A Physics-Based Model of Electro-ActivePolymer Actuators as the Basis for a
Gopinath-Style Motion State Observer
by
Christoph Michael Hackl
A Thesis presented for the degree of
Bachelor of Science
(Electrical Engineering)
Lehrstuhl für Elektrische Antriebssysteme
Technische Universität München, Germany
Professor Dr.-Ing. Dr.-Ing. h.c. D. Schröder
Department of Mechanical Engineering
Department of Electrical & Computer Engineering
University of Wisconsin-Madison, United States of America
Professor Ph.D. PE Fellow-IEEE R.D. Lorenz
December 2003
A Physics-Based Model of
Electro-Active Polymer Actuators as the Basis for a
Gopinath-Style Motion State Observer
by
Christoph Michael Hackl
Under the supervision of
Professor Dierk Schröder (Technische Universität München)
and
Professor Robert D. Lorenz (University of Wisconsin-Madison)
Approved by ____________________________________
Dierk Schröder, Date
____________________________________
Robert D. Lorenz, Date
A Physics-Based Model of
Electro-Active Polymer Actuators as the Basis for a
Gopinath-Style Motion State Observer
by
Christoph Michael Hackl
Submitted for the degree of Bachelor of Science
(Electrical Engineering)
December 2003
Abstract
In this thesis a physics-based model of an electro-active polymer (EAP) actuator is
developed. Measurements were carried out and system parameters were estimated.
Simulations of the derived nonlinear model were run to provide insight into the qual-
itative behavior of an EAP actuator. An operating point model is established and a
Gopinath-style motion state observer was developed, simulated and evaluated. The
research of this thesis represents a fundamental basis for a later implementation of a
closed loop control system using Gopinath-style observer theory to estimate the actual
displacement or motion of an electro-active polymer actuator.
Declaration
The work in this thesis is based on research carried out at the Department of Mechanical
Engineering and the Department of Electrical & Computer Engineering, University of
Wisconsin, Madison. No part of this thesis has been submitted elsewhere for any other
degree or qualification and it is all my own work unless referenced to the contrary in
the text.
Copyright c© 2003 by Christoph Michael Hackl.
“The copyright of this thesis rests with the author. No quotations from it should be
published without the author’s prior written consent and information derived from it
should be acknowledged”.
iv
Acknowledgements
I would like to thank my family and my friends for their love and support, encouraging
me over and over again during this often frustrating work.
I am deeply grateful for the help and the inspiring thoughts provided by Professor
Robert D. Lorenz. By his advice I am well prepared for all my future work.
I would like to thank Ray Tang, who made great efforts in preparing polymer spec-
imens and maintaining the test set-up. I appreciate the productive team work on this
most challenging project.
Finally, I am in debt to my supervisor Professor Dierk Schröder, who made all this
possible. I am very thankful for the oppertunity to gather experience abroad.
v
Contents
Abstract iii
Declaration iv
Acknowledgements v
1 Introduction 3
1.1 Observers and State Filters . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Comparison of Closed Loop Observer Topologies . . . . . . . . . . . . . 6
1.3 Electro-Active Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Project Overview and Preparatory Work 12
2.1 Overview of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Used Hardware Components . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Preparatory Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Physics Based Model of Polymer and Set-up 19
3.1 Dynamics Model of Polymer and Set-Up . . . . . . . . . . . . . . . . . 19
3.1.1 An Introduction to Finite Elasticity . . . . . . . . . . . . . . . . 19
3.1.2 Neo-Hookian Model of the Elastomer . . . . . . . . . . . . . . . 24
3.1.3 Prestrained Polymer by External Load . . . . . . . . . . . . . . 27
3.1.4 Demonstrative Example of an Elastic Deformation . . . . . . . . 28
3.1.5 Kelvin-Voigt Damping . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.6 Combined State Space Model of Test Set-Up Dynamics . . . . . 30
vi
Contents vii
3.2 Electrical Circuit Model Of Polymer . . . . . . . . . . . . . . . . . . . 32
3.3 Force Output of a Charged Polymer . . . . . . . . . . . . . . . . . . . . 34
3.3.1 Induced Pressure of Applied Electric Field . . . . . . . . . . . . 34
3.3.2 Calculation of the Electric Field Eel . . . . . . . . . . . . . . . . 35
3.3.2.1 Electric Field of a Charged Disc . . . . . . . . . . . . . 35
3.3.2.2 Electric Field of Two Parallel Compliant Electrodes . . 36
3.3.3 Pressure pz and Compressive Force Fz Acting Along the z-Axis 37
3.3.4 Electrostrictive Transduction . . . . . . . . . . . . . . . . . . . 37
3.4 Complete Nonlinear Model of Polymer Set-Up . . . . . . . . . . . . . . 40
3.5 Encountered Problems and Difficulties . . . . . . . . . . . . . . . . . . 42
4 Parameter Estimation and Measurement 45
4.1 Modulus of Elasticity E (Young’s modulus) . . . . . . . . . . . . . . . 45
4.2 Damping Coefficient CD . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Dielectric Constant ǫr of Polymer Material 3M VHB 4905 . . . . . . . 49
4.4 Resistance Rp + R and Inductance Lp . . . . . . . . . . . . . . . . . . . 49
4.5 Friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.1 Static Friction Fstatic . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5.2 Sliding Friction Fsliding . . . . . . . . . . . . . . . . . . . . . . . 52
4.5.3 Conclusion of the Friction Estimation . . . . . . . . . . . . . . . 54
4.6 Displacement Measurement for an Applied Voltage . . . . . . . . . . . 54
4.7 Unexpected Problems during Measurements . . . . . . . . . . . . . . . 55
5 Simulation of Nonlinear Polymer Model 57
5.1 Simulation of Charge and Capacitance . . . . . . . . . . . . . . . . . . 57
5.2 Simulation of Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Simulation of Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Simulation of Voltage-Displacement Relation . . . . . . . . . . . . . . . 60
Contents viii
6 Gopinath-Style Motion State Observer 62
6.1 Condition of Equilibrium (Operating Point) . . . . . . . . . . . . . . . 62
6.2 Operating Point Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Development of Gopinath-style Observer . . . . . . . . . . . . . . . . . 68
6.4 Evalution of Estimation Accuracy . . . . . . . . . . . . . . . . . . . . . 72
6.4.1 Influence of Parameter Errors of prestrained Length Lpre0 , Width
W pre0 and Thickness Hpre
0 . . . . . . . . . . . . . . . . . . . . . . 74
6.4.2 Influence of Parameter Errors of PWM Driver Gain Kpwm and
High Voltage Converter Gain Khvc . . . . . . . . . . . . . . . . 75
6.4.3 Influence of Parameter Errors of Polymer Resistance Rp and In-
ductance Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.4.4 Influence of Parameter Errors of Young’s Modulus E, Damping
Coefficient CD and Dielectric Constant ǫr . . . . . . . . . . . . . 76
6.4.5 Disturbance Estimation Accuracy . . . . . . . . . . . . . . . . . 77
6.4.6 Remarks and Observations . . . . . . . . . . . . . . . . . . . . . 78
7 Conclusion 79
Bibliography 81
Appendix 85
A List of Symbols 85
B Hardware Configuration 90
B.1 HardwareSetup.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
B.2 getEncoderData.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.3 OutputSerial.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.4 ControlPWM.c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
C Maple Commands 109
List of Figures
1.1 State Block Diagram of Physical System and General Structure of a
Real-Time Observer (Open and Closed Loop) . . . . . . . . . . . . . . 5
1.2 State Block Diagram of Physical System - DC Motor Drive . . . . . . . 6
1.3 State Block Diagram of Enhanced Luenberger-Style Observer . . . . . . 7
1.4 State Block Diagram of Enhanced Gopinath-Style Observer . . . . . . . 7
1.5 High Bandwidth Observers - FRF ω(s)ω(s)
with Paramter Errors for Kt =
(1 ± 0.25)Kt, Jp = (1 ± 0.30)Jpand Ke = (1 ± 0.25)Ke . . . . . . . . . . 8
1.6 Low Bandwidth Observers - FRF ω(s)ω(s)
with Paramter Errors for Kt =
(1 ± 0.25)Kt, Jp = (1 ± 0.30)Jp and Ke = (1 ± 0.25)Ke . . . . . . . . . 9
1.7 Disturbance Estimation - FRF Td(s)Td(s)
without Paramter Errors . . . . . . 10
2.1 Test Setup Components and Polymer . . . . . . . . . . . . . . . . . . . 13
2.2 Components Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Counter Chip Connection Diagram . . . . . . . . . . . . . . . . . . . . 17
2.4 PWM Driver Chip Connection Diagram . . . . . . . . . . . . . . . . . 18
3.1 Reference (Lagrangian) & Deformed (Eulerian) Configuration . . . . . 20
3.2 Cauchy Stress Components . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Deformation of Polymer as a Result of Applied Pressure pz . . . . . . . 28
3.4 Kelvin-Voigt Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Nonlinear State Block Diagram of Set-Up Dynamics . . . . . . . . . . . 31
3.6 Electrical Circuit of Polymer . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 State Block Diagram of Electrical Circuit . . . . . . . . . . . . . . . . . 34
ix
List of Figures x
3.8 Sketch of Charged Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.9 Electric Field of Parallel Plates . . . . . . . . . . . . . . . . . . . . . . 37
3.10 Complete State Block Diagram of Nonlinear Polymer System . . . . . . 43
3.11 Changes in Test Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.1 Tensile Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Nominal Stress Curvefit - Young’s Modulus E with Glued Compliant
Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Nominal Stress Curvefit - Young’s Modulus E without Compliant Elec-
trodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Measuring Points on Compliant Electrode . . . . . . . . . . . . . . . . 50
4.5 Results of Resistance Measurements . . . . . . . . . . . . . . . . . . . . 51
4.6 Results of Inductance Measurements . . . . . . . . . . . . . . . . . . . 51
4.7 Static friction measurement setup . . . . . . . . . . . . . . . . . . . . . 52
4.8 RLSQM - Excitation with Square Wave . . . . . . . . . . . . . . . . . . 53
4.9 Burned Polymer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Simluted Changes in Charge and Capacitance . . . . . . . . . . . . . . 58
5.2 Simulated Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Simulated Changes in Dimensions . . . . . . . . . . . . . . . . . . . . . 60
5.4 Simulated Relation of Displacement ux and Input Voltage ei . . . . . . 61
6.1 State Block Diagram of Operating Point Model of System and Gopinath-
Style Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Estimation Accuracy Frequency Response ux
uxfor Dimension Parameter
Errors Lpre0 = (1±0.1)Lpre
0 , W pre0 = (1±0.1)W pre
0 and Hpre0 = (1±0.25)Hpre
0 74
6.3 Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of
PWM Gain Kpwm = (1 ± 0.1)Kpwm and High Voltage Converter Gain
Khvc = (1 ± 0.2)Khvc . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
List of Figures xi
6.4 Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of
Polymer Resistance Rp = 1.6Rp and 0.6Rp and High Voltage Converter
Gain Khvc = (1 ± 0.2)Khvc . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.5 Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of
Dielectric Constant ǫr = (1 ± 0.05)ǫr, Young’s Modulus E = (1 ± 0.2)E
and Kelvin-Voigt Coefficient CD = 1.8CD and 1.4CD . . . . . . . . . . 77
6.6 Disturbance Estimation Accuracy Frequency Response − Fdist
Fdistwithout
Parameter Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.1 Measured hysteresis - deformation in length and width, when weight is
applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Thesis Outline
A brief overview of the chapter contents is presented to give an outline of this thesis.
It will help the reader to find their way through.
In chapter one the state of the art of modern sensor replacement methods and its
general idea are discussed. A comparison of two commonly used observer topologies is
illustrated to show their capabilities. To introduce the reader to electroactive polymer
actuators a review of the latest research in this field is added.
Chapter two describes the project set-up and lists the used hardware components.
Also the preparatory work with the encountered difficulties is outlined.
The core of this thesis is the developement of a physics based model for the used
electro-active polymer actuator set-up. The derivation of the model is clearly arranged
in chapter three. Commencing with modeling the elastic properties of an elastomeric
polymer and the electrical circuit, a combined nonlinear model is derived, also decribing
the force output of the polymer actuator.
Chapter four presents measurements and estimations for unkown parameters in the
model. Due to unpredictable circumstances the results in this chapter are not satisfying
so far and should be extended and repeated (not possible within the given time-frame).
In chapter five the developed nonlinear model is implemented and simulated in
Matlab/Simulink for a better understanding of the system behavior. The most
significant properties are illustrated in diagrams and interpretations are given.
In order to build and analyze the desired Gopinath-style observer topology, chapter
six begins to derive an operating point model of the nonlinear system at an equilibrium
condition. The Gopinath-style observer and the “real” physical model are implemented
1
List of Figures 2
in Matlab/Simulink and thus the estimation accuracy is evaluated and presented at
the end of chapter six.
The thesis concludes with chapter seven. The accomplished work is summarized
and the directions of future research are outlined.
Chapter 1
Introduction
1.1 Observers and State Filters
Control systems essentially need feedback signals normally provided by sensors to
achieve closed-loop feedback to make the system controllable. Especially in drive and
power electronic systems, signals are often needed for the control loop. These signals
are difficult to sense or only are measurable using very costly and special implemented
sensors, for example for flux, temperature and torque measuring. This undesired factor
led to the development of observers and state filters as “sensor replacement” [8] meth-
ods. Often these methods are misleadingly called “sensorless”, though sensors are still
needed to feed the control system. More affordable and more easily integrated sensors
are used to sense variables with which the desired system states are estimated. The
desired system states are often more difficult and more expensive to measure.
Observers can be described as real-time (mathematical) models of physical systems.
The same measured or commanded inputs are provided to both the observer and the
real system. Thus observers estimate the system response of the real system due to
its inputs. When a controller is additionally implemented to the real time system, the
observers are forced to converge on the measured states.
The observer theories of Luenberger [10] and Gopinath [11] are based on linear alge-
braic models of the mathematical representation of the real physical system. Normally
3
1.1. Observers and State Filters 4
the state space algebraic model is used, as shown:
x = Ax + BUm + DUd
y = Cx (1.1)
where A is the state feedback matrix, x is the (physical) state vector of the system,
B is the manipulated input coupling matrix, Um is the manipulated input vector, D
is the disturbance input coupling matrix, Ud is the disturbance input vector, C is the
measurement selection matrix and y is the measurement output vector.
For observers a similar model is used, only the estimated parameters and values
are indicated using “^”. The disturbances are not implemented in the real time model,
since those are generally not known or predictable. The general form of an open and
closed loop observer is shown in Figure 1.1.
The inputs used for open loop observers are only those fed to the physical system.
These command feedforward inputs allow the real time model to estimate states and
therefore track the states of the physical system dynamically. If the model reflects
the real system accurately, the open loop observer is tracking the system states with
zero lag [8]. Although model and parameter errors limit the zero lag property and
the estimation accuracy and disturbances are not estimated, open loop observers are
commercially used for stator and rotor flux estimation in induction motor drives [8].
Closed loop observers can improve the estimation behavior by adding an extra ref-
erence input and a controller K0 (see Figure 1.1) to the real time model. To maintain
the desired zero lag property, only deviations between reference inputs (e.g. measured
states) and the corresponding estimated states should be controlled. Not measured
states should not be included in the controller, this would result in a loss of the zero
lag property [8]. With a properly formed closed loop observer the parameter sensitivity
is reduced and thus estimation accuracy improved. Disturbance estimation is also per-
formed by the observer controller, but it is mostly depend on the observer bandwidth
1.1. Observers and State Filters 5
1
s
D
CB
A
mU
dU
xx
xx
−
++
y
1
s
mU
−
+ yB
A
C+
0K+
−
Actual Inputs
State Feedback
ReferenceInput to
Observer
Feedforward Inputto Observer
Estimated Disturbance
Estimated State Feedback
+
Physical System
Open Loop Observer
Closed LoopObserver
Figure 1.1: State Block Diagram of Physical System and General Structure of a Real-Time Observer (Open and Closed Loop)
and should be taken into account when tuning the observer controller gains [8].
A state filter is a simplified form of the closed loop observer. No feedforward input
is applied to state filters, therefore, in general, state filters have a phase lag problem
and no disturbances can be estimated [8].
The use of the estimation accuracy frequency response (FRF) has shown to be
helpful [9] in comparing and evaluating estimation accuracies of observers. Therefore,
the transfer function relating estimated state X(s) to actual state X(s) is plotted and
analysed, using a very common method in control engineering called Bode diagrams. A
linear scale in magnitude∣
∣
∣
X(s)X(s)
∣
∣
∣is prefered to make the deviation more evident [8,9]. If
the estimation is absolutely accurate the transfer function would be X(s)X(s)
= 1 and thus
the magnitude∣
∣
∣
X(s)X(s)
∣
∣
∣= 1 and the phase ∠
X(s)X(s)
= 0
1.2. Comparison of Closed Loop Observer Topologies 6
1.2 Comparison of Closed Loop Observer Topologies
Parameter sensitivity is still inherent in closed loop observers. The parameter and
bandwidth sensitivity of different observer topologies is demonstrated in this section,
a simple DC motor drive system1 (see Figure 1.2) is picked to analyze the different
estimation accuracies.
dT
+
−
1
sai ω−
+ae 1
s−
1
pL
Physical System
pR
ω 1
sTK
1
pJθ
eK
Feedforward Input Reference Input depending onObserver Topology
Closed Loop ObserverEstimates
Figure 1.2: State Block Diagram of Physical System - DC Motor Drive
Two different observer structures are build to estimate the velocity ω. The Luenberger-
style observer and the Gopinath-style observer are chosen in enhanced structure. In
general Luenberger observers estimate inner states using the measured outermost state
1This example is picked from Project 4, Lecture ME 746 by Prof. Robert D. Lorenz, “Dynamics ofControlled Systems” , University of Wisconsin, Madison [12]
1. Parameters: Armature Resistance Rp = 2.6Ω, Armature Inductance Lp = 0.02H , Back EMFConstant Ke = 0.14 V s
rad, Torque Constant KT = 0.14Nm
A, Moment of Inertia Jp = 15 · 10−6kg ·
m2, Disturbance Torque Td
2. Luenberger Gains: b0 = 0.0778Nms, Kso = 76.71Nm, Kio = 14420.8Nms
(high bandwidth),
b0 = 0.0777Nms, Kso = 0.767Nm, Kio = 1.44Nms
(low bandwidth),
3. Gopinath Gains: K1 = 0.00048NmsA
, K2 = 0.289NmA
, K3 = 14.285NmAs
(high bandwidth), K1 =
0.00024NmsA
, K2 = 0.0024NmA
, K3 = 0.0143NmAs
(low bandwidth)
1.2. Comparison of Closed Loop Observer Topologies 7
as reference input. In contrast, Gopinath observers approximate the outer states by
using measured internal states as reference and feedforward input. The observer shown
in Figure 1.3 is an enhanced Luenberger-style velocity observer which is fed with the
feedforward input voltage ea and the measured position θ (outermost state) as ref-
erence input. Figure 1.4 is depicting an enhanced Gopinath-style velocity observer.
The Gopinath-style observer is fed with the feedforward input voltage ea and with the
reference and feedforward input armature current ia (inner state).
