r w:
. .quu
, .35...
~ ma.
r.
r
b.
£
\
v3.
4‘“
4:.»
0.2 V) is applied. Since the nonlinear ODE cannot be explicitly solved, a systems per-
spective is taken to derive the analytical nonlinear mapping from the voltage to the charge.
It is verified by the numerical solution, and is practically useful in real-time control. A non-
linear circuit model is employed to capture the electrical dynamicsiof IPMC. It incorporates
nonlinear capacitance of IPMC derived from the nonlinear mapping function between the
charge and the voltage, ion diffusion resistance [13, 73], pseudocapacitance due to the elec-
trochemical process at the polyrner-metal interface [3], and nonlinear DC resistance of the
polymer [13]. Based on the electromechanical coupling effect, the curvature output can be
obtained fi'om the electrical dynamic model. The proposed model shows consistency with
the linear model when the voltage is small [64]. With definitions of the state variable, input,
and output, the model is further presented in the state space, which will be the starting point
for nonlinear control design. Parameters are measured or identified through experiments.
9
The proposed model is validated in experiments.
1.3.5 Compact sensing schemes for IPMC actuators
Compact sensing methods are desirable for feedback control of IPMC actuators to en-
sure precise and safe operation without using bulky, separate sensors. Extensive work has
been done in sensory actuators. It is intriguing to utilize the inherent sensory property of
an IPMC to achieve simultaneous actuation and sensing, like the self-sensing scheme for
piezoelectric materials [29]. However, this approach is difficult to implement due to the
very small magnitude of the sensing signal compared to the actuation signal (millivolts
versus volts) [79] and the nonlinear [14], dynamic [32] sensing responses. Newbury [66]
explored the idea of using two IPMCs, mechanically coupled in a side-by-side or bilayer
configuration, to perform actuation and sensing. The attempt was reported to be unsuccess—
ful since the sensing signal was buried in the feedthrough signal from actuation [66].
In this research, a novel sensing scheme for IPMC actuators is proposed by bonding
an IPMC actuator and PVDF (polyvinylidene fluoride) sensing films to achieve simulta-
neous sensing and actuation. TWO versions of IPMC/PVDF sensory actuator have been
developed. In the first version, a single-mode IPMC/PVDF sensory actuator has been built
by bonding an IPMC and a PVDF thin film with an insulating layer in-between. The in-
sulating layer thickness is properly designed to minimize the stiffness of the composite
IPMC/PVDF structure while reducing the electrical feedthrough coupling between IPMC
and PVDF. A distributed circuit model is developed to effectively represent the electri-
cal coupling dynamics, which is then used in real-time compensation for extraction of the
true sensing signal. Experimental results show that the developed IPMC/PVDF structure,
together with the compensation algorithm, can perform effective, simultaneous actuation
and sensing. As the first application, the single-mode IPMC/PVDF sensori-actuator has
been successfully used for performing and monitoring open-loop micro-injection of living
Drosophila embryos.
10
The second version of IPMC/PVDF structure adopts a differential configuration for
PVDF sensors. Two complimentary PVDF films are bonded to an IPMC actuator in a
sandwich fashion, and a differential charge sensing circuit is used to measure the sensor
output. As analyzed and verified experimentally, this design has a number of advantages:
1) it allows shielding of capacitive coupling between IPMC and PVDF, and eliminates
the fictitious, feedthrough sensing signal induced directly by the actuation signal; 2) it re-
moves the influence of thermal fluctuation and other environmental noises, which is often
the most challenging problem for PVDF sensors; 3) it compensates for asymmetric sensing
responses of a single PVDF film under compression versus extension; and 4) it minimizes
the internal stress at bonding interfaces, which could otherwise cause delamination of lay-
ers or spontaneous creep of the composite beam structure. The design of an IPMC actuator
incorporates both bending and force feedback. A force sensor with MN resolution is de-
signed by sandwiching a relatively rigid beam with two PVDF films. The force sensor is
then bonded to the tip of the IPMC actuator, (which itself is sandwiched by PVDF for sens-
ing of bending). The effectiveness of the proposed sensory actuator has been demonstrated
in feedback control experiments. Precise position tracking of the end-effector is achieved
through proportional-integral (PI) control based upon feedback fi'om the integrated sensor.
A laser distance sensor is used as an independent observer for validation purposes. The
capability of simultaneous force measurement is shown in experiments of piercing soap
bubbles. Interaction forces as low as uN are well captured.
1.3.6 Monolithic fabrication of IPMC actuators for complex deforma-
tions
Fabrication techniques for IPMCs have been developed by several research groups [79, 68]
based on commercially available ion-exchange membranes such as Nafion. A casting pro-
cess has been developed to fabricate thin (down to 20 um) IPMC actuators [48, 69]. Kim
11
and Shahinpoor fabricated thick IPMCs (up to 2 mm) by baking the mixture of Nafion
powder and solvent in a mold [50]. Plasma treatment was introduced to roughen the Nafion
surface and improve the actuation performance of IPMC [51]. Chung et al. applied silver
nano-powder in fabrication of IPMC to improve the adhesion between metal and polymer
[22]. Akle et al. developed direct assembly plating method to make electrodes and pro-
duced high-strain IPMC [2].
IPMC produced with the aforementioned fabrication methods can only generate bend—
ing motions. However, actuators capable of complex deformation are highly desirable in
many applications, such as contour control of space inflatable structures [90], biomimetic
robots [52, 93], and tunable mirror membranes [61]. Fig. 1.3 shows an inspiring example
from biology, where one can see that the pectoral fin of a sunfish produces sophisticated
conformational changes to achieve eflicient locomotion and maneuvering. The complex
shape change is enabled by multiple muscle-controlled, relatively rigid fin rays that are
connected via collagenous membrane. Several groups have assembled discrete compo-
nents to make such multiple degree-of-freedom (MDOF) actuators [52, 93]. However,
the performance of assembled actuators was limited by complex structure design, friction
among the components, and low energy efficiency in force transmission. While MDOF
IPMC actuators can be obtained by manually removing the metal electrode of IPMC in
certain areas [28], this approach is time-consuming and not amenable to miniaturization
and batch-fabrication. Monolithic fabrication of MDOF IPMC actuators with a simple
structure will be essential to improve the performance, reduce the'cost, and enable mass
production.
Efforts have been made in fabrication of patterned IPMCs by Jeon and coworkers [41,
43, 42], where they combined electroplating with electroless plating to selectively grow
platinum electrodes on a Nafion film. In their work, however, tapes were used as masks to
achieve patterning, which would not be conducive to microfabrication or batch-fabrication.
Lithography-based microfabrication of IPMC has also been pursued by several groups [89,
12
Pectoral fin
(a) Side view (D) Front view
Figure 1.3: Pectoral fin of a bluegill sunfish undergoing sophisticated shape changes during
maneuvering and locomotion [54].
97, 41], where metal layers are deposited directly on Nafion to form electrodes. Without the
critical ion-exchange and electroless plating processes, IPMCs fabricated this way typically
are not highly active, and the metal electrodes tend to peel off under large deformation
because of the poor metal-polymer bonding. Feng et a1. developed microfabrication-based
IPMCs with arbitrarily defined shapes, where the ion-exchange and electroless plating steps
were also incorporated [34]. However, the actuators fabricated with this approach have a
single degree of freedom and can perform bending only.
The contribution of this research is a new process flow for lithography-based, mono-
lithic fabrication of highly active, MDOF IPMC actuators. Inspired by the structure of the
sunfish pectoral fin, we propose to achieve sophisticated three-dimension (3-D) deforma-
tion by creating multiple IPMC regions that are mechanically coupled through compliant,
passive membrane. Both the IPMCs and the passive regions are to be formed from a same
Nafion film. There are two major challenges in fabricating such actuators. First, the pas-
sive areas can substantially constrain the motion of the active areas. An effective, precise
approach is needed for tailoring the stiffness of the passive areas. Second, Nafion films
are highly swellable in a solvent. Large volume change results in poor adhesion of pho-
toresist to Nafion and creates problems in photolithography and other fabrication steps.
13
To overcome these challenges, two novel fabrication techniques have been introduced: 1)
selectively thinning down Nafion with plasma etch, to make the passive areas thin and com-
pliant; 2) irnpregnating Nafion film with platinum ions, which significantly reduces the film
swellablity and allows subsequent lithography and other steps.
The developed fabrication process involves plasma etching, ion-exchange, lithography,
physical vapor deposition (PVD), and electroless plating steps. Reactive ion etching (RIE)
with oxygen and argon plasmas is used to selectively thin down the passive area of Nafion
membrane with a patterned aluminum mask. An etch recipe has been developed to achieve
a high etching rate of 0.58 rim/min without damaging the film. In IPMC fabrication, an
ion-exchange step is usually performed immediately before electrode plating. However,
we have found that impregnating Nafion with platinum ions through ion-exchange can also
increase its stiflhess by two to three times, and reduce its swellability in water and in
acetone by over 20 times. This has proven critical in successful photolithography-based
patterning in that it ensures good adhesion of photoresist to the Nafion film. A positive
photoresist, AZ9260, is used in photolithography to create thick patterns which are used
as the mask in the electroless plating process to selectively grow platinum electrodes in
IPMC regions. After IPMC is formed, we treat the sample with hydrochloride acid (HCL)
to undo the effect of the stiffening process, which softens the passive areas and enhances
the actuation performance.
As a demonstration, we have applied the proposed method and fabricated artificial pec-
toral fins, each with three IPMC regions. A characterization system consisting of a CCD
camera and image processing software has been set up to quantify the deformations gener-
ated by the fabricated samples. We have verified that, by controlling the phase differences
between the voltage signals applied to the IPMC regions, each artificial fin can produce
sophisticated deformation modes, including bending, twisting, and cupping. For example,
a twisting angle of 16° peak-to-peak can be achieved with actuation voltages of 3 V at 0.3
Hz, when the top and bottom IPMCs lead the middle IPMC in phase by i90°. We have
14
investigated the impact of the thickness of passive area, and verified that the thinner the
passive region, the larger the deformations. This has thus provided supporting evidence for
our approach of modulating mechanical stiffness through plasma etching. We have firrther
performed preliminary studies of the fabricated fins in underwater operation using a Digital
Particle Image Velocimetry (DPIV) system. Interesting flow patterns are observed when the
fin is actuated, which shows the promise of the fabricated fins in robotic fish applications.
1.4 Organization of This Dissertation
The remainder of this dissertation is organized as follows. Control-oriented modeling
of IPMC actuator is presented in Chapter 2. Modeling of IPMC sensors is discussed
in Chapter 3. Modeling of biomimetic robotic fish propelled by an IPMC actuator is
shown in Chapter 4. Nonlinear control-oriented model for IPMC is discussed in Chap-
ter 5. IPMC/PVDF sensory actuator is described in Chapter 6. Monolithic fabrication of
the IPMC actuator to achieve complex deformations is presented in Chapter 7. Conclusions
and future work are provided in Chapter 8.
15
Chapter 2
A Dynamic Model for Ionic
Polymer-Metal Composite Actuators
The remainder of the chapter is organized as follows. The governing PDE is reviewed in
Section 2.1. In Section 2.2, the electrical impedance model for IPMC actuator is derived by
exactly solving the PDE, with and without considering the surface resistance. This lays the
groundwork for deriving the full actuation model, which is described in Section 2.3. Model
reduction is discussed in Section 2.4. Experimental validation of the proposed model is pre-
sented in Section 2.5. Model-based Hoe controller design and its real-time implementation
are reported in Section 2.6. Finally, concluding remarks are provided in Section 2.7.
2.1 The Governing Partial Differential Equation
The governing PDE for charge distribution in an IPMC was first presented in [64] and then
used by Farinholt and Leo [32] for investigating the sensing response. Let D, E, 4), and
p denote the electric displacement, the electric field, the electric potential, and the charge
16
density, respectively. The following equations hold:
D
E—————V 2.1
¢, ( )
V-D=p=F(C+—C’), (2.2)
where IQ is the effective dielectric constant of the polymer, F is Faraday’s constant, and C+
and C‘ are the cation and anion concentrations, respectively. The continuity equation that
relates the ion flux vector J to C+ is given by,
8C+
V- =——.
J at
(2.3)
Since the thickness of an IPMC is much smaller than its length or width, one can assume
that, inside the polymer, D, E, and J are all restricted to the thickness direction (x-direction).
This enables one to drop the boldface notation for these variables. The ion flux consists of
diffusion, migration, and convection terms:
C+F C+AV
__ + _
J— d(VC + RT V¢+ RT
Vp) +C+v, (2.4)
where d is the ionic diffusivity, R is the gas constant, T is the absolute temperature, p is
the fluid pressure, v the free solvent velocity field, and AV is the volumetric change which
demonstrates how much the polymer volume swells after taking water. From (2.2), C+ can
be written as
1 _
0‘ = Fp +C , (2.5)
where C ‘ is homogeneous in space and time-invariant since anions are fixed to the polymer
backbone. Taking the gradient with respect to x on both sides of (2.5) and using (2.1) and
(2.2), one gets
VC+ = 7:3V2E. (2.6)
17
Darcy’s Law [26] is used to relate the fluid velocity v to the pressure gradient Vp [3 8],
v=k’(C_FE—Vp), (2.7)
where k’ denotes the hydraulic permeability coefficient. Neglecting the convection term
[32], i.e., assuming v = 0, leads to
Vp = C‘FE . (2.8)
Substituting (2.5), (2.6), and (2.8) into the original ion flux equation (2.4) and using th) =
—E, one can rewrite J as
x, 2 Ke(l—C‘AV) Fc-(r-C-AV)
2. _v _ V - — . .
J d(F E RT E E RT E (2 9)
Assuming KeVE < C ‘F (see [63] for justification), the nonlinear term involving VE - E in
(2.9) is dropped, resulting in
.__ £2 2 _£C_‘ _ —
J— d(FVE RT (1 CAV)E). (2.10)
Nextwriting
193-122 _ 123%). 12325
a: “Fat '— F at _ Faxat’
and using (2.3), one obtains the following equation involving E:
K}; 82E (ice 83E FC“
_ 8E
73x5: Fw—fifl—C AV)—). (2.11)
8x
Expressing (2.11) in terms of p = KeVE :2 Keg—g, we can get
233. _ 82p + deC‘
3t 8x2 KeRT
(1 —-C‘AV)p = 0, (2.12)
18
which is the governing PDE for the charge density p inside the polymer. Nemat-Nasser
and Li [64] assumed that the induced stress is proportional to the charge density:
a = aop, (213)
where 010 is the coupling constant.
Farinholt [33] investigated the current response of a cantilevered IPMC beam when
the base is subject to step and harmonic actuation voltages. A key assumption is that the
ion flux at any point on the IPMC electrodes is zero. This assumption, which serves as a
boundary condition for the governing PDEs, leads to
(if—F2039» —0
8x3 KeRTBx x=:l:h— '
(2.14)
The work of Farinholt was based on obtaining an analytical but approximate solution p (x, t)
to (2.12) with some assumptions on the solution. For the step response, the solution is as-
sumed to take a separable form p (x, t) = P(x)Q(t). For the harmonic response, the solution
is assumed to be the sum of forced and resonant components. These assumptions limit
the model’s applicability in real-time control where the model should work for arbitrary
voltage inputs.
2.2 Electrical Impedance Model
From (2.13), the stress induced by the actuation input is directly related to the charge den-
sity distribution p. Therefore, as a first step in developing the actuation model, we will
derive the electrical impedance model in this section. While the latter is of interest in its
own right, one also obtains the explicit expression for p as a byproduct of the derivation.
Consider Fig. 2.1, where the beam is clamped at one end (2 = 0), and is subject to an
actuation voltage producing the tip displacement w(t) at the other end (2 = L) . The neutral
l9
axis of the beam is denoted by x = 0, and the upper and lower surfaces are denoted by x = h
and x = —-h, respectively. To ease the presentation, define the aggregated constant
Figure 2.1: Geometric definitions of an IPMC cantiliver beam.
9: F de'
K
KeRT
(1 —C-AV) . (2.15)
Performing Laplace transform for the time variable of p(x,z,t) (noting the independence
of p from the y coordinate), one converts (2.12) into the Laplace domain:
829 (x,z,s)
8x2 +Kp(x,z,s) =0, (2.16)
sp (x,z,s) — d
where sis the Laplace variable. Define B (s) such that B2(s) = S :K. With an assumption
of symmetric charge distribution about x = 0, a generic solution to (2.16) can be obtained
as
p(x,z,s) = 2c2(z,s) sinh([3 (s)x) , (2.17)
where c2(z,s) depends on the boundary condition of the PDEs. Using (2.17) and the field
equations (2.1) and (2.2), one can derive the expressions for the electric field E and then
20
for the electric potential (1) in the Laplace domain:
E(x,z,s) = 2c2(z,s)c—OS-ltcl—e-gflflcgy—x2 +a1(z,s), (2.18)
¢(x,z,s) = —2c2(z,s)§%;£—8fl — a1(z,s)x+ az(z,s), (2.19)
where a] (z, s) and a2 (2, s) are appropriate functions to be determined based on the boundary
conditions on 4). Two different boundary conditions are discussed next, one ignoring the
surface electrode resistance and the other considering the resistance. In both cases it will
be shown that the final actuation current is proportional to the applied voltage input V(s),
and thus a transfer function for the impedance model can be derived.
2.2.1 Model ignoring the surface resistance
First consider the case where the surface electrodes are perfectly conducting, as was as-
sumed by Farinholt [33]. The electric potential is uniform across both surfaces x = ih,
and without loss of generality, the potential is set to be:
¢ (ih,z,s) = i V?) . (2.20)
Combining (2.19), (2.20) with (2.14), one can solve for a1(z,s), a2(z,s), and c2(z,s), and
then obtain E (h,z,s) from (2.18):
V(S) Y(S)(S+K)
E(hizis>=- 2;. (57(s)+manh(y(s>>>’
(2.21)
where y(s) 2 fl (s)h. The total charge is obtained by integrating the electrical displacement
D on the boundary x = h:
W L W L
Q(s)=/O /(;D(h,z,s)dzdy=/0 AireE(h,z,s)dzdy. (2.22)
21
Plugging (2.21) into (2.22), one can derive Q(s), which is linear with respect to the external
stimulus V (s). The actuation current i(t) is the time-derivative of the charge Q(t), and
hence [(3) = sQ(s) in the Laplace domain. The impedance is then derived as:
tanh s
Z ( )- V(S) -S+K YES D (2 23)
IS — 1(s) _ Cs(s+K) ’ .
WL
where C = Ice—27 can be regarded as the apparent capacitance of the IPMC.
2.2.2 Model considering distributed surface resistance
The surface elecuode of an IPMC typically consists of aggregated nanoparticles formed
during chemical reduction of noble metal salt (such as platinum salt) [49]. The surface re-
sistance is thus non-negligible and has an influence on the sensing and actuation behavior
of an IPMC [78]. In this chapter the effect of distributed surface resistance is incorporated
into the impedance model, as illustrated in Fig. 2.2. Let the electrode resistance per unit
length be r1 in z direction and r2 in x direction. One can further define these quantities in
terms of fundamental physical parameters: r1 = ’11 / W, r2 = #2 / W, with r"l and r’2 represent-
ing the surface resistance per {unit length - unit width} in z and x directions, respectively.
In Fig. 2.2, i p (2,5) is the distributed current per unit length going through the polymer due
to the ion movement, ik(z,s) represents the leaking current per unit length, and is(z,s) is
the surface current on the electrodes. Rp denotes the through-polymer resistance per unit
length, which can be written as RP = R1,, / W, with R}, being the polymer resistance per
{unit length - unit width}. Note that by the continuity of current, the current is(z,s) on the
top surface equals that on the bottom surface but with an opposite direction. The surface
current is(0,s) at z = 0 is the total actuation current i(s).
22
One unit
- - . E El t d L
...... a...) -129th--,_l'zsl§§l3-----.‘ifff- a:
x 5 .
s o ,
Li ofylrga‘mlcs D R Polymer Layer
2 E Movement p
...... ill-has) H
@(LS) r2 Electrode Layer
----------------------------------- rr
Figure 2.2: Illustration of the IPMC impedance model with surface resistance.
The following equations capture the relationships between is (2, s), i p (z, s), ik(z, 5), (pi (z, s):
aim: (er) __ rl -
—82 — :F Wls(zrs) :1 (2'24)
Liz”) = —(ip(z.s> + res» . (225)
82
. . . . V(s) . .
From the potential condrtron at 2 :: 0, 1.e., ¢i(0,s) = :i:—2—-, the boundary condrtrons for
(2.19) are derived as:
¢(ih,z,s) = ¢i(z,s) ¥ ip(z,s)r§/W . (2.26)
With (2.24) and (2.26), one gets
_ iV (s) 2 ti . "2 .
¢ (films) — —2—-— ; [0 W1. (r,s>dr— We (2,8) , 1227)
Combining (2.27) with (2.19), one can solve for the fimctions a1(z,s) and a2(z,s) in the
generic expression for ¢(x,z,s). With consideration of the boundary condition (2.14), one
23
can solve for cz (2,3). With a] (2,3), a; (2,3) and c2 (2,3), one obtains E (h,z,3) from (2. 18):
0 (has) 7(5) (s+K)
h V(s)s+Ktanh(r(s)) '
E (h,z,3) = — (2.28)
Define the actuation current along the negative x-axis direction to be positive. The current
i p due to the ion movement can be obtain as
ip (2,3) = —sWD(h,z,s) 2' —3WKeE (h,z,s) . (2.29)
The leaking current i], can be obtained as
1), (2,3) = (142,27; (2’s) . (2.30)
P
With (2.28), (2.29) and (2.30), one can solve the PDE (2.25) for the surface current is (2,3)
with the boundary condition is(L,3) = 0. The total actuation current 1(3) 2 i5(0,s) can be
obtained, from which the transfer function for the impedance can be shown to be
V(s) 2./B(s)
z 3 = = , 2.31
2” 1(5) A(s)tanh(¢F(TviL) ( )
where
A(3) 2 9(5) +2—W (2.32)
(1+I'§9(3)/W) Rg’
B(3) 2 %A(s>, (2.33)
E sWan00+K>
9“" " hisyts>+1' (2'34)
See Appendix A.l for the detailed derivation.
One can show that Z2(s) is consistent with Z1 (3), (2.23), when r’l —> 0, 1”2 -—> 0 and
R;,—>oo.
24
2.3 Actuation Model
First we derive the transfer fimction H (3) relating the free tip displacement of an IPMC
beam, w(L,3), to the actuation voltage V(s), when the beam dynamics (inertia, damping,
etc.) is ignored. From (2.13) and (2.17), one obtains the generic expression for the stress
0‘(x,z,s) generated due to actuation:
6(x,2,3) = 201062 (2,3) sinh([3 (s)x). (2.35)
Note that c2 (2, s) is available from the derivation of the impedance model. When consider-
ing the surface resistance, the bending moment M (2,3) is obtained as
h
M(z,s) = [xO'(x,z,3)de
'i
= /2010chz(z,s)sinh(fi(3)x)dx (2.36)
-h
_20l0KWKe (7(5) — mums)» ¢ 01,238)
(57(5) +Ktanh(Y(S)))
From the linear beam theory [37],
82w(z,3) = M (2,3)
322 Y]
_2aoKWKe (7(5) 43111100») ¢ (hizaS)
Y1(SY(S)+Ktanh(Y(S)))
_aoKWKe(Y(S) -tanh(l’(5)))
Y1(sr'(s) +Ktanh(7(s)))
Z I
V(s)—2/0%V1—is(t,3)dt
' 1+r’29(3)/W ’
(2.37)
where the last equality follows fiom (2.27) and (A.l), Y is the effective Young’s modulus
of the IPMC, and I = gWh3 is the moment of inertia of the IPMC. Solving (2.37) with
25
boundary conditions w(0,3) = 0 and w’ (0,3) = 0, one can get
_1aoWKKe(7(s)- -tanh(r(s)))
Y1 (y(s (s)+Ktanh(r(s)))
3)L2—4/0L fez/0W —is(,r3)dra'z’dz
1+r’9(s)/W
w(L,3)
Using (A7) and (A.8), one can show
L z 2’ ,J
1.2—4f / / -—lis(t',s)dt'dz'dz=2L2X(3)V(3),
0 0 o W
where X (3) is defined as:
2 _1—sech(\/WL)— tanh((/B—(s_)L) \/B—(3_)L
X(3) B (S) L, (2.38)
One thus obtains the transfer function H (3) 2 151(3)?)
_ £2010W K x470) - tanh(r(s))) 2X(s)
1“ )— ‘ 2Y1 <7s+1>> (1 +46 0) /W) ' (2'39)
H (3) for the case where the surface resistance is ignored can be derived in an analogous
and simpler manner, and it is omitted here for brevity. Note that the blocking force output
. 3Y1 .
F (3) at the tip can be derived vra F (3) = w(L,3)Ko, where K0 = 7 denotes the spring
constant of the beam.
Back to the free bending case, in order to accommodate the vibration dynamics of the
beam, we cascade G(3) to H (3), as illustrated in Fig. 2.3. As the output of 6(3) represents
the bending displacement (as that of H (3) does), 0(3) will have a DC gain of 1. Since the
actuation bandwidth of an IPMC actuator is relatively low (under 10 Hz), it often suffices to
26
capture the mechanical dynamics 0(3) with a second-order system (first vibration mode):
_ ‘03
— 32+2§wn3+003’
0(3) (2.40)
where a)" is the natural frequency of the IPMC beam, and 5 is the damping ratio. The natu-
ral frequency (0,, can be further expressed in terms of the beam dimensions and mechanical
properties [91].
Bending
Voltage displacement
H (3) 0(3) —_’
Figure 2.3: Actuation model structure.
