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Smart Mater. Struct. 21 (2012) 085002 (12pp) doi:10.1088/0964-1726/21/8/085002

Mitigating IPMC back relaxation throughfeedforward and feedback control ofpatterned electrodes

Maxwell J Fleming1, Kwang J Kim2 and Kam K Leang1,3

1 Electroactive Systems and Controls Laboratory, Department of Mechanical Engineering,University of Nevada, Reno, NV 89557-0312, USA2 Active Materials and Processing Laboratory, Department of Mechanical Engineering,University of Nevada, Reno, NV 89557-0312, USA

E-mail: kwangkim@unr.edu and kam@unr.edu

Received 17 March 2012, in final form 10 May 2012Published 13 July 2012Online at stacks.iop.org/SMS/21/085002

AbstractWith low driving voltage (<5 V) and the ability to be operated in aqueous environments, ionicpolymer–metal composite (IPMC) materials are quickly gaining attention for use in manyapplications including soft bio-inspired actuators and sensors. There are, however, drawbacksto IPMC actuators, including the ‘back relaxation’ effect. Specifically, when subjected to anexcessively slow input, the IPMC actuator will slowly relax back toward its original position.There is debate over the physical mechanism of back relaxation, although one prevalent theorydescribes an initial current, caused by the electrostatic forces of the charging electrodes, whichdrives water molecules across the ion exchange membrane and deforms the IPMC. Once theelectrodes are fully charged, however, the dominant element of the motion is the osmoticpressure, driving the water molecules back to equilibrium, thus causing back relaxation. Anew method to mitigate back relaxation is proposed, taking advantage of controlled activationof patterned (sectored) electrodes on the IPMC. By actuating sectors in opposing directions,back relaxation can be effectively canceled out. An integrated feedforward and feedbackcontroller is employed based on this concept, and is shown to minimize back relaxation, whilereducing the input voltage required, as compared to the case of the non-sectored IPMC.Experimental results show nearly an order of magnitude reduction in the tracking errorcompared to the uncompensated case, and that the IPMC’s position can be maintained over aperiod of 60 and 1200 s with minimal evidence of back relaxation.

(Some figures may appear in colour only in the online journal)

1. Introduction

The ionic polymer–metal composite (IPMC) material is aclass of innovative electroactive polymer that offers combinedsensing and actuating ability in a lightweight and flexiblepackage. Setting IPMCs apart from other active materials areits soft and flexible structure, low driving voltage (<5 V),and ability to operate in aqueous environments. These traitsmake IPMCs very attractive for underwater robotics [1–4].

3 Author to whom any correspondence should be addressed.

Biomedical applications are also possible, including artificialmuscles [5] and endoscopy [6]. IPMCs are typically used ina simple cantilever configuration to create bending motion;however, patterning of the IPMC electrodes has been exploredto alter the bending behavior. For instance, by applyingdifferent driving signals to different sectors of the IPMC,complex motion, specifically twisting, can be created [7, 8].

In certain applications, the benefits of IPMC actuatorsare overshadowed by unwanted dynamic effects andnonlinearities [9], the most obvious of which is ‘backrelaxation’, which can cause significant positioning error

10964-1726/12/085002+12$33.00 c© 2012 IOP Publishing Ltd Printed in the UK & the USA

Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 1. Concept: (a) independently controlling each sector to produce a net canceling effect and (b) corresponding experimental stepresponses.

when left uncompensated. When subjected to a DC inputvoltage, a Nafion R©-based IPMC in the traditional cantileverconfiguration experiences a relatively quick deflection towardthe anode side, followed by a slow period of relaxation towardthe cathode side [10]. It is widely accepted that the so-calledback relaxation effect is due to diffusion forces acting onthe solvent within the ion exchange membrane. This effectcan lead to significant positioning error and prevents IPMCactuators from being used in low-frequency applications. Forexample, IPMCs have been suggested for use as pectoralfins on robotic fish for enhanced maneuverability [11]. If alow-frequency input had to be used, back relaxation couldcompromise the maneuver if it was not compensated for. Aspreviously mentioned, IPMCs have also gained great attentionfor potential as artificial muscles, with one specific applicationbeing a hand prosthesis [5]. While IPMCs could potentiallyhandle dynamic muscle actions, back relaxation would makestatic tasks (like holding an object with artificial fingers)nearly impossible. In [12], it was proposed to use IPMCsto guide a cardiac catheter. In such a sensitive application,back relaxation could be catastrophic, causing the catheter tolose course and perhaps injure a patient. Thus, minimizing oreliminating back relaxation can enhance the performance ofsystems and devices that utilize IPMCs. Herein, a new methodto mitigate back relaxation is proposed, taking advantage ofcontrolled activation of patterned (sectored) electrodes on theIPMC.