Enhanced Luenberger-StyleVelocity Observer
1
sdT− +
−
1
sθ
ˆTK
ai
1
sioK
+
soK
ob
+ ˆlωˆ
lω
ˆRω
+
θ
+
+
+
Estimated Disturbance
Estimated Velocity
Reference Input,Position
Feedforward Input, Current
dT−
ω
1ˆ
pJ
1ˆ
pJ
Figure 1.3: State Block Diagram of Enhanced Luenberger-Style Observer
dT− +
−
1
s
ˆTK
ai
1
s+
+ ˆlωˆ
lω
ˆRω
++
+
+
Estimated Disturbance
Estimated Velocity
FeedforwardInput, Voltage
Reference & FeedforwardInput, Current
dT−
ω
1ˆ
pJ
1ˆ
pJ1K
2K
3K
ae
1
s−
+−
ˆpR
ai
ˆeK
Enhanced Gopinath-StyleVelocity Observer
1ˆ
pL
Figure 1.4: State Block Diagram of Enhanced Gopinath-Style Observer
1.2. Comparison of Closed Loop Observer Topologies 8
It has been shown that the Gopinath-style observers generally yield inferior per-
formance in estimation accuracy compared to the Luenberger-style observer [8]. This
inherent parameter sensitivity of the Gopinath structure can be explained with the “im-
plicit” estimation reference produced by the (virtual) open loop cancellation method [9],
which produces estimation errors even at low frequencies and within the observers band-
width due to a linear parameter sensitivity [8]. This undesired sensitivity is unavoidable,
because of the internal structure of this observer design.
The estimation accuracy frequency responses (FRFs) ω(s)ω(s)
of both observer topologies
can be seen in Figure 1.5 for high bandwidth and in Figure 1.6 for low bandwidth.
10−1
100
101
102
103
104
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
|what/w| Bode Diagram − Luenberger−Style Velocity Observer (high bandwidth)
10−1
100
101
102
103
104
−10
−5
0
5
10
15
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Kt+25%
Kt−25%
Jp+30%
Jp−30%
10−1
100
101
102
103
104
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
|what/w| Bode Diagram − Gopinath−Style Velocity Observer (high bandwidth)
10−1
100
101
102
103
104
−10
−5
0
5
10
15
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Kt+25%
Kt−25%
Jp+30%
Jp−30%
Ke+25%
Ke−25%
a) Luenberger-FRF b) Gopinath-FRF
Figure 1.5: High Bandwidth Observers - FRF ω(s)ω(s)
with Paramter Errors for Kt =
(1 ± 0.25)Kt, Jp = (1 ± 0.30)Jpand Ke = (1 ± 0.25)Ke
The Luenberger-style oberserver shows no parameter sensitivity within its band-
width for both low and high bandwidth configuration (see Figure 1.5a and Figure 1.6a).
Beyond the observers bandwidth the estimation errors scale linearly with the parameter
estimation errors. It also shows minimal phase deviations in estimation accuracy, even
beyond the observer’s bandwidth, as a direct result of the feedforward path. Within
the bandwidth the zero lag property is maintained by the proper design of the observer
controller, which forces the structure to converge on the measured input. The distur-
bance estimation frequency response is accurate in magnitude and phase within the
1.2. Comparison of Closed Loop Observer Topologies 9
bandwidth of the observer (see Figure 1.7), but due to the missing knowledge of the
disturbance and thus the lack of a feedforward disturbance input, the disturbance es-
timates lag beyond observer bandwidth frequencies. The disturbance is filtered within
the bandwidth and can be used for “disturbance input decoupling control” [8].
10−1
100
101
102
103
104
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
|what/w| Bode Diagram − Luenberger−Style Velocity Observer (low bandwidth)
10−1
100
101
102
103
104
−10
−5
0
5
10
15
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Kt+25%
Kt−25%
Jp+30%
Jp−30%
10−2
10−1
100
101
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
|what/w| Bode Diagram − Gopinath−Style Velocity Observer (low bandwidth)
10−2
10−1
100
101
−15
−10
−5
0
5
10
15
Pha
se [°
, lin
ear]
Kt+25%
Kt−25%
Jp+30%
Jp−30%
Ke+25%
Ke−25%
a) Luenberger-FRF b) Gopinath-FRF
Figure 1.6: Low Bandwidth Observers - FRF ω(s)ω(s)
with Paramter Errors for Kt =
(1 ± 0.25)Kt, Jp = (1 ± 0.30)Jp and Ke = (1 ± 0.25)Ke
In stark contrast, the Gopinath-style designed observer shows estimation errors even
at frequencies within its bandwidth (see Figure 1.5b and Figure 1.6b), because of the
observer structure with its inherent implicit reference [9]. This topology also shows a
greater sensitivity on the chosen eigenvalue placement, resulting in the observer band-
width. For low and high bandwidth Gopinath-style observers’ minimal phase deviations
can be maintained even beyond and near the configured bandwidth (see Figure 1.5b
and Figure 1.6b). Disturbance magnitude estimation is nearly accurate within the ob-
server bandwidth, but disturbance phase deviation mainly depends on the configured
observer bandwidth (see Figure 1.7a,b).
Although a Gopinath-style observer shows inferior estimation behavior compared to
the Luenberger-style oberserver, it is an attractive choice for selected control systems,
where several influence issues have to be balanced such as cost factors of used sensors or
1.3. Electro-Active Polymers 10
10−1
100
101
102
103
104
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]|Tdhat/Td| Bode Diagram − Disturbance Estimation Accuracy (low bandwidth)
10−1
100
101
102
103
104
−200
−150
−100
−50
0
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
LuenbergerGopinath
10−1
100
101
102
103
104
0
0.5
1
1.5
2
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
|Tdhat/Td| Bode Diagram − Disturbance Estimation Accuracy (high bandwidth)
10−1
100
101
102
103
104
−200
−150
−100
−50
0
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
LuenbergerGopinath
a) Low Bandwidth Observers b) High Bandwidth Observers
Figure 1.7: Disturbance Estimation - FRF Td(s)Td(s)
without Paramter Errors
structural constraints including small components which are needed, but the available
sensors are to big in size. A properly designed Gopinath-style observer is an adequate
and satisfying structure providing acceptable state estimates to build a closed loop
system. Due to the fact this topology uses easily measurable states like current or
voltage as reference inputs and those needed sensors are inexpensive, a control system
with a Gopinath-style designed observer is a cost-efficient solution.
1.3 Electro-Active Polymers
Polymers were subject of research in recent years, due to many attractive characteris-
tics. Polymers are lightweight, inexpensive, fracture tolerant, pliable and easily config-
urable [1]. Since the 1990s new polymers have been develeped, which have a significant
deformation response to electrical stimulation. Dielectric E lectro-Active Polymers
(EAP) [1] showed especially enormous deformation levels for applied electric fields of
≈ 100 Vµm
, e.g. the commercially available adhesive 3MTM V HBTM tapes are able to
produce planar strains of more than 300% [1]. These electro-active polymers became of
interest for more and more engineers and scientists of several disciplines, due to their
conceivable potentials such as the possibility of future use as artificial muscles. The
1.3. Electro-Active Polymers 11
high electric fields and breakdown strength of the air still constrain the application, but
research is going on to produce and manufacture better and more convenient E lectro-
S tatically S tricted Polymer (ESSP) actuators [1] which can be stimulated with lower
electric fields to master these challenges.
This thesis tries to combine the promising potentials in the use of EAP actuators
as linear actuators and the observer theory of Gopinath [11] to establish a basis for a
closed loop control system using state estimates as feedback signals. In respect to later
applications, where position or motion measurements with sensors are not desirable
solutions because these sensors are not easily integrated or are too big, the topology
of a Gopinath-style observer is a perfect fit. The observer will be fed only with signals
measurable “outside” the actual actuator system for example a commanded feedforward
signal provided by the controller and an easily gaugible signal of the power electronics
unit as reference input.
Chapter 2
Project Overview and Preparatory
Work
2.1 Overview of the Project
This overview outlines the purpose of this thesis and tries to introduce the problem to
the reader. More detailed descriptions and formulations will be found in later sections.
In Figure 2.1 the actual test set-up is shown. A slide is running with ball-bearings
on two bars of the base frame. The base frame is put up vertically to reduce friction
between the ball bearings and the slide rail (see Section 4.5). An optical encoder
module is attached to the slide to measure displacement. A linear motor armature is
fixed beneath the slide to inject disturbances by the linear motor, where the permanent
magnet is mounted to a counterpiece of the slide, fixated on the lower bars of the base
frame. A linear encoder stripe is also placed on the counterpiece to offer the necessary
fixed reference to the optical encoder modul, moving with the slide.
A polymer is mounted with one side to the slide and with the opposite to the
top of the base frame. Because of the gravity of the free-running slide the polymer
will be initially strained in length and contracted in thickness and width because of
the behavioral characteristics of the material, until the internal “spring” force of the
elastomer will compensate the gravity force. This situation is considered to be the
12
2.1. Overview of the Project 13
Figure 2.1: Test Setup Components and Polymer
initial (prestrained) condition of the set-up configuration.
The polymer is prepared with two compliant electrodes of carbon black dust, so it
can be regarded as a capacitor. When a high voltage is applied to these electrodes,
the capacitor is charged. The unlike charges will be electrostatically attracted and this
attraction force will cause the polymer to deform. As a result of this phenomen, the
polymeric elastomer will contract in thickness and expand in length and width. In
addition the capacitance will change due to the geometric deformation.
The displacement of the slide is measured by the optical encoder module, the pro-
duced pulses are interpreted in a counter chip. The actual position is transferred to the
microprocessor, which will convert the data and send it over a serial communication
interface to a host computer.
Furthermore the microprocessor is to generate PWM signals by its timer modules
2.1. Overview of the Project 14
output to the PWM drivers circuit, applying desired voltages to the linear motor and
high voltages (amplyfied by a high voltage converter) to the capacitor. The digital
control system will be implemented in the micro-controller, including a digital PID
controller and a Gopinath-style motion state observer. This observer will be used to
estimate the displacement of the slide. A motion or position sensor is undesirable, due
to its high cost and its integration problems in potential future micro-structured uses
such as artificial muscles.
Host Computer(Cross Development &
Data Collection)
PWM Drivers
Linear Motor(Disturbance)
Counter
Micro-Processor(Controller & Observer)
PWM Signals
Serial Connection Voltage
8-B
it B
us
CounterControl
Polymer(Actuator)
Slide
OpticalEncoderModul
(Sensor)
Analog MeasuredDisplacement
Signal
BaseFrame
Voltage
8-Bit Bus
Disturbance Force
High VoltageConverter High Voltage
Figure 2.2: Components Overview
Thus the later aim is to remove the optical encoder module and chip (removing
all gray components in Figure 2.2). The necessary signals to build a closed control
loop will be provided by the observer estimates. The Gopinath-style observer’s feed-
forward input will be the duty cycle signal as a output of the controller and for the
reference and feedforward input the measured charging current will be used. With this
observer structure the displacement and the disturbance force of the linear motor will
2.2. Used Hardware Components 15
be estimated.
This thesis focuses on developing a physics-based model of the above described test
set-up and estimating the unknown parameters. The introduced model is to be verified
by experminents and a Gopinath-style motion state observer is to be build at a desired
operating point due to the nonlinearities of the model. Unpredictable difficulties and
encountered problems are reported to descibe failures and to explain the seemingly
inadequate results.
2.2 Used Hardware Components
To complete the overview a brief list with specifications of the hardware components is
given.
Host Intel-Pentium II 200Mhz, RAM 96MByte, HD 2GByte, Windows 98
Microprocessor Hitachi H8/3664F, Model HD64F3664, 10Mhz, 16-Bit Architecture,
ROM 32kByte, RAM 2048Byte
PWM Driver Chip Texas Instruments TI SN754410NE, Quadruple Half-H Drivers,
max. Output Voltage 36V , max. cont. Output Current ±1.1A
Optical Encoder US Digital HEDS-9200 Module, Linear Stripe Module HEDS-9200-
360, Resolution 360lpi (lines per inch)
Multi Mode Counter LSI Computer Systems, Inc., Model LS7166, 24-Bit Quadra-
ture Counter, 5V Operation, TTL & CMOS I/O compatible, 8-Bit I/O-Bus to
Microprocessor
High Voltage Converter High Voltage Corporation EMCO, Model G30, Input Volt-
age Range 0 − 12V (with 0.7V turn on Voltage), max. Output Voltage +3000V
Polymer 3M VHB Tape, Model 4905 & 4910, Thickness 1mm & 0.5mm, acrylic ad-
hesive and pressure sensitive tapes
2.3. Preparatory Work 16
2.3 Preparatory Work
As no development host was available, one had to be assembled collecting unused com-
ponents from partly broken computers. A like-new motherboard, RAM, harddrive were
used. Because of the host’s low cpu power and its insufficient memory, Windows 98
was chosen as the operating system. All needed software was installed, including cross-
development environment and debugger.
For later measurements and data acquisition the microprocessor and its peripherials
were set up properly, including I/O-Ports, PWM drivers chip control and duty cycle
command output, counter chip control and data read out. Only a brief summary is
given about the hardware initialization of the micro-processor and the peripherials. For
more details the reader is asked to read the well documented source code of the main
initialization files (see Appendix B). The internal clock frequency of the microprocessor
is Φ = 9.84Mhz.
Interrupts Enabled for Timer A Module, Timer W Module, A/D Converter, SCI3
Module (Serial Communication Interface), Watchdog Timer, Software Interrupts
Timer A Module Software Interrupt Request with constant frequency Φ8
= 19.22kHz
(for data transfer over the SCI3 in identical time intervals)
Timer W Modul Configured as PWM signal output port, PWM counter frequency
is Φ8
= 1.23Mhz
Port 1 Configured as counter chip control (port pin connections:P16 to C/D, P15 to
R, P14 to W of counter chip)
Port 5 Configured as general I/O-port to read and write data on 8bit port
Port 7 Configured as PWM driver chip control (port pin connections: P75 to EN3, 4,
P74 to 1A and P74 to 2A of PWM driver chip)
Serial Communication Interface Baud Rate 9600bps, no Parity , 1 Stop bit, Data
Length 8bit
2.3. Preparatory Work 17
Counter Chip Binary Count Mode, 4x Quadrature Mode, Counter Preset to 1677721510
(Avoiding Negative Values, Raw Data Transferred to Host to Preserve CPU Power,
Data Evaluation in Host Computer)
The previously constructed circuit of the peripheral devices (including counter & PWM
driver chip and inverter gate) was connected to the microprocessor. The connection
diagrams for the 24-Bit Quadrature Counter and for the PWM Driver Chip are shown
in Figure 2.3 and Figure 2.4.
All components including serial communication interface (SCI3), interrupt service
routines (ISR), Counter chip communication, preset and read-out, PWM driver control
and output, were configured. Finally the whole microprocessor system with peripherals
was tested and debugged.
P16 (Pin 53)
5V
...
P57 (Pin 27)
P50 (Pin 13)
C/_D (Control/ Data Input)
PORT 5
PORT 1
P14 (Pin 51)
Modul
Encoder
Microprocessor
H8/3664
P15 (Pin 52)
Optiacl
...
(not used)
D2
D1
D0
B (Counter Input)
A (Counter Input)
VDD
(not used)
_CS (Chip Select)
GND
D7
D4
D3
D5
D6
(not used)
_WR (Write Input)
(not used)
_RD (Read Input)
1 20
1110
(24−Bit Quadrature Counter)
LSI LS7166
Figure 2.3: Counter Chip Connection Diagram
2.3. Preparatory Work 18
12V
5V
P75 (Pin 29)
P74 (Pin 28)PORT 7
FTIOB (Pin 38)
FTIOC (Pin 39)
GNDGND
Microprocessor
H8/3664
1 161,2EN
1A
1Y
2Y
2A
Vcc2
8 9
Vcc1
4A
4Y
3Y
3A
3,4EN
Voltage Converter)
(To High PWM
Signal
Output
Charging
Electrodes
Linear Motor
(Bidirectional Drive)
(PWM Driver Chip)
TI−SN754410
Figure 2.4: PWM Driver Chip Connection Diagram
Chapter 3
Physics Based Model of Polymer and
Set-up
3.1 Dynamics Model of Polymer and Set-Up
3.1.1 An Introduction to Finite Elasticity
In this section an outline is presented to approach the modeling of elastomer dynamics
and to introduce the reader to the nomenclature used throughout the thesis for finite
elasticity decribing large deformations.
It will be started with the consideration of a body, depicted in Figure 3.1a, with ref-
erence configuration B in the Lagrangian coordinate system ~X = (X, Y, Z). In general,
forces applied to the body B can result in translations, rotations and deformations.
Focusing on deformations, a deformed or current configuration B‘ of the body is intro-
duced in the Eulerian coordinate system ~x = (x, y, z), depicted in Figure 3.1b.
To describe a deformation, one has to choose a reference system. All quantities then
will be defined relative to the chosen system. When using the general Lagrangian for-
mulations the reference system is the initial, undeformed system (Figure 3.1a) with the
referential or Lagrangian coordinates ~X = (X, Y, Z) [3]. For the Eulerian formulations
the current, deformed system (Figure 3.1b) with the current or Eulerian coordinates
19
3.1. Dynamics Model of Polymer and Set-Up 20
X
Y
Z
B
N
dA
T
Deformation
x
y
z
B‘
n
da
t
a) Lagrangian Coordinate System b) Eulerian Coordinate System
Figure 3.1: Reference (Lagrangian) & Deformed (Eulerian) Configuration
~x = (x, y, z) is chosen [3]. These formulations have a relationship, called a configu-
ration map ~x = ~χ( ~X), or motion map ~x(t) = ~χ(t, ~X), when the deformation is time
variant [2–4]. The transformation between these coordinate systems can be achieved
by using the configuration gradient F of the configuration map
F =
(
∂xi
∂Xj
)
=∂~x
∂ ~X=
∂χ( ~X)
∂ ~X
which is generally nonsingular and nonsymmetric [2–4].
A current displacement u(t, ~X) =(
ux(t, ~X), uy(t, ~X), uz(t, ~X))T
can be stated [2–
4], related to an initial configuration χ(0, ~X) = ~X
u(t, ~X) = χ(t, ~X) − χ(0, ~X) = χ(t, ~X) − ~X
with this the true deformation gradient [2–4] is defined to
D =∂u
∂ ~X= F − 1
which describes the deformation related to the reference configuration and where 1 =
tr(1, 1, 1) is the unity matrix [2–4].
Looking again at Figure 3.1, the body has deformed due to a resultant (actual) force
3.1. Dynamics Model of Polymer and Set-Up 21
df acting on a surface element
df = t · da = T · dA
where t represents the Cauchy (or true) traction vector defined in the current configura-
tion ~x, while T represents the first Piola-Kirchhoff (or nominal) traction vector defined
in the reference configuration ~X, and both have the same direction [2]. In the future,
small letters will refer to the current configuration and capitel letters to the reference
configuration. N and n represent the respective normals on the surface elements dA
and da.
Cauchy’s stress theorem [2] states that tensor fields σ and P exist, so that
t(~x, t,n) = σ(~x, t)n (3.1)
T( ~X, t,N) = P( ~X, t)N (3.2)
where σ is the symmetric Cauchy (or true) stress tensor and P characterizes the first
Piola-Kirchhoff (or nominal) stress tensor. These tensors can be related to each other
by using the Piola transformation [2]
P = (detF)σ(F−1)T (3.3)
σ = (detF)−1PFT (3.4)
It is obvious that P is not symmetric in general and thus has nine independent compo-
nents [2]. For a better illustration one can return to the convenient matrix notation to
express the first statement of Chauchy’s stress theorem
t =
tx
ty
tz
= σ(~x, t)n =
σxx = σx σxy σxz
σyx σyy = σy σyz
σzx σzy σzz = σz
nx
ny
nz
(3.5)
where the true stress σ is symmetric and therefore reduces to six independent compo-
3.1. Dynamics Model of Polymer and Set-Up 22
Figure 3.2: Cauchy Stress Components
nents, as σxy = σyx, σxz = σzx and σyz = σzy [2]. In Figure 3.2 the stress components
of the Chauchy stress tensor σ are shown as a result of the traction vectors t1, t2 and
t3, acting on the faces of a cube pointing in the principal directions n1 = (1, 0, 0)T ,
n2 = (0, 1, 0)T and n3 = (0, 0, 1)T .
These fundamental equations characterize and hold for any continuum body for
all times. But they do not specify material properties of different deformable bodies.