2.4 Model Reduction
An important motivation for deriving a transfer function-type actuation model is its poten-
tial use for real-time feedback control. For practical implementation of feedback control
design, the model needs to be finite-dimensional, i.e., being a finite-order, rational ftmction
of 3. However, in the actuation model derived earlier, H (3) is infinite-dimensional since it
involves non-rational functions including sinh(-), cosh(-), (f, etc. A systematic approach
to model reduction is Padé approximation [7], where one can approximate H (3) with a ra-
tional function of specified order. However, the computation involved is lengthy and the
resulting coeflicients for the reduced model can be complex. Therefore, in this chapter
a much simpler, alternative approach is proposed for model reduction by exploiting the
knowledge of physical parameters and specific properties of hyperbolic functions.
27
For ease of presentation, decompose H (3) as
H(S) = f(S)-g(S)'X(S), where
_L2aoWKx.))
2Y1 (y(s)s+Ktanh(Y(s)))’
2
3(5) = l+r§9(S)/W
(2.41)
(2.42)
Based on the physical parameters (see Table 2.1 in Section 2.5), |y(3)| >> 10, and K > 10°,
which allows one to make the approximation in the low frequency range (< 100 Hz):
3
:1
E
22
1 , (2.43)
K
h(/; =. y. (2.44)
With (2.43) and (2.44), one can simplify f (3), 6(3) and g(3)as
i
i“,
22
__L2aoWKKe(Y- 1))
“5) 2” 2Y1 (ys+K) ’ (2'45)
3Wrcey(3+K)
0(3) hWHK) , (2.46)
~ 2h(y3+K)
g(s) ~ szxes(3~+1<)+h(ys+1<)' (2'47)
The Taylor series expansions of sinh(a) and cosh(a) will be used for approximating
X (3):
02n+2 02’!
1+ 2 .—
(2n + 1)! —(2n)!
X(s)~ m a2n+2 , (2.48)
":0 (2n)!
with a = (/B(3)L, for some finite integer m. When |3| is small (low—frequency range) and
,J
32—) is small (which is indeed the case, see parameters in Table . B (3) is small and
p .
(2.48) approximates X (3) well with a small integer m. Note that only even-degree terms
appear in (2.48), and hence (2.48) is a function of B(3)L2 instead of (/B(3)L. Finally, since
28
B(3) is a rational firnction of 6(3) and 9(3) is approximated by a rational function (2.46),
one can obtain an approximation to X (3) by a rational function of 3.
Combining (2.45), (2.47), and the approximation to X (3), one gets a rational approx-
imation to H (3). Since the mechanical dynamics 0(3) is already rational, one obtains a
finite-dimensional actuation model. Note that a reduced model is still a physical model. In
particular, it is described in terms of fundamental physical parameters and is thus geomet-
rically scalable. This represents a key difference from other low-order, black-box models,
in which case the parameters have no physical meanings and one would have to re-identify
the parameters empirically for every actuator.
2.5 Experimental Model Verification
2.5.1 Experimental setup
Fig. 2.4 shows the experimental setup. An IPMC sample is dipped in water and clamped at
one end. The IPMC is subject to voltage excitation generated from the computer (through
dSPACE DSl 104 and ControlDesk). A laser displacement sensor (OADM 2016441/SI4F,
Baumer Electric) with precision set to 3:0.02mm is used to measure the bending displace-
ment w(t). The IPMC actuation current is measured with a current-amplifier circuit.
2.5.2 Identification of parameters in impedance model
Table 2.1 lists the parameters obtained for the impedance model. Among them some are
physical constants (gas constant R and Faraday’s constant F), some can be measured di-
rectly (absolute temperature T, effective Young’s modulus Y [18], actuator dimensions,
surface resistance r1 in z direction and through-polymer resistance R p), and the others need
to be identified through curve-fitting. Since |C’AV| << 1 [64], we take 1 — C‘AV z 1.
The IPMC materials used in this work were obtained from Environmental Robots Inc., and
29
Contact
electrodes &
Computer &
claiming—T4]?
(__ IPMC
dSPACE
‘—‘ Tank 1
fl .6— Laser ' ‘
Curren
wate_r_
SCIISOI’ 1 '
measurement
Figure 2.4: Experimental setup.
the sample dimensions reported have an accuracy of :l:0.5 mm in the length and width
directions and :l:0.5 pm in the thickness direction.
Table 2.1: Parameters for the impedance model.
F R T R;
96487 C/mol 8.3143 J/mol . K 300K 0.37Q-m2
Y [18] h H, H,
5.71 x108 Pa 180 (um) 22.3 n 1.8 x 10-5 n- m
d C‘ K,
1.38 x10‘9 mz/s 1091 mOVm3 1.34x10‘6 F/m
A nonlinear fitting process is used to identify the diffusion coeflicient d, the anion con-
centration C“, the dielectric constant ice, and the surface resistance density r3 in x direc-
tion, based upon the empirical impedance response of an IPMC actuator with dimensions
37.0 x 5.5 mm. In particular, the impedance model Z2 ( j27r f ) predicts the magnitude and
phase response of the actuator at fiequency f, as a nonlinear fimction of the parameters.
The Matlab function f minsearch can be used to find the parameters that minimize the
squared error between the empirical frequency response and the model prediction. The
identified parameters are listed in Table 2.1, where the values of d and C’ are close to
what were reported in the literature [63, 33]. The value of ice, however, differs from those
30
reported in [63, 33] by several orders of magnitude. This could be attributed to different
materials and experimental conditions (e.g., in water versus in air). It should be noted that
the value of Ke in the relevant literature was also obtained through model fitting instead of
direct physical measurement. It is thus of interest to examine more direct measurement of
these parameters in the firture.
For independent verification of the proposed model, the identified parameters will be
used in predicting impedance behaviors of other IPMC actuators with different dimensions,
as will be seen in Section 2.5.3.
2.5.3 Verification of the impedance model
Impedance model verification will be conducted on two aspects. First, it will be shown
that the model considering the surface resistance is more accurate than the model ignoring
the resistance, by comparing them with the measured frequency response of an IPMC ac—
tuator. Second, the geometric scalability of the proposed model will be confirmed by the
agreement between model predictions and experimental results for IPMC actuators with
different dimensions.
Effect of surface resistance
In order to examine the difference between the impedance models Z 1 (3) and Z2(3), their
model parameters were identified separately through the nonlinear fitting process described
in Section 2.5.2. The experimental data were obtained for an IPMC actuator with dimen-
sions 37.0 x 5.5 x 0.360 mm3. Fig. 2.5 compares the predicted frequency response (both
magnitude and phase) by each model with the measured frequency response. It is clear that
the model considering the surface resistance shows better agreement than the one ignoring
the resistance. This indicates that the model incorporating the surface resistance is more
effective in capturing the actuation dynamics of IPMC, and thus it will be used for the
remainder of this chapter.
31
A 70 3‘ 57 73.7.. . - - -Model without surface resistance
% 60 1 )5 133;}; Model with surface resistance
v . ‘ 13133.; 3 -O—Experimental data
'3 50... ‘, "',$.s.;...:..z..:.,,,,, ..... Z...:..:..:.,,,,, ..... :.,.:..:,.:.:.,,2
.340). ......................................................
(U 30 ........................................ h... ..................
220 _
10'2 10'1 10° 101 1o2
3?
2
or
a:
E
m
(D
m
.r:
a 100 L Liiiiiii ; Li.....1 . iiiiliir ..
10"" 10‘1 10° 101 102
Frequency (Hz)
Figure 2.5: Comparison of experimental impedance responses with model predictions, with
and without consideration of surface resistance.
Geometric scalability of the dynamic model.
Three samples with different dimensions (see Table 2.2) were cut from one IPMC sheet,
and were labeled as Big, Slim, and Short for ease of referencing. The model parameters
were first identified for the Slim sample, as discussed in Section 2.5.2. Without re-tuning,
these parameters (except geometric dimensions) were plugged into (2.31), i.e., the model
Z2 (3), for predicting the frequency response for the Big and Short samples.
Table 2.2: Dimensions of three IPMC samples used for verification of model scalability.
IPMC beam length (mm) width (mm) thickness (,um)
Big 39.0 1 1.0 360
Slim 37.0 5.5 360
Short 27.0 5.5 360
32
Fig. 2.6 shows the Bode plots of the frequency responses for the Slim and Big samples.
It can be seen that for both samples, good agreement between the model prediction and the
experimental data is achieved. Fig. 2.7 compares the frequency responses of the Slim and
Short samples. Reasonable match between the model predictions and the empirical curves
is again achieved for both samples. These figures show that the model is geometrically
scalable.
Magnitude (dB)
Model prediction (big)
0: +Experimental data (big) ? ? ?? 3&7:
_ - - - Model prediction (slim)
................
I
N
O
Phase (degree)
8
-60 ~ . 3
_80 -E :22; ; '.;.I -
10'2 10'1 10° 10 102
Frequency (Hz)
Figure 2.6: Impedance model verification for the Big and Slim IPMC samples.
2.5.4 Verification of the actuation model
The actuation model has two modules serially connected, as shown in Fig. 2.3. All pa-
rameters of H (3) have been identified during identification of the impedance model except
the stress-charge coupling constant do. The natural frequency to" and the damping ratio 6
in 0(3) can be identified based on the measurement of damped oscillations of the IPMC
33
Magnitude (dB)
Model prediction (short)
0 '— —II— Experimental data (short)
-20 _. . - - - Model prediction (slim)
—>— Experimental data (slim)
Phase (degree)
is
-60 g
'80 . . . . .: . . .
10'2 10‘1 10° 101 102
Frequency (Hz)
Figure 2.7: Impedance model verification for the Slim and Short IPMC samples.
beam in the passive state. For the Big sample, we obtained (1),, = 28.9 rad/s, and 6 = 0.1.
Finally, (10, which is simply a gain parameter in the actuation model, was identified to be
a0 = 0.129 J/C using the magnitude of actuation response measured under a sinusoidal
voltage input.
The whole actuation model was verified in experiments by applying sinusoidal actua-
tion signals V(t) with amplitude 0.2 V and frequency from 0.02 Hz to 20 Hz. The laser
sensor was used to measure the bending displacement w(t) at the fiee end of the Big sample.
The magnitude gain and phase shift from the input V(s) to the output w(s) were obtained,
which show good agreement with the model prediction; see Fig. 2.8. Note that, from the
empirical Bode plot, the natural fl'equency of the IPMC beam in the active state (under ac-
tuation) is about 30 rad/s, and thus slightly higher than that measured in the passive state. It
indicates that the effective stiffness of an IPMC is influenced by the actuation input. How-
34
ever, such an effect is not significant for relatively low actuation voltages, and a detailed
discussion on this would be beyond the scope of the current chapter.
a :
E "8 .................................................. -
m ......
'3 916 Experimental data
E -100-- - - -Simulation data with original model : 1 : 1:2.
3’ —Simu|ation data with reduced model ; g g g 5;;
2 :i:::::: I'TT‘TIT: --.......
—120 - 1 4 - ...
101 10° 101
i? ‘3'
95’,
a) _.100 ......................................................................
E
o
g -200-
«C ..:.:...:
“- 300 ;;;;;;;; ; 3;;;;;;;
10'1 10°
Frequency (Hz)
Figure 2.8: Comparison of the measured actuation response with the proposed full and
reduced models for the Big sample.
Model reduction was then carried out for H (3) using the techniques discussed in Sec-
tion 2.4, where m = 2 was used. This resulted in a seventh—order model 191(3) for ap-
proximating H (3). The Matlab command reduce was further used to reduce 1-71 (3) to a
second-order function H(s), which leads to a fourth—order reduced model for the overall
actuation response for the Big sample:
_ 0.005s + 0.043 835
(S) (S) (S) 32+733+204 s2+5.78s+835
(2.49)
From Fig. 2.8, the reduced model also matches closely the empirical response. It will be
used for model-based controller design in the next section.
35
2.6 Controller-Design Example: Model-based Hoe Control
In this section we provide an example to illustrate the use of the proposed model in model-
based controller design. While other control design methodologies can be adopted, Hoo
control has been chosen to accommodate multiple considerations, including stability in the
presence of uncertainty, attenuation of the effect of sensing noise, and minimization of
control effort.
Consider Fig. 2.9, where the IPMC is represented by some nominal model P(3) with
an additive uncertainty A0. Let P(s) be the reduced model (2.49) for the Big sample. Then
A, represents the error between the hill actuation model and P(s) plus the unmodeled non-
linearities. The signals d1 and 6!; denote the actuation noise and the sensing noise, respec-
tively. One is interested in designing a controller K (3) which ensures closed-loop stability
and robust tracking performance in the presence of A0 and the noises 611 and d2 while taking
into account the consumed control effort. Standard Hoe control techniques [98] are used in
the following controller design.
"r--- m) 4.1: P(s) 4i We(s) _.22
Figure 2.9: Schematic of the closed-loop control system for an IPMC actuator.
To ensure the closed-loop stability in the presence of Ag, one needs to first obtain the
bound “Aalloo. Fig. 2.10 shows the modeling error - the difference between the measured
36
response and P(s), as well as a bound Wa(3) on the error, where
0.15
W, = . 2.50
(S) s2+37s+1318 ( )
Then ||Aa||°° g ||Wa(3)||°o = 1.65 x10‘4.
..80, ............... " .....
5332
ifo
_-,.
E 90 +Modeling error . ;; .333
7} --—Simulation data ofWa(s) 33f};
'e
U)
S
-110-
.120-.-; g;;; ,.: 1
10‘1 10° 101
Frequency (Hz)
Figure 2.10: The modeling error and its bound Wa(3).
The closed-loop system in Fig. 2.9 can be regarded as the feedback connection of A0
and Ms(3), as illustrated in Fig. 2.11. Ms(3) can be obtained by computing the transfer
function from the output of A0 to the input of A0 in Fig. 2.9:
K (S)
MS(S) : 1+P(3)K(3) ° (25])
From the small gain theorem [98], a sufficient condition for internal stability is
1
||M_,(s)||.. < M = 6038. (2.52)
37
+
+
0'2
Figure 2.11: The feedback connection of [145(3) and Aa(3).
To proceed with the controller design, define two artificial outputs 21 and 22 as in
Fig. 2.9, where the performance weight We(3) and the control weight Wu(3) are chosen
to be
3+ 124 _ 100(s+0.24)
We = , W, _. .
(S) 4(3+3.1x10‘3) (S) 3+3.4><105
See [98] for guidelines on choosing these weight firnctions. Now ignore the A0 block, and
design K (3) to minimize the H00 norm of the transfer fimction from {d1,d2}T to {21,22}T.
This would minimize the effect of the noises on the tracking performance and the control
effort. The resulting controller is
K (S) _ 29527 (3 + 2.569)
_ (3 + 0.00314) (3 + 4.952) °
From (2.51), one can calculate ||Ms (3)||°° = 5395, which satisfies the internal stability
condition (2.52) under the uncertainty.
The designed H00 controller was implemented for tracking control of the Big IPMC
sample, where the reference r used was
r(t) = 0.133 sin(0.027rt) +0.0665 sin(0.067rt) mm .
The laser sensor for measuring the tip displacement has a noise level of :l:0.02 mm. For
38
comparison purposes, a PI controller
K1(3) = 3000(1+%)
was also implemented together with a low-pass filter
_ 961
_ 32+623+961
F(3)
for the output measurement. Note that a PID controller was explored for IPMC actuators
by Richardson et al. [75].
Fig. 2.12 shows the IPMC tracking performance under model-based Hoe control and
Fig. 2.13 shows the tracking performance under PI control. Simulation results under both
PI control and H00 control are also shown in the figures. It can be seen that the tracking error
under H00 control is almost at the level of sensing noise, while the error under PI control
is about twice as large. The agreement between experimental and simulation results has
further validated the reduced model. Fig. 2.14 compares the controller output under Hoo
control and PI control in the experiments, which shows that the H00 control requires lower
control effort. Therefore, controller design based on the reduced model is effective.
2.7 Chapter Summary
In this chapter a dynamic model for IPMC actuators was developed by solving the physics-
goveming PDE analytically in the Laplace domain. It is distinguished from existing mod-
eling work of IPMC actuators in that it is amenable to model reduction and control design
while capturing fundamental physics. The modeling work bridges the traditional gap be-
tween the physics-based perspective and the system-theoretic perspective on these novel
but sophisticated materials. The model also incorporates the effect of surface electrode re-
sistance in an integrative manner. The compact, explicit, transfer-function representation of
39
E 0'3 ‘ Measurement
g 0.2 - - - - Reference
E
a) 0.1
E
8 0
g -81
.2
g -0.2 . + .
0 50 100 150 200
E 0.1 . ,
E
h
E
a)
or
.E
x
8
h
-o.1 . , '
'- 50 100 150 200
Time (s)
Figure 2.12: Experimental and simulation results on tracking of IPMC actuator under Hoo
control.
the physics-based model can be reduced to low-order models for real-time feedback control
purposes. A number of experimental results were presented to demonstrate the geometric
scalability of the model. Due to the physical nature of the model, the agreement between
model predictions and experimental results also provides insight into the underlying actu-
ation mechanisms of IPMC materials. An Hoe controller based on the reduced low-order
model has been designed and implemented in real-time tracking experiments. Experimen-
tal results have proven that the proposed model is faithful and suitable for control design.
Note that while this chapter is focused on a particular class of smart materials, pursuing
physics-based and control-oriented models could be a valuable approach to the design and
control of a variety of materials and manufacturing systems (see, e.g., [55]).
Future work will be focused on two aspects. First, the proposed actuation model will
be extended to incorporate material nonlinearities which become pronounced at large ac-
4O
E 0'3 T ' Measurement
v 0.2 - - - - Reference
'E
a) 0.1
as) 0
8
E. -0.1 — «
.9
D _O.2 1 1 l
0 50 100 150 200
9
..s
d
.0
c
or
-0.05
Tracking error (mm)
0
.5
so 100 150 200
Time (s)
0
Figure 2.13: Experimental and simulation results on tracking of IPMC actuator under PI
control.
tuation levels. The nonlinearities include nonlinear elasticity, hysteresis [19], and the de-
pendence of parameters (such as surface resistance) on the curvature output [74]. The
actuation model in this chapter was assumed to be a cascade of stress-generation module
H (3) and linear beam dynamics 0(3). However, the two-way coupling effects existing be-
tween the stress-generation module and the beam dynamics module, as indicated by the
curvature-dependent electrical parameters, introduce challenging nonlinearities in model-
ing and control of IPMC materials that require further study. The second direction of future
work is the application of the proposed modeling approach to control of micromanipulation
[18] and biomimetic robots [86, 58]. There the model has to be extended to account for
force interactions with external objects.
41
0.6 I I r
E 0.4
- 0.2
g
c 0
o
o -0.2
3:8 -0.4
0.6
E 0.4
'6 0.2
h
E o
8 -o.2
E —0.4
0 _ 50 100 150 200
Time (s)
Figure 2.14: Comparison of controller outputs under H00 control and PI control.
42
Chapter 3
A Dynamic Model for Ionic
Polymer-Metal Composite Sensors
This chapter is organized as follows. In Section 3.1 the exact solution and thus the model
are derived, with and without considering the surface resistance. Model reduction is dis-
cussed in Section 3.2. Experimental validation results are presented in Section 3.3. Finally,
concluding remarks are provided in Section 3.4.
3.1 A Dynamic Sensing Model
The governing PDE of sensing model is the same PDE as described in the modeling of
actuation (See Section 2.1). Farinholt and Leo investigated the short-circuit current (and
charge) sensing response of a cantilevered IPMC beam when the tip is subject to a step
displacement [32], as illustrated in Fig. 2.1. Their work is based on obtaining an analyti-
cal but approximate solution p(x,t) to (2.12), which is assumed to take a separable form
p(x,t) = P(x)Q(t). A key assumption in [32] is that the initial charge density at any point
on the IPMC surface along the length direction (denoted as z-direction) is proportional to
the induced stress at the same point. This assumption, which serves as an initial/boundary
43
condition for (2. 12), is made based upon that a similar assumption was used in modeling
the actuation response of IPMCs [64], and that IPMCs demonstrate reciprocity between
sensing and actuation [66]. Note that the solution p(x,t) has implicit dependence on the
length coordinate 2 due to the nonuniform stress profile on the surface.
The objective of this modeling work is to derive a sensing model for IPMCs that accom-
modate arbitrary mechanical stimuli (including step deformations as a special case). While
F arinholt and Leo assume perfectly-conducting surface electrodes [32], we also incorporate
the distributed surface resistance into the proposed model, which will be shown to produce
more accurate predictions in experiments. The model is based upon the exact solution to
(2. 12) subject to appropriate boundary conditions, which is made possible by converting it
to the Laplace domain. The latter also makes transfer function a natural representation for
the model.
Consider Fig. 2.1, where the beam is clamped at one end (2 = 0), and is subject to
an external force F (t) at the other end (2 = L) producing the tip displacement w(t). The
neutral axis of the beam is denoted by x = O, and the upper and lower surfaces are denoted
by x = h and x = —h, respectively. Performing Laplace transform for the time variable of
p (x,t), one converts (2.12) into the Laplace domain:
3p (x,3)— d-a-zpa(—:’S)-+Kp =0, (3.1)
where 3 is the Laplace variable. After rearranging, (3.1) becomes
329 __(__x S) (_S+K)
——a—x2 — d p ( s) (3.2)
2 S + K . . . . .
Define 3(3) such that [3 (3) = d . A generic solutron to (3.2) is obtarned as
P (xiS) = Ci (Sle'fi(s)x+cz(S)efi(S)xa (3-3)
44
for some appropriate firnctions c1(3) and c2(3). An assumption, analogous to the one in
[32], will be made to determine c1(3) and c2(3). In particular, it is assumed that the charge
density p(:l:h,s) at the boundary x = :l:h is proportional to the induced stress 0(ih,3):
CHI/1,3) = %P(ih,S), (3-4)
where 010 is the charge-stress coupling constant. From 0(h,3) = —0'(—h,3), one gets
p(h,S) +p(—haS) : 0:
which implies c1(3) = —c2(3) and thus
P(an) = 262(5) Sinh(B(S)x)- (35)
One can further relate 0(h,3) to the external stimuli. In the time domain,
M(t)h
o(h,t) = 1 ,
(3.6)
2
where M(t) is the bending moment, and I = §Wh3 is the moment of inertia with W being
the beam width (refer to Fig. 2.1). M(t) is related to the external force F (t) at z = L by
M(t) =F(t)(L-Z), (3.7)
where L is the beam length. The out—of-plane deflection w(t) at the tip can be related to the
force F (t) [72] by
L3F
W) = 3),?) , (3.8)
45
where Y denotes the Young’s modulus of the beam. Combining (3.6), (3.7), and (3.8) yields
3Y h L — 2
o(h,t) = —(L3———)w(t). (3.9)
Transforming (3.9) into the Laplace domain and combining with (3.4), one gets
31’ h L — 2
00715) = film), (“W
which, together with (3.5), implies
3Yh(L—z) (3.“)
62(5) 2 20101.3 sinh (p(s)h) W”
Note that c2 (3) depends implicitly on the length coordinate 2.
Using (3.5) and the field equations (2.1) and (2.2), one can derive the expressions for
the electric field E and then for the electric potential 4) in the Laplace domain:
E (x,3) 2 202(3) 00:25:?” +a1(3), (3.12)
sinh (13 (SW
1902(3)
¢(x,3) 2 —202(3)
—a1(3)x+a2(3), (3.13)
where a1 (3) and a2 (3) are appropriate firnctions to be determined based on boundary condi-
tions on 4). Two different boundary conditions are discussed next, one ignoring the surface
electrode resistance and the other considering the resistance. In both cases it will be shown
that the final sensing current is proportional to the applied deformation w(s), and thus a
transfer function relating the sensor output to the deformation input can be derived.
3.1.1 Model ignoring the surface resistance
First consider the case where the surface electrodes are perfectly conducting, as was as-
sumed by Farinholt and Leo [3 2]. In the short-circuit current (or charge) sensing mode, the
46
electric potential is uniform across both surfaces x = ih, and without loss of generality,
the potential is set to be zero:
¢(h,z,s) = ¢(-h,z,S) = 0. (3.14)
Note that the 2—dependence of q) is made explicit in (3.14) for clarity (but with abuse of
notation). Combining (3.14) with (3.13), one can solve for a1(3) and a2(3):
Sinh(l3((S)h))
= —2
a1(s) c2(3) hkeB2(S) 3
cats) = 0- (3.16)
(3.15)
The total induced sensing charge is obtained by integrating the electrical displacement D
on the boundary x = h:
W L W L
=/ / D(h,z,s)dzdy:A /0 KeE(h,Z,S)dZdyo (317)
0 0
Note that, again, the z—dependence of D and E (through (:2 (3)) is made explicit in (3.17)
by putting 2 as the second argument of D or E. All field variables ((1), D, and E), on-the
other hand, have no dependence on the width variable y. From (3.17), one gets
9(5) = [W [re [262“ ()Ccoshims)h)_, Us” whom] My
IQSN ) ’7 l32( )
= 2W([3(3)hcosh([3(2)h)—sinh( ()[3(3h) )))/01426“
’43 (S)
: 2W(fi(3)hcosh([i(3)h) —sinh([3(3 )h)) L 3Yh(L— z)w (3) dz
11520 ) 20105‘ 811111030011)
_ 3YW([3(3)hcoth(/3(3)h)—l)w(3) L -—2 z
‘“ aoL302 / (L )d
= 3YW ([3(3)hcoth(,8(3)h) — 1) w(3)
2aoL132(S) °
In the above the first equality is from (3.12) and (3 . 1 5), the second equality follows from
47
the independence of c2(3) from y, and the third equality is from (3.11). The short-circuit
current i (t) is the time-derivative of the charge Q(t), and hence i (3) = 3Q(3) in the Laplace
domain. The transfer fimction from the mechanical input w(s) to the sensing output i (3) is
then derived as
_i_(3_)_ = 33YW([3(3)hcoth([3(3)h) — 1)
1"“): w(s) 2042920)
(3.18)
3+K
with 0 (3) = d
3.1.2 Model considering distributed surface resistance
The surface electrode of an IPMC typically consists of aggregated nanoparticles formed
during cherrrical reduction of noble metal salt (such as platinum salt) [49]. The surface
resistance is thus non-negligible and has an influence on the sensing and actuation behavior
of an IPMC [78]. In this modeling work, the effect of distributed surface resistance is
incorporated into the sensing model, as illustrated in Fig. 3.1. An IPMC beam is clamped
at one end (2 = 0) and is subject to an applied displacement w(s) at the other end (2 = L).
