Linear modeling of elongated bending EAP actuator at large deformations

Abstract

This paper describes a linear dynamic model of an elongated bending Electroactive Polymer (EAP) actuator applicable with deformations of any magnitude. The model formulates relation of a) voltage applied to the EAP sheet, b) current passing through the EAP sheet, c) force applied by the actuator and d) deformation of the actuator. In this model only the geometry of EAP piece and four empirical parameters of the EAP material: a) bending stiffness, b) electromechanical coupling term, c) electrical impedance and d) initial curvature are considered. The contribution of this paper is introducing a model that can be used to characterize the properties of different EAP materials and compare them. The advantage of the model is its simplicity and ability to provide insights in to the behavior of bending EAPs. Additionally, due to linearity of the model, the real-time control is feasible. Experiments, using Ionomeric Polymer-Metal Composite (IPMC) sheet from Environmental Robotics Inc., where carried out to verify the model. The experimental results confirm the model is valid.

Full text

Linear modeling of elongated bending EAP actuator at large
deformations
Indrek Musta, Mart Antonab, Maarja Kruusmaaac, Alvo Aablooa
a IMS Lab, Institute of Technology, Tartu University, Nooruse 1, 50411 Tartu, Estonia
bDepartment of Electrical & Computer Engineering,
Michigan State University, East Lansing, MI 48824, USA
cCenter for Biorobotics, Tallinn University of Technology,
Akadeemia tee 15A, 12618 Tallinn, Estonia
ABSTRACT
This paper describes a linear dynamic model of an elongated bending Electroactive Polymer (EAP) actuator applicable
with deformations of any magnitude. The model formulates relation of a) voltage applied to the EAP sheet, b) current
passing through the EAP sheet, c) force applied by the actuator and d) deformation of the actuator. In this model only the
geometry of EAP piece and four empirical parameters of the EAP material: a) bending stiffness, b) electromechanical
coupling term, c) electrical impedance and d) initial curvature are considered. The contribution of this paper is
introducing a model that can be used to characterize the properties of different EAP materials and compare them. The
advantage of the model is its simplicity and ability to provide insights in to the behavior of bending EAPs. Additionally,
due to linearity of the model, the real-time control is feasible. Experiments, using Ionomeric Polymer-Metal Composite
(IPMC) sheet from Environmental Robotics Inc., where carried out to verify the model. The experimental results confirm
the model is valid.
Keywords: EAP, IPMC, ICPF, modeling, characterization, linear, dynamic, large deformations
1. INTRODUCTION
Bending Electroactive Polymers (EAP) are materials that are able to bend in response to electric stimulation and vice-
versa [1]. Because of their unique shape, large deformation, and the simple construction, they offer new opportunities for
designing devices [2-5]. There are many types of EAP materials with different properties. Different applications have
different requirements. To compare EAP materials and choose the one best suited for given application we need a
scalable model of an EAP sensor/actuator, where the behavior of different EAP materials with specific mechanical
construction and actuation mechanism would be defined in the same terms.
The need of unified characterization and modeling techniques has been recognized before and some steps have been
taken in that direction [6], [7]. However, no scalable general purpose model has been presented so far.
Several researchers have worked on finding dynamic scalable models for different specific EAP materials [8], [9].
However, these models do not consider large deformations. We consider deformation of EAP sheet large, if the
deflection angle is greater than 90 degrees. A large deformation model is presented in [10], [11]. These models are static
and nonlinear. To the best of our knowledge, no linear, dynamic, and large deformation model has been presented so far.
In this paper linear, dynamic, large deformation universal model of an EAP actuator is presented. The actuator consists
of EAP sheet and a rigid elongation attached on top of it. The advantages of using elongation are listed in [11], [12]. The
model considers initial curvature of EAP and enables concurrently varying load and position. We characterize a sample
of IPMC and use the model to estimate the behavior of that sample.

