Arm

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Measured Step Responses without Payload.
Kp-0.406, Kv-0.254 for the Aluminum Arm, and

Kp-0.403, er0.252 for the Composite Arm

0000000000000

Simulated Step Responses without Payload.
Kp-0.406, Kv90.254 for the Aluminum Arm, and

Kp-0.403, Kv-0.252 for the Composite Arm

Block—Diagram of the PD Output Feedback Control
System with the Disturbance of d(t)

Measured Step Responses of the Aluminum Arm
without Payload. Kp- 0.534, Kv- 0.267, and

3(t)-0.08 sin(3.25 Hz)t

Comparison of the Measured Step Responses of the
Aluminum Arm with and without the Disturbance.

xp-o.534, Kv-0.267 and d(t)-0.08 sin(3.25 Hz)t ....... 121

Comparison of the Simulated and Measured Step
Responses of the Aluminum Arm with the Disturbance.

Kp-0.534, Kv-O.267 and d(t)-0.08 sin(3.25 Hz)t ....... 122

Measured Step Responses of the Composite Arm
without Payload. Kp-0.292, Kv-0.146 and

3(c)-0.08 sin(4.75 Hz)t

Comparison of the Measured Step Responses of
the Composite Arm with and without the Disturbance.

Kp-0.292, xveo.145, and 3(t)-0.08 sin(4.75 Hz)t ...... 125

Comparison of the Simulated and Measured Step
Responses of the Composite Arm with the Disturbance.

xp-o.292, Kv-O.146 and d(t)-0.08 sin(4.75 Hz)t ....... 126

Measured Step Responses of the Composite Arm
without Payload. Kp- 0.292, Kv- 0.146, and

3(t)-o.4 sin(34.5 Hz)t ................................ 127

xiii

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure
Figure
Figure

Figure

Figure

Figure
Figure

Figure

Figure
Figure

Figure

Figure

.25

.26

.27

.28

.29

.30

.10

.11

.12

.13

.14

In-Position Band of the Servo-Drive System ........... 129
Measured Step Responses of the Aluminum Arm

without Payload. Kp-6.2 and Kv-0.4 ................... 131
Simulated Step Responses of the Aluminum Arm

without Payload. Kp-6.2 and Kve0.4 .................. 132
Measured Step Responses with Feedback Gains of Kp-6.2

and KV-0.4 for both Aluminum and Composite Arms ...... 133
Measured Step Responses of the Composite Arm

without Payload. Kp- 11.8 and Kv- 0.4 ............... 134
Simulated Step Responses of the Composite Arm

without Payload. Kp- 11.8 and Kv- 0.4 ............... 135
Electra-Rheological Fluid in Liquid

and Solid State ...................................... 140
Interaction between Charges on Electrodes and Those

in Electro-Rheological Fluid ......................... 141
Photomicrograph of Electra-Rheological Fluid .......... 142
Smart Cantilevered Beam Structure .................... 146
Mode Shapes of the Smart Cantilevered Beam ........... 147
Measured Frequency Response of the Smart ER Arm

without Payload ...................................... 155
Governing Characteristics of the Employed Electro-
Rheological Fluid Actuator ........................... 157
Block-Diagram of the Hybrid Control System ........... 159
Bang-off-Bang Type Input Voltage ..................... 161
Experimental Set-Up for the Frequency Response

Measurement of the Smart Cantilevered Beam ........... 163
Frequency Response of the Smart Cantilevered Beam ..... 165
Schematic Diagram of the Smart ER Arm ................ 166
Flow Diagram for Fabrication Procedures of the

Smart ER Arm ......................................... 167
Photograph of the Smart ER Arm ....................... 168

xiv

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

.15

.16

.17

.18

.19

.20

.21

.22

Measured Step Responses with Feedback Gains

of K - 0.5 and K - 0.25 .............................. 172
p v

Simulated Step Responses with Feedback Gains

of K - 0.5 and K - 0.25 .............................. 173
p v

Measured Step Responses with Feedback Gains

of K - 1.4 and K - 0.25 .............................. 174
p v

Simulated Step Responses with Feedback Gains

of K - 1.4 and K - 0.25 .............................. 175
p v

Measured Step Responses with Bang-off-Bang Type Input

Voltage. Kp-1.4 and Kv-0.25 ......................... 176

Measured Step Responses with the Disturbance.

Kp- 0.5, Kve 0.25 and a(:)-o.08 sin(4.1 Hz): ......... 178

Simulated Step Responses with the Disturbance.

Kp- 0.5, Kv- 0.25 and a(:)-o.08 sin(4.1 Hz): ......... 179

Measured Step Response with the Disturbance.

xp- 0.5, Kv- 0.25 and a(:)-o.a sin(16.8 Hz): ......... 180

Parameter Variation of the Composite Arm

with respect to the Payload .......................... 211

Parameter Variation of the Composite Arm

with respect to the Lamina Angle ..................... 211

Open-Loop Tip Displacement of the Nominal System ...... 216

Comparison of Step Responses with the Linear

and the Nonlinear Controller ........... .............. 218

Comparison of Step Responses with (-0.1 and e-0.003 ... 221

XV

CHAPTERI

INTRODUCTION

1 . 1 Motivation

Most of current day industrial robots depend on bulky design and can
handle objects no heavier than five percent of their own weight. The
bulkiness is mainly due to an inherent design requirement to minimize
structural vibration by increasing the mechanical stiffness of each
component. This massive structural design makes the robot slow and heavy;
hence, requires large actuators and high mounting strength and rigidity,
which consequently leads to high energy consumption and high overall
operating cost [26].

The insatiable demand in the international marketplace for high
performance articulating robotic systems has triggered a vigorous
research thrust in various multi-disciplinary areas such as design and
control. The high performance of robots may be quantified by high-speed
of operation, high end-point positional accuracy, lower or elimination of
friction and backlash incorporating indirect drive mechanism, and lower
overall cost. In order to achieve these high performances, light-weight,
flexible robotic arms have been proposed and developed.

One can simply reduce the weight of the links of the robot in order
to achieve these goals. However, it is not possible to move lightweight
arms quickly without the onset of structural vibration which is a

manifestation of inadequate structural damping as well as lower

1

2

structural strength and stiffness. Attempts have been and are being made
to overcome these deficiencies through optimization of the stiffness-to-
weight ratio by varying the length and cross section of each metallic
link [67]. A more innovative approach employs advanced composite
materials in the construction of high-speed manipulators to achieve
larger structural damping and higher stiffness—to-weight ratio [38, 118,
145, 146, 147]. Recent studies [38, 146] have demonstrated both
analytically and experimentally that transverse deflection can be reduced
substantially when links are made of advanced composite materials.
Moreover, to increase structural damping, optimization approaches have
been developed [82] . This is usually called a passive damping method or
passive control. However, present studies indicate that the increased
damping due to passive means is not sufficient by itself to eliminate
structural deflection [24]. Therefore, active control method, i.e. , the
design of control algorithms that take into consideration of the flexible
dynamics is needed.

A variety of dynamic modeling and control strategies for flexible
robotic manipulators, fabricated from conventional isotropic materials
such as steel and aluminum have been developed, and a few of them have
been experimentally implemented as described in the literature review.
However, up to now, the research areas on dynamic modeling and control of
flexible robotic manipulators fabricated from advanced composite
materials are considerably rare.

This study addresses the dynamic modeling and feedback control of a
single-link flexible manipulator fabricated from advanced composite
materials by undertaking an analytical and experimental investigation.

The advantages of employing composite materials relative to the

3

conventional isotropic materials can be divided into three categories;
design, control, and economics and safety. The advantages of the design
area are well presented in [145, 146]. The advantages of control area may
be described as a larger bandwidth due to the high stiffness-to-weight
ratio of the material, an increase of the stability margin due to larger
structural damping, a decrease of the control spillover effects due to
the inherent high natural frequency of the unmodelled dynamics.
Furthermore, more appropriate sampling time associated with observed and
controlled modes can be achieved by adjusting the high frequency and
large damping properties during the implementation [24] . The advantages
of the last category will include lower energy consumption, lower overall
cost due to high productivity on rate, and safer operation due to reduced
inertia forces. Some of these advantages relating control and economic
area will be observed both analytically and experimentally in this study.

It is also addressed in this study that the closed-loop system with
the conventional output feedback controller associated with two colocated
output sensors is very sensitive to an external disturbance which.has a.
forcing frequency close to the cantilevered-mode natural frequency of the
flexible arm. This attempt is made here to introduce a single-link
flexible manipulator fabricated from smart materials incorporating
electro-rheological fluids. The. robustness of the conventional output
feedback controller to the disturbance is to be expected by employing the
distributed parameter actuator which is characterized by embedded
electro-rheological fluids in the structure. It has been verified [41,56]
that the vibrational characteristics such as natural frequencies and
damping capacities of the flexible structures incorporating electro-

rheological fluids can be controlled in a very short time by imposing

4
electric potentials, hence providing the possibility of the elimination
of resonance phenomenon of the structures or systems.

Another important issue to be considered for controlling of flexible
robotic manipulators is system parameter variations such as natural
frequencies which may always arise in practice due to a wide spectrum of
various conditions associated with manufacturing of flexible arms,
dynamic modeling, and operating environments. The parameter variations
are more serious in the robotic systems featuring composite-based
flexible arms than those in the robotic systems featuring metallic-based
flexible arms. The reason is that the composite-based arms involve more
design variables, more complicated manufacturing process, and also higher
sensitivities to environmental conditions such as temperature and
moisture. In order to respond this problem, a nonlinear state feedback
controller robust to parameter variations caused by the varying payload
which may be encountered in the operation, and the mismatched lamina
angle which may be encountered in the manufacturing process, is developed
in this study.

These proposed observations which may provide significant impact on
the diverse marketplace such as automobile industry and aerospace
industry, coupled with the dearth of relevant research areas on the
control of flexible robotic manipulators fabricated from advanced
composite materials and smart materials have motivated the comprehensive

research program reported in this study.

1 . 2 Literature Review

Although this research program discusses on the dynamic modeling and

control of a single-link flexible manipulator, some of the major

5
contributions on the dynamic modeling and control of multi-link flexible
manipulators will be also reviewed here. FUrthermore, some research works
on the modeling and active vibration control of the large flexible
structures will be reviewed in order to provide a flavor of applicability
of the proposed research to the relevant structural systems. Finally, the
literature on the smart materials and smart structures will be reviewed
in order to introduce the proposed single-link flexible manipulator
fabricated from smart materials incorporating electro-rheological fluids.
It is tuited that the materials employed for the fabrication of flexible
arms in the following references are assumed to be conventional isotropic

materials such as aluminum and steel unless specific statement is made.

1 . 2 . 1 Dynamic Modeling

Schmitz [127] derived governing equations of motion and associated
boundary conditions for a single-link flexible manipulator by employing
the Hamilton's principle. The flexible link assumed to be governed by
Euler-Bernoulli beam theory. The characteristic equation for the
determination of natural frequencies and the eigenfunction for the
determination of mode shapes are obtained for both cantilevered-mode and
pinned-mode. Two different reduced-order models; reduced-order model with
system modes and cantilevered modes are derived from finite modal
expansion. The system model parameters such as natural frequencies and
modal slope coefficient are verified through experimental identification,
and shows good agreement between two results. Low [88,89,90] applied the
Hamilton's principle in order to derive the coupled, nonlinear integro-
<iifferentia1 governing equations for a manipulator containing both rigid

and flexible links. The method accounts for the distribution of the mass

6
and flexibility and its nonlinear dynamic effects. The formulation is
based on expressing the kinetic and potential energies of the robotic
system in terms of a set of independent generalized coordinates. A two-
1ink manipulator with one rigid link and one flexible link [90], and a
three-link flexible manipulator with three revolute joints [88] are
adopted to illustrate the proposed methodology.

Book [26,27] employed the transfer matrix method [114] to describe
the dynamic model in the frequency domain for a two-link planar elastic
arm. The method requires linear assumptions including small motions about
an equilibrium position with no Coriolis, centrifugal or gravity forces.
The model also was developed in the time domain by employing Lagrange's
equation, and showed good agreement to the results obtained from the
frequency domain. Book [22,23] also used 4x4 coordinate transformation
matrix [49] to derive the nonlinear dynamic equations of motion for
flexible manipulator arms containing of rotary joint and pairs of
flexible links. The computations resulting from the Lagrangian
formulation of the dynamics are reduced to recursive form similar to that
which has proved so efficient in the rigid-link case. The preliminary
programs were written [23] to evaluate computational efficiency showing
that the proposed method requires 2.7 times as many computations as the
most efficient rigid formulation with the same number of degrees of
freedom. Singh [134] used the 4x4 coordinate transformation matrix
incorporating a Lagrange approach to formulate a dynamic model for PUMA
type robot which has two rigidclinks and one last flexible link.

The finite element method has been applied by many researchers
[16,46, 105,135,142,153] to produce an accurate model of a flexible

manipulator. The justification for employing this method is three folds.

7
First, the finite element method is well known by many investigators.
Second, the method is generally versatile, hence the model can be easily
adapted to include as many, or as few, degrees of freedom as desired, and
also the structural damping can be accommodated more easily than some
other approaches such as Hamilton's principle. Third, the method can be
easily incorporated with different material properties, different
boundary conditions and change in cross section in a simple and
straightforward manner. Bayo [16] used a finite element method
incorporating Euler-Bernoulli beam theory to derive the discretized
equations of motion for a single-link flexible arm. The discretized
equations of motion in the time domain were reformulated in the frequency
domain to calculate the required input torque using Fast Fourier
Transformation (FFT). Dado [46] utilized the finite element technique to
derive a general dynamic equations for multi-link flexible manipulators.
In the forward analysis, the inertial loads are determined using the
known acceleration vector, while the acceleration of the driven
coordinates are determined by solving the dynamic equilibrium equations
of the system in the inverse analysis. Naganathan [105] obtained a
finite-element-based nonlinear model of flexibility effects in
manipulators. The governing equations of motion include rotary inertia,
shear deformation and the effects of gross nonlinear motion of each of
the links. The model was applied to the planar 2-R manipulator and
demonstrated that the nonlinear interactions contribute significantly to
the displacement errors, and hence they should not be ignored in the
modeling process. Skaar [135] used a finite element discretization of
distributed structures with fractional viscoelastic constitutive model

for the purpose of approximating the governing equations of motion. The

8
distributed structures are characterized by rotating viscoelastic beam
with rigid hub or rigid body with axial deflection of viscoelastic rod.

Usoro [153] developed a nonlinear dynamic model for a two-link
flexible manipulator by utilizing the finite element method, Lagrangian
method and the generalized inertia matrix. The modeling methodology
involves lumping each link into a number of elements, determining the
Lagrange functions for the overall system and invoking variational
principles to derive the governing dynamic equations. The finaI model
derived is extremely nonlinear and complex. It reflects the variation in
the effective inertia of the system as the manipulator configuration
changes as well as the interaction between the rigid-body dynamics and
vibration modes of the link. The axial deformation, viscous damping at
the joint and the flexibility of power train are ignored in the modeling
process. Sunada [142,143] developed a dynamic model for industrial
manipulators including the effects of distributed mass and flexibility,
and their dynamic nature. The individual link dynamic models are obtained
from standard finite-element formulation. Then these link models are
combined with the descriptions of the manipulator joint, expressed by 4x4
coordinate transformation matrix, to obtain complete dynamic model of a
flexible industrial manipulator system. Chang [33] used 4x4 coordinate
transformation matrix to describe kinematics for the large deformation,
and the finite element method to discretize the deformations.

Rakhsha [117] obtained a dynamic model for a single-link flexible
robot arm with a rigid mass attached to the tip by employing a Newton-
Euler method. The flexibility effects are modelled as an internal
disturbance torque generated by lateral bending which affects the rigid-

body motion. The method of constrained modes [148] is utilized to

9

determine the natural frequencies and mode shapes. Wang [155] developed a
dynamic model for a flexible robot arm with moving slender prismatic beam
by applying Newton's second law, and the model was approximated by
Galerkin method [119]. It is found that the extending and contracting
tundons have destabilizing and stabilizing effects on the vibratory
motions. Truckenbrodt [150] developed a dynamic model so-called hybrid
multibody model for the manipulator with rigid and flexible elements. The
model includes the effect of bending and torsion, and the Ritz-
Kantorovitch series expansion of the distributed deformation coordinates
was employed to derive the governing equations of motion which are
nonlinear, ordinary differential equations. The experimental verification
of the proposed dynamic model was undertaken by considering a single-link
fabricated from aluminum. Good agreement between theory and experiment
was shown by comparing the first four natural frequencies. Davis [47]
developed a class of hybrid robot arm model to study tracking problem
without the necessity for a priori approximation in the flexibility model
for the link. The model includes a distributed model of the dynamics of
the structural elements of a robot link, combined with a discrete model
for the actuator and load dynamics.

Asada [5] derived the equations of motion for two-link flexible arms
based on Lagrange method, then transformed to the virtual link coordinate
systems which consider the virtual links that are rigid and connecting
adjacent arm joints by straight line. Actuator torques required to track.
a given trajectory in the plane are computed by a simple and efficient
method using virtual link coordinate system through inverse dynamics. It:
is shown that the virtual link coordinates are not only convenient for

solving the inverse dynamics problem, but also advantages to reduce

10

computational complexity. Barraco [14] developed two methods for dynamic
modeling of multi-link flexible manipulators based on the choice of
reference configuration. In the first approach, part of unknown variables
represent the finite rigid-body moment while the remaining part deals
with the infinitely small deformation of members. In the second
formulation, an incremental form of the virtual principle is derived by
subtracting virtual work equations written at two successive time steps.
Nicosia [110] used a generalized Lagrange approach to derive the
equations of motion for single-link and double-link flexible
manipulators, and Streit [141] applied Lagrange approach to obtain
governing equations of motion containing parametric excitation terms for
a two-link flexible manipulator which has one translational and one
revolute joint. The dynamic model is converted to the first order form in
order to facilitate the stability analysis by using Floquet theory. It is
shown in the illustrative example that the stability diagrams are
significantly influenced by change in joint stiffnesses and mean
operating position.

Everett [51] utilized a variational principle to provide an integral
formulation of the governing differential equations, which is easily
discretized incorporating with the finite element method. A functional
expression is obtained by introducing appropriate Lagrange multipliers
and using vector techniques, and the variation of the functional was
undertaken before the discretization. Thompson [146,147] developed a
dynamic model for industrial robotic manipulator constructed from
composite materials by utilizing a variational theorem and a finite
element method. The variational theorem was formulated by considering

employed composite materials as linear viscoelastic homogeneous

11
materials. In the finite element formulation, the deflections of the
elastic members are assumed to be governed by the Euler-Bernoulli beam
theory. The difference of variational formulation from the work done in
[51] is that the vectors are transformed into a fixed reference
coordinate before the variation is computed, while Everett [51] present a

technique whereby this transformation need not be applied.

1 . 2 . 2 Controller Design

Cannon [31] designed a linear optimal state feedback control law
associated with noncolocated tip position sensor and experimentally
implemented the control law on a single-link flexible manipulator. The
experimental arm made of aluminum has a length of 1.13 m, a width of 6.6
cm and a thickness of 1 mm. It is shown through experiments that the tip
position feedback is much more sensitive to model errors than one that
uses only joint-angle feedback. It is also shown that the closed-loop
bandwidth achieved with tip position feedback is two times the first
cantilevered-mode natural frequency, which corresponds to a factor of
four improvement over typical bandwidth achieved with the joint-angle
feedback. Finally it has been investigated that the position feedback
yields substantially better disturbance rejection to end-point external
forces than does joint-angle feedback. Chassiakos [34] formulated a
Linear Quadratic Gaussian (LOG) controller with a torque actuator and
optimally located force actuator. The optimal location for force actuator
is chosen so as to minimize the bending during the arm motion. An
illustrative numerical example is given to demonstrate the proposed
methodology showing that the step response of the robot arm with the

torque and force actuator exhibits superior performances over the robotic

12

arm with only the torque actuator, such as faster settling-time and
increment of attenuation of flexible vibration mode. Hastings [64,65]
designed linear optimal state feedback control laws with a prescribed
degree of stability and experimentally implemented the control laws on a
single-link flexible robot arm. The experimental arm made of aluminum has
a length of 1.22 m, a width of 1.905 cm and a thickness of 4.76 mm.
Angular position from the potentiometer, angular velocity from the
tachometer and strain from the strain gages mounted on the root and
midpoint of the link are used to estimate actual states through
Luenberger reduced order observer. It is shown in the experimental
investigation [65] that the controller cycle time affects significantly
to the step response of a colocated controller associated with joint
position and joint velocity, resulting the increment in the excitation of
the third flexible mode with larger cycle time. It is also shown that the
optimal modal controller with large prescribed degree of stability yields
instabilities in the higher flexible modes.

Nelson [106] presented a method adjusting the control gain to
maintain the desired arm response by using an on-line estimator which
provides periodic updates of load mass estimates to a parametric optimal
controller. It is shown through an illustrative example that the proposed
method provides capability of achieving rapid, smooth and accurate motion
of flexible robotic arm under varying load conditions. Matsuno [91] and
Sakawa [123] designed linear optimal state feedback control laws with
prescribed degree of stability. 1.05 m long, 1.5 mm thickness of
stainless steel experimental arm [123] is used to investigate system
performances. The output sensors associated with optical encoder for

angular position, tachometer for angular velocity and strain gage mounted

13

on the root of the link for strain are employed to provide information
for estimating the actual states. The first two dominant modes are
controlled, and third mode was estimated in order to estimate the first
two modes to be controlled. It has been demonstrated that the
experimental implementation of the proposed control schemes performed
well. Book [27] and Neto [108] proposed three different linear state
feedback control schemes : joint angle and velocity feedback with (GRC)
and without (IJC) cross joint feedback, and feedback of flexible state
variables (FFC) . It is shown through simulation for a two-link flexible
arm that the bandwidth of GRC and IJC is limited to the fundamental
natural frequency by inadequate damping, and the PFC is very sensitive to
parameter perturbations and somewhat higher torque requirement. Fukuda
[53] considered a flexible robot arm with two degrees of freedom such as
a SCARA type of robotic arm for precise position control. The state
feedback control laws are designed and experimentally implemented. The
feedback gains could be adjusted adaptively in order to improve transient
performance due to nonlinearities and varying unknown payloads. Collision
and Contouring control using flexibility of the arm was formulated and
implemented successfully. And also a reliable control structure was
implemented on the basis of the fault-tolerant configuration, employing
dual-processor control systems. Chalhoub [32] designed the integral plus
state feedback controller based on the linearized equations of motion. It
is shown in experimental investigation that the proposed feedback
controller by considering rigid-body and first flexible mode exhibits
superior responses to that designed by considering only rigid-body.

Wang [157] proposed an output feedback controller for a single-link

flexible manipulator. The two colocated output sensor associated with

14

angular position and angular velocity are used as feedback sensors.
Feedback gains guaranteeing the stability of the closed-loop system are
obtained by the least square method and the minimum norm pseudo-inverse
method [28]. It is shown in the simulation work that the proposed output
feedback controller performs well and robust to parameter variations such
as change of natural frequencies. Wang [158] also conducted an
experimental investigation of the position control capability of a
single-link flexible robot arm by employing an output feedback
controller. The employed output sensors include a shaft encoder for the
measurement of hub angle, and a laser beam for the measurement of the
deflection slope of the flexible link. It was shown that the performance
of the overall closed-loop system was satisfactory and the controller was
robust to parameter variations such as different payload. Siciliano [131]
designed an output fast feedback controller which includes a fixed order
dynamic compensator based on convergent numerical algorithm for
calculating 112G optimal gains [102]. A composite control is adopted such
that a slow mode control which is of the same order as that of a rigid
arm, and a linear fast state control which models the deflections of the
arm. The slow mode controller is first designed to track a desired joint
trajectory, and fast state controller is subsequently designed to
stabilize the deformations along the trajectory.

Meldrum [98] proposed a direct model following adaptive controller
for a single-link flexible robot arm to control the torque at the pinned
end of the arm so as to command the free end to track a prescribed
sinusoidal motion. The control law was formulated by employing positive
realness condition which is easily satisfied for colocation but not for

noncolocation of the sensors and actuators. It is shown through

15

simulation work that the precise tracking of the reference model
trajectory is achieved by colocating the actuator and sensor to satisfy
the Command Generator Tracker [11] condition which assumes ideal
intermediate system. Siciliano [132] and Yuan [162] developed a nonlinear
robust adaptive control laws based on Lyapunov function for a single-link
flexible arm. This is accomplished by separating the system dynamics into
two parts ; linear and nonlinear (uncertainty in the linear model).
Simulation work shows that the proposed adaptive control laws perform
better than the typical linear optimal controller in spite of
nonlinearities or uncertainties, as it contributes to obtain better
tracking accuracy, smaller vibration amplitudes. Rovner [122] performed
experiments for the system identification with the aim of implementing
adaptive controller on a single-link flexible manipulator associated with
tip position and angular velocity sensors. The adaptive controller
maintains accurate position control of the manipulator while it picks and
places varying loads. The identification algorithm employed is a filtered
version of the recursive least squares algorithm.

Lee [78] proposed a nonlinear state feedback controller in order to
control the tip position of a single-link flexible manipulator. The form
of feedback is products of state variables to obtain a lightly damped
system in the beginning of a step response and a more heavily damped
system when it is close to the commanded position. It is shown from the
illustrative example that the proposed controller provides reduced
settling-time by 32 percent without increasing the percent overshoot,
comparing to a linear optimal controller. Singh [134] derived a nonlinear
feedback controller for the PUMA type of robot which has two rigid links

and one last flexible link. The control law decouples the joint angle

16

motion from the flexible motion and asymptotically decomposes the elastic
dynamics into two subsystems, representing the transverse vibrations of
the elastic link in two orthogonal planes. In the simulation, the
complete closed-loop system is shown to be globally asymptotically stable
and robust to uncertainty in' system parameters. Siciliano [129]
introduced a singularly perturbed model in order to design a nonlinear
composite controller which consists of a slow control and a fast control.
A slow control is designed for the slow subsystem, which is shown to be
the model of the equivalent rigid-link manipulator. Then a fast control
is designed to stabilize the fast subsystem around the equilibrium
trajectory set up by the slow subsystem under the effect of the slow
control. A single-link flexible robot arm is chosen as a case study, and
the comparison work with typical linear state feedback was undertaken
showing satisfactory system performances.

Goldenberg [60] presented a computed torque technique to control the
rotary joint of a single-link flexible robot arm. The nominal torque,
required for a given desired trajectory, is calculated from the rigid-
body model of the robot, and this provides a feedforward compensator.
Proportional plus derivative (PD) feedback controller associated with two
colocated angular position and velocity sensors as well as feedforward
one provide almost pole-zero cancellation of the dominant closed-loop
poles of the system, hence resulting in better transient response and
smaller steady-state error comparing with only PD feedback controller.
Schmitz [126] used a PD feedback controller associated with joint angle
and joint rate sensors for controlling of two-link flexible manipulator.
It is shown in the simulation work that with increasing shoulder position

gain, two rigid-body poles travel towards a pair of zeros which are very

17

close to the first locked-joints (cantilevered-mode) vibration frequency
which represents a limit on the maximum bandwidth attainable with
colocated feedback controller. Skaar [135] used a classical root-locus
technique to design a controller for viscoelastic systems. The
controllers are formulated based on the fractional derivatives of the
constitutive law of the material itself, which makes the proposed
feedback control laws too difficult to be implemented physically. Also it
is found that time-domain behavior of the stabilized closed-loop system
is not self-evident.

Wang [156] proposed a method for synthesizing open-loop control
inputs to move a flexible robot arm precisely along a given desired
trajectory. Proportional-integral-derivative (PID) controller for three-
axis Cartesian manipulator is formulated to minimize excessive residual
vibrations at the end of a movement. The computed reference error
adjustment technique was also introduced to obtain substantial reduction
in output error. Meckl [93] designed a open-loop controller for a three-
axis Cartesian manipulator using bang-bang control function. The function
provides optimal-time response to eliminate residual vibration of a robot
arm at the end of a movement. Book [25] developed a minimum time control
of a two-link flexible arm with actuator constraints and without
constraints on the flexible modes. The near minimum time trajectories for
robotic manipulators moving along the pre-defined path are modified such
that the flexible vibrations will be bounded. It is shown in simulation
that flexible manipulators with same actuator capabilities are faster
than rigid-body manipulators. Morimoto [103] used transfer function data
of the input-output relation to predict vibratory motion of the end

effector of the flexible manipulator which was commanded to move

18
according to a trapezoidal velocity pattern. The transfer function data
were obtained through experimental modal analysis technologies. It is
shown that the proposed method is very effective for suppression of the
flexibility-induced vibration of the robot arm.

Korolov [73] developed a robust controller for a one-ldxflt flexible
robot arm subjected to uncertainties. The control scheme was formulated
by treating the natural frequency variation, which may be possibly
occurred in practice, as bounded uncertainty. The proposed robust
controller consists of a linear part which is optimal state feedback
controller with a reduced order observer, and a nonlinear part which
depends on the boundedness of the parameter variations. The reduced order
observer for the state estimator was designed independent of the control
input. Nemir [107] presented a self-tuning type algorithm for the control
of a single rotating compliant link. The self-tuning control is devised
by treating angular position obtained from first mode information of
flexibility measured from strain gage and hub position measured from
position encoder, as belonging to an equivalent rigid link. The angle of”
the hub to tip projection (the pseudo-link) is presented using an auto
regressive exogeneous type time series model. The laboratory experimental
investigation shows that the proposed control leads to an improved
performances over a control which ignores compliance. Liegeois [83]
introduced a calibration procedure based on a learning technique a:
identify the structural parameters of the system. The damping of
oscillations induced by fast transient motions is carried out by means of
the measurement of cable deformations and their derivatives using the

strain gages.

19

Choi [37] proposed an integrated strategy for both the optimal
structural and controller design of a single-link flexible manipulator
fabricated from composite laminates. The formulation of the optimal
structural design is accomplished by integrating composite laminate
design variables such as stacking sequence, fiber volume fraction and
lamina thickness which characterize the mass, stiffness and damping
properties of the structure, and an objective function which
characterizes the elastodynamic response of the closed-loop system. It is
shown in an illustrative example that the optimally designed arm exhibits
superior performance characteristics such as the smaller control
spillover effects and faster rise-time without increasing the overshoot“
Mesquita [99] considered structural optimization of the laminated
composite materials to maximize the structural frequencies subjected to
frequency separation constraints and an upper bound on weight. The
control problem was solved by employing the independent modal space

technique introduced in [96].

1.2.3 Related Works on Modeling and Active Control
of Large Flexible Structures

Large flexible beam-like or truss-like structures are a
characteristic feature of many spacecraft appendages, such as space
station, for example. While these large structures are a prerequisite for
attaining the goals and objectives of these pioneering space projects,
the inherent low stiffness characteristics of these structures have
triggered research efforts focused on the development of dynamic modeling

and active vibration control strategies for these large flexible

20

structures which represent similar characteristics occurred in the case
of flexible manipulators. Two most important quantities which
characterize the vibrational behavior of flexible structures are the
natural frequencies and inherent damping capacities of these systems.
Since these continuum structures are characterized by an infinite number
of vibrational modes, modal truncation must be employed in the modeling
process. Hughes [66] discussed the existence and the selection of
critical modes of the flexible structures or systems. He suggested that
the appropriate number of modes should be selected based on the the order
of natural frequency as well as dynamical grounds of the system such as
boundary conditions. Balas [10] and Johnson [68] summarized on the
dynamic modeling methodologies and active vibration strategies developed
so far. Some of them will be reviewed in this section.