In the literature, many approaches have been proposed toaddress the back relaxation effect. For instance, one approachis using a different polymer membrane, such as Flemion.In Flemion-based IPMCs, after a fast deflection when a DCvoltage is applied, the actuator slowly continues to bend inthe same direction [10]. Of suitable membranes for IPMCs,however, Nafion R© is the most commercially available, makingit the material of choice. The effect of solvents has also beenexplored, and it has been shown that the proper combinationof solvent and cation can eliminate back relaxation [13].Unfortunately though, most practical IPMC applicationswould require that the actuator be immersed in water, meaningthe IPMC would have to be sealed to retain a different solvent.

Different electrode development processes, for example theinsertion of a palladium buffer between the ion exchangemembrane and platinum, have shown promise in reducingback relaxation [14]. Sensor-based feedback control has alsobeen used to counter back relaxation [15, 16]. With traditionalfeedback, however, the input voltage must be constantlyincreased to maintain a position. This is undesirable primarilybecause a voltage threshold will eventually be crossed atwhich the solvent in the actuator will undergo electrolysis. Asthis can permanently damage the actuator, it is best to keepthe input below the electrolysis point. The typical potentialassociated with electrolysis of water at 25 ◦C and a neutralpH is 1.23 V. For IPMCs, however, this potential might belarger, as dictated by the experimental conditions and materialproperties. In [17] the electrolysis potential for an IPMCwas identified as 1.8 V. Past this point the resulting reactionsplits water into hydrogen and oxygen and results in a highercurrent draw than is necessary for IPMC actuation. This isessentially wasted current, and leads to inefficient actuation.The degree to which electrolysis occurs in the actuation cycleis determined by the magnitude of the applied voltage andthe shape or dynamic properties of the input. Increased inputvoltage also undermines one of the most attractive qualities ofthe IPMC: low driving voltage. Greater input voltages lead tounnecessary power consumption and greater risk of damage ofboth the actuator and the surrounding environment, especiallyin biomedical applications.

The contribution of this work is a new method tomitigate back relaxation, utilizing sectored IPMCs. Thenew method described herein involves driving differentsectors of the IPMC in opposing directions, essentiallycanceling out the back relaxation. Figure 1 illustrates thebasic concept, displaying the relaxation experienced by eachsector when supplied with opposing inputs (u1 and u2),and the net canceling effect. A model-based feedforwardcontroller is developed based on this principle, and isshown to significantly reduce tracking error due to backrelaxation, improving the tracking error of a steady referencetrajectory by approximately 85%. To further improve theresponse, feedback control is integrated with the feedforward

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 2. IPMC model: (a) an electromechanical input–output model, where the input voltage u(t) is mapped to charge q(t), and then thecharge maps to curvature k(t), and (b) a resistor–capacitor model of the electrical behavior between u(t) and q(t).

controller to account for nonlinearities, disturbances, andother undesirable behaviors not captured in the model.The integrated feedforward/feedback controller brings thetracking error improvement to nearly 97%, while significantlyreducing the required input voltage, as compared to the purefeedback case.

The remainder of the paper is organized as follows.First, a general actuation model is discussed, and is thenadapted for sectored IPMCs in section 2. The developmentof the integrated feedforward/feedback controller is given insection 3. The experimental setup is described in section 4.The experimental results of these control strategies arepresented in section 5. Finally, concluding remarks are madein section 6.