2. THE PROPERTIES OF EAP MATERIAL
There are many different types of EAP actuators with different construction and actuation mechanisms [1]. In this
section four empirical parameters are introduced, which can describe the behavior of EAP material. The parameters are
independent from the geometry of the actuator.
2.1 Normalized bending stiffness
All EAP sheets bend in response to applied force moment. Lets denote change in curvature with kΔ and bending
moment with
M
. A bending stiffness is defined as
M
kΔ. The bending moment is proportional to the width of the sheet
w. In this paper, the bending stiffness is normalized with the width to describe the EAP material and is defined as
M
Bkw
=Δ⋅ . (1)
In the case of homogeneous material with Young modulus E and thickness d, the normalized bending stiffness can be
calculated from (2). An EAP actuator is usually made from a composite material. Even though, the material is not
homogeneous it can still be treated using (2), where E is the equivalent or effective Young modulus.
3
12
Ed
B⋅
= (2)
EAP-s can also have viscoelastic properties. The viscoleastic properties of Conductive Polymers (CP) is addressed in [9],
[13].The viscoelastic properties of IPMC-s have been studied in [14], [15]. To account with viscoelasticity normalized
bending stiffness can be a function of frequency.
M
and kΔ may be then looked at as phasors in complex form.
2.2 Normalized electromechanical coupling
Although the actuation mechanisms of different EAP materials has dispersed physico-chemical background, applied
voltage generates a bending moment in the material. There are different models developed for CP-based actuators [16]
and IPMC-s [8, 17] We consider electrically induced bending moment e
M
to be proportional to the input voltage U. It
is reasonable to assume that e
M
is proportional to the width of the sheet w. The normalized electromechanical coupling
K
is defined as
e
M
KUw
=⋅ (3)
Electrically induced bending moment my be frequency dependant [8], [9]. Therefore, normalized electromechanical
coupling is a function of frequency.
2.3 Normalized electrical impedance
We simplify the model, so that the current
()
I
s through the material is proportional to the voltage
()
Us, where
s
is
complex frequency. It is reasonable to assume that
()
I
s is inverse proportional to the area
A
of the sheet. The
normalized electrical impedance is defined as
() ()
()
Us A
Zs
I
s
⋅
= (4)
The normalized electrical impedance is related to effective impeditivity
()
X
s of the material with thickness d as
follows:
() ()
X
s
Zs d
= (5)

2.4 Initial curvature
An EAP sheet may have an initial curvature. This can be caused by non-symmetrical manufacturing or treatment. Ionic
EAP-s may also have hysteresis because they contain one or more layers of porous materials [18]. The hysteresis in
IPMCs is studied in [19], [20].
3. THE MODEL
In this section we present a model to describe a bending EAP actuator.
3.1 The actuator
The actuator consists of a rectangular piece of EAP sheet in cantilever configuration. One end of the EAP sheet is
clamped and a rigid elongation is attached to the other end (see Fig. 1). The rest of the EAP sheet can bend while a
voltage is applied through clamps. No inertial nor friction forces are considered in our model.
free par t
of the EAP
sheet
rigid
elongation
fixed part
of the EAP
sheet
clamps for
electrical stimulation
of EAP sheet
force
acting on
the actuator
Fig. 1. Perspective view of the system.
3.2 Formulation of the task
Our model will formulate the relation of:
1) voltage applied to the EAP sheet,
2) current passing through the EAP sheet,
3) force applied by the actuator to resist the outer force acting on the actuator and
4) deformation of the actuator.
The model also considers the geometry of the actuator and properties of the EAP material as discussed in the previous
section. Please refer to Table 1 for notations for the parameters.
We denote force applied by the actuator
()
F
s. A force
()
F
s− is applied to the actuator at a fixed distance from a point
located in front of the contacts. The point is called “joint of the EAP sheet” and would be exactly in the middle of the
free part of the EAP sheet in case of a straight sheet (See Fig. 2). The line segment between this point and the point
where the force is applied to is called arm of the actuator. The length of the arm is denoted as R. The deformation of the
actuator is defined by the angular deflection of the arm. The force applied by the actuator (force output)
()
F
s is always
perpendicular to the arm.