Most of purposes it is satisfactory to neglect shear deformation and
rotary inertia of the beam-like structures in the modeling process as
described by Archer [4]. While the Euler-Bernoulli beam theory leads to
very accurate model for the lower modes of long and thin beam,
significant error may be caused if no account is taken of shear
deformation and rotary inertia for higher modes of any beam structures,
or thick beam. Hence some references relating to Timoshenko beam theory
are reviewed for appropriate dynamic modeling of large flexible
structures. Davis [48] formulated a Timoshenko beam element which
includes the effects of shear deformation and rotary inertia. The
convergence tests are performed for a simply-supported beam and a
cantilevered beam. It is shown that the element is converged to an exact
solution of the elasticity equations for a simply-supported beam provided

that the correct value of the shear coefficients is used. Gamache [54]

21

compared the Timoshenko and the Euler-bernoulli beam theories by modeling
of planar linkage mechanism. The four-bar linkage with flexible coupler
and rocker links is chosen for simulation work and it is shown that the
percent error between two theories is function of the aspect ratio, i.e.
the thickness to length ratio of the linkages. Kapur [71] formulated a
beam element incorporating Timoshenko beam theory which is applicable to
uniform and nonuniform beams with any boundary conditions. Thomas [144]
developed Timoshenko beam element by considering three degrees of freedom
at each of two nodes and applied to a cantilevered beam to calculate
natural frequencies.

Bodden [21] proposed a minimum modification strategy for structure
and control parameter optimization for the vibration arrests of flexible
structures. Linear output feedback gains are determined using a novel
optimization strategy, and it is shown that the sensor and actuator
locations, and feedback gains can be modified to achieve satisfactory
results. Hale [62] discussed an optimization problem for maneuvering
flexible spacecraft in order to achieve both structural parameters and
active control forces. A specific cost functional which consists of a set
of structure parameters and control forces is chosen as an objective
function which is supposed to be minimized. Khot [72] developed an
integrating strategy for the structure and control design to reduce the
structural response from a disturbance. The structural modification of
some nominal finite element model, which is controlled in an optimal
fashion by a linear regulator, was employed to increase the active modal
damping factor. The objective function is the weight of the structure
with a constraints on the damping parameters of the closed-loop system.

Onoda [112] presented an approach to the simultaneous optimal design of

22

structure and control system for large flexible beam-like spacecrafts by
choosing the total cost of structure and control system as an objective
function with constraints represented by a given disturbance involving
both rigid-body and elastic modes. Direct feedback and linear quadratic
optimal control laws are used.with both inertial and noninertial
disturbing forces, and it is shown that the simultaneous opthmization of
the structure and control system is indeed useful.

Arbel [3] showed that when an output feedback gain matrix is chosen
as symmetric negative-definite, a linear quadratic optimization problem
can be stated and solved in closed-form via an inverse optimal control
formulation. This, in turn, allows to be used the well-studied properties
of the linear quadratic regulator for stability problem at hand. Balas
[9] formulated a direct velocity output feedback control that may
guarantee that all vibration modes remain stable when the active control
is in operation. However, it is pointed out that the proposed feedback
controller has some stringent restrictions, such as an equal number of
colocated force actuators and velocity sensors, and a non-negative
definiteness of the feedback gain matrix in order to achieve
stabilization.of all vibration modes. Elliott [50] proposed a robustness
of the control system which arises from the fact that the stability of
the system is dependent only on the controller and is not a function of
the system model parameters. It is pointed out that even though sensors
and actuators are colocated, it is not required that the signal from the
sensors drive only the colocated actuator, and the sensor signal may
drive all of the actuators. Lin [85] surveyed on the output feedback
control method for possible application to vibration control of large

flexible space structures ; output feedback control via modal decoupling,

23

output feedback control by pole assignment, optimal output feedback
control and suboptimal output feedback control. The underlying theory,
assumptions made, techniques used, computations required, and strength
and weakness associated with above methods are investigated in details.
Lin [86] derived an output feedback control associated with displacement
and velocity sensors. However, the controller has restrictions such that
the number of sensors and actuators should be same and the sensors should
be located with force actuators.

Baruh [15] developed a natural control based on a set of admissible
functions instead of the actual eigenfuctions in order to achieve the
robustness of controller to both parameter uncertainties and
discretization errors. It is shown that if natural control is considered
in conjunction with modal filters and an adequate number of sensors,
implementation of the control action by admissible function leads to a
stable closed-loop system under uncertainties. Joshi [70] discussed the
robustness properties of two types of control laws for large space
flexible structures that use colocated sensors and actuators. The two
types of controllers are attitude controller which uses negative-definite
feedback of measured attitude and rate, and damping enhancement
controller which uses only velocity feedback. McLaren [92] used a
positive real approach for robust controller design which leads to
guaranteed closed-loop system stability in the presence of erroneous
modal information and unmodelled modes. It is shown that the closed-loop
stability can be maintained in the event of sensor and/or actuator
failure. Preumont [116] developed a robust nonlinear saturation
controller based on the second method of Lyapunov to eliminate the

spillover instability associated with the uncertainties such as neglected

24
dynamics in the modeling process. Simulation work shows that the
instability is alleviated by a local modification of the control in the
vicinity of the equilibrium, which reinstates the asymptotic stability
properties of the open-loop system.

Bar-Kane [12] proposed a direct multivariable modal reference
adaptive control scheme for large flexible structural systems such as
beams, plates and frames. The adaptive control law is formulated based on
the positivity condition which depends on the location of actuators and
sensors, and inherent damping of the structure. It is shown that when
colocated actuators and combined position and rate sensors are used, the
error approaches to zero, provided the ratio of the position-to-rate
output is limited by a function dependent on the damping ratio and the
lowest structural frequency. Mufti [104] formulated a model reference
adaptive technique for active control of large flexible structural
systems based on the Command Generator Tracker (CGT) procedure and
Lyapunov stability theory. It is improved from Bar-Kana [12] that the
ratio of position-to-rate output is limited by twice the product of the
damping ratio and the lowest natural frequency. The control law in the
form of integral, proportional and relay adaptations along with the
integral of the output error is also proposed. Balas [7,8] used a linear
optimal state feedback controller with state estimator for a finite
number of modes of the flexible system, and applied it to a simply-
supported Euler-Bernoulli beam with a single actuator and sensor. It is
shown that the control and observation spillover due to the residual
modes causes the instability of the closed-loop system. Some remedies for
spillover, including a straightforward phase-locked loop prefilter, are

suggested to remove the instability mechanism. Meirovitch [96,97]

25

introduced an independent modal space control method, which features
invariant natural coordinates in the presence of feedback control, to
provide a unique and globally optimal closed-form solution to the linear
optimal control problem for the distributed parameter system. The
proposed distributed control can be approximated by finite-dimensional
control through a process that does not require truncation of the plant.

Meirovitch [95] implemented experimentally the independent modal
space controller developed in [96,97] in conjunction with a nonlinear,
on-off controller for the modal forces on a free-free uniform aluminum
beam which has a length of 3.657 m, a width of 15.24 cm and a thickness
of 4.76 mm. Four electromagnetic force actuators and nine displacement
sensors are employed. It has been demonstrated that the proposed
independent controller are very effective in suppressing the motion of
the beam, even though there is about a 50 percent difference between the
actual and computational natural frequencies. Schaechter [124] tested
experimentally the static shape control, active dynamic control, adaptive
control and optimal control laws by implementing these control laws on a
pinned-free stainless steel flexible beam which has a length of 3.8 m, a
width of 15.24 cm and a thickness of 0.794 mm. Excellent control
performances are obtained with these control laws through experimental
investigation and it is pointed out that the unmodelled torsional modes,
sensor and actuator nonlinearilities, computational delays are only a few
of difficulties to be overcome in future research. Schafer [125]
implemented experimentally a direct velocity feedback controller on a
clamped-free flexible beam in order to study the active vibration
control. Sensing is provided by a purely optimal displacement sensor,

while an electrodynamic force system provides the actuating forces. The

26
experimental beam was made of nonmagnetic stainless steel with a length
of 2.9 m, a width of 10 cm and a thickness of 1 mm. It is shown that the
closed-loop performance is satisfactory for both the colocated and
noncolocated actuator/sensor positions.

Lindberg [87] presented a new formulation of independent modal space
control which can provide an analytical solution for the optimal control
law for very high dimensional systems. An optimal actuator locations are
developed using open-loop and closed-loop approach in order to minimize
the actual control effort. Omatu [111] established a criterion for the
optimal sensor location to minimize the trace of the optimal filtering
error covariance function by employing the existence and uniqueness
theorem concerning a solution of the optimal filtering error covariance
function for the pointwise observation case. West-Vukovich [159] proposed
a decentralized robust servomechanism problem with constant disturbance
and set points for the large flexible space structures. Simulation work
shows that the controller eliminates the spillover problem which cause

instability of the closed-loop system.

1.2.4 Related Works on Smart Materials and Structures

Recently great attention has been focussed on the development of
technologies associated with 'smart' or 'intelligent' materials and
structures. This terminology has been used to describe or classify
materials and structures which have their own sensing, actuating, tuning,
controlling and computational capabilities. Typically, smart materials
and structures are employed to control the static and elastodynamic

responses of distributed parameter systems. This may be accomplished by

27
controlling the mass distribution, the stiffness and the energy
dissipation characteristics of the structures. Recently significant
progress has been made in the development smart materials and structures
which feature piezoelectric materials, electro-rheological fluids, shape
memory alloys and optical fibers. The piezoelectric materials are aptly
named as smart due to the capabilities of actuating and sensing through
the polarization a polymer. Electra-rheological (ER) fluids are adaptive
in that definition since it can be either fluid or solid state, as need
be for certain objectives. The shape memory alloys. are also aptly named
as smart for they remember own shape history and can recover the
memorized shape with appropriate heat supply. The optical fibers have own
merits for being named smart since they are one of the potential
candidates of sensor technologies for applications in smart materials and
structures. The distinct characteristics, possible applications and
existing problems of these smart materials and structures are well
summarized by Gandhi and Thompson [55].

Bailey [6], Burke [30], Miller [100] and Plump [115] used a
distributed parameter actuator incorporating piezoelectric polymer for
active vibration damper. The dynamics of the proposed actuator are
derived for the simplest possible damper configuration which features a
layer of piezoelectric polymer bonded to one side of the cantilevered
beam. The nonlinear control law was formulated based on the Lyapunov
second method. A Lyapunov functional of the system is the sum of the
squares of the curvature and velocity. Hence, integrating of the
functional along the length of the beam provides a measure of how far the
beam is from its equilibrium position or a measure of the energy stored

in the system. Minimizing the time derivative of this functional is then

28
equivalent to trying to bring the system to equilibrium as fast as
possible. The controller was successfully implemented for a cantilevered
'beam [6,115] and a simply-supported beam [30] yielding increment of loss
factor for small amplitude vibrations. Crawley'[£H5] presented an
analytical and experimental investigation of piezoelectric actuators as
elements of smart structures. A static and dynamic models for segmented
piezoelectric actuators, which are either bonded to an flexible structure
or embedded in composite laminates, are derived based on the force
equilibrium and the strain-displacement relationships. Some suggestions
for the selection of actuator location and actuator material are
'presentedu The actuators should be placed at the locations to excite the
primary vibrational modes effectively. The actuator materials should be
carefully chosen based on the property of piezoelectric-mechanical
coupling effectiveness, mechanical properties of the original structures
and the type of actuator, i.e., a surface-bonded actuator or an embedded
actuator. Three experimental cantilevered beams : an aluminum beam with
surface-bonded actuators, a glass-epoxy beam with embedded actuators and
a graphite-epoxy beam with embedded actuators, are used to investigate
the steady-state resonant vibration. It has been shown in the static test
that the ultimate strength of the glass-epoxy beam was reduced.by'
approximately 20 percent, although the modulus of the beam was not
affected significantly. Hanagud [63] discussed the effects of the
location or placement of the surface-bonded or embedded actuators and
sensors incorporating piezoelectric materials. It is shown through
simulation work that the actuator and sensor location is very important
in controlling flexible structures resulting in that the centralized

controls are effective on the basis of selective locations.

29

Baz [17] developed a dynamic model and control method for an
composite beam which consists of a surface-bonded actuator, bonding layer
and the original beam structure. The stiffness matrix is obtained by
integrating the mechanical properties of the three sub-structures :
piezoelectric actuator, bonding layer and original beam. The mass matrix
are determined using the lumped mass method in which the mass and
rotational inertia of each element is distributed among the nodes. A
modified independent modal space control is presented to select the
optimal locations of the piezoelectric actuators, control gains and input
voltage to the actuators. This method accounts for the spillover from the
controlled modes into the uncontrolled modes due to the use of fewer
actuators than the modelled modes. The effectiveness of this proposed
control method in conjunction with the effect of bonding layer material,
actuator location and number of finite elements are investigated in
details through simulation work. Lawson [77] investigated a frequency
characteristics of infinite piezoelectric plates vibrating between two
grounded infinite electrodes. It is shown that the frequencies depend on
the piezoelectric constants as well as the elastic constants of the
piezoelectric crystal. It is also found that the frequencies of free
vibration depend on the gap of the electrodes, and this dependence is
not linear. Mindlin [101] derived the approximate equations so as to
accommodate the frequencies of the shear modes of thick rectangular
plates incorporating piezoelectric crystal. The two-dimensional equations
of coupled flexural and extensional motion of crystal plates are deduced
from the three-dimensional equations of linear elasticity based on the

series expansion method and variational method. The uniqueness of

30
solution and orthogonality condition are also discussed in the deriving
process of the approximate equations.

Winslow [160] described for the first time some phenomena of the
induced fibration of small particles in liquid suspension. Various
properties of these fluids are discussed regarding to the micromechanism
of the induced fibration and its application in slip clutch and hydraulic
devices. It is mentioned that the observed values of induced shear
resistance are very similar to those predictable from elastic and
magnetic cases. Block [20] reviewed literatures on electro-rheological
(ER) fluids by describing the nature of these fluids and the influence of
important parameters such as shear rate, field intensity, stability of
the fluids, temperature and fluid composition. This review also includes
the dielectric characteristics of dispersions in flow, various modeling
methodologies and practical applications that have currently and may have
in the future. Stangroom [140] introduced some chemical and mechanical
properties of ER fluids as well as their possible applications to the
practical devices such as clutch and servo-valves. It is also mentioned
about the minimum requirement for ER activity such that the base liquid
must be hydrophobic and the solid must be hydrophilic and porous so that
it can contain an appropriate amount of absorbed water which greatly
influence the properties of the final ER fluids. It is suggested that
insulating materials for use ER fluids require good surface and bulk
dielectric strength and resistivity as well as chemical and mechanical
stability. Adriani [2] developed a microscopic model for an ER fluid by
considering a concentrated suspension of hard spheres with aligned field-
induced electric dipole moments. The presence of dipole moments causes

clustering of the particle into an anisotropic suspension characterized

31
by a particle probability distribution function. It is shown that dynamic
viscosity is insensitive to dipole strength, while elastic shear modulus
increases significantly with dipole strength indicating the relative
importance of the particle distribution to elastic properties.

Bullough [29] proposed a vibration isolation system for the
investigation of the usefulness and effectiveness of a controllable and
nonlinear damper incorporating ER fluids. This is accomplished by
considering the performance of a simple model suspension system which is
an idealized form of a prototype damper controlled by imposing an
electric field on the associated electroviscous valves. The proposed
suspension model consists of two springs, one linear viscous damper and
one nonlinear Coulomb damper. Simulation result shows that it is possible
to control this idealized suspension system so as to achieve much
improved performance characteristics. Stevens [137,138] performed
experimental investigation of ER clutch to examine the utilization of ER
fluids for torque transmission, and also to provide guidelines for the
development of practical ER devices. The proposed ER clutch which
involves a horizontal shaft with a totally enclosed reservoir for the ER
fluids, and the adjustable space of the clutch plates was proposed, built
and tested. Experimental results show that the amount of transmitted
torque can be made a linear function of the available stressed area of
the ER fluids and can be controlled by the electric field intensity. It
has been also found that as the ER fluid temperature increases, the
viscosity of the base fluid (thus the viscous shear force) decreases,
leading to an overall decrease in transmitted torque. Stevens [139] also
conducted experimental testing to obtain mechanical properties of ER

fluids such as shear stresses. The influence of the critical dimensions

32

such as spacing, electrode area and the viscosity of the base fluid on
the shear stress are investigated with respect to externally applied
electric field. Stanway [136] proposed an approach for identifying the
damping law of an ER fluid in vibration. The damping law is achieved by
employing a nonlinear sequential filter to estimate the system parameters
associated with an nth-power velocity model of the damping mechanism. It
is shown in experimental study that the proposed method performed well at
the lower electric field strength (less than 1 kV/mm) because the ER
damping force did not depart significantly from a viscous
characteristics, however, some trial and error was required to obtain an
estimate of the actual state at 1 kV/mm and above.

Choi [39] described possible applications the ER fluids to the
machine and robotic systems. A robot with structural member incorporating
ER fluids, ER hydraulics and conceptual design of an ER-based hydraulic
servomotor are discussed in terms of dynamic modeling. Gandhi [58]
developed a phenomenological dynamic modeling for an ER-based
articulating robotic manipulator. The dynamic modeling is accomplished
based the variational theorem characterizing the linear viscoelastic
homogeneous materials, and the finite element formulation. It is shown
through simulation work that the smaller vibration amplitude, faster
settling time is achieved by employing the robotic system featuring smart
structures incorporating ER fluids. Choi [41] conducted experimental
investigation of beam-like structures fabricated from ER fluid to examine
the change of natural frequencies and damping properties of the structure
with respect to electric potential. The experimental studies were
undertaken under diverse conditions such as different ER fluids,

different interface condition at the boundary between solid and fluid

33
domains, different ER volume fraction, different electrode area and
different embedded ER locations. Gandhi [56] proposed a new generation of
revolutionary, intelligent, ultra-advanced composite materials featuring
ER fluids for active vibrational control of structures and mechanical
systems. The graphite-epoxy composite beam filled with ER fluids are
fabricated and tested in a variety of different operating condition in
order to provide a basis for evaluating the controllability of these
structures in real time. It is shown that the first mode natural
frequency and damping ratio of the cantilevered beam was increased by 25
percent and 45 percent respectively with 2 kV/mm of electric field. Choi
[40] proposed an active vibration tuning methodology for smart flexible
structures incorporating ER fluids through proof-of-concept
investigation. The tuning methodology was developed by employing a pseudo
proportional-p1us-derivative feedback control which was deduced from the
phenomenological equations of motion incorporating modal analysis. A free
vibration of cantilevered beam filled with ER fluids was tested and
showed good correlation between the predicted and measured results.
Gandhi [57] performed experimental test to investigate the forced
vibrational response of slider-crank mechanism featuring a smart
connecting-rod incorporating ER fluids. This is the first time to
experimentally demonstrate the ability of smart materials to tailor their
vibrational behavior by approximately selecting the voltage imposed on
the embedded ER fluids in the member when it is subjected to forced
excitations. It has been shown that the magnitude of the midspan
deflection was reduced by 15-73 percent with 2 kV/mm of electric field,

depending on the crank angle, at the mechanism operating speed of 95 rpm.

34

Rogers [120] proposed a shape memory alloy reinforced composite for
active modal modification and active strain energy tuning. The active
modal modification uses the shape memory alloy's capability of changing
its stiffness properties during a temperature activated, reversible,
phase transformation modifying the modal response of the composite
structure. The active strain energy tuning concepts induces from the
shape memory alloy’s ability to impart large distributed loads through
the material to alter the stored strain energy within the composite
structure. This active strain energy tuning allows for greater authority
and versatility of control to be exercised over active modal modification
to vary the natural frequencies and to change the mode shapes of the
structure. It is shown the experimental test that the natural frequency
of the composite beam changes from 2.2 Hz without actuators to 3.8 Hz
with six actuators featuring nitinol-based shape memory alloy. Yaeger
[161] developed a one-point three-quarter-inch-stroke linear actuator
using a spring type of nitinol-based shape memory wire. The design
includes prevention from ancillary jams.

Claus [42] described potential applications of optical fiber sensors
for measurement of phase,polarization, mode, strain, chemical
concentration, resin cure, acoustic vibration, crack propagation and
impact. The optical fiber sensors deserve their merits as a candidate of
sensor technologies due to the small size, low weight, large bandwidth,
lower power, multiple mutiplexing options and dielectric property. The
fiber sensors are either bonded on the external surfaces of structures or
embedded within materials. Rogowski [121] used a laser-radiated optical
fiber to measure static and dynamic strain in composite materials. This

was accomplished by integrating technologies of modal domain

35

interference, optical time domain reflectometry, an optical phased-locked
loop and the laser radiation. Waite [154] used a polarization maintaining
optical fiber embedded in a woven glass-epoxy composite three point
bending specimen for the detection of damage and strain of the structure.
The experimental optical system consists of a helium-neon laser which
injected light into the optical fiber, two photodiode detectors for
signal recording and a microcomputer controlled stepper motor. Various
fiber surface treatments such as stripped and silane treated, stripped
and release agent treated, and buffered are tested and compared.

The research fields on dynamic modeling and control of single-link
or multi-link flexible manipulators, dynamic modeling and control of
large flexible structures, and smart materials and structures have been
reviewed. Numerous research studies have been done on.tflua dynamic
modeling and control of flexible manipulators fabricated from
conventional isotropic materials such as aluminum and steel. However,
none of the current literature did attempt to model and control of
flexible manipulators fabricated from advanced composite materials or
smart materials incorporating electro-rheological fluids. Moreover, it is
clearly evident from this literature review that only a few of
researchers conducted experimental implementation of control laws for
controlling flexible manipulators. Therefore, this research program
combining analytical, computational and experimental work will provide a
complete quantitative and qualitative in a sense comparison of control
performances of the flexible manipulators fabricated from aluminum,
composite materials and smart materials. In addition, this study will
furnish viable guidelines and directions for future work in this or

relevant research fields.

36

l . 3 Thesis Organization

In Chapter II, the fourth-order partial differential equation and
associated boundary conditions for a single-link flexible manipulator
fabricated from advanced composite laminates are derived. The governing
equation of motion is developed by employing the Hamilton's principle.
The equation is then discretized using a finite element method
incorporating the Euler-Bernoulli beam theory. The inherent damping of
the structure is modeled in the finite element formulation based on the
hypothesis that the damping matrix is proportional to the stiffness
matrix of the structure. The coupled discretized equation of motion is
decoupled through modal analysis and expressed in the state space
representation which is convenient form for controller design.

A detailed description of the experimental apparatus of the proposed
robotic system featuring an output feedback controller associated with
two colocated output sensors is presented in Chapter III. The apparatus
is divided into five functional groups; flexible arms, a DC servo-motor,
a servo-positioning assembly, sensors and a microprocessor. After a
description of the apparatus, an experimental identification of the
system model parameters such as natural frequencies is undertaken, and
the measured parameters are compared with the predicted ones.

Chapter IV presents an output feedback control associated with two
colocated output sensors. The proportional-plus-derivative feedback
controller is designed using the root-locus technique and the Lyapunov
method. The controller is then experimentally implemented by
incorporating the apparatus described in Chapter III. Experimental
results such as step responses to a commanded step angular position, and

the characteristics of the system instability are presented and compared

37
between two robotic systems featuring the flexible arm fabricated from
aluminum and composite laminates respectively. The corresponding
simulated results are also presented.

In Chapter V, a hybrid controller which consists of the output
feedback controller implemented in Chapter IV and a pseudo-state feedback
controller featuring the distributed electro-rheological (ER) fluid
actuator is presented. A brief background on the electro-rheological
fluids is presented prior to developing a phenomenological equation of
motion of the robotic system featuring a flexible arm fabricated from ER
fluids. The phenomenological equation of motion is developed based on the
experimental observation of the modal characteristics of the smart beam
structure. Then, the hybrid controller is formulated by incorporating an
empirical model of the distributed ER actuator. The experimental
implementation of the proposed hybrid controller is undertaken, and the
results are presented by considering step responses of the system
subjected to no disturbances and disturbances.

Chapter VI presents a nonlinear robust state feedback controller for
a flexible manipulator fabricated from composite laminates by introducing
uncertain parameters which may cause system parameter variations such as
natural frequencies. The varying payload encountered in the operation and
the mismatched lamia angle encountered in the manufacturing process are
considered as uncertain parameters. The controller design is accomplished
by utilizing the properties of uniform and uniform ultimate boundedness
of the solution. A so-called matching condition which says that the
uncertain quantities can not enter arbitrary into the state equations is

excluded in the process of controller design. An illustrative example is

38

taken and simulated to demonstrate the robustness of the proposed

nonlinear controller in the presence of uncertain parameters.

Finally, conclusions of this research program and recommendations

for future work are given in Chapter VII.

1 . 4 Principal Contributions

1.

The output feedback controller featuring two colocated angular
position and velocity sensors to one actuator was designed and
experimentally implemented in order to investigate and compare the
control performances of single-link flexible manipulators fabricated
from aluminum (aluminum arm in short) and composite laminates
(composite arm in short) respectively. So far the author is aware,
this is the first experimental investigation on complete quantitative
comparison of control performances between two flexible manipulators.

It has been demonstrated through the analytical and experimental
investigation that the composite arm had superior control
performances such as faster settling-time, smaller input torque,
smaller overshoot and a superior increment of stability relative to
the aluminum arm. It was obtained from the measured step responses
that with certain position and velocity feedback gains the delay time
was 0.26 second for the aluminum arm, while that was 0.21 second for
the composite arm, and the maximum overshoot was 1.02 degree for the
aluminum arm, while that was 0.3 degree for the composite arm. It was
also observed that the reduction of the energy consumption by 40-50
percent for the commanded motion, and the decrease of the tip
deflection up to 50 percent were achieved by employing the composite

arm.

39

It has been also shown that the implemented output feedback
controller was very sensitive to the external input (considered as
disturbance), which has a forcing frequency close to the
cantilevered-mode natural frequency of the arm, regardless of the
aluminum and composite arm. This attempt has been made here to
introduce a single-link flexible manipulator fabricated from smart
materials incorporating electro-rheological fluids (smart ER arm in
short) for the purpose of obtaining the robustness of the output
feedback controller to the imposed disturbance with the help of
distributed parameter actuator, which is characterized by embedded
electro-rheological fluids in the structure.

The phenomenological equation of motion of the robotic system
featuring the smart ER arm has been developed based on experimental
observations of the modal characteristics of the smart'bemn
structure. The hybrid controller which consists of the output
feedback controller and the pseudo-state feedback controller was
formulated by employing the empirical dynamic model of the electro-
rheological fluid actuator. It has been shown through the
experimental implementation that the robustness of the employed
conventional output feedback controller was significantly enhanced by
employing the proposed pseudo-state feedback controller in the
presence of the disturbance which has a forcing frequency close to
the cantilevered-mode natural frequency of the robot arm.‘The author
strongly believes that this work is unique and initial

accomplishment.

40

The nonlinear robust state feedback controller for the composite
arm was designed for the first time to overcome uncertain parameters
associated with the varying payload encountered in the operation and
the mismatched lamina angle encountered in the manufacturing process.
The controller design was accomplished by utilizing the properties of
uniform and uniform ultimate boundedness of the solution. A matching
condition which says that the uncertain quantities can.not enter
arbitrary into the state equations was excluded in nature in the
process of controller design from the point of dynamic
characteristics of the proposed flexible manipulator. It was
demonstrated through the simulation that the proposed nonlinear
controller could tolerate larger system parameter variations than the
conventional linear controller, showing much faster settling-time and

smaller overshoot with the nonlinear controller.

CHAPTERII

MODELING OF A SINGLE-LINK FLEXIBLE MANIPUIATOR
FABRICATED FROM ADVANCED COMPOSITE LAMINATES

2 . 1 Introduction

Two most important quantities which characterize the control
performance of flexible manipulators are the natural frequencies and the
inherent damping capacities of these systems. Since these continuum
structures are characterized by an infinite number of vibrational modes,
in the modeling process it is essential to choose a suitable number of
modes based on dynamic conditions encountered in practice and practical
limitations other than the order of natural frequencies alone [66] . This
modal truncation procedure invariably results in performance trade-offs
such as control and observation spillover effects when designing a
controller for distributed parameter systems [8] . However, in most cases
this modal truncation is justified by the bOundness of energy, and the
fact that most of actuators and sensors cannot operate in the high
frequency range [31,74]. Another issue which should be carefully
considered in the modeling process of flexible robotic manipulators is
the effects of rotary inertia and shear deformation on the vibrational
behaviour of the systems. In other words, the problem follows that
whether the system should be modeled based on Euler-Bernoulli or
Timoshenko beam theory. Up to now, many of related works on the dynamic
modeling of the single-link flexible manipulator have verified that the

41

42
model based on Euler-Bernoulli beam theory agreed well to experimental
results [65,127].

In this chapter, the fourth-order partial differential equation and
associated boundary conditions for a single-link flexible manipulator
fabricated from composite laminates (composite arm in short) are derived.
The governing equation of motion is developed by employing the Hamilton's
principle. The equation is then discretized using a finite element method
incorporating the Euler-Bernoulli beam theory. The inherent damping of
the structure is modeled in the finite element formulation based on the
hypothesis that the damping matrix is proportional to stiffness matrix of
the structure. The coupled discretized equations of motion is decoupled
through the modal analysis and expressed in the state space.

representation which is convenient form for controller design.

2.2 Derivation of Equations of'Mbtion

Consider the horizontal motion of a single-link flexible manipulator
as shown in Figure 2.1. The arm is modeled as a continuous and uniform
beam of length L whose moment of inertia about the root is IB, with an

additional lumped inertia I at the actuator assembly (hub). The payload

H

of mt is attached to the other end of the arm. The ox’ is the fixed

reference line and ox is the tangential line to the beam's neutral axis
at the hub. The total displacement of any point along the beam's neutral

axis at a distance x from the root of the beam is the sum of the small

elastic deflection. W(x,t) and.hub angle 9(t) which can be arbitrary
large. Before determining the kinetic and potential energy of the

proposed system, following assumption is made.

43
Assumption 2.1 : (1) Elastic deflection is small. (2) Axial
deflection is neglected. (3) Rotary inertia and shear deformation are
ignored (Euler-Bernoulli beam model). (4) Internal damping forces are

neglected (they will be included later on in the finite element

formulation).

Under this assumption, if the total displacement w(x,t) (arc length

measured from the fixed reference line ox') is defined as
w(x,t) - §(x,:) + (x+r)9(t) (2.1)

then, the total kinetic energy of T can be given by

L ~ 2 .2 2
T - l m (Qflifiegl) dx + l I e (:) + l m (QELLegl) (2.2)
o H 2 : a:

where m is the mass per unit length of the beam and (.) denotes time

derivative.

 

fit

 

Figure 2.1 Schematic Diagram of a Single-Link Flexible Manipulator

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

hm
hm-

 

middle surface

 

bl

 

 

 

 

 

 

Section A-A

Figure 2.2 Composite Laminate Configuration

In order to determine the potential energy, consider the composite

laminate as shown in Figure 2.2. It is seen from this figure that h

is the vertical distance from the laminate middle surface (z-o) to the

th

upper surface of the k lamina and h is the total thickness of the

laminate. The strain-displacement equations of the laminate are given

from the classical lamination theory [69], hence:

45
e - e + zn

e - e + zx (2.3)
Y y Y

e - e + 25
xy XY XY

0 O O O C
where Ei’ £1 and ”i are strains at arbitrary location (x,y,z), strains
on the middle surface (x,y,o) and curvature changes, respectively.