2. Modeling

2.1. Conventional unsectored IPMC model

In the modeling of IPMCs, it is convenient to separate theelectrical and mechanical characteristics of the material. Asdepicted in figure 2(a), an electrical model (E) can be used torelate input voltage, u(t), to the surface charge, q(t), and thena separate electromechanical model (M) can relate the inputcharge to the mechanical output, curvature, k(t).

Simple resistive–capacitive (R–C) models have been usedto describe the electrical response of IPMCs [1, 18]. Moreadvanced models have also been developed representing theIPMC as a series of R–C circuits to take into accountthe voltage drop down the length of the IPMC [19, 20].A thorough nonlinear model was developed by Chen andTan [21]. For simplicity, the R–C model developed in [18] isadapted here to describe the electrical response of the IPMC,as shown in figure 2(b). In this representation, u(t) is thedriving voltage and R0 is the internal resistance of the voltagesource. The R1−C branch accounts for the capacitive behaviorof the IPMC, while R2 accounts for the current that continuesto flow even after an electrical steady state is reached. Inthis representation, the charge on the IPMC surface can be

determined from the following equation:

Rdq(t)

dt= αu(t)−

q(t)

C, (1)

where

R = R1 +R0R2

R0 + R2, (2)

and

α =R2

R0 + R2. (3)

Converting to the Laplace domain and assuming zeroinitial conditions, the relationship between the input voltageU(s) and surface charge Q(s) can be given by the followingtransfer function:

Q(s)

U(s)=

α/R

s+ 1/τ1, (4)

where τ1 = RC is the time constant of the circuit.Although many models have been developed to describe

the actuation of IPMCs, most do not account for theback relaxation behavior. Models have been developedusing differential equations to predict back relaxation withreasonable accuracy [6]; however, the structure is best suitedfor finite element analysis (FEA) simulation. A simpledescription of back relaxation is given in [18]. In this model,the IPMC motion is governed by three factors: (1) the chargeaccumulated on the electrodes, (2) the rate of change ofsurface charge, and (3) the curvature. If an IPMC is subjectedto a step voltage, there is a quick initial forward motion whichis dominated by the ionic flux through the IPMC. Mobilecations are drawn to the charging cathode, and carry watermolecules across the membrane. This leads to swelling onthe cathode side of the IPMC and bending toward the anode.This ionic current is related to the surface charge rate, andas such, the forward motion fades as the electrodes becomefully charged. At this point, back relaxation sets in, caused bythe pressure gradient due to the water imbalance inside theIPMC. The water pressure lessens as the curvature decreasesand the water molecules move closer to equilibrium. The back

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 3. Sectored IPMC: (a) cantilever configuration with twoinputs u1(t) (Sector 1) and u2(t) (Sector 2) and (b) the input–outputmodel in the Laplace domain.

pressure is opposed by the lingering electrostatic force ofthe surface charge acting on the hydrated cations. When theelectrostatic force balances the water pressure, motion ceases.This is one reason why the steady state position of the IPMCis rarely the same as the starting position. In this description,the motion of the actuator is modeled by

dk(t)

dt= K1

dq(t)

dt−

1τ2[k(t)− K2q(t)], (5)

where τ2 is the time constant of the relaxation, and K1

and K2 are coefficients weighting the effects of the surfacecharge rate and of the surface charge, respectively. Thissingle equation effectively describes the IPMC’s actuationbehavior, and needs only the surface charge and charge rate asinputs. Referring to the simple block diagram in figure 2(a),equation (5) represents the electromechanical model, M.

Converting equation (5) into the Laplace domain andassuming that the initial curvature is zero, that is k(t = 0) = 0,the linear charge solution equation (4) can be substituted forQ(s) to produce the transfer function between curvature andinput voltage:

K(s)

U(s)=

KV1

τ1

s+ KV2KV1τ2

s2 +

(1τ1+

1τ2

)s+ 1

τ1τ2

, (6)

where KV1 = αCK1 and KV2 = αCK2. For a step input ofmagnitude a, the time domain solution is

k(t) = a

(KV2 −

KV1τ2 − KV2τ1

τ2 − τ1e−

tτ1

+τ2(KV1 − KV2)

τ2 − τ1e−

tτ2

). (7)

Nonlinear models such as the one given in [21] can beused; however, a linear model may be more convenient forcontroller design and synthesis.