Let the resistance per unit length be r = r0/ W, with re representing the surface resistance
per unit length and unit width (a parameter independent of IPMC dimensions). For each
section A2 of the IPMC, a current i p (2,3)Az is generated inside the polymer and then joins
the surface current is (2,3). Note that by the continuity of current, the current is (2,3) on the
top surface equals that on the bottom surface but with an opposite direction. The surface
current is(0,3) collected at z = 0, where ¢(h,0,3) = ¢(—h,0,3) = 0, is the short-circuit
sensing current i(s).
The following equations capture the relationships between the surface current is (2,3),
the current density i ,, (2,3) within the polymer, and the electric potential ¢(ih,2,3) on the
48
is(z,s) ¢(h,z,s)
x:+h _
i,(0,s)
x = - h —
I 15(Z,S) $011,215) |
Figure 3.1: Illustration of the distributed surface resistance for the IPMC sensing model.
surfaces:
3¢(h,2,3) _ . _ m.
T — rlS(Z’S) — WIS(Z’S)’ (3'19)
a¢(—h,z,s) _ _ . _ __r_9_.
——82 _ r15(z,3) —- WIS(ZrS)r (3'20)
8is(z,3) _ _.
82 ._ rp(z,3). (3-21)
From the short-circuit condition at 2 = 0, i.e., ¢(h,0,3) = ¢(—h,0,3) = 0, the boundary
conditions for (3. 13) are derived as:
¢(h.z.s> = jo $441,042 (3.22)
¢(_h,Z,S) : —/0‘zr—p£is(rrs)dt- (3.23)
Combining (3.23) and (3.23) with (3.13), one can solve for the functions a1(3) and a2(3) in
the generic expression for ¢(x,z,3):
fzrois(t',s)d’t sinh(jB(s)h)
a1(S) = r 0 kW “202(Sl—Wa
02 (S) = 0. (3.25)
(3.24)
49
Next we will eliminate i p (2,3) in (3.21) so that the equation involves is (2,3) only, which
can then be solved for the sensing output is(0,3). Note that the generated sensing charge
on a Az section can be expressed as D(h,z,3)WAz, i.e.,
ip(2,3)Az
s = D(h,z,s)WAz = KeE(h,Z,S) WAZ,
implying
ip(z,3) = SKeE(h,Z,S)W. (3.26)
Evaluating E (h,z,3) using (3.12) with (3.11) and (3.24) plugged in for c2(3) and a1(3),
respectively, one gets (after simplification):
ip(z,3) = A(3)(L —2) — B(3) /Ozis(t,3)dt’, (3.27)
where
33YW(B(3)hcoth(l3(3)h) — 1)w
aoL3l32(S)
SKero
B(3) = h . (3.29)
(s), (3.28)
Plugging (3.27) into (3.21), one obtains an integro-differential equation for is (2,3):
axis) 2 _A(,)(L _z) + 3(5) [024.301. (3.30)
Eq. (3.30) can be solved analytically through (yet another) Laplace transform, this time for
the 2 variable.
We introduce the unilateral Laplace transform for frmctions of the length coordinate
z. The new Laplace variable will be denoted as p since 3 has already been used for the
50
transform of time firnctions. For instance, the transform of is (2,3) will be defined as
13(p,3)é/0 is(z,3)e_Pzdz.
Now perform the Laplace transform with respect to the 2 variable on both sides of (3.30).
Using properties of Laplace transforms [23, 36], one gets
prim) — i.(0.s) = -—-A(s) (£- -— "1517) ”(3)5932. (3.31)
Solving for 15(p,s), one obtains
13(pa5) : p2——EE(—S)is(0,5) '- A(S)r (3-32)
which can be rewritten through partial fraction expansion as:
1 l l .
1s(PaS) —- 5 (mm + m) 15(0,S) +
_Q_l(_S_) (12(5) 43(5) S 333
( P +p- B(S)+p+\/B(S))A()’ (' )
with
l l—L(/B(3) 1+L(/B(3)
NSF—m, ‘12“): 23(3) ’q3(s): ZB(S)
The surface current i 3 (z, 3) is then obtained from (3.33) using the inverse Laplace transform
51
of15(p,s):
e‘/B(s)z + e-‘/B(s)z .
iS(Z,S) : 2 15(0 S)+
__ +1————‘/B(S)e\/_L B()+sz 1+‘/B(S)Le‘ ZA()
B(s)+ zB(_—s—) 2B(s) S
= cosh( B(s)z )is(0,s)+
(—1 +cosh(\/B(s)z) —B(\/ s)Lsinh (B(\/ ()z)) 3—8. (3.34)
Refer to Fig. 3.1. Since the circuit is open at z = L, the following holds:
is(L,s) = O. (3.35)
Plugging z = L into (3.34) and using (3.35), one obtains the sensing current is(0,s) as
A(s) (1 — cosh(. /B(s)L) + \/B(s)Lsinh(\/—B(s_)L))
W’s) ___ mambo/W) (3'36)
In particular, the short-circuit sensing current i(s) = is(0,s) is obtained as
i(s) _ 3sYW([3(s)hcoth(fl (s)h) — 1) .
_ aoL3l32(S)
(I — cosh(N/B(s)L) + \/B(s)Lsinh(\/B(3)L)) ( ) (3 37)
B(s) cosh( B(s)L)
The transfer function from the mechanical input w(s) tothe sensor output i (s) is then
H _ _i(s_) _ 3SYW (B (s)hcoth(fi (3)/1) — 1) .
2(3) _ W(S) — 00L3fi2(5)
(1 -—cosh(\/B(s)L) + ./B(s)Lsinh(./B(s)L)) 3 38
B(s)cosh(‘/B(s)L) ’ ( ' )
where B(s) is as defined by (3.29).
52
Note that the model H2 (3) incorporating surface resistance is consistent with the model
H1 (s) ignoring surface resistance. Indeed, H2(s) degenerates to H1(s) when the resistance
r0 —> 0. To see this, fi'om (3.18) and (3.38), one can write
2 (1 -— cosh(\/B(s)L) + \/B(s)Lsinh(\/B(s)L))
H”) 2 '57 B(s)cosh(\/B(s)L) 'H‘ m
When r0 —-> O, t/B(s) = \/SK'er0/h —> 0. From l’Hopital’s rule,
. a _
Taking \/B(s) to be a in (3.39), one obtains
lim H(s)—3 L—2H(s)--H(s)
ro-)0 2 —L2 2 l — l O
3.2 Model Reduction
An important motivation for deriving a transfer function-type sensing model H (s) is its
potential use for real-time feedback control and for sensor signal conditioning. In the case
of feedback control, knowing the sensor dynamics is essential to the controller design [36].
For pure sensing applications (such as structural health monitoring), the knowledge of sen-
sor dynamics allows one to correctly reconstruct the original mechanical stimulus w(s)
based on the sensor output i (.9), either online or offline. This can be done through inversion
of the sensor dynamics:
w(s) =Hinv(5)i(5)a (3.40)
where Him,(s) represents the inverse dynamics
Hinv(5) =
H(s). (3.41)
53
For practical implementation of sensor conditioning or feedback control design, the
model H (3) needs to be finite-dimensional, i.e., being a finite-order, rational function of
s. However, the sensing models derived earlier, H1(s) and H2(s) ((3.18) and (3.38)), are
infinite-dimensional since they involve non-rational functions including sinh(-), cosh(), (f,
etc. A systematic approach to model reduction is to use Padé approximation [7], where one
can approximate H1 (.9) or H2 (3) by a finite-dimensional transfer function with any specified
order.
In the following we discuss reduction of the IPMC sensing models by exploiting their
specific properties. The method is less general than Padé approximation, but it is also
simpler. The discussion will be focused on H2(s) since it covers H1(s) as a special case.
For ease of presentation, decompose H2 (3) as
H2 (S) = f (S) °g(S),
with
3sYW (B(s)hcoth(l3(s)h) —— 1)
f(s) aw, (”LS (3.42)
1 —— cosh ((/B(s)L) + (/B(s)Lsinh ((/B(s)L)
g(s) - . (3.43)
B(s) cosh (ML)
Based on the physical parameters (see Table 3.1 and Table 3.2 in Section 3.3), the
. K . . . . .
composrte constant 2 IS in the order of 1012, which implies
3+K
d
|fi(S)| =
|>106,
for s 2 fix), where (1) denotes the angular frequency of any sinusoidal input. Since the
54
thickness h of an IPMC is typically bigger than 1 x 10’4 m, it can be seen that
lfi(S)h| >> 10,
which allows one to make the approximation
coth([3(s)h) z 1. (3.44)
With (3.44), one can simplify (3.42) as
s+Kh l
~3sYW(/3(s)h-1) _ 3sYWd d —
atofi2 (S)L3 “05‘ S + K
3SYW\/c_1(\/s +Kh — (A?)
a0L3(s +K)
f (S)
One can further approximate \/s + K by its Taylor series abouts = O. For instance, consid-
ering up to the second-order terms results in the following approximation to f (s):
3sYW\/a (hx/EU + %) — x/E)
1101.3 (3+K)
f (s) z . (3.45)
The following Taylor series expansions of sinh(a) and cosh(a) will be used for g(s):
3 5
. a a
smh(a)=a+-:-S-!+§+~-,
a2 a4
cosh(a)=l+-2—!+-4—!+~-,
55
with a = ( /B(s)L. This results in
(1 — %) B(s)L2 + (% — 217) Bz(s)L4 + (% — %) 193(9)].6 + ~ --
B(s) (1+ B(ZLZ + 82$)“ + - - -) .
g(s) = (3.46)
SKero
h
Recall B (s) =
Truncation of the series in (3.46) leads to a finite-order approximation. For example,
keeping the first three terms in each series yields
)~ (I — 21—!) B(s)L2 + (31? — 4i!) 32(s)L4 + (.51.! _ %) 33(S)L6
~ 3.47
W B 1 B(s)L2 132 (5)134 ( )
(S) + 2! + 41
Combining (3.45) and (3.46) leads to a fourth-order model for IPMC sensors:
3sYWx/c-i (In/Eu + —S—) — x/a)
1:] (S) = 3 2K
00L (s+K)
1 2 1 1 2 4 1 1 3 6
1—— B(s)L+ ——— B(s)L+ ——— B(s)L
2! 3! 4! 5! 6!
- . (3.48)
B(s)L2 32(s)L4
B(s) (1+ —2! + 4!
Although H(s) is an improper rational function, i.e., the numerator is of higher order than
the denominator, it is not a concern for feedback or sensing applications. This is because
the inverse dynamics (3.41), which is what matters in implementation, will be proper.
Note that a reduced model like (3.48) is still a physical model. In particular, it is de-
scribed in terms of fimdamental physical parameters and is thus geometrically scalable.
This represents a key difference fiom other low-order, black-box models, in which case the
parameters have no physical meanings and one would have to re-identify the parameters
empirically for every sensor.
56
3.3 Experimental Verification
3.3.1 Experimental setup
An experimental setup was built to produce periodic mechanical stimulus with controllable
frequency. The schematic of the setup is shown in Fig. 3.2(a), while Fig. 3.2(b) shows its
picture. A crank-slider mechanism is used to convert the rotational motion generated by a
DC motor (GM8724SOO9, Pittman) into the linear, oscillatory motion of the slider, which
slides on a fixed rail. A rigid bar connects the slider and the rotating disk, and by changing
the distance from the bar end to the disk center, the amplitude of translational motion can be
adjusted. The oscillation frequency is controlled by tuning the voltage input to the motor,
and it is measured through an optical switch. The setup can provide periodic excitation
from 1 to 20 Hz.
The free end of a cantilevered IPMC beam is constrained to a slit in the slider, which
correlates the slider motion directly with the tip-bending deformation w(t) of the IPMC.
In particular, w(t) can be calculated explicitly based on the bar-disk configuration and the
optical switch signal. A differential current-amplifier circuit is used to measure the IPMC
sensing current generated under the mechanical stimulus. Data acquisition and processing
are performed through a PC running on a real-time Linux kernel.
3.3.2 Parameter identification
In the dynamic sensing models H1(s) and H2 (s), some parameters are physical constants
(gas constant R and Faraday’s constant F), some can be measured directly (absolute tem-
perature T, Young’s modulus Y, sensor dimensions, and surface resistance density r0), and
the others need to be identified through curve-fitting. Table 3.] lists the physical constants
and the parameters obtained through direct measurement. Since IC’AVI << 1 [64], we take
1 — C‘AV = l. The IPMC materials used in this work were obtained from Environmental
57
Differential current
amplifier
Clamp and/
electrodes
IPMC —°
Connecting bar
on a'l
r I Rotating disk
(a)
Slider assembly Fixed end IPMC
1:
Rail
(b)
Figure 3.2: (a) Schematic of the experimental setup; (b) picture showing a cantilevered
IPMC under mechanical deformation.
Robots Inc., and the thickness is 360 um, i.e., h = 180 pm. For an IPMC that is 22 mm
long and 7 mm wide, the surface resistance was measured to be 19 9, corresponding to
r0 = 6.05 52.
The parameters that remain to be determined include the diffusion coefficient d, the
anion concentration C‘, the dielectric constant ice, and the charge-stress coupling constant
a0. The empirical fi'equency response of an IPMC sensor with dimensions 22 x 7 mm2
58
Table 3.1: Physical constants and directly-measured parameters.
F R T Y[18] h r0
96487 C/mol 8.3143J/mol-K 300K 5.71x108Pa 180(um) 6.0552
was used to identify the remaining parameters through a nonlinear curve-fitting process, as
described next.
Fix an excitation fi'equency f, and acquire the mechanical deformation w(t) and the
IPMC sensor output i (t). Fast Fourier transforms are then performed on w(t) and i(t) to
extract their amplitudes and phases, based on which one can compute the magnitude gain
and the phase shift of the sensor dynamics (fi'om the mechanical input to the Sensing out-
put) at that particular frequency. Repeat this process for other excitation frequencies that
are available from the experimental setup (1-20 Hz), which produces the empirical Bode
plots for the fi'equency response. Extra care is required to ensure consistent experimental
conditions at different frequencies. This is because the sensing behavior of an IPMC de-
pends on the hydration level of the material, and the latter decreases over time when the
sensor is operated in air. To handle this problem, the IPMC sample is soaked in water for
several minutes to get fully hydrated, and is then mechanically excited in air for ten minutes
to get rid of excessive water before collecting data for each frequency.
One can then tune the unknown parameters of the sensor model H; (s) to fit the empiri-
cal frequency response. In particular, H2 ( j27r f) predicts the magnitude and phase response
of the sensor at frequency f, and it is a nonlinear function of the parameters. The Matlab
fimction f minsearch can be used to find the parameters that minimize the squared error
between the empirical frequency response and the model prediction. The identified param-
eters are listed in Table 3.2, which are close to the values reported in literature [64, 32].
For independent verification of the proposed model, the identified parameters will be used
in predicting behaviors of other IPMC sensors with different dimensions, as will be seen in
Section 3.3.3.
59
Table 3.2: Identified parameters through curve-fitting.
d C“ K.» ao
3.32x10‘11m2/s 1091movm3 1.88x10’3F/m 104J/C
3.3.3 Model verification
Model verification will be conducted on three aspects. First, it will be shown that the model
considering the surface resistance is more accurate than the model ignoring the resistance,
by comparing them with the measured frequency response of an IPMC sensor. Second, the
geometric scalability of the proposed model will be confirmed by the agreement between
model predictions and experimental results for IPMC sensors with different dimensions.
Third, the performance of the reduced model will be illustrated through its prediction of
the time-domain sensing responses under a damped, oscillatory excitation and a step defor-
mation, respectively.
Effect of surface resistance.
In order to examine the difference between the models H1 (s) and H2(s), their model pa-
rameters were identified separately through the nonlinear fitting process described in Sec-
tion 3.3.2. The experimental data were obtained for an IPMC sensor with dimensions
22 x 7 x 0.36 mm3. Fig. 3.3 compares the predicted frequency response (both magnitude
and phase) by each model with the measured frequency response. Both models show good
agreement with the experimental data on the magnitude plot. On the phase plot, however,
it is clear that the model considering the surface resistance shows better agreement than the
one ignoring the resistance. This indicates that the model incorporating the surface resis-
tance is more effective in capturing the sensing dynamics of IPMC, and thus it will be used
for the remainder of this chapter.
60
— - - Model without surface resistance 5
Magnitude (dB)
-60 Model with surface resistance 4
i E E ': F I E + Experimental data 3
-70, JI_LLI,1 ........ 1 ..... ;"';'"';"i";";”;'i' .......... L' ..... in?
10° 101
Phase (degree)
Frequency (Hz)
Figure 3.3: Performance of the models with and without consideration of surface resistance.
Geometric scalability of the dynamic model.
Three samples with different dimensions (see Table 3.3) were cut fiom one IPMC sheet,
and were labeled as Big, Slim, and Short for ease of referencing. The model parameters
were first identified for the Slim sample, as discussed in Section 3.3.2. Without re-tuning,
these parameters (except geometric dimensions) were plugged into (3.38), i.e., the model
H2 (5), for predicting the frequency response for the Big and Short samples.
Table 3.3: Dimensions of three IPMC samples used for verification of model scalability.
IPMC beam length (mm) width (mm) thickness (um)
Big 22 14 360
Slim 22 7 360
Short 1 1 7 360
Fig. 3.4 (a) shows the Bode plots of the fiequency responses for the Slim and Big sam-
ples. It can be seen that for both samples, good agreement between the model prediction
61
and the experimental data is achieved. Furthermore, since the two samples differ only in
width, the model (3.38) predicts that their magnitude responses will differ by 2010g2 = 6
dB uniformly in frequency, while their phase responses will be the same. Both predictions
are confirmed in the figure: the experimentally measured magnitude responses are parallel
to each other with a difference about 6 dB, and the measured phase responses overlap well.
Fig. 3.4 (b) compares the frequency responses of the Slim and Short samples. Rea-
sonable match between the model predictions and the empirical curves is again achieved
for both samples. The figure also indicates the need to incorporate the surface resistance
in modeling. Since the two samples differ only in length, the model ignoring the surface
resistance (3.18) will predict that the magnitude responses of the two samples would differ
just by a constant while their phase responses would be identical. But the empirical mag-
nitude curve for the Slim sample rises with frequency by approximately 14 dB per decade,
while that for the Short sample rises by roughly 18 dB per decade. Moreover, the empirical
phase curves clearly do not overlap. All these subtle trends, however, are captured well by
the model considering the surface resistance, as can be seen in Fig. 3.4.
Verification of the reduced model.
Experiments were further conducted to verify the effectiveness of the model reduction ap-
proach presented in Section 3.2. Two mechanical stimuli, which were different fi'om pe-
riodic signals, were used to demonstrate the wide applicability of the proposed model. In
the first experiment the cantilevered Slim IPMC sample was allowed to freely vibrate upon
an initial perturbation on the tip. In the second experiment the tip of the cantilevered Big
sample was subject to a step displacement and then held there. A laser displacement sensor
(OADM 2016441/S14F, Baumer Electric) was used to record the tip displacement trajec-
tory w(t). The fourth-order reduced model (3.48) was adopted to predict the short-circuit
current sensing response for each case, where the model parameters from the identification
62
Magnitude (dB)
— Model prediction (big)
+ Experimental data (big)
g , ;_ - - - Model prediction (slim)
g, - e — Experimental data (slim)
3 . ' g..: ...............
33
m 40 .. ........
.c
n- 20
10°
Magnitude (dB)
—- Model prediction (short)
+ Experimental data (short)
.1.
O
1
’5,‘ >- - _ ~l>~ j 3 - - - Model prediction (slim)
“a” 30 .. . I j 2. '- .Z — B» — Experimental data (slim)
q) " s . : : : : : t
g _ ........... I.. *5... 1...: ..............
a: 60 4"“
g 40 ...... ‘ ’ '. ,
.1:
0' 20 . ~ .
10° 101
Frequency (Hz)
(b)
Figure 3.4: Frequency responses of the sensing dynamics. (a) For the Big and Slim samples;
(b) for the Slim and Short Samples.
experiment in Section 3.3.2 were used directly.
Fig. 3.5 compares the predicted sensing response with the experimental measurement
63
under the damped, oscillatory stimulus. The beam has a natural frequency of about 30 Hz,
which is outside the frequency range of the signals used in parameter identification, yet the
reduced model is able to predict the sensor output well. Fig. 3.6 shows the model prediction
of the sensing response against the experimental data for the case of a step tip-deformation.
It can be seen that satisfactory agreement between the predicted and the measured curves
is again achieved.
3.3.4 Impact of hydration level on sensing response
Finally, the proposed dynamic model is explored to gain insight into the dependence of
IPMC sensing response on the hydration level of the sample. Such dependence is clearly
seen from the experimental curve in Fig. 3.7. In this experiment the Slim IPMC sample was
first soaked in water, and then subject to continuous, 6 Hz periodic mechanical excitation
in air. Data (both the stimulus and the sensor output) were collected for about 40 minutes.
The fi'equency response of the sensor at 6 Hz at each minute was then extracted from the
data, including the magnitude gain and the phase shift. The evolution of this response is
plotted in Fig. 3.7 (note the horizontal axis represents time instead of fiequency), showing
that the sensor output gets weaker over time.
The time-varying response is believed to arise from the water evaporation of the IPMC
sample. In order to correlate this phenomenon with the proposed model (3.38), we assume
that all parameters except the diffusion coefficient at are constant over time. Taking the
values identified in Section 3.3.2 for these fixed parameters, we identify the evolving value
of d by fitting the measured magnitude and phase response at each minute. The dashed
line curves in Fig. 3.7 show the results of model fitting with a time—dependent d. The
time-trajectory of the identified at is plotted in Fig. 3.8, and its decaying trend is consistent
with the decreasing hydration level. This might suggest that the hydration level impacts the
sensing behavior of an IPMC through its influence on the ion diffusivity. However, since d
cannot be measured directly to confirm the identified trajectory in Fig. 3.8, more research
64
P
u:
-o.5 >-
Displacement (mm)
o 0.1 0.2 0.3 0.4 0.5
Time (s)
(a)
40 T r 1
_ —— Experimental data
I_ . _ . . . _ _:---Modelprediction
10k ,
-10......-‘... .. _, i...,
Sensing current (uA)
0.1 0.2 0.3 0.4 0.5
Time (s)
(b)
Figure 3.5: The sensing response of the Slim sample under a decaying, oscillatory mechan-
ical stimulus. (a) The tip displacement trajectory; (b) prediction of the sensing response
versus experimental measurement.
is needed before a conclusive statement can be made.
65
Displacement (mm)
o 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Time (s)
(a)
70 . r
, “ —Experimentaldata
' , \ . ---Modelprediction
60» .. . . ~ . .
3 5°’
'3' 4O
'5
o 30»
8’
'5 20-
c:
a:
CD 10_...
0 __L__
_10 1 1 i 1‘; i_ 1
0 0.05 0.1 0.15 0. 0.2 0.3 0.35
Time (s)
(b)
Figure 3.6: The sensing response of the Big sample under a step stimulus. (a) The tip
displacement trajectory; (b) prediction of the sensing response versus experimental mea-
surement.
3.4 Chapter Summary
In this chapter, a dynamic model for IPMC sensors has been developed by solving the
physics-goveming PDE analytically in the Laplace domain. The model accommodates
66
a ; : : '--Modelfitting
E-40~~ . ‘ ,.: ....... g -+-—Expen‘mentaldata..
.8 I I f i
3 _45_ ..... .1
:2'.‘
5
to -50 .
2 V . . ; V
_55 1 1 1 J m 1 1
5 10 15 20 25 30 35 40 45
N
O
888
Phase (degree)
8
N
O
01
10 15 20 25 30 35 4O 45
Time (min)
Figure 3.7: The sensing response at 6 Hz for the Slim IPMC sample over time.
1
4-5x 0 v ' 1 1 7 f .'
4...
A 3.5
E
N
E 3
‘5
E 2.5
0
8 2
C
.9
w 1.5
5
D 1
0.5 *1
0 i '1 '1 i 1 i i
5 10 15 20 25 30 35 40 45
Time(s)
Figure 3.8: Identified diffusion coefficient versus time.
the surface electrode resistance in an integrative manner. The mechanical stimulus enters
as a boundary condition for the PDE, and the sensing output is related to the mechani-
67
cal input linearly. This leads to a compact, explicit, transfer-function representation of
the physics-based model, which can be further reduced to low-order models for real-time
sensing and feedback control purposes. A number of experimental results were presented
to demonstrate the geometric scalability of the model, as well as its applicability to arbi-
trary mechanical inputs. Due to the physical nature of the model, the agreement between
model predictions and experimental results also provides insight into the underlying sens-
ing mechanisms of IPMC materials.
68
Chapter 4
Modeling of Biomimetic Robotic Fish
Propelled by An Ionic Polymer-Metal
Composite Caudal Fin
This chapter is organized as follows. The robotic fish is described in Section 4.1. The
proposed model is presented in Section 4.2. Experimental results on model validation are
presented in Section 4.3. Finally, concluding remarks are provided in Section 4.4.