Therefore additional equations have to be established, describing the material response
and the material behavior itself. These “constitutive laws” [2] should give an approxi-
mation of the observed physical behavior of a real material. In recent years researchers
have made gradual progress in develeping models for elastomeric materials, in spite
of this enormous complex issue. Mostly these models are based on Strain Energy
Function Ψ (SEF, also called Helmholtz free energy function [2]) and Finite Strain
(FS) theories [2, 4]. These SEF models describe the elastic properties of elastomers
(without damping and hysteresis) and are based on the extension ratios or principal
stretches λi, i = x, y, z [3]. The principal stretches λi descibe the length deformation of
unit vectors parallel to the principal axis x,y and z. Rivlin proposed [7], that the SEFs
should only depend on the strain invariants I1 = λ2x + λ2
y + λ2z, I2 = λ2
xλ2y + λ2
xλ2z + λ2
yλ2z
and I3 = λ2xλ
2yλ
2z. The finite strain elasticity of Rivlin is developed with a generalized
3.1. Dynamics Model of Polymer and Set-Up 23
Hooke’s law, where no assumptions about small deformations are made [4]. The finite
strain elastic theory is derived directly from the SEF by relating the finite strains ǫx,
ǫy and ǫz to the extension ratios λx, λy and λz.
The neo-Hookian model with SEF
Ψ(λx, λy, λz) = C1(I1 − 3) = C1(λ2x + λ2
y + λ2z − 3) (3.6)
is most appropriate for elastomers, where incompressibility can be assumed [3]. The
value
C1 =G
2(3.7)
is constant, where G represents the shear modulus. The choice of the neo-Hookian
model is also motivated by the derivation of its SEF from statistical theory, where
the material is regarded as a three-dimensional network of cross-connected long-chain
molecules [2]. This is exactly how a polymer can be viewed in its micro-structure.
The shear modulus G can be related to the Young’s modulus (or general modulus of
elasticity) E by
G =E
2(1 + ν)(3.8)
with the poisson ratio ν. For incompressible bodies the poisson ratio is ν = 0.5 [18]
and this simplifies the relation [3] to
G =E
3(3.9)
So these results can be related to the constant C1 by substituting Equation (3.9) in
Equation (3.7)
C1 =1
2
E
3(3.10)
Although this is the most appropriate choice for modeling elastomers, it will not
perfectly approximate the elastomeric material behavior [3].
3.1. Dynamics Model of Polymer and Set-Up 24
3.1.2 Neo-Hookian Model of the Elastomer
For homogenous pure strains (ǫi 6= 0 with i = x, y, z) in the principal directions x,y
and z and thus no shear (ǫxy = ǫxz = ǫyz = 0 in xy-, xz- and yz-direction), the normal
components of finite strain ǫi can be related to the principal stretches [3, 4] by
λx = 1 + ∂ux
∂X= 1 + ǫx (3.11)
λy = 1 + ∂uy
∂Y= 1 + ǫy (3.12)
λz = 1 + ∂uz
∂Z= 1 + ǫz (3.13)
and the configuration map reduces to a linear matrix multiplication
~x =
λx 0 0
0 λy 0
0 0 λz
~X (3.14)
not surprisingly in this case the configuration gradient is
F =∂−→x∂ ~X
=
λx 0 0
0 λy 0
0 0 λz
= FT (3.15)
and therefore
F−1 =
1λx
0 0
0 1λy
0
0 0 1λz
= (F−1)T (3.16)
and the deformation gradient is given by
D = F− 1 =
λx − 1 0 0
0 λy − 1 0
0 0 λz − 1
(3.17)
3.1. Dynamics Model of Polymer and Set-Up 25
In this thesis only homogenous pure strain will be considered.
Furthermore the material is assumed to be isotropic and thus the generalized mod-
ulus of elasticity is homogenous in all directions
Ex = Ey = Ez ≡ E (3.18)
and the material is incompressible [1, 4], which results in the constraint
detF = λxλyλz = 1 (3.19)
It was noted earlier, that for neo-Hookian materials a SEF Ψ(λ1, λ2, λ3) = C1(I1 − 3)
should be used for electro-active polymers. This energy function is used to derive the
Cauchy true stress tensor. To obtain the true stress σ, one has to evaluate
σi = −p0 + λi
∂Ψ
∂λi
for i = x, y, z (3.20)
for the principal stress components σx, σy and σz [2]. The incorporated scalar p0 is
fundamental to maintain the assumption of incompressibility [2,3]. It will be determined
for boundary conditions, which constrain the dynamic behavior of the material. When
Equation 3.20 is evaluated for the neo-Hookian model (Equation 3.6) with the constant
C1 = 12
E3, the Cauchy stress tensor σ [3] is given by
σ =
σx 0 0
0 σy 0
0 0 σz
=
−p0 + E3λ2
x 0 0
0 −p0 + E3λ2
y 0
0 0 −p0 + E3λ2
z
(3.21)
where E is the Young’s modulus and p0 is a to be determined hydrostatic pressure to
maintain incompressibility.
When a voltage is applied to the compliant electrodes on top and bottom of the
polymer, a pressure pz (see also Section 3.3.1) is induced by the attraction force of
3.1. Dynamics Model of Polymer and Set-Up 26
the unlike charges. According to this external (negative) stress −pz in the principal
direction of the z-axis, the polymer will contract in thickness and expand in area [16].
Thus the polymer actuator is constrained in the z-direction by the pressure pz and no
external stresses (px = py = 0) in the principal directions of y and x can be assumed.
This can be formulated in equations
σx = px = 0 (3.22)
σy = py = 0 (3.23)
σz = −pz 6= 0 (3.24)
With those, the hydrostatic pressure p0 can be determined by combining Equations
(3.21) and (3.24) to
σz = −p0 + E3λ2
z = −pz (3.25)
⇒ p0 = pz + E3λ2
z (3.26)
also the relation of the principal stretches λx and λy can be derived, when (3.22) is
equated with (3.23)
σx = σy (3.27)
−p0 +E
3λ2
x = −p0 +E
3λ2
y (3.28)
⇒ λx = λy (3.29)
When applying this result to the constraint for the assumed incompressibility (Equation
3.19) and proposing the definition of the stretch
λ ≡ λx = λy (3.30)
A relation for the principal stretches λz can be given by
3.1. Dynamics Model of Polymer and Set-Up 27
λz =1
λxλy
=1
λ2(3.31)
Finally, when substituting the hydrostatic pressure p0 with (3.26) and using (3.30) and
(3.31), the true stress component σx in the principal direction of x can be stated by
σx = −p0 +E
3λ2
x
= −pz −E
3λ2
z +E
3λ2
x
= −pz +E
3
(
λ2 − 1
λ4
)
(3.32)
3.1.3 Prestrained Polymer by External Load
As the setup is used in a vertical position, the mass of the slide Mslide (or any additional
attached mass madd) causes the polymer to elongate in length and due to the relation
of incompressibility (3.19) to contract in width and height. The initial dimensions of
the polymer will change. The inital length L0 and the initial width W0 will become the
prestrained length Lpre0 and prestrained width W pre
0 . Respectively, the initial thickness
H0 will be changed to the prestrained thickness Hpre0 . Prestrained length and width
can be measured, and thus the corresponding stretches λprex =
Lpre0
L0and λpre
y =W
pre0
W0
can be calculated. The thickness is hardly to measure, but it can be computed with
the assumption of incompressibility λprez = 1
λprex λ
prey
. According to this relation the
prestrained thickness is Hpre0 = λpre
z H0. As from now, this prestrained configuration is
referred to be the initial prestrained condition of the polymer. The changing dimensions
in the model will always be related to these prestrained dimensions Lpre0 , W pre
0 and
Hpre0 and the deformation due to the gravity force of the mass Mslide +madd will not be
considered in the further development of the physics based model.
At the moment an additional prestrain is essential for the functionality of the ac-
tuator. By the prestrain, an adequate adjusted thickness must be obtained with which
the attraction force is actually capable of deforming the polymer material. Later manu-
factured and prepared polymer actuators should have the correct thickness and should
3.1. Dynamics Model of Polymer and Set-Up 28
be directly functional.
3.1.4 Demonstrative Example of an Elastic Deformation
This example should help to illustrate an elastic deformation to the reader. Consider
an elastic polymer cuboid with the inital (or prestrained) length Lpre0 , width W pre
0 and
the height Hpre0 depicted in Figure 3.3. When a pressure pz (negative stress in z) will
be applied , the polymer begins to deform until the internal spring force balances the
external force/pressure. The polymer will contract in thickness and expand in area, so
length and width elongate as indicated with red arrows at the bottom of Figure 3.3.
The new dimensions will be l, w and h. It will be assumed that the polymer is isotropic
and incompressible.
Figure 3.3: Deformation of Polymer as a Result of Applied Pressure pz
3.1. Dynamics Model of Polymer and Set-Up 29
In the shown example, the point P (X, 0, 0) is displaced by ux to P ‘(x, 0, 0). When
considering a point at the end of the cuboid for example Pend(L0, 0, 0), the extension
ratio λ ≡ λx = lL
pre0
can be formed with the current length l and the inital length Lpre0 .
Further on, the current length is l = Lpre0 + ux|L0
= (1 + ǫx)Lpre0 with the finite strain
ǫx = ∂ux
∂X
∣
∣
Lpre0
. The current dimensions can be given by using Equations (3.11), (3.30)
and (3.31)
l = λxLpre0 = λLpre
0 (3.33)
w = λyWpre0 = λW pre
0 (3.34)
h = λzHpre0 =
1
λ2Hpre
0 (3.35)
The volume will stay constant, this can be shown with (3.30) and (3.31)
V pre0
V=
Lpre0 W pre
0 Hpre0
lwh=
Lpre0 W pre
0 Hpre0
λxLpre0 λyW
pre0 λzH
pre0
=1
λxλyλz
=1
λλ 1λ2
= 1 (3.36)
for the above depicted example. Finally, it is assumed that the cross-sectional areas Axy
(in the xy-plane), Axz (in the xz-plane) and Ayz (in the yz-plane) of the polymer can
be described as rectangles throughout the thesis, with (3.30) and (3.31) the relations
can be established to
Axy = l · w = λxLpre0 λyW
pre0 = λ2Lpre
0 W pre0 (3.37)
Axz = l · h = λxLpre0 λzH
pre0 =
1
λLpre
0 Hpre0 (3.38)
Ayz = w · h = λyWpre0 λzH
pre0 =
1
λW pre
0 Hpre0 (3.39)
3.1.5 Kelvin-Voigt Damping
Examination of the principal behavior of the polymer showed, when a simple uniaxial
strain resulted from a weight being hung on the end of the polymer, that it will elon-
gate slowly with a creeping process until it reaches an equilibrium state. This noticable
creeping process can be modeled with the Kelvin-Voigt Model [2, 6]. This model is
3.1. Dynamics Model of Polymer and Set-Up 30
depicted in Figure 3.4 consisting of a spring in parallel to a dashpot. The spring rep-
resents the elastic behavior already described with the neo-Hookian model (see Section
4.5). The dashpot tries to model the damping with a damping coefficient CD.
By introducing the differentiated term using (3.30) and (3.11)
λ =∂λ
∂ux
· ∂ux
∂t=
ux
Lpre0
(3.40)
with the displacement velocity ux and the prestrained length Lpre0 , a damping stress
σdamping [3] is given by
σdamping = CD · λ = CD · ux
Lpre0
(3.41)
DC
σ σ
Figure 3.4: Kelvin-Voigt Model
3.1.6 Combined State Space Model of Test Set-Up Dynamics
Finally a combined dynamics model of the whole set-up can be presented. The initial
condition is the prestrained polymer configuration, thus the weight which resulted in
the extension force is not included in this model. As a dynamics model is of interest,
the stresses have to be transformed to actual acting forces, therefore the stresses must
be multiplied with the corresponding cross-sectional areas. With (3.32), (3.39) and
(3.41) the neo-Hookian force FneoHookian and the Kelvin-Voigt damping force FDamping
can be derived to
3.1. Dynamics Model of Polymer and Set-Up 31
FneoHookian = Ayzσx =1
λW pre
0 Hpre0
(
−pz +E
3
(
λ2 − 1
λ4
))
(3.42)
FDamping = Ayzσdamping =1
λW pre
0 Hpre0 · CD · ux
Lpre0
(3.43)
The slide and half of the polymer mass will be moved by the translation force F effx and
a unknown disturbance force Fdist, which influences the motion. The balance of forces
can then be established (friction is neglected, refer to Section 4.5)
M · ux = F effx + Fdist − FneoHookian − FDamping (3.44)
where M = Mslide + 12Mpolymer + madd represents the inertia mass to be moved. It is
assumend, that in a lumped model half of the polymer mass Mpolymer is also accelerated.
To be consistent the additionally attached mass madd is included, but will be set to zero
(madd = 0) for future considerations. The additional attached mass allows a desired
prestraining of the polymer.
1
s1
M+
text 24
1
3z
Ep λ
λ
− + −
0
1preLDC
+
+
0 0
1 pre preH Wλ
−
++
1
sxu
λ
Dynamics Model
Neo-Hookian Model and Kelvin-Voigt Damping
distF
+eff
xF xu
xu
zp
0
1 xpre
u
L+
Figure 3.5: Nonlinear State Block Diagram of Set-Up Dynamics
3.2. Electrical Circuit Model Of Polymer 32
By proposing the state vector x2
x2 =
vux
ux
with the displacement velocity vux= ux and the displacement ux a nonlinear state
space model is introduced with (3.42) and (3.43)
x2 = f2(
x2, Feffx , FDamping, Fdist
)
=
f3
(
vux, ux, F
effx , FDamping, Fdist
)
f4
(
vux, ux, F
effx , FDamping, Fdist
)
=
1M
[
F effx − FneoHookian − FDamping + Fdist
]
vux
(3.45)
=
1M
[
F effx − 1
λW pre
0 Hpre0
(
−pz + E3
(
λ2 − 1λ4
)
+ CDλ)
+ Fdist
]
vux
(3.46)
where the stretch λ can be replaced with λ = λ(ux) = 1 + ux
Lpre0
and the periodic change
of the stretch λ with λ = λ(vux) = vux
Lpre0
= ux
Lpre0
. In Figure 3.5 the state block diagram
of the dynamic model is depicted. Within this thesis, nonlinear blocks are tagged by
using double bordered boxes.
3.2 Electrical Circuit Model Of Polymer
The prestrained polymer with applied compliant electrodes on the top and bottom side
can be considered as a capacity CP (l, w, h) depending on its geometry. The capacity
CP (l, w, h) will change, when the dimensions (length l, width w, thickness h) of the
polymer alter. The area of the comliant electrodes Ace is assumed to be the cross-
sectional area Axy and therefore with (3.37) can be set to
Ace = l · w ≡ Axy = λ2Lpre0 W pre
0 (3.47)
3.2. Electrical Circuit Model Of Polymer 33
x
y
z
R+R L
e (t)
e (t)
C
l
w
h
i(t)
c
p
pp
i
Figure 3.6: Electrical Circuit of Polymer
Furthermore the compliant electrodes and the special high voltage cables can be treated
as a serial connection of resistances Rp +R and an inductance Lp. The electrical circuit
is depicted in Figure 3.6. With Kirchhoff’s Voltage Law [18] the circuit can be described
by
ei(t) = Lp · i(t) + (Rp + R) · i(t) + ec(t) (3.48)
replacing ec = 1Cp(l,w,h)
∫
i(t)dt = 1Cp(l,w,h)
q(t) and defining the system states
x1 =
i
q
= f1 (i, q, ei) =
f1 (i, q, ei)
f2 (i, q, ei)
(3.49)
a nonlinear state space model of the electrical circuit can be derived to
x1 =
i
q
=
−RP +RLp
· i − 1Lp·Cp(l,w,h)
· q + 1Lp
· ei
i
(3.50)
where the capacitance is related to the dimensions l, w and h and therefore to the
stretch λ. This can be expressed with Equations (3.33), (3.34) and (3.35) for the
current dimensions
Cp(l, w, h) = ǫ0ǫrl·wh
= ǫ0ǫr
Lpre0 W pre
0
Hpre0
· λ4 = Cp(λ) (3.51)
3.3. Force Output of a Charged Polymer 34
The block diagram of the polymer circuit is depicted in Figure 3.7.
1
s
( )ie t +
−
1
pL
1
s−
( )i t ( )q t
( )ce t
Electrical Circuit Model of Polymer
pR R+
( , , )p
q
C l w h
Figure 3.7: State Block Diagram of Electrical Circuit
3.3 Force Output of a Charged Polymer
3.3.1 Induced Pressure of Applied Electric Field
When a voltage ei is applied to the polymer circuit, the polymer capacitor will be
charged, the opposite charges q on both compliant electrodes will be attracted. The
attraction force will force the polymer to contract in thickness and expand in area. The
phenomenological induced pressure pz in principal direction of −z depending on the
electric field Eel between the electrodes can be described with the following relation
[1, 15, 16]
pz = ǫ0ǫrE2el (3.52)
where ǫr is the relative dielectric constant of the used polymer and ǫ0 = 8.854 ·10−12 AsV m
is the permittivity of free space [18]. This pressure is twice the stress normally induced
on two rigid, charged capacitor plates [15]. It can be regarded as an effective stress,
being a result of both a compressive stress component acting in direction of thickness
and tensile stress components acting in planar directions [15].
3.3. Force Output of a Charged Polymer 35
3.3.2 Calculation of the Electric Field Eel
Assuming an electrostatic model, the electric field interior the electrodes is zero, there
will be only a electric field Eel perpendicular to the area of the electrodes. Further on,
the charges will have moved to a steady state position and thus a constant surface charge
distribution σ can be assumed constant over an area dA of the compliant electrodes.
First deriving the electric field Ecd of a charged disk with radius R, with a constant
charge distribution σ per unit area dA = 2πrdr and thus a differential charge dq = σdA,
the electric field Eel will be approximated later by considering two parallel infinite planes
with a dielectric in between.
3.3.2.1 Electric Field of a Charged Disc
Consider a charged disc, depicted in Figure 3.8 surrounded by a dielectric with a relative
dielectric constant ǫr. The electric field Ecd of the charged disc will be calculated at
(x, 0, 0), by integrating the differential expression for an electric field dE = 14πǫ0ǫr
· xd3 ·
dq [18], only interested in x-direction, as the other components will be cancelled by
symmetry. The electric field can be given by
Ecd =
∫ q
0
dE =1
4πǫ0ǫr
∫ q
0
x
d3dQ =
1
4πǫ0ǫr
∫ R
0
x
d32πσr · dr (3.53)
where d is the actual distance between the disc and the point (x, 0, 0), so d =√
r2 + x2.
When this is integrated (see Table of Integrals in [19])
Ecd =σx
2ǫ0ǫr
∫ R
0
r
(r2 + x2)3
2
· dr =σx
2ǫ0ǫr
[
− 1√r2 + x2
]R
0
(3.54)
and evaluated for the intergration limits, the electric field of a charged disc is
Ecd(R, x) =σ
2ǫ0ǫr
·(
x
|x| −x√
R2 + x2
)
(3.55)
3.3. Force Output of a Charged Polymer 36
x
y
dr
R
r
dQ
d
x
z
Figure 3.8: Sketch of Charged Disc
3.3.2.2 Electric Field of Two Parallel Compliant Electrodes
The electric field Eip of one inifinite extended charged plane can be described by in-
creasing the radius R to infinity in the above derived equation for the electric field Ecd
of a charged disc
Eip = limR→∞
Ecd =
σ2ǫ0ǫr
, ifx ≥ 0
− σ2ǫ0ǫr
, ifx < 0(3.56)
Considering a set-up depicted in Figure 3.9, one of the planes is charged positive and
the other negative, so the electric fields between the two plates can be added, while
outside of the planes the electric fields are cancelled. Thus a relation for the electric
field Eel of two infinite parallel planes can be derived, where E+ = E− = Eip and the
surface charge distribution is σ = q
A
Eel = E+ + E− = 2Eip =σ
ǫ0ǫr
=q
ǫ0ǫrA(3.57)
Although the above presented result is only valid for two charged plates of inifinite
length and width, it is a satisfying approximation for an electric field between the finite
3.3. Force Output of a Charged Polymer 37
-
-
-
-
-
-
+
+
+
+
+
+
E-E- E-
E+ E+E+
Figure 3.9: Electric Field of Parallel Plates
extended electrodes of the polymer, due to the fact, that the thickness is (very) small
compared to the area A = Ace of the compliant electrodes.