Figure 4. Feedforward control architecture for sectored IPMC.

2.2. A sectored IPMC model

Next, the model described above is expanded to describe asectored IPMC, that is an IPMC with patterned electrodessuch as those described in [8]. Consider as an example anIPMC that has been partitioned into three electrically isolatedregions as shown in figure 3(a). The electrodes are patternedsuch that the surface is symmetric about the longitudinalaxis. For the IPMC shown, the outer portions are used asone sector (Sector 1) and are driven by input u1(t), whilethe middle portion is used as a second sector (Sector 2) andis driven by input u2(t). Thus, the IPMC consists of twocontrollable sectors. In this two-sector case, the IPMC ismodeled as a dual input single output (DISO) system, in whichthe sectors of the IPMC are assumed to be decoupled and canbe treated as independent systems. Although the curvatureof the whole IPMC must physically be the same if simplebending is assumed, a ‘virtual’ curvature can be obtained foreach individual sector, defined as

Ki(s) = Ei(s)Mi(s)Ui(s), (8)

where Ki(s),Ei(s),Mi(s), and Ui(s) are the curvature,electrical model, electromechanical model, and input voltagefor the ith sector, respectively. When each sector is activated,the sum of the virtual curvatures produces the net responseof the complete IPMC, as is depicted in the model shown infigure 3(b). In this diagram, the final block for the ith sector,Pi(s), represents the conversion from curvature, Ki(s), to tipdisplacement, Yi(s). Another transfer function could be usedhere; however, in this study, the relationship is assumed tobe linear, and the gain parameters in the electromechanicalmodel of the ith sector (Mi(s)) are tuned to produce tipdisplacements. From this point forward, the ith sector willonly be considered as a single transfer function model, Gi(s),replacing Ki(s) with Yi(s) in equation (6). For a general IPMCwith n number of sectors, the total tip displacement is thengiven by

Y(s) =n∑i

Ui(s)Gi(s). (9)

3. Sector control strategy

3.1. Feedforward approach

To control the behavior of the sectored IPMC, thefeedforward architecture in figure 4 is used. In this block

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 5. Simulated step responses of an IPMC’s total motion,forward motion component, and back relaxation component.

diagram, C1(s) and C2(s) are the controllers associated withSector 1 and Sector 2, respectively, and Yd(s) is the desiredtrajectory. This structure is an open loop version of themaster–slave control scheme given in [22]. This layout takesadvantage of the assumption, described in section 2.2, thatthe IPMC sectors are decoupled. One advantage of the openloop feedforward control approach is sensor feedback isnot required; thus simplifying implementation. However, asdescribed below, feedback control can be added to furtherimprove performance, where feedback information can comefrom laser sensors, strain-based sensors [23], or other methodssuch as integrated PVDF films [20]. The specific layout of themaster–slave structure can vary, but in this work, the inputproduced by C1(s) (the master controller) is fed into C2(s)(the slave controller). By inspection, the open loop transferfunction for this architecture is

TOL(s) ,Y(s)

Yd(s)= C1(s)[G1(s)+ C2(s)G2(s)]. (10)

To aid in the design of the controllers, C1(s) and C2(s),the IPMC model for each sector, Gi(s), is first brokeninto two separate transfer functions, one accounting for thequick forward movement of the IPMC (GFi(s)), and theother for the slow back relaxation (GBi(s)), composed of theconstants (KV1i,KV2i, τ1i, τ2i) associated with the ith sector.The transfer functions add to produce Gi(s), as follows:

Gi(s) = GFi(s)+ GBi(s) =Ai

s+ 1τ1i

+Bi

s+ 1τ2i

. (11)

Equating coefficients, the following is obtained:

Ai =KV2i −

KV1iτ2iτ1i

τ1i − τ2i(12)

and

Bi =KV1i

τ1i+

KV1iτ2iτ1i− KV2i

τ1i − τ2i. (13)

To illustrate this concept, figure 5 shows simulated stepresponses using a general Gi(s) (total motion), GFi(s)(forward motion component), and GBi(s) (back relaxationcomponent).