Table 1. Notations of the model parameters.
Type Meaning Notation Unit
Length of free part of the EAP sheet l m
Total length of the fixed part of the EAP sheet c
l m
Width of the EAP sheet w m
Dimensions
of the EAP
Actuator
Arm length of the actuator R m
Normalized bending stiffness of EAP
()
Bs
N
m⋅
Normalized electromechanical coupling of EAP
()
K
s 1
NV
−
⋅
Normalized electrical impedance of EAP
()
Z
s 2
mΩ⋅
The parameters
of EAP material
Initial curvature of EAP 0
k 1
m−
Angular deflection of the arm
()
s
α
rad
Voltage applied to the EAP sheet
()
Us V
Force output of the actuator
()
F
s
N
Signals
in frequency
Domain
Electric current passing through the EAP sheet
()
I
s A
free part
of the EAP
sheet
rigid
elongation
joint of
the EAP sheet
modeled as hinge
arm of
the actuator
Fig. 2. The geometric definition of parameters. This is the top view of the system. Please consider that the elongation is
close to but not strictly parallel to the arm.

In our model, we assume that the bending moment generated by the outer force, curvature, and all four parameters of the
free part of EAP sheet are uniform. In [21] it is shown, that IPMC actuator can be modeled as a hinge with joint in the
middle. The error of such approximation is calculated. The results can be easily extended to other bending EAP
materials. Based on that we may assume, that the radius of curvature of EAP sheet can be approximated as
()
l
s
α
.
3.3 Solution
In [11] it is shown that the mean bending moment caused by an external force is equal to the bending moment in the
center of the bendable section. That bending moment is approximately
()
RFs−⋅ . This gives us the opportunity to
utilize the theory in [11] for IPMC actuators with constant EIBM, which leads to the derivation of (6).
() () () () ()
0
skR
Us Ks Bs Fs
ls w
α

⋅= − +

 (6)
From (4) and taking into account that area of the sheet is
()
c
A
wll=⋅+ we get the electrical model of EAP actuator
()
()
()
()
c
I
sUs
wll Zs
=
⋅+ (7)
3.4 Inferences
From (6)
()
F
s and
()
s
α
can be revealed.
() () () () ()
0
sk
w
F
sUsKs Bs
Rls
α


=⋅−−





. (8)
() () () () ()
0
k
lR
sUsKsFs
Bs w s
α

=⋅−+


(9)
From (7)
()
I
s and
()
Us can be revealed.
() () ( )
()
c
Us w l l
Is Zs
⋅⋅+
= (10)
() () ()
()
c
I
sZs
Us wll
⋅
=⋅+ (11)