4.1 Description of IPMC-propelled Robotic Fish
Fig. 4.1 shows a prototype of the robotic fish, which is an upgraded version from that re-
ported in [86]. The fish is designed to be fully autonomous and serve as a mobile, aquatic
sensing platform. It consists of a rigid body and an IPMC caudal fin. Two gold-coated
copper electrodes are used for IPMC to reduce corrosion of the electrodes in water. Cor-
rosion of the electrodes results in high resistance of the contact, which would reduce the
actuation performance of the IPMC tail and consume more electrical power. The IPMC
actuator is further covered by a passive plastic fin to enhance propulsion. The rigid shell
69
of the fish was custom-made to reduce the wetted surface while having enough interior
room to house rechargeable batteries and various electronic components for control, sens-
ing, wireless communication, and navigation. All of these components are contained in a
water-proof packaging with necessary wires and pins exposed for charging batteries and
driving IPMC actuator. Mthout the tail, the fish is about 20 cm in length and 5.7 cm in
diameter. Total volume is about 180 cm3. The fish body is in a water drop shape, which is
expected to lead to good hydrodynamic efficiency. The Reynolds number of the swimming
robotic fish is given by UDo / v, where U is the speed of the robot, Do is the body diameter,
and v = 10‘6 mz/s is the kinematic viscosity of water. With a speed U of about 0.02 m/s
(see Section 4.3.3), the Reynolds number of the robot is at the order of 1000. The total
weight of the robotic fish is about 290 g. The shape and configuration of the robot put it
into the category of carangiform fish.
4.2 Model
In this section we first review Lighthill’s theory on elongated-body propulsion (Section 4.2.1).
IPMC beam dynamics in fluid is discussed next, considering general force and moment in-
puts (Section 4.2.2). This is followed by detailed consideration of actuation-induced bend-
ing moment in the model, as well as the load contribution to the IPMC beam from the
passive fin (Section 4.2.3). Finally, the model for computing the speed of IPMC-propelled
robotic fish is obtained by merging Lighthill’s theory and the hybrid tail dynamics (Sec-
tion 4.2.4).
4.2.1 Lighthill’s theory of elongated-body propulsion
A body is considered elongated if its cross-sectional area changes slowly along its length.
The robotic fish described in Section 4.1 is thus elongated and Lighthill’s theory [56] ap-
plies.
70
Side View Passive fin
w(z, t)
Top View
I'1
Figure 4.1: (a) Schematic of the robotic fish; (b) prototype of the robotic fish.
71
Suppose that the tail is bending periodically with the bending displacement at z denoted
by w(z,t). Refer to Fig. 4.1(a) for notation. At the steady state, the fish will achieve
a periodic, forward motion with some mean speed U. In the discussion here, the word
“mean” refers to the average over one period. The mean thrust 7 produced by the tail can
T: [_((a_;_>)(9_;_2))] (.1)
where z = L denotes the end of tail, (-) denotes the mean value, m is the virtual mass density
be calculated as
at z = L, expressed as
1
m = 2iusfipwp. (4.2)
In (4.2), Sc is the width of the tail at the end 2 = L, pw is the fluid density, and B is a non-
dimensional parameter close to 1. Eq. (4.1) indicates that the mean thrust depends only on
BW
82
flow conditions, will experience a drag force FD:
the lateral velocity 65—? and the slope at the tail end. A cruising fish, under inviscid
_ CDpwUZS
F
D 2 1
(4.3)
where S is the wetted surface area, and CD is the drag coefficient. At the steady state, the
mean thrust T is balanced by the drag FD, from which one can solve the cruising speed U:
m(8w§z,t))2
t
\ Cbpws.m(agg2)2
Since the speed of the fish is related to the lateral velocity and the slope of the trailing edge,
(4.4)
.. z=L
one needs to fully understand the actuation dynamics of the tail.
72
4.2.2 IPMC beam dynamics in fluid
In order to obtain the full actuation model of IPMC, we start with a fourth-order PDE for
the dynamic deflection function w(z, t) [24]
82w(z,t)
4
a w(z,t) +C19w(z,t) 7’2— 2““), (4.5)
Y1 324 at
+ pmA
where Y, I, C, pm, and A denote the effective Young’s modulus, the area moment of inertia,
the internal damping ratio, the density, and the cross-sectional area of the IPMC beam,
respectively, and f (z, t) is the distributed force density acting on the beam.
Converting (4.5) into the Laplace domain, we get
84w (2, s)
"—324
+ Csw(z,s) + pmAszw (z, s) = F (2,5). (4.6)
The force on the beam consists of two components, the hydrodynamic force thdm from
water, and the driving force Fm,e due to the actuation of IPMC:
F(Z,S) =thdro (ZaS)+Fdrive (zas)- (4-7)
The hydrodynamic force acting on the IPMC beam can be expressed as [76]:
11...... (2,3) = —pw§W2s2r1(w)w(z,s), o < z < L, (4.8)
W is the width of the IPMC beam, F1(w) is the hydrodynamic function for the IPMC
beam subject to an oscillation with radial frequency a), and pw is the density of fluid. The
hydrodynamic function for a rectangular beam can be represented as [76]:
1+
n (w) = Q(R.) (4.9)
WEKO ("i iRe)
73
4iK1 (—i 1R.) ]
where the Reynolds number
p... W 2 a)
R =
e 4 n
3
K0 and K1 are modified Bessel functions of the third type, Q(Re) is the correction function
associated with the rectangular beam cross-section [76], n is the viscosity of fluid, and W
is the width of the IPMC beam.
With (4.7) and (4.8), the beam dynamics equation (4.6) can be written as:
84w (2,3)
Y1m824
+ Csw(2,s) + (Hm + mdF1)szw (2,3) = Fdn've (z,s), (4.10)
where rm; 2 ng W 2 is the added mass and pm 2 pmA is the mass of IPMC per unit length.
Under harmonic oscillation with frequency a), we can denote
“v : Hm+de€(r1), (4-11)
C. = C—mdem(F1), (4.12)
where 1.1,, is the equivalent mass of IPMC per unit length in water, and Cv is the equivalent
damping coefficient of IPMC in water. Re(-) and Im(-) are the functions that get the real
part and the imaginary part from a complex value, respectively. Eq. (4.12) means that the
damping of IPMC vibration in water includes both the internal damping in IPMC and the
frequency-dependent external damping caused by fluid. With (4.11) and (4.12), Eq. (4.10)
can be written as [16]:
34w (2,3)
-—a—Z4—— + CvSW(Z,S) + Liv-92W (2,3) = Fdrive (Z,S). (4'13)
Y1
According to the mode analysis method, we can express the solution to (4.13) as the
74
sum of different modes [5 7]:
co
W(z,S) = 2 (Pi (2)611 (S), (4.14)
i=1
where ¢,- (2) is the beam shape for the i-th mode, and q,-(s) is the corresponding generalized
coordinate. The mode shape ¢,(z) takes the form
(p,- (z) = cosh (1,2) — cos (1,2) — [3,- (sinh (1,2) — sin (1,2)) , (4.15)
where 1,- can be obtained by solving
1 + cos (1,L) cosh (1,L) = 0,
and
fi' _ sinh (1.3L) — sin (14L)
' — cosh (1,L) + cos (1,L)’
The generalized coordinate q,-(s) can be represented as
611(5) =fi(S)Qi(S)a (416)
where f,-(s) is the generalized force,
l
1' = , 4.17
Q (S) 32 + 25,015 + (OS-2 ( )
and the natural frequency a), and the damping ratio .5,- for the i-th mode are
C? Y!
a), = —' ————, 4.18
L2 “V(wi) ( )
. = __ 4.19
75
C, = 1,L. Noting that F1(w) is almost a constant value in the frequency region around (1),,
one can consider uv(a),) as a constant in Eq. (4.18). So a), can be obtained approximately.
Then with (1),, 5,- can be obtained from Eq. (4.19). The generalized force f,(s) is obtained
from Fdn'vei
L
f.- (s) = 114“. [Fave (2.5)101(Z)dz,
0
where M,- is the generalized mass
L
M (5) = film-2 (2)612 : “vL- (4'20)
0
The next step is to derive the generalized force f,(s) from the moment generated by
IPMC actuation, and from the hydrodynamic force acting on the passive fin but transmitted
to the IPMC beam.
4.2.3 Actuation model of the tail
In our earlier work [21], we fiilly investigated the electrical dynamics of IPMC to obtain
the moment generated within IPMC, but there the beam dynamics in water was not con-
sidered. In the following we will incorporate both electrical dynamics and hydrodynamic
interactions into a firll actuation model for IPMC hybrid tail in water.
The ion movement inside an IPMC, under the influence of an applied electric field, leads
to a distribution of net charge density along the thickness direction of IPMC. A physics-
based model for IPMC proposed by Nemat-Nasser and Li [64] relates the actuation-induced
(axial) stress proportionally to the charge density, through electromechanical coupling.
Variation of the actuation-induced stress along the thickness direction thus results in a
bending moment at each point along the length direction. Our work on physics-based,
control-oriented modeling of IPMC actuators [21] further incorporates the effect of dis-
tributed surface resistance. The latter leads to non-uniform potential difference along the
76
length direction, which in turn leads to actuation-induced bending moment that varies along
the length direction, referred to as distributed bending moment in this section.
As shown in Section 2.3, with distributed surface resistance, we can relate the actuation-
induced internal bending moment MmMc (2, s) at point z to the actuation voltage V(s) by an
infinite-dimensional transfer function [21]:
aOWKke(y(S) -—tanh(y(s)))
”PMC(Z’S)= (sy1s)+Ktanh(y>)
cosh ((/B(s)2) — sinh ((/B(s)z) tanh ((/B(s)L)
' 1+ 9 VW’
r2 (S)
(4.21)
with
A Wkesy(s)(s+K)
9(9) ‘ h = Minions). (4.23)
The rationale of this replacement is that these components can generate the same bending
moment as MIpMc(z,s). See Appendix B.1 for the details of this justification. With (4.21),
it can be verified that Fc(L, s) E 0. Then the generalized force can be obtained as [57]:
L
fli(S) = $1 ([11:05) (Pi (4)61Z+11’1(I«S)‘P.f (U) - (4-24)
0
Fig. 4.2 shows that the original moment is replaced by a distributed force density and a
concentrated moment. With the above replacement, we can derive the models for the
b1
11--
---------r
Figure 4.2: The original moment in (a) is replaced by a distributed force density and a
concentrated bending moment in (b).
78
IPMC tail only case and the hybrid tail case.
With IPMC tail only
With (4.22) and (4.23), the generalized force (4.24) can be written as
f1: (S) = ”/1 (S) V(S)a (425)
where
1 (a—b)(aL+bL—cL—dL)
111(5) = “'—
f 21% 431(0- 1?) (0L - bL +jCL -J'dL)
aoWKke (7(s) — tanh (1(0)) «p; (L)
M(sy(s)+Ktanh()/(s))) (1+r29(s))cosh(cL)’
(4.26)
and
a = aoWKke(r(s) —tanh(7(s))) B(s) (4 27)
(SY(S)+Ktanh(Y(S))) 1+r29 (S)’ '
b = atanh(\/B(s)L), c = \/B(s), (4.28)
01 = 3mh(c(:::li)L), b1 = Slnh(c(C_-A:~i)L)’ (4.29)
c, = sinh(c(::ii«-)L), d, = Sinh (5:12)“. (4.30)
See Appendix B.2 for the derivation of H f,- (s).
From (4.14), one can then get the transfer function H1(z,s) relating w(z,s) to V(s):
= 2 (p,(z)Hf,-(s)Q,-(s). (4.31)
We can also derive the transfer function H ”(2, 5) relating the slope of the beam 8w 2’ S
79
to the input voltage V(s):
aw(z,s) 0°
Hides): 32 = Z Hf.-(s)Q.-(s). (4.32)
V(s) i=1
Hybrid Tail
From (4.1) and (4.2), the tail width S, at the end has a significant impact on the speed U.
One could increase S, by simply using a wider IPMC beam. Due to the IPMC actuation
mechanism, however, a too wide beam (i.e., plate) will produce curling instead of bending
motion and is thus not desirable. Therefore, it has been chosen to increase the edge width
by attaching a passive plastic piece, as illustrated in Fig. 4.3. While such a hybrid tail
is expected to increase the thrust, one has to also consider that the extra hydrodynamic
force on the passive fin adds to the load of IPMC and may reduce the bending amplitude.
Therefore, it is necessary to model these interactions carefully.
Figure 4.3: Illustration of an IPMC beam with a passive fin. The lower schematic shows
the definitions of dimensions.
The hydrodynamic force acting on the passive fin can be written as [76]
ft... (2.5) = {0.st (z)2 r2(w)w(z,s), £0 < z < 1., (4.33)
80
where 13(0)) is the hydrodynamic function of the passive fin. Note that the hydrodynamic
force acting on the active IPMC beam has been incorporated in (4.10), so only the hydro-
dynamic force on the passive fin needs to be considered here. Since the passive fin used is
very light, its inertial mass is negligible compared to the propelled virtual fluid mass and
is thus ignored in the analysis here. Considering that the passive fin is rigid compared to
IPMC, its width b(z) and deflection w(z,s) can be expressed as
b —b
b (2) = ‘ ° (2 —Lo) + be, (434)
L1 —L0
aw(Lo,S)
<92
W(2,8) = W(L0,S) + (z-Lo), (435)
where b0, b1, L, L0, L1 are as defined in Fig. 4.3. Then one can calculate the moment
introduced by the passive fin: for L0 g 2 g L1,
M... (2,.) = [12.1 (as) (r—zwr
L0
L1 L1
= [1.105) (T—Lo)dT+(Lo —2) [16.1 (r.s)d1.
L0 L0
(4.36)
If we define
L1 L1
Mtail (S) = /f1a11(T,S) (T—Lo)d’€, 1”111110) = fftail(TaS)dT, (437)
L0 L0
then (4.36) can be written as
Mfin (2,5) = Mraii (S) +Ftail (S) (L0 - Z)- (438)
Fig. 4.4 shows the forces and moments acting on the hybrid tail.
81
.Y---
Figure 4.4: Forces and moments acting on the hybrid tail.
One can get the generalized force as:
Lo
fzi (S) = 1117, (de(ZaS)¢i(Z)dZ+k.+)l sin(wt + 411042)), (4.49)
3%9121 = 4. 111.04»! sin(wt + 411.041». (4.50)
where 2(.) denotes the phase angle and H (s) and Hd(s) are the transfer functions relating
the bending displacement w(L,s) and the slope w’ (L,s) to the voltage input V(s), respec-
tively, which can be obtained as in Section 4.2.3. From (4.4), one can then obtain the
steady-state speed U of the robotic fish under the actuation voltage V(t) 2 Am sin(a)t) as
_ \/ 21243.4)2 IHl2
chpWS+ mAi. 1111.041)? '
(4.51)
One can easily extend (4.51) to periodic signals of other forms. For instance, the prototype
in Fig. 4.1(b) uses square-wave voltage signals for ease of implementation. To derive the
speed U, we can write out the Fourier series of a square wave. Then the velocity of the fish
actuated under a square wave voltage with amplitude A m can then be obtained as
8w2A2. °° ,
m' 7:2 m 2 |H(J"0>)|2
n=1,3,5---
U: 8A2 00 H ’a) 2. (4.52)
n
\ CDPWS'i'm'7én 1% 5 J—dLh—z—ZL
n: , , ..-
Derivation of (4.52) is omitted here due to the space limitation.
84
4.3 Experimental Verification and Parameter Identifica-
tion
In this section, three different types of experiments have been carded out for model iden-
tification and validation: l) drag coefficient identification (Section 4.3.1); 2) identification
and validation of the actuation model for IPMC underwater with and without the passive
fin (Section 4.3.2); and finally, 3) validation of the model for fish motion with different tail
dimensions (Section 4.3.3).
4.3.1 Drag coefficient identification
The most important parameter related to the fish body is the drag coefficient CD, which
depends on the Reynolds number, the fitness ratio of the body, and the properties of the
fish surface and fluid. In order to identify CD, the fish was pulled with different velocities,
and metric spring scales were used to measure the drag force FD. With the measured drag
force, velocity, and surface area of the fish, the drag coefficient CD was calculated from
(4.3). Fig. 4.5 illustrates the experimental setup for drag force measurement.
‘ DC motor
K Metric spring scale
fish
Water
Figure 4.5: Experimental setup for the drag force measurement.
85
Through drag force measurement, one can get the plot of the drag force versus velocity.
Based on (4.3), one can fit the experimental data with simulation data through the least
squares method to identify the drag coefficient. Fig. 4.6 shows the drag force versus the
velocity of the fish. Table 4.1 shows the parameters related to the drag force.
0.5 . 1 r
0.45 i- 0 I
0 Experimental data I ’
0.4 - - - Simulation data ,
08 O ,
O
0.35 - I
I
A o 3 '
Z, i 08/ O
I
g o 25 S
o O ’
LI. 0.2 ’ ’0 O
I
0 15 ’0’ 8 00 1
I
0.1 - ’ 1
I I I O 0 O
0.05 - Q ’
«B
0 M 1 1 1 1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Velocity (m/s)
Figure 4.6: Drag force versus velocity of the fish.
Table 4.1: Parameters related to the drag force.
S pw CD
218 x10’4m2 1000 kg/m3 0.12
4.3.2 Fish tail model verification
To investigate the parameters related to the IPMC beam dynamics, the natural vibrations of
IPMC in water and in air were measured without an actuation voltage applied to the IPMC.
86
They were also used to verify Q (s) for the first mode vibration. To investigate the hydro-
dynamic effect of passive fin on the IPMC beam, the frequency responses of the tail subject
to voltage input were measured for both without and with the plastic fin. They were also
used to verify the actuation models of IPMC. Fig. 4.7 shows the schematic of experimental
setup. In the natural vibration testing, the tail was fixed in water and a mechanical impulse
was applied at the tip to make the beam vibrate. The first-mode vibration was measured by
a laser sensor (OADM 2016441/Sl4F, Baumer Electric). In the fiequency response testing,
the fish tail was fixed in the water by a frame arm and a sequence of sinusoid voltages with
amplitude 3.3 V and frequency ranging from 0.05 Hz to 10 Hz were applied to the IPMC.
The lateral displacement of the IPMC beam was captured by a laser sensor and actuation
voltage was measured by a dSpace system (DS1104, dSPACE).
Contact
electrodes &
damn—2:"
*—— IPMC Ta11 Computer &
dSPACE
wate_r_, "'— Tank l
.6— Laser 1
sensor 1 ;
r
Figure 4.7: Experimental setup for identification and verification of IPMC actuation model
in water.
Beam dynamics identification
Since the actuation bandwidth of IPMC actuators is relatively low (up to a few Hz), it
suflices to consider the first mode of the beam motion. The parameters related to the beam
dynamics can be identified through passive vibration tests of IPMC in water. The first mode
87
vibration related to the step response of the second-order system Q1 (s) (see Eq. (4.17)) is
y(t) =y(0) e'glwltcos (wltfl l —— 512).
In the experiment, we tapped the tip of the cantilevered IPMC beam (submerged in water)
and recorded the tip trajectory with the laser sensor as the beam underwent passive, damped
oscillations. Fig 4.8 shows both the simulation data and the experimental data on the tip
displacement of the vibrating IPMC beam, where the beam dimensions were L = 23 mm,
W215 mm.
Experimental data
- - - Simulation data
O
l
Bending displacement (mm)
3:.
0 0.2 0.4 0.6 0.8 1
Time (s)
Figure 4.8: Vibration of IPMC beam in water.
From Fig. 4.8, the natural frequency and damping ratio in water were identified to be
001 = 54 rad/sec, £1 = 0.14. Hydrodynamic function of IPMC beam 11(0)) can be simu-
lated based on (4.9). The correction function (2(a)) for rectangular shape beam reported in
[76] is used in the simulation. Around the natural fiequency, one can pick Re(F1) = 1.07
and Im(T‘l) = 0.04. Based on (4.11), one can get 11,. Based on (4.18), (4.19), one can
88
obtain Y and C... Table 4.2 shows all the parameters related to the beam dynamics.
Table 4.2: Parameters in IPMC beam dynamics.
C. 11. W L
3.17 0.2 kg/m 15 mm 23 m
h C1 Y F;
11511111 1.8751 2.91x108Pa 1.07+0.04j
Fish tail model identification
In the fish tail model, some parameters can be directly measured, such as dimensions,
temperature, resistance, and density of IPMC. Some parameters are physical constants,
such as R, F, and pw. Since IC'AVl << 1 [64], we take 1 - C‘AV = 1. Some parameters,
such as kg, 010, r2, need to be identified through fitting the frequency responses with model
simulation, which was discussed in [21]. F2 can be identified through fitting the frequency
response of hybrid tail with simulation data. Table 4.3 shows the parameters related to the
electrical dynamics of IPMC. The dimensions of IPMC-only tail are shown in Table 4.2.
The dimensions of hybrid tail Tail 1 are shown in Table 4.3.3.
Table 4.3: Parameters related to the electrical dynamics of IPMC.
F T R c—
964870.161 300K 8.3143 7.1.1.11 1091:110va
r1 r2 d K},
2109/11. 0.04Q-m 5.39 x10-9m/s 2.48x10-5F/m
R (10
P
38 Q- m 0.08 J/C
The actuation model of IPMC with and without passive fin is verified. We applied
sinusoidal voltage signals with amplitude 3.3 V and different fiequencies to IPMC. Both
the voltage input and the bending displacement output at the tail tip were measured to obtain
the empirical frequency responses. In the case of an IPMC beam only, the displacement
89
measurement was made at the beam tip; in the case of a hybrid tail, the displacement was
measured at the tip of passive fin. In the simulation of the actuation models, only the first
mode was taken into account, because the frequencies used were below or close to the
first-mode resonant frequency. Fig. 4.9(a) compares the Bode plot of H1 (L,s) (Eq. (4.31)
in Section 4.2) with its empirical counterpart, and the agreement is good in both magnitude
and phase. The cut-ofl frequency is estimated to be about 8.6 Hz, which is consistent
with the IPMC beam’s natural frequency in water, as identified from the free vibration
experiment shown in Fig. 4.8. Fig. 4.9(b) compares the Bode plot of H3 (L1 ,5) (Eq. (4.48)
in Section 4.2) and the measured fi'equency response from voltage input to the tail tip
displacement for the hybrid tail. As can be seen in the figure, the cut-off frequency of
the hybrid tail is much lower than that of an IPMC alone. This can be explained by the
additional mass effect at the IPMC tip, introduced by the fluid pushed by the passive fin.
4.3.3 Speed model verification
To validate the speed model of the robotic fish, the velocities of the fish propelled by the
IPMC under square-wave voltage inputs with amplitude 3.3 V and different frequencies
were measured. In this experiment, the robotic fish swam freely in a tank marked with start
and finish lines, and a timer recorded the time it took for the fish to travel the designated
range after it reached the steady state. Fig. 4.10 shows a snapshot of fish swimming in the
tank.
The capability of the model in predicting cruising speed was verified for different op-
erating frequencies, for different tail dimensions. The speed model for square wave input
(Eq.(4.52)) was applied to the robotic fish as described in Section 4.1. In the simulation of
(4.52), we took the first three terms in each infinite series, which provided a good approx-
imation to the sum of infinite series. Four different hybrid tails were investigated, shown
in Table 4.3.3. The identified hydrodynamic firnctions F2 are shown for Tail 1 and T ai12 in
Fig 4.11. It can be seen that, while the hydrodynamic functions are qualitatively close to
90
— i I U —*— Experimental data
-100 _ . . _ . - ' TSlmUlallOn data :
Magnitude (dB)
8
10'1 10° 101
3
9
U)
0
E
d)
m
(B
1:
'3- n 1 - .
10'1 10° 101
Frequency (Hz)
(a)
8
E
d)
‘O
P.
'E
U)
(U
2 a
10‘1 10°
A o l
11> _
9 2 9
8-100 +
B .
g :
,5 -200 . 3 -
c . .
n- i l I
10'1 10°
Frequency (Hz)
(b)
Figure 4.9: Validation of model for IPMC operating underwater. Shown in the figure are
the Bode plots for (a) model H1(L,s) without passive fin; and (b) model H2(Lo,s) with
passive fin.
each other, they are shape-dependent. The predicted speeds match the experimental data
well, as shown in Fig. 4.12 and Fig 4.13. Intuitively, within the actuation bandwidth of
91
Figure 4.10: Snapshot of robotic fish in swimming test.
IPMC, the speed achieved increases with the actuation frequency. As the frequency gets
relatively high, the bending amplitude of IPMC decreases. Thus for each tail, there is an
optimal frequency under which the fish reaches the highest speed. Both the optimal fi'e-
quency and the corresponding highest speed depend on the dimensions of both IPMC and
passive fin, which can be predicted by the speed model.
Table 4.4: ’ ' ' ' " ' ' variables).
Lo(mm) Mm!!!) W (Inn!) bo(mm) b1(mm) 00min)
Tail] 18 23 15 20 40 40
Tai12 18 23 15 20 50 30
Tai13 18 23 20 20 65 25
Tail4 18 23 20 20 50 30
4.4 Chapter Summary
In this chapter, the modeling of steady-state cruising motion was presented for an IPMC-
propelled robotic fish. The model incorporates both IPMC actuation dynamics and by-
drodynamic interactions, and it firrther considers the effect of a passive fin on the robot
92
5
For Tail1
4 - - - For Tail2
3
2
1 4L
-
------------------------
4 T
3..
N
[5" 2r
2
11—
o l l l
0 0.5 1 1.5 2
Frequency (Hz)
Figure 4.11: Identified F2 for Tail 1 and T ai12.
performance. The model was verified in experiments for robotic fish with different tail di-
mensions. The model will be useful for design and control of the robot to meet the tradeoff
between locomotion speed and energy consumption.