The slave controller, C2(s), is designed to counteract theback relaxation of Sector 1 by inducing back relaxation in

Figure 6. Block diagram of integrated feedforward/feedbackcontrol scheme.

the opposite direction for Sector 2. With the inputs properlyweighted, the back relaxation effect of each sector roughlycancels. Then, C2(s) is responsible for setting U2(s) inproper proportion to U1(s), such that the sectors relax thesame amount. Since GBi(s) is responsible for the amount ofrelaxation in each sector, the following constraint must besatisfied for the relaxations to cancel,

U1(s)GB1(s)+ U2(s)GB2(s) = 0, (14)

and thus, C2(s) is given as

C2(s) =U2(s)

U1(s)= −

GB1(s)

GB2(s). (15)

Although C2(s) theoretically removes the back relax-ation, the response is left offset, due to the opposing forwardmotions of the two sectors. To account for this, U1(s) must begained by the master controller, C1(s), such that the forwardmotions add up to the desired level, dictated by Yd(s), that is

U1(s)GF1(s)+ U2(s)GF2(s) = Yd(s). (16)

Rearranging and using equation (15) to substitute forU2(s),C1(s) becomes

C1(s) =U1(s)

Yd(s)=

1GF1(s)+ C2(s)GF2(s)

. (17)

For C1(s), the numerator is of a higher degree than thedenominator, making it an improper transfer function. Toremedy this, a pole is added in the form of a low pass filter,

CL(s) =a

s+ a, (18)

where a is the cutoff frequency, and is chosen to be muchgreater than the other poles of C1(s), to minimize the effecton the low-frequency response. Without this filter in place,the open loop transfer function is equal to unity, meaningthe feedforward controller would ideally provide a perfectresponse below the cutoff frequency of the filter, if the modelused is perfect.

3.2. Adding feedback control

To account for disturbances, nonlinearity, and other undesir-able factors not captured in the model, a feedback controller(C0(s)) is integrated with the feedforward controller. Asshown in figure 6, the feedback loop is set up such thatthe feedforward and feedback inputs (Uff(s) and Ufb(s),respectively) are added to yield U1(s). Although this is not

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 7. IPMC electrode patterning techniques: (a) masking and (b) surface machining.

Figure 8. Experimental setup, including an IPMC with two effective sectors.

typical for a master–slave controller, this structure is chosenso the feedforward controller is still active even if the feedbackcontroller is disabled. This can be seen in the closed looptransfer function for the feedforward/feedback system:

TCL(s) ,Y(s)

Yd(s)=

C0(s)[G1(s)+ C2(s)G2(s)]

1+ C0(s)[G1(s)+ C2(s)G2(s)]

+C1(s)[G1(s)+ C2(s)G2(s)]

1+ C0(s)[G1(s)+ C2(s)G2(s)]. (19)

If C0(s) is set equal to zero, equations (10) and (19) areidentical. The feedback controller C0(s) can be chosen asdesired provided it yields a stable closed loop system. Giventhe structure shown in figure 6, the feedback controller canbe considered to be in series with a single plant, G1(s) +C2(s)G2(s), and can be designed to improve the response ofthis system.

In this work, however, it is important to comparethe integrated feedforward/feedback strategy with simplefeedback control. To give a fair comparison, the feedbackcontroller is designed to control the IPMC as if it wasunsectored, without the additional controller, C2(s). A modelis developed, Gtot(s), for the response of the IPMC withthe sectors actuated in unison and used to design a simpleproportional–integral (PI) feedback controller,

C0(s) = Kp +Kint

s, (20)

where Kp and Kint are proportional and integral gains,respectively. The controller is designed to achieve zero steadystate error with less than 5% overshoot. The overshoot (OS) isused to determine the desired damping ratio (ζ ) of the closedloop system, as follows:

ζ =|ln(OS)|√|ln(OS)|2 + π2

. (21)

The desired natural frequency (ωn) and proportional gain (Kp)are then determined by equating coefficients of the desiredcharacteristic polynomial for the closed loop system underproportional control:

s2+

(1τ1+

1τ2

)s+

1τ1τ2+ Kp = s2

+ 2ζωns+ ω2n. (22)

To limit the effect on the transient response, the integralgain (Kint) is typically chosen to be much lower than theproportional gain. As the focus of this work is the trackingof low-frequency signals, the integral gain can be chosen tobe relatively high. The ratio of Kint to Kp in this work rangesfrom five to ten times greater than ζωn. Both gains producedusing this method can be rather high, so to reduce excessivesensitivity to disturbances, the gains are scaled down whilemaintaining their relative proportions.