3.3.3 Pressure pz and Compressive Force Fz Acting Along the
z-Axis
Combining the above presented results, especially (3.47), (3.52) and (3.57), the pressure
pz can be rewritten to
pz = ǫ0ǫrE2el =
1
ǫ0ǫr
(
q
Ace
)2
=1
ǫ0ǫr
(
q
Lpre0 W pre
0
)21
λ4(3.58)
and with (3.47) and (3.58) the force Fz can be derived to
F effz = pz · Ace =
q2
ǫ0ǫrAce
=1
ǫ0ǫr
q2
Lpre0 W pre
0
1
λ2(3.59)
where λ = 1 + ux
Lpre0
is the principal stretch related to the displacement ux.
3.3.4 Electrostrictive Transduction
It is of interest to calculate the effictive stress or force output in the translation direc-
tion (principal axis of x), hence the main goal will be to control the displacement of
the polymer. The effective force ouput can be explained with the electrostrictive trans-
duction [17]. Assuming there will be no thermal losses in the polymer material and it
is fully isochoric and isotropic, the force in the x-direction can be derived using energy
3.3. Force Output of a Charged Polymer 38
conservation for a deformable capacitor C(x, y, z) = ǫ0ǫrxy
zwith the area Ace = xy
perpendicular to the z-axis and the thickness z. The energy stored in an capacitor for
a constant charge q [18] is
U =q2
2C=
q2 · z2ǫ0ǫrAce
=q2 · z
2ǫ0ǫrxy(3.60)
by differentiating partially with respect to x, y and z, the change in stored energy
related to a deformation in the principal directions can be expressed and regarded as a
force acting in the considered direction [17]. The following relations can be found
∂U
∂x= −Fx = − q2
2ǫ0ǫrxy· z
x(3.61)
∂U
∂y= −Fy = − q2
2ǫ0ǫrxy· z
y(3.62)
∂U
∂z= −Fz =
q2
2ǫ0ǫrxy(3.63)
These forces help to explain the deformation as a result of the charging process more
precisely. It was already explained that unlike charges on the compliant electrodes are
attracted. This attraction force is represented by F effz = pz · Axy (3.59) and causes
the polymer to contract in thickness h and to expand in area Axy prependicular to the
thickness (z-axis). Since the material is incompressible, the force F effz is “transformed”
in the above shown components. Thus the same deformation would be obtained, when
the forces Fx, Fy and Fz were applied on the corresponding cross-sectional areas Ayz ,
Axz and Axy. It is emphasized that physically, the attracted charges are inducing only
the pressure pz and thus the force F effz .
Since a translation in the direction of the x-axis is of interest, the effective force
F effx will be derived. Conservation of energy postulates that there is no loss in energy
at all times, so a change in energy by the work dU (charging the compliant electrodes)
will result in a deformation, the potential energy in the elastomer and the capacitor
3.3. Force Output of a Charged Polymer 39
will change
dU = Fx · dx + Fy · dy + Fz · dz (3.64)
The new dimensions of the capacitance can be calculatd to x = λLpre0 , y = λW pre
0 and
z = 1λ2 H
pre0 , and thus an effictive force F eff
x in the x-direction can be derived by relating
dz and dy to dx. It is obvious that dzdλ
= − 2λ3 ·Hpre
0 , dy
dλ= W pre
0 and dxdλ
= Lpre0 , herewith
the relations are given by
dz
dx=
dzdλdxdλ
= − 2
λ3· Hpre
0
Lpre0
= −2 ·1λ2 H
pre0
λLpre0
= −2z
x⇒ dz = −2
z
x· dx (3.65)
anddy
dx=
dy
dλdxdλ
=W pre
0
Lpre0
=y
x⇒ dy =
y
x· dx (3.66)
Applying these results to (3.64)
dU = Fx · dx + Fy · dy + Fz · dz
=q2
2ǫ0ǫrxy
(
−z
xdx − z
y· y
xdx − 2
z
xdx
)
= −2 · q2
ǫ0ǫrxy· z
x· dx (3.67)
the effective force output F effx in the x-direction is found to
F effx = −∂U
∂x= 2 · q2
ǫ0ǫrxy· z
x(3.68)
The negative sign is easy to interpret, due to the fact that charges on the polymer
capacitor will induce a pressure in negative z-direction and therefore the polymer will
deform, increasing in area Ace = xy and decreasing in thickness z, the stored energy in
the capacitor reduces with ∂U < 0 and is transformed in potential energy stored in the
polymer. As the force output at the edges is of interest, the equation can be simplified
to
3.4. Complete Nonlinear Model of Polymer Set-Up 40
F effx
∣
∣
x=λLpre0
,y=λWpre0
,z= 1
λ2H
pre0
= 2 · Hpre0
ǫ0ǫrWpre0
·(
q
Lpre0
)2
· 1
λ5(3.69)
and this can be related to the force F effz (Equation 3.59) or to the pressure pz (Equation
3.58) by
F effx = 2
Hpre0
Lpre0
1
λ3· F eff
z = 2Ayz · pz = 21
λW pre
0 Hpre0 · pz (3.70)
This effective force F effx will act in the principal direction of x and thus in the translation
direction of the slide. It accelerates the slide.
3.4 Complete Nonlinear Model of Polymer Set-Up
To conclude this chapter, a complete and combined nonlinear state space model is pre-
sented. The complexity and highly nonlinear behavior is obvious. Based on simulations
(See Chapter 5) the developed physics based model is capable of mirroring the behavior
of the real system.
When the state space representations (3.46) and (3.50) are combined, the fully
fourth order single-input-single-ouput state space model with the system states
x =
x1
x2
=
i
q
vux
ux
(3.71)
is given by
3.4. Complete Nonlinear Model of Polymer Set-Up 41
x = f (x, ei, Fdist) =
f1 (x, ei, Fdist)
f2 (x, ei, Fdist)
(3.72)
y = g (x) = ux (3.73)
and explicitly by
x =
i
q
vux
ux
=
−Rp+R
Lp· i − 1
Lp
1Cp(λ)
· q + 1Lp
· ei
i
1M
(
F effx − FneoHookian − FDamping + Fdist
)
vux
(3.74)
When looking closer at the state derivatives i and vux= ux, a more precise description
is found. Replacing the capacitance Cp(λ) in (3.74) with Equation (3.51), the explicit
long form of
i = −Rp + R
Lp
· i − 1
Lp
Hpre0
ǫ0ǫrLpre0 W pre
0
1
λ4· q +
1
Lp
· ei (3.75)
can be given. Further on, when F effx is substituted with (3.68), FneoHookian with (3.42),
FDamping with (3.43), Ayz with (3.39) and pz with (3.58), a more detailled relation of
the acceleration ux = vux
vux=
1
M
(
F effx − FneoHookian − FDamping + Fdist
)
=1
M[2Ayz · pz − Ayz (σneoHookian + σDamping) + Fdist]
=1
M
[
2Ayz · pz − Ayz
(
−pz +E
3
(
λ2 − 1
λ4
)
+ CDλ
)
+ Fdist
]
=1
M
[
Ayz
(
3pz −E
3
(
λ2 − 1
λ4
)
− CDλ
)
+ Fdist
]
=1
M
[
1
λW pre
0 Hpre0
(
3pz −E
3
(
λ2 − 1
λ4
)
− CDλ
)
+ Fdist
]
(3.76)
can be derived. Combining these results (3.75) and (3.76) and inserting pz (3.58) a
3.5. Encountered Problems and Difficulties 42
complete nonlinear state space model is introduced by
x =
−Rp+R
Lp· i − 1
LP
Hpre0
ǫ0ǫrLpre0
Wpre0
1λ4 · q + 1
Lp· ei
i
1M
[
Fdist + 3H
pre0
ǫ0ǫrWpre0
(
q
Lpre0
)21λ5 − W pre
0 Hpre0
(
E3
(
λ − 1λ5
)
+ CDλλ
)
]
vux
(3.77)
Still the principal stretch λ and its “velocity” λ should be replaced with λ = λ(ux) =
1 + ux
Lpre0
and λ = λ(vux) = vux
Lpre0
, but to retain clearness this step will be abandoned. A
block diagram is given in Figure 3.10. It depicts the overall developed physics based
model of the polymer set-up. Again nonlinear blocks are tagged by double bordered
boxes.
3.5 Encountered Problems and Difficulties
The test set-up was changed several times, thus the model had to be adjusted each
time. In Figure 2.1 the sketches of the four considered test set-ups are illustrated. Test
set-up 1 and test set-up 2 (Figure 2.1a & b) were discarded. In Set-up 1 (see 2.1a) the
slide should be moved by two polymer pieces mounted on both sides and attached to
the base frame. In theory one of the polymers could be charged with unlike charges
and the other with like charges. The polymer with like charges will expand in thickness
due to the repulsion force of the charges on the electrodes and therefore contract in
area. The polymer charged with unlike charges will contract in thickness due to the
attraction force of the unlike charges and expand in area. The slide will be pushed
by the unlike charged polymer piece and pulled by the like charged piece. Because
of difficulties in building the power drive circuit to charge the polymers and not yet
functionable polymer pieces, test set-up 1 was modified to test set-up 2.
In test set-up 2 (see Figure 2.1b) the polymer piece is prestrained by a spring. By
increasing or decreasing unlike charges on the polymer compliant electrodes the slide is
3.5
.Encounte
red
Pro
ble
ms
and
Diffi
cultie
s43
1
s1
M+
0
1preLDC
+
+
−
++
1
sxu
λ
λ
λ
Dynamics Model
Neo-Hookian Model , Kelvin-Voigt Damping
distF
+eff
xF xu
xu
0
1 xpre
u
L+
text 24
1
3z
Ep λ
λ
− + − q
PWM gain High-Voltageconverter
1
sie +
−
1
pL
1s−
i
ce
pR R+
hvcKpwmK
Electrical Circuit & Electrostrictive Transduction
cd zp
zp
λ
2
40 0 0
1 1pre pre
r
q
W Lε ε λ
0
40 0 0
pre
pre prer
H q
L Wε ε λ0 0
1 pre preH Wλ
0 02 pre pre zpW H
λ⋅
Figu
re3.10:
Com
plete
State
Blo
ckD
iagramof
Non
linear
Poly
mer
System
3.5. Encountered Problems and Difficulties 44
Slide
Base Frame
Polymer
Slide
Base Frame
PolymerSpring
a) Set-up 1 b) Set-up 2
Base Frame with Pulley
PolymerSlide
Thread with Mass
Base Frame (vertically)
Polymer
Slide
Thread with Mass
c) Set-up 3 d) Set-up 4
Figure 3.11: Changes in Test Set-up
moved. However, for the necessary prestrain an appropriate spring should be applied.
The desired spring kit, providing several springs to vary the prestrain of the polymer,
was to expensive to order. Thus set-up 3 was composed.
In set-up 3 the polymer can be prestrained by attaching different weights to the
slide with a thin thread running over a pulley. Problems arose also from set-up 3 (see
Figure 2.1c). Due to friction estimations (see Section 4.5) this configuration had to be
adjusted again and finally set-up 4 was developed. This set-up was eventually used.
Test set-up 4 (see Figure 2.1d) is similar to configuration 3, only the pulley is
removed and the base frame is arranged vertically to reduce friction. Again the prestrain
can be varied by adding different weights.
Chapter 4
Parameter Estimation and
Measurement
4.1 Modulus of Elasticity E (Young’s modulus)
To estimate the Young’s modulus E, a tensile machine depicted in Figure 4.1 was
utilized. This machine is straining the specimen with a slow constant displacement
speed ux = dǫx
dt= const along the principal axis x and is measuring the corresponding
nominal (first Piola-Kirchhoff) stress component Px referred to the initial cross-sectional
area Ayz of the specimen. Because of the slow speed, damping can be neglected.
In this configuration the new-Hookian model has to be adjusted, since a stress is
only acting along the principal axis x. Recalling Equation (3.21) and with the boundary
conditions for this setup σx = px 6= 0 and σy = σz = 0, the hydrostatic pressure p0 can
be derived [3] to
p0 =E
3λ2
y =E
3λ2
z (4.1)
where λy = λz. With the constraint of incompressibility (3.19), the principal stretches
also can be related for this configuration to
λy = λz =1√λx
(4.2)
45
4.1. Modulus of Elasticity E (Young’s modulus) 46
Figure 4.1: Tensile Machine
When the results (4.1) and (4.2) are substituted to Equation (3.21), the principal true
stress component σx [3] is given by
σx = −p0 +E
3λ2
x =E
3
(
λ2x −
1
λx
)
(4.3)
For an adequate curve fitting method, this has to be transformed to the first Piola-
Kirchhoff component Px, as the machine is measuring with reference to the initial
cross-sectional area Ayz0 = H0W0. With the Piola transformation (3.3), (3.16) and
(3.19), the nominal stress component Px is
Px = σx
1
λx
=E
3
(
λx −1
λ2x
)
(4.4)
The measured data was curve-fitted using Matlab/Simulink and the results can
be seen in Figure 4.2 and 4.3. The noise in the captured data is an indication for
the statistic theory, which regards elastomeric materials as 3-dimensional networks of
cross-connected long chain molecules [2]. When for example a long-chain polymeric
molecule is overstretched, it will snap, this will reduce the spring force suddenly and
for a small amount. This means, one of the “internal springs” of the polymeric material
is torn, and will not contribute to the accumulated spring force. This has to be taken
4.1. Modulus of Elasticity E (Young’s modulus) 47
into account when the polymer is used as a linear actuator, this “noisy” behavior is
not desired. The polymer should be strained and released several times before usage to
reduce snapping effects of “internal springs”.
0 0.5 1 1.5 2 2.5 3−0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
5 Stress−strain curve of 3M VHB4905 (with glued compliant electrodes )
strain (epsilon) in [m/m]
stre
ss (
sigm
a) in
[N/m
2 ]
measured curveestimated curve with E =255957.8488 N/m2
Figure 4.2: Nominal Stress Curvefit - Young’s Modulus E with Glued Compliant Elec-trodes
The Figures 4.2 and 4.3 illustrate the influence of the attached compliant electrodes
on the elastic behavior of the polymer. In Figure 4.2 the polymer was prepared with
compliant electrodes, which were made up of a mixture of carbon black dust and super-
glue sprayed on the surface. A stiffer response is the result with an estimated Young’s
modulus E = 256 · 103 Nm2 . This stiffness is an undesired side effect of glued compliant
electrodes, because the effective force output will be reduced. A greater pressure is
required to obtain the same displacement than with a less stiff polymer specimen.
Hence other methods were explored to build polymers with compliant electrodes,
where the impact of the attached electrodes is negligible (Ray Tang’s work). The result
of several tests showed that compliant electrodes made out of carbon black dust have
very little influence on the elastic behavior. Although these polymers were not very
durable, due to the fact that the electrodes will vanish slowly, this configuration is
4.2. Damping Coefficient CD 48
chosen for the test set-up. A Young’s modulus
E = 96 · 103 N
m2(4.5)
can be approximated for a specimen without compliant electrodes (see Figure 4.3). This
estimated parameter can be used as a good approximation for the Young’s modulus,
even for polymers prepared with compliant electrodes made out of only carbon black
dust.
0 1 2 3 4 5 6 7 8 9−0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5x 10
5 Stress−strain curve of 3M VHB4910 (without compliant electrodes)
strain (epsilon) in [m/m]
stre
ss (
sigm
a) in
[N/m
2 ]
measured curveestimated curve with E =95980.2034 N/m2
Figure 4.3: Nominal Stress Curvefit - Young’s Modulus E without Compliant Electrodes
4.2 Damping Coefficient CD
To estimate the damping coefficient CD a dynamic mechanical analyzer (DMA) of
MTSM Research Group1 was likely to be utilized. But the analyzer was not avail-
able ultimately. Thus only roughly observations can be presented. The elastomeric
polymer showed aperiodic transient behavior. When a mass was applied, the polymer
1Department of Chemical and Biological Engineering, Molecular Thermodynamics and StatisticalMechanics Research Group, UW Madison
4.3. Dielectric Constant ǫr of Polymer Material 3M VHB 4905 49
elongated slowly, until it reached the equilibrium. No overshoot was noticable. There-
fore a high damping coefficient CD can be deduced. Due to simulations (see Chapter
5) the Kelvin-Voigt damping coefficient CD should be greater than 10000Nsm2 .
4.3 Dielectric Constant ǫr of Polymer Material 3M
VHB 4905
By consulting recent research papers about the material used (3M VHB 4905 & 4910
Tape), the dielectric constant ǫr was already measured before. Refering to [15], the
value of the dielectric constant is ǫr = 4.8(±0.5).
4.4 Resistance Rp + R and Inductance Lp
The resistance Rp and inductance Lp were measured using a LCR-meter2 at the fre-
quencies f1 = 100Hz, f2 = 120Hz and f3 = 1kHz. The measurements were made
for an unstrained and a strained polymer specimen with compliant electrodes of car-
bon black dust. The unstrained polymer had a length of L0 = 7.3cm and a width of
W0 = 5.4cm. The strained polymer was l = 11.2cm long and w = 4.3cm wide. The
measuring points are shown in Figure 4.4.
The values for the resistances Ri→4, Ri→5 and Ri→6 and the inductances Li→4, Li→5
and Li→6between the points i → 4, i → 5 and i → 6 were determined for all points on
the left side (i = 1, 2, 3). Mean values of the resistances
Ri→4,5,6 =1
3(Ri→4 + Ri→5 + Ri→6) (4.6)
and the inductances
Li→4,5,6 =1
3(Li→4 + Li→5 + Li→6) (4.7)
2used LCR-meter: Hewlett Packard Model 4263A
4.4. Resistance Rp + R and Inductance Lp 50
Figure 4.4: Measuring Points on Compliant Electrode
were computed. Although the measured values were noisy, averaged and rounded,
when reading the meter display, they allow a qualitative statement to be made about
the sensitivity of the resistance and inductance based on the spreaded density of carbon
black dust.
The results for the mean resistances are plotted in Figure 4.5 and for the mean
inductances in Figure 4.6. The plots show for both the resistance and the inductance
frequency sensitivity. Also, the figures reveal a analogous spreaded compliant electrode,
the resistance and the inductance vary depending on the chosen measuring points.
Thus it is obvious that the resistance and the inductance differ between unstrained and
strained configuration (compare a & b in Figure 4.5 and Figure 4.6), the density of
carbon black dust per unit area will decrease for the strained specimen, resulting in a
greater resistance and inductance.
To reduce these effects, future compliant electrodes should be applied with a higher
and more homogeneous density of carbon black dust, so that the deviations between
unstrained and strained specimen are negligible.