Although a focus of the modeling work here is to understand how the steady-state speed
of the robot depends on the fin design and actuation input, the approach to modeling IPMC
fins in underwater operation holds promise for understanding general motions and maneu-
vers of the robotic fish. We will extend the presented model to investigate steady turning
motion under periodic but asymmetric (left versus right) actuation of the IPMC, as well as
unsteady motions such as the acceleration and deceleration of the robot. Path planning and
control of the robotic fish will also be examined. In addition, we are exploring the use of
IPMC as flow sensors for robotic fish control.
We note that other ionic-type electroactive polymers, in particular, conjugated poly-
mers, have also been explored as propelling mechanisms for robotic fish [87, 4, 59]. The
93
0.025
0.02
E 0 015
g .
E"
o
o
a 0.01
>
0.005
0
0.025
0.02
E 0 015
g .
.é‘
8
a 0.01
>
0.005
0
Figure 4.12: Verification of motion model for the fish with Tail 1 and T ai12. (a) With Tail 1;
(b) with Tai12.
r I
- - - Simulation data
0 Experimental data
. or \
I .\
\
’ \
’ \
. r .
o
’ \
e I \
’ \
I
. I \0
I s
I s
I ‘ ‘
I ~ ~
l- I ~ ~
I ‘ ~ - _
I
I
I l l J
0 0.5 1 1.5
Frequency (Hz)
(a)
- - - Simulation data
0 Experimental data
’ -
_ I . ‘05 e
f \
I
’ \
I s
- I ‘
O I ‘
I ‘ o
, \
O I ‘
I ‘ ‘
b- ’ \
I x
I ‘ s
I ~
I
~I
l
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Frequency (Hz)
(b)
94
0.025 I T T T fl I l I I
- - - Simulation data
0 Experimental data
0.62. l
A . -
$ ’ ’ 0‘ "
E 0 015 " . I \ -1
V I I x
.Z‘ O ’ \ s
'6 I s
o I \
35 0.01 - I \ ‘ -
> ’ s
I s
7 ‘ s
S
‘ ~
0.005 '- § ‘4
0 l l l l l l 1 l l
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency (Hz)
(a)
0.025 I I I f I - 1* l l l
- - - Simulation data
0 Experimental data
0
0.02 - , 6 — ‘ v 5
, s
I V
I \
1;; I 0 \
\ b I \ 5
E 0 015 ’ \
a? , ’ s
8 n \
TD 0.01 - , ’ t \ .
> I s ‘
I
I ‘ 4
I
0.005 ”I 7
I
o l l l l L l 1 l J
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Frequency (Hz)
03)
Figure 4.13: Verification of motion model for the fish with T ail 3 and T ail4. (a) With T ail 3;
(b) with T ail4.
95
focus of the current chapter is on incorporating both IPMC actuation dynamics and hy-
drodynamics into modeling. The comparison of IPMC-enabled robotic fish with those of
conjugated polymer-enabled ones is outside of the scope of this work, although a clear
distinction is that a conjugated polymer fin needs either to be encapsulated or to work in
electrolyte for long-term operation [59]. However, the presented approach to coupling actu-
ation and hydrodynamic effects can be potentially extended to conjugated polymer-enabled
robotic fish by using corresponding actuation models, e.g., [31].
96
Chapter 5
Nonlinear Control-oriented Model for
Ionic Polymer-Metal Composite
This chapter is organized as follows. The governing nonlinear PDE is reviewed in Sec-
tion 5.1. Section 5.2 shows numerical and analytical analysis of the nonlinear PDE at the
steady state. In Section 5.3, a nonlinear circuit model is introduced to capture the electrical
dynamics of IPMC. The derivation of curvature output of IPMC and a nonlinear control-
oriented model are also shown in Section 5.3. Experimental validation of the proposed
model is presented in Section 5.4. Finally, concluding remarks are provided in Section 5.5.
5.1 Governing Nonlinear PDE
Many papers [64, 63, 33] have mentioned that the nonlinear term Rkfi-(l — C‘AV)—aa% -E
can be ignored in the flux (2.9), based on the assumption
p(x) =ke%§ < 8, decrease x0 = x0 — 81
99
and go to Step 2; ifE(—h) - E(h) < —8, increase x0 = x0 + £1 and go to Step 2;
Step 4: Do integration — ffhE (x) dx to get ¢(x) with ¢(—h) = 0;
Step 5: If |¢(h) — V| < 82, then go to Step 6; If ¢(h) — V > 82, then increase E0 2
E0 +83 and go to Step 1; If¢(h) — V < —£2, then decrease E0 2 E0 — £3 and go to Step 1;
Step 6: Calculate p (x) = KeE’ (x) and do integration Q = f: p (x) de, then stop.
0
In the steps above, 8, £1, 82, 82 are small positive constants, S = W L is the surface area
of IPMC. All the physical parameters in the PDE are listed in Table 5.1 in Section 5.4.
Fig. 5.1, Fig. 5.2, and Fig. 5.3 show the nrunerical simulation results when V = 2.61V.
Fig 5.1 shows the asymmetric charge distribution along the thickness direction. Two inset
figures show the details at the turning points of the curve. The negative charge density near
the anode approaches the saturation value C ‘F . The charge density distribution also shows
that (5.1) will not hold in the region close to the boundaries. In other words, the linear PDE
will not hold when a high voltage is applied.
Numerical solution offers us insight into the charge distribution, electrical field and
electrical potential along the thickness direction for a given step voltage, but it does not
provide us an overall picture of the induced charge versus applied voltage. Moreover, the
numerical solution takes recursive steps to find proper initial conditions, and requires much
computation which is not practical for control purposes. An analytical solution is practical
in real-time implementation. It is also the starting point in the derivation of nonlinear
capacitance of IPMC.
5.2.2 Analytical solution of the PDE at the steady state
A A ,
Define y = E and p = E . Eq. (5.4) becomes
pdp
(up + b)
= ydy. (5.7)
100
x10
2.5 . . .
—- Simulation data
- - - Bound of negative charge density
2 L
”A
E
9, 1.5
7
a? X 10 x 107
5 1 0 8
3 6
or _
0.5 -
0 ~10 .. .. - 2
_ _ 0
0 18° ‘60 178 180
_[. ........... .
-150 -100 -50 0 50 100 150
x (11 m)
.11 A
N -§
I I
E (V/m)
0.8 ~
0.6
T
0.4 -
I
0.2
t - .l
-150 -100 -50 0 50 100 150
X(um)
Figure 5.2: Numerical solution of the nonlinear ODE for V = 2.61V: Electrical field.
101
-150 -100 -50 0 50 100 150
X(um)
Figure 5.3: Numerical solution of the nonlinear ODE for V = 2.61V: Electrical potential.
We integrate both sides of (5.7). On the left-hand side we integrate from p(xo) to p, while
on the right-hand side, we integrate from E (x0) to y:
p b a _1 E(xo)2
2'351”(3P+1)—§y2‘ 2 ° (5'8)
Let
b
yég(p)=\/2(§—;ln(%p+l))+E(xo)2. (5.9)
Since it is difficult to get an explicit function of p = g‘1 (y) , one cannot continue to solve
the ODE equation to get an explicit firnction E (x). We will take a systems perspective to
solve this problem. What we are really concerned about is how the total charge is ana-
lytically related to the input voltage, so it is not necessary to know the explicit function
E (x).
102
By integrating both sides of (5.3) fi'om x = —h to x = h, we get
h h h
E” (x) dx — aE' (x)E (x)dx — bE (x)dx = 0. (5.10)
l l 1
With (5.6), Eq. (5.10) can be written as
E’(h)—E’(—h)—g(E2(h)—E2(—h)) —bV=O. (5.11)
From (5.5) and (5.11), one can get
V = i—(E’M) —E’(—h)). (5.12)
The total charge Q can be obtained by integrating p (x) from x = x0 to x = h, where p (x0) =
0:
h
/ p(x)de= Q= (E (h) —E(xo))5' 0, there exist two roots (p1 , p2) for (5. 1 6) such that —5 < p1 < 0 and
p2 > 0. Furthermore, if k =2 0, then pl 2 p2 = 0.
See Appendix CI for the proof of Proposition 1.
In order to get the mapping function fi'om V to Q, we need find out how k is related
to the distance of two roots I p1 — pzl. As shown in Fig 5.4, the roots of (5.16) are the
intersection points (p1 and p2) of the following two curves:
0 = %m(gp+1)=f(p),
n = p 1:16.).
a
Figure 5.4: Illustration of solving for p1 and p2.
. . . . . b b .
From Fig. 5.4, the negatrve root p1 w1ll never hit the line p = —— because p = —; 1s
a
the asymptote of the logarithmic function of n = ln(-:- p + 1). The physical explanation of
104
this is the following. Since the negative ions cannot move and the negative ion density are
uniform in IPMC, % = C’F is the bound of the negative charge density. So p > —2
implies that the layer of depleted positive charges will not fornr, although the positive
charge density can be very close to zero.
Finally, we can write k in terms of the voltage input V
_ F V ,V>0
k=r(V)-§- ( ) (5.17)
0,V=0
where
A b aV aV
and 1" is continuous at V = 0. See Appendix C.2 for the derivation of (5.17). With (5.15)
and (5.17), one can get the total charge Q as a firnction of V:
Q=S1ce 2F(V). (5.19)
Note that the nonlinear mapping function from input voltage is similar to the one reported
in [73], which Porfiri obtained by solving the nonlinear PDE with matched asymptotic
expansions.
When V —> 0, one can approximate F(V) using its Taylor series expansion around V 2
0.
_ V 2
l"( V) 2 fléL. (5.20)
VS
Then Q :2: \fi) 2 K8 , which is consistent with the charge generated in the linear case [64].
Fig. 5.5(a) shows the simulation results of charge versus voltage at the steady state. It
includes the results based on the analytical solution of the nonlinear ODE, the numerical
solution of the nonlinear ODE, and the numerical solution of the linear ODE (which ignores
the nonlinear term), respectively. One can see that the analytical solution matches well the
105
numerical solution of the nonlinear ODE. When the voltage is small, one can ignore the
nonlinear term in the nonlinear PDE. However, if a relatively large voltage (> 0.2V) is
applied, the error between the nonlinear model and the linear one becomes significant.
Fig. 5.5(b) shows the charge densities at the boundaries of IPMC based on the analytical
solution. Note that these densities correspond to the solutions p1 and p2 of (5.16), which
can be analytically obtained as shown in Appendix C2. The figure shows that p(—h)
cannot go lower than —C“F, but p(h) can go arbitrarily high. The inset figure shows
the zoom-in view around the turning comer where the negative charge density approaches
saturation. It is consistent with the numerical simulation results in Fig. 5.1. The voltage
corresponding to the turning comer is about 0.2 V. The above analytical analysis of PDE at
the steady state will be helpfirl for deriving the nonlinear capacitance of IPMC, which will
be discussed next.
5.3 Nonlinear Circuit Model
The above analysis can only capture the nonlinear capacitance of IPMC. A dynamic model
needs to capture the transient processes in IPMC as well. We propose a nonlinear cir-
cuit model, as shown in Fig. 5.6. It incorporates the nonlinear capacitance of IPMC C1,
pseudocapacitance Ca due to the electrochemical adsorption process at the polymer-metal
interface, ion diffusion resistance RC, electrode resistance R,,, and nonlinear DC resistance
of polymer Rdc.
5.3.1 Nonlinear capacitance of IPMC
The nonlinear capacitance can be obtained by taking derivative of (5.19),
_iQ_ r"(V)
'dV‘SWH—(o’
106
C1 (V) (5.21)
u-L
O
.1
9 ..
8 .-
7 ..
6 _
E, 6
:32 5*
‘5“ 4 _
0
3 _
2 _ -—0— Analytical solution of nonlinear PDE
- - - Numerical solution of nonlinear PDE
1 . -I— Numerical solution of linear PDE
0 0l5 1 1:5 1 2:5 3
Voltage (V)
(a)
9
15 x 10
E .
E 10
1:: 5 -
Q.
0 1 L 4, 1 1
0 7 0.5 1 1.5 2 2.5 3
0x10
—— Simulation data
E x 107 — ~ — Bound of negative charge density
E -5 '6
’E -8
CL, 10 _ _ -
-10 ~ 0.1 0.2
0 0.5 1 1.5 2 2.5 3
Voltage (V)
(b)
Figure 5.5: (a) Charge versus voltage at the steady state; (b) charge density at the bound-
aries versus voltage input.
where T"(V) is the first derivative of F(V). From (5.18),
(5.22)
aV_ V_ __ aV
T"(V)=g(l—e l)e" l aVe .
aV (eaV _ 1)2
107
Figure 5.6: Circuit model of IPMC.
The proposed analytical solution of nonlinear capacitance captures the fundamental physics
in the IPMC. It is represented by a firnction in terms of physical parameters and dimensions,
and is geometrically scalable.
5.3.2 Pseudocapacitance due to adsorption
For an electrochemical surface process, e. g., the so-called underpotential deposition of H
[25], the following holds:
K
M+H30+ + e : MHads +H20, (5.23)
where M is the substrate (usually a noble metal, Pt, Rh, Ru or Ir). Since IPMC has Pt as
electrode and some electrolyte in the polymer, the underpotential deposition process should
be incorporated into the model [3]. The adsorption current due to this electrochemical
process can be represented by [25]:
(5.24)
where
_V,,F
A qlSF K1CH+8 RT
C (Va) = a (5-25)
" RT _£ 2
chH++e RT
Va is the voltage on the pseudocapacitance, q1 is some constant (For H on polycrystalline
k
Pt, q1 = 210 11C / cm2 [25]), K1 = 75-, k1, L] are the chemical rate constants for forward
and reverse directions of (5.23), and 6"“ is the concentration of H +.
5.3.3 Nonlinear DC resistance
The current response under a step voltage input will not vanish at the steady state [13]
because of the DC resistance of polymer. One can approximate the DC current by a series
of polynomial firnctions Y (V). In this modeling work, we use a third-order polynomial
function:
A .
1.. = Y(V)=SIgn(V)(Y1|VI + YleI2 + WP). (526)
Note that 101,. is supposed to be an odd function of V. That is why sign(V) appears in (5.26).
5.3.4 Curvature output
The induced stress is proportional to the charge density [64]:
a = 0401) . (527)
where 010 is the coupling constant. The moment generated by IPMC can be written by:
h h
M = / O'dox= / W aoxp (x)dx. (5.28)
-h -h
109
dE
Since p(x) = ice—CE,
h
M = [WaoxexdE
-h
h
= Wang (xEI’L, — [5.5) . (5.29)
—.h
H:
Since E(—h) = E(h) = 3%, with (5.19) and / de =2 V, (5.29) can be written as
—h
M=Waoxe (sign(V)2h(/2F(|Vl)—V>. (5.30)
If one takes V ——> O,
M—> Waoer (h\/l_)— 1) z WaoKthx/b, (5.31)
which is consistent with the moment reported in the linear case [64]. One can then obtain
the curvature output via
K = — (5.32)
2 . .
where I = 3 W h3 is the moment inertia and Ye 15 the equivalent Young’s modulus of IPMC.
With (5.30), Eq. (5.32) can be written as
A 30101:. (sign(V)2h(/2F(|V|) — V)
7 211.113
K = ‘P(V) . (5.33)
We note that one could also use nonlinear elasticity theory to model the mechanical
output under large deformation [30]. This is outside of the scope of the current work.
110
5.3.5 Nonlinear control-oriented model
The objective of this work is to derive a control-oriented nonlinear model which can be
used in controller design. Based on the circuit model (Fig. 5 .6) and the curvature output
(5.30), the model structure is shown in Fig. 5.7.
U - l 1 + I 1 1 V K
r——.(‘ )———q - TV —>
"’ Ra+Rc _ Cull”) S ( )
I
g); " C.(V) s ‘
+
I
dc Y(V) :
Figure 5.7: Model structure.
From (5.24) and (5.26), one can get
U — V
—— — Y V
9K 2 Ra+Rc ( ). (5.34)
dt C1(V)+Ca(V)
Defining the state variable x = V, the control input u = U, and the system output y = K,
one can obtain a first-order nonlinear dynamic model in the state space:
* = ‘(ci(x)+ca)
_4 L L 1 1 1
0 50 100 150 200 250 300
Experimental data
40 r r . -' - - Simulation data
A
<
E
H
5
t
3
O
_40 1 1 1 1 1
0 50 100 150 200 250 300
Time (s)
Figure 5.11: Current response under a sinusoid voltage input.
115
nonlinear PDE at the steady state shows an asymmetric charge distribution along the thick-
ness. A systems perspective was taken in analytical analysis of nonlinear PDE to obtain a
nonlinear mapping from the voltage to the induced charge, which represents the nonlinear
capacitance of IPMC. A nonlinear circuit model is employed to capture the electrical dy-
namics of IPMC, including the nonlinear capacitance of IPMC, the ion diffusion resistance,
the pseudocapacitance due to the electrochemical process at the polymer-metal interface,
and nonlinear DC resistance of polymer. The curvature output is obtained fiom the cir-
cuit model. The proposed model is described in the state space, which will be the starting
point of nonlinear control of IPMC. The proposed model is validated experimentally by the
electrical responses of IPMC.
116
Chapter 6
Integrated Sensing For IPMC Actuators
Using PVDF Thin Films
Compact sensing methods are desirable for ionic polymer-metal composite (IPMC) actua-
tors in microrobotic and biomedical applications. In this chapter, a novel sensing scheme
for IPMC actuators is proposed by integrating IPMC and PVDF (polyvinylidene fluoride)
thin films. This chapter is organized as follows. Section 6.] introduces the IPMC/PVDF
with single mode sensing configuration and its biological application. Section 6.2 discusses
an IPMC/PVDF structure with differential configurations and its validation in feedback
control. Finally, concluding remarks are provided in Section 6.3.
6.1 IPMC/PVDF Single-Mode Sensory Actuator and Its
Biological Application
6.1.1 Design of IPMC/PVDF single-model structure
To construct the IPMC/PVDF sensori-actuator, a PVDF film (30 pm thick, Measurement
Specialties Inc.) is bonded to an IPMC (340 pm thick, Environmental Robotics Inc.) with
117
an insulating layer (Polyvinyl chloride PVC film, 30 pm thick) in between. The Fast-
Cure Elastic Epoxy (Polysciences Inc., Warrington, PA) is used in bonding. The design
of the IPMC/PVDF structure is illustrated in Fig. 6.1, where a picture of the IPMC/PVDF
prototype is shown at the bottom.
An IPMC using water as solvent needs to be hydrated to work in air. For the IPMC/PVDF
structure, it is not recommended to immerse the sample in water due to the presence of the
PVDF. Instead, one can place a damp paper towel on top of the IPMC side of the struc-
ture and at the same time apply a uniform compressive stress. This will facilitate uniform
hydration of the IPMC while preventing the delamination of the layers. The effect of the
humidity level on PVDF sensing is not appreciable [60]. The hydrating procedure is not
required if one adopts IPMC samples that use ionic fluids [11] or other non-water—based
solvents.
Side view Front view
Insulating layer IPMC
./
.E /
/ . .1 ' 1
ElectrodeSM £
Top view E: _ ]
Picture
Figure 6.1: Design of the IPMC/PVDF composite structure.
When the IPMC/PVDF structure is bent due to IPMC actuation or external forces,
charges are generated on the PVDF, which can be measured by a charge amplifier. Fig. 6.2
shows a differential charge amplifier which can minimize the common-mode noise. The
118
transfer function of the charge amplifier is described by:
V0(s) _ 2R]s IE
Q(s) _ 1+R1CisR2’
(6.1)
V 0 (S ) 2R3
—) —. However,
Q(s) C1 R2
in the circuit implementation, R1 cannot be infinitely large because the bias current of
which is a high-pass filter. As R1 ——> co, the transfer function
the operational amplifier will saturate the signal output. To accommodate the actuation
bandwidth of IPMC (typically below 10 Hz), the R1 and C. values in the circuit are properly
chosen so that the cutoff frequency of the charge amplifier is sufliciently low. By picking
R1 = 5000 MS) and C1 = 1350 pF, a cutoff frequency of 0.023 Hz is achieved.
R
—-L—_L_]-—
LL
R
Q(s) C: R —‘i’—
2
_——::_jr>— Vo
PVDF {LA‘> R,
-Q(S)I - R
1 R39
———l::l——
! _
l
r
1
Figure 6.2: Design of the charge amplifier.
Basically, the charge Q(s) generated by the PVDF is proportional to the bending dis-
placement Z (s) of the beam [82]:
Q(s) = GZ(S)7 (6-2)
where the constant G depends on the transverse piezoelectric coefficient d3] , the geometry
of the composite beam, and the Young’s moduli of individual layers. By combining (6.1)
and (6.2), one can obtain the transfer function from Z (s) to V0 (s). A laser displacement
sensor (OADM 2016441/Sl4F, Baumer Electric) is used for both calibration of the PVDF
119
sensor and validation of the sensing approach. In order to test the charge amplifier cir-
cuit, the IPMC/PVDF beam with one end fixed is tapped and then the laser sensor is used
to detect the damped vibration of the beam. The measured vibration frequency is 42 Hz,
which is much higher than the cutofl’ frequency of the charge amplifier. Fig. 6.3(a) shows
the charge output of PVDF corresponding to the damped vibration, and Fig. 6.3(b) demon-
strates that the charge signal is almost linear with respect to the bending displacement.
These experimental results have validated the performance of the charge amplifier circuit.
6.1.2 Impact of the stiffening effect and design of the insulator thick-
BESS
The additional PVDF and insulating layers make the composite beam stiffer than the IPMC
layer itself. It is of interest to understand the impact of this stiffening effect on the bending
performance since this will be useful for the optimal IPMC/PVDF structure design. The
investigation is conducted by combining analytical modeling, finite element computation,
and experiments. Design optimization here is concentrated on the thickness of the insulat-
ing layer, but the approach can be used for the design of other parameters, such as the type
of material for the insulating layer and the dimensions for IPMC and PVDF.
Fig. 6.4 illustrates the schematic of the IPMC/PVDF structure and the used notation in
the following discussion. The beam stiffness can be characterized by its spring constant
K = —, (6.3)
zmax
where F is a quasi-static transverse force applied at the free end of the cantilever beam
and zmax is the corresponding displacement at the acting point. The spring constant can be
calculated analytically using composite beam theory [37]. In Fig. 6.4, the position of the
120
0.016 3
0.015 * i
0.014 '
Charge (uC)
O
3
O)
0.012 r
0.01 1 *
0.01 ‘ ‘ r
0 0.05 0.1 0.15
Time (s)
(a)
0.016
0.015 -
0.014 *
Charge (uC)
O
S
c»
0.012 ~
0.011 * ‘
0.01
5 5.5 6
Displacement (mm)
(b)
Figure 6.3: (a) Charge output of the PVDF corresponding to the damped vibration; (b)
charge output versus the bending displacement.
121
mechanical neutral axis of the composite beam is given by:
2,11 Elm-Ci
ho = .
2:, EH.-
(6.4)
Here E1, E2 and E3 are the Young’s moduli of IPMC, insulating layer, and PVDF, respec-
tively. H1, H2 and H3 are the thicknesses of those layers. C1, C2 and C3 are the positions of
the central axes of the layers, which can be calculated as:
C1%H1/2,C2 2H1 +H2/2,C3 =H1+H2+H3/2. (6.5)
The distance between the central axis and the neutral axis can be written as:
di=ICi-ho|, for i-—-l,2,3. (6.6)
The moment of inertia of each layer is:
I,-=éWIL-3+WH,-d,2, for i=1,2,3. (6.7)
From the moment balance equation [37],
where p(x) is the radius of curvature. For small bending, the radius of curvature can be
given by:
1 dzz
36‘; = a}? (6-9)
where Z(x) denotes the deflection of the beam along the length x. With the boundary con-
122
dition 2(0) = 0 and 2(0) = 0, one gets
F sz x3
—— —-——). (6.10)
2?:1Ei1i( 2
200 = 6
Evaluating 2 at x = L yields the expression of spring constant
F 321 15.1,-
K = __ = ___,_1 , 6.11
w
e 1
Neutral hO IPMC H1
aXIS .................................................... .- l +
Figure 6.4: Bending of IPMC/PVDF composite beam.
Experiments are conducted to measure and compare the spring constants of the IPMC
and IPMC/PVDF beams. The IPMC or IPMC/PVDF beam is clamped at one end and is
pushed by the tip of a calibrated micro-force sensor which is mounted on a linear actuator.
The sensitivity of the micro-force sensor is 9.09 mV/uN :l: 6.5% and its spring constant
is 0.264 N/m. A laser displacement sensor measures the bending displacement of the
beam 2m under the pushing force F. A 20X microscope (FS60, Mitutoyo) is used to
monitor the experiments. Fig. 6.5(a) illustrates the diagram of the experimental setup, while
123
Fig. 6.5(b) shows the actual picture. Measurements are conducted for an IPMC beam and
two IPMC/PVDF beams which have insulating layers in different thickness (IPMC/PVDF]
and IPMC/PVDFZ). Detailed beam dimensions can be found in Table 6.1. Fig. 6.6 shows
the measured displacement versus force data and the linear approximations, from which
the spring constants can be calculated. From the experimental data, the Young’s moduli
of individual layers can be identified using (6.11): E1 = 0.571 GPa, E2 = 0.73 GPa, E3 =
1.96 GPa. These values are within the ranges reported in the literature [83, 77].
Table 6.1: Dimension and spring constant of different beams.