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 9. Model validation: (a) input voltage for all tests, (b) individual and simultaneous sector actuation, (c) individual sector responsesand corresponding simulation results, (d) simultaneous sector actuation and sum of corresponding simulations, and (e) error betweensimultaneous sector actuation and sum of corresponding simulations.

4. The experimental system

4.1. IPMC fabrication

For this study, all IPMCs are fabricated in-house, usinga platinum reduction method similar to that described byKim and Shahinpoor [24]. The ion exchange membraneused is Nafion R© (1100 EW) with a thickness of 0.5 mm.To increase surface area for the reduction process, themembrane is roughened with sandpaper. All sanding is donein the transverse direction, to promote greater bending. Aftersoaking the membrane in deionized (DI) water for severalhours, it is cleaned in successive baths of hydrogen peroxide(H2O2) and sulfuric acid (H2SO4), followed by two baths inDI water. The plating process begins by introducing platinumions to the membrane by soaking it in a platinum complexsolution, tetraamineplatinum chloride hydrate (Pt[NH3]4Cl2),overnight. Metallic platinum is formed on the surface via areducing agent, sodium borohydride (NaBH4). Ammoniumhydroxide (NH4OH) is also added during the reductionprocess to regulate the pH of the solution. After the reduction,the plated membrane is washed in sulfuric acid and two DIwater baths. The plating process is repeated until a desirablesurface resistance is reached. The plated IPMCs are cut downto sheets with dimensions of approximately 50 mm× 10 mm.

Finally, the platinum electrodes are patterned as necessary.The outer electrode portions are each roughly 3 mm across,and the middle section is approximately 4 mm across. Thereare several methods available to pattern the electrodes. Onepopular method involves masking the Nafion R© during theplating process, to prevent the reduction of platinum betweenthe desired electrode sectors (see figure 7(a)). Alternatively,the electrodes can be patterned after the plating processby removing the necessary platinum down to the Nafion R©

layer using a laser or CNC machine (see figure 7(b)). Asharp blade can also be used to scrape away the unwantedplatinum, which is the method employed in this work [8, 25,26]. After a great amount of testing, the surface resistancesof the IPMCs degrade. When this happens, the IPMCs arerevitalized by adding a layer of gold over the platinum layer.While immersed in a gold solution, a low voltage is appliedbetween the IPMC and a piece of stainless steel, leading toelectroplating of gold onto the IPMC [11].

4.2. IPMC test setup

A custom-designed voltage/current amplifier is used to drivethe IPMCs in this study [23]. To provide a strong clampingforce and resist corrosion, nickel plated neodymium magnets

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Table 1. Gain parameters for model validation.

Sector KV1 KV2 τ1 τ2

1 0.430 0.146 0.531 35.5432 0.243 0.115 0.652 19.224

are used to conduct power to the IPMCs [11, 27]. A mountfor the magnets is designed to hold the magnets in place,so each would only contact one sector of the IPMC surfaceelectrode (see figure 8). In this case, the IPMC is dividedinto three sections, meaning six magnets are used. Theouter two sections are actuated in unison, however, makingthem effectively one sector. MATLAB, in conjunction withSimulink and xPC target, is used to operate a 12-bit dataacquisition card, which serves to output the voltage signals,as well as record the IPMC tip displacement, via a laserdisplacement sensor (SUNX Microlaser Sensor LM10, witha resolution of 1 µm). An additional amplifier and filter areused to improve the signal-to-noise ratio of the laser outputsignal. All tests were conducted in DI water, in a small tank,at room temperature.