. th . . .
The stress-strain relation of the k lamina is also given by

 

«k
”1 Q11 Q12 0 ‘1

-Q Q 0 911M} (24)
”2 12 22 ‘2 Q ‘ 1-2 '
‘12 0 0 Q66‘ ‘12

where the Qij’ so-called reduced stiffnesses, are

Q _ 1 Q _ ”12E2
ll l-vlzu21 l2 l-u12V21
E2 (2.5)
Q - _ , Q = G
22 1 u12u21 66 12

In equation (2.4) and (2.5), o and a

1 2 represent stress in principal

material (fiber) and its perpendicular direction respectively, atui 012

denotes shear stress in the 1-2 plane. 6

e and 6 define strain

1’ 2 12

correspondin to a , a and a res ectivel . E and E denote Youn 's
3 1 2 12 P y 1 2 g

modulus in principal material and its perpendicular direction

respectively, and C12 defines shear modulus in 1-2 plane. u represents

ij

Poisson's ratio and is defined by

46

Vij - - 6 (2.6)

And the Young's modulus E E and the shear modulus G are given by

1' 2 12
E1 - Efo + Eme
52 - EfEm/(Efvm + Eme) (2.7)

612' Gme/(vam + vaf)

where Ef and Em are Young's modulus of the fiber and matrix, Vf and Vm

are volume fraction of the fiber and matrix, and Gf and Cm are shear

modulus of the fiber and matrix respectively.

. . th . .
The stress-strain relation for the k lamina of arbitrary
orientation can be obtained by employing the transformation matrix

relating 1-2 coordinate system and x-y coordinate system“ tflnich has the

 

form
2 2 ,
cos 0 sin 0 2sin0cos€
[T1] - sin20 c0526 -2$in0c030 (2.8)
_-sin0cos€ sinocosfl coszfl-sinzod

 

where 9 represents fiber orientation (lamina angle), which is defined as
positive in the counter-clockwise direction. This matrix provides the

stress-strain relation in the x-y coordinate, that is

r

 

0X 6X
-1 k -T A - k
«ay - [T1] [Q] [T11 ey - [Q] I‘Ix-y (2.9)
a 6
L 33' X)’

47

where [O]krepresents the transformed reduced stiffness matrix of the kth

lamina. Then the constitutive equation of the laminate is given by

 

 

 

 

 

 

PF 1 r- . "I r EOI
x ' X
F A z B 60
y ‘ y
F 3 e°
xy ° KY
4000}:- eeeeeeeeeeeeeeeeeeeeee 40000) (2.10)
M o n
x - x
M B i D n
Y ° Y
M , 1 7c
LXYJ t. . .. L XYJ

where the matrices A, B and D denote extensional, extensional-bending
coupling and bending stiffnesses of the laminate, respectively, and their

elements are defined by

- - n - k

l 2 h2 l h3 h3 2 11
(hk’hk-1' 2(hk' k-l)’ 3( k' k-1)) I ' I
where n is the total number of lamina. And in equation (2.10), the F1

and M1 describe the resultant forces and moments respectively.

The potential energy for the laminate can be obtained from the

strain energy and takes the form of

n
V - i 2 [IA Jhk {oxex+o e + ax 6x } dsz1 (2.12)
k_1 1 hk_1 y y y y

where A1 refers to the area of the laminate. The substitution of the

strain-displacement relationship (2.3), stress-strain relations [(2.9),

and constitutive equation (2.10) for the symmetric laminate*with respect

48
to tfiuatniddle surface (Eij-O from the equation (2.11)) into the equation

(2.12) yields the following potential energy.

2

A 2 6x 2 6x 12 6x 6y
1

-£ 2‘22 522222 522.22”
+ D16 [ 6x6 + 5— 6x ] + -E- [ ] + [ 8y ] (2.13)

Y X By —2_
_ _ 2, _ - 2-
+11” [g§%+g—§y]+fi" [g§%$+%;§y]
- 2 , _ _ 2
+2..<2[§3] 22(2)

where u and v are the middle surface displacement with respect to x and

y direction respectively, and d, B are defined by

5,“:2 B__2+2

ax , 6y (2.14)

where w is the displacement along the z direction.
Thus, by considering the proposed orthotropic composite laminate
that has no twisting coupling effect, and one-dimensional beam model

shown in Figure 2.1, the equation (2.13) simply reduces to

L 2... 2
- % I b 6,, [ 2.212;£1 ] dx (2.15)
0 6x

It is noted that the displacement w in equation (2.14) is equivalent to

the deflection denoted by w(x,t) in Figure 2.1.

The work done by the nonconservative external torque u is given by
w - 6(t)e(t) (2.16)

The dissipated energy due to the inherent damping property of the
structure is not considered at this point, however the damping matrix
which is proportional to the stiffness matrix will be augmented in the
finite element formulation.

Now, the substitution of equations (2.2),(2.15) and (2.16) into the
Hamilton's principle

t2
6 I ( T - V + W ) dt - 0 (2.17)
t1

and integrating by parts yield following equations of motion and

associated boundary conditions.

4 2
bfill W'F aw _ 0 (2.18)
8x at

m(x

L 2 ..
I + r) Q-Ei§+£l dx + I e(c)
0 at H

2
+ m (L + r) Q—31%+91 - E(t)
t at

§(o,t) - o - (2.19)

50

 

a§(o.t) _ 0
6x

2
bf)11 W Z,‘ t - 0

6x x-L

3 2

Q w(x,t) w X t
bD11 3 ' mt 2
8x x=L at x-L

 

In equation (2.17), t1 and t2 are two arbitrary times (t2 > t1), amnd the

virtual displacements 6w(x,t) and 66(t) are independent and taken equal

to zero at x-O and x=L.

2.3 Finite Element Formulation

There are several ways to solve the governing equation (2.18) in
order to obtain system model parameters such as natural frequencies and
mode shapes which will be used for designing of feedback controllers.
Herein, a finite element formulation incorporating Euler-Bernoulli beam
theory is developed to obtain system model parameters. 12ma<displacement
(xf the governing partial differential equation (2.18) can be discretized

as follows.
W(x,t) - [N] {U} (2.20)

where [N] is the row vector containing shape function which is dependent
upon the position only and {U} is the column vector containing the nodal

variables U1, U2,U3 and U, which is function of time only. In this study,

two-node element was employed with two degrees of freedom at each node as

shown in Figure 2.3. And a cubic polynomial is chosen to develop shape

51

 

 

 

 

 

Figure 2.3 Two-Node Beam Element

function which satisfies the conditions for admissibility. The polynomial
is given by
2 3
w(x,t) - a0 + a,x + a2x + aax (2.21)
After some manipulations, following coefficients of the cubic

polynomial are determined.

 

 

 

 

 

 

a0 2 o o 0 U,
2
a, _ 1 0 2 0 0 U2
: 2 £2 4 + (2.22)
82 '3 '22 3 ‘£ U3
a 2 1 L2 1 U
1 3. .2 2 . . 4.
A
- [A] {U}

‘wherwa 2 is the length of the element. Then the shape function [N] is

given by

[N] "' [K] [A] " [N1 N2 N3 N4] (2-23)

where

[22] -

2 2 2 3
N, - *2 [2 - 3x + 2 x

2

1 2 2 3
N2 - ‘2 [2 x - 22x + x

2

2 3
N3 - l2 [3x - 2 x ]
2

2

1 2 3
N4 - ‘2 [-2x + x ]

2

The effective mass density of the kth

obtained by
k
p - pm Vm + pf Vf

where pm is the density of matrix,

52

1 (2.24)

lamina shown in Figure 2.2 is

mass per unit length of the beam is expressed by

~ n
m - b 2 p
k—l

k
(hk ‘ hk-l)

Now, the local mass and stiffness matrix is given by

A L
[M] - a J [N]T [N] dx

0

A

[K] - b

where

[B]

[B] - Z,[-3 + g x
2

L
IO [BIT 13“ [B] dx

is the strain transformation matrix which is

-22 + 3x

(2.25)
pf is the density of fiber. Then, the
(2.26)
(2.27)
(2.28)
given by
2
3~ 2 x -2 + 3x] (2.29)

53
It is generally known that an appropriate modeling of the damping
matrix is the most difficult in the modeling process of the flexible
structures. The model for the damping matrix, what is considered here, is
based on the hypothesis that damping is proportional to the stiffness

matrix, i.e.

[81 - 2 [1?] (2.30)

where 19 is constant damping coefficient and it is very similar to
effective Rayleigh damping coefficient. It is noted that the hypothesis
made here is only valid for small damping or the beam-like structures
which possess well separated eigenvalues (natural frequencies) [94]. The
proportionality of the damping matrix to the stiffness matrix allows the
governing equations to be decoupled as shown in the next section.

Now by incorporating the boundary conditions described by equation
(2.19) and using the transformation matrices relating the local and
global reference frames yield the global finite element equation,

governing the motion of the proposed robotic system, which has the form

of
[Mltfn + {01113} + mm =- {Q} (2.31)

where the global mass, damping and stiffness matrices are denoted by [M],

[C] and [K], respectively, and {Q} represents the generalized forces.

2.4 State Space Representation of the Model

Since the governing equation of the motion described by equation

(2.31) is coupled, it is convenient to decouple and represent in the

54
state space for controller design. By introducing modal coordinate {n}

and modal matrix [¢], i.e.
{U}- [¢] {0} (2.32)

the governing equation of motion can be transformed imnua a set of
decoupled modal equations and then written in state space representation

as follows.

 

 

 

 

 

 

 

 

 

 

E - X; + 33
2 -2 (2.33)
y - Cx
where
' "o l ’ o 1 o o . . . o o 1
50 o 0 o o . . - o o
n, o o o 1 . - - o o
o 2
n1 0 O -wl-2§lwl- - . O O
;-< O L, K. O O I O O O O O O
"n o o o o - - - o 1
' o o o o . - - -w -2 w
, "n , _ C .
d¢<o> d¢<o> T
~ —1— 1 . . . n
B I o 1 0 dx 0 dx (2.34)
T
1 o d¢1(0) 0 . . . d¢n(0) 0
dx dx
6 _ o 1 o d¢1(o). . . o d¢n(o)
dx dx
L L o ¢1(L) o - - - ¢n(L) o .

 

 

55

In equation (2.34), mi and Ci are the natural frequency and damping ratio

of the ith mode respectively, I is the total moment of inertia (hub,

T

beam and payload), d¢i(o)/dx is the modal slope coefficient and ¢i(L) is

the tip sensor gain. It is noted that the output matrix ‘6' includes the
gains associated with two colocated sensors - hub angle (first row) and
hub angular velocity (second row), and one noncolocated sensor - tip
position (third row).

Since the dynamical model given by the equation (2.33) contains many
modes in general, all of modes cannot be controlled in practice. Hence,
the equation (2.33) should be decomposed into two subdynamic equations by
considering the rigid-body mode and the first few critical flexible modes
as primary modes, and the remaining modes as residual modes. Two

subdynamic equations are

Si-KSE+EG
P P P P
~ ~ ~ (2.35)
- C x
yP P P
and
31-3.; +§E
r r r r
~ ~ ~ (2.36)
yr - Crxr

where subscript (p) denotes the primary modes and (r) represents the
residual modes. In general, there is no certain rule differentiating the
primary and residual modes. The differentiation heavily depends upon the
geometrical and material properties of the flexible arm and the dynamic
characteristics of the employed experimental apparatus such as actuators

and sensors [74].

56

2.5 Summary of the Chapter

The governing partial differential equation and associated boundary
conditions for the single-link flexible manipulator fabricated fnmn
composite laminates were derived from the Hamilton's principle, and the
equation was discretized using the finite element method incorporating
the Euler-Bernoulli beam theory. Then, the decoupled linear dynamic model
was (flytained.through the modal analysis and presented in the state space
representation, which is convenient form for controller design. It is
‘noted that the relative simplicity of the dynamic model for the proposed
composite arm disappears when considering multi-link flexible
manipulators. However, the approach employed in this study could be
extended to the case of multi-link flexible manipulators or other
complicate flexible structures such as helicopter blades and appendages

of the satellites fabricated from advanced composite materials.

(HHHHER.III

EXPEKHflflHml.APHMMUflEIANDIUMQHHITCAIION
OF SiEflfllHDDELIHMMflflflEBS

3.1 Introduction

A detailed description of the experimental hardware employed for the
proposed single-link flexible robotic system featuring the aluminum or
the composite arm will be presented in this chapter. Also, an
experimental identification of the system model parameters such as
natural frequencies and damping ratios will be undertaken. The measured
parameters will be compared with the predicted ones. This chapter
establishes a point of reference and sets a physical scale of the
experiment for the subsequent chapter, which presents the design and
implementation of the output feedback controller associated with two

colocated sensors for the proposed robotic system.

3.2 Experimental Apparatus

The experimental hardware of the proposed single-link flexible
robotic system can be separated into five functional groups; flexible
arms (aluminum or composite arm), a DC motor and a servo-drive, a servo-
positioning assembly, sensors, and a microprocessor for controller
implementation. Figure 3.1 presents a photograph showing these functional

groups, and Figure 3.2 shows a schematic diagram of overall robotic

57

58

 

Figure 3.1

W

 

 

Photograph of Overall Robotic System

 

 

Flexible Arm

 

 

 

Payload

    

Strain Gage

 

+ Servo-Drive

 

 

 

f

    

Tachometer

 

Status

 

Servo-Positioning
Assembly

 

 

 

l

 

Figure 3.2

Decoder
x 4

 

Commanded

Microprocessor

 

 

 

Position

Incremental
Encoder

     

 

Schematic Diagram of Overall Robotic System

59
system consisting of these functional groups. In this section, The

detailed description of each functional group will be presented.

3 . 2 . 1 Flexible Arm

3.2.1.1 Aluminum Am

The aluminum arm is a 0.812 m long, flexible beam which can be
easily bended in the horizontal plane, but it is designed to be hard to
bend in the vertical and torsional direction to prevent or reduce the
torsional effects which were neglected in the modeling process. The arm

is clamped on a rigid-body hub which has a radius of 4.3 cm and a moment

2
of inertia of 0.00496 kg-m, and is mounted directly on the vertical

 

Figure 3.3 Photograph of the Experimental Aluminum Arm

6O
shaft of the DC servo-motor as shown in Figure 3.3. Therefore the
rotatfixnl of the arm in the horizontal plane is activated by the torque
applied to the motor without speed reduction factor. 'It is seen from
Figure 3.3 that the two extreme travel limit switches and one home
position limit switch are mounted on the plate of the fixture supporting;
the motor. The purpose of the installation of the limit switches is
prevent overtravel in the emergency case such as instability of the
system. When the arm touches one of the limit switches, the main power
for the whole system will be turned off automatically. The home position
switch is installed.in.order'to obtain high repeatability of the
experimental results. With this home position detective switch tfiua'robot
arm can always start to move from the same position like zero-position in
typically industrial robotic manipulators. The payload made of aluminum

is attached to the free end of the arm. The 30 mm (length) x 40 mm

Table 3.1 Material and Geometrical Properties of the
Aluminum Arm

 

 

 

 

 

 

 

 

Properties Specifications
Beam Material A2 606-T6511-QQA200/8
Young's Modulus 64.3 GPa
Beam Length 0.812 m
Beam Width 19.02 mm
Beam Thickness 3.14 mm
Beam Mass 0.132 kg

 

 

 

 

61
(width) x 8.56 mm (thickness) size of payload is used and its mass is
0.061 kg. Table 3.1 presents the material and geometrical properties of

the experimental aluminum arm.

3.2.1.2 Composite Arm

The design and fabrication of the composite arm is not as simple as
the case of the aluminum arm. Once the geometrical and material
properties of the aluminum arm has been determined, the design of the
composite arm can be started based on some design constraints and design
criteria which depend upon the objective of experimental investigation
and the adaptability of the experimental apparatus. Herein, since the
objective of the experimental investigation is to compare quantitatively
the control performances between the aluminum and composite arm,

following design constraints and a criterion are imposed.

Design Constraints:

beam length(L); (L)al - (L)com

(3.1)
beam w1dth(b) ; (b)a1 - (b)com
Design Criterion;
flexural rigidity(FR); (FR)al - (FR)com (3.2)

where subscripts (a1) denotes the aluminum arm and (com) represents the
composite arm. The hub which was employed for the aluminum arm can be
easily used for the composite arm with the constraint of same width. The
selection of the design criterion, i.e., flexural rigidity (bending

stiffness) is somewhat rational way rather than logical manner. Someone

62

 

evaluation of design criterion
for the aluminum arm

 

 

 

 

 

]

selection of composite materials

 

 

 

 

V

 

 

imposition of design
constraints

 

 

 

 

7

 

determination of design variables: _‘
fiber orientation and no. of ply

 

 

 

 

 

V

 

evaluation of design criterion
for the composite arm

 

 

 

   

comparison

 
    
   

 

 

 
 

 

of design
criterion unsatisfactory
satisfactory r- -------- 1
..... .. p: fabrication ‘

|—-________

Figure 3.4 Design Methodology of the Composite Arm

63

can choose other design criteria such as beam geometry. However, it seems
to be reasonable start to choose such design constraints and a criterion,
since it is well-known that the composite materials has higher strength-
and stiffness-weight ratio than that of the aluminum, hence resultilu; in
significant reduced inertia forces in the dynamic motion of the system.

Once the design constraints and design criterion have been decided,
the design procedure can be accomplished based on a design methodology
presented in Figure 3.4. The design variables are restricted to the fiber
orientatdxnl (lamina angle) and the number of the prepreg ply, since the
manufacturing company generally provides the prepreg which has a certain.

fixed fiber volume fraction. The flexural rigidity of the composite arm

 

 

 

 

 

 

 

 

 

300
— PRESSUR=85 PSI, VACUUM=28 IN HG

* PRESSURE=1OO PSI, VACUUM=VENT
(Do 200—
if - .................................
0:
D
l— 4
<1
0:
“J 1
“2-
L13 100-]

O ‘Y I I I Y Y j r Y r Y 1 f l I I T Y I
0 60 120 180 240 300

TIME(sec)

Figure 3.5 Curing Cycle for AS4/3501 Graphite-Epoxy

64
is calculated from the finite element analysis by changing the design
variables to satisfy the imposed design criterion. Several combinations
of the design variables were determined through simulation. However, in

o of lamina angle was chosen for simplicity of

this study, the 0
fabrication.
Magnamite graphite prepreg tape, type AS4/3501-5A, manufactured by
Hercules Inc., was employed to fabricate the composite arm. The arm was
fabricated and cured by Auto Air Corporation, Lansing, Michigan, using
the hand lay-up technique, vacuuming and autoclaving based on the curing

cycle provided from Hercules Inc. as shown in Figure 3.5. The cure cycle

depends upon the resin used, and the material properties are dependent

Table 3.2 Material and Geometrical Properties of the
Composite Arm

 

 

 

 

 

 

 

 

 

 

 

Properties Specifications
Beam Material AS4/3501 - 5A
Young's Modulus 115.0 GPa
Beam Length . 0.812 m
Beam Width 19.02 mm
Beam Thickness 2.5 mm
Beam Mass 0.06 kg

0
Fiber Orientation 0
Ply Thickness 0.12 - 0.13 mm
Number of Ply 20

 

 

 

 

65

 

Figure 3.6 Photograph of the Experimental Composite Arm

annn the curing process provided from the manufacturer. The cure process

also suggests a post-cure to allow for final cross linking of the polymer

chains. The post-cure process has been done at 300°F for three hours with
ambient pressure and no vacuum before the cutting laminate with the
constrained geometry. Table 3.2 presents the geometrical and material
properties of the composite arm, and Figure 3.6 shows the photograph of
the composite arm mounted on the rigid hub.

In order to experimentally evaluate the imposed design criterion, the
load-deflection characteristics of the both aluminum and composite arms
were obtained through the static testing. One end of the arm was clamped
on the working table and monotonically increasing load was imposed on the
other end of the arm, and the corresponding end-point (tip) deflection

was measured subsequently.

66

 

 

 

 

 

 

 

 

 

4o
. AHA ALUMINUM
, o—e COMPOSITE

E .

g 30~

Z .J

O

.: .

0

LIJ .

_J

[:3 20—

D d

[—

g 2

O

,L .

C', 10—

z 4

LL]

0 r T I I I T I I I f T 1 I I I T I j I

0 1o 20 3o 40 50

LOAD(grom)

Figure 3.7 Load-Deflection Curve for the Aluminum
and Composite Arm

Figure 3.7 presents the load-deflection curve of the both arms. It
is seen from this figure that the flexural rigidity (bending stiffness)
of the composite arm is little smaller than that of the aluminum arm. The
flexural rigidity of the composite arm is approximately 92 percent of
that of the aluminum arm. It is recognized that the error is attributed
to the manufacturing process such as curing cycle. From this figure, the
Young's modulus of the aluminum was distilled by 64.3 GPa, and the
Young's modulus in the fiber direction of the composite material was

distilled by 115.0 GPa.

3.2.2 DC Motor and Servo-Drive

The DC servo-motor shown in Figure 3.1 is a permanent magnet field,

step-free torque motor, model 1338B-AV20, manufactured by Allen-Bradley.

67
The motor has some standard features such as the capability of 100
percent rated torque output at stall, the minimization of torque ripple
and motor cogging by employing a skewed armature. The peak stall torque

of this motor is 19.7 N-m and the peak stall current is 50 amperes at the

temperature of 40°C. It is noted that the fundamental flexural bending
and torsional natural frequencies of the motor shaft are 4.9 kHz and 2.1
kHz respectively, which are of course far beyond the natural frequencies
of the aluminum and composite arm. The servo-motor is driven by a servo-
drive, model l388-series B, manufactured by Allen-Bradley. This servo-
drive is a pulse width modulated (PWM) driving system which consists of a
150V DC power supply, a transistorized power amplifier, and isolated
current sensing and performance testing points such as current feedback
signal, current error signal to the power amplifier, and tachometer

signal.

3.2.3 Servo-Positioning Assembly

The servo-positioning assembly consists of a servo-controller
module, model l773-M3 and a servo-expander module, model l77l-ES, both
manufactured by Allen-Bradley. The commanded position data from the
microprocessor is sent to the servo-controller module, and the update
status from the servo-expander is sent to the microprocessor. The servo-
controller sends axis motion commands to the servo-expander which
commands the axis motion by sending an analog signal to the servo-drive
through D/A converter which has a 12 bits resolution. The servo sampling
period is 2.4 milliseconds and the servo output voltage is ilOV DC

maximum .

68

3.2.4 Sensors

3.2.4.1 Angular Position Sensor

An optical incremental encoder was mounted on the shaft of motor
using a flexible encoder coupling adapter,model l326DP-MOD-Ml-C2,
nunnifactured by Allen—Bradley, in order to measure hub angular position.
The incremental digital encoder, model 845N-SJH-N4—CRY2, manufactured by
Allen-Bradley, has 2500 lines and provides feedback signals that indicate
the magnitude and direction of any change of axis position. Also the
phase relationship of the encoder output signals allows the decoding
circuit to count either 1,2 of 4 feedback pulses for each line of the

encoder, hence increasing the resolution of the feedback signal.

3.2.4.2 Angular velocity Sensor

The tachometer mounted on the shaft of the motor was used to measure
hub angular velocity feedback signal. The tachometer has voltage gradient
of 0.3 vwsec/rad and 2.0 percent of ripple. The tachometer has 20 turn
potentiometer that scales the tachometer feedback signal, and the gain of
potentiometer is adjustable through the pot which has ten graduated
increments. In order to verify the voltage gradient provided from the
manufacturer, the joint was rotated at constant velocity, and the output
voltage of the tachometer was measured, then the corresponding velocity
was computed. Figure 3.8 presents the relationship between the velocity
and the measured output voltage. It is distilled from this figure that
the voltage gradient is 0.298 v-sec/rad which is almost same as the

value provided by the manufacturer.

69

 

 

 

 

15
’o‘ ..
Q)
U)
\
‘55 10a
E
U
o -4
_l
LIJ
>
§ 5—
D
0
Z
< .1
0 T I r r I T r I I
O 1 2 3 4 5

TACHOMETER OUTPUT(volt)

Figure 3.8 Relationship between the Velocity and the Measured
Voltage of the Tachometer

3.2.4.3 Tip Deflection Sensor

The strain gages, type WD-DY-2SOBG-350, manufactured by
Micromeasurement Group Inc., were mounted on the root of the arm in order
to measure tip deflection, even though the signal was not used as a
feedback signal in this study. The half-bridge circuit was arranged to
measure the strain. The measured strain was conditioned and amplified
through Wheatstone bridge conditioner/amplifier, model 2100, manufactured
by Micromeasurement Group Inc. In order to obtain calibration factor
between the amplified voltage (strain) and tip deflection, the series of
load was imposed monotonically on the free end of arm and the

corresponding deflection and amplified voltage were measured. Figure 3.9

70
shows the test results of this calibration for the both aluminum and

composite arms.

3 . 2 . 5 Microprocessor

The microprocessor, model PLC-2/l6, manufactured by Allen-Bradley
was employed for the implementation of the output feedback controller
associated with two colocated output sensors - angular position and
velocity.ffluaudcroprocessor has features such as 3K words of memory
size, computation capability up to 12 digits and conversion binary to
binary coded decimal (BCD) which is a numbering system used to express

individual decimal digits (0-9) in four bit binary notation. The personal

 

 

 

 

 

 

 

 

 

40

, A--A ALUMlNUM

_ o—e COMPOSITE
E .
\E’ 30-
z .
Q
5 ~
Lu ..
.1
L5 20—«
O u
l-
; .4
o
l -4
CL 10—
z
LIJ '1

O I I I I I I F
0.0 0.5 1.0 1.5 2.0

AMPLIFIER 0UTPUT(voIt)

Figure 3.9 Relationship between the Amplified Voltage
and the Tip (End-Point) Deflection

71
computer, model ZVC-lS9-2, manufactured by Zenith Data Systems, was used
as a program operating system. The program was written as relay-like

language developed by ICOM Inc. [152].

3.3 Identification of System.Hbde1 Parameters

In this study, the rigid-body mode and the first two flexible modes

are retained as the primary modes. Hence, the equation (2.35) becomes

 

 

~=K~+E~
xp pxp pu (3.3)
”=6;
yP P P
where
P 0 0 -
l 0
K-
p -w1 ~2§1w1 0 O
0 0 0 l
_ 0 0 -w2 -2§2w2‘

- 1 d¢1(o> d¢2(o) T
Bp'T ° 1 0 'Ti" 0 —E_

T

' d¢ (o) d¢ (o) '

__1___ __2___
1 0 dx 0 dx 0

Op ‘ d¢1<o> d¢2<o)

50 1 0 ft?— 0 dx _

 

 

It is noted that the output matrix Cp includes sensor gains associated

with two colocated output sensors - angular position and velocity. In

72

this section, the system model parameters such as wi, (i and d¢ go)/dx

will be identified through the simulation and measurement.

3.3.1 Predicted

For the calculation of the system model parameters, the governing
equation of motion given by the equation (2.31) was solved without

considering the damping and generalized forcing terms,i.e.,
[M]{U} + [K]{U} = {0} (3 4)

The two-node element with two degrees of freedom at each node as shown in
Figure 2.3 was used and the number of element was ten. Table 3.3
presents the system model parameters obtained from the inhousing finite
element program for the both aluminum and composite arms without payload.
The geometrical and material properties given in Tables 3.1 and 3.2 were

employed for the aluminum and composite arm respectively. The 0i in the

Table 3.3 represents the natural frequency of the ith cantilevered-mode
(clamped-mode). It is observed from Table 3.3 that the pinned-mode

fundamental natural frequency (col) of the composite arm is increased by

7.5 percent relative to that of the aluminum arm, while the clamped-mode

fundamental natural frequency (01) of the composite arm is increased by

29.2 percent. This increment of the clamped-mode fundamental natural
frequency can increase the equivalent rigid-body mode control bandwidth

of the closed-loop system which is limited by 01 with a certain

compensator zero as will be observed in the subsequent chapter.
Table 3.4 presents the system model parameters for the aluminum and

composite arm with payload of 0.061 kg. It is interesting to observe that

73

Table 3.3 Comparison of the System Model Parameters of the
Aluminum and Composite Arm without Payload

 

 

 

 

 

 

 

 

Parameters Aluminum Arm Composite Arm
w1(Hz) 8.961 9.688
w2(Hz) 25.232 34.147
01(Hz) 3.738 5.269
02(Hz) 23.425 33.024
Sflggl 2 1534 1 5385

dx . .
Effiégl 1.0135 0.4978

 

 

 

 

 

Table 3.4 Comparison of the System Model Parameters of the
Aluminum and Composite Arm with Payload of 0.061 kg

 

 

 

 

 

 

 

 

 

 

Parameters Aluminum Arm Composite Arm
w1(Hz) 7.425 7.424
w2(Hz) 20.337 25.835
01(Hz) 2.201 2.318
02(Hz) 18.068 24.337
(1451(0)

dx 3.1545 3.0273
d¢2<°7

dx 1.8573 1.1858

 

 

 

 

 

74

the pinned-mode fundamental natural frequency of the aluminum arm is
almost same as that of the composite arm, even though the second mode
frequency of the composite arm is larger than that of the aluminum arm.
And also the clamped-mode fundamental natural frequency of the composite
arm is larger than that of the aluminum arm by only 5 percent. The
reason follows from the ratio of the payload to the beam mass. It is seen
from Tables 3.1 and 3.2 that the ratio of payload to beam mass is 46.2
percent and 101.6 percent for the aluminum and composite arm
respectively.

Figure 3.10 presents the first pinned-mode natural frequency (col)

variation (with respect to the first cantilevered-mode natural frequency

(01) as a function of the the ratio of the hub moment of inertia (IH) to

the beam moment of inertia (IB). It is noted that the IB of the aluminum

arm is 0.029 kg-mz, while that of the composite arm is 0.0132 kg-m2. It
is apparent from this figure that even a small ratio of the hub moment of
inertia to the beam moment of inertia has significant effect in
decreasing the pinned-mode natural frequency. The pinned-mode natural
frequency approaches to the cantilevered-mode natural frequency as the
ratio increases as expected. Figure 3.11 shows the fundamental natural

frequency variation with respect to the ratio of the payload (mt) to the
beam mass (mb). It is noted that the mass of the aluminum arm is 0.132

kg, while that of the composite arm is 0.06 kg. The effect of the payload
on the frequency variation is not as significant as the effect of the hub
moment of inertia, even though the frequency noticeably decreases in the

small ratio of the payload to the beam mass.

75

Figure 3.12 shows the first three pinned-mode shape for the aluminum
and composite arm without payload. The slopes of the mode shapes at the
hub (x-O) have been chosen positive by convention. It is observed that
the slope at the hub decreases by increasing the mode due to the presence
of lumped inertia of the hub. Thus, the pinned—mode natural frequency
becomes closer to the clamped-mode natural frequency by increasing the
mode. It is also interesting that the modal magnitude at the end of the
arm (x/L) has alternating sign from one mode to next mode. This
phenomenon is interpreted as nonminimum phase of the tip position which
should be carefully considered in controller design associated with the
tip position sensor.

Figure 3.13 presents the calculated Bode plot of open-loop transfer
function for the angular velocity of the aluminum and composite arm
without payload. The damping ratio of the aluminum arm was chosen by
0.005 for each flexible mode, while that of the composite arm was chosen
by 0.007 for each flexible mode. The peaks correspond to the pinned-mode

natural frequencies mi, and the valleys correspond to the clamped-mode
natural frequencies Oi. At the pinned-mode, the output signal of angular

velocity is maximum (equivalent to saying that the strain energy of the
root of the beam is minimum), and at the clamped-mode, the angular
velocity output signal is minimum (equivalent to saying that the strain
energy of the root of the beam is maximum). It is seen from this figure
that the output magnitude at the second mode natural frequency of the
composite arm is less than that of the aluminum arm by 40 percent. This
implies that the output signal effect induced from the second mode of the
composite arm is not as significant as the, case of the aluminum arm.