4.5. Friction 51
100 200 300 400 500 600 700 800 900 10004.4
4.6
4.8
5
5.2
5.4
5.6
5.8x 10
5
Frequency [Hz]
Res
ista
nce
[Ohm
]Resistance of Unstrained Polymer (L0=7.3cm, W0=5.4cm)
Mean Resistance from 1−>4,5,6
Mean Resistance from 2−>4,5,6
Mean Resistance from 3−>4,5,6
100 200 300 400 500 600 700 800 900 10007
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12x 10
5
Frequency [Hz]
Res
ista
nce
[Ohm
]
Resistance of Strained Polymer (l=11.2 cm, w=4.3cm)
Mean Resistance from 1−>4,5,6
Mean Resistance from 2−>4,5,6
Mean Resistance from 3−>4,5,6
a) Unstrained Polymer b) Strained Polymer
Figure 4.5: Results of Resistance Measurements
100 200 300 400 500 600 700 800 900 10001
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Frequency [Hz]
Indu
ctan
ce [H
]
Inductance of Unstrained Polymer (L0=7.3cm,W0=5.4cm)
Mean Inductance from 1−>4,5,6
Mean Inductance from 2−>4,5,6
Mean Inductance from 3−>4,5,6
100 200 300 400 500 600 700 800 900 10006
8
10
12
14
16
18
20
22
24
Frequency [Hz]
Indu
ctan
ce [H
]
Inductance of Strained Polymer (l=11.2cm, w=4.3cm)
Mean Inductance from 1−>4,5,6
Mean Inductance from 2−>4,5,6
Mean Inductance from 3−>4,5,6
a) Unstrained Polymer b) Strained Polymer
Figure 4.6: Results of Inductance Measurements
4.5 Friction
4.5.1 Static Friction Fstatic
The static friction force Fstatic which seems unimportant at first, cannot be disregarded
due to made measurements. Tests were performed to determine the minimum amount
of force necessary to move the slide. Therefore a small bag (with a mass mbag ≈ 0.2g)
4.5. Friction 52
was connected to the slide by a thin thread running over a pulley. Weights were added
gradually by steps of madd = 0.2g until the slide started to move. The setup is illustrated
in Figure 4.7. The measurement results indicated a static friction force of
Fstatic ≤ Fweight = (∑
madd + mbag)g ≈ 0.0032kg · 9.81m
s2≈ 0.031N (4.8)
This is likely to be related to the static friction coefficient µ0 = Fstatic
FN≈ 0.046, where
FN = Mslide ·g is the normal force of the slide with the gravitational constant g = 9.81ms2
and the mass of the slide Mslide = 0.0692kg.
Figure 4.7: Static friction measurement setup
4.5.2 Sliding Friction Fsliding
The value of the sliding friction coefficient µ has been estimated by using least square
methods, included in the optimization toolbox of Matlab/Simulink and a self im-
plemented recursive LSQM [13].
To estimate sliding friction, the slide with ball bearings running on two horizontally
bars was moved by an excitation force Fexcitation(t) = Cf · i(t) of a linear motor induced
by an applied current i(t) and the actual displacement was measured. The current-
force transformation coefficient Cf was roughly approximated by measuring the current
4.5. Friction 53
imeasured ≈ 13mA, needed to overcome static friction. Thus, Cf could be estimated to
Cf = Fstatic
imearsured≈ 2.38N
A.
The experiment was made with different excitation signals, such as sinusoidal and
square waves at varying frequencies in the range of f = [1, 10Hz]. The assumed behav-
ior of the moving slide can be expressed with a balance of forces
Mslide · x = Fexcitation − µ · FN (4.9)
0 200 400 600 800 1000 1200 1400-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
step
-mu,
mea
n er
ror
Recursive LSQ with SQUAREWAVE EXCITATION, SLIDING FRICTION: Voltage Amplitude 3.49V , Frequency: 10.26Hz
-mu [Ns/m] = 0.043559mean error = 2.0674e-005
Figure 4.8: RLSQM - Excitation with Square Wave
The experiments showed a small impact of sliding friction on the movement. The
least square methods allow values for the friction coefficient µ ≈ 0.044 to be estimated,
as seen in Figure 4.8. This behavior was expected due to the assumption that in general
sliding friction is smaller than static friction (µ < µ0). The sliding friction force can be
given by
Fsliding = µ · Mslide · g ≈ 0.03N (4.10)
It is also very small.
4.6. Displacement Measurement for an Applied Voltage 54
4.5.3 Conclusion of the Friction Estimation
The measurements presented in this section are inaccurate, nevertheless they do illus-
trate that friction is very small, which is inherent for ball bearings, and that it should
be negligible in normal applications. However for this particular setup, where the EAP
actuator produces only a very small output force itself (see Chapter 5), the friction
should be considered. However, static and sliding friction are unattractive to model,
and therefore the base frame was set up vertically to reduce friction further. With this
configuration the nonlinear friction effects can be neglected.
4.6 Displacement Measurement for an Applied Volt-
age
Due to unpredictable problems (see Section 4.7) the measurements presented in this
section are imprecise. For the experiment a polymer specimen was mounted on the
base frame and on the slide (only one available). The initial, unstrained dimensions
of the polymer piece were L0 = W0 = 25.4mm and H0 = 0.5mm. The polymer was
prestrained in width to W pre0 = 31mm when preparing with the compliant electrodes
and in lengths to Lpre0 = 35mm by the gravity force of the slide mass Mslide = 69.2g.
Both prestrains were measured with a ruler. Since the thickness Hpre0 is difficult to
measure, it can be calculated using the assumption of incompressibility (3.19) resulting
in a constant volume (3.36)
V pre0
V0
=Lpre
0 W pre0 Hpre
0
L0W0H0
≡ 1 (4.11)
By solving the (4.11) for Hpre0 , the prestrained thickness can be computed to
Hpre0 =
L0W0
Lpre0 W pre
0
· H0 ≈ 0.3mm (4.12)
4.7. Unexpected Problems during Measurements 55
The prestrained dimensions are found. An input voltage ei of 3000V was applied and
the deformed length l and width w were recorded. The measurement results were
l ≈ 39mm (4.13)
w ≈ 34.5mm (4.14)
After the voltage was removed (ei = 0), the polymer redeformed to a length L ≈35.5mm and a width W ≈ 31.5mm close to the initial, prestrained dimensions. This
deviation may indicate an inherent hysteresis. The voltage was induced again, but
before measurements could be made, the polymer was destroyed by arcing (see Figure
4.9).
Unfortunately only one representative measurement could be carried out, because
of difficulties in the preparation of new functional polymer specimens. A more detailed
description of the encountered problems is given in the next section.
4.7 Unexpected Problems during Measurements
The first functional polymer prepared with compliant electrodes was available after
about 13 weeks of experimentation (Ray Tang‘s work). Compliant electrodes made out
of conductive grease show a satisfaying conductivity, but the spreading is difficult and
the grease is expensive. Besides the grease sticks nicely to the polymer surface. This is
a favored feature to have electrodes which are durable and adherent.
To reduce costs and still retain good adherent properties, new electrodes were devel-
oped, consisting of a mixture of carbon black dust and superglue. Because of the glue in
the mixture, this type of electrodes was not conductive. For this reason the amount of
carbon black dust had to be increased and the portion of superglue decreased. Despite
an improved conductivity, the inherent greater stiffness (see Section 4.1) of this glued
electrode configuration made its application undesirable.
Hence conductivity and low stiffness are the most desired properties of compliant
4.7. Unexpected Problems during Measurements 56
electrodes, the durability was not taken into account. Only carbon black dust is spread
on the polymer to preserve low stiffness and acceptable conductivity.
Although these prepared polymers were sufficiently conductive on both sides, when
high voltage was applied, arcing occured at the edges. The arcing burned much of
the polymer specimen and rendered it unusable, because of a short circuit induced by
the arcing at the edges. With wider margins the arcing could be minimized. Though
arcing occured arbitrarely over the area of the compliant electrodes and the polymer
specimens were burned by arcing. A burned and destroyed polymer is shown in Figure
4.9.
Figure 4.9: Burned Polymer
Finally the microprocessor or one of peripheral devices was broken. The ordering
of new components took too long and the few possible measurements were made using
a normal ruler and therefore are inaccurate.
Because of the difficulties in preparing, the late availability of very few functional,
but easily destroyable polymer specimens and the lack of precise measurement pos-
sibilities, the measurements of only poor results can be presented to determine the
voltage-displacement relation. Nevertheless they help verify the capability of the devel-
oped physics based model to reflect the actual system behavior qualitative (see Chapter
5).
Chapter 5
Simulation of Nonlinear Polymer
Model
The developed nonlinear model was implemented1 in Matlab/Simulink. The system
behavior is analyzed and illustrated by simulation plots. The qualitative behavior of
an electro-active polymer is shown. The simulations help to understand the properties
and the behavior of a linear EAP actuator.
5.1 Simulation of Charge and Capacitance
When a high voltage ei of 3000V is applied to the polymer, the attached compliant
electrodes representing a capacitor will be charged. The attraction force of the op-
posite charges q on the top and bottom eletrode causes the pliable polymer material
to contract in thicknes. Due to its incompressibility the material will expand in area.
This deformation in area and thickness induces a change in the capacitance Cp(l, w, h)
depending on the dimensions l, w and h. The change of the capacitor is depicted in
Figure 5.1b. Since the capacitance is increased and the induced voltage is kept (nearly)
1Used Parameters : Polymer Resistance Rp = 500kΩ, Cable Restistance (negligible) R = 0Ω,Polymer Inductance Lp = 1.3H , Mass of Polymer Mpolymer = 0.2g, Mass of Slide Mslide = 69.2g,Dielectric Constant of Polymer ǫr = 4.8, Young’s Modulus E = 96kN
m2 , Damping Coefficient CD =
14kNsm2 , Prestrained Dimensions L
pre0
= 35mm, Wpre0
= 31mm, Hpre0
= 0.3mm
57
5.2. Simulation of Forces 58
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−1
0
1
2
3
4
5
6
7
8x 10
−7 Charge when High Voltage (3000V) is impressed and removed
Time [s]
Cha
rge
[As]
Disconnecting High Voltageat Time 1.1s
Impressing High Voltage (3000V)at Time 0.1s
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.81.4
1.6
1.8
2
2.2
2.4
2.6x 10
−10 Capacitance of Polymer with Compliant Electrodes
Time [s]
Cap
acita
nce
[As/
V]
Impressing High Voltage (3000V)at Time 0.1s
Disconnecting High Voltageat Time 1.1s
a) Change in Charge b) Change in Capacitance
Figure 5.1: Simluted Changes in Charge and Capacitance
constant (ei = 3000V = const) by the power drive circuitry, more charges can be stored
on the compliant electrodes until the input voltage ei and the voltage of the capacitor
ec are balanced. A steady-state is reached.
ei = ec = Cp · q (5.1)
The charging time depends mainly on the resistance Rp, inductance Lp and deforming
capacitor Cp. Compared to the slow deformation and thus the slow change of the
capacitance, the influence of the resistance Rp and the inductance Lp on the charging
rate is negligible. The charging process is depicted in Figure 5.1a. After a very fast
charging, the amount of charges increase at a slower rate, due to the slow deformation
of the polymer.
5.2 Simulation of Forces
In Figure 5.2 the forces acting on the slide are depicted. It is important to mention that
the effective force F effx produced by the linear EAP actuator is very small. Simulations
5.3. Simulation of Dimensions 59
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2Force Simulation of Nonlinear Model
Time [s]
For
ce [N
]
Fxeff−FneohookianFdampingImpressing High Voltage (3000V)
at Time 0.1s
Disconnecting High Voltageat Time 1.1s
Figure 5.2: Simulated Forces
show a maximum force output of
F effx ≈ 0.12N (5.2)
for an impressed voltage of ei = 3000V . Also the high damping force Fdamping is
depicted, which results in an aperiodic transient behavior. The neo-Hookian force
FneoHookian = Ayz · σx (3.42) represents the internal spring force of the elastomeric
polymer. This spring force changes, when an external pressure pz (induced by charges)
is applied, because on one hand the neo-Hookian force depends on the cross-sectional
area Ayz = 1λW pre
0 Hpre0 and on the other hand directly on the pressure pz. Both factors
will decrease the force FneoHookian, when a voltage is impressed and therefore a pressure
pz is induced and will increase the neo-Hookian force, when the voltage is disconnected,
repectively. In steady-state the forces F effx and FneoHookian are balanced.
5.3 Simulation of Dimensions
In Figure 5.3a the alterations of the length and width are depicted. Figure 5.3b shows
the change in thickness. The length and width increase as expected. The thickness
decreases. Incompressibility is retained.
5.4. Simulation of Voltage-Displacement Relation 60
With these simulation results and the measurements presented in Section 4.6, the
developed nonlinear model can be verified qualitatively. The measured length l ≈ 39mm
and the simulated length ≈ 39.3mm nearly match. The same can be stated for the
measured width w ≈ 34.5mm and the simulated width ≈ 34.8mm. The deviations may
be traced back to inaccurate estimated parameters. This fact is not surprising, recalling
the inaccurate parameter estimation results shown above. Even so the physics based
model and its nonlinear simulation demonstrate a qualitatively correct approximation
of the polymer behavior in reality.
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.031
0.032
0.033
0.034
0.035
0.036
0.037
0.038
0.039
0.04
Time [s]
Leng
th [m
] and
Wid
th [m
]
Elongation in Length and Width when High Voltage (3000V) is impressed and removed
LengthWidth
~39.3 mm
Disconnecting High Voltageat Time 1.1s
Impressing High Voltage (3000V)at Time 0.1s
~34.8 mm
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 22.3
2.4
2.5
2.6
2.7
2.8
2.9
3x 10
−4 Thickness when High Voltage (3000V) is impressed and removed
Time [s]
Thi
ckne
ss [m
]
Disconnecting High Voltageat Time 1.1s
Impressing High Voltage (3000V)at Time 0.1s
a) Changes in Length and Width b) Changes in Thickness
Figure 5.3: Simulated Changes in Dimensions
5.4 Simulation of Voltage-Displacement Relation
Due to observations during the experimentation phase in preparing functional polymer
specimen, a displacement ux is first noticable when a voltage ei greater than ≈ 1000−1300V is applied. This property can be reflected by the presented nonlinear model. In
simulations the input voltage ei was increased slowly for 500V per second (to maintain
a steady-state condition) and the displacement ux was recorded. The final plot can be
seen in Figure 5.4. Not until the voltage ei reaches a value of ≈ 1300V , a displacement
ux greater than 0.5mm is noticable. For higher voltages ei the relation becomes more
5.4. Simulation of Voltage-Displacement Relation 61
significant, since the slope ∂ux
∂eiis steeper. Therefore a greater displacement is expected
for higher voltages.
0 500 1000 1500 2000 2500 3000−1
0
1
2
3
4
5x 10
−3 Simulated Relation of Displacement and Input Voltage
Voltage ei [V]
Dis
plac
emen
t ux [m
]
Noticable Displacement ux > 0.5mm
for Voltages ei > ~1300 V !!!
Figure 5.4: Simulated Relation of Displacement ux and Input Voltage ei
Chapter 6
Gopinath-Style Motion State Observer
6.1 Condition of Equilibrium (Operating Point)
To develop an operating point model, the system will be assumed to remain in an
equilibrium condition, this will be the operating point. In this section the derivation of
the equilibrium is shown. The derivation is only given in short form. The system will
be in an idle position [14] for
x = f(x∗, e∗i , F∗
dist) ≡ 0 (6.1)
y∗ = g(x∗) (6.2)
where x∗ =(
i∗, q∗, v∗
ux, u∗
x
)Trepresents the steady-state condition state vector with the
equilibrium or reference solutions [14] of the current i∗, the charge q∗, the displacement
velocity v∗
uxand the displacement u∗
x. All reference solutions will be indicated with “∗”.The system is fed with a constant input voltage e∗i = const. The unknown influence of
the disturbance is set to zero (F ∗
dist = 0). With these definitions and terms, it is easy
to find the equilibrium values of i∗ and v∗
uxby (6.1)
q ≡ 0 ⇒ i∗ = 0 (6.3)
ux ≡ 0 ⇒ v∗
ux= 0 (6.4)
62
6.1. Condition of Equilibrium (Operating Point) 63
By (6.4) the relation for λ∗ can be found to
λ∗ =v∗
ux
Lpre0
= 0 (6.5)
With the general condition of equilibrium (6.1) a system of equations can be intro-
duced by evaluating and simplifying the nonlinear state space model (3.10) for the
state derivatives i and vuxapplying the reference values e∗i = const, F ∗
dist = 0, i∗ = 0
and v∗
ux= 0
i ≡ 0| · Lp (6.6)
⇒ − Hpre0
ǫ0ǫrLpre0 W pre
0
1
(λ∗)4 · q∗ + e∗i ≡ 0 (6.7)
vux≡ 0| · M (λ∗)5
Hpre0 W pre
0
(6.8)
⇒ 3
ǫ0ǫr
(
q∗
W pre0 Lpre
0
)2
− E
3
(
(λ∗)6 − 1)
≡ 0 (6.9)
with the equilibrium stretch
λ∗ = 1 +u∗
x
Lpre0
(6.10)
When Equation (6.9) is transformed and solved for (λ∗)4, the following relation can be
derived
(λ∗)4 =(
(λ∗)6)2
3 =
(
9
ǫ0ǫrE
(
q∗
Lpre0 W pre
0
)2
+ 1
)2
3
(6.11)
Replacing (λ∗)4 in Equation (6.7) with (6.11) and several trivial transformations, a
polynomial can be given which relates the charge q∗ to the voltage e∗i
9 (e∗i )3
2
ǫ0ǫrE (Lpre0 W pre
0 )2
(√q∗)4 −
(
Hpre0
ǫ0ǫrLpre0 W pre
0
)3
2(√
q∗)3
+ (e∗i )3
2 = 0 (6.12)
6.2. Operating Point Model 64
This polynomial of√
q∗ can be solved numerically and the equilibrium value of the
charge q∗ is given by
q∗ =(√
q∗)2
(6.13)
Applying the computed value q∗ to (6.9) and solving the equation for the stretch λ∗
3
ǫ0ǫr
(
q∗
W pre0 Lpre
0
)2
=E
3
(
(λ∗)6 − 1)
(6.14)
(λ∗)6 =9
ǫ0ǫrE
(
q∗
Lpre0 W pre
0
)2
+ 1 (6.15)
and extracting the root, finally the equilibrium stretch λ∗ and with (6.10) the equilib-
rium displacement u∗
x can be calculated to
λ∗ = 6
√
9
ǫ0ǫrE (Lpre0 W pre
0 )2 · (q∗)2 + 1 (6.16)
u∗
x = Lpre0 (λ∗ − 1) = Lpre
0
(
6
√
9
ǫ0ǫrE (Lpre0 W pre
0 )2 · (q∗)2 + 1 − 1
)
(6.17)
Hereby all values are presented to build an operating point model, which will be derived
in the next section.
6.2 Operating Point Model
A linear system can be described with a state space algebraic model [8,14]. The system
described here is a single input single output (SISO) [14] system with four states. Thus
a fourth order state space model is given by
x = Ax + bei + dFdist (6.18)
y = cTx (6.19)
6.2. Operating Point Model 65
with the physical state vector x = (i, q, vux, ux)
T , the state feedback matrix A =
(aij)|i,j=1..4 (and its components aij), the input coupling vector b = (b1, b2, b3, b4)T ,
the disturbance input coupling vector d = (d1, d2, d3, d4)T and the measurement (or
output) selection vector cT = (c1, c2, c3, c4).