4.3. Model validation

A key assumption in modeling the sectored IPMC is thatthe sectors are decoupled. To validate this assumption, thesectors of an IPMC are actuated individually with a step inputof 1 V (see figure 9(a)). Then, the same step input is givento both sectors at the same time and the tip displacementis compared to the sum of the tip displacements for theindividual actuation tests, as shown in figure 9(b). The resultsare in good agreement, providing confidence in the modelingassumption.

To determine the various gain parameters for thepreviously described model, each sector of the IPMC issubjected to a step input, and the displacement is recorded.A step input is used for the model fitting process to exciteas much of the dynamics of the IPMC as possible. Using aleast squares curve fit tool in MATLAB, the model, Gi(s),is fit to the data for each sector. The gain parameters hadto be adjusted periodically as the response of the IPMCchanged; however, the gain parameters used for this modelvalidation experiment are given in table 1. The model isthen implemented in Simulink to carry out all simulations, aswell as the experiments in conjunction with the xPC targetsystem. Figure 9(c) compares experimental sectored IPMCstep responses and the corresponding simulations. Finally,the decoupled assumption is further verified by comparingthe sum of the model responses to the experimental IPMCresponse when both sectors are actuated simultaneously. Asshown in figure 9(d), the model represents the true responsereasonably well. The corresponding error, defined as

e(t) =

[|y(t)− yd(t)|

max[yd(t)] −min[yd(t)]

]× 100%, (23)

is given in figure 9(e), and is found to be less than 10% for themajority of the test. The root-mean-square (RMS) error over

Figure 10. Open loop, feedforward, and feedforward/feedbackresponses to an exponential reference: (a) tip displacement,(b) tracking error, and (c) input voltages.

the total testing time, T , is defined as

eRMS =

√

1T

∫ T0 [y(t)− yd(t)]2 dt

max[yd(t)] −min[yd(t)]

× 100%, (24)

and is found to be 3.73%, providing good confidence in themodel.

5. Experimental results

Although step inputs were used to identify the modelparameters, an alternative signal is chosen for theseexperiments. It is desirable to have a smooth referencetrajectory, to avoid large spikes in the control inputs. Anexponential function of the form yd(t) = D(1 − exp(−τ t))is used as a reference trajectory, where D is the steady statevalue of the function, and τ is the time constant. In these tests,D was chosen to be relatively small (0.4 mm) to keep the inputvoltages low and within the more linear range of operation ofthe IPMC. Furthermore, for these tests, τ = 2.

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 11. Feedback response to an exponential reference with an imposed voltage threshold: (a) tip displacement and (b) input voltages.Response without the voltage threshold: (c) tip displacement and (d) power consumption.

Table 2. Feedforward and feedforward/feedback control results.

Openloop Feedforward

Improvement overopen loop

Feedforward/feedback

Improvement overopen loop

emax (%) 77.20 13.85 82.06 9.25 88.02eRMS (%) 53.03 8.13 84.67 1.71 96.78

First an open loop response is taken as a control,supplying the input

u(t) =1

ymaxyd(t), (25)

to both sectors, where ymax is the modeled maximum value ofa simultaneous step response for both sectors. To determineymax, equation (6) is fitted to an experimental simultaneousstep response for both sectors, producing the single model,Gtot(s), for the whole IPMC. Using the resulting gainparameters, the derivative of equation (7) is taken. Settingthe derivative equal to zero, the time of the maximumdisplacement, tmax, is found. This time is then plugged backinto equation (7) to find the maximum value, ymax. The openloop response and corresponding tracking error are given infigures 10(a) and (b), respectively.

Next, the feedforward control alone is implemented forthe same reference trajectory, significantly improving theresponse (see figures 10(a) and (b)), as compared to the openloop case. The maximum error, found by taking the maximumof equation (23) and RMS error, calculated with equation (24),are given in table 2. In this case, the RMS error is improved byapproximately 85%. Furthermore, the input voltages suppliedto the IPMC (see figure 10(c)) do not grow rapidly, implying

that the position of the IPMC can be maintained for extendedperiods of time.