Hence somehow the second mode can be neglected in the modeling process

81)

 

 

[AUflWNUM ]

 

€10—

 

 

 

 

. I .
0.00 0.35 0.70 1.05 1.40
IH/IB

 

 

- [COMPosqg]

”1/“1

 

 

 

 

I I I
0.00 0.75 1.150 2.25 3.00
IH/IB

Figure 3.10 Frequency Variation with respect to the Ratio of the
Hub Moment of Inertia to the Beam Moment of Inertia

77

L120~———_

 

 

[ALUNMNUNIJ

 

 

 

 

L00 . , , I , I .7
0.0 2.0" 4.0 6.0 8.0

mt/m

 

2L280

 

TCOMPosnE]

 

1.672—

L368—

 

L064-

 

 

 

0750 ., . . , . . . , . . . sr., . .
0.0 4.0 8.0 12.0 ' 16.0

Figure 3.11 Frequency Variation with respect to the Ratio of the
Payload to the Beam Mass

78

 

 

 

 

12

« —— rigid—body mode

~ —-— first elastic mode ALUMINUM ARM
8 “ second elastic mode

 

PHl(x)

 

 

 

 

"8 I I r l I r I l I I I l I I I I r I
0.0 0.2 0.4 0.6 0.8 1.0
NORMALIZED LENGTl-Kx/ L)
12

 

 

l

— rigid-body mode
-— first elastic mode
second elastic mode

I

COMPOSITE ARM

 

 

 

 

 

I I I

 

 

0.0 012 ' 0:4
NORMAUZED LENGTH(x/L)

I I I
0:6 0.8 1.0

Figure 3.12 First Three Pinned-Mode Shape for the Aluminum
and Composite Arm

PHASE(deg)

79

 

 

* ALUMINUM
' ~ —— COMPOSITE

 

 

 

MAGNITUDE(dB)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

“40 . . . . 1 . - . . 1 . . . ' l . . ' . 1 . T . .
0 50 100 1 50 200 250
FREQUENCY(rad / sec)
1 50
1 ' AUUMHWJM
: -—- COMPOSWE
100;
50-? i E {l
05 § é
-505
...-100 I I I I I I I r I I r j I r l I T j I I T I I I1
0 50 100 1 50 200 250
FREQUENCY(rad / sec)

Figure 3.13 Calculated Open-Loop Transfer Function for Angular
Velocity without Payload

PHASE(deg)

80

 

 

‘ - ALUMINUM
‘ —— COMPOSWE

 

 

 

MAGNITUDE(dB)

 

 

 

 

—40 I I I I l I I I

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I I I I I I I
0 50 1 00 1 50 200
FREQUENCY(rad/ sec)
150
7 ° ALUMINUM
‘—— COMPOSWE
1 004
503- 5 I
0 l
—509 ' 5 5 3
’ : 1 5 3
“—J .Ka—5 . ' I. ... ' ...............
—1 00 I I I T l I I I I l T I I I l I I I 1
0 50 100 1 50 200
FREQUENCY(rad/sec)

Figure 3.14 Calculated Open-Loop Transfer Function for Angular
Velocity with Payload of 0.061 kg

81
without causing serious error. Herein, the second mode of the both
aluminum and composite arms was maintained for reasonable quantitative
comparison. It is also interesting to observe from Figure 3.13 that the
phase margin corresponding to the first gain crossover at w - 16.3
rad/sec for the aluminum arm and w - 25.8 rad/sec for the composite arm
is 90 degree. However, the phase margin will be lower for any practical
system due to the unmodelled dynamics of the actuators and sensors as
well as sampling delays of the output feedback signals. This phenomenon
vfill be investigated in the next section. Figure 3.14 shows the
calculated Bode plot of the open-loop transfer function for the angular

velocity of the aluminum and composite arm with payload of 0.061 kg.

3.3.2 Measured

In order to measure the natural frequency and damping ratio of each
mode for the aluminum and composite arm, a sinusoidal input voltage was
applied to the servo-drive and its frequency response was swept in the
range of interest (0.8 Hz-32 Hz for the aluminum arm and 0.8 Hz-40 Hz for
the composite arm). The 12 MHz programmable function generator, model
275, manufactured by Wavetek, was employed to provide the sinusoidal
input and the output signal (magnitude and phase) from the tachometer was
measured using the oscilloscope, model T912, manufactured by Tektronix.

Figure 3.15 shows the calculated and measured frequency response of
the aluminum arm without payload from the servo-drive input to the
tachometer output. It is seen from this figure that the cantilevered-mode
natural frequencies (valleys) match very well between the calculated and
measured values, while the pinned-mode natural frequencies (peaks) do not

‘match well. The poor agreement of the pinned-mode natural frequencies is

82
attrflnufid motim motor joint friction which may produce significant
effect to the pinned boundary conditions. This joint friction may exit as
a Coulomb friction and is hard to model ”incorporating with the linear
model employed in this study. It is also observed from the measured phase
response that the phase margin corresponding to the first gain crossover
at w - 16.1 rad/sec is 63 degree which is much lower than the calculated
value of 90 degree. This reduced phase margin limits the stability of the
system in practice whose stability is guaranteed theoretically with the
infinite gain margin and 90 degree of the phase margin. The damping
ratio of the proposed structural system may be obtained from this
frequency response. Herein, a half-power point method [148] was employed

to determine the damping ratio of each mode. The damping ratio §1= 0.0224
and {2a 0.0143 were distilled from the Figure 3.15.

Figure 3.16 presents the calculated and measured frequency response
of the composite arm without payload from the servo—drive input to the
tachometer output. Same phenomena as those of the aluminum arm are
observed from this figure. The phase margin corresponding to the first
gain crossover at w = 25.0 rad/sec is 81 degree which is larger than that
of aluminum arm by 22.2 percent, hence implying the increment of the
stability margin of the practical system featuring the composite arm. The

dampin ratio f =0.0237 and g - 0.0148 were distilled from this frequenc
8 1 2 Y

response by using a half-point method. Tables 3.5 and 3.6 present a
comparison of the calculated and measured natural frequencies of the
aluminum and composite arm without and with payload.

The phase lag observed in the Figures 3.15 and 3.16 was arisen from

the inherent dynamic characteristics of the employed velocity sensor -

PHASE(deg)

MAGNITUDE(dB)

83

 

 

 

~ - - calculated
—o— rneasured

 

 

 

 

 

 

 

 

I I l 1 I I I I I I I I r I I I I
0 50 100 150 200 250
FREQUENCY(rad/sec)
150
1 - calculated
2 + measured
100—

 

 

 

 

 

 

I I I

 

I I I I I
o 50 160 150
FREQU ENCY(rad / sec)

r I I I I I

Figure 3.15 Calculated and Measured Frequency Response of the

Aluminum Arm without Payload

PHASE(deg)

84

 

 

. --~ calculated
+ measured

 

 

 

MAGNITUDE(dB)

 

 

 

 

I I I I I I I I I l I I I I
0 5‘0 100 150 200 250
FREQUENCY(rod / sec)

I I

150

 

 

- - - calculated
0 measured

 

 

 

O
O

01
O

llllllllllljllllLllLLil

l
(J1
O

 

 

 

_200 d I I I I 1 T r I I j I I l I
O 50 100 150
FREQUENCY(rad / sec)

 

I I I I fl

I I I
200 250

I

Figure 3.16 Calculated and Measured Frequency Response of the
Composite Arm without Payload

85

Table 3.5 Comparison of the Calculated and Measured Natural
Frequency without Payload

 

 

 

 

 

 

 

 

Frequency Aluminum Arm Composite Arm
(Hz) Calculated Measured Calculated Measured
wl 8.961 9.15 9.688 10.20
w2 25.232 26.30 34.147 35.50
01 3.738 3.73 5.269 5.22
02 23.425 23.40 33.024 33.09

 

 

 

 

 

Table 3.6 Comparison of the Calculated and Measured Natural
Frequency with Payload of 0.061 kg

 

 

 

 

 

 

 

 

 

Frequency Aluminum Arm Composite Arm
(Hz) Calculated Measured Calculated Measured
wl 7.425 7.83 7.428 7.89
“2 20.337 21.10 25.835 26.70
01 2.201 2.12 2.318 2.29
02 . 18.068 18.10 24.337 25.5

 

 

 

 

 

 

86

tachometer. Ideally, the angular velocity is given by 8(5) - 59(5) where
s is Laplace variable. However, it is hard to say in practice that the
actual hub rate is exact derivative of the hub angle. Hence, there exist

always unmodelled sensor dynamics. For more clear observation, let the

angular velocity have a relationship with the angular position by 8(5) =
(s/ (s/a + 1)) 9(5) so that the phase lag of the response can be caused
depending on the value of a. Figure 3.17 presents the simulated open-
loop transfer function for the angular velocity with three different
values of a. It is clearly observed from this figure that the phase lag

increases as the value of a decreases, and furthermore, as the value of a

 

 

 

 

 

150,

100{

50%
@j
SCI
83
§ —50:
0.

 

 

-—100{‘5f31:4

 

 

 

 

I I 7 I I I y t t

I I I I I I
0 50 100' 150 200 250
FREQUENCYOUd/sec)

Figure 3.17 Calculated Open-Loop Transfer Function for the Angular
Velocity with é(s)-(s/(s/a+1))e(s)

87

goes to infinity,the response becomes to the ideal case, iӣ3., 8(5) =
58(5). Hence, the larger value of a, the smaller phase lag. For the
employed velocity sensor in this study, the corresponding value of a is
approximately 145 which is equivalent to 6.89 milliseconds of time delay.
Therefore, attention should be paid to the unmodelled dynamics of sensors
or actuators in order to predict the stability of the practical system.
Now, the system parameters of the dynamic model given by the
equatflxni (3.3) were identified through the calculation and measurement.
Since all parameters were not available from the experimental
measurement, herein, the available data from the both calculation and
measurement were adopted to provide the necessary numerical values in the
model. The measured data about the natural frequencies and damping ratios
and the calculated data about the modal slope coefficients at the hub

(d¢i(o)/dx) were adopted.

3.4 Summary of the Chapter

The detailed description of the experimental apparatus of the
proposed robotic system featuring the output feedback controller
associated with two colocated output sensors has been presented. Each
functional group of the overall robotic system was described in terms of
the manufacturers and specifications as well as verification of the
performance requirements. The system model parameters such as natural
frequencies and damping ratios of the flexible modes have been identified
Iflucugh the calculation and measurement. Good agreement between two

results was achieved.

CHAPTERIV

OUTPUT FEEDBACK CONTROL ASSOCIATED WITH
TWO COIDCATED OUTPUT SENSORS

4.1 Introduction

A proportional-plus-derivative (PD) output feedback controller
associated with two colocated output sensors for angular position and
velocity is designed based on the reduced-order model containing the
primary modes only. Then, the controller is experimentally implemented in
order to investigate and compare the control performances of the
flexible manipulators fabricated from aluminum (aluminum arm) and
composite laminates (composite arm). The advantages of the
implementation of such a simple feedback controller are that some
guidelines for the comparison of control performances between the
aluminum and the composite arm can be provided easily and the dynamic
model parameters can be readily identified and compared between
experimental and theoretical results.

In this chapter, the PD output feedback controller will be
formulated using the root-locus technique and the Lyapunov method.
Experimental results such as step responses to a commanded step angular
position, step responses in the presence of the sinusoidal disturbances
which have forcing frequencies close to the clamped-mode natural
frequencies of the arms, and the instability of the employed system will

be presented and compared between the aluminum and composite arm. The

88

89
simulated results corresponding to the measured resultS'withmzalso

presented.

4.2 Controller Design

The dynamic model given by the equation (3.3) is employed for the
controller design. Consider the block-diagram of the closed-loop system
as shown in Figure 4.1 which features a proportional-plus-derivative
output feedback controller associated with two colocated sensors. The

output equation in (3.3) can be rewritten as
._ '5' '5 ..
y - ~1 ] a 1 J x (4.1)

where the output §1 represents angular position and §2 denotes angular

velocity. 61 and CZ represent the output matrix associated with §1 and §2

respectively. In Figure 4.1, E represents the commanded angular position,

KP is the position feedback gain and RV is the velocity feedback gain.

 

~

r+ K +
[1:— P - p pp P ~ P
Y1

 

Cl

 

 

 

 

Xl‘
I
>l
XI
+
an
El
“<l

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.1 Block-Diagram of the Output Feedback
Control System

90
The openéhxn>transfer function from the input torque to angular

position is given by

YI(S) A _1

= 60(5) = c1 (51 - AP ) B (4.2)

5(5) p

where s is the Laplace variable. The control torque 3(5) is given by
u(S) - Kp( r(5) - y1(S)) - Kvy2(8) (4.3)

Assuming the angular velocity sensor output is the exact derivative of

the angular position sensor, the equation (4.3) becomes
u(S) - KP ( r(S) - y1(3)) - sKvy1(S) (4.4)
Now, the closed-loop transfer function of the system is obtained by

§1(s) é o ( ) _ KD 60(5)
c2 3 1 + Kv(s + Kp/Kv) 60(5)

 

(4.5)
Ecs)

Therefore, the closed-loop poles of the system are roots of the following

characteristic equation given by

1 + xv (s + KP / xv) 60(5) - o (4.6)

The shape of the loot-locus of the closed-loop system, that is, the

closed-loop system performance depends on the location of the compensator
zero, defined by zcoé -Kp/KV. With the certain compensator zero, the

rigid-body closed-loop poles may move from the origin of the s-plane (two
open-loop poles are zeros) toward the open-loop zeros of the system which
are characterized by cantilevered-modes, or toward the real axis of the

s-plane resulting in sluggish motion.

IMAGINARY

IMAGINARY

91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

390
~ A open—loop poles
- o open—loop zeroes ALUMINUM ARM
" I compensator zero
260—
- A .
l30~
“ A
0 —I
s A
—130~
o
. A
—260 I I I r I I I r l I I I I l I I I
—4.0 —3.0 —2.0 —-1.0 0.0
REAL
2590
~ A open—loop poles
- o Open—loop zeroes COMPOSITE ARM
‘ - compensator zero
260-
I ‘ -
130-
- A
.. O
0 I
- o
.. A
—130-
- A .
“-260 I T I I I I I I l I I I m I I I I
-4.0 —3.0 —2.0 —1.0 0.0
REAL
Figure 4.2 Open-Loop Pole-Zero Location with the Compensator

Zero 2 - -1
co

92
Figure 4.2 shows the open-loop pole-zero location with the

compensator zero Zoo: -1 for the experimental aluminum and composite arm

whose material and geometrical properties are presented in Tables 3.1 and
3.2 respectively. It is seen from this figure that all non-zero poles are
located left to the given compensator zero. It is noted that open-loop
zeros and poles physically represent the clamped-mode and pinned-mode
natural frequencies of the arm respectively. Figure 4.3 presents the
root-locus of the closed-loop system obtained by increasing the feedback

gain RV (or Kp, since the ratio is fixed) with the compensator zero

 

 

 

 

 

 

 

 

 

Zco=-l. Since the root-locus is symmetric with respect to real axi5,only
250
: --- ALUMINUM
- -— COMPOSITE < :
2004 W
: mm mm
’3 q , 4 ' ‘ poms“): 901.239) 3
(D 150'“ “---0
Q _ 0.0, 0 0 0.0, 0.0
'o . 0.0, 0.0 0.0, 0.0
o . -1.288, 57 47 4.519. 66.07
L q -1.288,-57 47 -1.519,-66.07
E.” 100- —2.363, 165.23 -3.256, 223.03
a: _ -2.363,-165.23 -3.256,-223.03
2 : mosh): zsnosu»
o . —————— 1- -0.525, 23.4.3 -0.777 32 79
g 50: ’ -0.525,-23.43 —o.777,-32.79
_ q ', -2.102, 147.01 -3.036, 207.95
. . __________ 4 -2.102,-147.01 -3.036.-207.95
0.: ..- fl IIOOMPENSATOR mam-1.0.0.0)
“'50 I I I I l I I I I I
-120 -80 -40 O
REAL(rad/sec)
Figure 4.3 Root-Locus with the Compensator Zero Z - -1
co

 

93
upper plane of the locus is presented. It is interesting to observe that
the open-loop poles located at the origin (rigid-body mode) start to move
toward to have imaginary part of the poles, and quickly move toward to
have real part only as the feedback gain increases for the both aluminum
and composite arms. The one rigid-body pole goes to the negative infinite
on the real axis, while the other rigid-body pole ends at the given
compensator zero as the feedback gain goes to the infinity. This
phenomenon will results in the sluggish response of the closed-loop
system without generating large overshoot to the step commanded input. As
the feedback gain increases, the other open-loop poles move toward close
open-loop zeros which correspond to the finite two cantilevered-mode
natural frequencies. The same type of root-locus shape is obtained with

the compensator zero Zco=-2 which is located between two non-zero poles

on the real axis.
Now, for the same dynamic model, consider the pole-zero plot on the

s-plane with the compensator zero Zco--15 which is positioned far left to

the all open-loop poles and zeros as shown in Figure 4.4. Figure 4.5

presents the root-locus as a function of the feedback gain Kv with the
compensator zero Zco--15. Two poles located at the origin (rigid-body

mode) move toward close two open-loop zeros which correspond to the
fundamental cantilevered-mode natural frequencies. Therefore, it may be
said that the bandwidth of the rigid-body mode is limited by the

fundamental cantilevered-mode natural frequency {ll-3.73 Hz and I'll-5.22

Hz for the aluminum and composite arm, respectively. The maximum
bandwidth of the composite arm is larger than that of the aluminum arm by

28.5 percent. This larger bandwidth results in faster response to the

94

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

390
.1 A open—loop poles
~ 0 open—loop zeroes ALUMINUM ARM
‘ I compensator zero
260~
. A.
g 130—
< -I
Z; .
(D A
< ...
Q3 0 -~I
: 1
—130-— ,
a A
“—260 I I I I I I I I I I I I I I I I
~18 ~15 ~12 ~9 ~6 ~3
REAL
390
~ A open~loop poles
- o open—loop zeroes COMPOSITE ARM
‘ I compensator zero
260~
j ‘6
E 130—
E u
__ ~ A
((9 . 0
2 O a
3 O
1 A
~130~
1
- A?
“-260 I I I I I I I I I T I I I I I I
~18 ~15 ~12 ~9 ~6 ~3
REAL

Figure 4.4 Open-Loop Pole Zero Location with the Compensator
Zero zco- -15

95

 

 

 

 

 

 

 

 

 

 

 

250
: --- ALUMINUM
. - COMPOSWE
200— W
1 mm EMS—HP 1T3
I; ‘ POLES(A): POLES(A):
0 150~
m _ O£.0£ mo,mo
\\ . 0b,0b mo,mo
13 . -1.288, 57.47 -1.519, 64.07
L_ q -;.§g§,-57.47 -1.519,-64.07
\_, _d - . , 165.23 -3.256. 223.03
E 100_ -2.363,-165.23 -3.256,-223.03
:21:- : zanosu ): zaaosu ):
(p - -0.525, 23.43 -o.777, 32.79
<2: 50— -o.525,-23.t.3 —o.777,-32.79
__ . -2.102, 147.01 -3.036, 207.95
. —2.102.-147.01 -3.036,-207.95
O_ __ _ 1 = I :COHPENSATOR ZERO(-15.0,0.0)
-50 I I I I I I I I I I I I
~120 ~80 ~40 O
REAL(rod/sec)
Figure 4.5 Root-Locus with the Compensator Zero Z ~ -15
co

commanded step input. From these characteristics of the closed-loop
system depending upon the certain fixed compensator zero, appropriate
output feedback gains have to be determined to evaluate the control
performances for the aluminum and composite arm.

It is seen from Figures 4.3 and 4.5 that any choice of the feedback

gains KP and Kv assure the stability of the closed-loop system. This can

be interpreted that the control law is equivalent to a mechanism
consisting of the torsional spring and dashpot mounted at the pinned end
of the arm. Therefore this dashpot always provides a damping to the
system ensuring the stability. However, this guaranteed stability is no

longer valid in practical system, since unmodelled dynamics of the

96
flexible link, actuators and sensors, delays caused by measuring and
sampling feedback signal and the inherent limitation of servo-drive are
always present. This instability will be investigated through the
experimental test.
Another useful way to determine feedback gains which guarantee
theoretically the stability of the system is to use the Lyapunov method.

Consider a system described as

x + fii + 6x = 0 (4.7)

~

According to the Lyapunov theorem for the stability, if both matrices P

and a are symmetric and positive-definite, the system (4.7) is

asymptotically stable. This theorem can be easily proved by choosing a

Lyapunov function candidate V by

l

v = 5 <qu x + I? .2) (4.8)

Then time derivative of V becomes

V - - iT ? i (4.9)

Hence, this satisfies the stability condition; V > 0 and V < 0 for all x

#0.

The dynamic equation (3.3) can be reformulated in the form of the
equation (4.7) using the modal coordinate given in the equation (2.34).

Hence, the equation (3.3) becomes

0 + 25 + on - Bpfi (4.10)

97

2 2
where 0 (= diag (O,w1,w2)) is the modal frequency matrix, 2 (= diag (O,

2§1w1,2§2w2)) is the modal damping matrix and Bp is the control influence

matrix which is given by

A 1 d¢1(0) d¢2(0) T
p I dx dx

 

 

(4.11)

Also the output matrix given by equation (4.1) can be rewritten as

y1=clfl
A . (4.12)
y2=C2W
where
c — c - I B T (4.13)

Since the system described by equation (4.10) is controllable, the

PD output feedback controller is given by
u - -pr1 - KVy2 (4.14)
where KP is the constant position feedback gain and RV is the constant

velocity feedback gain. The substitution of the equations (4.12) and

(4.14) into the equation (4.10) yields

0 + (z + BpKv02)n + (a + BPKPC1)n = 0 (4.15)
It is noted that the matrices Z and 0 are symmetric and positive semi-
definite due to the rigid-body mode which is characterized by zero

rmtural frequencies. Thus, the feedback gains KP and Kv in equation

A A

(4.15) should be carefully chosen such that the matrices (Z + BpKvCZ) and

98

A A

(O + BprCl) are symmetric and positive definite. Then the resulting

closed-loop system (4.15) will be asymptotically stable.
The following proposithniis stated without a proof before

determining the feedback gains.

Proposition 4.1 : Let P and Q be any two symmetric matrices with the
same dimension. If P is the positive definite and Q is the positive semi-

definite, then (P+Q) is the positive definite.

Now, under the condition stated in the Proposition 4.1, the feedback

gains KP and RV can be chosen such that the closed-loop system be

asymptotically stable. For example, let

A (4.16)

where P and Q are arbitrary symmetric and positive definite matrices.

Then the feedback gains KP and Kv are obtained as follows.

A T A -1 A T A AT A AT -1
K = B B B P c c c
p 1 p pJ p 1 [ 1 1]
(4.17)
A T A -1 A T A AT A AT -1
RV - [Bp Bp] Bp Q 02 [02 c2]

It is noted that the minimum norm solution(1east effort solution)
and least error solution method [28] were employed to obtain the feedback
gains. It is also noted that both matrices Z and 0 in equation (4.15)

remain as symmetric and positive semi-definite, even though the model

99
parameters such as natural frequencies and damping ratios are perturbed
in a certain threshold. Therefore, the perturbed system will be still
asymptotically stable so long as the output feedback gains are chosen
based on the above manner. Thus, it can be stated that the proposed
output feedback controller assures the robustness of the system
stability. The feedback gains determined from the afbrementioned two

approaches will be used in the following experimental implementation.

4.3 Implementation Results and Discussions

Figure 4.6 presents the experimental set-up for the output
measurement. All experimental results presented in this section were

obtained based on the following measurement procedures. The angular

position feedback

 

velocity feedback

r 1 j

strain gage

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

servo
positioning assembly " servo drive " motor ' flexible arm]
commanded input torque velocity
position
status
microprocessor analog filter Str232 Egggiéigger
1
1
A/D data
D/A c~ acquisition - analog filter

system

 

 

 

Figure 4.6 Experimental Set-Up for Output Measurements

3:“ out

 

 

100
+ channel 1 F——0 + channel 2
IO‘ out
V
+ {IL

 

_J_

 

 

 

 

 

 

 

 

 

 

 

 

channel 1 1"}

in _01

V+ channel 2 +0—
1 in ~ 0—

Circuit Diagram of Two Channel Operational

 

 

 

 

Figure 4.7
Amplifier

position data stored in the microprocessor as digital values were
microprocessor to the data acquisition system

transferred from the
through D/A converter, model l771-OFE1, manufactured by Allen-Bradley.

The D/A converter has output range of O to 10 volts DC. Since the data
stored in the microprocessor was isolated from the main computer network

which is employed for the data postprocessing and plotting the results,
this transfer procedure was necessary.

The employed data acquisition system, model MicroVMS II/GPX 21,
manufactured by Digital Equipment Corporation, has 15 channels of A/D
converter. The data sampling rate of 150 Hz, which is far beyond the
considered natural frequencies of the arm, was employed for the

measurement of all output signals except the second mode exciting

response where 800 Hz of sampling rate was used. The input torque and

101
angular velocity from the servo-drive were amplified through the
operational amplifier, since the actual output signals were relatively
small resulting in significant effect of the computer noise of the data
acquisition system to the actual output signals. Two channel operational
amplifier producing the amplification factor of 100 was assembled as
shown in Figure 4.7.

The amplified signals then were fed to a dual analog filter, model
432, manufactured by Wavetek Inc., in order to eliminate high frequency
components such as noises. The 100 Hz of cutoff frequency was employed
before the filtered analog signals were digitized by A/D data acquisition
system. The output signal from the strain gage mounted on the root of the
flexible arm was conditioned and amplified through Wheatstone bridge
amplifier, model 2100, manufactured by Measurement Groups Inc.,before the
conditioned and amplified signal was fed to the low-pass analog filter

with the cutoff frequency of 100 Hz.

4.3.1 Step Responses without Disturbances

For the first comparison of the closed-loop performances between the

aluminum and composite arm, the small compensator zero Zco--2 was chosen.
With this compensator zero, the feedback gains KI) and RV were decided

such a manner that the damping ratio of the closed-loop poles of the
rigid-body mode is equal to 1 (critical damping). This means that the
closed-loop poles of the rigid-body mode has no imaginary part for the
first time as the feedback gain increases. From the root-locus analysis,

the feedback gains Kp- 0.534 N-m, Kv- 0.267 N-m-sec and Kp- 0.292 N-m,

Kv - 0.146 N-m-sec were obtained for the aluminum and composite arm

102

respectively. Table 4.1 the presents closed-loop poles of the system with

these feedback gains.

Table 4.1 Closed-Loop Poles with the Compensator Zero Zco= -2

 

 

 

Aluminum Arm (Kp — 0.534) Composite Arm (Kp - 0.292)
Real Imaginary Real Imaginary
-3.820 0 -3.638 0
-4.425 0 -4.538 O

-l9.736 52.19 -lO.948 62.26

-l9.736 -52.19 -10.948 -62.26
-6.119 164.03 -4.279 222.89
-6.119 -l64.03 -4.279 -222.89

 

 

 

 

 

 

Figure 4.8 shows the measured step responses to a commanded step
angular position of 7.2 degree. The hub angular position reaches its
commanded position within 2.0 second for the both aluminum and composite
arms. No overshoot in the angular position response is observed as
expected by the employed feedback gains. The steady-state error between
the commanded and actual position was 0.1 degree for the aluminum arm and
0.08 degree for the composite arm. This steady-state error is attributed
to the joint friction and can be eliminated by applying the integral
control action or manual calibration method. The significant different
response between two arms is observed from the input torque response. The
applied input torque for the composite arm is only 53 percent of the
torque required for the aluminum arm. This amount of reduction of energy

consumption will provide tremendous benefit of the economic area on the

103

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106

real production line. It is also observed from the tip deflection
response that the magnitude of the vibration amplitude of the aluminum
arm was reduced up to 30 percent by employing the composite arm. The
second mode effect to the total tip deflection is very small for both
aluminum and composite arms. The response decayed to zero within 2 second
for the aluminum arm and 1.8 second for the composite arm respectively.
It is noted from the angular velocity response that the ratio of the

maximum angular velocity to the first pinned-mode natural frequency,

i.e., 0 is 0.0033 and 0.0029 for the aluminum and composite arm

max /wl’
respectively, hence eliminating the possibility of the resonance
phenomenon due to the input torque.

Figure 4.9 shows the simulated step responses corresponding to the
measured response presented in Figure 4.8. The agreement between two
results is excellent, even though there exit small mismatch in the
response of tip tip deflection. For the comparison of the step responses
between the open-loop and closed-loop system, the open-loop step input
torque was introduced to the model. Figure 4.10 presents the simulated
open-loop responses of the aluminum arm for the commanded angular
position of 7.2 degree. The magnitude and period of the step input torque
was determined to have almost same delay time (0.42 second) as the case
of the closed-loop system. It is clear from this figure that the angular
position and velocity responses exhibit oscillations as expected, and the
settling-time of the tip deflection is increased by 45 percent comparing
with the closed-loop response shown in Figure 4.9.

For the second comparison of the control performances between the

aluminum and composite arm, the compensator zero Zco- - 5 was chosen and

107

same feedback gains were employed to the both aluminum and composite
arms. Table 4.2 presents the closed-loop poles of the system with

feedback gains of Kp- 1.0 N-m and KV- 0.2 N-m-sec for both arms. From

this table, the damping ratio of rigid-body mode was obtained by 0.633

and 0.855 for the aluminum and composite arm respectively.

Table 4.2 Closed-Loop Poles with the Compensator Zero zco- -5

 

 

 

Aluminum Arm (Kp = 1.0) Composite Arm (Kp - 1.0)
Real Imaginary Real Imaginary
-2.831 4.645 -5.526 5.159
-2.831 —4.645 -5.586 -5.159

-15.233 55.47 -14.523 61.18

-15.233 -55.47 -l4.523 -6l.l8
-5.307 164.58 -4.630 222.80
-5.307 -164.58 -4.630 -222.80

 

 

 

 

 

 

Figure 4.11 presents the measured step responses to a command step
angular position of 7.2 degree with above feedback gains. From the
angular position response, the delay time was obtained by 0.26 second for
the aluminum arm and 0.21 second for the composite arm. It is also
observed from the angular position response that the maximum overshoot of
the aluminum arm is 1.02 degree, while that of the composite arm is 0.3
degree. From the input torque response, it is also seen that the actual
input torque required for the composite arm during the commanded motion
is only 60 percent of the input torque required for the aluminum arm. The
tip deflection of the aluminum arm induced from the step motion was

decreased up to 50 percent by employing the composite arm. From these

108

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110

results, it may be summarized that with same feedback gains, the system
performances of the composite arm have been remarkably enhanced over
those of the aluminum arm, exhibiting the smaller delay time by 20
percent, the smaller maximum overshoot by 70 percent, the less energy
consumption by 40 percent, and the smaller tip deflection up to 50
percent. It is observed from the tip deflection response that the tip
deflection is not zero even after the angular position damped out to
reach its commanded position. This phenomenon arises from the increased
hysteretic joint friction due to the relatively high employed position
gain. Figure 4.12 shows the simulated step responses corresponding to
the measured results shown in Figure 4.11. It is observed from Figures
4.11 and 4.12 that two corresponding responses agree well except for the
first 0.2 second of the motion.