In the neighborhood of an equilibrium condition, a nonlinear system can be approx-
imated by an operating point model [14]. Therefore the Jacobian matrices are to set up
to be evaluated at the reference solutions x∗, e∗i and F ∗
dist [14]. The Jacobian matrices
are defined for the nonlinear system (3.77) presentend in Section 3.4 as follows. The
state feedback matrix A is given by
A = ∂f(x,ei,Fdist)∂xT
∣
∣
∣
∗
=∂fi(x, ei, Fdist)
∂xj
∣
∣
∣
∣
∗,i,j=1..4
(6.20)
and the input coupling vector b and the disturbance input coupling vector d are
b = ∂f(x,ei,Fdist)∂ei
∣
∣
∣
∗
=∂fj(x, ei, Fdist)
∂ei
∣
∣
∣
∣
∗,j=1..4
(6.21)
d = ∂f(x,ei,Fdist)∂Fdist
∣
∣
∣
∗
=∂fj(x, ei, Fdist)
∂Fdist
∣
∣
∣
∣
∗,j=1..4
(6.22)
The output selection vector c is represented by
c =∂g(x)
∂xT
∣
∣
∣
∣
∗
=∂g(x)
∂xj
∣
∣
∣
∣
∗,j=1..4
(6.23)
Here only the final results are presented. It is trivial to derive the correct partial
differentation terms. At first the components of the state feed back matrix A are
presented. The matrix components of the first row are given by
6.2. Operating Point Model 66
a11 ≡ ∂f1
∂i
∣
∣
∗= −Rp + R
Lp
(6.24)
a12 ≡ ∂f1
∂q
∣
∣
∣
∗
= − 1
Lp
Hpre0
ǫ0ǫrLpre0 W pre
0
1
(λ∗)4 (6.25)
a13 ≡ ∂f1
∂vux
∣
∣
∣
∗
= 0 (6.26)
a14 ≡ ∂f1
∂ux
∣
∣
∣
∗
=∂f1
∂λ· ∂λ
∂ux
∣
∣
∣
∣
∗
=4
Lp
Hpre0
ǫ0ǫr (Lpre0 )
2W pre
0
q∗
(λ∗)5 (6.27)
The second row consists of the components
a21 ≡ ∂f2
∂i
∣
∣
∗= 1 (6.28)
a22 ≡ ∂f2
∂q
∣
∣
∣
∗
= 0 (6.29)
a23 ≡ ∂f2
∂vux
∣
∣
∣
∗
= 0 (6.30)
a24 ≡ ∂f2
∂ux
∣
∣
∣
∗
= 0 (6.31)
The third row of the matrix can be represented by the following components
a31 ≡ ∂f3
∂i
∣
∣
∣
∣
∗
= 0 (6.32)
a32 ≡ ∂f3
∂q
∣
∣
∣
∣
∗
= 6 · Hpre0
ǫ0ǫrWpre0 (Lpre
0 )2
q∗
(λ∗)5
1
M(6.33)
a33 ≡ ∂f3
∂vux
∣
∣
∣
∣
∗
=∂f3
∂λ· ∂λ
∂vux
∣
∣
∣
∣
∣
∗
= −W pre0 Hpre
0
Lpre0
CD
M
1
λ∗(6.34)
a34 ≡ ∂f3
∂ux
∣
∣
∣
∣
∗
=∂f3
∂λ· ∂λ
∂ux
∣
∣
∣
∣
∗
=
=Hpre
0 W pre0
M · Lpre0
(
−15
ǫ0ǫr (λ∗)6
(
q∗
Lpre0 W pre
0
)2
− E
3
(
1 +5
(λ∗)6
)
+CDλ∗
(λ∗)2
)
(6.35)
The fourth row of the matrix has the components
6.2. Operating Point Model 67
a41 ≡ ∂f4
∂i
∣
∣
∗= 0 (6.36)
a42 ≡ ∂f4
∂q
∣
∣
∣
∗
= 0 (6.37)
a43 ≡ ∂f4
∂vux
∣
∣
∣
∗
= 1 (6.38)
a44 ≡ ∂f4
∂ux
∣
∣
∣
∗
= 0 (6.39)
Further on, the components of input coupling vector b can be given by
b1 ≡ ∂f1
∂ei
∣
∣
∣
∗
=1
Lp
(6.40)
b2 ≡ ∂f2
∂ei
∣
∣
∣
∗
= 0 (6.41)
b3 ≡ ∂f3
∂ei
∣
∣
∣
∗
= 0 (6.42)
b4 ≡ ∂f4
∂ei
∣
∣
∣
∗
= 0 (6.43)
and the disturbance input coupling vector d with its components
d1 ≡ ∂f1
∂Fdist
∣
∣
∣
∗
= 0 (6.44)
d2 ≡ ∂f2
∂Fdist
∣
∣
∣
∗
= 0 (6.45)
d3 ≡ ∂f3
∂Fdist
∣
∣
∣
∗
=1
M(6.46)
d4 ≡ ∂f4
∂Fdist
∣
∣
∣
∗
= 0 (6.47)
6.3. Development of Gopinath-style Observer 68
Finally the output selection vector c can be derived to
c1 ≡ ∂g
∂i
∣
∣
∗= 0 (6.48)
c2 ≡ ∂g
∂q
∣
∣
∣
∗
= 0 (6.49)
c3 ≡ ∂g
∂vux
∣
∣
∣
∗
= 0 (6.50)
c4 ≡ ∂g
∂ux
∣
∣
∣
∗
= 1 (6.51)
Combining all these results the linear algebraic system model is found, valid for a small
neighborhood around the equilibrium condition x∗ =(
i∗, q∗, v∗
ux, u∗
x
)Tand y∗ = g(x∗)
presented in Section 6.1. The operating point model will only consider small signal
responses, the output “offset” value y∗ is not included. Based on this “linear” operat-
ing point model a Gopinath-style motion state observer is developed and estimation
accuracy evalution is simulated in Matlab/Simulink.
6.3 Development of Gopinath-style Observer
The Gopinath-style observer is fed by the PWM duty cycle signal dc as feedforward
input provided by the system controller and by the measured charging current i as
reference input. To force the controller error ce to zero a PI-controller is used. According
to the introduction and [8, 9] the Gopinath-style motion state observer controller is
formed properly using only the deviation of the measured state i (charging current)
and the corresponding estimate i as controller input. Thus the zero lag property can
be maintained. The overall state space model is given by
xall = Aallxall + Balldc + DallFdist (6.52)
yall = Callxall (6.53)
6.3. Development of Gopinath-style Observer 69
where xall =(
i, q, vux, ux, i, q, vux
, ux, h, qdist
)T
represents the overall state space vector
of the 10th grade system. The estimated states are the current i, the charge q, the
velocity vuxand the displacement ux. Additional states h1 and qdist are introduced,
where h1 is the integrated current of the observer controller and qdist represents the
“disturbance charge” to estimate the disturbance force Fdist.
In Figure 6.1 the overall simulation set-up with physical system and Gopinath-style
observer is depicted, where a11, a12, a14, a32, a33 and a34 are the estimated components
of the state feedback matrix, Kpwm and Khvc are the estimated PWM driver and high
voltage converter gains and b1 is the estimated input coupling factor. The parameter
estimates are represented by the previously introduced relations in Section 6.2 for aij ,
only that here the estimated values for the dimensions Lpre0 , W pre
0 and Hpre0 , the Young’s
modulus E, the dielectric constant ǫr, the damping coefficient CD, the polymer resis-
tance Rp, the cable resistance R and the polymer inductance Lp are used for calculation.
The estimate b1 can be calculated respectively.
In this model the PWM driver gain Kpwm = 125V and the high voltage converter
gain Khvc = 250 are included. A duty cycle dc of 100% represents a 5V output voltage
and therefore the maximum output voltage of the PWM driver circuit is 12V . The high
voltage converter module provides a maximum output of 3000V .
The Gopinath-style observer is built with a simple PI-controller with the propor-
tional gain K1 and the integration gain K2. By inspecting the state block diagram of
the overall system model shown in Figure 6.1, the 10th order state space model can be
given by
6.3
.D
evelo
pm
ent
ofG
opin
ath
-style
Obse
rver
70
Physical System
1
s+ 1
sxu
distF
+
effxF xu
xu
q1
s
ie + 1
sicd
pwm hvcK K⋅ 1b+
+
12a
11a
+
14a
32a
33a
34a
+
+
3d
1
s+ 1
s
ˆxu
+
ˆ effxF ˆ
xu
ˆxu
q1
s
ie + 1
si
+
+
+
+
ˆ ˆpwm hvcK K⋅ 1b
11a
12a
32a
33a
34a
14a
1K
2K
iReference Input
Charging CurrentFeedforward Input
Duty Cycle
+−
++ +
+
Closed LoopGopinath-Style Observer1
s
OberserverController
ec
disti
1
s32a
ˆdistq1
3d − distF−
h
Figu
re6.1:
State
Blo
ckD
iagramof
Operatin
gPoin
tM
odel
ofSystem
and
Gop
inath
-Sty
leO
bserver
6.3. Development of Gopinath-style Observer 71
xall =
a11 a12 0 a14 0 0 0 0 0 0
1 0 0 0 0 0 0 0 0 0
0 a23 a33 a34 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0
0 0 0 0 a11 a12 0 a14 0 0
1 − K1 0 0 0 K1 0 0 0 K2 0
0 0 0 0 0 a32 a33 a34 0 0
0 0 0 0 0 0 1 0 0 0
−1 0 0 0 1 0 0 0 0 0
−K1 0 0 0 K1 0 0 0 K2 0
xall +
+
KpwmKhvcb1
0
0
0
KpwmKhvcb1
0
0
0
0
0
dc +
0
0
d3
0
0
0
0
0
0
0
Fdist (6.54)
with the output or measurement vector yall
yall =
ux
ux
Fdist
=
0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 a32
d3
xall (6.55)
Here ux is the displacement of the “real” system, ux represents the corresponding esti-
6.4. Evalution of Estimation Accuracy 72
mate of the displacement and Fdist represents the estimated disturbance force. With
the state space representation of the system, the transfer function vector Fei(s) relating
the output vector yall to the input voltage ei can be derived by using the formula [14]
Fei(s) =
yall
ei
(s) =
ux
ei(s)
ux
ei(s)
Fdist
ei(s)
= Call (s1 − Aall)−1
Ball (6.56)
with the unity matrix 1. With the disturbance force Fdist as input to the system, the
transfer function vector FFdist(s) can be found respectively [14]
FFdist(s) =
yall
Fdist
(s) =
ux
Fdist(s)
ux
Fdist(s)
Fdist
Fdist(s)
= Call (s1 − Aall)−1
Dall (6.57)
The transfer functions can be derived in general with Maple, but are not presented
due to their complexity. The Maple commands to compute the results are included
in Appendix C. The Figures 6.2,6.3,6.4,6.5,6.6 shown below are more illustrative, they
deliver insight into the system behavior and show the influence of paramter estimation
errors on the estimation accuracy of the observer structure.
6.4 Evalution of Estimation Accuracy
Evaluation accuracy frequency responses were simulated in Matlab/Simulink. The
10th order state space model (6.54) was used1 and evaluated for different parameter
1The model implemented in Matlab/Simulink was set up with following parameters:
1. Simulation Parameters of Physical System: Polymer Resistance Rp = 500kΩ, Cable Restistance(negligible) R = 0Ω, Polymer Inductance Lp = 1.3H , Mass of Polymer Mpolymer = 0.2g,Mass of Slide Mslide = 69.2g(measured), Dielectric Constant of Polymer ǫr = 4.8, Young’sModulus E = 96kN
m2 , Damping Coefficient CD = 14kNsm2 , PWM Drive Gain Kpwm = 12
5V , High
Voltage Converter Gain Khvc = 250, Prestrained Dimensions Lpre0
= 35mm, Wpre0
= 31mm
and Hpre0
= 0.3mm, Operating Point Model for Input Voltage e∗i = 2500V (see Calculations forequilibrium condition in Section 6.1)
6.4. Evalution of Estimation Accuracy 73
estimates with errors. The transfer functions used to plot the diagrams were
Fux(s) =
ux
ei(s)
ux
ei(s)
=ux
ux
(s) (6.58)
to relate displacement estimate ux to actual displacement ux and the disturbance trans-
fer function
FFdist(s) = − Fdist
Fdist
(s) (6.59)
When the transfer functions computed in Maple (see Appendix C) are evaluated for
low frequencies, especially for s = 0, the steady state relations of ux
ux(s)can be derived
to
Fux(s)|s→0 =
ux
ux
(s)
∣
∣
∣
∣
s→0
=a32KpwmKhvcb1
a32KpwmKhvcb1· a12a34 − a14a32
a12a34 − a14a32(6.60)
and the disturbance estimation − Fdist
Fdist(s) is given by
FFdist(s)∣
∣
s→0= − Fdist
Fdist
(s)
∣
∣
∣
∣
∣
s→0
= − a32
d3
· a14d3
a12a34 − a14a32(6.61)
Both steady state funcions show significant sensitivity to the accuracy of the estimated
parameters. (6.61) produces a disturbance estimation error FFdist(s)∣
∣
s→06= 1 at all
times. Even for a correct estimated parameter d3 = d3, the ratio a14a32
a12a34−a14a32is not
equal to 1. Unfortunately, the Gopinath-style observers only offers a poor estimation
value −Fdist for the disturbance force Fdist. The observer controller gains were chosen
by trail and error, without specifying the observer bandwidth. This should be done
later, when the closed loop system is built and evaluated for the actual control system.
2. Controller Gains of Low Bandwidth Gopinath-style Observer : K1 = 0.001 and K2 = 0.0005 1
s
(Oberserver Eigenvalues: s1 = −3.84605260472244 ·105, s2,3 = −28.031868761±14.586659646j,s4 = −3.922356207, s5 = −0.553519505)
3. Controller Gains of High Bandwidth Gopinath-style Observer : K1 = 0.1 and K2 = 0.05 1
s
(Oberserver Eigenvalues: s1 = −3.83600318017794 · 105, s2 = −1014.246030495, s3 =−40.040085367, s4 = −10.695515591, s5 = −0.500436230)
6.4. Evalution of Estimation Accuracy 74
6.4.1 Influence of Parameter Errors of prestrained Length Lpre0 ,
Width W pre0 and Thickness Hpre
0
In Figures 6.2a and b, the estimation accuracy frequency response ux
uxof the build
Gopinath-style observer is depicted for parameter estimation errors of the dimensions.
Since the estimated values of the prestrained dimensions directly influence the steady
state transfer function Fux(s)|s→0 (6.60), it is obvious that estimation errors occur
even at low frequencies. Due to the properly formed observer controller using only
the deviation of reference input current i, the corresponding current estimate i and
the feedforward path, the desired zero lag property can be retained. The influence of
parameter errors for the dimensions on the estimation accuracy are depicted in Figure
6.2 for observers with low and high bandwidths. The parameter errors of the estimated
length, width and thickness show nearly the same sensitivity within and beyond the
bandwidth of the slow and fast observers.
10−2
100
102
104
106
108
1010
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.001 [1] and K
2=0.0005 [1/s]
10−2
100
102
104
106
108
1010
−15
−10
−5
0
5
10
15
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Length L0pre*1.1
Length L0pre*0.9
Width W0pre*1.1
Width W0pre*0.9
Thickness H0pre*1.25
Thickness H0pre*0.75
10−2
100
102
104
106
108
1010
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.1 [1] and K
2=0.05 [1/s]
10−2
100
102
104
106
108
1010
−15
−10
−5
0
5
10
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Length L0pre*1.1
Length L0pre*0.9
Width W0pre*1.1
Width W0pre*0.9
Thickness H0pre*1.25
Thickness H0pre*0.75
a) Low Bandwith b) High Bandwidth
Figure 6.2: Estimation Accuracy Frequency Response ux
uxfor Dimension Parameter
Errors Lpre0 = (1 ± 0.1)Lpre
0 , W pre0 = (1 ± 0.1)W pre
0 and Hpre0 = (1 ± 0.25)Hpre
0
6.4. Evalution of Estimation Accuracy 75
6.4.2 Influence of Parameter Errors of PWM Driver Gain Kpwm
and High Voltage Converter Gain Khvc
The impact of the parameter errors of PWM driver gain Kpwm and high voltage con-
verter gain Khvc on the estimation accuracy of ux
uxis shown in Figure 6.3a and b for low
and high bandwidth observer. Again, a steady state error is obvious, since the estima-
tion accuracy FRF depends on Kpwm and Khvc. The zero lag property is maintained
again for both low and high bandwidth observer configurations. However, for these dis-
cussed parameter errors a significant difference in the estimation accuracy is obvious.
The low bandwidth observer estimation will be exact after about 10Hz, whereas the
high bandwidth observer will always have an estimation error, even after about 100Hz.
Thus, when the PWM driver gain Kpwm and the high converter gain Khvc are not very
well know, the slow observer should be the first choice.
10−2
100
102
104
106
108
1010
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.001 [1] and K
2=0.0005 [1/s]
10−2
100
102
104
106
108
1010
−6
−4
−2
0
2
4
6
8
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
PWM Gain Kpwm
*1.1PWM Gain K
pwm*0.9
HV Converter Gain Khvc
*1.2HV Converter Gain K
hvc*0.8
10−2
100
102
104
106
108
1010
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.1 [1] and K
2=0.05 [1/s]
10−2
100
102
104
106
108
1010
−6
−4
−2
0
2
4
6
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
PWM Gain Kpwm
*1.1PWM Gain K
pwm*0.9
HV Converter Gain Khvc
*1.2HV Converter Gain K
hvc*0.8
a) Low Bandwith b) High Bandwidth
Figure 6.3: Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of PWM
Gain Kpwm = (1 ± 0.1)Kpwm and High Voltage Converter Gain Khvc = (1 ± 0.2)Khvc
6.4. Evalution of Estimation Accuracy 76
6.4.3 Influence of Parameter Errors of Polymer Resistance Rp
and Inductance Lp
In this section the influence of the estimated values of the polymer resistance Rp and
inductance Lp is shown. The bode diagram of the FRF ux
uxis depicted in Figure 6.4.
For the given tuned observer gains, a minor parameter error sensitivity is found for
both, low and high bandwidth observers. Though the low bandwidth observer provides
especially good estimation accuracy. The maximum estimation error within the shown
frequency range is 0.1%. Thus compared to the high bandwidth configuration with a
maximum error of 10%, the low bandwidth designed observer should be preferred.
10−2
100
102
104
106
108
1010
0.999
0.9995
1
1.0005
1.001
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF − K1 = 0.001 [1] and K
2=0.0005 [1/s]
10−2
100
102
104
106
108
1010
−0.02
−0.01
0
0.01
0.02
0.03
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Resistance Rp*1.6Resistance Rp*0.6Inductance Lp*3Inductance Lp*0.5
10−2
100
102
104
106
108
1010
0.9
0.95
1
1.05
1.1
1.15
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]FRF for K
1 = 0.1 [1] and K
2=0.05 [1/s]
10−2
100
102
104
106
108
1010
−2
−1
0
1
2
3
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Resistance Rp*1.6Resistance Rp*0.6Inductance Lp*3Inductance Lp*0.5
a) Low Bandwith b) High Bandwidth
Figure 6.4: Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of Poly-
mer Resistance Rp = 1.6Rp and 0.6Rp and High Voltage Converter Gain Khvc =(1 ± 0.2)Khvc
6.4.4 Influence of Parameter Errors of Young’s Modulus E,
Damping Coefficient CD and Dielectric Constant ǫr
In Figure 6.5 the estimation accuracy frequency response ux
uxfor parameter errors of the
young’s modulus E, damping coefficient CD and dielectric constant ǫr is depicted. The
estimation accuracy sensitivity does not depend on the choice of the observer eigen-
6.4. Evalution of Estimation Accuracy 77
values nor therefore the observer bandwidth. Both plots show similar results for these
parameter errors in magnitude and phase of the FRF ux
uxfor low and high bandwidth ob-
servers. The sensitivity within the observer bandwidth is small for parameter errors of
the estimated damping coefficient CD and dielectric constant ǫr, whereas a parameter
error of the approximated Young’s modulus E shows a significant effect on the esti-
mation accuracy even within the observer bandwidth. The high bandwidth observer
configuration has higher phase deviations.