The integrated feedforward/feedback controller is thenimplemented to further reduce the error. The feedback gainsare developed as previously described, and then scaled downto much lower values to ensure that the feedback controlleris not dominant. The gains are varied as the model is refitted,but for this example, Kp = 0.5 and Kint = 3.41. The responseis again improved (see figures 10(a) and (b)). In this case, theRMS tracking error is improved by nearly 97%, as comparedto the open loop case. All tracking errors are given in table 2.

Finally, to illustrate the improvement this technique offersover traditional feedback control, a longer test is conducted,comparing the integrated feedforward/feedback controllercase to a simple feedback control case. In the latter case,the previously developed feedback controller is used withoutthe feedforward controller, treating the sectored IPMC as anon-sectored IPMC. The tip displacement and input voltage(identical for each sector in the simple feedback case) aregiven in figures 11(a) and (b), respectively. It is observedthat simple feedback produces an increasing input voltage asexpected to maintain a constant position in the presence ofback relaxation. A voltage threshold is imposed to preventdamage to the IMPC, and once it is reached, the IPMC quickly

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 12. Sectored IPMC with a triangular reference: (a) open loop and pure feedback control, and (b) feedforward andfeedforward/feedback control.

relaxes. For this sort of feedback to be effective, the voltagemust be able to rise freely, but this can lead to serious damageof the actuator, as previously discussed.

Another advantage of the integratedfeedforward/feedback control method over traditional feed-back is a significantly reduced power consumption. Todemonstrate this reduction, the previously described voltagethreshold was removed, allowing the feedback controller tofreely raise the control input voltage as needed to maintain thetip position (see figure 11(c)). As shown in figure 11(d), thepower consumption in the pure feedback case ramps up overtime and becomes much greater than the power consumptionfor the integrated feedforward/feedback case. In these tests,the RMS error is lower for the feedforward/feedback case(3.94%, compared to 6.88% for pure feedback). Without thevoltage threshold, there is a greater risk of damage to theIPMC, so the steady state value and time constant of thereference signal were changed to D = 0.3 mm and τ = 0.5,respectively, to reduce the feedback control input.

To further verify the effectiveness of this method, morecomplex waveforms are also used as reference trajectories.The IPMC’s responses under open loop, feedforward,

feedback, and feedforward/feedback control to triangular andtrapezoidal waveforms with a non-zero DC component aregiven in figures 12 and 13, respectively. In these tests, itis noted that although the feedback control input is notalways greater than the other control inputs, it does appearto increase at a greater rate. Finally, the IPMC is subjectedto the previously described exponential reference underfeedforward/feedback control for a period of 20 min (1200 s).With the proposed control technique, the IPMC is able tomaintain a steady position for the entire period with boundedinputs to both sectors, as shown in figure 14. It should be notedthat under open loop conditions, the IPMC in this study beginsto relax, after approximately 10 s. In this control test, therewas no sign of relaxation for a period 120 times longer.

6. Conclusions

This paper explored controlled activation of patternedelectrodes on IPMCs to mitigate back relaxation. Afeedforward controller was developed, based on the principlethat IPMC sectors, driven in opposite directions, cansignificantly reduce the back relaxation effect and improve

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

Figure 13. Sectored IPMC with a trapezoidal reference: (a) open loop and pure feedback control, and (b) feedforward andfeedforward/feedback control.

Figure 14. Sectored IPMC with a twenty-minute exponentialreference: (a) response and (b) input.

tracking error. Performance was further improved byintegrating a feedback controller with the feedforwardcontroller, yielding nearly a 97% improvement in trackingerror as compared to the open loop case. The primarylimitation of this work is that the model used is linear,and cannot perfectly capture the behavior of an IPMC overa large voltage range. Future work should include the useof an accurate nonlinear model, which would allow thefeedforward controller to be more effective, even withoutintegrated feedback.

Acknowledgments

Authors gratefully acknowledge support from the Officeof Naval Research, grant number N000140910218. Theauthors would like to thank Viljar Palmre and David Pugalof the Active Materials and Processing Laboratory fortheir help with fabricating the IPMC actuators used in theexperiments; and Joel Hubbard of the Electroactive Systemsand Controls Laboratory for his help with setting up thecontrol experiments.

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Smart Mater. Struct. 21 (2012) 085002 M J Fleming et al

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