Next comparison of the control performances was undertaken with the
payload of 0.061 kg which is 46.2 percent and 101.6 percent of the mass

of the aluminum and composite arm respectively. The compensator zero Zoo-
-1 was chosen with the feedback gains of Kp-0.48 N-m for the aluminum
arm and Kp-0.24 N-m for the composite arm based on the same manner as

in the first case. Table 4.3 presents the closed-loop poles of the
system with these feedback gains, and from this table, it can be seen
that the closed-loop poles of the rigid-body mode has no imaginary part
for both arms, hence expecting the sluggish motion and no overshoot.
Figure 4.13 presents the measured step responses to a commanded
angular position of 7.2 degree for both arms with the payload of 0.061
kg. As expected, it took 3.5 second to reach its commanded position

without occurring overshoot. The required energy of the composite arm for

111

this movement was only 48 percent that of the aluminum arm, and tip
deflection of the aluminum arm was reduced by up to 60 percent by
employing the composite arm. With smaller energy consumption, the more
accurate end-position was obtained. It is expected that with the same
ratio of the payload to the arm mass, the better control performances of
the composite arm can be achieved with appropriate feedback gains. Figure
4.14 shows the simulated step responses corresponding to the measured
responses presented in Figure 4.13. Again the correlation between the
measured and simulated responses is impressive.

The employment of the Lyapunov method is also another useful
technique to determine the feedback gains which guarantee asymptotical

stability of the closed-loop system as discussed earlier. Herein, the

A A

following symmetric and positive definite matrix P and Q in the equation

(4.16) were chosen for the both aluminum and composite arms.

Table 4.3 Closed-Loop Poles with the Compensator Zero Zco- -1

 

 

 

Aluminum Arm (Kp - 0.48) Composite Arm (Kp - 0.24)
Real Imaginary Real Imaginary
-1.234 0 -1.679 0
-6.418 0 -2.578 0

-36.197 28.93 -20.125 44.14

-36.197 -28.93 —20.125 -44.14
-9.656 126.82 -S.l7l 166.96
-9.656 -126.82 -5.l7l —166.96

 

 

 

 

 

 

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0
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114

A

(4.18)

A

, and the output matrices C1

and C without the payload, the feedback gains of KP - 0.406 N-m, K‘,-

2

0.254 N-m-sec for the aluminum arm, and K1) =0.403 N-m, Kv = 0.252 N-m-sec

for the composite arm were obtained from the equation (4.17). Table 4.4

presents the closed-loop poles of the system with above feedback gains.

As expected, all closedeloop poles fOr both arms have negative real part

assuring the stability of the system.

Table 4.4 Closed-Loop Poles with Kp-0.406, Kv-0.254 for
the Aluminum Arm, and Kp- 0.403, Kv- 0.252 for

the Composite Arm

 

Aluminum Arm

Composite Arm

 

 

 

 

 

 

Real Imaginary Real Imaginary
-2.303 0 -l.843 0
-5.560 0 -13.430 0

-18.806 52.60 -17.268 58.17
-18.806 -52.60 -17.268 -58.17
-5.960 164.13 -4.937 222.64
~5.960 ~164.13 -4.937 -222.64

 

 

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117

Figure 4.15 presents the measured step responses to a commanded
angular position of 7.2 degree with above feedback gains. It is observed
form the angular position response that there is no overshoot and it
takes about 2.5 second to reach its commanded position for the both
aluminum and composite arms. From the input torque response, it is
distilled that the control input torque required for the composite arm is
approximately 60 percent of that of the aluminum arm. It is also seen
from the tip deflection response that the maximum tip deflection of the

aluminum arm was reduced up to 35 percent by employing the composite arm.

A

It should be noted that the different choice of the matrices P and £2
will produce different control performances of the system. Figure 4.16
shows the simulated step responses corresponding to the measured results
shown in Figure 4.15. The agreement between two results is again
excellent.

It is noted from these results that no controller/observer spillover
was occurred. This is attributed to the relatively high unmodelled
frequencies (truncation effect is very small), large inherent damping of
the system, and no state observer in the control system (no state

estimation error, hence fast servo-sampling time).

4.3.2 Step Responses with Disturbances

It is generally known that the closed-loop system associated with
the PD type of the output feedback controller is very sensitive to
uncertainties such as parameter variation, internal and external
disturbances. Herein, the investigation of this sensitiveness was

accomplished by considering external inputs, which have forcing

118

frequencies close to the cantilevered-mode natural frequencies of the
arm, as the disturbances. This attempt is made here to introduce a
single-link flexible manipulator fabricated from smart materials
incorporating electro-rheological fluids which will be discussed in the
subsequent chapter. The robustness of the PD controller to the
disturbances is to be expected with the help of the distributed
parameter actuator which is characterized by embedded electro-rheological
fluids in the structure.

With the introduced disturbance, the dynamic model given by the

equation (3.3) becomes

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x-K§+§~+i§5t
p pp p‘1 p” (4.19)
”-5;
yP PP
;+ K L~~~~~~ yp
p x- x+Bu+B d(t) 4»
— PPPP P
3'2
Pi
Kv-t

 

 

 

 

 

 

Figure 4.17 Block-Diagram of the PD Output Feedback Control
System with the Disturbance of d(t)

119

where d(t) is the imposed disturbance. Since the disturbance is applied

to the servo-drive, the disturbance influence matrix is same as the input
influence matrix F . Figure 4.17 presents a block-diagram of the closed-
loop system with the PD output feedback controller subjected to the

disturbance d(t). This distmubance is supplied from the function
generator, model 275, manufactured by Wavetek, to servo-drive.

Figure 4.18 shows the measured step responses of the aluminum arm
without the payload to a commanded step angular position of 7.2 degree

with the imposed disturbance of 0.08 sin flt N-m where f1= 3.25 Hz. The

feedback gains of Kp- 0.534 and Kv= 0.267 used for Figure 4.8 were

employed, and the disturbance was applied to the servo-drive from the

starting of the motion. It is noted that the forcing frequency f1=3.25 Hz
is very close to the clamped-mode fundamental natural frequency 01- 3.73

Hz in Table 3.5. As expected, the response of hub angular position and
tip deflection are oscillated resulting near resonance phenomenon of the
system. If the amplitude of the disturbance increase, the system response
will be grown up resulting in the instability of the system.

Figure 4.19 presents a comparison of step responses of the aluminum
arm with and without the disturbance. It is interesting to observe from
the angular position response that the reached angle after 2 second is
7.2 degree with the disturbance, while it is 7.1 degree without the
disturbance. The reason is that the disturbance helps to overcome joint
friction so that the arm can reach to the commanded position with the
tradeoff the induced oscillation.

Figure 4.20 shows a comparison of step responses between the

ANGULAR POSITION(deg)

TIP DEFLECTION(cm)

9i)

120

 

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Figure 4.18 Measured Step Responses of the Aluminum Arm

without Payload. Kp- 0.534, Kv- 0.267, and
d(t)-0.08 sin(3.25 Hz)t

ANGULAR POSITION(deg)

TIP DEFLECTION(cm)

Figure 4.19 Comparison of the Measured Step Responses of the

121

 

 

 

 

 

 

 

 

 

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with disturbance
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xp-o.53a, Kv-O.267 and d(t)-0.08 sin(3.25 Hz)t

 

122

 

 

 

simulated
rneasured

 

 

 

 

 

 

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Figure 4.20

 

 

 

TlME(sec)

Comparison of the Simulated and Measured Step
Responses of the Aluminum Arm with the Disturbance.

Kp-0.534, vao.267, and d(t)-0.08 sin(3.25 Hz)t

123
simulation and measurement with the imposed disturbance. The agreement
between two results is satisfactory except for the beginning the motion.
The reason of this mismatch at the beginning of the motion is that the
motion is delayed due to the applied disturbance which induces the motion
in the opposite direction to the commanded position.
Figure 4.21 presents the measured step responses of the composite

arm to a commanded angular position of 7.2 degree with feedback gains of
Kp-0.29l6 N-m, Kv= 0.1458 N-m-sec and the disturbance of d(t) = 0.08 sin

flt N-m where fl- 4.75 Hz, which is close to the fundamental clamped-

mode natural frequency of 01= 5.22 Hz of the composite arm. As in the

aluminum arm, near resonance phenomenon has been occurred even though the
small oscillation amplitude is observed from the angular position
response.

Figure 4.22 shows a comparison of the measured step responses of the
composite arm with and without the disturbance. Similar characteristics
to the aluminum arm are observed. Figure 4.23 presents a comparison of
the measured and simulated step response in the presence of the
disturbances. The agreement of two results is excellent except first
portion of the angular position response. Again, the delay of the
response arises from the imposed disturbance which hinders the motion in
opposite direction to the commanded step position.

Figure 4.24 shows the measured step responses of the composite arm

with the same feedback gains used for the Figure 4.21 and the disturbance

d(t)=0.4 sin f2t where f2 = 34.5 Hz, which is very close to the second

clamped-mode mode natural frequency 0 - 33.09 Hz of the composite arm.

2

ANGULAR POSITION(deg)

TIP DEFLECTION(cm)

124

 

9.0

 

 

TlME(sec)

 

 

 

-1.2 I l I r

 

 

TlME(sec)

Figure 4.21 Measured Step Responses of the Composite Arm
without Payload. Kp-O.292, Kv-O.l46 and

d(t)-0.08 sin(4.75 Hz)t

ANGULAR POSITION(deg)

TIP DEFLECTION(cm)

Figure 4.22

125

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

121)
with disturbance
" — without disturbance
81)—
4a0‘
.
000 PI I I I I I I T I l I I I I —‘I I I fl I
0 1 2 3 4
TlME(sec)
1.5
2 with disturbance
J — without disturbance
1 0—1
0.5—
—o.54 ’ ‘
1
—1.0 I I I I I I I I I I I I I I I I I I
O 1 2 3 4

TlME(sec)

Comparison of the Measured Step Responses of

the Composite Arm with and without the Disturbance.
xp-o.292, Kv-O.146, and d(t)-0.08 sin(4.75 Hz)t

ANGULAR POSITION(deg)

TIP DEFLECTlON(cm)

126

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12X)
simulated
‘ ——— rneasured
81)—
413—
J:
000 I I I i I I I I I r I fl I 1 I I I I
0 ‘l 2 3 4
TlME(sec)
1.5
‘ - simulated
1 ——- nneasured
1.0-3
1
.1
0.5-4
0.05
1
—o.5-3
4
1
—1.0 I 1 I fl 1 I I I I I I F fl I I i I fi I
0 1 2 3 4
TlME(sec)

Figure 4.23 Comparison of the Simulated and Measured Step

Responses of the Composite Arm with the Disturbance.
Kp-0.292, KV-O.l46, and 3(:)-o.03 sin(4.75 Hz):

ANGULAR POSITION(deg)

HP DEFLECflON(cn0

127

 

 

 

 

 

 

 

 

 

9.0
6.0—
3.0— /
0.0 I I I l I I I l I I l I I I I l I
O 1 2 3 4
TlME(sec)
1.2
0.6—
O 0.: .,‘nfl11$IISI$Lw‘wtvxlI-v III £1H I:1"113W111MWINUIHI1M1attiimihy WU”
—O.6-
_1.2 l l I I I I I I 1 I I I I I I I I I I
O 1 2 3 4
TlME(sec)

Figure 4.24 Measured Step Responses of the Composite Arm

without Payload. Kp- 0.292, Kv- 0.146 , and

E(:)-o.a sin(34.5 Hz):

128

As observed from the step responses of the closed-loop system in the
presence of the disturbance which has a forcing frequency close to the
clamped-mode natural frequency of the arm, the responses are very
sensitive to the imposed disturbance regardless of either the aluminum or
the composite arm. Even though significant advantages of the control
performances with the composite arm in the absence of disturbance were
accomplished, there is a limit to improve the overall control system
especially in the presence of uncertainties such as introduced
disturbances.

In order to circumvent this problem, a single-link flexible
manipulator fabricated from smart materials incorporating electro-
rheological fluids is proposed and investigated in the subsequent
chapter. Since the natural frequencies of the arm is controllable in real
time by employing an external voltage to the embedded electro-rheological
fluid domains, the robustness of the PD output feedback control system

is to be expected.

4.3.3 Instability of the System

As discussed in Section 4.2, the closed-loop system with the PD type
of the output feedback controller associated with two colocated angular
position and velocity sensors is theoretically stable for any choice of
feedback gains. However, this is not true in practice since the unmodeled
dynamics such as actuator and sensors, the neglected higher flexible
modes, the presence of time delay in the controller implementation, and
the limited inherent servo-characteristics of the closed-loop system may
cause the system instability. In this section, the instability of the

closed-loop system caused by the limited inherent servo-characteristics

129
will be discussed through the experimental investigation. The servo-drive
system employed in this experimental investigation has an in-position
band which defines the maximum allowable position from the commanded
angular position (in other words, the allowable positional accuracy) as

shown in Figure 4.25. In Figure 4.25, the T1 and T2 represent reacting

torques to force the actual position to the in-position band. The
magnitude of these torques depend on the employed feedback gains, and the
geometrical and material properties of the arm and the hub. The actual
position in general remains within the in-position band with the small

feedback gains (small T1 and T2 due the small gains, small spring and

inertia force). However, as the feedback gains increase, the reacting

Commanded Position

\ / In-Position Band

 

Hub

Figure 4.25 In-Position Band of the Servo-Drive System

130

torques T1 and T2 become larger in order to force the actual position to

the in—position band. Hence, with certain critical feedback gains, the
reacting torques continuously increase resulting in the system
instability. The experimental test was undertaken in a such a manner that
the position feedback gain was increased until the instability of the
system was occurred, while the velocity feedback gain was fixed. Since
the instability may cause serious injury to persons or damage to adjacent
hardwares of the system, the emergency switch has been activated after
just 4 seconds of the commanded motion.

Figure 4.26 presents the measured step responses of the aluminum arm
to the commanded angular position of 7.2 degree with feedback gains of

Kp- 6.2 N-m and Kv- 0.4 N-m-sec. With these feedback gains, the system

became unstable resulting in the growing responses as time goes on. It is
interesting to observe from the tip deflection response that the
magnitude of oscillation does not grow up noticeably until 2 second,
however after then, the response grows up rapidly. Figure 4.27 presents
the simulated step responses corresponding to the measured responses
given in Figure 4.26. As expected, the system is stable with these
feedback gains. These feedback gains may be defined as maximum allowable
feedback gains of the practical system featuring the aluminum arm. It is
noted that the maximum allowable feedback gains have different

combination of KP and Kv’ In other words, if the different velocity

gain is chosen, the different maximum position feedback gain, which
causes the instability of the closed-loop system, is determined.
From these results, it may be recognized that the controller design of

the system should be based on both analytical analysis and experimental

ANGULAR POSITION(deg)

TIP DEFLECTION(cm)

l3l

 

 

 

 

TlME(sec)

 

 

 

 

 

I I ' ' 1 I 'T ' ' ' I
O 1 2 3 4
TlME(sec)

Figure 4.26 Measured Step Responses of the Aluminum Arm
without Payload. Kp-6.2 and Kv-0.4

132

 

 

ANGULAR POSITION(deg)
CD
I

 

 

 

 

 

 

 

O I I ' I I I I ‘ T l I U I I l I l I I
O 1 2 3 4
TlME(sec)
3.0
A 1.5-1
E
8 .
z
9 .
B
m 004
d
w 4
o
E -1
'- —1.5—
_30 I T I I I I r I m I I I I I I I r I I
O 1 2 3
TlME(sec)

Figure 4.27 Simulated Step Responses of the Aluminum Arm
without Payload. Kp-6.2, Kv-O.4

 

133

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

12
q - ALUMINUM
—— COMPOSITE
”a.
m -1
3 .'
z . '
0 —~ w~fi~ ;. {a e As— +~ ~ A
E .
(f)
O
0.
Cl:
5
I)
(.9
Z
<
I I I f I I I I I l. I I I I
2 3 4
TlME(sec)
12
~ ALUMINUM
'-—- COMPOSWE
8..
E c1
8 . ..
z _ '.
2 4 =
'- ' :
o . ,. .
LIJ e'. .
_l . . .
u. ' 3
LLJ 4'
D . '. :
o. . -' : E
i z.
-4... o
l
—8 I I I I I I I I I I I I I I I I I I I
O 1 2 3 4

TlME(sec)

Figure 4.28 Measured Step Responses with Feedback Gains of Kp-6.2
and Kv-O.4 for both Aluminum and Composite Arms

134

 

ANGULAR POSITION(deg)
CD
1

 

 

TlME(sec)

 

 

 

 

 

 

TIP DEFLECTION(cm)
O
l

 

 

 

 

 

 

 

TlME(sec)

Figure 4.29 Measured Step Responses of the Composite Arm
without Payload. Kp- 11.8, Ky? 0.4

 

TIP DEFLECTION(cm)

135

 

 

ANGULAR POSITION(deg)
O)
J

 

 

 

 

 

 

 

 

O —I I I I —T n n I m I I I I I I I I I I
O 1 2 3 4
TlME(sec)
3.0
1.5-
0.0-I v
-1.5-
—30 I I I I T I I I I I I I 1 I I If I I I
O 1 2 3 4
TlME(sec)

Figure 4.30 Simulated Step Responses of the Composite Arm
without Payload. Kp- 11.8, Kv- 0.4

 

136
identification of the employed system. Figure 4.28 shows a comparison of
the measured step responses of the aluminum and composite arm with

feedback gains of Kp- 6.2 N-m and KV- 0.4 N-m-sec. With these feedback

gains, the composite arm exhibits stable closed-loop responses of the
system, hence shows significant improvement of the stability of the
employed practical system. It is recognized that the reduced spring and
inertial force of the composite arm is contributed to this result.

The position feedback gain with the same fixed velocity feedback

gain of Kv= 0.4 N-m-sec was increased until the instability of the

composite arm was occurred. Figure 4.29 shows the measured step
responses of the composite arm to a commanded angular position of 7.2

degree with the position gain of KP- 11.8 N-m. From the response of the

tip deflection, it is also observed that the response quickly grows up
after 2 second as the case of the aluminum arm. Figure 4.30 presents the
simulated step responses corresponding to the measured responses shown in
Figure 4.29. The closed-loop system is stable with these feedback gains
as expected. From this experimental investigation, it may be summarized
that the stability margin of this employed particular robotic system
featuring the aluminum arm can be enhanced up to 45 percent by employing
the composite arm. This is also another significant advantage of the

composite arm over the aluminum arm.

4.4 Sumary of the Chapter

The output feedback controller associated with two colocated sensors
was designed utilizing the root-locus technique and the Lyapunov method,

and experimentally implemented for the aluminum and the composite arm.

137
The comparative experimental investigation was undertaken to evaluate the
characteristics of the control performances of the aluminum and composite
arm. It has been demonstrated from the analytical and experimental
investigation that the control performances of the overall closed-loop
system for the both aluminum and composite arms were satisfactory, and
the composite arm exhibited some superior control performances relative
to the aluminum arm such as faster settling-time, smaller input torque,
smaller tip deflection (better positional accuracy), auui superior
increment of the stability of the practical system. These experimental
results clearly indicate the significant payoffs associated with
employing the composite materials featuring superior strength- and
stiffness-to weight ratios to the commercial metals on the flexible
manipulators. It has been also demonstrated that the aluminum and
composite arm exhibited the inevitable sensitivity to the imposed.
disturbance which has a forcing frequency close to the cantilevered-mode
natural frequency of the arms. This attempt was undertaken here in order
to introduce a single-link flexible manipulator fabricated from smart
materials incorporating electro-rheological fluids for the purpose of
enhancing the robustness of the output feedback controller to the
disturbance. Subsequent chapter addresses this new class of flexible

manipulator fabricated from smart materials.

CHAPTERV

HYBRID CONTROL FEATURING A DISTRIBUTED
ELECTRO-RHEOIDGICAL FLUID ACTUATOR

5 . 1 Introduction

As observed in the previous chapter, the closed-loop system
associated with PD output feedback controller is very sensitive to the
disturbance, which has a forcing frequency close to the clamped-mode
natural frequency of the arm, regardless of the aluminum or the composite
arm. In order to circumvent this problem, a single-link flexible
manipulator fabricated from smart materials incorporating electro-
rheological (ER) fluids (smart ER arm in short) will be investigated by
proposing a hybrid control methodology. The hybrid controller consists of
the output feedback controller employed for the aluminum and composite
arm in the previous chapter, and a pseudo-state feedback controller
characterized by a distributed ER fluid actuator [40]. A crucial
ingredient in the development of an appropriate control algorithm for
this class of smart materials is the development of an appropriate system
(plant) model. Since the phenomenological behavior associated with the
interaction of the structure and the embedded ER fluid is fairly complex,
simplifying assumptions have been incorporated in the modeling process in
order to furnish equations of motion which are analogous in mathematical
structure to those encountered in beams fabricated from homogeneous

monolithic materials. A phenomenological equation of motion, based on

138

139
these simplifying assumptions and experimental observations, admits a
pseudo-state feedback controller to be implemented. The pseudo-control
force (external input voltage) associated with the pseudo—state feedback
controller accounts for the change in the global damping and stiffness
properties of the structure due to the electrical potential applied to ER
fluid domains.

Since the natural frequencies and damping ratios of the flexible
smart structures are tunable on real time by employing the external input
voltage to the embedded ER fluid domain [41,56], the resonance phenomenon
due to the imposed disturbance can be avoided by applying the pseudo-
control forces. Hence, the robustness of the employed PD type output
feedback controller is to be expected. In this chapter, a brief
background on the ER fluids will be introduced prior to developing a
phenomenological equation of motion of the proposed robotic system. Then
a hybrid controller will be formulated based on the empirical model of
the distributed ER fluid actuator. Finally, an experimental
implementation of the proposed hybrid controller will be presented by
investigating step responses of the system subjected to no disturbances

and disturbances .

5 . 2 Background on Electra-Rheological Fluids

Electra-Rheological (ER) fluids are typically suspensions of micron-
sized hydrophilic particles suspended in the suitable hydrophobic carrier
liquids, which undergo significant instantaneous reversible changes in
material characteristics when subjected to electrostatic potentials.
References [20,41,55,56,57,l36,137,138,140,151,160] provide a flavor of

the research activities in this field. The most significant change in the

140
material characteristics of ER fluids is associated with the bulk
viscosity of the suspension, which varies dramatically upon applying an
electrical field to the fluid domain. This dramatic reversible change
from a liquid-like state, as shown in Figure 5.1 (a) to a solid-like
state as shown in Figure 5.1 (b) by the imposition of an appropriate
electrical potential can be usefully exploited in vibration-control

applications.

 

I
, . _ ’ w: _
a. ,» . ‘ ., ./ I. '. .

(a) liquid state (b) solid state

 

Figure 5.1 Electra-Rheological Fluid in Liquid and Solid State

Figure 5.2 presents a mechanism showing that how this reversible
change works in terms of interaction between charges placed on electrodes

and those in the particles in the ER fluid. The charges in the fluid may

141

electrode

ER particle

electrode

 

(a) without voltage (b) with voltage

Figure 5.2 Interaction between Charges on Electrodes and Those
in Electra-Rheological Fluid

be either positive or negative, and free to move in the material in the
absence of charge on the electrodes as shown in Figure 5.2 (a). When a
voltage is applied to the electrodes, the charges in the fluid form a
chain-like formulation in milliseconds as shown in Figure 5.2 (b).

Figure 5.3 presents photomicrographs of an ER fluid subjected to
electrical field intensities of 0 kV/mm and 2 kV/mm respectively. The
photomicrographs were taken in the Biothermal Sciences Laboratory at
Michigan State University using a Zeiss universal phase-contrast
microscope with X40 magnification and a Chinon camera. The black regions
at the top and bottom of the photographs are images of the electrodes
employed to generate the electrical field in the ER fluid. Figure 5.3 (a)
clearly illustrates the random structure of the suspension which imparts

nominally isotropic global mechanical properties to the fluid mixture

142
when a potential difference is not generated between the electrodes.
Figure 5.3 (b) clearly shows the truly dramatic change in the structure
of the suspension upon developing a potential difference between the
electrodes of magnitude 2 kV/mm. Under this condition, the particles in
the suspension orientate themselves in relatively regular chain-like
patterns to form a mixture with globally anisotropic mechanical
properties. These columnar structures of particles restrict the fluid
motion, thereby increasing the energy-dissipation or viscous
characteristics of the suspension, in addition to increasing the

stiffness characteristics of the fluid mixture. Thus, by imposing an

 

(a) 0 kV/mm (b) 2 kV/mm

Figure 5.3 Photomicrograph of Electro-Rheological Fluid

143

electric field upon an ER fluid, the stiffness and energy dissipation, or
damping, characteristics of the electro-viscous suspension are changed.
When the field returns to a state of zero potential, upon switching off
the electrical energy supplied to the electrodes, the particles return to
a state of random orientation in the carrier fluid as presented in Figure
5.3 (a).

The voltages required to activate the phase-change in ER fluids are

typically in the order of 4 kV per millimeter of fluid thickness, but

since current densities are in the order 10 pA/cm2 or less, the total
power required to trigger this phenomenon is quite low. Furthermore, the
response of these ER fluids to the excitation voltage is almost
instantaneous since the fluids can respond to voltage pulses with
frequency up to approximately 11 kHz, which is, of course far beyond the
requirements of most practical devices. ER fluid suspensions typically
have viscosities of the order of 50 cP in the uncharged state, and some
fluids are able to withstand a nonvanishing yield stress up to 60 kPa and
shear rates up to 400/3. When an ER fluid suspension is stressed below
this threshold, it behaves as a solid, but as the stresses are increased,
then fluid flow occurs but the yield stress remains nominally constant.
Since the characteristic aspects of ER fluids have been recognized
by Winslow [160] in 1949 for the first time, numerous applications of ER
fluids in engineering practice have been and being proposed and
developed. Possible ER fluids devices and mechanisms proposed and
developed so far are well documented in [128]. Typical applications
include ER actuators, ER active damper systems, ER clutches, ER hydraulic

valves, ER brakes, ER suspension systems, ER shock absorbers, rotary ER

144
viscometers, tunable ER isolators, and wide-band-high power ER

vibrators.

5.3 Phenomenological Equation of Motion

Phenomenologically, the global finite element equation governing the
motion of the proposed smart ER robotic system has the following

functional form [40,41,56,58]
IMIII‘II + [C(V(t))]n‘n + [K(V(t))]{U} - {Q(t)l (5.1)
where the global mass,damping and stiffness matrices are denoted by [M],

[C(V(t))] and [K(V(t))] respectively. Clearly the stiffness and damping

A

matrices are functions of the external voltage V(t) applied to the ER

fluid domains within the robotic arm. It must be emphasized that the

damping matrix [C(Il(t))] in the equation (5.1) is, in general, not
proportional to theistiffness and/or mass matrix. The nodal displacement
is represented by the column vector {U} and the generalized force is
denoted by the column vector {Q(t)}. Without loss of generality, the

equation (5.1) may be reformulated as follows:
[Mllfil + ([C]+[AC(V(t))]){fJ} + ([K]+[AK(V(t))l)lUl - {Q(t)l (5.2)
where [K] and [C] are the global stiffness and damping matrices

associated with zero electric intensity, and [AK(V(t))] and [AC(V(t))]

are the changed global stiffness and damping matrices due to the change

A

in voltage from zero volts to V(t).

145
Consider now the modal analysis as did in Section 2.4, and in
addition, assume that the ER fluid domains within the arm are not

subjected to an external voltage. Then the equation (5.2) becomes

{mm} + [CIII'II + [K]{U} = {Q(t)} (5.3)
Now, by introducing modal coordinates {I7} and the modal matrix [<I>] , i.e. ,

{Ul-[‘P]lnl (5.4)
the equation (5.3) becomes

{II} + IZIIéI + [mm = IBIS (5.5)

2 2

where [0] (- diag (0,w1, ...,wn)) is the modal frequency matrix with the
pinned-mode natural frequencies wi, [Z] ( - [<I>TC <1>]) is the transformed
modal damping matrix, [B](- % [1 d¢1(o)/dx , . . . . , d¢n(o)/dx]T)

T

is the input influence matrix with the modal slope coefficient d¢i(o)/dx,

and ii is the input torque. The transformed modal damping matrix [2] is
coupled, since the damping matrix [C] is not assumed to be proportional
to the stiffness and/or mass matrix. This coupling manifests itself in
the non-zero values of the off-diagonal terms of the matrix [2]. In order
to investigate the magnitude and hence significance of these terms, an
experimental program was undertaken by considering a smart beam structure
which has a similar modal characteristics of the proposed smart robotic
arm structure. Figure 5.4 presents the schematic diagram of the employed
smart cantilevered beam whose fabrication procedure is presented in

details in Section 5.5.

146
Figure 5.5 presents the measured first and second mode shapes
(magnitude and phase) of the proposed cantilevered beam. In Figure 5.5,
the symbols and lines are the measured values and the curve-fitted
interpolation respectively, and the magnitude of mode shapes is
normalized as well as the beam length. These experimental values were
obtained from the frequency response data utilizing the procedures

described in Section 5.5. It is clearly evident from the phase of the

mode shapes that the phases are all near 00 or 1800 at both 0 kV and 3
kV. Thus, the mode shapes are close to real [52], except in the region

close to the nodal points where the phase transition occur. This is

A
/ 223.5 m

T / ./“’"““

 

 

 

 

 

 

190 mm 1/ 27.5

i L.

 

Silicone Rubber Aluminum ER Fluid

1.36 mm

 

Section A—A

Figure 5.4 Smart Cantilevered Beam Structure

rnognfiude ofrnode shope

phose ofrnode shape (degree)

147

 

 

00 0 WV
" 0.- 3.0 kv ,'

 

 

 

I
1 O- ”‘ " '7
I \ I
I \
z ‘ I
I ‘\ I
I I
.. i ‘- I
I \ "‘
mode 2 I a I ‘ o ’
\ I
‘ /

l \ ’
I \ ”- I
'1 / V l
.4
I ’z \ I
l ' ‘ ’
A I I’ \ ’
(IS I , ‘ " r
, ,I . \ 1
I ” I I
.. I 1’ - I
I I I "
/ ,’ I
I O I
I I
0 4—— , I’ ’
, I I O I -_‘
I I I
I z’ I
// ” l
- 7 I
I . .. I -
I
I I
I ‘ I
O 2— I, mode 1 I ,’ —
i \
-". \ l
_ ,- ‘I
’d’ d
I

 

 

 

OJ) CI2 0J4 C16 (18 1.0

 

27o.-..'...m.. '
0,00k
0,-3.0kv -

 

 

 

 

 

 

 

 

Figure 5.5 Mode Shapes of the Smart Cantilevered Beam

1148
particularly true for the employed cantilevered beam structure or the
proposed smart robotic arm structure, since the distribution of all the
eigenvalues of these structures are well separated. However, this fact
may not be true for the complicated structures such as truss-like
structures which have clustered eigenvalues. It is, therefore, reasonable
to assume that each mode is uncoupled from the others. Hence, the off-
diagonal terms in the transformed modal damping matrix [Z] can be
ignored without causing serious errors.

Assuming that the mode shapes are real, the coherence factor
introduced in [75] is adopted here to investigate the viability of
employing the modal matrix in the absence of externally-imposed voltage
on the ER fluid domains, and also the situation where a non-zero voltage

is imposed on these domains. The coherence factor is defined by

({¢§}T{¢§}>2

ID

 

coh (5.6)

i3 ({¢§}T{¢§})- ({¢§}T{¢§})

where coherence of 1 means two modes of i and j are identical, while

coherence of 0 represents an orthogonal set of two modes. In equation

(5.6), <I>c and <I>C represent the ith and jth cantilevered-mode modal

i .1
vector respectively. Table 5.1 presents the experimentally-determined
coherence factors for the first two flexible modes of the cantilevered
beam structure investigated herein at four discrete state of voltage.
Upon reviewing these data it is clearly evident that there is an
insignificant difference between the coherence factor in the 0 kV state
and the other states. Clearly, therefore the mode shapes are identical

for the range of voltages considered in this study.