4. THE EXPERIMENTS
In this section the system setup for the experiments is presented and details of implementation are discussed.
4.1 Experimental device
The experimental device consists of a rigid clamp with electrical contacts made of gold attached to an EAP test sample
(See Fig. 3). A light-weight rigid elongation made of PC and carbon fiber is attached to the opposite end of sample. The
free length of the sample can be freely adjusted by varying a length of EAP between the securing plates on the
elongation. An isometric transducer MLT0202 from AD instruments is softly attached to the elongation at a desired
distance. The strain is measured approximately perpendicular to the arm (and to the elongation) of the actuator. The
actuator is oriented so that it bends at the horizontal plane, therefore gravity does not affect the measurements. In our
system setup not the elongation but the clamp can be moved on a circular trajectory to achieve a desirable curvature of
the test sample.
The angle of the clamp is controlled via rotary solenoid actuator type GDRX-035. The voltage is applied to EAP through
1Ω resistor, and EAP current is calculated from resistor voltage drop. Input voltages for EAP and solenoid are input
signals of the system. The Strain, current, and solenoid position are outputs of the system. Input signals are generated
and outputs are measured using NI PCI-6036 data acquisition device connected to NI SC-2345 connector block. Inputs
are controlled and outputs read by NI LabVIEW [22] program.
Fig. 3. The system setup. Diagram of the system setup (a) and close-up of the actuator (b).
A series of experiments can be generated in such a way that one or more parameters are variables. The system stabilizing
and data acquisition minimum-time can also be specified. An average-cycle report is automatically generated after each
test cycle.
4.2 The EAP material used in experiments
A type of ionic EAP called Ionomeric Polymer-Metal Composite (IPMC) was used in the experiments. The sample was
purchased from Environmental Robots Inc.. Initially the sample was ionic-liquid based, but the solvent has been replaced
by water. All experiments were performed with the same IPMC sheet, where the width of the IPMC sheet was
19.0 mmw=, total length 12.2 mm
c
ll+= , and thickness 0.28mmd=.
The IPMC sample is plated with a thick platinum coating to minimize the effect of surface resistance.

In order to keep the hydration level constant, while avoid immersing the test instrumentation into the water, a
recirculation water pump (windscreen washer pump) system was used to wet the test sample (see Fig. 4).
Fig. 4. Photo of the System setup. To keep the hydration level of IPMC constant water is pumped on it.
4.3 Measurement methodology
In this subsection the methodology used to identify the parameters of the material and the experiments to verify the
model are discussed.
The parameters
K
and
Z
were determined by exciting the system with sinusoidal voltage input and measuring the
output force and current. The amplitude of voltage was 1.6 V. The frequency of the input signal ranged from 0.037 Hz to
25 Hz. The angle of the actuator arm was maintained at 0.
Our experiments are mostly done in 0–20 Hz range where the viscoelasticity is not significant [15]. Therefore,we may
only consider the static bending stiffness. It means that in our case the normalized bending stiffness is just a real-valued
constant
()
Bs B=. The parameters B and 0
k were determined by measuring the output force of the actuator at two
different deflection angles.
The experiments were carried out by varying free part and arm lengths. In case of each geometry also experiments at
random voltages, angles, and frequency were performed. The total time consumed for the measurements in each
geometry was about 40 minutes. All experiments were made at room temperature.
When the input signal starts, the system acquires a time to be stabilized. The stabilizing time on each
Z
and
K
measurement was 4 seconds or 1 period minimum, and measuring time was 6 seconds or 6 periods minimum. The
stabilizing time on each random experiment was 15 seconds or 1 period minimum,and measuring time was 8 seconds or
8 periods minimum. The stabilizing time on each B and 0
k measurement was 10 seconds, and measuring time was 4
seconds.

5. RESULTS
Experiments were performed to characterize the material described in section 4.2 and to verify the model presented in
section 3. Please see Table 2 for details about the different dimensions of the actuators used in series of experiments.
Note that the experiments with the first set of dimensions of the actuator are repeated at the end. In each series
normalized electromechanical coupling
K
and normalized electrical impedance
Z
were measured. Please see Fig. 5 for
plots of measured average parameters
K
,
Z
and standard deviation. The measured
K
,
Z
are coherent with the results
presented in [8].
In Fig. 6 the filtered averaged cycles of signals from 4.series are presented. The electrical model of IPMC is known to be
nonlinear [23]. Our results also show that there is a notable nonlinearity in the electrical response at low frequencies..
Table 2. The geometries used in different series of experiments.
Series l c
l wR
1. 6mm 6.2mm 19mm 35mm
2. 6mm 6.2mm 19mm 60mm
3. 8mm 4.2mm 19mm 35mm
4. 8mm 4.2mm 19mm 60mm
5. 6mm 6.2mm 19mm 35mm
(a) (b)
Fig. 5. Plots of transfer functions a) normalized electromechanical coupling and b) normalized impedance.