Beams IPMC IPMC/PVDF] IPMC/PVDF2
W(mm) 7.3 8.2 7.6
L (m) 37.2 36.0 33.0
H1 (pm) 355 340 350.0
H2 (pm) 30.0 100.0
H3 (11111) 30.0 30.0
Km (N/m) 0.906 2.283 4.676
KFEA(N/m) 0.908 2.286 4.647
To validate the linear analysis above, more accurate finite-element computation is con-
ducted using CoventorWare, where the identified parameters are used together with the
given geometric dimemsions. The spring constants are calculated based on the free-end
deflection of beams when they are subjected to an external force F = 20 pN at the tip.
Table 6.1 lists the spring constants obtained through experimental measurement (Kmea) and
finite element analysis (KFEA), for the different beams. The close agreement between Kmea
and KFEA validates the model and analysis.
As shown in Table 6.1, the thicker the insulator, the stiffer the IPMC/PVDF structure. In
order to optimize the bending performance of the IPMC/PVDF structure, one should select
the elastic insulating layer as soft and thin as possible. However, thinner insulating layer
may result in stronger electrical feedthrough coupling. In our design, the thickness of the
insulating layer is chosen to be 30 um to achieve tradeoff between the two considerations.
124
(— Linear actuator
Clamp Micro Force
(— Sensor
IPMC/PVDF
(— Laser Sensor {7
Computer _) |:l
=
Figure 6.5: (a) Experimental setup for spring constant measurement; (b) picture of the
experimental setup.
125
* Experimental data
()3 Linear approximation *
‘5 025* IPMC/PVDF1
3 0.2 IPMC
«’5 4
..s \
0.15 .
i
0.1- fl 1,
_ it a x
°-°5‘ * IPMC/PVDFZ
I I . . ,
0 200 400 600 800
Force (uN)
Figure 6.6: Spring constant of IPMC/PVDF beams.
6.1.3 Electrical feedthrough coupling and model-based real-time com-
pensation
The feedthrough coupling effect
Since the PVDF film is closely bonded to the IPMC with a very thin insulating PVC
film, the coupling capacitance between the IPMC and the PVDF results in the electrical
feedthrough effect during simultaneous actuation and sensing. When the actuation signal
is applied to the IPMC, the actuation voltage generates coupling current going through the
insulating layer and then induces coupling charge on the PVDF. As a result, the charge
amplifier gathers both the sensing and coupling charges from the PVDF. The presence
of feedthrough coupling is illustrated by applying a 0.4 Hz square-wave actuation input
(peak-to-peak 1.4 V). In the experiment, the humidity is 34% and the temperature is 23 °C.
Fig. 6.7(a) shows the bending displacement detected by the laser sensor, while Fig. 6.7(b)
126
shows the output from the charge amplifier. The spikes in the PVDF sensor output arise
from the capacitive coupling between the IPMC and PVDF layers when the actuation volt-
age jumps.
Modeling of the coupling effect
A complete circuit model of the IPMC/PVDF structure is developed to understand and
capture the feedthrough coupling dynamics. As shown in Fig. 6.8, the model includes the
equivalent circuits for individual layers and their natural couplings. Due to the nonneg-
ligible resistances resulted from the porous surface electrodes of the IPMC, the voltage
potential is not uniform along the IPMC length. A distributed transmission-line type model
is thus proposed. The overall circuit model is broken into discrete elements along its length
for parameter identification and simulation purposes. In this work, the circuit model is
chosen to have four sections of identical elements. The surface resistance of IPMC is rep-
resented by RS] , while other key electrodynamic processes (e.g., ionic transport, polymer
polarization, and internal resistances) are reflected in the shunt element consisting of resis—
tor Rcl and capacitor Cpl. The polymer resistance is described by Rpl- In the circuit model
of the insulating layer, R p2, sz, R62 are resistances and capacitances between the IPMC
and PVDF. In the circuit model of the the PVDF, R53 is the surface resistance of PVDF and
R p3, Cp3 represent the resistance and capacitance between the electrodes of the PVDF.
In order to identify the circuit parameters, the impedances are measured at multiple fre-
quencies. The impedances of each layer are nonlinear fiinctions of the resistances and ca-
pacitances involved. The parameters are identified using the Matlab command nl inear f it,
which estimates the coeflicients of a nonlinear function using least squares. Table 6.2 lists
the identified parameters.
The proposed circuit model will be validated by comparing its prediction of the feedthrough
coupling signal with experimental measurement. We first explain a simple method for mea-
suring the coupling signal. We observe that, due to the low surface resistance of PVDF (see
127
0.08
P .° P
o o o
c N h c»
Displacement (mm)
I
P
o
N
-0.04 -
-0.06 ‘ *
Time (s)
(a)
spike due
to coupling '
0.3L s '. . ,
r / ‘. .~
PVDF sensor output (V)
O
0 2 4 6
T lme (s)
(b)
Figure 6.7 : (a) Bending displacement detected by the laser sensor; (b) sensing output from
the PVDF, showing the spikes fiom electrical feedthrough.
128
i Levelt Level 4
'(3) +1: ;_.:
+ R81
R1
]p
H
H
4F—
] IPNK:
Fist “ICp1
J: .2
vmn E RcL
J—l
- pd-—---H----—d---
O
O
r 0
vii
‘ -----— -----------
L:
r
I
‘4
r1“.
1
4
’i,
r
'1."
3",; ——E —|i
l
: . .i
§Rp2 [1E 092: a Insulating layer
_i.... _
.117
To charge 5 RS3
amplifier
“‘+—L1
§Rp3 PVDF
. :45“ . . 4 47$"; 4 5: . I“ «L2
3 R33 : 9
Figure 6.8: Circuit model of the IPMC/PVDF structure.
Table 6.2: Identified parameters in the circuit model.
IPMC layer Insulating Layer PVDF layer
R51 17 Q sz 500 M!) RS3 0.1 Q
RC1 30 Q sz 42 pF Rp3 600 M9
Cpl 3 mF R02 4.5 M!) C13 290 pF
RP] 25 KS2
129
Table 6.2), the electrode layer L1 in Fig. 6.8 shields the coupling current from reaching the
electrode layer L2. This means that the feedthrough coupling signal does not exist in V _,
which is related to the charge fiom the layer L2. This statement is supported by the mea-
surement, shown in Fig. 6.9(a), where spikes only appear in Vp+. Since only Vp+ has the
coupling component while the sensing components in Vp- and Vp+ have a phase shifl of
180°, the coupling signal is obtained as:
Vc=Vp++V-. (6.12)
Fig. 6.9(b) shows the extracted coupling signal.
Fig. 6.10 compares the Pspice simulation results based on the circuit model with exper-
imental results when a 1 Hz square-wave actuation voltage is applied. Good agreement is
achieved for both the actuation current in IPMC (Fig. 6.10(a)) and the coupling voltage Vc
(Fig. 6.10(b)).
The transfer function from the actuation voltage to the coupling voltage can be derived
fi'om the circuit model. Since there are 14 capacitors in the circuit model, the transfer
firnction will be 14th-order, which is not easy to implement in real time. After an order-
reduction process, the transfer function of the coupling dynamics can be approximated by
a 5th-order system:
T _ —(509s4+72s3+ 1.5 x104sz+2203s)
C _ s5 +9525s4 +1.5 x10483 +2.9 x10582 +4.5 x105s+6 x104.
(6.13)
To further verify the coupling model, a sequence of sinusoidal voltage signals with
frequency ranging from 0.01 Hz to 20 Hz are applied to the IPMC. Actuation voltages are
measured and coupling signals are effectively extracted from Vp+ and Vp— for the purpose
of obtaining the empirical Bode plots of coupling dynamics. Fig. 6.11 shows that the Bode
plots of the derived transfer function (6.13) match up well with the measured Bode plots.
130
0'2 Spike due *1“
" ‘ ~to coupling
Time (s)
(a)
0.08
0.06 :
0.04 -
0.02 .
ch
-0.02 i
-0.04 -
-0.06 ' ‘
0 2 4 6
Time (s)
(b)
Figure 6.9: (a) Vp+, Vp- sensing signals; (b) extracted coupling signal Vc.
131
— Experimental data
- - - Simulation data
2'."
E .
E
2
I-
3
O
C
.2
iii
3
‘6
< I
0 1 2 3 4
Time(s)
(a)
0.06 7 -— Experimental data
5 - - - Simulation data
0.04. ........ .......... ..... ..
E ' .
5’ r
5 l
3 .
> 01 _______ ...
U) I
s .
Q .
0 l
-01“. ......... - .......... . ........
l _ t ' :
I q I
-0.06 #
0 1 2 3 4
Time(s)
(b)
Figure 6.10: Comparison of model prediction and experimental measurement. (a) Actua-
tion current; (b) coupling signal.
132
‘20 t7: 7 7* * *7 ‘‘‘‘‘ ff?
6 f”. ,3;
3 :25:
82-40’. ...__. . .::I:::I . Iiilifll j
2 5f . :572225 +EXPerimentaldatai iii???
; g ; ';i;i§i; ---Simulationdata
_6O -2 L 1111.1;1-1 L 111111110 1 IZZZTITI1 I I :1211112 2 . . .1. 3
10 1O 10 10 10 10
300 7 7”??? 7 777???? . if???
a . :..‘::.L. I i. i: I ..:.:..I
3 250-
8 9
2 200’
0- :j
150 . ....1 “.1 ....l ...i ....
10'2 10‘1 10° 101 102 103
w(rad/s)
Figure 6.11: Verification of the coupling dynamics.
Real-time compensation in simultaneous actuation and sensing
There are several possible schemes to get rid of the coupling signal. Inserting another
conductive layer between the IPMC and PVDF to shield the feedthrough coupling is one
potential solution, but at the cost of increased stiffness and fabrication complexity. Another
solution is to just use Vp— as the sensing signal, but this single-mode sensing scheme is
sensitive to the common-mode noise in practice. Since the coupling dynamics has high-
pass characteristics, one might also try to eliminate the coupling component with low-
pass filtering. However, the relatively low cut-off frequency of the coupling dynamics,
comparing to the actuation bandwidth (See Fig. 6.12), makes this approach infeasible.
In this reseach, a model-based real-time compensation scheme is proposed to remove
the feedthrough coupling component. The coupling charge is calculated from the coupling
circuit model (6.13). By subtracting it from the measured charge of the PVDF, the sensing
charge can be extracted. Fig. 6.13 illustrates the compensation scheme. Fig. 6.14 compares
the displacement measurement obtained from the PVDF sensor with that from the laser
133
Mag (dB)
<1)
01
l
.h
01
—+—Bode plotofcoupling dynamics 3
j i 1.3.};3- 0 -Bode plotofactuation dynamics g g 1.33;;
_55 L..;;;;;i ;;;;;;;;, 2;;;;;;;; ;;;;;;;;; '
10'2 10‘ 100 101 10
I
01
O
A ALLA
(0 (rad/s)
Figure 6.12: Bode plots of coupling dynamics and actuation dynamics.
sensor when a 0.4 Hz square-wave actuation input is applied. It is seen that the spike related
to the electrical coupling is removed by the compensation scheme. Although there is about
12% error shown in Fig. 6.14, the amplitudes and the phases agree well. Investigation is
under way to further improve the measurement accuracy of the PVDF sensor.
6.1.4 Application to micro-injection of living drosophila embryos
The developed IPMC/PVDF sensori-actuator is applied to the micro-injection of living
Drosophila embryos. Such operations are important in embryonic research for genetic
modification. Currently this process is implemented manually, which is time-consuming
and has low success rate due to the lack of accurate control on the injection force, the
position, and the trajectory. The IPMC/PVDF structure is envisioned to provide accurate
force and position control in the micro-injection of living embryos, and thus to automate
this process with a high yield rate. In this research, an open-loop injection experiment with
134
. IPMC/PVDF
structure
Actuation signal
? i ,9 sensing slggpl
- ' I sensing + + E + -
Feedthrough , i
Figure 6.13: Diagram of the real-time compensation.
0.08
9 9
Displacement (mm)
-0.06 :
-0.08
O
I I
p p
c c
-h N
coupling dynamics :
5*; Feedthrough :
5 coupling model g, _____ Coupling
I : prediction
----------------------------
—— Measured by laser sensor
- - - Measured by PVDF
1‘”
I
I I
I I
l
\. 3
L
4
Time (s)
Figure 6.14: Comparison of displacement measurements by laser sensor and PVDF sensor.
135
the IPMC/PVDF sensori-actuator is conducted, and the process of the injection behavior is
captured by the PVDF sensor.
The developed IPMC/PVDF micro-force injector is illustrated in Fig. 6.15. A mi-
cropipette with an ultra-sharp tip (1.685 pm in diameter and 2.650 in angle), is mounted
at the end point of a rigid tip attached to the IPMC/PVDF structure. The Drosophila
IPMC/PVDF
Mounting l E
Connector Z
\
'\ Micro
/ ii" Force
Micropipette
Ineedle X
Figure 6.15: Illustration of the IPMC/PVDF micro-force injector.
embryos are prepared as described in [81]. The dimensions of the embryos are variable
with an average length of 500 um and a diameter of about 180 um. Fig. 6.16(a) shows
the diagram of the experimental setup for embryo injection, while Fig. 6.16(b) shows the
photo. A 3-D precision probe station (CAP-945, Signatone), which is controlled by a 3-D
joystick, moves the needle close to an embryo and then a ramp voltage, which starts from 0
V and saturates at 2 V, is applied to the IPMC. The IPMC drives the beam with the needle
to approach the embryo. After the needle gets in contact with the membrane of the embryo,
136
Microscope
Lens
L—A Micropipette/n
- Micropipette eedle (20X
Halocarbon 700 011 i Ineedle image)
Adhesive tape
' 3-D
Glass Slrde 3-D
Probe _
\ Station Joystick
Micro-Force
Sensor
(8)
Laser sensor
(b)
Figure 6.16: (a) Diagram of experimental setup for embryo injection; (b) picture of exper-
imental setup.
137
the latter will be deformed but not penetrated due to its elasticity. At this stage, the needle
is still moving until the reaction force between the needle and embryo reaches the pene-
tration force. The needle stops at the penetration moment for a while (about 0.2 ms) due
to temporary force balance. After that, the embryo membrane is penetrated and the needle
moves freely into the embryo.
Fig. 6.l7(a) shows the snapshots of the successful injection progress. Fig. 6.l7(b)
shows both the displacement of the needle detected by the laser sensor and by the com-
pensated PVDF sensing signal. It is concluded that the predicted displacement reflects the
movement of the needle, and the process of injection can be monitored by the PVDF.
138
0.12 - -
-— Measured by laser sensor
- - - measured by PVDF , V
0.1 i z
E 0.08- , -
T: I
C I
0 I
E 0 06 , .
O I
o I
4g .
.2 0'04 - a- “ 5 ’ 1
n I ‘ ’
a I
0.02 - ,’ Injection -
r “ f ’
"o 6
Time (s)
(b)
Figure 6.17: (a) Snap shots captured during the embryo injection; (b) bending displacement
during the injection measured by both the laser sensor and the integrated PVDF sensor.
139
6.2 IPMC/PVDF Differential-Mode Sensory Actuator and
Its Validation in Feedback Control
6.2.1 Design of IPMC/PVDF differential-mode sensory actuator
Integrated differential-mode sensor for bending output
Fig. 6.18 illustrates the design of the integrated differential mode bending sensor for an
IPMC actuator. Two complementary PVDF films, placed in opposite poling directions, are
bonded to both sides of an IPMC with insulating layers in between. In our experiments, we
have used 30 um thick PVDF film from Measurement Specialties Inc., and 200 pm thick
IPMC fiom Environmental Robots Inc. The IPMC uses non-water-based solvent and thus
operates consistently in air, without the need for hydration. Scrapbooking tape (double-
sided adhesive tape,70 um thick) fi'om 3M Scotch Ltd. is used for both insulating and
bonding purposes. A picture of a prototype is shown at the bottom of Fig. 6.18. Since we
are focused on demonstrating the proof of the concept in this work, the materials used are
chosen mainly based on convenience. However, the models to be presented later will allow
one to optimize the geometry design and material choice based on applications at hand.
The differential charge amplifier, shown in Fig. 6.19, is used to measure the PVDF
sensor output. In particular, the inner sides of two PVDF films are connected to the common
ground, while the outer sides are fed to the amplifiers. Let Q1(s) and Q2(s) be the charges
generated on the upper PVDF and the lower PVDF, respectively, represented in the Laplace
domain. The signals Vp+ and Vp‘ in Fig. 6.19 are related to the charges by
R15
— —mQ2(S),
iPMC VDF
Eiectrode—+ Insulating
layers
PVDF
Figure 6.18: Design of the IPMC/PVDF composite structure for sensing of bending output
(force sensor not shown).
and the sensor output V0 equals
_ R1R3S
Vo(S) - m(Q1 (S) — Q2(S))- (6-14)
Let the bending-induced charge be Q(s) for the upper PVDF, and the common noise-
induced charge be Q,,(s). If the sensor response is symmetric under compression ver-
sus tension (more discussion on this in Section 6.2.2), one has Q1(s) = Q(s) + Qn(s),
Q2(s) = —Q(s) + Qn(s), which implies
2R1R3S
Vo(S) = mg”, (6-15)
and the effect of common noises (such as thermal drift and electromagnetic interference)
is eliminated fiom the output. The charge amplifier (6.15) is a high-pass filter. To accom-
modate the actuation bandwidth of IPMC (typically below 10 Hz), the R1 and C] values
in the circuit are properly chosen so that the cutoff frequency of the charge amplifier is
sufficiently low. By picking R1 = 5000 MD, C1 = 1350 pF and R2 = R3 = 10 k9, a cutoff
141
frequency of 0.023 Hz is achieved.
:1 R
Q1 C —i.'_}:l——
PVDF1 I ]P°""9
direction
PVDF—$133231... mam
-Q2 R1 R39
IF _
91
Figure 6.19: Differential charge amplifier for PVDF sensor.
A model is developed for predicting the sensitivity of the bending sensor in terms of the
design geometry and material properties. Refer to Fig. 6.20 for the definition of geometric
variables. Suppose that the IPMC/PVDF beam has a small uniform bending curvature with
tip displacement 21; without external force, the force sensor beam attached at the end of
IPMC/PVDF appears straight with tip displacement 22. One would like to compute the
_Q_
22 ’
effector displacement 22. With the assumption of small bending for IPMC/PVDF beam,
sensitivity where Q represents charges generated in one PVDF layer given the end-
the curvature can be approximated by [19]
1 z — (6.16)
p
where p represents the radius of curvature. As H3 << 0.5H1 +H2, we assume the stress
inside the PVDF to be uniform and approximate it by the value at the center line of this
layer:
0.5H H 0.5H
032E3£=E3 ‘+ p” 3 (6.17)
where E3 is the Young’s modulus of the PVDF. The electric displacement on the surface of
142
PVDF is
Ds=d3103. (6.18)
where d3] is the transverse piezoelectric coefficient. The total charge generated on the
PVDF is then
Q=/DSdS=DsL1W1. (6.19)
With (6.16), (6.17), (6.18) and (6.19), one can get
_ 2d31E3W1(0.5H1+H2 +0.5H3)Z1
6.20
Q L] ( )
The end-effector displacement 22 is related to 21 by
22:21+Lzsin arctan(a) zz1(l+& , (6.21)
L1 L1
Combining (6.20) and (6.21), one can get the sensitivity
S: Q = 2d31E3W1(0.5H1+H2+0.5H3). (6.22)
22 L1 '1' 2L2
Table 6.3 lists the parameters measured or identified for our prototype. The sensitivity
is predicted to be 1830 pC/mm, while the actual sensitivity is characterized to be 1910
pC/mm using a laser distance sensor (OADM 2016441/Sl4F, Baumer Electric). With the
charge amplifier incorporated, the sensitivity 2:3 at frequencies of a few Hz or higher is
measured to be 2.75 V / mm, compared to a theoretical value of 2.71 V/mm.
Force Sensor for End-effector
The structure of the force sensor is similar to that of IPMC/PVDF sensory actuator. As
illustrated in Fig. 6.2], two PVDF films are bonded to the both sides of a relatively rigid
beam. In our experiments, we have used 200 pm thick Polyester from Bryce Corp. for the
143
Table 6.3: Parameters identified for the IPMC/PVDF sensory actuator prototype (including
force sensor).
W1 L1 H1 H2 H3
10 mm 40 mm 200 um 65 um 30 um
W2 L2 h] h2 113
6 mm 30 mm 200 um 65 um 30 um
Er Ez E3 6131
5 GPa 0.4 GPa 2 GPa 28 pC/N
Net-Lital axis “ 91PMC ’ “ a"?
O Y
., -e.-. “-
,-., ' .
insulating layer
L1
Z ' IPMC/PVDF
PVDF force sensor
-------------.
Figure 6.20: Geometric definitions of IPMC/PVDF sensory actuator.
144
beam. An end-effector, e.g., a glass needle in microinjection applications, is bonded the
tip of the force sensor. An external force experienced by the end—effector will cause the
composite beam to bend, which produces charges on the PVDF films. Another differential
charge amplifier as in Fig. 6.19 is used for the force sensor. The whole force-sensing beam
is attached to the front end of the IPMC/PVDF beam.
P
Bottom View
Figure 6.21: Design of the force sensor for the end-effector.
Refer to Fig. 6.22. The sensitivity model for force sensing, QFI, is provided below.
Here Q f represents the charges generated in one PVDF in response to the force F exerted
by the end-effector. The beam curvature can be written as
1 F L —
_ = #1 (6.23)
P (x) 24:0 EiIi
where p(x) denotes the radius of curvature at x, E], E2, E3 are the Young’s moduli of the
Polyester film, the bonding layer, and PVDF respectively. 11, 12 and 13 are the moments of
inertia for those layers, which are given by
1 3
11 — l—z‘thla
1
I2 = -6-W2hg+
l
13 = 6W2h§+
W2h2(h1+h2)2
2
,
W2h3(h1 + 2h2 + h3)2
145
The stress generated in the PVDF is approximately
h] + 2112 + h3
0307) = E383 (x) = E3 (6-24)
210 (x)
With (6.18), (6.23) and (6.24), one can get the electric displacement in PVDF,
h] +2112 +h3 F(L2 -x)
D3 (3‘) = 6131030?) = E36131 - (6-25)
2 23:05:]:
The total charge generated in the PVDF can be written as
L2 d E W L2 11 2h h
QfZ/ D3(x)W2dx= 3‘ 3 2 2(3 1+ 2+ 3)F (6.26)
0 42,-:oE11i
Then the sensitivity of the force sensor is
d E W L2 h 2h h
szgiz 313 2 2( 1+ 2+ 3). (6.27)
F 42:05:11
Relevant parameters for the force sensor in our prototype can be found in Table 6.3.
Theoretical value of S f is computed to be 0.456 pC/IIN, which is close to the actual value
0.459 pC/IIN from measurement. With the charge amplifier circuit, the sensitivity of the
overall force sensor @- at high fiequencies (several Hz and above) is characterized to be
0.68 mV/uN, compared to the model prediction of 0.67 mV/IIN.
The integrated IPMC/PVDF sensory actuator and the charge sensing circuits are placed
in conductive plastic enclosures (Hammond Manufacturing) to shield electromagnetic in-
terference (EMI) and reduce air disturbance and thermal drift. A slit is created on the
side of the shielding box enclosing IPMC/PVDF so that the end-effector protrudes out for
manipulation purposes. Fig. 6.23 shows the picture of the overall system.
146
Figure 6.22: Geometric definitions of the PVDF sensor.
.,_Jii .: __ Conductive boxes“
. Sensing circuit ’ ‘
l for position sensor
Q 4 —. 4. "1—~:.~——
Sensing circuit
for force sensor ; - ,i
,
\ .
i .‘ :.;"" ’ VA ‘-
Isl; 48.»:\.-~
my: :.$ ,- 7.. ‘
Figure 6.23: IPMC/PVDF sensory actuator and sensing circuits in shielding enclosures.
147
6.2.2 Experimental verification of sensor robustness
In this section we experimentally verify the robustness of the proposed sensory actuator
with respect to the following undesirable factors: 1) feedthrough of actuation signal, 2)
thermal drift and other environmental noises, and 3) asymmetric PVDF sensing responses
during compression versus tension. The discussion will be focused on the PVDF sensor
for IPMC bending output, since the problems associated with the PVDF force sensor are
similar and actually simpler (no need to worry about actuation feedthrough).
Feedthrough Coupling
Close proximity between IPMC and PVDF results in capacitive coupling between the two.
Fig. 6.24 illustrates the distributed circuit model for the composite IPMC/PVDF beam.
Suppose an actuation signal V,-(s) is applied to IPMC. If one connects both sides of a single
PVDF film to a differential charge amplifier, as done typically in Section 6.1, the output will
pick up a signal that is induced by the actuation signal via electrical coupling. Fig. 6.7 in
Section 6.1 illustrates the traditional feedthrough problem. While one can attempt to model
the feedthrough coupling and cancel it through feedforward compensation, the complexity
of such algorithms and the varying behavior of coupling make this approach unappealing
to real applications.
In the new charge sensing scheme proposed in this design, the inner sides of the two
PVDF sensors are connected to a common ground (see Fig. 6.19). Since the surface elec-
trode resistances of PVDF films are very low (< 0.152), the inner layers L2 and L3 in
Fig. 6.24 will effectively play a shielding role and eliminate the feedthrough coupling sig-
nals. This analysis is verified experimentally, where a square-wave actuation voltage with
amplitude 2 V and frequency 0.1 Hz is applied to the IPMC. Fig. 6.25(a) shows that the
charge amplifier output V0 contains no feedthrough-induced spikes. The definitions for V0,
V}, and Vp‘ in the figure can be found in Fig. 6.19. Furthermore, the bending displacement
148
PVDF1
Insulating
layer
To charge
amplifier
—_l-_
4)
IPMC
k ‘
insulating
layer
PVDF2
Section 1 Section n
Figure 6.24: Distributed circuit model of IPMC/PVDF beam.