149

 

 

Table 5.1 Coherence Factor of the First and Second Mode at
Different Voltage
Mode 0 kV 1 kV 2 kV 3 kV
First 1 0.993 0.993 0.993
Second 1 0.987 0.984 0.983

 

 

 

 

 

 

 

Ihmethese experimental observations, the modified equations of
motion for the proposed robotic system in the presence of an electrical

field intensity can be expressed as
{n} + [c1{a} + [K]{n} - [BIG (5.7)
where
A T A .
[Cl-[ZIH‘I’] [AC(V(t))][‘P]. [Z]=dlag(0. 2§1w1"”' 2§nwn)
A T A
[Kl-[OHM] [AK(V(t))][<I>]
Furthermore, from the experimental results presented in Table 5.1

[¢]T[A6(V(t))][¢]-[Z(§(t))]

T A A (5.8)
[é] [AK(V(t))][<I>]=[0(V(t))]

where [2(V(c))1 (- diag (o, 2c1<v<t))w1<V(t))..... 2cn<V(t))wn(v<c>)) is

A

the changed modal damping matrix due to the electric field and [0(V(t))]
2 2
( - diag (0, w1(V(t)), ., wn(V(t))) is the changed undamped natural

frequency matrix; namely

150

§.(V(t))w.(V(t)) - (§.w-)A - (§.w.)A

1 1 1 1 V(t)¢0 1 1 V(t)=0

2 " 2 2 (5.9)
wi(V(t)) = (wi)x -(w.)x

V(t)#0 1 V(t)=0

Equation (5.7) can be expressed in a state-space representation as

follows.

= A; + K(V(t))§ + ES

X10

(5.10)

~

y =6;
where state variable x, plant matrix K in the absence of an external

~

inaltage, input matrix 8, and the output matrix C is same as that defined

in equation (2.33), respectively. And the changed plant matrix X(V(t))

due to the electrical potentials is given by

 

O 0 0 0 o o o o O
o o o o . - - o o
0 O 0 O o o o O 0
K(V(c))- 0 0 '121 '811 ° ' ° 0 O (5.11)
0 O O O o o o 0 O
_0 0 0 0 . - . -a2n 181“;

 

where

2" 2"
821- w1(V(t)). azn= wn(V(t))

311' 2:1(V<t>>w1(v<c>). aln- 2cn<V(t))wn(v<c>>

It is noted that only flexible modes are voltage dependent.

151

S.h Controller Formulation

As did in Section 2.4, the equation (5.10) can be decomposed into
two subdynamic equations by considering the rigid—body mode and the first
few critical flexible modes as primary modes and remaining modes as

residual modes. The two subdynamic equations are

§=K§ +XVt§+Eu
P P P P( ( )) P P

~ ~ ~ (5.12)
= C x

yP P P

and

XI. = rxr + A (V(t))x + Bru

~ ~ ~ (5.13)

yr = Crxr

where subscript (p) denotes the primary modes and (r) represents the
residual modes.
Before proceeding further, consider the stability problem of the

system (5.12). Suppose the information about the changed plant matrix

A

KP(V(t)), which will be later termed as a pseudo-control force after

reformulating it based on experimental observations, is not.awailable at

A

this moment. Then, the matrix KP(V(t)) can be considered as a time-

A

varying linear perturbation matrix due to the voltage V(t) with respect
to the system in the absence of electrical potentials, that is, nominal
system. Then following theorem for the stability can be established based

on the Lyapunov function approach.

152
Round: 5.1 : In the following, the vector norm is taken to be

Euclidean and the matrix norm is the corresponding induced norm, hence

1 2
”A” - “max (AT A)) / where A (.) represents the operation of

max(min)

taking the maximum(minimum) eigenvalue.

Lem 5.1 : Let Q be a symmetric and positive definite matrix. Then,

.. 2 ~T .. ~ 2
Amin(Q)||xp|| s xp Q xp 5. Amax(Q)| prII (5.14)

for any 32p (see [35] for the proof).

Theorem 5.1 : Suppose the system (5.12) in the absence of electrical

potentials is stable with a linear output feedback controller 5. Then,

the system is stable if

A (Q)

.. A 1
"Ap(v(t))ll < 21m n(P)

max

(5.15)

where P is a symmetric and positive definite matrix which is the solution

of the Lyapunov equation

~ ~T
PAc+AcP+Q-O (5.16)

where Q is a given positive definite matrix and
K -— K + E i2 ‘6 (5.17)
C P P P

where I? is output feedback gains associated with the controller 6.

 

 

 

153
Proof : With the equation (5.17), the state equation in (5.12) can be

rewritten as

N10

=3§+x Gt; 518
p Cp ¢<>>p < >

Now, choose Lyapunov function candidate as

~ ~T ~
V x — x P x 5.19
( p) p p ( )

It is noted that V(xp)> 0 for all xp# 0 and V(xp) 4 w as IprII 4 m. Now,

9(xp) < 0 is required to guarantee the stability of the system (5.12).

w;)=§p; +flp;
P P P P
~T ~T ~ ~ ~T~TA ~
— x A P + P A x + x A V t P x
p(C p p p ¢<)) p
+flpx(&a);
P P P
~T ~ ~T~TA ~
= - x x + 2 x A V t P x 5.20
p Q p p p( ( )) p ( )
Hence,
a; <01£
( p)
~T ~ ~T ~T A ~
x x > 2 x A V t P x 5.21
p Q p p p( ( )) p ( )
Therefore, from the Lemma 5.1 and also
~T30%))?” IIKdMDIIHPIIHWI2 (5m)
x t x S t x .
P P P P P

it follows that the inequality (5.21) is satisfied if

 

 

15h

A .(Q)

||Kp(V(t))|| < 2Am1“(P>

max

Note that IIPII = AmaX(P).

Physically, this theorem states that if the bound of the unknown

A

perturbed plant matrix KP(V(t)) due to the electrical potentials remains

within the certain threshold value characterizing the closed-loop system:
in the absence of the voltage, then the system (5.12) is stable. However,
this theorem can be modified once the perturbed plant matrix is defined
as a distributed ER fluid actuator, and the characteristics of the ER
fluid actuator are identified through the experimental observation.
Later'cn1 in this section, a modified theorem will be established for the
stability of the system featuring the proposed hybrid controller which
consists of the output feedback controller employed in the previous
chapter and a pseudo-state feedback controller associated with the ER
fluid actuator.

The philosophy of developing the proposed hybrid control methodology
for the system given by the equation (5.12) isrmnflwated by the
complementary experimental program which is focussed on the development
of an empirical actuator dynamics associated with the uniformly

distributed ER fluid actuator. The empirical actuator dynamics should be

A

expressed as an explicit function of the applied voltage V(t) in order to
formulate an appropriate control form. This can be accomplished by
establishing the relationship between the stiffness and energy

dissipation change and the applied electric field through the

 

 

155
experimental test. In order to obtain this relationship, frequency
response was measured at various voltage levels.

Figure 5.6 presents the measured frequency response of the smart ER
arm without payload at two different discrete voltage levels. This
frequency response was obtained by applying random torques (white noise)
input from the dynamic analyzer to the servo-drive, and measuring output
signals from the strain gage mounted at the root of the smart ER arm. In
Figure 5.6, the peaks correspond to the clamped-mode natural frequency
having the maximum strain energy at the root of the arm, and the valleys
represent the pinned-mode natural frequency having the minimum strain
energy. It is noted that this phenomenon is opposite to the case shown in
Figure 3.15. It is clearly evident from Figure 5.6 that the natural
frequencies of the arm, and also the arm's energy dissipation

characteristics are both voltage dependent.

 

 

‘ '“ 0.0 kV
‘ '—- 5£)kV

 

 

 

MAGNITUDE(dB)

 

 

 

 

FREQUENCY(Hz)

Figure 5.6 Measured Frequency Response of the Smart ER Arm
without Payload

 

156
The increment of the natural frequency and the damping ratio of each
mode at prescribed voltages relative to the no voltage state was obtained
from Figure 5.6. And the distilled results suggest that the empirical

actuator dynamics is governed by the following linear relationships of

 

 

the form
ci<V<t)>wi(V<c)> .
w - aliv(t)
‘1 i
2 A (5.23)
2 = a21V(t)
(.0.
1
where 0:11 and (121 are experimentally determined constants which

characterize the increment of the damping ratio and natural frequency of

A

each mode as an explicit function of applied voltage V(t). Figure 5.7
presents this governing relationship for the first two flexible modes. In
Figure 5.7, symbols are the measured values and the lines are the least-
square error curves. From the slope of these curves,a

-0.418, a -0.656,

11 12

a21 - 0.303 and a22- 0.666 were distilled. It should be noted that the

governing relationship described by the equation (5.23) may be modified
once a viable microstructural-level model of the system is developed.
This microstructural-level model would include the different chemical
composition of the ER fluids, the interface conditions between.t1u3 solid
and fluid layers, the conductivity of the face materials (electrodes),
and also the dielectric properties of the insulation between the
electrodes. The integration of all this information may provide a viable
model for the distributed "ER fluid actuator dynamics, which is a function

of the electric field intensity. And also, the precise investigation on

 

 

157

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.8
0 FIRST MODE
1 A SECOND MODE A.-
N... 1.2-~ .4!
S
3‘ _
(E .
NéH .
O.6~ 5A1
0
.1 .A
" o
0.0 . I I l I I l l
0.0 1.0 2.0 3.0
ELECTRIC FIELD (RV/mm)
1.8
0 FIRST MODE
1 A SECOND MODE A,
.3: 1.2—4
k
<2:
9:4 .1
< E 0.6“ .‘A.
u .1 9.,
000 I I r I I I l I
0.0 1.0 2.0 3.0
ELECTRIC FIELD (kV/mm)
Figure 5.7 Governing Characteristics of the Employed Electro-

Rheological Fluid Actuator

 

158
the ER effect to the different excitation amplitude will improve the
accuracy of the dynamics.
Now with the equation (5.23), the system equation (5.12) may be

reformulated as follows.

 

 

§ - K § + E u + E V(t)
p p p p p
~ ~ ~ (5.24)
y - C x
p p p
where
r0 0 0 o - . . o o ‘
o 0 o o - . - o o
O O O O o o o 0 0
E _ o o -k21 -k11 - - - o o (5.25)
p 0 O O O O O O O O
0 O O 0 o o o O o
_0 o o o - - . -k2p ~k1p‘
k11‘ 2"‘11§1“’1' klp- 2°1p§pwp
2 2
k21' “zlwl' kzp‘ a2pwp

V(t) - § V(t)
P
Clearly the equation (5.24) admits a hybrid control featuring the

output feedback control {1 and the form of state feedback control V(t) .
Even though external input voltage is imposed to an ER fluid domain in a
fashion of the open-loop control, the distributed ER fluid actuator
possesses a characteristic aspect showing the pseudo-state feedback
controller which produces the associated pseudo-control force in a
fashion of the closed-loop control. In fact, this pseudo-control force
accounts for the increased stiffness and damping properties of the

structure due to the imposed external voltage. Figure 5.8 presents a

 

159

 

 

 

71-
H
{>1
X!
+
U)
C
+
7:
<
A
f?
V

 

 

 

 

 

 

 

<?
A
('T
V
XI
'0

 

 

Y2 Y1

 

 

Gm ~

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.8 Block-Diagram of the Hybrid Control System

block-diagram of the proposed hybrid control system. In Figure 5.8, 5,1

represents angular position output, 3'72 denotes angular velocity output,

A

it stands for the commanded angular position and V(t) denotes applied

external input voltage. Hence, the closed poles Ad of the system is
obtained from the following characteristics equation

det(Ad - Adl) - O (5.26)

where A-K+§RC+EVt
d (p p 9 pH)

7“

-[-1<p -Kv]

Now, as mentioned earlier in this section, the Theorem 5.1 can be

modified for the proposed hybrid control system as followings.

 

 

160

Theorem 5.2 : If the position gain Kp and the velocity gain Kv are

chosen such that the closed-loop system (5.24) is asymptotically stable

in the absence of electrical potentials, the closed-loop system is

A

asymptotically stable for any positive input voltage V(t) which does not
exceed the limit considered in this experimental investigation (see

Figure 5.7).

Sketch of the Proof : Let X be the eigenvalues of the original plant

matrix KP in equation (5.24). Then, Re(Xi) s O for all i. The equality

holds for the rigid-body mode. Since experimental observations indicate
that the natural frequency (pinned-mode and clamped-mode) and damping

ratio of each mode increase as the input voltage increases [41,56] , the
eigenvalues ; of the matrix (Kp+ Rp§(t)) becomes Re(;i) - Re(xi)+1i(t) s
0 for all i, and 71(t) s 0. The variable 11(t) may be defined as the
degree of the stability and it is absolutely dependent upon the ER
actuator characteristics which are given by the matrix RP. Hence, it cart

be shown that with the feedback gains of Kp and Kv the closed-loop system

A

(5.24) is asymptotically stable for any positive input voltage V(t). It
is noted that since the rigid-body mode is not affected by the input

voltage, the corresponding 71(t)=0.

Therefore, once the output feedback gains of Kpand Kv are determined

in a same manner mentioned in Section 4.2, the external input voltage

161

V(t) can be chosen based on the degree of the stability, i.e., 11(t).
However, the choice of the 71(t) is limited by the maximum applicable

electric field which can be imposed on the smart ER arm without causing
any insulation problems. The input voltage can be constant during whole
commanded motion or can be bang-off-bang type as shown in Figure 5.9.
This bang-off-bang type voltage can be easily implemented in practice
since the response time of the ER fluids is typically in the order of
milliseconds, and a direct current (DC) pulse square input voltage can be
readily imposed on the ER fluid domains of the arm [140]. With this type

of the input voltage, the system given by the equation (5.24) becomes

INPUT VOLTAGE

 

 

Figure 5.9 Bang-off-Bang Type Input Voltage

162

 

4 forOStSr1
"-632
ryP PP
. . (5.27)
:2 -"SE +Bu+KV(t)
P PP P pm

1 forr StSr
~ ~~ -l m

-=Cx
ryP PP

 

Clearly the closed-loop poles of the system during the motion vary

depending upon a certain period of Ti. Hence, the system can respond at

any time to the desired requirements of the system performance by simply
changing the input voltage. This is one of the salient features of the
proposed hybrid control methodology. The robustness of this hybrid
control system to the disturbance which has a forcing frequency close to
the clamped-mode natural frequency of the robot arm is also to be

expected. This will be investigated in the subsequent section.

5.5 Experimental Implementation
5.5.1 Apparatus and Instrumentation

Most of experimental apparatus and procedures for the implementation
of the hybrid controller are same as those described in Chapter III.
Hence, only experimental procedures for the measurement of frequency
response of the cantilevered beam shown in Figure 5.5, the application

procedure of the external input voltage (pseudo-control force) and the

163
fabrication procedure of the smart ER arm will be presented in this

section.

Figure 5.10 presents a schematic diagram of the experimental
cantilevered beam shown in Figure 5.4 and the associated instrumentation

for nunmitoring and recording the frequency response. The random external

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

HP 6255A Kaman P3410
DC Power Supply Power Supply
V
Standard Energy HP 3456A Kaman
Model SC 30 Digital KB 2400
DC-DC Converter J Multimeter Proximeter
Displacement
Probe
Displacement
Triggering
Mechanism
V
——>@ ER Beam l
I I B 8 K 8200
Ling D.S. Force Transducer
V411
Shaker I
» B & K 2635
,7" Charge Amplifier
HP 35660A
Hafler P500 5‘
Dynamic
Signal Amplifier

 

 

 

 

Signal Analyzer

 

 

 

 

Results

Figure 5.10 Experimental Set-Up for the Frequency Response
Measurement of the Smart Cantilevered Beam

164

forcing function (white noise) was applied from a dynamic signal
analyzer, model 35660A, manufactured by Hewlett-Packard, to the
structure. This forcing signal was passed through an amplifier, model
P500, manufactured by Hafler and then employed to energize a shaker,
model V411, manufactured by Ling Dynamic Systems Limited. The amplitude
of this excitation was kept below a certain level to reduce the influence
of nonlinearity of the ER fluids and other parasitic effects such as air
damping, cable drag,etc. A force transducer, type 8100, manufactured by
Bruel and Kjaer was mounted on the head of the shaker in order to
determine the input force to the beam, whereas the displacement probe was
positioned at several locations along the beam in order to measure the
output signal which was monitored by a non-contacting displacement
transducer, model KD2400, manufactured by Kaman Corporation. Then these
two signals were processed by the signal analyzer to obtain the frequency
response (magnitude and phase) corresponding to each location.

In order to reduce the influence of the noise and get more consistent
results ,eulaverage of 40 times measurements was used. Figure 5.11
presents one of the frequency responses of the cantilevered beam at two
discrete voltage levels using the displacement probe mounted on the tip
of the beam. The phase-change in the ER fluid was initiated by energizing
the upper and lower faces of the beam specimen. The various voltages were
generated by a Hewlett-Packard dc power supply, model 6255A, and
amplified through a dc-to-dc converter, model SC30, manufactured by
Standard Energy. The desired input voltage was measured using a dc high
voltage probe, model 3411A and digital multimeter, model 3566A which are

both manufactured by Hewlett-Packard.

165

 

   

 

 

 

40
:18 20.5 Hz _ 0 RV
--"‘ 3.42 kV
1
10
dB -
Q/div _J
3
E _
E 176.0 52
L- ..........................
-4O 1 1 1 1 1 1 1 l I
Start: 15 Hz Stop: 215 HZ

FREQUENCY (Hz) RMS‘ 4°

Figure 5.11 Frequency Response of the Smart Cantilevered Beam

The detail-design features of the smart ER arm are schematically
presented in Figure 5.12, and the salient features of the fabrication
process for the smart robot arm incorporating the ER fluid are
highlighted in the flow diagram presented in Figure 5.13. RTV silicone
rubber adhesive was selected as an insulator in order to provide both
excellent bonding between the electrodes and to also furnish a high
dielectric constant. The arm was fabricated by employing an appropriate
die to generate the hollow section and also to control both the thickness
of the ER fluid layer and the insulator . The die was subsequently
removed after partial curing of the sealant, and the arm was maintained
at room temperature for 24 hours in order to achieve the full strength of

the silicone rubber adhesive. During this time an air blower,

166
manufactured by Air Control Installations, Ltd. , U.K. , was employed to
remove the acid from the silicone rubber by exposing the arm to an air

stream for two hours. After curing at room temperature, the arm

subsequently cured at 65°C for 2 hours in a hygrothermal chamber, model
TH27S, manufactured by BMA Inc. The arm was then exposed to room
temperature for 24 hours before checking the dielectric behavior. After
checking the dielectricity of the arm, an acrylic sheet was employed at
one end where it needs to be inserted into the hub. The hollow arm, which

featured a cavity of uniform rectangular cross-section, was filled with

A
__ / 72.4 cm

__ ll ——— .//——O'5 cm

 

 

 

 

 

 

 

 

. 70 cm 1/ l ,l.4_cm|

Silicone Rubber Aluminum ER Fluid

2,2um1

 

Section A-A

Figure 5.12 Schematic Diagram of the Smart ER Arm

167

I Preparation of Structural Components

6

Bonding

I

l ItfitialPartialCuring l

 

 

 

 

 

 

 

- 20 minutes at 245°C

Runovfng Die
6 4
Air Drying
Intermediate :aItial Curing
- 24 hours at 245°C
- 2 hours at 65°C
- 24 hours at 245°C

 

 

 

 

 

 

 

 

 

 

 

  

Not
Acceptable

  
 

Check
Dielectric

 

 

  

 

Filling Beam with ER Fluid
and Scaling of Entry Port

 

 

 

- Final Curing ’
- 30 hours at 245°C

Figure 5.13 Flow Diagram for Fabrication Procedures of the
Smart ER Arm

 

 

“.~.
\A‘
, §
. I“. n ~

Figure 5.14 Photograph of the Smart ER Arm

the silicone-oil-based ER fluid. The ER fluid volume fraction to the
total volume was 36 percent. Finally a Measurements Group strain gages,
type WD-DY-ZSOBG-350 were mounted at the root of the arm. It is noted
that the fabrication procedure of the cantilevered beam shown in Figure
5.4 is same as that for the smart ER arm. Figure 5.14 presents a
photograph of the smart ER arm clamped on the rigid hub.

From the measured frequency response shown in Figure 5.6, system
model parameters such as natural frequencies and damping ratios were

identified. The modal slope coefficient d¢i(o)/dx was determined as
followings. The open-loop transfer function G°(s) in the equation (4.2)

is also given by [127]

169

l0

2
1 p (s /0. + 2§is/Oi + 1)

 

NH

2 (5.28)
I 5 i=1 (5 /w. + ZCiS/wi + l)

H
1"

where p is the number of primary mode, 0i is the ith clamped-mode natural
frequency and wi is the ith pinned-mode natural frequency. (hi the other

hand, the 60(3) in equation (4.2) can be expressed as

1 1 P 2 2 2
G (s) = ——*—3‘ + —_— 2 (d¢.(o)/dx) (l/(s +2§.w.s+w.)) (5.29)
o I 5 IT i-l 1 i i 1
T

Therefore, the modal slope coefficient d¢(o)/dx can be determined by

equating equations (5.28) and (5.29). Table 5.2 presents the measured

Table 5.2 Measured System Model Parameters of the Smart
Robotic Arm

 

 

 

 

 

 

 

 

 

 

 

 

Parameters Values
wl (Hz) 15.2
”2 (Hz) 30.5
01 (Hz) 4.3
02 (Hz) 17.1
{1 0.0526
{2 0.0426
d¢1(0)

dx 3.12
d¢2(0)
dx 5.38

 

 

 

 

170
system model parameters of the smart ER arm in the absence of the
electrical field. In this investigation, rigid-body mode and first two

flexible modes were considered as the primary modes.

5.5.2 Results and Discussions

Figure 5.15 presents the measured step responses to a commanded step

angular position of 7.2 degree with the compensator zero Zc--2. The
feedback gains of Kp=0.5 and Kv=0'25 were employed. The external input

voltage (pseudo—control force) of 2.0 kV/mm was employed continuously
throughout the motion to the ER fluid domain. It is clear from this
figure that the responses with and without input voltage are basically
same except the tip deflection response of the arm. This aspect may be
expected since the mass of the robotic arm remains same regardless of the
input voltage, and the pseudo-control force does not affect the rigid-
body motion. The amplitude of the tip deflection of the robot arm in the
absence of the electric field was reduced up to 35 percent by employing
the input voltage of 2.0 kV/mm. It is also observed from this response
that the settling-time (decaying time) is about 1.3 second with the input
voltage, while it is about 1.6 second without the input voltage. This can
be interpreted as the enhancement of the positional accuracy of the
robotic system. No second flexible mode effect to the total tip
deflection is observed for both cases. The corresponding simulated
results are presented in Figure 5.16 which compare very favorably with
the measured results shown in Figure 5.15.

For the second comparison of control performances with and without

the input voltage, the compensator zero ZCO=-5.6 was chosen. Figure 5.17

171

shows the measured step responses with feedback gains of Kp-l.4 and
KV-0.25. From the angular position response, the rise time was obtained

by 0.42 second for both cases, and the maximum overshoot was about 0.89
degree without the input voltage and 0.93 degree with the input voltage.
This is basically same for both cases, and the input torque is alt“) same
for both cases as expected. The tip deflection without the input voltage
was reduced up to 20 percent by employing the input voltage for whole
duration of the response. With these feedback gains, it is also observed
jittering phenomenon as in the case of the aluminum or composite arm (see
Figure 4.11) (hue to the hysteretic joint friction of the motor. Figure
5.18 presents the simulated step responses corresponding to the measured.
results shown in Figure 5.17.

In order to implement the bang—off-bang type input voltage shown int
Figure 5.9, on-off switched response was obtained by imposing a non-zero
input voltage at certain time. Figure 5.19 presents the measured step
responses with the same output feedback gains used for the results in
Figure 5.17. The input voltage was imposed from the beginning up to the
peak time (0.61 second) of the angular position response in Figure 5.17,
and was released throughout the response. It is observed that the tip
deflection without the input voltage was reduced up to same amount as
achieved in Figure 5.17. This experimental result illustrates the speed
with which the smart ER.arm can.respond to the imposed input voltage
(pseudo-control force) in order to synthesize the desired vibrational
response of the flexible robot arm. It is noted that the determination of
the best switching time depends on various parameters such as rise time,
peak time and settling-time, and also depends upon the system performance

requirements such as minimum time or minimum effort problem.

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Figure 5.19 Measured Step Responses with Bang-off-Bang Type Input
Voltage. Kp-l.4, Kv-0.25

177

'The proposed hybrid controller gets its distinguished credit when
the system is subjected to a disturbance which has a forcing frequency
close to the clamped-mode natural frequency of the arm. In other words,
the robustness of the employed PD type output feedback controller to the
imposed disturbance can be improved by employing the pseudo-control force
which accounts for the increased stiffness and energy dissipation of the
robot arm. With the introduced disturbance, the system given by the

equation (5.24) becomes

><l°

-A§ +BE+Bd(t)+EV(t)
P P P P P P

(5.30)

XI

”-6
yp pp

where d(t) is the imposed disturbance. This disturbance is supplied from
the function generator, model 275, manufactured by Wavetek, to the servo-
drive.

Figure 5.20 presents the measured step responses to a commanded step
angular position of 7.2 degree with the disturbance d(t)=0.08 sin flt N-m

where f1 - 4.1 Hz. The feedback gains used for the results in Figure 5.15

were employed and the disturbance was applied from the beginning the

commanded motion. It is noted that the forcing frequency f1-4.l Hz is
almost same as the fundamental clamped-mode natural frequency 01 = 4.3 Hz

in the absence of the electrical field as shown in Table 5.2. AS
expected, the response of the hub angular position and tip deflection are
oscillated resulting in near resonance phenomenon of the system. However,
the amplitude of the oscillated tip deflection was reduced by 40 percent

by employing the external input voltage. The reduction of this amplitude

ANGULAR POSITION(deg)

TIP DEFLECTION(mm)

178

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10.0
~ input voltage=0.0 kV/mm
* -— input voltage=2.0 kV/mm
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6.0~
4.0--I
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1
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O 1 2 3 4
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- — input voltage=2.0 kV/mm
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0.0-3
1
-613: .5 .:
“12.0 d 1 I r 1 I 1 r 1 1 I 1 1 I 1 I 1 r 1 1
O 1 2 3

TlME(sec)

Figure 5.20 Measured Step Responses with the Disturbance.
Kp- 0.5, xv- 0.25, d(t)-0.08 sin(4.1 Hz)t

 

ANGULAR POSITION(deg)

TIP DEFLECTION(mm)

179

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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‘ -—- Inpuft voltage=2.0 kV/mm
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Figure 5.21 Simulated Step Responses with the Disturbance.
xp- 0.5, va 0.25, d(t)-0.08 sin(4.l Hz)t

180

 

 

input voltoge=0.0 kV/mm
1 ~— input voltage=2.0 kV/mm

 

 

 

   

 

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Figure 5.22 Measured Step Response with the Disturbance.
xp- 0.5, Kv- 0.25, 3(:)-o.4 sin(16.8 Hz):

is attributed to both increased stiffness and energy dissipation due to
the imposed electric field. A quantitative measure of the contribution of
each effect can be inferred from the frequency response presented in
Figure 5.6. Figure 5.21 shows the simulated step responses corresponding
to the results presented in Figure 5.20. The agreement between two
results is satisfactory.

Figure 5.22 presents the measured step responses with the

disturbance d(t)-0.4 sin f2t where f2-16.8 Hz. The forcing frequency f2

is very close to the second clamped-mode natural frequency of 02 - 17.1

Hz in the absence of electrical field. The amplitude of oscillation

181
without input voltage was reduced by 80 percent by employing the input
voltage of 2.0 kV/mm. Since the increment ratio of the second mode due to
the electrical field is larger than that of the first mode, the larger
reduction of the amplitude is obtained.

It is clear from these results that the proposed hybrid controller
is highlighted by enhancing the robustness of the conventional PD type
output feedback controller to the imposed disturbance. Even though the
pseudo-control force was applied in the fashion of the open-loop control
in this study, it is possible to set up the closed-loop algorithm once
the dynamics of the distributed ER actuator are identified more
accurately, and the information about the disturbance are available from
the output measurement of the system. More details on this matter are

documented in Section 7.2 as a future work.

5.6 Summary of the Chapter

The phenomenological equation of motion was developed for the smart
ER arm based on the experimental observation of the modal characteristics
of the beam structure. The hybrid controller featuring the output
feedback controller employed for the composite arm in the previous
chapter, and the pseudo-state feedback controller obtained from the
empirical dynamics of the ER fluid actuator has been formulated. 1}“; two
types of pseudo-control forces (input voltage), namely constant and bang-
off-bang type, were applied to the distributed ER fluid actuatoru iIt has
been shown through the experimental implementation that the robustness of
the employed output feedback controller subjected to the disturbance

which has a forcing frequency close to the clamped-mode natural frequency

182
of the robot arm was impressively enhanced by employing the pseudo-
control force. This methodology can be extended to the control of other
flexible structures such as helicopter blades, aircraft wings and.
appendages of satellites. These potential applications can be exploited
by the successful incorporation with other existing control strategies
such as optimal feedback and nonlinear feedback controllers. It is
anticipated that the successful transition of this embroynic technology
into an emerging technology would significantly accelerate the evolution
of this innovative class of multi-functional, dynamically-tunable,
intelligent materials for military, aerospace and advanced manufacturing

environments of the future.

CHAPTERVI

NONLINEAR STATE FEEDBACK CONTROL ROBUST TO
SYSTEM PARAMETER VARIATIONS

6 . 1 Introduction

The variations of system model parameters such as natural
frequencies, damping ratios and model slope coefficients may arise in
practice due to a wide spectrum of various conditions associated with
design and manufacturing process of the flexible manipulators, dynamic
modeling, and operating environments. These parameter variations will be
more serious in the robotic system featuring the flexible arm fabricated
from composite materials than those in the robotic system featuring the
flexible arm fabricated from conventional materials. The reason follows
that the flexible robot arm fabricated from composite laminates involve
more number of design variables, more complicated manufacturing process,
and also higher sensitivities to the environmental conditions such as
temperature and moisture. Hence, a robust control algorithm for the
position control of a single-link flexible manipulator fabricated from
composite laminates (composite arm in short) with parameter variations
will be developed in this chapter.

A robust controller which guarantees stable performances of the
system for all possible variations of the parameters is designed by
treating the variations as uncertainties. In other words, the robust
controller which guarantees that the response of the uncertain system

183

184

enters and remains within a particular bounded set after a finite
interval of time is to be formulated. Relating works on the robust
control to system uncertainties can be found in [13,36,44,6l,79,80, 81,
133]. In these references, the controllers were designed based on
possible bound of uncertainties, and it has been assumed that so-called
matching conditions are satisfied except references [13,36,133]. The
matching conditions simply say that the uncertain quantities can not
enter arbitrary into the state equations. The robust controllers without
the matching conditions were considered first in [13], and [36,133] later
on. The basic approach without matching conditions is same for the case
with matching condition except establishing more stringent bound on
uncertain parameters by introducing so-called a measure of mismatch and a
mismatch threshold. The system dynamics of the flexible robotic
manipulators lend themselves to the dynamics which can not formulated to
be assumed that the matching conditions are satisfied as will be seen in
the next section, while the dynamics featuring the rigid-body arms only
are adapted in nature to satisfy the matching conditions. And also, it
will be observed in the next section that one uncertain parameter causes
the variation of all flexible modes in the system dynamics of the
flexible manipulators, while it is not true in the system dynamics of the
rigid-body manipulators.