10−2
100
102
104
106
108
1010
0.4
0.6
0.8
1
1.2
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.001 [1] and K
2=0.0005 [1/s]
10−2
100
102
104
106
108
1010
−15
−10
−5
0
5
10
15
20
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Dielectric Constant epsilonr*1.05
Dielectric Constant epsilonr*0.95
Youngs Modulus E*1.2Youngs Modulus E*0.8Damping Coefficient C
D*1.8
Damping Coefficient CD
*1.4
10−2
100
102
104
106
108
1010
0.4
0.6
0.8
1
1.2
1.4
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
FRF for K1 = 0.1 [1] and K
2=0.05 [1/s]
10−2
100
102
104
106
108
1010
−20
−10
0
10
20
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
Dielectric Constant epsilonr*1.05
Dielectric Constant epsilonr*0.95
Youngs Modulus E*1.2Youngs Modulus E*0.8Damping Coefficient C
D*1.8
Damping Coefficient CD
*1.4
a) Low Bandwith b) High Bandwidth
Figure 6.5: Estimation Accuracy Frequency Response ux
uxfor Parameter Errors of Di-
electric Constant ǫr = (1± 0.05)ǫr, Young’s Modulus E = (1± 0.2)E and Kelvin-VoigtCoefficient CD = 1.8CD and 1.4CD
6.4.5 Disturbance Estimation Accuracy
In Figure 6.6 the disturbance estimation accuracy frequency response − Fdist
Fdistis depicted.
The diagram is plotted without any parameter estimation errors, though an inherent
inaccuracy is obvious. Recalling the (steady-state) transfer function (6.61) this behavior
is unavoidable. The estimation accuracy would degrade even when parameter errors
would be assumed. Both observer configurations show lagging problems beyond their
bandwidth. Therefore only for low frequencies disturbance input decoupling would be
effective.
6.4. Evalution of Estimation Accuracy 78
10−2
10−1
100
101
102
103
104
0
0.2
0.4
0.6
0.8
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]Disturbance FRF for K
1=0.001 and K
2=0.0005
10−2
10−1
100
101
102
103
104
−300
−250
−200
−150
−100
−50
0
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
10−2
10−1
100
101
102
103
104
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Frequency [Hz, log.]
Mag
nitu
de [l
inea
r]
Disturbance FRF for K1=0.1 and K
2=0.05
10−2
10−1
100
101
102
103
104
−300
−250
−200
−150
−100
−50
0
Frequency [Hz, log.]
Pha
se [°
, lin
ear]
a) Low Bandwith b) High Bandwidth
Figure 6.6: Disturbance Estimation Accuracy Frequency Response − Fdist
Fdistwithout Pa-
rameter Errors
6.4.6 Remarks and Observations
The above presented estimation behavior analysis reveals acceptable estimation accu-
racy for the displacement ux for both the low and high bandwidth observer designs.
However the build observer structure has severe problems estimating the disturbance
force Fdist.
Recalling the parameter estimation results for the polymer resistance Rp and induc-
tance Lp, which were inherently sensitive to the density of the spread carbon black dust
making the estimates of these parameters inaccurate, a low bandwidth observer should
be chosen. The PWM driver gain Kpwm and the high voltage converter gain Khvc were
assumed to be constant, but this may not hold for all frequencies or the complete in-
put voltage range, and therefore significant estimation errors for these parameters can
be expected. Based on the analysis above, a low bandwidth observer would fit more
adequate and would show minor parameter sensitivity.
Chapter 7
Conclusion
This thesis presented the development of a physics based model of electroactive polymer
actuators. Rough parameter estimation was shown. The derived model was simulated
and verified with the fewest possible and imprecise measurements. Better measure-
ment results could not be gathered due to the described problems encountered while
working on this most challenging and often frustrating project. However, the devel-
oped nonlinear model seems to be capable of mirroring the qualitative behavior of the
system.
An operating point model was used to build a Gopinath-style motion state observer.
The overall system model was implemented in the simulation tool Matlab/Simulink.
Simulations were run and estimation accuracy frequency reponses were presented and
the inherent properties of the Gopinath observer topology are illustrated for several
parameter errors. The estimation accuracy of the motion was satisfying, whereas the
disturbance estimation accuracy analysis reveals poor results.
In the future there will be plenty of work and research to finish this project. First,
reliable, durable and functional polymer specimen have to be prepared and manufac-
tured. With those, parameter estimation is to be completed and the developed nonlinear
model is to be verified by more exact measurements. The developed Gopinath-style ob-
server is to be implemented in the microprocessor and its accuracy is to be evaluated.
The observer controller gains have to be adjusted to the actual real system with a
79
Chapter 7. Conclusion 80
closed-loop displacement controller.
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.530
40
50
60
70
80
90
Force [N]
Leng
th [m
m]
Change in Length due to applied Weight
Weight increased
Weight decreased
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.523
24
25
26
27
28
29
30
31
32
33
Force [N]
Wid
th [m
m]
Change in Width due to applied Weight
Weight increased
Weight decreased
a) Length Deformation b) Width Deformation
Figure 7.1: Measured hysteresis - deformation in length and width, when weight isapplied
As observations indicate, a term describing hysteresis should be included in the
physics based model. The hysteresis effects can not be neglected for this setup. In
Figure 7.1 the hysteresis is adumbrated. A (unprepared) polymer specimen was strained
by the gravitational force of applied weights and the strain was recorded. Thereafter,
the weights were gradually removed and again the remaiming strain was measured.
Apparently, the elastomer shows the characteristic of a hysteresis, with a maximun
deviation ≈ 5mm between both paths (see Figure 7.1). Thus it is obvious that for the
used EAP actuator with an approximated maximal displacement of ≈ 4− 5mm a term
describing hysteresis should be included in the model. In [5] a possibility for elastomers
is proposed.
Finally, it was shown that the force output of the actuator is very low due to
simulations. If frames were attached to the polymer actuator, the displacement force
acting in the other planar direction could be exploited and transformed in the desired
translation direction. Besides, frames represent an additional challenge to the nonlinear
modelling of the polymer actuator.
Chapter 7. Conclusion 81
Bibliography
[1] Bar-Cohen, Yoseph, “Electroactive polymer (EAP) actuator as artificial mus-
cles - reality, potential, and challenges”, SPIE-The International Society for Optical
Engineering, Bellingham, 2001
[2] Holzapfel, Gerhard A., “Nonlinear Solid Mechanics”, John Wiley & Sons Ltd.,
Chichester, 2000
[3] Banks, H.T., Nancy Lybeck “Modeling Methodology for Elastomer Dynamics,
Invited lecture”, Conference on Math Theory of Network and Systems (MTNS), 1996
[4] Banks, H. T., N. J. Lybeck, M. J. Gaitens, B. C. Munoz, L. C. Yanyo,
“Modeling the Dynamic Mechanical Behavior of Elastomers”, presented at a meeting
of the Rubber Division, American Chemical Society, Kentucky, 1996
[5] Banks, H.T., Gabriella A. Pinter, L.K. Potter, “Modeling of Nonlinear
Hysteresis in Elastomers”, Center of Research in Scientific Computation, Technical
Report, CRSC-TR99-09, North Carolina State University, Raleigh, 1999
[6] MSC.Software Corporation, “Nonlinear Finite Element Analysis of Elas-
tomers”, Technical Paper, Provided by MSC.Software Coorporation, California,
USA, 2000
[7] Rivlin, R.S., “Large elastic deformations of isotropic materials I, II, III”, Phil.
Trans. Roy. Soc. A, 240, 1948, pp. 459-525
82
Bibliography 83
[8] Lorenz, R.D., “Observers and State Filters in Drives and Power Electronics”,
Keynote paper, IEEE IAS OPTIM 2002, Brasov, 2002
[9] Jansen, P.L. and R.D. Lorenz, “A Physically Insightful Approach to the Design
and Accuracy Assessment of Flux Observers for Field Oriented Induction Machine
Drives”, in IEEE Trans. on Ind. Appl., 1994, pp.101-110
[10] Luenberger, D.G., “An Introduction to Observers”, IEEE Trans on Aut. Con-
trol., AC-16, 1971, pp. 596-602
[11] Gopinath, B., “On the Control of Linear Multiple Input-Output Systems”, The
Bell System Technical Journal, Vol. 50, No. 3, 1971, pp. 1063-1081
[12] Lorenz, R. D., “Dynamics of Controlled Systems (ME 746)”, Lecture, Departe-
ment of Mechanical Engineering, University of Wisconsin - Madison, 2003
[13] Schröder, D., “Intelligente Verfahren für mechatronische Systeme”, Vor-
lesungsskript, Lehrstuhl für Elektrische Antriebssysteme, Technische Universität
München (TUM), 2002
[14] Schmidt, G., “Regelungs- und Steuerungstechnik 2”, Vorlesungsskript, Lehrstuhl
für Steuerungs- und Regelungstechnik, Technische Universität München (TUM),
2001
[15] Pelrine, R. , R. Kornbluh, Q. Pei, J. Joseph, “High-Speed Electrically
Actuated Elastomers with Strain Greater Than 100%”, Report, SRI International,
Menlo Park, 1999
[16] Pelrine, R. , R. Kornbluh, G. Kofod, “High Strain Actuator Materials Based
on Dielectric Elastomers”, in Advanced Materials, Wiley-VCH, Issue 12, 2000, pp.
1223-1225
[17] Wang, Q., “Principle of Electromechanical Sensors and Actuators (ME 2082)”,
Lecture, Department of Mechanical Engineering, University of Pittsburgh, Spring
2003
Bibliography 84
[18] stöcker, H., “Taschenbuch der Physik”, 3. überarbeitete und erweiterte Auflage,
Verlag Harri Deutsch, Frankfurt am Main, 1998
[19] stöcker, H., “Taschenbuch mathematischer Formeln und moderner Verfahren”,
4. korrigierte Auflage, Verlag Harri Deutsch, Frankfurt am Main, 1999
Appendix A
List of Symbols
A State Feedback Matrix
Aall Final State Feedback Matrix (System & Observer)
Ace Cross-Section Area of Compliant Electrodes[
m2]
Axy Cross-Section Area Perpendicular to z-Axis[
m2]
Axz Cross-Section Area Perpendicular to y-Axis[
m2]
Ayz Cross-Section Area Perpendicular to x-Axis[
m2]
aij Components of State Feedback Matrix
aij Estimated Components of State Feedback Matrix
B Input Coupling Matrix
Ball Final Input Coupling Matrix (System & Observer)
bij Components of Input Coupling Matrix
bij Estimated Components of Input Coupling Matrix
CD Kelvin-Voigt Damping Coefficient[
Nsm2
]
CD Estimated Kelvin-Voigt Damping Coefficient[
Nsm2
]
Cp Capacitance of Polymer[
F = AsV
]
C Output or Measurement Selection Matrix
Call Final Output or Measurement Selection Matrix (System & Observer)
85
Appendix A. List of Symbols 86
cij Components of Output or Measurement Selection Matrix
cij Estimated Components of Output or Measurement Selection Matrix
ce Gopinath-Style Controller Error [A]
D Disturbance Input Coupling Matrix
Dall Final Disturbance Input Coupling Matrix
D Deformation Gradient
dc Duty Cycle Output of Controller
dij Components of Disturbance Input Coupling Matrix
detF Determinant of Configuration Gradient F
E Young’s Modulus or General Modulus of Elasticity[
Nm2
]
E Estimated Young’s Modulus or General Modulus of Elasticity[
Nm2
]
Eel Electric Field[
Vm
]
Ecd Electric Field of Charged Disc[
Vm
]
Eip Electric Field of Infinite Planes[
Vm
]
ei Input Voltage [V ]
ec Capacitor Voltage [V ]
ǫr Dielectric Constant of Polymer
ǫr Estimated Dielectric Constant of Polymer
ǫ0 Permettivity of Free Space[
8.854 · 10−12 AsV m
]
F Configuration Gradient
Fdist Disturbance Force [N ]
Fdist Estimated Disturbance Force [N ]
FN Normal Force [N ]
Feffx Effective Force in x-Direction [N ]
Feffz Effective Force in z-Direction [N ]
FneoHookian Neo-Hookian Force [N ]
FDamping Damping Force [N ]
Appendix A. List of Symbols 87
Fsliding Sliding Friction Force [N ]
Fstatic Static Friction Force [N ]
FeiTransfer Function Vector respectively the Input Voltage ei
FFdistTransfer Function Vector respectively the Input Disturbance Force Fdist
FuxEstimation Accuracy Transfer Function ux
ux
FFdist
Disturbance Estimation Accuracy Transfer Function Fdist
Fdist
G Shear Modulus[
Nm2
]
g Gravitational Constant[
9.81ms2
]
H0 Initial Thickness of Polymer [m]
Hpre0 Prestrained Thickness of Polymer [m]
h Current Thickness of Polymer [m]
i Charging Current [A]
i Estimated Charging Current [A]
idist “Disturbance” Current [A]
K1 Proportional Gain of Gopinath-Style Observer Controller [1]
K2 Integrator Gain of Gopinath-Style Observer Controller[
1s
]
Kpwm Gain of PWM Driver Chip [V ]
Khvc Gain of High Voltage Converter
L Redeformed Length of Polymer [m]
Lp Inductance of Compliant Electrodes [H]
L0 Initial Length of Polymer [m]
Lpre0 Prestrained Length of Polymer [m]
l Current Length of Polymer [m]
λ Principal Stretch or Deformation Ratio of Developed Model
λ Periodic Change of the Principal Stretch λ[
1s
]
λx Principal Stretch or Deformation Ratio along x-Axis
λy Principal Stretch or Deformation Ratio along y-Axis
Appendix A. List of Symbols 88
λz Principal Stretch or Deformation Ratio along z-Axis
M Inertia Mass of System [kg]
Mslide Mass of Slide[kg]
Mpolymer Mass of Polymer [kg]
madd Mass of Weights added to Bag [kg]
mbag Mass of Bag [kg]
µ0 Static Friction Coefficient
µ Sliding Friction Coefficient
n Normal Vector in Eulerian System
N Normal Vector in Lagrangian System
ν Poisson Ratio
pz Pressure in z-Direction[
Nm2
]
p0 Hydrostatic Pressure (Retain Incompressibility)[
Nm2
]
P First Piola-Kirchhoff or Nominal Stress Tensor[
Nm2
]
Pij Nominal or First Piola-Kirchhoff Stress Component[
Nm2
]
Ψ Strain Energy Function (SEF) or Helmholtz Free Energy Function
q Charge [As]
q Estimated Charge [As]
qdist Estimated “Disturbance” Charge [As]
Rp Resistance of Compliant Electrodes on Polymer [Ω]
Rp Estimated Resistance of Compliant Electrodes on Polymer [Ω]
R Resistance of Cables [Ω]
R Estimated Resistance of Cables [Ω]
s Laplace Frequency s = δ + jω ∈ C
si Eigenvalue i of System
σ Cauchy or True Stress Tensor[
Nm2
]
σij Cauchy or True Stress Components[
Nm2
]
Appendix A. List of Symbols 89
t Time [s]
t Cauchy or True Traction Vector [N ]
T First Piola-Kirchhoff or Nominal Traction Vector [N ]
ux Displacement along x-Axis [m]
ux Estimated Displacement along x-Axis [m]
uy Displacement along y-Axis[m]
uz Displacement along z-Axis[m]
vux Displacement Velocity[
ms
]
vux Estimated Displacement Velocity[
ms
]
W Redeformed Width of Polymer [m]
W0 Initial Width of Polymer [m]
Wpre0 Prestrained Width of Polymer [m]
w Current Width of Polymer [m]
~x Coordinate Vector in Eulerian System
~X Coordinate Vector in Lagrangian System
x State Space Vector
x1 State Space Vector of Electrical Circuit
x2 State Space Vector of Neo Hookian Model with Damping
χ Configuration or Motion Map of Deformation
y Measurement Output (Scalar)
yall Final Measurement Output Vector
Appendix B
Hardware Configuration
The main initialization files for the hardware set-up of the microprocessor and its pe-
ripherials are presented. The well documented files outline the amount of accomplished
preparatory work and help to expedite the proceedings of future changes in the hard-
ware configuration.
B.1 HardwareSetup.c
/**********************************************************************
*
* FILE : Hardware Setup
* DATE : September 09, 2003
* DESCRIPTION : Polymer Control Hardware Definitions
* CPU TYPE : H8 Tiny/Super Low Power
*
* Author: Christoph Hackl
*
/*******************************************************************/
#include <machine.h>
#include "iodefine.h"
#include "sci3.h"
static struct SCI_Init_Params SCI_Init_Data=BRR_9600,P_NONE,1,8;
/*******************************************************************/
/* INTERNAL FUNCTION DECLARATION */
void HardwareSetup(void);
/*******************************************************************/
void HardwareSetup(void)
short wait = 20;
90
B.1. HardwareSetup.c 91
/*******************************************************************/
/* Setting up System*/
/* Stick to initial values for these Registers IEGR1,2 , IRR1, IWPR*/
/* Stick to initial values for these Registers SYSCR1,2 */
/*MASK ALL INTERRUPTS BEFORE HARDWARE SETUP => NO INTERRUPT
REQUEST POSSIBLE*/
set_imask_ccr(1);
/* Interrupt Edge Select Register 1(IEGR1)*/
/*
7 NMIEG 0 NMI Edge Select ... falling
6 --- 1
5 --- 1
4 --- 1 reserved
3 IEG3 0 IRQ3 Edge Select ... falling
2 IEG2 0 IRQ2 "
1 IEG1 0 IRQ1 "
0 IEG0 0 IRQ0 "
*/
IEGR1.BYTE = 0x70;
/* Interrupt Edge Select Register 2(IEGR2)*/
/*
7 --- 1
6 --- 1 reserved
5 WPEG5 0 WPEG5 Edge Select ... falling
4 WPEG4 0 "
3 WPEG3 0 "
2 WPEG2 0 "
1 WPEG1 0 "
0 WPEG0 0 "
*/
IEGR2.BYTE = 0xC0;
/* Interrupt Enable Register 1*/
/*
7 IENDT = 0 Direct transfer interrupt DISABLED
6 IENTA = 1 Timer A interrupt ENABLED
5 IENWP = 0 Wakeup interrupt DISABLED
4 ------ = 1 RESERVED
3 IEN3 = 0 IRQ 3 DISABLED
2 IEN2 = 0 IRQ 2 DISABLED
1 IEN1 = 0 IRQ 1 DISABLED
0 IEN0 = 0 IRQ 0 DISABLED
*/
IENR1.BYTE = 0x50;
/* Module Standby Control Register 1 (MSTCR1) */
/*
7 --- = 0
6 MSTIIC = 1 IIC DISABLED
5 MSTS3 = 0 SCI3 ENABLED
4 MSTAD = 0 A/D Converter ENABLED
3 MSTWD = 0 Watchdog Timer ENABLED
2 MSTTW = 0 Timer W Module ENABLED
1 MSTTV = 0 Timer V MOdule ENABLED
B.1. HardwareSetup.c 92
0 MSTTA = 0 Timer A Module ENABLED
*/
MSTCR1.BYTE = 0x40;
/**************************************************************/
/**************************************************************/
/* SECTION PORT 1 as EncoderControl */
/* Need access to C/D- , RD-, WR-, CS- */
/* Setting up 4 pins as general I/O (used P17,P16,P11,P12)*/
/* so P15 (IRQ1), P14 (IRQ0), P10(TMOW) are reserved and still available*/
/* Port Mode Register 1 for Port 1 AND Port 2*/
/*
7 IRQ3 = 0 general I/O
6 IRQ2 = 0 general I/O
5 IRQ1 = 0 "
4 IRQ0 = 0 "
3 --- = 1
2 --- = 1
1 TXD = 1 TXD Output Pin
0 TMOW = 1 TMOW Output Pin
*/
IO.PMR1.BYTE = 0x0f ;
/* set those as output */
/* Port Control Register 1
7 PCR17 = 1 output (not used)
6 PCR16 = 1 " => CONTROL/_DATA (C/_D)
5 PCR15 = 1 " => _READ (_RD)
4 PCR14 = 1 " => _WRITE (_WR)
3 ---
2 PCR12 = 0
1 PCR11 = 0
0 PCR10 = 0
*/
IO.PCR1.BYTE = 0xf0;
/*SECTION PORT 1 END*/
/**************************************************************/
/**************************************************************/
/* SECTION PORT 7 as general I/O (all Pins) */
/* Setting P75, P74 AS OUTPUTS*/
IO.PCR7.BIT.B5 = 1; /* P75 connected to EN3,4 => Charge Electrodes*/
IO.PCR7.BIT.B4 = 1; /* P74 connected to 1A, _P74 (inverted by
external gate) connected to 2A
=> Direction of LM */
/* setting values to zero for startup*/
IO.PDR7.BIT.B5 = 0;
IO.PDR7.BIT.B4 = 0;
/* SECTION PORT 5 END*/
/**************************************************************/
/**************************************************************/
B.1. HardwareSetup.c 93
/* SECTION PORT 5 as general I/O (all Pins) */
/* Port Mode Register 5
7 WKP7 -- P57
6 WKP6 -- P56
5 WKP5 = 0 general I/O P55
4 WKP4 = 0 " P54
3 WKP3 = 0 " P53
2 WKP2 = 0 " P52
1 WKP1 = 0 " P51
0 WKP0 = 0 " P50
*/
IO.PMR5.BYTE = 0x00;
/* Port 5 as Input*/
IO.PCR5.BYTE = 0x00;
/* Reset Port Data Register 5 to Zero*/
IO.PDR5.BYTE = 0x00;
/* CAUTION !!! have to set P57 and P56 as general I/O with ICCR*/
/* disable I2C Module*/
IIC.ICCR.BYTE = 0x00;
/* Port Pull-UP Control Register 5 (PUCR5)*/
/* all off-state */
IO.PUCR5.BYTE = 0xff;
/* SECTION PORT 5 END*/
/**************************************************************/
/**************************************************************/
/* SECTION TIMER A */
/* Timer Mode Register A (TMA)*/
/* Timer Mode A Register
7 TMA7 0
6 TMA6 0 (not too important for us, as would be output to TMOW)
5 TMA5 0 Clock output select (internal clock PHI/32)
4 --- 1 Reserved
3 TMA3 0 Internal clock select => count outputs of prescaler S
2 TMA2 0
1 TMA1 1
0 TMA0 1 Internal Clock Select/ Clock Input to TCA (when TMA3 = 0)
chosen PHI/512
*/
TMRA.TMA.BYTE=0x13;