149
obtained from the PVDF output V0 correlates well with the actual bending displacement
measured by the laser distance sensor, as shown in Fig. 6.25(b). Note that the PVDF output
V0 is related to the bending displacement 21 through the charge amplifier dynamics (6.15)
and the proportional relationship (6.20). Since (6.15) represents a high-pass filter, at rela-
tively high frequencies (determined by the cut-off fiequency), the correlation between V0
and 21 can be approximated by a constant; however, at lower fiequencies (including the
step input, in particular), the dynamics (6.15) has to be accommodated to obtain the dis-
placement trajectory from the raw PVDF signal V0. The latter has been adopted throughout
the chapter, whether the inverse of the charge amplifier dynamics is implemented digitally
to retrieve 21.
Thermal Drift and Environmental Noises
PVDF sensors are very sensitive to ambient temperatures and electromagnetic noises. Such
environmental noises could significantly limit the use of PVDF bending/force sensors, es-
pecially when the operation frequency is low (comparing with the fluctuation of ambient
conditions). Refer to Fig. 6.19. Let noise-induced charges be Q"l and Q"2 for PVDF1
and PVDF 2, respectively. Suppose that no actuation signal is applied, and thus bending-
induced charge Q(s) = 0. The voltage signals can then be expressed as
+ _ R15
_ Rls
Vp (S) —an2(S), (6-29)
_ R1R3S
V00) — R20 +R1Cls)(Q..(s) —Qn.(s)). (6.30)
Inside a conductive shielding enclosure, thermal and EMI conditions are relatively steady
and uniform. This implies Q"I (s) z Qn2(s) and the influence of environmental noises on
the sensor output V0 is negligible.
Two experiments have been conducted to confirm the above analysis. In order to isolate
150
—— V
o
3 t - - . Vp+
- - - V
2 p-
2 1 ’ .1
a II 5 I '
g o -’_ "'I‘ is, ,’ u ,s ’I " I”
g 1 r \i ’ ‘ I, : ‘ I!" ‘1 ’
‘ i \ .I I " I 1
’r\ ‘1‘" 'I \ ‘I‘: I, K \‘ i
-2 4 l ‘. ‘ \ I ‘. \ ‘ ‘ , \i
N ' ‘ \\ I ‘. \ '
\ I s i ‘ '
..3 i- \' ~ ‘ I 4
0 5 10 15 20 25 30
Time (s)
(a)
Measured by laser sensor
; - - - Predicted by PVDF sensor
Bending displacement (mm)
0
_2 l
20 5 10 15 20 25
E . -
E , ,
8 ;
t
m -2 1 1 1 1
0 5 10 15 20 25
Time (s)
(b)
Figure 6.25: Experimental results showing elimination of feedthrough signal. (a) Raw
PVDF sensing signals under a square-wave actuation input (2V, 0.1 Hz); (b) Comparison
between the bending displacements obtained from the PVDF sensor and from the laser
sensor.
151
the effect of noises, no actuation signal is applied. In the first experiment, the IPMC/PVDF
beam was exposed to ambient air flows and electromagnetic noises. Fig. 6.26(a) shows the
complementary sensing outputs Vp+ and VP“ and the resulting charge amplifier output V0.
From (6.28)-(6.30), the discrepancy between Vp+ and VP“ indicates that the noise-induced
charges Q”l and Q"2 on the two PVDF layers can be significantly different, leading to
relatively large sensing noise in V0. In contrast, Fig. 6.26(b) shows the results from the sec-
ond experiment, where the IPMC/PVDF sensory actuator was placed inside the conductive
shielding enclosure. In this case, while V; and Vp‘ could still vary over time individually,
their trajectories are highly correlated and close to each other. Consequently, V0 remained
under 1 mV, compared to about 20 mV in the first case. These experiments have confirmed
that the proposed differential sensing scheme, together with the shielding enclosure, can
effectively minimize the effect of thermal drift and other common noises.
Asymmetric Sensing Response during Extension versus Compression
Because of its compliant nature, a single PVDF film does not produce symmetric charge
responses when it is under tension versus compression. In particular, it is diflicult to ef-
fectively introduce compressive normal stress into the flexible film. As a result, the charge
response of a PVDF layer under extension can faithfully capture the beam motion while
the response under compression cannot. This is illustrated by experimental results shown
in Fig. 6.27(a): each of the sensing signals Vp+ and Vp‘ fiom the two PVDF layers is asym-
metric under a symmetric, sinusoidal actuation input. With the differential configuration
of two PVDF films, however, the asymmetric responses of individual PVDF films combine
to form a symmetric output V0, as seen in Fig. 6.27(a). This is because when one film is
in compression, the other is in tension. Fig. 6.27(b) shows that the bending displacement
obtained based on the PVDF signal V0 agrees well with the laser sensor measurement. We
have further examined the performance of the proposed integrated sensing scheme under
other types of actuation inputs, including the step inputs. From Fig. 6.28, the bending
152
Voltage (V)
0 5 10 15 20 25 30
Time (s)
(a)
0.5
0 4 ‘5 .f’c . _
—.q’.:'\"‘ |Pi ~..~_T_f:.~~...
"ti/II 1i “ " " " ~ ::=’s‘5‘~‘r
E 0.3 ~ ’ 1 t .J
o i
a:
i -—Vo
> 0.2 ! - - -V J
p+
- -.Vp_
0.1 ‘
WW
0 L 1 J
0 5 10 15 20 25 30
Time (s)
(b)
Figure 6.26: (a) Sensing noise when IPMC/PVDF placed in open field; (b) sensing noise when
IPMC/PVDF placed inside conductive shielding enclosure.
153
trajectory under a step input (2V) can be captured well by the PVDF sensor.
Another advantage of adopting two complementary PVDF films is that it alleviates
the effect of internal stresses at bonding interfaces. When bonding a single PVDF to
IPMC, mismatch of internal stresses at the PVDF/IPMC interface could lead to delami-
nation and/or spontaneous creep of the composite beam. While this problem could be less—
ened by using appropriate bonding technologies, it was found that the proposed scheme
can effectively maintain the structural stability of the composite beam, without stringent
requirements on bonding.
6.2.3 Feedback control based on the integrated sensor
The practical utility of the proposed IPMC/PVDF sensory actuator has been demonstrated
in feedback control experiments. Trajectory tracking experiments are first performed,
where no tip interaction force is introduced. Simultaneous trajectory tracking and force
measurement are then conducted to examine both integrated bending and force sensors.
Feedback control of bending displacement
Fig. 6.29 illustrates the closed-loop system for the control of IPMC bending displacement.
Here P(s) represents the actuation dynamics for the IPMC/PVDF composite structure, H (s)
is the bending sensor dynamics, K (s) is the controller, r is the reference input, u is the ac-
tuation voltage, and 22 is the bending displacement of the end-effector. In experiments data
acquisition and control calculation are performed by a dSPACE system (DSl 104, dSPACE
Inc.); for real applications such tasks can be easily processed by embedded processors, e. g.,
rnicrocontrollers. A laser sensor is used as an external, independent observer for verifica-
tion purposes.
In general K (s) can be designed based on a nominal model of the plant P(s) and various
objectives and constraints. An example of H00 control design can be found in [20], where a
physics-based, control-oriented model is also developed for IPMC actuators. Since IPMC
154
“V
0
--IV
2 13+ _
---V
p—
2 1"
a)
a)
g 0»
o
> ¢\ ¢' ’a¢‘\ " _-\
I 3 .\ 'I. I“, " ' —
I - '
-1r I i v" I .' " \ ‘
I I \ ‘ ' .1
\ 1 ' \ I . I /\ 1.“
I I l ‘ ’ \ ‘
\ \ , ‘ , \ l -I \ I \ -\
’ ' ' 1 I ‘3. ' \
’ \. I \ \ I \ -
*2‘ 'I \ " \l \, I! \
v v ‘I V
0 5 10 15 20
Time(s)
(a)
Measured by laser sensor 4
- - - Predicted by PVDF sensor
0.5 ................ ................ . .............................
Bending displacement (mm)
C
-0.5
E 05. ............. ................................. .1
..E, 3 -j
8 3 i
LU : g
0 5 10 15 20
Time (s)
(b)
Figure 6.27: Self-compensation of asymmetric tension/compression sensing response. (a)
Raw PVDF sensing signals under a sinusoidal actuation signal (0.2 Hz, amplitude l V); (b)
Comparison between the bending displacements obtained fi'om the PVDF sensor and from
the laser sensor.
155
A 1 r
E
:7 0.8~
r:
o
E 0.6 p . 1
§ 04 p - .. ~ Measuredbylasersensor 1‘
'8'. ' -— Predicted by PVDF sensor
:6 ‘ : .
or 0.2
.E
.2 0 ......
0 i 1 1 1 '1
m 0 10 20 30 40 50 60
A 1 I V 1 I I
E
E .
r 0,,— -
8
0 10 20 30 40 50 60
Time (s)
Figure 6.28: Comparison between the bending displacements obtained from the PVDF
sensor and fi'om the laser sensor, when a 2V step input is applied.
Bending
r + K(s) u P(S) displacemen:
3 ' z2
PVDF
sensing signal
H (s)
H "‘(s) 4, ,
IPMC/PVDF” ------------------------ =
Figure 6.29: Closed-loop system for control of IPMC bending displacement.
156
modeling and control design are not the focus of this chapter, we have identified the plant
model P(s) empirically and used a simple proportional-integral (PI) controller for K (s) to
validate the integrated sensing scheme. In particular, the empirical frequency response of
the IPMC/PVDF sensory actuator has been obtained by applying a sequence of sinusoidal
actuation inputs (amplitude 0.2 V, frequency 0.01 Hz to 10 Hz) and measuring the corre-
sponding bending response. It has been found that the measured dynamic behavior could
be approximated by a second order system, the parameters of which have been further
determined using the Matlab command “fitsys”. The resulting P(s) is
P(s) _ 2.73 +20
_ 1000(82 + 33.4S + 18.9) '
The sensing model is obtained from (6.15) and (6.20):
181505
H =———.
(S) 6.57s+1
The following reference trajectory is used: r(t) = sin(0.37tt) mm. Based on the mod-
els and the reference, a PI controller K (s) = 1000 (40 + ?) is designed to achieve good
tracking performance while meeting the constraint |u| < 2 V. Fig. 6.30(a) shows the ex-
perimental results of tracking the bending reference. It can be seen that the PVDF sensor
output tracks the reference well; firrthermore, the actual bending displacement, as observed
by the laser sensor, has close agreement with the PVDF output. The actuation voltage u,
shown in Fig. 6.30(b), falls within the limit {—2, 2] V.
Feedback bending control with simultaneous force measurement
It is desirable in many applications to have both displacement and force feedback. With the
proposed IPMC/PVDF sensory actuator, one can perform feedback control of the displace-
ment while monitoring the force output, as well as perform feedback control of the force
157
—- Displacement from laser sensor
,4 1.5- '- - -- Displacement from PVDF sensor
E - - - Reference
E 1 -
0
i
g 0.5 ~
Q
.2
'0
.E .
“g .
(an) -0.5
-1 . ’
0 5 10 15 20
Time (s)
(a)
2
1.5 -
1 _
E 0.5 -
O
> -0.5~
-1 .
-1.5
..2 1 1 .
0 5 10 15 20
Time (s)
(b)
Figure 6.30: Experimental results on feedback control of bending displacement using inte-
grated PVDF sensor. (a) Bending displacement; (b) actuation voltage.
158
output while monitoring the displacement. In the following experiment we will demon-
strate the feedback bending control with simultaneous force measurement.
To mimic the force level often encountered in bio and micromanipulation applications,
we have attached a sharp glass needle as an end-effector at the tip of force-sensing beam
and used it to pierce soap bubbles. Fig. 6.3 1(a) shows the experimental setup. A number of
bubble-penetrating experiments were conducted to get an estimate of the rupture force by
moving a bubble manually toward the needle until it breaks, when no actuation voltage was
applied. Fig. 6.31(b) shows the force sensor response during a typical run. It can be seen
that the response first rises from zero to a peak value, and then starts decayed oscillations.
Since the PVDF sensor measures essentially the bending of the passive beam, its output
can be interpreted as an interaction force only when the end-effector is in contact with a
foreign object. Thus for the response in Fig. 6.3l(b), only the first rising segment truly
represents the force, after which the membrane ruptures and the beam starts oscillating.
Hence we take the peak value of such responses as the penetration force. Fig. 6.32 shows
the penetration force measured in 26 independent experiments. Overall the measurements
are consistent with an average of 11 MN. The variation is believed due to the randomly
created bubbles that might have different thicknesses. Note that for many real applications,
such as microinjection of embryos or cells [18], the end-effector will maintain contact with
the object under manipulation, in which case the output of PVDF force sensor would truly
represent the interaction force at all times.
A feedback bending control experiment with force monitoring has been conducted,
where the reference for the end-effector displacement r(t) = 0.2 sin(0.47rt) mm. During
the experiment, the end-effector penetrated two soap bubbles at t = 9.32 and t = 15.72
seconds, respectively. Fig. 6.33(a) shows the estimated end-effector displacement based on
the integrated PVDF bending sensor (sandwiching IPMC), under the assumption that the
force-sensing PVDF beam is not deflected. The estimated displacement trajectory follows
closely the reference, with slight perturbations at the moments when penetrations occur,
159
Penetration force
10-
’z‘
3 5- A .
E U m- . V A A A A A. A A.
a - , vvvvvvv
_5_ True force ‘
-10-
_.,= 1 ‘ Beam oscillations _
"’0 0.5 1 1.5 2 2.5 3
Time(s)
(b)
Figure 6.31: Measurement of the micro force in piercing soap bubbles. (a) Experimental
setup; (b) PVDF sensor response during and afier penetration.
160
.15 ' T ' ' m
10"
Penetration force of bubble p. N)
5 r .
o 1 1 1 1 1
0 5 10 15 20 25 30
Samples
Figure 6.32: Measured forces during penetration of soap bubble membranes.
indicating that the feedback control was in effect. Fig. 6.33(b) shows the output of the
integrated force sensor, where the two penetrations were captured clearly. Note that, as
explained earlier and illustrated in Fig. 6.31(b), only the first rising segment of the trajectory
during each penetration truly represents the interaction force, while the remaining portion
of the signal arises from oscillations following penetration. The control output (actuation
voltage) is shown in Fig. 6.34, where one can see that feedback is in action to suppress the
disturbance caused by penetration.
Note that the end-effector displacement 22 predicted by the PVDF bending sensor alone
(Fig. 6.33(a)) does not capture the true displacement d when the end-effector interacts with
objects. To obtain the true displacement, one can combine the bending sensor output 22 and
the force sensor output F:
d = 22 +F/k, (6.31)
where k is the stiffness of the force-sensing beam. For our prototype, k = 0.067 N/m.
Fig. 6.35 compares the end-effector displacement obtained from (6.31) and that observed
by the laser sensor, which shows that indeed the end-effector position can be monitored by
161
I; l
- - - Reference
03 . '- - -- Measured by PVDF sensor
0.2 -
0.1"
Bending displacement (mm)
O 5 10 15 20
Time (s)
(a)
15
10"
Tip force ()1 N)
O
-10..
_15 1 1 1
0 5 10 15 20
Time (s)
(b)
Figure 6.33: Experimental results on bending feedback control with tip force measurement.
(a) Displacement of the end-effector estimated based on the integrated PVDF bending sen-
sor alone; (b) PVDF force sensor output.
162
0.5
Control output (V)
O
0 5 10 15 20
Time (s)
Figure 6.34: Actuation voltage generated by the feedback controller.
combining the integrated bending and force sensors.
6.3 Chapter Summary
In this chapter, a novel scheme was proposed for implementing integrated sensors for an
IPMC actuator, to achieve sensing of both the bending displacement output and the force
output. In the first design, an IPMC is bonded with a PVDF sensing film in a single-
mode sensing configuration. The stiffening effect and the electrical feedthrough coupling
are investigated. In the second design, two thin PVDF films are bonded to both sides of
an IPMC beam to measure the bending output, while a passive beam sandwiched by two
PVDF films is attached at the end of IPMC actuator to measure the force experienced by
the end-effector. The differential configuration adopted in both sensors has proven critical
in eliminating feedthrough coupling, rejecting sensing noises induced by thermal drift and
EMI, compensating asymmetric tension/compression responses, and maintaining structural
stability of the composite beams. For the first time, feedback control of IPMC has been
163
0.3 ’
0.2 * ~ -' , l-
0.1 i l l
-0.1 ~
-0.3r
-0.4 -
-0.5 ‘ ‘ '
0 5 10 15 20
Time (s)
End-effect displacement (mm)
— Laser sensor
- - - Integrated sensors"
Figure 6.35: Estimation of true end-effector displacement by combining the integrated
bending and force sensors, and its comparison with the laser sensor measurement.
successfully demonstrated using only integrated sensors, showing that one can simulta-
neously regulating/tracking the bending displacement and monitoring the force output (or
vice versa).
164
Chapter 7
Monolithic Fabrication of Ionic Polymer
Metal Composite Actuators Capable of
Complex Deformations
This chapter is organized as follows. The fabrication process for MDOF IPMC actuators is
presented in Section 7.1. In Section 7.2 we investigate the change of stiffness and swella-
bility of Nafion films caused by the ion-exchange process. The performance of fabricated
artificial pectoral fins is characterized in Section 7.3. Finally, concluding remarks are pro-
vided in Section 7.4.
7.1 FabricationProcess
In this section, we outline the overall fabrication flow first and then discuss the individual
steps in more details. Fabrication of an artificial pectoral fin is taken as an example. As
illustrated in Fig. 7.1, the major process steps include:
o (a): Create an aluminum mask on Nafion with e-beam deposition, which covers the
intended IPMC regions;
165
o (b): Etch with argon and oxygen plasmas to thin down the passive regions;
0 (c): Remove the aluminum mask and place the sample in platinum salt solution to
perform ion-exchange. This will stiffen the sample and make the following steps
feasible;
0 ((1): Pattern with photoresist (PR), where the targeted IPMC regions are exposed
while the passive regions are protected;
0 (e): Perform the second ion-exchange and reduction to form platinum electrodes in
active regions. To further improve the conductivity of the electrodes, 100 nm gold is
sputtered on the sample surface;
0 (t): Remove PR and lift off the gold on the passive areas. Soften the passive regions
with HCL treatment (to undo the effect of step (c));
o (g): Cut the sample into a desired shape.
7.1.1 Aluminum mask deposition
Since plasma will be used to selectively thin down the passive areas of the Nafion film,
the first step is to make an aluminum mask to protect the active areas from being etched.
Two shadow masks made of transparency films are used to cover both sides of the Nafion
film, in such a way that the passive areas are covered and the active areas are exposed. The
sample is then put into an e-beam deposition system, where aluminum can be deposited
at room temperature. Aluminum fihns of 200 nm thick are deposited on both sides of the
Nafion film. When the transparency masks are removed, the aluminum masks stay on the
active regions and the sample is ready for plasma etching.
166
Nafion Transparency mask
9 1
¢ o 4
E-beam aluminum deposition
Cross-section view Top view
(a) Deposit aluminum mask on both sides of Nafion film;
Plasma etching
:21...) an m
(c) Remove aluminum mask and perform
(b) Thin down passive area
ion—exchange to make Nafion stiffer;
with plasma etch;
((1) Deposit PR and then pattern (e) Perform another ion-exchange and
PR through lithography; electroless plating of platinum to create
IPMC electrodes; 2
Top view
(0 Remove PR and perform (g) Cut the patterned IPMC into a
final treatment; fin shape.
Cl N afion - Transparency Aluminum
Photoresist (PR) - Platinum - Gold
Figure 7.1: The process flow for monolithic fabrication of an MDOF IPMC actuator.
167
7.1.2 Plasma etching
Plasma treatment has been used for roughening the Nafion surface to increase the capaci-
tance of IPMC [51]. In this research, we use plasma to thin down the passive areas in the
MDOF IPMC actuator. The plasma etching system used in this research is Plasmaquest
Model 357. Experiments have been conducted to study the etching rates with different
recipes of gas sources. The etching rates with different recipes are shown in Table 7.1.
Table 7.1: Plasma etching with different recipes.
No. Ar 02 RF power Microwave power Etching rate
R1 20 sccm 30 sccrn 70 W 300 W 0.28 um/ min
R2 0 seem 50 sccm 70 W 300 W 0.22 um/ min
R3 50 sccm 30 sccm 70 W 300 W 0.51 um/ min
It can be seen that the combination of oxygen and argon plasmas can achieve a higher
etching rate than using the oxygen plasma alone. The oxygen plasma performs chemical
etching, which can oxidize the polymer and break Nafion into small molecules. The higher
oxygen flow rate, the higher etching rate but also the higher temperature on the Nafion film.
We have found that too strong oxygen plasma will damage the film because of overheating.
The argon plasma performs physical etching with high-speed, heavy argon ions. It can
remove the small molecules created by the oxygen plasma and roughen the Nafion surface,
which creates larger contact area for the oxygen plasma and thus accelerates the chemical
etching. The combination of oxygen and argon plasmas can achieve a high etching rate
with a low resulting temperature, which is critical to maintaining the original properties of
Nafion. In this research, the recipe R3 is adopted.
After several hours of plasma etching, the passive areas are thinned down to the desired
value. Then the sample is boiled in 2 N hydrochloride acid solution at 90 °C for 30 minutes
to remove the aluminum mask and to remove impurities and ions in Nafion. After that, the
membrane is further boiled in deionized (DI) water for 30 minutes to remove acid. Fig. 7.2
shows the Scanning Electron Microscope (SEM) picture of a selectively etched Nafion
168
sample, where the thinnest region is 48 um (down from the original 225 pm). The etched
sidewall is almost vertical (the angle is 88°). To roughen the active areas, the sample is
treated with plasma for 5 minutes. This roughening process can enlarge the metal-polymer
contact areas, which enhances the actuation performance of IPMC [51].
300 Olim
I
15.0kV15.3mm x180 SEiMl7/15f20091314 SOOum
Figure 7.2: SEM picture of a plasma-etched Nafion film.
7.1.3 Stiffening treatment
It is difficult to perform lithography on a pure Nafion film because Nafion swells in the de-
veloper, which usually contains water and organic solvent, and the swelling force can eas-
ily destroy the photoresist pattern. To address this challenge, we perform an ion-exchange
process to impregnate Nafion with large platinum complex ions (Pt(NH3)i+). Usually,
this ion-exchange process is used to absorb platinum complex ions for electroless platinum
plating [78]. However, we have discovered that, after the ion-exchange, the Nafion film
becomes stiffer and virtually non-swellable in water or acetone, which makes lithography
and the subsequent patterned electroless plating possible.
169
To perform ion-exchange, 25 ml of aqueous solution of tetraammine-platinum chloride
[Pt(NH3)4]C12 (2 mg Pt/ml) is prepared with 1 ml of ammonium hydroxide solution (5%)
to neutralize. The sample is immersed into the solution at room temperature for one day.
The formula of reaction is
[Pt(NH3 )4]2+ + 20H” + 2H+ (Nafion) 2°—°‘§ [Pt(NH3 )4]2+ (Nafion) + 2H20.
The impact of stiffening treatment will be studied in detail in Section 7.2.
7.1.4 Patterning of Nafion surface
AZ 9260 positive photoresist is selected to create thick PR patterns to protect the passive
areas of the actuator from electroless platinum plating. We spin-coat PR at 1000 rpm to
get 17 pm thick film and then bake it in oven at 90 °C for 2 hours. Since the Nafion film
alone is not rigid enough for spin-coating and it is easy to deform when baked in oven, an
aluminum frame is used to support and fix the Nafion film (25 mm by 25 mm), as shown
in Fig. 7.3. UV light with power density of 20 mw/cm2 is used to expose the sample with
the pattern mask for 105 seconds. After the sample is exposed, the PR is developed with
AZ 400K developer.
7.1.5 Electroless platinum plating and gold sputtering
The electroless platinum plating process is used to create thick platinum electrodes on the
active areas, which results in strong bonding between the metal and the polymer. Three
steps are taken for this process. First, another ion-exchange is performed to absorb more
platinum complex ions. Second, the sample is put into a bath with DI water at 40 °C.
Third, we add 1 ml of sodium boronhydride solution (5 wt% NaBH4) into the bath every
10 minutes and raise the bath temperature up to 60 °C gradually. After 30 minutes of
reduction, about 10 pm thick platinum electrodes will grow on the surface of the active
170
Sprew Aluminpm frame
o 0‘.
Q Nafion film 0
(25 mm by 25 mm)
Top view
I I l :1 1 Picture
Cross section view
Figure 7.3: Nafion film fixed by an almninum frame.
areas. To further improve the electrode conductivity, 100 nm thick gold is sputtered on
both sides of the sample. The surface resistance can be reduced by half with this gold-
sputtering step. Since PR patterns are still on the surface, the passive areas are protected.
When the PR is removed with acetone, the gold on the passive areas will be lified off.
7.1.6 Final membrane treatment
After electroless plating, the Pt complex ions in active regions are reduced to platinum
metal. But the Pt complex ions in the passive regions are still there, which makes the
passive regions stiff. To facilitate 3-D deformation, we need to undo the eflect of step (c)
to replace the Pt complex ions with H+. This can be achieved by simply boiling the sample
in 2N hydrochloride (HCL) acid [50]. The formula of reaction is
[Pt(NH3)4]2+(Nafion) + 2H+ 1&0 [Pt(NH3)4]2+ + 2H+(Nafion).
171
After the sample becomes flexible, it is put into sodium or lithium ion solution (1 N)
for one day to exchange H+ with Na+ or Li+ ions, to enhance actuation of IPMCs. A
fabricated IPMC membrane is shown in Fig. 7.4.