In this chapter, a nonlinear robust controller for the composite arm
will be designed by utilizing the properties of uniform and uniform
ultimate boundedness of the solution. The main difference of this
approach from those considered in current literature is that the effect
of each uncertain parameter as well as the effect of the interaction of

all employed uncertain parameters to the total system performance are

185

considered in this study. Hence, a certain measure of allowable magnitude
of each uncertain parameter can be achieved. The advantage of this
approach may come from that the considered uncertain parameters can be
properly weighed by adjusting the magnitude of the uncertain quantities.
Therefore, the system can be prevented from serious conditions associated
with the system instability caused by the uncertain parameters. In the
subsequent section, the problem formulation for the composite arm will be
presented, and Section 6.3 presents the robust controller design based on
the problem formulation. Finally, an illustrative example will be taken
and simulated in Section 6.4 to demonstrate the viability of the proposed
robust controller.

The author believes that this is the first work to design a robust
controller for a single-link flexible manipulator fabricated from
advanced composite laminates which accounts for serious uncertainties.
Even though the proposed controller is not experimentally implemented in
this study, the experimental realization of this controller may be
accomplished by incorporating the advanced high speed of microprocessors

as well as other necessary experimental hardwares such as modal filters.

6 . 2 Problem Formulation

The parameter variations of the composite arm may be arisen from
three different processes. First, the variation may arise from the
mismatch between the design and manufacturing process of the composite
arm associated with the factors of lamina angle, curing cycle, fiber
volume fraction, delamination, cracks, voids, etc. Since it is hard
maintain the designed lamina angle in the manufacturing process, there

always exists a mismatch of the lamina angle. And also, since curing

186

facilities are in general different at different place, there exists an
error regarding to material and geometrical properties as observed in
Section.3.2..A recent study [149] shows a frequency variation of the
composite laminates with respect to the delamination of the laminate.
Second, the variation may come from the dynamic modeling process. Since
the dynamic model is developed in general based on some assumptions such
as small elastic deflection, there exists an error whenever the motion of
the robot is over a certain level of these assumptions. Furthermore, the
nonlinear joint friction is often neglected in the modeling process
resulting in significant difference of the system performance between the
predicted and measured values in certain situations. Finally, the
variations may arise from the operating conditions such as varying
payload, varying temperature and moisture, and existence of obstacle in
the operation. Suppose that the robot is assigned to work for welding job
under high temperature condition, or oil exploring under the ocean. Then,
the parameter variations caused by the high temperature and moisture
should be accounted for the desirable system performance requirements.

All these uncertain parameters may cause system parameter variations
such as natural frequencies and input influence coefficients. In this
study, two most important factors; the varying payload and the mismatched
lamina angle are considered as uncertain parameters to develop the robust
controller. However, the proposed methodology can be exploited to account
for other uncertain parameters such as delamination and temperature. 1}“:
varying payload is encountered in operating environment and the
mismatched lamina angle is encountered in the manufacturing process of

the composite arm.

187

Now rewrite the state equation in (2.35) by

i = A x + B u
P P P

 

 

 

 

P
_ (6.1)
xp(t0) — xp0
where
I"o To 1 o o o o
"0 O O 0 ...
.71 o o o 1
. ‘2
x _ ‘01} ’ A = O 0 -w1 -2§lwl - - - 0 0
P P
O O 0 O 0.. O 1
nq 2
' o o 0 o -2
."q. _ ”q {qua
(6.2)
B =[0 b 0 b - - - blT
p 0 l (1
b0- l/IT , bi- (d¢i(o)/dx)/IT , i - 1, ..., q

where subscript (q) is the number of primary flexible modes,.and xp0 are

arbitrary initial conditions. Let

p I .
w - w .+ Aw + Aw. , 1 = 1, ..., q
i 01 i 1 (6.3)
b - b + Abp + Abfi i - 0 q
i 0i i i ’ ’ °'°’
where w and b are nominal natural frequency and control influence

Oi Oi
coefficient respectively under the conditions that (i) no payload, (ii)
known fixed lamina angle, (iii) satisfied all other assumptions made in

the modeling and manufacturing process as well as operating conditions.

The Aw}; and Ami are possible frequency deviation due to the varying

188

paylxuui and the mismatched lamina angle respectively. And the Ab? and

1 . . . . . . .
Abi are the corresponding dev1ations of the input influence coeff1c1ent.

Defining relative variation as

 

 

 

 

p 2
p A A“’i 2 A A“’i
(0.: , (0'3 11:1! 9q
r1 wOi r1 wOi
(6.4)
p 2
p A Abi b2 A Abi 0
ri g b ’ ri = b ’ 1 B ' ’ q
Oi Oi
the deviations can be expressed as follows.
A”: ' ”Eiwgi ' A”: ‘ wfiiwéi ' 1 T 1' ' q
(6.5)
Ab? = bp.bp. , Ab? - b£.b£. , i - o, ..., q
1 r1 01 1 r1 01
Thus, the state equation (6.1) becomes
x-A+AA +AA£+AA ,2):
p (0 (p) (> (13>)p
+ (80+ AB(p) + AB(£))u (6.6)
xp(to) - xp0
where
To 1 o o - - . o o '
0 0 0 0 . - . 0 0
0 0 0 1 o o o 0 0
2
0 O -w01 -2§1001- - 0 0 0
A... o o o o o O 0 0 (6.73)
o C O O O O C O O C
o O 0 o 1
2 2
o O 0 'w ' w
. 0q (q 0q.

 

 

189

 

 

 

 

 

 

To 0 o o - - - o 0
o o o o - . - o o
0 O O O o o g 0 O
O 0 p11 -p21 - . . O 0
AA(p) E O O O O O O O O O (6.7b)
o o o o . - - o o
.0 o o o - - - -plq -p2q -
2
p p _ p
p11 wOlwr1(wrl+ 2) ' p21 2i1“’01“’r1
2
1 p p _ p
plq ququ(qu+ 2) , p2q zngqurq
”o o o o . - . o o I
O 0 O 0 o o o 0 O
O O O 0 o o o O 0
o o -211 -221 . . . o o
AA(£) = o o o o o o o o o (6.7C)
O O O O o o o o o
o o o o . . . - -
L 1Q 2Q -
2 - 02 2 ( 2 + 2) 2 - 2 w mg
11 01°r1 ”:1 ' 21 ‘1 01 r1
2 2 2
2 - w w w + , 2 - 2 w w
lq Oq rq( rq ) 2Q {q 0q rq
’o o o o . . - o o ‘
o o o 0 . . - o o
o o o o - . . o o
o 0 -p21 0 - - - o o
AA(p’2)- o o o o o o o o o (6_7d)
o o o o - . - o o
o o o o - - - -p2 o
. q .
2 2
1 p 3 a 9
p21 2w01wrlwrl , pflq 2w0qququ
B - [o b o b - - - o b ]T (6 8a)
0 oo 01 Oq '
- P P . . . P T
AB(p) [o boobr0 o b01br1 o boqbrq] (6.8b)

2 2 2 T
_ . . . .8
AB(2) [O bOObrO O bOlbrl O qubrq] (6 c)

It is clear from the equations (6.7) and (6.8) that the uncertainty
Inatrices AA(-) and AB(-) can not be decomposed into a manner such that

the matching conditions are met, i.e.,
AA<-> - Bo D(.)

AB(°) " BO E(°)

Where D(-) and E(-) will be defined in the subsequent section. Therefore,
without consideration of the matching conditions, a nonlinear robust
state feedback controller which guarantees the stability of the system

(6.6) for arbitrary initial conditions x will be designed in the next

p0

section.

6.3 Controller Design

For generality, consider following dynamical system described by the

state equation
i<t> - (A0 + AA(01(t)) + AA(02(t)) + AA<a1<t).a2(t))> x(t)
+ (B0 + AB(ol(t)) + AB(02(t))) u(t) (6.9)

x(to) - x for t e [t0,t

o' 1]

where x(t) 6 RF is the state, u(t) E R? is the input, and A0, B0 are the
prescribed nominal system matrices of appropriate dimensions. In equation

(6.9), AA(al(t)) and AA(02(t)) are plant uncertain matrices associated

with uncertain parameters 01(t) and 02(t) respectively, and AA(al(t),

191

02(t)) is the plant uncertain matrix associated with the both parameters
01(t) and 02(t). AB(al(t)) and AB(a2(t)) are input uncertain matrices
depending on parameters 01(t) and 02(t), respectively. Now following

assumptions and definitions are made before proceeding further.

Assumption 6.1 : The uncertain matrices AA(-) and AB(-) are

prescribed functions which are continuous on RP.

Assumption 6.2 : Uncertain parameters

.14 p
01( ) . R. F1 C R

.1_, p
02().R 1‘ch

(6.10)
are Lebesgue measurable, where 1‘1 and I‘2 are prescribed compact subsets

of appropriate spaces.

Assumption 6.3 : The pair (AO,BO) is controllable. This assumption

in fact can be released for.the system (6.6).

Assumption 6.4 : All state variable x(t) are available from direct

measurements or state estimators.

Remark.6.1.: In this chapter, the vector norm is taken to be

Euclidean and the matrix norm is the corresponding induced norms, hence

1 2
[IAII - (A )(ATA)) / where A )(.) denotes the operation of

max(min max(min

taking the maximum (minimum) eigenvalue.

192
Definition 6.1 : Given a solution x(o): [t0,t1] n RP, x(to) ==xo for

the equation (6.9), it is said that the solution is uniformly bounded if'

there is a positive constant 6(x0) < 00, possibly dependent on initial

but not on initial time t such that

value x0, 0,

||x(:)|| s 6(x0) for all : e 1:0,:1]
Definition 6.2 : Given a solution x(o): [t0,w) 4 RP, x(to) = x0 for

time equation (6.9), it is said that the solution is uniformly ultimately
bounded with respect to a set S if there is a non-negative constant

and S, but not on t such that

1(xo S) < m, possibly dependent on x 0,

0

x(t) 6 S for all t 2 t0+ r(xo,S)

Definition 6.3 : The equation (6.9) is said to be decomposable if

there exist continuous matrix functions
D(al(t)): Planpx“
D(02(t)): r21RPX“
D(al(t),02(t)): (r1,r2)422x“ (6.11)
E(al(t)): rlenpx“
E(02(t)): r2422x“

and AA(°) and AE(-) having following properties.

AA<a1(t)> - Bon(a1<t)) + AX(al(t>)

193

AA(02(t)) = BOD(02(t)) + AK(02<c))
AA<a1(t).a2<t)) = B0D(al(t),02(t)) + AK<al<t).a2<t>> (6.12)
AB(al(t)) = BOE(01(:)) + A8(al(t))

AB(02(t)) — BOE(02(t)) + AE(a2(t))

Remark 6.2 : The argument t is often omitted throughout this section

when no confusion can arise.

It is observed from the equation (6.12) that the uncertain matrices

AA(-) and AB(-) are decomposed into two parts ; the matched part BOD(-)

and BOE(-), and the mismatched part AA(-) and AE(-). It is noted that

this decomposition is highly nonunique. In some sense, a 'good'
decomposition may be achieved by lumping uncertainties as much as into
the matched part, hence minimizing the portion of the mismatched part.
One way to obtain a 'good' decomposition is to apply the least-square

error method [28]. For example, by letting
AA<~> = BOD(-)

AB(o) =BOE(°) (6.13)

following decomposition can be accomplished in the least-square error

sense.

T -l T
D(01) - (B0 BO) BO AA(01)

T -1 T
D(a2) - (B0 BO) BO AA(02)

194

T -1 T

D(01,a O

2) 2)

T -l T
E(01) - (BO BO) BO AB(ol)

3(02) - (33 BO)'1 Bo AB(oz) (6.14)
AA(01) - AA(01) - BO D(al)
AA(02) - AA(02) - BO D(02)

AA(01,02) - AA(a B D(a

1’02) ' o 1'02)

AE(01) - AB(al) - BO E(al)

AB(OZ) - AB(02) - B0 E(02)

The purpose of this problem is to design a state feedback controller
which assures that every solution of the equation (6.9) is uniformly
bounded and uniformly ultimately bounded with respect to a set S to be
specified, no matter what the uncertainties and initial conditions are.

For the sake of generality, suppose that the nominal plant matrix A0 is

not stable. However, there exist a constant feedback gain matrix K such

that

AC-A +B K (6.15)

is stable, since the pair (A0, B0) is assumed to be controllable. This
implies that the real parts of all eigenvalues of Ac are negative. Now,

consider a state feedback controller which has the form of

u - Kx + p(x) for all x 6 H2 (6.16)

195

where the function p(-): 22422 is such that

 

I 'r
B0 PX . T
- [IBT lel (p1(x) + p2(X) + p3(X)). 1f IIBO PXII>€
0
p(x)-< (6.17)
B3 Px T
- -———:—-—- (p1(x) + p2(x) + p3(x)), if [[30 PxIISe

where e is a prescribed positive constant, P is the solution of the

Lyapunov equation
AT 9 + P A = -Q (6 18)
c c '

for a given constant positive definite (nxn) matrix Q, and pi(-): RF 4 R1

will be specified subsequently.
Substituting equations (6.12) and (6.16) into the equation (6.9)

yields
x - Acx + Bop(x) + Bo(em1(x,t) + em2(x,t) + em3(x,t))

+ 31(x,:) + 52(x,:) + 33(x,:) (6.19)

where

em1(x,t) - D(al)x + E(01)Kx + E(al)p(x)
em2(x,t) - D(02)x + E(02)Kx + E(02)p(x)

em3(x,t) - D(al,az)x

E1(X,t) " (AZ(01) + A§(OI)K)X + AB(gl)p(x) (620)

32(x,:) - (AA(02) + AE(02)K)x + AE(02)p(x)

196

E3<x.t> = (AK<al.a2>>x

for all (x,t) 6 RP x R1. In equation (6.19), BOemi(x’t) and ei(x,t) are
called as the matched error vector and mismatched error vector

respectively. From the equation (6.20)

Ilem1(x,t)|| 5 max ||D(al)x|| + max IIE(01)KxII

aleFl alEFI

+ max IIE(al)IIp1(x) 9 p1(x)

aleFl
Ilem2(x,t)ll s max IID(02)xII + max IIE(02)KxII (6.21)
0 EF 0 EP
2 2 2 2
+ max ||E(02)||p2(X) g p2(X)
026F2
A
llem3(x,t)ll s max IID(01,02)x|| = p3(x)
a GP
lerl
”22

The defined parameter p (x) in equation (6.21) can be solved provided

max ||E(al)|| < 1

OIEFI (6.22)

max ||E(02)|| < l

026F2

Thus, with conditions (6.22), pi(x) are obtained as follows.

p1(x) - [1 - max IIE(01)I|]-1[ max ||D(al)xll + max I|E(ol)Kx||

alePl OIEFI alefl

p2(x) - [1 - max ||E(02)II]-1[ max ||D(02)xll + max ||E(02)lel (6.23)
02€P2 02€P2 026P2

197

p3(x) - max I|D(al,02)xll

a GP
1 1
02€F2

Now define 31(x): RP* R: by

~ A
P1(x) - 01 IIXII
~ A
P2(x) - 02 llxll

~ A
p3(x) 3 03 lell
where

a1 - [1 - max ||E(al)||]'1[ max ||D(al)|| + max I|E(01)KII
6P 0 61‘ a 61‘
”1 1 1 1 1 1

a2 - [1 - max IIE(02)II]-1[ max ||D(02)|| + max IIE(02)KII

azerz 02€P2 026F2
a3 - max I|D(al,02)ll
a GP
OIEPI
2 2

Thus, from the equations (6.23) and (6.24)

p1(X) S 31(X)

IA

p2(X) 32(X)

p3(X) S 33(X)

for all x 6 RP. Furthermore, from the equations (6.21) and (6.26)

llem1(X.t) + em2(X.t) + em3(X.t)I|
s Ilem1(x,t)|| + Ilem2(x,t)|| + llem3(x,t)ll

S p1(X) + p2(X) + p3(X)

(6.24)

(6.25)

(6.26)

198

s Zl<x) + 32(x> + E3<x> = (a1+ a2+ a3)||x|| <6-27>

Now in order to handle the mismatched error vector in equation

(6.20), following definitions are introduced.

Definition 6.4 : Let the measure of mismatch be defined as follows.

:2
I

max IIAK(01)|| + max ||A§(01)K|| + a max IIA§(01)|I

1 1
aleFl alefl aleFl
m2 é max IIAK(02)II + max ||A§(02)K|| + a2 max ||a§(a2)|| (6 28)
6P a 6P 0 GP
”2 2 2 2 2 2
A ~
M3 - max IIAA(01,02)I|
a GP
lerl
”2 2

Definition 6.5 : Define the mismatch threshold as

 

A (Q)
* A min
M - 2A (P) (6.29)
max

Definition 6.6 : Define the mismatch index as

A (”1 + M2
fl - x
M

+ M3) (6.30)

It is noted in Definition 6.4 that M1 and M2 account for the mismatch
associated with uncertain parameters 01(0) and 02(-) respectively, and

M3 accounts for the mismatch associated with both parameters. From the

equations (6.20) and (6.28)

199

||51(x.t>|| 5 M1 ||X||
||32(x.t)|| 5 M2 leII (6.31)
||53(X.t)|| 5 M3 IIXII

Thus ,

||31<x.t) + 32(x.c) + E3<x,t)ll

IA

”2:16.61: + ||52<x.t>ll + l|53(x,t)ll

5 (M1 + M2 + M3) ||x|| (6.32)

x
The mismatch threshold M defined in (6.29) implies a limit on the
amount of the mismatched uncertainties, defined as the measure of

mismatch Mi’ which can be tolerated. The mismatch index [3 in (6.30)

indicates the normalized total measure of mismatch (M1+ M2+ M3) ‘flith
respect to the mismatch threshold. Now consider the uniform and uniform
ultimately bounded sets which satisfy the conditions given in Definitions

6.2 and 6.3. With a hypothesis that the total measure of mismatch is not

two large, i.e.,
fi<1

or (M1 + M + M3) < u* (6.33)

2

form the scalar function h(-): [0, w)+R given by
A 2
h(n) - (Amin(Q) - ZAmax(P)(M1+ M2+ M3))n -((al+ a2+ a3)6/2)n (6.34)
Then, letting
A

5 max ( n : h(n) s o } (6.35)

200
yields

(a1+ a + a3)e

2

fl ' (6.36)
2(Amin(Q) - 2Amax(P)(M1+ M2+ M3))

It is noted that the E in (6.36) implies the radius of the largest closed

ball which is wholly contained within the bounded set (see Theorem 6.1).

Remark 6.3 : It is observed from the condition (6.33) that the
*
mismatch threshold M depends on eigenvalues of the matrices Q and P. The

* *
larger'li tolerates the larger uncertainties. The maximization of the M

will be discussed later in this section.

Since Ac in equation (6.15) is stable and the matrix Q in equation

(6.18) is positive definite, and hence the matrix P is positive definite,

following Lyapunov ellipsoids can be constructed.

X(k) - { x e R? | xTP x s k ) (6.37)

where k is positive constant. Now define the smallest ellipsoid

containing ball 8(5) as

2
km 9 min ( k | 3(5) g X(k) ) = Amax(P) ‘5 (6.38)

And also, for iven an E > km,and an initial condition x X(E),define
8 Y Y 0

the constant

P x ' (6.39)

and

201
... ~ 1 2 1 2
c(x0,k) 9 min ( h(n) : (k/Amax(P)) / s n s (kO/Amin(P)) / } (6.40)

Finally, following theorem regarding to the boundedness of the solution

of the equation (6.9) can be established.

Theore-.6.1 : Consider the dynamical system given by equation (6.9).
And suppose that the conditions given by the equations (6.22) and (6.33)

are satisfied. Then, the controller given by the equation (6.16) assures
the uniform boundedness of every solution x(-):[t0, t1] 4 Rh, and uniform

ultimate boundedness of every solution x(-) extended over [t0,m).

~

m . . . .
Furthermore, given any k > k , every solution corresponding to initial

condition (x0,to) E Rpx R is uniformly bounded with

F 1/2 ...
(Amax(P)/Amin(P)) leoll for xo ¢ X(k)
6(xo) - 1 (6.41)

 

.. 1/2 ..
L(k/Amin(P)) for x0 6 X(k)

and is uniformly ultimately bounded with respect to a set S e X(E) with

r

(k0- E)/c(xo, E) for xo ¢ 3
?(xo,3) - < (6.42)

0 for x0 6 S

 

Proof : Denote the right-hand side of the equation (6.19) by
A
f(x,t) - Acx + B0p(x) + Bo(em1(x,t) + em2(x,t) + em3(x,t))

+ 31(x,t) + 52(x,c) + 33(x,c) (6.43)

202

where f(x,t) is Caratheodory functhn1; f(x,t) is continuous in x and

x -
Lebesgue measurable in t. Now let X(k ) be a superset of X(k) containing

all initial conditions. Then for given initial conditions, f(x,t) can be

*
uniformly bounded over the domain D e X(k ) x R from the observation

IA

llf<x.t>|| IlAcll lell + (.1. a2+ a3>|lBoll llxll

+ IIBO(em1(x,t) + em2(x,t) + em3(x,t))||

+ ||51<x.t> + 52(x,t> + E3cx.c>||

IA

IIACII llxll + 2<a1+ .2. a3>||Boll llxll (6.44)

+ (Ml+ M2+ M3)||x||

lll>

g(X)

*
Since g(-) is continuous [19] on X(k ),there exist a finite maximum value

A

k - max g(x) (6.45)
xeX(k*)

A

Thus, it is clear that k plays a role of a uniform bound for.f(x,t) over

domain D. Therefore,it follows from [43] that there exist a solution x(-)
:[to,t1] 4 RP. The extension of this solution will be shown later of this
proof.

Now, introduce a Lyapunov function candidate given by

V(x) - xTP x for all x 6 RP (6.46)

where V(o):RP4R is continuously differentiable. To show that it is indeed
a Lyapunov function for the system (6.9) with controller (6.16), consider

the Lyapunov derivative as follows.

203

T

9(x,t) - x P x + xTP x

- - xTQ x + 2(33? x)T (p(x) + em1(x,t) + em2(x,t)
+ em3(x,t)) + 2 xTP (31(x,t) + 22(x,c) + 33(x,t))

- xTQ x + 2(33? x)T (p(x) + ((ng x/||ng x||) (6.47)

IA

(p1<x> + p2<x) + p3<x>>> + 2 xTP (51(x.t>

+ 32(x,c) + E3(x,c))
It is observed from the equation (6.17) that, if IIBg P xII > e, the term
of p(x) + (BE P x / IIBg P x||) (pl(x) + p2(x) + p3(x)) vanishes, and if

[[83 P xll < 6, then it has maximum value of e (p1(x) + p2(x) + p3(x))/2.
Hence, from the equations (6.27) and (6.32), the equation (6.47) becomes

V(x,t) s - xTQ x + ((a1+ a2+ a3)e/2)I|x||

2
+ 2Amax(P)(M1+ M + M3)I|xll (6.48)

2
Thus

v(x,t) < 0 (6.49)
for all t E R; and all x such that
TQ x - ((a + a + a )6/2) x
x 1 2 3

2
- 2Amax(P)(M1+ M + M3)||x|| > 0 (6.50)

2

However, from the Lemma 4.1 in the previous chapter

204

T

2 2
Amin(Q)||x|| s x Q x s Amax(Q)||x|| (6.51)

where Amin(Q) > 0, since Q >0. Thus,the condition (6.49) is satisfied for

all t e R; and all x such that

2
(Amin(Q) - 2Amax(P)(M1+ M2+ M3))||xll

- ((a1+ 62+ a3)e/2)llx|| > 0 (6.52)

Hence, it can be said that the inequality given by the equation (6.49) is

guaranteed for all te R;and all x ¢'B(3),where E is given by the equation
(6.36), provided.that the condition (6.33) is satisfied. It should be
noted, however, that the condition (6.52) is only a sufficient condition.

Now consider the smallest Lyapunov ellipsoid that contains the ball
8(3), that is, X(km), where km is given by the equation (6.38). From the
Lemma 4.1,

2 T 2
Amin(P)||x|| s x P x s Amax(P)||x|| (6.53)

where Amin(P) >0, since P >0. The condition (6.49) implies boundedness of

all solution. That is, if x(o):[t0,tl]4RP with initial condition x(tO)-xo

is a solution of the equation (6.9) with the controller given by the

equation (6.16) and allowable uncertainties, then there are following two
cases to be considered: if x0 Q'X(E), that is, x(t) e X(ko) for all t e
[to,t1],then following inequality can be obtained in view of the equation

(6.53).

205

2 T T 2
0 < Amin(P)||x(t)|| s x (t) P x(t) 5 x0 P x0 5 Amax(P)||xO|| (6.54)

If xo 6 X(E), that is x(t) e X(E) for all t e [t0,tl], then following

< <

Therefore, 6(x given by the equation (6.41) can be achieved from the

0)

equations (6.54) and (6.55) for x0 ¢ X(E) and x e X(E) respectively. And

0

from this turLform boundedness and existence of the solution, it follows

that a solution x(o): [t0,t1] 4 RP can be extended over time interval

[to.m>.
The proof of ultimate boundedness follows again from the condition

(6.49). If x0 6 X(E), it is an immediate consequence of the boundedness
result, hence it suffices to take ?(x0,3) - 0. If xO ¢ X(E), the V(x(t))

decreases as long as x(t) Q X(km) and x(t) reach the boundary 6X(E) in a

finite time. From the equations (6.54) and (6.55)

~ 12 12
(k/Amax<r>> / s llx(t>|| s (ko/Amin<P>) / (6.56)

whenever x(t) 6 X(kO)/X(E)>. Hence, from the equations (6.48) and (6.52)

v(x,t)

IA

'h(llx(t)||)

. ~ . ~ 1/2 ~ 0 1/2
S -m1n {h(n). (k/Amax(P)) S n S (k /Xmin(P)) }

--e(xo,E) < 0 (6.57)

206

where
2
h(||x(t)|| - (2(M1+ M2+ M3)||P|| - Amin(Q))||x|I

+ ((al+ 02+ a3)e/2)||x|| (6.58)

~ * * - ~
Suppose x(t) g X(k) for all t e [t0,t ], where t - (kO- k)/c(x0,k). Then

from the equation (6.57)

' E)t* s - (k0- E) (6.59)

V(x(to+ t*)) V(xo) s - c(x

o,

k. Then this contradicts the supposition. Thus

IA

so that V(x(to- t*))

A

x A ~

there exist a time t S t such that x(t) E X(k) = S. Therefore, from the
x * ~

boundedness and x(t) e S for all t 2 t, t - r(xo,S). This concludes the

proof of the Theorem 6.1.

Now, consider the maximization of the mismatch threshold M* given by
the equation (6.29) as noted in Remark 6.3. It is clear from this
equation that the threshold depends on the eigenvalues of the positive
definite matrices Q, and P which is a solution of the Lyapunov equation

associated with closed-loop eigenvalues of the nominal system. Hence, the

*
maximization of M can be accomplished by considering either given matrix

Q or the linear feedback gains in equation (6.16). Herein, the

*
maximization of the M is achieved by considering the matrix Q with fixed

feedback gains. Before establishing a theorem regarding to the

*
maximization of M , following lemma is introduced.

207

lemma 6.1 (see [113] ) : Let P be the unique positive definite

solution of the Lyapunov equation

T ~ ~ -
Ac P + P Ac = -Q = - 1Q (6.60)

where 1 is positive scalar constant, and Ac and Q are in equation (6.18).
Then

* .. ..
M = Amin(Q)/24max(P) - Amin(Q)/2Am (P) (6.61)

3X

where P is the solution of the Lyapunov equation (6.18).

Proof : Rewrite the equation (6.60) as
T ~ ~
AC (P/v) + (P/1)Ac - - Q (6 62)

Since P >0 and P >0, and also there is an unique solution of the equation

(6.18) and (6.60) respectively, it follows from the equations (6.18) and

(6.62), P - P/V . Hence
Amax(P) = Amax(P)/7 (6.63)

Thus the equation (6.61) is satisfied

Theorem 6.2 : Suppose the linear feedback gains in equation (6.16)

*
are fixed. Then the mismatch threshold M is maximum when the positive

definite matrix Q - I (identity matrix).

208

Proof : Let the matrix Q in equation (6.60) be arbitrary positive

0 O O - ~ 3 ~ 0 0 th
definite matrix and let 1 l/Amin(Q). Then Amin(Q) 1, where Q is 111 e

equation (6.60). The solution of the equation (6.60) is given [18] by

~ AZC ~ Act
P = I” e Q e dt (6.64)
0
And also consider
T A A
AP+PA=-I (6.65)
c c
with the solution
A AZt Act
P - Im e I e dt (6.66)
0
Now define
~ A ~
M - l/Amax(P)
" A " (6.67)
M l/Amax(P)
Then from the equations (6.64) and (6.66)
~ a Act ~ Act
P - P - [m e (Q - I) e dt (6.68)

0

However Q-IZO since all eigenvalues of the Q greater or equal to 1. Hence

P - P 2 0, A (P) 2 A (P) and M 2 M from the equation (6.67). Since M
max max

is invariant under any choice of the positive scalar 1 by the Lemma 6.1”

M 2 M holds for all Q. This completes the proof of the Theorem 6.2

209
Therefore, the mismatch threshold defined by the equation (6.29) can

be modified as

* A 1
M - 2A (P) (6.69)
max

 

by choosing Q - I. Other quantities involving Amin(Q) are modified by

replacing Amin(Q) by 1. For example,the radius of the largest closed ball

5 defined by the equation (6.36) becomes

(a1+ a + a e

2 3)

5 = -
2 4Amax(P) (M1+ M + M3)

2

6 . 4 Illustrative Example

In order to demonstrate the robustness of the developed nonlinear
state feedback controller, the composite arm, which has geometrical and

material properties given in Table 6.1, is considered. It is clear from

this table that the nominal system has no payload and 200 of lamina angle
(fiber orientation) for each lamina. These two parameters will be
considered as uncertain parameters in this example. The parameter
variations of the proposed robotic system associated with the varying
payload and the mismatched lamina angle are presented in Figures 6.1 and
6.2 respectively. It is observed from Figure 6.1 that the modal slope

coefficient d¢1(o)/dx is very sensitive to a whole range of the payload,

while the first mode natural frequency is sensitive to only a small range
of the payload. And it is clear from Figure 6.2 that both the first and

second mode natural frequencies are very sensitive to the lamina angle.

Table 6.1

210

Geometrical and Material Properties of the

Nominal Composite Arm

 

Parameters and Properties

Specifications
(Graphite-Epoxy)

 

 

 

 

 

Beam Length (L) 1.5 m
Beam Width (b) 3.0 cm
Beam Thickness (h) 1.8 mm
Payload (mt) 0.0 kg

 

Hub Moment of Inertia (1“)

0.00496 kg-m2

 

 

 

 

 

 

 

 

Number of Lamina 6
di - 0.3 mm
Lamina Thickness
(1 - 1. .6)
01 - 20°
Fiber Orientation
(1 - 1. .6)
v; - 0.60
Fiber Volume Fraction
(i - l, .,6)
Efiber - 220 GPa
Young's Modulus
E . -4.27 GPa
matrix
-1750 k /m3
pfiber 5
Density 3
”matrix.1200 kg/m
v - 0.20
Poisson's Ratio fiber
u - 0.34

 

matrix

 

 

w1(rad/sec)

w1(rad/sec)

 

25

 

 

 

 

13
01)

I I I r I I I I I

(15

Fi

I I I I
1 .0
PAYLOAD(kg)

gure 6.1

I I I

1.5

I I

I

21)

 

N
O
l

U]
l

 

 

IIIII

I I I I

60

r I I I

LAMINA ANGLE(deg)

Figure 6.2

 

90

211

20
mi
3; a
5...