/* SECTION TIMER A END*/
/**************************************************************/
/**************************************************************/
/* SECTION PWM , TIMER W*/
/* Using Timer W -> FTIOB...C in PWM mode*/
/* Timer mode register W: TMR
7 CTS = 0 no count at start up
6 --
5 BUFEB = 0 GRD as input capture / output compare reg.
4 BUFEA = 0 GRC as input capture / output compare reg.
B.1. HardwareSetup.c 94
3 --
2 PWMD = 1 PWM mode D
1 PWMD = 1 PWM mode C
0 PWMD = 1 PWM mode B
*/
TMRW.TMR.BYTE = 0x0f;
/* Timer Control Register W: TCR
7 CCLR = 1 TCNT cleared by compare match GRA
6 CKS2 ....
5 CKS1 .... 011 , 000 => PHI, 001 => PHI/2 , 010 => PHI/4 , **011 => PHI/8**
4 CKS0 .... all 3 bits allow to select clock source and frequency
3 TOD = 1
2 TOC = 1
1 TOB = 1 .... all output values 1 for B..D
0 TOA = 0 GRA initial output 0
*/
TMRW.TCR.BYTE = 0xbe;
/* Timer Interrupt Enable Register W: TIER
7 OVIE = 1 Timer Overflow interrupt enabled
6 ---
5 ---
4 ---
3 IMIED = 1
2 IMIEC = 1
1 IMIEB = 1
0 IMIEA = 1 ... enable all A...D Input capture/output compare interrupts
*/
TMRW.TIER.BYTE = 0x0f;
/* Timer Status Register W: TSR
7 OVF
6 ---
5 ---
4 ---
3 IMFD
2 IMFC
1 IMFB
0 IMFA ... Input Capture/ Output Compare Match flags
**** clear all pending interrupts at startup *****
*/
TMRW.TSR.BYTE = 0x00;
/* Timer I/O Control Register 0 W: TIOR0
7 ---
6 IOB2 = 0 IO Control B2 = output compare register
5 IOB1 = 1
4 IOB0 = 0 IO Control B1/0 , will use 1 output at GRB compare match
3 ---
2 IOA2 = 0 see above just for A
1 IOA1 = 1
0 IOA0 = 0
*/
TMRW.TIOR0.BYTE = 0x22;
/* Timer I/O Control Register 1 W: TIOR0
B.1. HardwareSetup.c 95
7 ---
6 IOD2 = 0 IO Control D2 = output compare register
5 IOD1 = 1
4 IOD0 = 0 IO Control D1/0 , will use 1 output at GRB compare match
3 ---
2 IOC2 = 0 see above just for C
1 IOC1 = 1
0 IOC0 = 0
*/
TMRW.TIOR1.BYTE = 0x22;
/* => ALL ARE OUTPUT COMPARE REGISTERS */
/* don´t forget to set GRA...D => set in main or function*/
/* SECTION PWM END (TIMER W)*/
/**************************************************************/
/**************************************************************/
/* SECTION SCI3*/
/* Serial Communication Interface 3*/
InitSCI3(SCI_Init_Data); /* initialise serial port */
/* SECTION SCI3 END*/
/**************************************************************/
/**************************************************************/
/* SECTION ENCODER INITIALIZE*/
/* EXTERNAL ENCODER CONTROL / INITIALIZATION*/
/* USING PORT 1 & 5 for Control Setting in Registers*/
/* Do nothing right now */
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 1
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 1;
/*USING PORT5 as output to WRITE to ENCODER*/
/*Setup Port 5 as Output*/
IO.PCR5.BYTE = 0xff;
/* initialize Port 5 with 0x00*/
IO.PDR5.BYTE = 0x00;
/**************************************************************/
/* MCR Master Control Register
7 0
6 0 ... accessing MCR 00
5 1
4 1
3 0
2 1
1 0
0 1 ... RESET ALL
B.1. HardwareSetup.c 96
*/
IO.PDR5.BYTE = 0x35;
/* WRITE TO MCR */
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 0;
/* MCR END*/
/**************************************************************/
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/**************************************************************/
/* ICR INPUT Control Register
7 0
6 1 ... accessing ICR 01
5 0 initialize pin 3 -> CNTR load input
4 0 initialize pin 4 -> reset input
3 1 enable inputs A/B
2 0 NOP
1 0 NOP
0 0 A = up count input, B = down count input
*/
IO.PDR5.BYTE = 0x48;
/* WRITE TO ICR */
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 0;
/* ICR END*/
/**************************************************************/
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
B.1. HardwareSetup.c 97
/**************************************************************/
/* OCCR Output Control Register
7 1
6 0 ... accessing OCCR 10
5 0
4 0 pin 16 = _CY, pin 17 = _BW
3 0 BCD or binarycount mode
2 0 normal count mode
1 0 normal count mode
0 0 binary count mode
*/
IO.PDR5.BYTE = 0x80;
/* WRITE TO OCCR */
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 0;
/* MCR END*/
/**************************************************************/
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/**************************************************************/
/* QR quadrature Control Register
7 1
6 1 ... accessing QR 11
5 0
4 0
3 0
2 0
1 1
0 1 enable 4x quadrature mode
*/
IO.PDR5.BYTE = 0xC3;
/* WRITE TO ICR */
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
B.1. HardwareSetup.c 98
IO.PDR1.BIT.B4 = 0;
/* ICR END*/
/**************************************************************/
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/**************************************************************/
/* PR Preset Register
7 0
6 0 ... adressing MCR
5 0
4 0
3 0
2 0
1 0
0 1 Reset PR/OL address pointer
*/
IO.PDR5.BYTE = 0x01;
/* NEED 3 WRITE CYCLES, starting PR0 (LSB) ... PR2(MSB)*/
/* WRITE TO PR*/
/* with B7 = (_CS)= 0
B6 = C/_D = 1 first write to MCR
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 0;
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/* SETTING INITIAL VALUE OF CNTR with 0x7fffff (half of highest value)*/
/* with B7 = (_CS)= 0
B6 = C/_D = 0 write to data register PR
B5 = _RD = 1
B4 = _WR = 1
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 0;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 1; // right now do nothing
/* Set LSB = PR0 = 0xff*/
IO.PDR5.BYTE = 0xff;
B.1. HardwareSetup.c 99
IO.PDR1.BIT.B4 = 0; // write value to PR, pointer auto incremented
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/* Set PR1 middle 0xff */
IO.PDR5.BYTE = 0xff;
IO.PDR1.BIT.B4 = 0; // write value to PR, pointer auto incremented
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/* Set MSB = PR2 = 0x7f*/
IO.PDR5.BYTE = 0x7f;
IO.PDR1.BIT.B4 = 0; // write value to PR, pointer auto incremented
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/* TRANSFER PR -> CNTR */
/* with B7 = (_CS)= 0
B6 = C/_D = 1 write to control register MCR
B5 = _RD = 1
B4 = _WR = 1
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 1; // right now do nothing
/*transfer PR to CNTR, therefor write command into MCR => control register*/
IO.PDR5.BYTE = 0x08; // in MCR Transfer PR to CNTR
IO.PDR1.BIT.B4 = 0;
while(wait--);
wait = 20;
/*stop writing, reset port 5 data*/
IO.PDR1.BIT.B4 = 1;
IO.PDR5.BYTE = 0x00;
while(wait--);
wait = 20;
/* PR END*/
/**************************************************************/
/*stop writing, reset port 5 data, port 5 as input*/
IO.PDR1.BIT.B4 = 1; // _WR = 1
B.1. HardwareSetup.c 100
IO.PDR5.BYTE = 0x00; // reset
IO.PCR5.BYTE = 0x00; //port 5 input
/* SECTION ENCODER END*/
/**************************************************************/
/**************************************************************/
/* UNMASK ALL INTERRUPTS => Enable interrupts to CPU*/
set_imask_ccr(0);
/**************************************************************/
B.2. getEncoderData.c 101
B.2 getEncoderData.c
/***************************************************************************
*
* FILE : getEncoderData.c
* DATE : September 09, 2003
* DESCRIPTION : Polymer Control getEncoderData
* CPU TYPE : H8 Tiny/Super Low Power
*
* AUTHOR : Christoph Hackl
*
***************************************************************************/
#define ENC_MASK 0x000000ff
#include <machine.h>
#include "iodefine.h"
extern unsigned int dutyB;
extern unsigned int dutyC;
extern unsigned int dutyD;
extern unsigned int period;
extern unsigned int timerA;
extern long encoder;
/*******************************************************************/
/* INTERNAL FUNCTION DECLARATION */
void getEncoderData(void);
/*******************************************************************/
/* This function will read data from the counter chip,
* therefor to the Master Control Register (MCR) of
* the counter chip has to be written
* => RESET Preset Register (PR) and Output Latch (OL) pointers
* and transfer Counter (CNTR) to OL (24 bits),
* but data bus is 8 bit wide, so the read out will be in 3 cycles */
void getEncoderData(void)
short int i = 0; /* dummy variable for loop*/
unsigned long counter = 0; /*counter variable*/
B.2. getEncoderData.c 102
short wait=20;
/**********************************************************/
/* WRITE 3 -> MCR of Encoder at least 60ns*/
/*Setup Port 5 as Output*/
IO.PCR5.BYTE = 0xff;
/*SET port 5 data register*/
/* Reset PR/OL address pointer, Transfer CNTR to OL (24 bits)*/
IO.PDR5.BYTE = 0x03;
/* with B7 = (_CS)= 0
B6 = C/_D = 1
B5 = _RD = 1
B4 = _WR = 0
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 1;
IO.PDR1.BIT.B5 = 1;
IO.PDR1.BIT.B4 = 0;
/* _WR = 1 disable write*/
IO.PDR1.BIT.B4 = 1;
/********************************************************/
/********************************************************/
/* READ Data from D0...D7 , C/_D = 0, _CS = 0, _RD = 0, _WR = 1*/
/* read 3 timer (24 bit) => loop starting with LSB ,
* auto increment address pointer by encoder*/
/* Set Port 5 to 0, just to be sure*/
IO.PDR5.BYTE &= 0x00;
/*Set Port 5 as input*/
IO.PCR5.BYTE = 0x00;
timerA = TMRA.TCA;
/* 3 cycle read out*/
for (i = 0;i < 3; i++)
unsigned long buffer = 0;
/* READ CYCLE */
/* with B7 = (_CS)= 0
B.2. getEncoderData.c 103
B6 = C/_D = 0
B5 = _RD = 0
B4 = _WR = 1
*/
IO.PDR1.BIT.B7 = 0;
IO.PDR1.BIT.B6 = 0;
IO.PDR1.BIT.B5 = 0;
IO.PDR1.BIT.B4 = 1;
buffer = IO.PDR5.BYTE & ENC_MASK;
buffer = (buffer < < (i*8)) ;
counter += buffer;
/* _RD = 1 stop read till next cycle*/
IO.PDR1.BIT.B5 = 1;
/****************************************************************/
/*return read 24-bit value*/
encoder = counter;
B.3. OutputSerial.c 104
B.3 OutputSerial.c
/***************************************************************************
*
* FILE : Serial Output
* DATE : September 09, 2003
* DESCRIPTION : Serial Communication / Data Transfer
* CPU TYPE : H8 Tiny/Super Low Power
*
* AUTHOR : Christoph Hackl
***************************************************************************/
#include "sci3.h"
#include <stdio.h>
#include <stdlib.h>
extern unsigned int dutyB;
extern unsigned int dutyC;
extern unsigned int dutyD;
extern unsigned int period;
extern unsigned int timerA;
extern unsigned long encoder;
/*******************************************************************/
/* INTERNAL FUNCTION DECLARATION */
extern char *ultoa(unsigned long value, char *string, int radix);
/*******************************************************************/
/*******************************************************************/
/* INTERNAL FUNCTION DECLARATION */
void outputSerial(void);
/*******************************************************************/
/* outputSerial will convert the desired data to strings and
* send those to the host by using the SCI3 interface
* => it must be adjusted for the desired data
* => here only time and encoder data are transmitted */
void outputSerial(void)
char b1[16];
B.3. OutputSerial.c 105
char b2[16];
/* char b3[16]; just need 2 buffers for ENCODER & TIME
char b4[16];
char b5[16]; */
char *show;
/*****************************************************/
/* ENCODER VALUE date transfer (24 bit) */
/* conversion from unsigned long into char pointer */
show = ultoa(encoder,b1,10);
/* transmit over SCI3*/
PutStr((unsigned char*)show);
PutStr((unsigned char*)"\t");
/*****************************************************/
/*****************************************************/
/* COUNTER (GRA) data transfer */
/* conversion from unsigned int into char pointer*/
show = ultoa((unsigned long)timerA,b2,10);
/*transmit over SCI3*/
PutStr((unsigned char*)show);
PutStr((unsigned char*)"\t");
/*****************************************************/
/*new line*/
PutStr((unsigned char*)"\r\n");
/*****************************************************/
B.4. ControlPWM.c 106
B.4 ControlPWM.c
/***************************************************************************
*
* FILE : ControlPWM.c
* DATE : September 09, 2003
* DESCRIPTION : Main C source file
* CPU TYPE : H8 Tiny/Super Low Power
*
* AUTHOR : Christoph Hackl
*
***************************************************************************/
#include "iodefine.h"
/*******************************************************************/
/* INTERNAL FUNCTION DECLARATION */
void enableDCDC(void);
void disableDCDC(void);
void turnMotorRight(void);
void turnMotorLeft(void);
void stopMotor(void);
void setPeriod (unsigned int newPeriod);
void setDutyCycleLinearMotor (unsigned int newDCLM); // GRB
void setDutyCycleElectrodes (unsigned int newDCE); // GRC
void setGRD(unsigned int newGRD);
/*******************************************************************/
/* ENABLE 3,4 (P75) on PWM chip, thus 3Y (not used) & 4Y (Charging electrodes)
are enabled as outputs*/
void enableDCDC(void)
IO.PDR7.BIT.B5 = 1;
/* DISABLE 3,4 (P75) on PWM chip, thus 3Y (not used) & 4Y (Charging electrodes)
are disabled as outputs*/
void disableDCDC(void)
B.4. ControlPWM.c 107
IO.PDR7.BIT.B5 = 0;
/* SET DIRECTIONS OF linear motor*/
void turnMotorRight(void)
IO.PDR7.BIT.B4 = 0; /* P74 connected to 1A, _P74 (inverted by
external gate) connected to 2A */
void turnMotorLeft(void)
IO.PDR7.BIT.B4 = 1; /* P74 connected to 1A, _P74 (inverted by
external gate) connected to 2A */
/* STOP linear motor => no disturbance*/
void stopMotor(void)
setDutyCycleLinearMotor(0x0000);
/*CHANGE PERIOD of Timer W counter register GRA, it will affect all PWM signals */
void setPeriod (unsigned int newPeriod)
/* Changing value of GRA will change Period for all PWM B...D*/
TMRW.GRA = newPeriod;
/*CHANGE DUTY CYCLE OF linear motor, this will change
* the (average) output voltage of the GRB (FTIOB) */
void setDutyCycleLinearMotor (unsigned int newDCLM)
/* Changing value of GRB => changing duty cycle for PWM B */
/* PWM Signal for linear motor, FTIOB connected to 1,2 EN on PWM chip*/
TMRW.GRB = newDCLM;
/*CHANGE DUTY CYCLE OF linear motor, this will change
* the (average) output voltage of the GRC (FTIOC) */
void setDutyCycleElectrodes (unsigned int newDCE)
B.4. ControlPWM.c 108
/* Changing value of GRC => changing duty cycle for PWM C*/
/* PWM Signal for Charging Polymer, FTIOC connected to 4A on PWM chip,
* the PWM driver output voltage
* is amplified by the HIGH VOLTAGE CONVERTER */
TMRW.GRC = newDCE;
Appendix C
Maple Commands
The Maple commands are presented in chronological order to compute the explicit
transfer function vectors Feiand FFdist
of the 10th order system.
• Define Matrix (s1 − Aall) ∈ R10x10:
AallS:=Matrix([
[s-a11,-a12,0,-a14,0,0,0,0,0,0],
[-1,s,0,0,0,0,0,0,0,0],
[0,-a32,s-a33,-a34,0,0,0,0,0,0],
[0,0,-1,s,0,0,0,0,0,0],
[0,0,0,0,s-a11hat,-a12hat,0,-a14hat,0,0],
[-1+K1,0,0,0,-K1,s,0,0,-K2,0],
[0,0,0,0,0,-a32hat,s-a33hat,-a34hat,0,0],
[0,0,0,0,0,0,-1,s,0,0],
[1,0,0,0,-1,0,0,0,s,0],
[K1,0,0,0,-K1,0,0,0,-K2,s]]);
• Invert Matrix (s1 − Aall)−1 ∈ R10x10:
with(LinearAlgebra):Ainv :=MatrixInverse(AallS);
• Define Matrix Call
109
Appendix C. Maple Commands 110
Call := Matrix([
[0,0,0,1,0,0,0,0,0,0],
[0,0,0,0,0,0,0,1,0,0],
[0,0,0,0,0,0,0,1,0,a32hat/d3hat]]);
• Defince Vector Ball
Ball := Vector([Kpwm*Khvc*b1,0,0,0,Kpwmhat*Khvchat*b1hat,0,0,0,0,0]);
• Define Vector Dall
Dall := Vector([0,0,d3,0,0,0,0,0,0,0]);
• Compute Transfer Function Vector Fei= yall
ei(s)
Call.Ainv.Ball;
• Compute Transfer Function Vector FFdist= yall
Fdist(s)
Call.Ainv.Dall;