A Physics-Based Model of Electro-Active Polymer Actuators ... · A Physics-Based Model of Electro-Active Polymer Actuators as ... I would like to thank my family and my friends

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


by


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


by


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)


(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


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



Polymer Resistance Rp = 1.6Rp and 0.6Rp and High Voltage Converter

Gain Khvc = (1 ± 0.2)Khvc . . . . . . . . . . . . . . . . . . . . . . . . . 76



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)


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 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 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


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


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


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


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)


(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


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


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


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


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)


(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


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


Pha

se [°

, lin

ear]

LuenbergerGopinath

10−1

100

101

102

103

104

0

0.5

1

1.5

2


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


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


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


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


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-


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


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].


3.1.2 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)


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


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


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


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


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


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


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].


3.3.2 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)


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


-

-

-

-

-

-

+

+

+

+

+

+

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


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


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

(


)

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

(


)

=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



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)




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



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)




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]



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


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



~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

]



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


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


0 )2 · (q∗)2 + 1 (6.16)

u∗

x = Lpre0 (λ∗ − 1) = Lpre

0

(

6

√

9


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)


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


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


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)


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


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)


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)


6.4.1 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


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


Pha

se [°

, lin

ear]

Length L0pre*1.1

Length L0pre*0.9

Width W0pre*1.1

Width W0pre*0.9

Thickness H0pre*1.25


10−2

100

102

104

106

108

1010

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4


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


Pha

se [°

, lin

ear]

Length L0pre*1.1

Length L0pre*0.9

Width W0pre*1.1

Width W0pre*0.9



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


6.4.2 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


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


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


Mag

nitu

de [l

inea

r]


2=0.05 [1/s]

10−2

100

102

104

106

108

1010

−6

−4

−2

0

2

4

6


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



uxfor Parameter Errors of PWM

Gain Kpwm = (1 ± 0.1)Kpwm and High Voltage Converter Gain Khvc = (1 ± 0.2)Khvc


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


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


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


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


Pha

se [°

, lin

ear]

Resistance Rp*1.6Resistance Rp*0.6Inductance Lp*3Inductance Lp*0.5



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-


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


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


Pha

se [°

, lin

ear]

Dielectric Constant epsilonr*1.05


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


Mag

nitu

de [l

inea

r]


2=0.05 [1/s]

10−2

100

102

104

106

108

1010

−20

−10

0

10

20


Pha

se [°

, lin

ear]



Youngs Modulus E*1.2Youngs Modulus E*0.8Damping Coefficient C

D*1.8

Damping Coefficient CD

*1.4



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.


10−2

10−1

100

101

102

103

104

0

0.2

0.4

0.6

0.8


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


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


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


Pha

se [°

, lin

ear]


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 ]


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


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

]


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




*/

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


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*/

/**************************************************************/

/**************************************************************/


/* 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.


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


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*/


/**************************************************************/

/* 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


*/


/* 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;


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

*/


/* 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;


IO.PDR1.BIT.B4 = 1;


while(wait--);

wait = 20;


/**************************************************************/

/* 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

*/


/* 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;


IO.PDR1.BIT.B4 = 1;


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;


IO.PDR1.BIT.B4 = 0;

/* ICR END*/

/**************************************************************/

while(wait--);

wait = 20;


IO.PDR1.BIT.B4 = 1;


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

*/


/* 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;


IO.PDR1.BIT.B4 = 1;


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;


IO.PDR1.BIT.B4 = 0; // write value to PR, pointer auto incremented

while(wait--);

wait = 20;


IO.PDR1.BIT.B4 = 1;


while(wait--);

wait = 20;

/* Set PR1 middle 0xff */

IO.PDR5.BYTE = 0xff;


while(wait--);

wait = 20;


IO.PDR1.BIT.B4 = 1;


while(wait--);

wait = 20;

/* Set MSB = PR2 = 0x7f*/

IO.PDR5.BYTE = 0x7f;


while(wait--);

wait = 20;


IO.PDR1.BIT.B4 = 1;


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;


IO.PDR1.BIT.B4 = 1;


while(wait--);

wait = 20;

/* PR END*/

/**************************************************************/

/*stop writing, reset port 5 data, port 5 as input*/

IO.PDR1.BIT.B4 = 1; // _WR = 1


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


* DESCRIPTION : Polymer Control getEncoderData


*

* AUTHOR : Christoph Hackl

*

***************************************************************************/

#define ENC_MASK 0x000000ff

#include <machine.h>


extern unsigned int dutyB;

extern unsigned int dutyC;

extern unsigned int dutyD;

extern unsigned int period;

extern unsigned int timerA;

extern long encoder;

/*******************************************************************/


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*/


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)*/


/* 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


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


* DESCRIPTION : Serial Communication / Data Transfer


*


***************************************************************************/

#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;

/*******************************************************************/


extern char *ultoa(unsigned long value, char *string, int radix);

/*******************************************************************/

/*******************************************************************/


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


* DESCRIPTION : Main C source file


*


*

***************************************************************************/


/*******************************************************************/


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)


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)


/* 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;

Documents

A Physics-Based Model of Electro-Active Polymer Actuators ... · A Physics-Based Model of Electro-Active Polymer Actuators as ... I would like to thank my family and my friends