0

0

w2(rad/sec)

 

 

 

 

IFIIIIIIITrI

.0 (l5

1 .0
PAYLOAD(kg)

[W I

1.5

Parameter Variation of the Composite Arm
with respect to the Payload

I I

213

 

70

30~

10

 

 

IIIII

LAMINA ANGLE(deg)

with respect to the Lamina Angle

Parameter Variation of the Composite Arm

 

212

The system model parameters of the nominal system are presented in Table

6.2. For the determination of these parameters,

two-node element was

employed with two degrees of freedom at each node, and.the number of

elements was ten.

In this simulation, the rigid-body mode and the first flexible mode

are considered as the primary modes. Hence the system matrices in

equation (6.6) become

 

 

 

 

 

 

 

(6.70a)

(6.70b)

_ l 0 0
0 0 0
A0-
0 1
_ -539.9 -O.3253‘
. 0 -
10.2135
B0-
0
_30.7323_
Table 6.2 System Model Parameters of the Nominal System
d¢i
Mode wi(Hz) E);— (0) ¢i(L)
1 3.698 3.009 -0.960
2 8.603 2.798 0.993
3 19.800 1.139 -1.156
4 37.931 0.566 1.180
5 62.484 0.338 -1.188

 

 

 

 

 

 

213

 

 

 

 

 

 

 

 

-0 0 -
AA(p) - 0 0 (6.70c)
0 0
-0 0 'p11 'p213
a p p = 9
p11 539.9wr1(wr1+ 2), p21 0.32536r1
[0 0 0 '
0
AA(£) - (6.70d)
0
_0 0 -211 -221_
2 2 2
£11- 539.9wr1(wr1+ 2), £21= 0.325360r1
[0 0 0‘
AA(p,2) - 0 0 0 (6.70e)
0 0
L 0 -p21 0,
_ 9
p21 1079.9wr1wr1
- 0 -
p .
AB(p) - 10°2135br0 (6.70f)
0
30.7323bp
- r1-
AB(2) - 0 (6.70g)

It is noted that the damping ratio of the first flexible mode was chosen
by 0.007. And the input uncertain matrix AB(2) was chosen by 0, since the
variation of the lamina angle does not affect the variation of the

control influence coefficient.

214
Now the decomposition of the system uncertain matrices can be

accomplished in view of the equation (6.14) as follows.

- _ p P _ p

D(p) [ 0 0 15.82wr1(wr1+2) 0.00953er]
2 2 2
0(2) - [ 0 0 -15.82wr1(wr1+2) -0.00953er] (6.71)
D(p 2) — [ 0 0 -31 64wp wi 0]
’ ' r1 r1

_ p p

E(p) 0.0994615r0 + 0.9br1

The feedback gains K in equation (6.15) were chosen by
K - [ -35 -5 ~50 -4 ] (6.72)

Hence with the positive definite matrix Q-I, the solution of the

Lyapunov equation (6.18) was obtained as

' 8.683 0.730 1.295 -0.242 ‘
P _ 0 730 0.189 1.397 -0.0548 (6.73)
1.295 1.397 27.133 -0.343

_-O.242 ~0.0548 -0.343 0.0194‘

 

 

The variation of the control influence coefficient was assumed to be

bounded with -0.06 s bios 0,

and -0.05 S bgls 0 so that the condition
(6.22) is satisfied. Then the p(x), which is nonlinear portion of the
controller, is obtained as follows in view of the equations (6.23) and

(6.26)

p(x) - -<p1<x> + p2(x) + p3(x)) sgn(BgPX). (6.74a)

if |0.00143x + 0.2461x + 3.7167x + 0.0376x4I > e

l 2 3

or

215
p(x) - -(p1(x) + p2(x) + p3(x))(0.00143x1+ 0.2461x2

+ 3.7167X3+0.0376X4)/6, (6.74b)

if I0.00143x1+ 0.2461x2+ 3.7167x3 + 0.0376x4I s e

where
p p 2
pl(x) - msx (1.05374 (15.82(2.0|wr1| + (wrl) )Ix3|
wrl
+ 0.051(35|x1| + 5Ix2| + 50Ix3| + 4|x4l))}
(x) - max {15 82(2 0|m£ | + (m2 )2)|x | (6 75)
92 ' ° r1 r1 3 '

2
r1

w
+ 0 00953lw£ | |x I)
' r1 4

2
p3(x) - mgx {31.64(|w:1| Iwr1|)|x3l}
wil
r1

0:

and sgn(Bng) is defined by

 

[ 1, if Bng > 0
T T
sgn(8o P x) - < 0, if BOPx - 0 (6.76)
L-1, if Bng < 0

Figure 6.3 presents the open-loop tip displacement of the nominal
system. The vibration of the first flexible mode was invoked by taking

the initial conditions as
T
x(O) - { 0 0 0.06 0} (6.77)

It is clear from the figure that the amplitude decays with only assumed

damping ratio of the first mode.

216

 

 

 

 

 

 

 

 

 

 

 

 

TIP DISPLACEMENT(cm)
53
O
L

 

 

 

 

 

_8.0 r I I f I I I T
0.0 1.0 2.0 3.0 4.0

TlME(sec)

I T I i I r I

Figure 6.3 Open-Loop Tip displacement of the Nominal System

Figure 6.4 presents the simulated step responses for the commanded
tip position of 15 cm with the linear and nonlinear controller. Herein
the linear controller is given by u - Kx, and the nonlinear controller
is defined by equation (6.16). The typical value of e - 0.003 was chosen,

and the actual bound on the variation of the uncertain parameters were

chosen by
-0.2 5 6:1 5 0
(6.78)
-0.4 s w! .<_ 0

P

The variation of wrl

corresponds to the variation of payload from 0 kg to

about 0.17 kg. It is noted that the mass of the flexible arm is 0.124 kg.

217

. 2 . .
0n the other hand, the variation of wrl corresponds to the variation of

o o
the lamina angle from 20 to approximately 48 . For the larger variation

. 2 .
of the uncertain parameters, both wgl and wrl were taken as negative.

The situation of the positive mil and wfil, which correspond to an

increment of the fundamental natural frequency is not considered in this
study. In fact, an increment of the natural frequency due to the payload

can not be happened in practice. Furthermore, choosing the positive of

w makes the system be undergone with the smaller perturbation due to

rl

P

2
r1 and positive wrl'

the compensation between the negative w

It is observed from the tip displacement response in Figure 6.4 that
the response with the nonlinear controller settles in the commanded
position in 5.8 seconds (settling-time) inducing maximum overshoot of 9
cm, while the response with the linear controller settles in the
commanded position in 12 seconds (settling-time) inducing maximum
overshoot of 13.8 cm. This shows that the nonlinear controller have a
much better capability for regulation of the response than the linear
controller in the presence of system uncertainties. This can be also
interpreted that the nonlinear controller can tolerates larger system
parameter variations than the linear controller. It is also interesting
to observe from the input torque response that the required input torque
of the nonlinear controller is about 60 percent that of the linear
controller for the simulated response time.

Table 6.3 presents a comparison of the tip displacement response

(settling-time and maximum overshoot) between the linear and nonlinear

INPUT TORQUE(N.m)

TIP DlSPLACEMENT(cm)

218

 

 

 

—- NONUNEAR

UNEAR

 

 

N
.p.
l l L l l

  

(I)
l

.1
.0.

 

I I

 

_6 I I I r I I
0.0 3.0 6.0

TlME(sec)

 

12.0

 

 

 

—- NONUNEAR

UNEAR

 

 

 

 

I— I

 

"0.3 I I r I T 1*
0.0 3.0 6.0

TlME(sec)

9.0

 

12.0

Figure 6.4 Comparison of Step Responses with the Linear

and the Nonlinear Controller

ANGULAR VELOCITY(rad/sec)

219

 

ANGULAR POSITION(deg)

 

 

 

IJNEAR
-— NONUNEM?

 

 

 

0..
o 0 on
o ...-... ‘00....-

 

 

3.0

I I I l I

. I '
6.0 9.0 ' 12.0
TlME(sec)

 

.0
O
l l l l l l l l l

 

l
.0
01

 

 

IJNEAR
-— NONUNEAR

 

 

 

 

 

.0
0

Figure 6.4

I I I I I

I
61) 9&) 121)
TlME(sec)

(Continued)

220

Table 6.3 Comparison of the Tip Displacement Response (Settling
Time and Maximum Overshoot)

 

 

 

 

 

 

 

 

Variation Settling Time (sec) Maximum Overshoot (cm)
Bound Nonlinear Linear Nonlinear Linear
451- 0 2.9 5.8 7.1 11.3

-o.isw1r’lso 4.5 8.6 8.2 12.6

025651; 5.8 12.0 9.4 13.8

03545150 11.6 22.1 10.8 15.5

$456550 24.3 44.5 12.4 17.0

 

 

 

 

 

 

 

controller under various bounds of oil. The variation of the lamina angle

was fixed as -0.4 S mils 0. It is clear from the table that the settling-
time with the nonlinear controller is half of that with the linear
controller in all variation bounds. The bound -0.3 S ”:15 0 corresponds

to the varying payload from 0 kg to about 2.0 kg which is about 16 times

of the arm mass. Similar results are anticipated from tflua'various

. . 2 . . . .
‘variations of the wrl With the fixed variation of the wgl. Therefore, it

is possible to adjust the variation bound of each uncertain parameter for
achieving the required system performances.
Next, in order to observe the sensitiveness of the controller

parameter' c in the equation (6.17), the different value of e - 0.1 was

221

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

36
- c=0.1
‘ — e=0.003
30“
E -
3
I.—
2
Lu
2
Lu
Q
1’)
O.
Q
Q
E
}_
'—'6 I I I I I I I I I I
0.0 3.0 6.0 9.0 12.0
TlME(sec)
0.6
=04
- -— e=0.003
’23 _
E; 0.3
LL] .
:3 ..
0
C!
O
I—
f—
D
0. ...... 11
.2.
"0.3 I l I I l I I I I I
0.0 3.0 6.0 9.0 12.0
TlME(sec)
Figure 6.5 Comparison of Step Responses with 6-0.1 and 6-0.003

222
chosen. Figure 6.5 presents the comparison of the step responses with two
different values of c. It is clear from this figure that the smaller 6

produces the better system responses. This fact is obvious from the
largest ball I; defined by the equation (6.36). For stabilization

problem, it is desirable to have I). as small as possible, hence
increasing the robustness of the system stability. As 6 4 O, n 4 0.
However, the smaller 6 may produce the larger unwanted vibration in
practice due to the larger chattered input torque as shown in Figure 6.5.

It is finally noted that even though two uncertain parameters;the
varying payload and the mismatched lamina angle were considered in this
study, the approach developed herein can be extended to the case which
accounts for many uncertain parameters. The manner of choosing uncertain
parameters should be dependent on the physical environments conditions.
For example, if the robotic system is supposed to operate under high
temperature or moisture environment, these uncertain parameters should be

taken into consideration.

6.5 Sui-nary of the Chapter

The nonlinear robust controller for the composite arm was designed by
utilizing the properties of the uniform and uniform ultimate boundedness
of the solution. The varying payload and the lamina angle were chosen as
uncertain parameters. It has been demonstrated that the nonlinear
controller had much better capability for regulation of the response than
the linear controller in the presence of the uncertain parameters. This
implies that the larger tolerance of the fabrication error (lamina angle)

of the composite arm can be allowed for the nonlinear control system, and

223
also the larger bound of the unknown payload is allowed with guaranteed
system stability. Even though the proposed nonlinear controller was not
experimentally implemented in this study, the realization of this
controller would accelerate the relevant researdh area in the future. The
extension of this controller to flexible multi-link robotic systems, and
other flexible structure systems fabricated from composite materials are
remained as further research areas as well as the experimental realization

of this controller.

CHAPTER'VII

CONCLUSIONS AND RECOMMENDATIONS

7.1 Conclusions of the Thesis

Three different control schemes for a single-link flexibLe
manipulator fabricated from composite laminates or smart materials
incorporating electro-rheological (ER) fluids were developed in this
thesis ; the output feedback control, the hybrid control, and the
nonlinear state feedback control robust to system parameter variations.

The output feedback controller featuring two colocated angular
position and velocity sensors was designed and experimentally implemented
in order to investigate and compare the control performance of the
flexible manipulator fabricated from aluminum (aluminum arm) and
composite laminates (composite arm). The controller design was
accomplished by utilizing the root-locus technique and the Lyapunov
method. In the root-locus analysis, the effect of the compensator zero
was investigated by considering various location of the compensator zero.
When aLlIunhzero open-loop poles and zeros were located left to the
,given compensatory the system exhibited a sluggish motion without
inducing overshoot. However, when the compensator zero was positioned far
left to all open-loop poles and zeros, the system exhibited a fast motion
with tradeoff the overshoot.

It has been demonstrated through the analytical and experimental

224

225

investigation that the composite arm had superior system performances
such as faster settling-time, smaller input torque, smaller overshoot and
a superior increment of stability relative to the aluminum arm. It was
obtained from the measured step responses that with certain position and
velocity feedback gains the delay time was 0.26 second for the aluminum
arm and 0.21 second for the composite arm, and the maximum overshoot was
1.02 degree for the aluminum arm and 0.3 degree for the composite arm. It
was also observed that the reduction of energy consumption by 40-50
percent for the commanded motion, and the reduction of the tip deflection
up to 50 percent was accomplished by employing the composite arm. These
experimental results clearly indicate the significant payoffs associated
with implementing composite materials featuring superior strength- and
stiffness-to-weight ratios to the commercial metals on the flexible
manipulators.

It has been also shown that the implemented output feedback
controller was very sensitive to the external disturbance which has a
forcing frequency close to the cantilevered-mode natural frequency of the
arm, regardless of the aluminum and composite arm. This attempt was
undertaken herein to introduce a single-link flexible manipulator
fabricated from smart materials incorporating electro-rheological fluids
(smart ER arm) to obtain the robustness of the output feedback controller
to the imposed disturbance with the help of distributed parameter
actuator, which is characterized by embedded electro-rheological (ER)
fluids in the structure.

As a sequel, the hybrid controller featuring the output feedback
controller employed for the aluminum and composite arm, and the pseudo-

state feedback controller associated with the distributed ER fluid

226

actuator was proposed and implemented on the smart ER arm in the Chapter
V. Before developing the hybrid control scheme, the phenomenological
equation of motion of the smart robotic system has been developed based
on the experimental observations of the modal characteristics of the
smart beam structure. Then, the hybrid controller was formulated based on
the phenomenological equation of motion and the empirical dynamic model
of the ER fluid actuator. The pseudo-control force (input voltage)
associated with the pseudo-state feedback controller accounts for the
change in the global damping and stiffness properties of the smart
robotic arm due to electrical potentials imposed on the ER fluid domains.
The stability criterion has been established for the robotic system by
considering the changed properties as the plant perturbation due to
electrical potentials. Moreover, this stability criterion has been
modified by incorporating the empirical dynamic model of the ER fluid
actuator. The modified criterion says that the stable closed-loop system
with the output feedback controller remains as stable for positive input
voltage considered in this investigation. It has been shown through the
experimental implementation that the robustness of the employed
conventional output feedback controller was significantly enhanced by
employing the proposed pseudo-state feedback controller in the presence
of the disturbance.

The nonlinear robust state feedback controller for the composite arm
was developed for the first time to overcome uncertain system parameters
associated with the varying payload encountered in the operation and the
mismatched lamina angle encountered in the manufacturing process. The
controller design has been accomplished by utilizing the properties of

uniform and uniform ultimate boundedness of the solution. The major

227

difference of this approach from the currently existing approaches was
that the effect of each uncertain parameter as well as the effect of the
interaction of the uncertain parameters to the total system performance
was considered in this study, hence providing a certain measure of the
magnitude of each uncertain parameter. It has been demonstrated from the
simulation that the proposed nonlinear controller could tolerate larger
system parameter variations than the conventional linear controller,
showing much faster settling-time and smaller overshoot with the
nonlinear controller. It has been also shown that the system performance
was very sensitive to control parameters such as e.

The author believes that the research prosecuted in this thesis will
provide significant impact on the diverse marketplace such as automobile
industry and aerospace industry. And also, this complementary
theoretical, computational and experimental investigation on the control
methodologies for the flexible manipulators fabricated from composite
laminates or smart materials incorporating ER fluids will furnish
fundamental and important guidelines, and precious information for the
development of the next generation robotic system characterized by high-
speed of operation, high end-point positional accuracy, and also high

adaptability to unstructured environments.

7 . 2 Reco-endations for Future Work

As observed in Chapter V of this thesis, the proposed hybrid
controller was implemented on the smart robot arm by applying the pseudo-
control force (input voltage) in the fashion of the open-loop control.
Hence, the information of the disturbance such as a forcing frequency

should be identified prior to applying the proper input voltage. However,

228

it is possible to construct a closed-loop control system of the pseudo-
state controller by employing appropriate output sensors to obtain the
information of the imposed disturbance, and by identifying the system
parameters as a function of the input voltage in the frequency domain.
From the disturbance information and identified system parameters, the
appropriate pseudo-control force can be obtained and supplied from the
microprocessor to the embedded ER domains in real time. Thisiclosed-loop
system will highlight the distinct merit of employing the smart robotic
system which can respond to unstructured environmental conditions.

Another potential future work is to develop a flexible robotic
system featuring a high level of hybrid controller associated with
conventional actuators and various type of actuators incorporating smart
materials. The high level of controller consists of more than three
actuators which provide own individual actuating force for the different
purpose and also in a different manner. Since each smart material such as
ER fluids, piezoelectric materials and shape memory alloys has its own
distinct characteristic as observed in the literature review in Chapter
i1, it is possible to construct the high level of controller which can
adaptively respond to highly unstructured environments. As an example,
consider a flexible robotic system featuring a conventional actuator, an
ER fluid actuator and a piezoelectric actuator. When the system needs to
be settled very fast, then the control force associated with the
conventional actuator and the piezoelectric actuator are supposed to be
activated to increase damping properties. However, if the system is
subjected to a disturbance which has a forcing frequency close to the
natural frequency of the robot arm, the control force associated with the

ER actuator and the conventional actuator are supposed to be activated in

229

order to avoid the resonance phenomenon of the system. This high level of
hybrid controller will provide tremendous advantages over the
conventional controller in real world. The potential advantages include
the self-controllability of the plant, the robustness of the controller
to the uncertainties such as external and internal disturbances, the
flexibility of the trajectory or work space, and the flexibility of all
control parameters.

In parallel with above mentioned future work, it may be useful to
develop a hierarchical control methodology for the smart flexible
structures. The principal idea behind the hierarchical control
methodology is that the overall control system is divided into more than
two sub-control system ; for example, global controller, intermediate
controllers and regional controllers in the flexible structure. By
considering each finite segment actuator incorporating ER fluids or
piezoelectric materials as a regional controller, sum of several segment
actuators as an intermediate controller, and sum of all intermediate
controllers as a global controller, the total system can be controlled
for the desired system performances. This control scheme will be very
useful especially for the very large flexible structures such as space
station. Since there always exist wave propagation delay during the
sensing of the outputs in the large flexible structures, the system
instability may be arisen.due to the observation spillover caused by the
delay. This observationxspillover problem can be eliminated by employing
the hierarchical control methodology. The optimal placement of the
appropriate actuator associated with each regional controller is one of
the problem to be solved in the process of formulating the hierarchical

controller.

230
Finally, the extension of the nonlinear robust state feedback
controller developed in Chapter VI to the multi-links flexible
manipulators, and experimental implementation of this controller are

remained as a future work.

APPENDIX

APPENDIX

List of Publications

Thesis

1.

S.B. Choi, " On Robust Tracking in Uncertain Systems - A Variable
Structure Approach," Master Thesis, Department of Mechanical
Engineering, Michigan State University, June 1986.

S.B. Choi, " Control of Single-Link Flexible Manipulators Fabricated
from Advanced Composite Laminates and Smart Materials Incorporating
Electra-Rheological Fluids," Ph D Thesis, MEL/IMSL, Department of
Mechanical Engineering, Michigan State University, June 1990.

 

Refereed Journal Elications

1.

M.V. Gandhi, B.S. Thompson, S.B. Choi and S. Shakir, " Electro-
Rheological-Fluid-Based Articulating Robotic Systems, " ASME Jougnal

9f Meghanisns, Transmissions, and Automation in Design, Vol.111, No.
3, 1989, pp. 328-336.

S.B. Choi, M.V. Gandhi, B.S. Thompson and C.Y. Lee, " An Experimental
Investigation of an Articulating Robotic Manipulator with a Graphite-

Epoxy Composite Arm," ournal of Robotic Systems, Vol.5, No.1, 1988,
pp. 73-79.

M.V. Gandhi, B.S. Thompson and S.B. Choi, " A New Generation of
Innovative Ultra-Advanced Intelligent Composite Materials Featuring
Electra-Rheological Fluids: An Experimental Investigation," Jouznal

9f Qonngsige Materials, Vol.23, 1989, pp. 1232-1255.

M.V. Gandhi, B.S. Thompson and S.B. Choi, " A Proof-of-Concept
Experimental Investigation of a Slider-Crank Mechanism Featuring a
Smart Dynamically-Tunable Connecting-Rod Incorporating Embedded

Electra-Rheological Fluid Domains," Journal of Sound and Vibration,
Vol. 135, No.3, 1989, pp. 511-515.

S.B. Choi, C.Y. Lee, B.S. Thompson and M.V. Gandhi, " Elastodynamic
Characteristics of Smart Beams Incorporating Electra-Rheological

F1uids.," AIAA Journal of Guidance, Control, and Dynamics (under
review).

231

10.

ll.

12.

13.

14.

232

S.B. Choi, A. Magolan, B.S. Thompson and M.V. Gandhi, " An
Experimental and ,Theoretical Investigation of the Static and
Elastodynamic Responses of an Industrial Robotic Manipulator
Featuring a Graphite-Epoxy Composite Arm," ournal of Robotic Systems
(under review).

S.B. Choi and R.L. Tummala, " Modeling and Control of a Flexible
Single-Link Robotic Arm Fabricated from Composite Materials," lEEE

Iransgggions on Robotics and Automation (under review).

S.B. Choi, B.S. Thompson and M.V. Gandhi, " Smart Structures
Incorporating Electra-Rheological Fluids for Vibration-Control and
Active-Damping Applications: An Experimental Investigation," M
Joninnl 9f Vibration, Acousticfis, Stress and Reliability in Design

(under review) .

 

S.B. Choi, B.S. Thompson and M.V. Gandhi, " Modeling and Position
Control of a Single-Link Flexible Robotic Manipulator Fabricated from
Advanced Composite Materials," Journal of Robotic Systems (under
review).

S.B. Choi, B.S. Thompson and M.V. Gandhi, " Modeling and Output
Feedback Control of a Single-Link Flexible Robotic Manipulator
Featuring a Graphite-Epoxy Composite Arm,” W
Systens, Measnrenent, and Control (under review).

" A Nonlinear Control Robust to System Parameter Variations with
Application to a Single-Link Flexible Manipulator Fabricated from
Composite Laminates," (in preparation).

" Modeling and Control of Smart Flexible Structures Incorporating
Piezoelectric Materials," (in preparation).

" Active-Vibration-Tuning of a Simply-Supported Beam Featuring
Electra-Rheological Fluids,“ (in preparation).

" Modeling and Control of a Single-Link Flexible Robotic Manipulator
Fabricated from Smart Materials Incorporating Electra-Rheological
Fluids, " (in preparation) .

Confergngg Elications

1.

S.B. Choi and S. Jayasuriya, " A Sliding Mode Controller
Incorporating Matching Conditions Applied to Manipulators,"

Pinceedings oi the 10th IFAC World Congress, Munich, West Germany,
July 1987.

S. Jayasuriya and S.B. Choi, " On the Sufficiency Condition for
Existence of a Sliding Mode," 0 eed o Ame a o
Coniegence, Minneapolis, Minn., 1987, Vol.1, pp. 84-89.

10.

11.

233

S.B. Choi, F. Sun, B.S. Thompson and M.V. Gandhi, " An Experimental
Investigation of the Static and Dynamic Responses of an Industrial

Robotic Manipulator," Proceedings of 1982 ASME Design Automation

Conferance, Boston, Mass., Sept. 1987, ASME Paper No. 87-DAC-50, DE-
Vol.10-2, pp. 421-427.

M.V. Gandhi, B.S. Thompson, S.B. Choi and S. Shakir, " Electro-
Rheological-Fluid-Based Articulating Robotic System,” Proceedings of

l982 ASME Design Automation Conference, Boston, Mass. , Sept. 1987,
ASME Paper No. 87-DAC-55, DE-Vol.10-2, pp. 1-10.

M.V. Gandhi, B.S. Thompson and S.B. Choi, " Smart Ultra-Advanced
Composite Materials Incorporating Electra-Rheological Fluids for
Military, Aerospace, and Advanced Manufacturing Applications,"
Proceedings 9f ghe Bid Annual ASM/ESD Advanced Composiga Confeijence
and Expositign, Detroit, MI, Sept. 1987, pp. 23-30.

S.B. Choi, B.S. Thompson and M.V. Gandhi, ” Electra-Rheological
Fluids Technology Stimulates a New Generation of Robotic and Machine

Systems," Pioceedings of the 10th Annlied Mechanism Conference, New
Orleans, La., December 1987, Vol.1, pp. 3B.1-381.8.

S.B. Choi, M.V. Gandhi, B.S. Thompson and C.Y. Lee, " An Experimental
Investigation of an Articulating Robotic Manipulator with a Graphite-

Epoxy Composite Arm," Pioceedings 9f the l0§h Annliea Mechanism
Conteranae, New Orleans, La., December 1987, Vol.1, pp. 3B2.1-3B2.8.

M.V. Gandhi, B.S. Thompson and S.B. Choi, " Ultra-Advanced Composite
Materials Incorporating Electra-Rheological Fluids," Prageedings 9i

ghe 4th Japan-US Composite Conference, Washington D.C., June 1988,
pp. 875-884.

M.V. Gandhi, B.S. Thompson and S.B. Choi, " Smart Materials
Incorporating Electra-Rheological Fluids: A Theoretical and

Experimental Investigation," Proceedings of the 18th Annual Meeting
of ghe Eina Particle Society, 1987 Particle/Power Technology,

Microcontamination and Biotechnology Exhibition and Technology Forum.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " An Experimental
Investigation on the Active-Damping Characteristics of a Class of
Ultra-Advanced Intelligent Composite Materials Featuring Electro-
Rheological Fluids," Proceedings of the Damping '89 Conferenge, West
Palm Beach, Fl. , February 1989, Organized by the Flight Dynamics
Laboratory of the Air Force Wright Aeronautical Laboratories, Wright
Patterson Air Bases, Ohio.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " An Active Vibration-
Tuning Methodology for Smart Flexible Structures Incorporating
Electra-Rheological Fluids: A Proof-of-Concept Investigation,”

Proceedings of 1989 American Control Confegenae, Pittsburgh, Pa. ,
June 1989.

 

12.

l3.

14.

15.

16.

17.

18.

234

S.B. Choi, B.S. Thompson and M.V. Gandhi, " Smart Structures
Incorporating Electra-Rheological Fluids for Vibration-Control and
Active-Damping Applications: An Experimental Investigation,"
roceedi s of the t B ennial ASME Con ere ce 0 echa ical
Vibration and Noise, Montreal, Canada, Sept. 1989.

S.B. Choi, A. Magolan, B.S. Thompson and M.V. Gandhi, ” An
Experimental and Theoretical Investigation of the Static and
Elastodynamic Responses of an Industrial Robotic Manipulator
Featuring a Graphite-Epoxy Composite Arm,“ P oceed n s o the lst

Natipnal Qpnference on Applied Mechanisms ana Robotics, Cincinnati,

Ohio, Nov. 1989, Vol.2, pp. 8B5.1-8B5.11.

S.B. Choi, L.P. Chao, B.S. Thompson and M.V. Gandhi, " An Integrated
Design Strategy for the Optimal Structural Fabrication and the
Optimal Control of a Single-Link Flexible Manipulator Fabricated
from Advanced Composite Laminates," Proceedings of the 1st National

Conference an Applied Mechanisms and Robotics, Cincinnati, Ohio, Nov.
1989, Vol.2, pp. 8B4.1-8B4.10.

S.B. Choi, M.V. Gandhi and B.S. Thompson, " An Experimental
Investigation of the Elastodynamic Responses of a Slider-Crank
Mechanism Featuring a Smart Connecting-Rod Incorporating Embedded

Electra-Rheological Fluid Domains," Proceedings of the lst National

Conference on Applied Mechanism and Robotica, Cincinnati, Ohio, Nov.
1989, Vol.2, pp. 9B3.1-9B3.7.

 

C.Y. Lee, S.B. Choi, B.S. Thompson and M.V. Gandhi, " A Variational
Formulation for the Finite Element Analysis and Control of Linkage
and Robotics Systems Featuring Smart Materials Incorporating

Piezoelectric Materials," Proceedings of the lst National Conference

an Applieg Mechanisms and Robotics, Cincinnati, Ohio, Nov. 1989,
Vol.1, pp. 2C1.1-2C1.10.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " A Theoretical and
Experimental Investigation on the Control of a Single-Link Flexible
Robotic Manipulator Fabricated from Composite Materials," to ha

ppesanpaa a; tha 1990 American Control Conference, San Diego, CA, May
1990.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " Modeling and Feedback
Control of a Single-Link Flexible Robotic Manipulator Featuring a

Graphite-Epoxy Composite Arm," to be presented at the l990 lEEE
lntepnapional Conference on Robotics and Autonation, Cincinnati,

Ohio , May 1990.

Technical Mr};

1.

S. Shakir, L.P. Chao, S.B. Choi, M.V. Gandhi and B.S. Thompson, "
Smart Materials Incorporating Electra-Rheological Fluids Technology
for Articulating Systems and Advanced Structures," MSU Machinery

Elaapodynanias Lab, Report, Report No. 870108801, Feb. 1987.

2.

235

S.B. Choi, B.S. Thompson and M.V. Gandhi, " The Synthesis of Smart
Materials Incorporating Electra-Rheological Fluids for Military,
Aerospace and Advanced Manufacturing Applications: An Interim

Report," MSQ Maahinegy Elaagpaynaniaa Lab, gepopt t9 Laid
Winn, NC, Report No. 87082501, July 1987.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " The Synthesis of Smart
Materials Incorporating Electra-Rheological Fluids for Military,
Aerospace and Advanced Manufacturing Applications: An Interim

Report," U Ma iner Elastod n m cs ab e art to Lo (1
Winn, NC, Report No. 87082502, December 1987.

S.B. Choi, B.S. Thompson and M.V. Gandhi, " The Synthesis of Smart
Materials Incorporating Electra-Rheological Fluids for Military,
Aerospace and Advanced Manufacturing Applications: A Final Report,"
SU a e E a tod am cs a e o to Lord Cor oratio , NC,
Report No. 87082503, March 1988.

BIBLIOGRAPHY

10.

11.

BIBLIOGRAPHY

Abou-Hanna,J.J. and Evces,C.R. , " Dynamics and Control of Flexible

Manipulators," Proceedinga pi ASME Winger Annual Meeting, Boston,
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