Fig. 6. Filtered averaged cycles of signals from 4. series.
In each series 24 experiments with random parameters were performed. The same set of randomly generated parameters
was used each time to make results comparable. The limits of the parameters are given in Table 3. Before each
experiment the parameters B and 0
k were measured – 24 times in each series. The normalized bending stiffness and the
initial curvature measurements are presented in Fig. 7. Eventhough, the same IPMC piece was used for all the
experiments in all series, B and 0
k vary notably.
The variation of the normalized bending moment can be explained by changes in the hydration process [24-27]. Using
(2) from the mean normalized bending moment 0.71mN mB=⋅ and thickness 0.28mmd=, the equivalent Young
modulus 388 MPaE= can be calculated. This is in the range of other equivalent Young modulus for IPMC reported in
the literature.
In Fig. 7 (b) initial curvature measurements are compared with angle amplitude before the measurements. There is a
strong correlation between changes in the angle and in the initial curvature. This can be explained by the hysteresis in
IPMC material [19-20].
Table 3. Limits of the parameters in random experiments.
Parameter Min Max
Voltage – direct component -0.01V 0.2V
Voltage – alternating amplitude 0V 1.57V
Deflection angle – direct component -15.1deg 24.4deg
Deflection angle – alternating amplitude 0 20.9deg
Frequency 0.0607Hz 27.8Hz
The expected values for the random test outputs according to input signal parameters and previously measured material
parameters were calculated on the basis of theoretical model and the result were compared to the measured output
values. The difference of corresponding signals was expressed as relative deviation - root mean square deviation divided
by mean root mean squares of signals. The relative deviation of the output force and current are presented in Fig. 8.

(a) (b)
Fig. 7. The normalized bending stiffness and initial curvature measurements. (a) normalized bending stiffness after
experiments and (b) initial curvature after experiments.
(a) (b)
Fig. 8. Relative deviation of (a) output force and (b) current.
6. CONCLUSIONS AND DISCUSSION
Actuators with different geometries were made using the same IPMC piece. For each geometry, parameters of the IPMC
material were measured. The variation in the normalized electromechanical coupling and normalized electrical
impedance were small. The normalized bending stiffness varied notably, but that can be explained with changes in the
hydration level. The noticeable variation in the initial curvature can be explained by hysteresis in IPMC. Experiments
with randomly selected parameters were conducted and the results were compared with the model prediction. Only a
small difference was found. We conclude that the model proposed in this paper is scalable and valid.
The electromechanical response of some IPMC actuators is known to be in nonlinear [28]. However no nonlinearity was
observed in electromechanical response of given IPMC sheet in the given frequency range. At the same time electrical
response was notably nonlinear.
Bending EAP actuators are know for large deformations and linear models for well established feedback control
techniques. Yet to the best of our knowledge no linear model for any bending EAP actuator that would enable large