7.2 Impact of Stiffening Treatment
Stiffening treatment is an important step in fabrication of MDOF IPMC actuators. In this
section, we study the impact of the ion-exchange process on the stiffness and swellability
of Nafion films, and its implication in lithography and the overall fabrication process.
7.2.1 Stiffness change
The experimental setup for the stiffness measurement is shown in Fig. 7.5. The film is
fixed at one end by a clamp. A load cell (GSO-10, Transducer Techniques) is mounted
on a moving stage (U-SVRB-4, Olympus), which can be manipulated by hand to generate
smooth horizontal motion. When the stage is moved, the load cell pushes the Nafion film
to bend, and the restoring force at the tip of the film is measured. A laser displacement
sensor (OADM 2016441/Sl4F, Baumer Electric) is used to measure the corresponding tip
displacement. The resolution of the laser sensor is 20 um and the resolution of the load
cell is 0.05 mN. The setup is placed on an anti-vibration table (LW3048B-OPT, Newport).
A dSPACE system (DS1104, dSPACE) is used for data acquisition. The spring constant of
the Nafion film is calculated as
k:—
d,
where F and d are the tip force and the tip displacement, respectively.
We have measured the stiffness of Nafion—l 17 (183 um) and Nafion-l 110 (240 um)
before and after the ion-exchange process. The ion-exchanged Nafion fihns are dried in
air before the experiments. Fig. 7.6 shows the force and displacement data for each case,
172
Platinum electrode
10 >1 Ill
Passive area
Active area
300 0pm
l‘fi—F—r—l—‘lfi—lfi—W—‘I
15.0kV15.3mm x180 SE(M) 7/1 512009 13:34 300um
(e)
Figure 7.4: Patterned IPMC membrane. (a) Top view picture; (b) planar dimensions of the
membrane; (c) SEM picture of the cross section.
173
Di.
Laser sensor '
Moving direction
m *“r
Horizon indicator
Load cell
Moving stage
Anti-vibration table
Figure 7.5: Experimental setup for measuring the stifliiess of a Nafion fihn.
together with the results using linear fitting. The spring constant of the cantilever film is
_YWfl
_ 4L3’
where Y, W, L, and T are the Young’s modulus, width, length, and thickness of the Nafion
film, respectively. Based on the measured spring constant and dimension parameters, we
can calculate the Young’s modulus. Table 7.2 shows the spring constant, dimensions, and
Young’s moduli of the Nafion films. It can be seen that the stiffness of an ion-exchanged
Nafion film increases by about 2-3 times, compared to that of pure Nafion.
Table 7.2: Spring constant, dimensions, and Young’s moduli of the Nafion films in stiffness
testing. The ion-exchanged films are identified with an asterisk.
k W L T Y
Nafion—II7 1.7N/m 26mm 20mm 183 pm 355 Mpa
Nafion—117* 6.1N/m 26mm 19mm 183 pm 1.05 Gpa
Nqfion —N1110 4.5 N/m 25 mm 20 mm 254 pm 354 Mpa
Nafion—~N1110* 10.7 N/m 25 mm 21 mm 254 pm 966 Mpa
174
100 . . i i . .
90 - * Experimental data for Naflon-N1110' a
80 _ ' ' " Simulation data for Nafion-N1110' _
D Experimental data for Naflon-Nf110 ’ .
70 - - - - Simulation data for Nafion-N1110 , ’ -
2 , ’
E 60 - t ’ -
v ’ I
g 50 b ’ I I ‘1
.9 ’i‘
E 40 ’ ’ ’ a; "
30 ,. I i a ‘_ a “a
I ' an
I ’ a
20 i- ’ ’ :- * ' I’ ’ ‘. -l
I rr
10’- r’it’ .. '0‘ flélwca ‘
2’2“ ‘ '
or 1 J 1 1 1 1
0 1 2 3 4 5 6 7
Tip displacement (mm)
(a)
60 r L ,
* Experimental data for Naflon-N117' ‘
50 _ " ' ' Simulation data for Naflon-N117' I I ’ _
:1 Experimental data for Nation-11 7 4’
- - - Simulation data for Naflon-N117 I ’
A 40 - I -
Z I
e ,«’
g 30 * fi’ ’* "i
l- 20 . 1" I a
I
* I ’ ’ a
I a
10 _ I ’ a .. er I T T U 4
I Q— I
’ 'FW 9 f]
at i ..U- " ’3'
w "Lr ? ’ l l l l l l l
0 1 2 3 4 5 6 7 8 9
Tip displacement (mm)
(b)
Figure 7.6: Results of stiffness measurement. (a) Nafion—Nl l 10; (b) Nafion-l 17.
175
7.2.2 Swelling capacity change
We have further investigated the swelling capacity of Nafion before and after the ion-
exchange step. Four samples as listed in Table 7.3 have been tested. Measurements are
taken in the following steps. First, the surface areas of the samples in the dry condition
are measured. Second, the samples are immersed in water for 5 minutes and then taken
out for surface area measurement. Third, the samples are dried with paper towel and in air
before being immersed in acetone. After 5 minutes, the samples are taken out of acetone
and their surface areas are measured again. Table 7.3 shows the percentages of surface area
change comparing to the original size, for the four samples, after being soaked in water and
in acetone, respectively.
Table 7.3: Surface area changes of Nafion films in water and acetone. The ion-exchanged
films are identified with an asterisk.
With water With acetone
Nafion — 11 7 +19.1% +59.4%
Nafion — I 1 7* +07% +08%
Nafion —N1110 +20.1% +55.9%
Nafion—N1110* +1.2% +1.6%
It can be seen that while pure Nafion can expand by about 20% and 60% when soaked in
water and in acetone, respectively, ion-exchanged Nafion experiences only 1-2% expansion
under the same conditions. In other words, ion-exchanged Nafion has very low swellability
in solvents, which is important in lithography based patterning of the film. One possible
reason for the significantly reduced swelling capacity is that the film becomes stiffer and
the swelling force is unable to enlarge the volumn of the film. But the precise explanation
of the phenomenon requires further study.
176
7.2.3 Impact of ion-exchange on lithography
The impact of ion-exchange process on lithography has been investigated. First, we per—
formed lithography on pure Nafion. When the sample was put into the developer, the fihn
swelled and the PR patterns were destroyed by the swelling force, as shown in Fig. 7.7(a).
Then we performed lithography on ion-exchanged Nafion. The patterning result was sharp,
as shown in Fig. 7.7(b). It thus has demonstrated that the lithography of Nafion can be
dramatically improved by the stiffening treatment using ion-exchange.
7.3 Characterization of Fabricated Artificial Fins
7.3.1 Characterization method
We have characterized the performance of fabricated MDOF IPMC actuators on producing
sophisticated shape change. To capture the deformation, one may use multiple laser sen-
sors to detect the bending displacement of active areas [43]. However, when the actuator
generates large, complex deformation, some laser sensors can lose measurements. This
approach is also expensive. Another approach is to use a CCD camera to capture the video
of the actuator movement and then use image processing to extract the actuator movement
[17].
The experimental setup we use to characterize MDOF IPMC actuators is shown in
Fig. 7 .8. The fabricated IPMC membrane is cut into a pectoral fin shape and its base is
fixed by a multi-electrode clamp. The dimensions of the pectoral fin is shown in Fig. 7.8.
To minimize the contact resistance, gold foils (0.1 mm thick) are used to make the contact
electrodes. A CCD camera (Grasshopper, Point Grey Research) is oriented toward the
edge of the fin. Sinusoid voltage inputs of the same amplitude and frequency but different
phases are generated by the dSPACE system and applied to individual IPMC regions. The
tip bending displacements of active areas are detected using the image edge detector (Vision
177
- 'rmi
(b)
Figure 7.7: Lithography results. (a) With pure Nafion; (b) with ion-exchanged Nafion.
178
Assistant 8.5, National Instruments).
40mm:
'<-—>n
dSpace
Computer
NI Vision
Assistant 8.5
h ---------------------
Figure 7.8: Experimental setup for characterizing MDOF IPMC actuators.
7.3.2 Demonstration of twisting and cupping
Actuation experiments are first conducted in air. The phase differences between the actu-
ation signals applied to the three IPMC regions could be arbitrary. Since the goal here is
to demonstrate sophisticated deformation and not to optimize the control input, we have
restricted ourselves to the following particular class of phase patterns: the top IPMC leads
the middle IPMC in phase by 4), while the bottom IPMC lags the middle IPMC in phase by
4). For such a phase pattern, we call it phase «1: in short.
With all three IPMCs receiving inputs of the same phase (i.e., 0° phase), the fin gen-
erates bending. As this is not surprising (a single IPMC produces bending), we will not
present the detailed results on bending here. An example of twisting is shown in Fig. 7.9,
where the artificial fin has the same Nafion thickness, 85 pm, in the active and passive
areas. The voltages applied are sinusoidal signals with amplitude 3.0 V, frequency 0.3 Hz,
179
and phase 90°. The actuator clearly demonstrates a twisting motion. In order to quan-
tify the twisting deformation, we define the twisting angle 0, which is formed by the line
connecting the tips of top and bottom IPMCs with the vertical line m-n, as illustrated in
Fig. 7.10(a). Note that the tip displacements d1, d2, d3 of the IPMC regions are extracted
from the video. Fig. 7.10(b) shows the time trajectories of the displacements and the corre-
sponding twisting angle. The twisting angle achieved is 16° peak-to-peak, showing promise
in robotic fish applications.
We have also verified the actuator’s capability to generate the cupping motion. Fig. 7.1 1(a)
illustrates the definition of the cupping angle a, formed by the two lines connecting the tip
of the middle IPMC to the tips of the top and bottom IPMCs. Fig. 7.11(b) shows the tra-
jectory of the cupping angle for the same sample mentioned above, where the actuation
voltages have amplitude 3.0 V, frequency 0.3 Hz, and phase 180°.
7.3.3 Impact of the thickness in passive and active areas
To study the effects of thicknesses in active areas and passive areas on the actuation perfor-
mance, we have fabricated 5 samples with different thicknesses in active areas and passive
areas. Table 7.4 shows the thicknesses of the MDOF IPMC actuators. All actuators have
the same planar dimensions as specify in Fig. 7.4(b). To compare the actuation perfor-
mance of actuators with different thicknesses in the active areas, we have fabricated three
samples (SI, SZ, S5), where each sample has the same thickness in its active and passive
areas. They are fabricated from Nafion-l 110, Nafion—l 17, and Nafion-1135, respectively.
To study the effects of different thickness in passive areas, we have fabricated three samples
(S2, S3, S4) with the same thickness in the active areas (170 pm) but different thicknesses
in the passive areas.
The twisting angles generated by different samples with 0.3 Hz, 3 V and 90° phase volt-
age signals are shown in Fig. 7.12. From Fig. 7.12(a), for samples with uniform thickness,
the thinner the sample is, the larger the deformation. This can be explained by that, under
180
Figure 7.9: Snapshots of an actuated MDOF IPMC actuator, demonstrating the twisting
motion.
181
E
E
‘ ........
F/p1 .3
o
,
9 x’ 0 Y
1"]. l g
“d2 p2 X 8’
: E
x 1
p3 " g ‘3:
4.-.s E
.1 d3 s E
n5 .2
.2
Front cross section view of pectoral fin Time (s)
(a) (b)
Figure 7.10: (a) Definitions of the tip displacements and the twisting angle; (b) trajectories
of the tip displacements and the twisting angle corresponding to the voltage inputs as in
Fig. 7.9.
181 . . f .
180
’5?
/ 2
~x/pr 3179- ~~
1' O 3
_ y 2
l g 178 -
a
m x E177 . ,
x _ P2 :3;
/ 176-
/Pa 175 i
0 1 2 3 4 5 6
Front cross section view of pectoral fin Time (s)
(a) (b)
Figure 7.11: (a) Definition of the cupping angle; (b) the trajectory of the cupping angle
for a sample with 85 um thickness in both active and passive areas (voltages: 3 V, 0.3 Hz,
180°).
182
Table 7.4: Thicknesses of MDOF IPMC actuators.
Thickness in active area Thickness in passive area
S1 240 pm 240 um
S2 170 pm 170 um
S3 170 pm 120 um
S4 170 pm 60 um
SS 85 pm 85 um
the same voltage, a thinner sample experiences higher electrical field, and that a thinner
sample is more compliant. From Fig. 7 . 12(b), it can be seen that for samples with the same
thickness in active areas, with thinner passive regions, the deformation gets larger under
the same voltage inputs. This has thus provided supporting evidence for our approach of
modulating mechanical stiffness through plasma etching.
7.3.4 Impact of actuation signals
We have further experimented with actuation signals with different phases and amplitudes.
Table 7.5 shows all the actuation signals we have used in the experiments. We have se-
lected SI and S5 as the test samples. All the signals have the same frequency 0.3 Hz. To
study the effects of phase on the actuation performance, the voltage signals from Control
#1 to Control #5 have the same amplitude (3 V) but different phases. To understand the
performance of an MDOF IPMC actuator under different voltage levels, the voltage signals
from Control #5 to Control #7 have the same phase (90°) but different amplitudes.
Fig. 7.13(a) shows the twisting angle of SI actuated under different voltage levels but
with the same phase and frequency. While as expected, the higher the voltage, the larger the
twisting, the gain in deformation does not appear to be linearly growing with the voltage
level. Such nonlinearities will be examined in our future work. Fig. 7.13(b) shows the
twisting angle of S5 actuated by the same voltage amplitude but different phases. From the
figure, 900 appears to be the best phase to generate the twisting motion, the reason of which
183
Twisting angle (degree)
Time (s)
(a)
Twisting angle (degree)
-6 1
Time (s)
Figure 7.12: Twisting angles generated by different samples with 0.3 Hz, 3 V and 90° phase
sinusoid voltage signals. (a) MDOF IPMC actuators with different thickness in both active
areas and passive areas; (b) MDOF IPMC actuators with the same thickness in active areas
(170 pm) but different thicknesses in passive areas.
184
Table 7.5: Actuation signals.
Frequency Amplitude Phase
Control #1 0.3 Hz 3 V 180°
Control #2 0.3 Hz 3 V 120°
Control #3 0.3 Hz 3 V 60°
Control #4 0.3 Hz 3 V 0°
Control #5 0.3 Hz 3 V 90°
Control #6 0.3 Hz 4 V 90°
Control #7 0.3 Hz 8 V 90°
will be explored in our future modeling work.
7.3.5 DPIV study on MDOF IPMC actuation in water
In the interest of robotic fish applications, we have also conducted preliminary study of
underwater operation of the MDOF IPMC actuators. Digital Particle Image Velocimetry
(DPIV) system is used to observe fluid motion generated by the actuator. In a DPIV system,
small particles are dispersed in a fluid and a laser sheet is created in the fluid to illuminate
the particles. Processing of images taken in quick successions can reveal the movement of
particles and provide information about the flow fluid. We have tested sample S5 in water,
by applying voltage signals (4 V, 0.3 Hz, and 90° phase) to the actuator. Fig. 7.14(a) shows
the snapshots of the MDOF IPMC actuation in water. Fig. 7.14(b) shows the velocity field
of the water around the actuator. It demonstrates that the MDOF IPMC actuator can make
3-D deformation in water and can generate some interesting flow patterns around it. The
connections between the flow patterns and the actuator deformations are a subject of future
investigation.
185
2 . . r
1.5_ '---‘V=3Volt
- - -V=4 Volt
’ \
if 1- 1’ 7'“ —V=8 Volt ,
2 I \ . ' '
8) ’ \\ I
‘o 0.5' I ' \ ’ ’
V I I ‘ ~"'s, \ "r
2 l ." ‘ ‘ .”
g o» ’ ‘ ,
(u I \s “.a
C) ‘ v - '
s —o.5~ ‘\ ' .’
ZS \ I
E ‘1 \\ ’ I ’
-1.5-
-2 I
0 1 2 3 4
Time (s)
(a)
8 . . . "'""Phase=0°
' ,.\ "“Ph - °
‘1 .‘ ase-60 ~"\
6’ L' ' ‘1 ""‘Phase=90° ! ' -
' ‘ _ o .I
3 4_ r p 1‘. Phase—120 !
2 i. \_ Phase=180° v
a) .
a)
3
.03
or
r:
to
c»
E
E
E
3
Time (s)
(b)
Figure 7.13: (a) SI actuated by voltage signals with difl‘erent amplitudes but the same
frequency (0.3 Hz) and phase (90°); (b) S5 actuated by voltage signals with the same am-
plitude (3 V) and frequency (0.3 Hz) but difierent phases.
186
(a)
Figure 7.14: DPIV study of MDOF IPMC actuator operating in water. (a) Actuation of the
actuator in water (the edge of the actuator is highlighted); (b) Velocity field of the fluid.
187
7.4 Chapter Summary
In this chapter we have presented a new process flow for monolithic, batch-fabrication of
MDOF IPMC actuators. The methodology effectively incorporates standard techniques
for IPMC fabrication and lithography-based micromachining processes. The actuator con-
sists of multiple IPMC islands coupled through a passive membrane. The size and shape
of each IPMC island are defined through photolithography and thus the approach is scal-
able. A key innovation in the developed process is to stiffen the Nafion film through an
ion-exchange step, which virtually eliminates swelling in solvents and enables success-
ful patterning. Tailoring the thickness and thus the rigidity of the passive area is another
novel aspect of the proposed fabrication method. The fabrication method has been applied
to manufacture prototypes of biomimetic fins, with demonstrated capability of producing
complex deformation modes such as twisting and cupping. I
The prototypes fabricated in this work have dimensions of centimeters because of our
interest in biomimetic actuation for robotic fish applications. The developed fabrication
process, however, can be directly applied to produce microscale devices since it is based
on microelectromechanical systems (MEMS) processes. We also note that, while commer-
cially available Nafion films have been used as the base material in our work, the approach
can be easily extended to Nafion membranes obtained through casting, and to other types
of ion-exchange membranes (e.g., Flemion).
Future work includes understanding and modeling of MDOF IPMC actuators, by ex-
tending the authors prior work [21] on modeling of IPMC beams. The interactions between
the active IPMC regions and the passive regions will be a focus of study. We will also in-
vesti gate systematically the hydrodynamics in underwater operation of such MDOF IPMC
actuators using combination of analytical modeling, computational fluid dynamics (CFD)
modeling, and DPIV studies, and explore the use of these actuators in biomimetic robotic
fish.
188
Chapter 8
Conclusions and Future Work
8.1 Conclusions
In this work, a systems perspective has been taken to address the challenges in realizing
IPMC-based smart microsystems. To obtain a faithful and practical mathematical model for
IPMC, we have developed a physics-based, control—oriented modeling approach for IPMC
sensors and actuators. The effects of distributed surface resistance have been considered
in the models. The models are amenable to model reduction, geometrically scalable, and
practical in real-time control. The actuation model of IPMC has been applied to the model-
ing of robotic fish propelled by IPMC, which has further validated the modeling approach.
Since traditional sensors cannot be embedded easily into bio/micro systems, we have de-
veloped a compact sensing scheme for IPMC. We have further developed a process for
monolithic fabrication for IPMCs capable of generating complex deformations. Artificial
pectoral fins are fabricated and characterized, which have demonstrated bending, twisting,
and cupping motions.
189
8.2 Future Work
Future work can be pursued in the following three directions. First, the fabrication of the
artificial pectoral fins can be improved through enhanced lithography and electrode plating
techniques. The analysis of DPIV results for the pectoral fins will be another integral part
of this work, which can lead to the control strategies for the fin in robotic fish applica-
tions. Second, it will be interesting to extend the developed planar fabrication technology
to 3-dimensional fabrication technology which can be used in microfabricating more com-
plex bio-inspired IPMC materials, such as artificial lateral lines. The lateral line will have
multiple micro IPMC sensing hairs standing on the substrate to sense fluid flow. Third,
nonlinear models are necessary when a relatively high voltage is applied and a large de-
formation is generated. Many nonlinearities such as hysteresis, nonlinear elasticity of the
polymer, and nonlinear capacitance of IPMC are involved in IPMC. Model-based nonlin-
ear control should be explored to effectively control IPMC in applications involving large
deformations.
190
1
Appendix A
Appendix for Chapter 2
A.1 Derivation of impedance model (2.31)
From (2.28) and (2.29),
i(z s) : ¢(hh,(zs) sWKey(s)(s+K)
p ’ )(SS+Ktanh(1’()))
___ (1:3) /—z,rsdr——IP(ZS)) (S),
where the second equality is from (2.27) and (2.34). This results in
ip(z,s) 2 (VS) ——/Oz-;—l/is (T,s)dt') T123275.
From (2.26) and (2.30), one obtains
4) (h,Z,S) _ 4) (_hizas) +2lp(Z,S)l'§/W
Rip/W
2d) (h, z ,s)+2ip(z, s)r’2/W
2 213:0? Warm),
ik (275) :
191
(Al)
(A2)
where the last equality is from (2.27).
Combining (2.25), (A.l), (A.2), one gets
_8is (2,5) : A(s)V(s) _
T __2_ 3(5) [0‘ is(t,s)d1', (A3)
where A(s) and B(s) are as defined in (2.32) and (2.33).
Eq. (A3) is an integro-difierential equation for is. To solve this equation, we intro-
duce the unilateral Laplace transform for functions of the length coordinate z. The new
Laplace variable will be denoted as p since 3 has already been used for the transform of
time functions. For instance, the transform of is(z,s) will be defined as
Is(p,s)§/0 is(z,s)e-pzdz.
Now perform the Laplace transform with respect to the 2 variable on both sides of (A.3).
Using properties of Laplace transforms, one gets
p1.5—(-f,fl. (AA)
Solving for 15(p,s), one obtains
_ p . __ 1 A(S)V(S)
1.(p,s)—p———2_B(S)z.(o,s) p2_3(s) 2 , (A5)
which can be rewritten through partial fraction expansion as:
15(p,S) :
2 p—W+m
0.5 0.5
130,5 ————- —-——-— , A.6
+ ‘ ’b—WUWW) ‘ ’
192
A(s>V ( q. (s) q2(s> )
with
1
‘11 (S) = )3 q2(S
2\/:3( 2‘ / 8(5)
The surface current is(z, s) is then obtained from (A.6) using the inverse Laplace transform
OfIs (p,S)I
. _ . A (S) V (S) .
13 (2,5) _ 1s (O,s)cosh( B(s)z) —— m)— smh( B(s)z) . (A.7)
Using the boundary condition iS(L,s) = 0, one obtains:
V (3)/1 (s) tanh (ML)
is (0’s) 2 2 B(s)
(A.8)
193
Appendix B
Appendix for Chapter 4
B.1 Derivation of M(L,s), FC(L,s), Fd(z,s) in Section 4.2.3
Based on the principle of replacement in Section 4.2.3, one gets, for 0 g 2 S L,
L
Mme (23S) = / E: (r) (r -z> dr +F.(L,s> + M(L,s).
2
At 2 = L, one gets
M(L,S) = MPMC(L73):
which is (4.23). Then one takes the derivative with respect to 2 on both sides of (4.22):
3M1mc (2,5) __ L 3(Fd(T)(T—Z))
az _ f2 32 dt—FC(L,s)
L
= —/ Fd(t)d'r—FC(L,s). (B.1)
Letting z = L in ( B.l), one gets
m.» = J’Mngt’S) 12:. = o.
194
Finally, Eq. (4.22) is obtained by taking the derivative with respect to 2 on both sides of
(8.1).
B.2 Derivation of H f,- (5)
With (4.22), (4.21), and definitions of a, b, c in (4.27) and (4.28), Eq. (4.24) can be written
as
f1,(s [Ag—)fl (acosh (cz)— -bsinh (02)) Q. [<14 Malawian/ch] .111...) 1
M’ + [(1), (Lo) kb + (P1(L0)kc] W (Lo,S)
i=1
197
(3.8)
(13.9)
(B.10)
(13.11)
(13.12)
and with (4.32), the slope can be written as
MH215) = Hld(5)V(3) -
i Mm; (Lo)Q1 (S) [ a. ((2.1)
Let’s start with the case when k > 0. Because 1(0) 2 —k < 0, 1(1)) is continuous in
(—-IZ,+°°) and
a
b
there exist pl 6 (—Z’O)’ p2 E (O,+oo) such that x(p1) = O and x(p2) = 0. Since
= P
ap+b’
X'(P) (CZ)
withp> —g,wecmgetx’(p)>0whenp>0andx’(p)<0when0>p> ~53. Sox(p)
is monotonically increasing in (0, +00), and monotonically decreasing in ( — — , 0). Then p,
a
and p2 should be unique. When k = O, and 75(0) = 0, then p1 2 p2 = 0. El E1
199
C.2 Derivation of Eq. (5.17)
D
Define Dépz —p1. Eq. (5.12) can then be written as V = b° From Fig. 5.4,
Pz-Pl
= a, C3
772 - Th ( )
where n, and 172 are the n-coordinates corresponding to p1 and p2, respectively, in Fig. 5.4.
Eq. (C.3) implies
b a D a = a. (C4)
;2' (“1(131’2+ ‘) ‘1“(‘5P' + 1))
With p2 = D + p, and (C4), one can solve p, in terms of D,
7'5— - 5,1) > 0
P1 = eb _ 1 - (C-5)
O,D=O
When D -—> O, with 1’ Hospital’ 3 rule,
a
_ 1 b
So p1 is still a continuous function of D. Since
3111(3 +1)=fl— (C6)
from (C5), one can get
b D 1
71m 3 a —b +1 =— GD 3 —k. (07)
a b -D a -D a
eb — 1 eb — 1
200
Since D = bV, one can get Eq. (5.17) from Eq. (C.7). Note that with 1’ Hospital’ 8 rule,
one gets
So k is a continuous function of V.
201
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