deformations has been presented so far. In this paper such a model is presented and it is suitable for all bending EAP
actuators. In addition the model considers dynamic behavior, initial curvature of EAP and enables concurrently varying
load and position.
This model only holds when we consider all the parameters along the sheet to be approximately uniform. The bending
moments generated along the sheet by outer force can be considered uniform only if lR. Also in IPMC, the high
resistance of the surface electrodes can cause the voltage distribution on the sheet to be non-uniform [8, 25, 29, 30]. The
surface resistance also varies with curvature [31]. When the IPMC sheet is
1. sufficiently short,
2. surface conductivity is high enough, and
3. current is low enough,
electrically induced bending moment can be considered uniform along its length [11].
In the future for each type of EAP, the 4 empirical parameters should be modeled. At this stage, the normalized bending
stiffness and initial curvature had to be measured before every experiment. It is also if the changes in could modeled and
estimated. For the future applications, if inertial forces need to be considered, corresponding terms should be added to
model. Also electrical model should be improved to consider the nonlinearities. Many EAP actuators can be used as
sensors. A similar model that was presented in this paper for EAP actuators should be derived for EAP sensors.
ACKNOWLEDGEMENTS
The financial support by Tartu University Foundation and Estonian Science Foundation (grant #6765) is gratefully
acknowledged. We would also like to thank Anoosheh Niavarani for reading the text and giving us suggestions.
REFERENCES
[1] Bar-Cohen Y 2004 Electroactive Polymer (EAP) Actuators as Artificial Muscles: Reality, Potential, and Challenges
(Bellingham, WA: SPIE Press)
[2] Mart Anton, Andres Punning, Alvo Aabloo, Madis Listak, Maarja Kruusmaa, "Towards a Biomimetic EAP Robot",
in Proc. of TAROS 2004, "Towards Autonomous Systems Systems", University of Essex, 6.-8. Sept. 2004.
[3] A. Punning, M. Anton, M. Kruusmaa, A. Aabloo ,”A Biologically Inspired Ray-like Underwater Robot with
Electroactive Polymer Pectoral Fins,” IEEE Confrence “Mechatronics and Robotics 2004” (MechRob04); Aachen,
Germany; 13.–15.09.2004. Aachen: Eysoldt, 2004, (2), 241–245.
[4] E. Mbemmo, Z. Chen, S. Shatara, X. Tan, "Modeling of Biomimetic Robotic Fish Propelled by An Ionic Polymer-
Metal Composite Actuator," in Proceedings of the 2008 IEEE International Conference on Robotics and
Automation, Pasadena, CA, pp. 689-694, 2008
[5] Chen Z, Shen Y, Xi N and Tan X 2007 Integrated sensing for ionic polymermetal composite actuators using PVDF
thin films Smart Mater. Struct. 16 S262–71
[6] Bar-Cohen, Y., X. Bao, S. Sherrit, S. Lih, “Characterization of the electromechanical properties of Ionomeric
Polymer-Metal Composite (IPMC),” Proc. of SPIE, v 4695, p 286–293, 2002.
[7] Diego Fernandez, Luis Moreno, and Juan Baselga, “Toward standardization of EAP actuators test procedures,“
Proc. of SPIE 5759, 274 (2005), DOI:10.1117/12.599106
[8] Z. Chen, X. Tan, "A Control-oriented and Physics-based Model for Ionic Polymer-Metal Composite Actuators,"
IEEE/ASME Transactions on Mechatronics, Vol. 13, No. 5, pp. 519-529, 2008
[9] Y. Fang, X. Tan, Y. Shen, N. Xi, G. Alici, "A Scalable Model for Trilayer Conjugated Polymer Actuators and Its
Experimental Validation," Materials Science and Engineering C: Biomimetic and Supramolecular Systems, Vol. 28,
pp. 421-428, 2008

[10] Bao X, Bar-Cohen Y, Chang Z and Sherrit, Stewart 2004 Characterization of bending EAP beam actuators Proc.
SPIE Int. Soc. Opt. Eng. vol 5385 p 388–94
[11] Anton M, Aabloo A, Punning A and KruusmaaM 2008 A mechanical model of a non-uniform ionomeric polymer
metal composite actuator Smart Mater. Struct. 17 1–10
[12] Andres Hunt, Andres Punning, Mart Anton, Alvo Aabloo, and Maarja Kruusmaa, “A multilink manipulator with
IPMC joints,” Proc. SPIE 6927, 69271Z (2008)
[13] Seong-Hun Song, Kang-Min Park, Woo-Sik Kim, Sang-Mok Chang, Analysis of the characteristics of
electrochemically polymerized polypyrrole films by using QCA and AFM, Materials Science and Engineering: C,
Volume 24, Issues 1-2, 14th Molecular Electronics and Devices Symposium, 5 January 2004, Pages 225-227, ISSN
0928-4931, DOI: 10.1016/j.msec.2003.09.066.
[14] Newbury K and Leo D J 2003 Linear electromechanical model of ionic polymer transducers—part I: model
development J. Intell. Mater. Syst. Struct. 14 333–42
[15] Newbury K and Leo D J 2003 Linear electromechanical model of ionic polymer transducers—part II: experimental
validation J. Intell. Mater. Syst. Struct. 14 343–57
[16] Gursel Alici, Brian Mui, Chris Cook, Bending modeling and its experimental verification for conducting polymer
actuators dedicated to manipulation applications, Sensors and Actuators A: Physical, Volume 126, Issue 2, 14
February 2006, Pages 396-404, ISSN 0924-4247, DOI: 10.1016/j.sna.2005.10.020.
[17] Choonghee Jo, Hani E Naguib and Roy H Kwon, "Modeling and optimization of the electromechanical behavior of
an ionic polymer–metal composite," Smart Materials and Structures, Vol. 17, pp. 065022 (13pp), December 2008
[18] J. Carmeliet and K. Van Den Abeele, “Mesoscopic approach for modeling the nonlinear hysteretic response of
damaged porous media in quasi-static and dynamic loading: Effects of pressure and moisture saturation,” in
Proceedings of the 4th International Conference on Fracture Mechanics of Concrete and Concrete Structures, 2001.
[19] Z. Chen, X. Tan, M. Shahinpoor, "Quasi-static Positioning of Ionic Polymer-Metal Composite (IPMC) Actuators,"
Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Monterey, CA, pp.
60-65, 2005
[20] Chul-Jin Kim, Hyun Woo Hwang, No-Cheol Park, Hyun-Seok Yang, Young-Pil Park, Kang-Ho Park, Hyung-Kun
Lee, and Nak-Jin Choi “Preisach modeling of IPMC-EMIM actuator” Proc. SPIE 6927, 692725 (2008)
[21] Mart Anton, PhD thesis “Mechanical modeling of IPMC actuators at large deformations” [supervisors: Maarja
Kruusmaa, Alvo Aabloo, Jan Villemson ; Faculty of Mathematics and Computer Science, University of Tartu,
Estonia] Tartu : Tartu University Press, c2008, 123 p
[22] http://www.ni.com/labview/
[23] Bonomo C, Fortuna L, Giannone P, Graziani S and Strazzeri S 2007 A nonlinear model for ionic polymer metal
composites as actuators Smart Mater. Struct. 16 1–12
[24] Characterization and dynamic modeling of ionic polymer-metal composites (IPMC): artificial muscles Ashwin
Mudigonda and Jianchao J. Zhu Proc. SPIE 6168, 616815 (2006)
[25] Nemat-Nasser S andWu Y 2003 Comparative experimental study of ionic polymer–metal composites with different
backbone ionomers and in various cation forms J. Appl. Phys. 93 5255–67
[26] Newbury K 2002 Characterization, modeling, and control of ionic polymer transducers Dissertation Virginia
Polytechnic Institute and State University
[27] Porfiri, M., 2009: "An electromechanical model for sensing and actuation of ionic polymer metal composites",
Smart Materials and Structures, 18(1), 015016
[28] C. Kothera, Characterization, Modeling, and Control of the Nonlinear Actuation Response of Ionic Polymer
Transducers. PhD thesis, Virginia Polytechnic Institute and State University, September 2005.
[29] Punning A, KruusmaaM and Aabloo A 2006 Surface resistance experiments with IPMC sensors and actuators
Sensors Actuators A 133 200–9
[30] Shahinpoor M and Kim J K 2000 The effect of surface-electrode resistance on the performance of ionic polymer–
metal composite (IPMC) artificial muscles Smart Mater. Struct. 9 543–51
[31] Punning A, Anton M, Kruusmaa M and Aabloo A 2006 Empirical model of a bending IPMC actuator Proc. SPIE
6168 61681V