Modeling and Inverse Compensation of the Quasi-static Voltage-StrainLonely Stroke and Hysteresis in Supercoiled Polymer Artificial Muscles

Revanth Konda1 and Jun Zhang1

Abstract— Supercoiled polymer (SCP) actuator is a recentlydeveloped thermally-driven artificial muscle that has shownlarge strain, high power density, and strong promise in roboticsand intelligent systems. Termed as lonely stroke, the first cycleof SCP actuators is inconsistent with subsequent cycles thatare repeatable and exhibit hysteresis. The lonely stroke notonly affects SCP actuators’ performances, but also presentscoupling with hysteresis. It is thus crucial to capture the lonelystroke and hysteresis of SCP actuators to fully unleash theirpotential. However, the existing modeling and control strategiesof SCP actuators fail to consider the lonely stroke nonlinearity.In this study, a modeling strategy is proposed to capture thequasi-static voltage-strain lonely stroke and hysteresis in SCPactuators. This is realized by expanding the input range of thePreisach operator, a widely used hysteresis model, to physicallyinfeasible region to account for the lonely stroke. An iterativealgorithm is proposed to compensate for the lonely strokeand hysteresis by approximately inverting the proposed model.For comparison purposes, a conventional Preisach operatorand a polynomial model are considered. The modeling andcontrol performance of the proposed approach is evaluated inexperiments, and the superiority of the proposed scheme isdemonstrated.

I. INTRODUCTION

Artificial muscles are soft or compliant actuators whichgenerate actuation in different modes through their shape-altering ability under external stimuli, such as heat or elec-tricity [1]. Traditional electromagnetic motors and hydraulicactuators, although highly accurate and powerful, are oftenheavy, rigid, bulky, and potentially dangerous. In contrast, ar-tificial muscles are inherently compliant and safe, have highpower-to-weight and force-to-weight ratios [1], which makethem strongly promising in various robotic applications suchas biomimetic robots, robotic prosthetics and exoskeletons,soft robots, and medical robots [2]–[4].

Super-coiled polymer (SCP) actuators belong to a recentlydiscovered class of artificial muscles which generate linearactuation when thermally activated [5]. SCP actuators gen-erate 15-20% strain and exhibit a power density of 27 W/g[1]. Different types of motions, such as linear contraction,twisting, and bending, have been generated using theseactuators [5], [6], consequently leading to great promise inrobotic applications such as robotic fingers, robotic hands,soft grippers, and assistive robots, among others [7]–[9].

Amidst all the successes, it remains a challenge to ac-curately model and control SCP actuators considering theirnonlinearities caused by the polymer materials’ properties.

1Revanth Konda and Jun Zhang are with the Department of MechanicalEngineering, University of Nevada Reno, 1664 N. Virginia St., Reno, NV89557, USA rkonda@nevada.unr.edu, jun@unr.edu

Input

Outpu

t

Imin ImaxInput

Out

put

imin imax

Lonely S

troke

Repeatable Hysteresis

Fig. 1: Illustration of lonely stroke and hysteresis.

Hysteresis is a common nonlinearity which occurs in varioussmart materials and artificial muscles. The output of ahysteretic system not only depends on the current input,but also the history of inputs [10], [11]. The hystereticrelations of load-strain, temperature-strain, and temperature-load have been captured [7], [12]–[14]. Phenomenology-based models [15] and physics-based models [16] havebeen proposed. Due to the inconvenience of temperaturesensing for SCP actuators that have small diameters, quasi-static voltage has been used as a temperature surrogate [7].Different control schemes like feedforward control, propor-tional–integral–derivative control, and robust and adaptivecontrol have been realized [17]–[21]. However, the modelingand control studies reported thus far have not consideredanother nonlinear behavior which was investigated in ourrecent work [22] – during the application of mechanical orelectrical input oscillation sequences, the first strain cycle isinconsistent with subsequent cycles that are repeatable andexhibit hysteresis. This first cycle is termed as “lonely stroke”in this work (Fig. 1).

Due to material properties and residual stress, the lonelystroke behavior has been encountered in other artificialmuscles, such as pneumatic actuators [23], McKibben ac-tuators [24], silicon elastomers [25], and NiTi alloys [26].However, no mathematical analysis has been reported sofar. The lonely stroke behavior may result in around 40%discrepancies of the total output range [25], [26]. Artificialmuscles may operate in the repeatable region by eliminatingthe lonely stroke. However, eliminating the lonely strokerequires specialized and accurate input cycles and conditions:start from the minimal input, monotonically increase tothe maximum input, then monotonically decrease to theminimum input (Fig. 1 under constant external conditions).Small errors of the input may result in imperfect eliminationand unpredictable behaviors. Furthermore, lonely stroke may

not be easily eliminated in cases where external conditionschange during normal operation. For example, when the loadof SCP actuators changes to different values during actuationoperation, the lonely stroke will appear between temperatureand strain each time when the load changes.

It has been a common practice to eliminate the lonelystroke and obtain models to only capture the remainingrepeatable cycles of hysteresis [7]. Although this has been acommon strategy, without capturing the first cycle, the behav-ior of SCP actuators cannot be fully predicted which could bepotentially dangerous [22]. The inclusion of the lonely strokein modeling SCP actuators is critical, especially consideringtheir applications in robotics, where non-repetitive tasks inunstructured and uncertain environments are often expected[1]. The outputs of existing hysteresis models are invariableat physically feasible minimum and maximum inputs [10].However, due to lonely stroke, although the invariance ofoutput at maximum input is unaffected, based on the inputhistory, the outputs obtained at physically feasible minimuminput are different. As illustrated in Fig. 1, the output maybe different when the input is at imin, depending on the inputhistories. As a result, the conventional hysteresis models failto capture the variability of output, consequently producingconsiderable amount of errors.

In this work, a modeling strategy is proposed to capture thelonely stroke and hysteresis in the quasi-static voltage-strainrelationship of SCP actuators. The proposed model, whichis based on the Preisach operator, is developed through theexpansion of input range to physically infeasible region. Theeffects of model parameters on the modeling performanceare studied. Furthermore, an iterative inverse compensationalgorithm is proposed for quasi-static open-loop positioncontrol of SCP actuators. The inverse compensation algo-rithm predicts the required input sequence for a desiredstrain output sequence. The proposed modeling and controlmethods are applicable to other types of artificial muscleswhich exhibit lonely stroke and hysteresis behavior.

II. REVIEW OF THE PREISACH OPERATOR

In this section, a brief review of the Presiach operator ispresented. The Preisach operator has been extensively usedto capture hysteresis in different areas and scenarios [10],[11], [27], and is thus adopted in this study. The Preisachoperator is expressed as a weighted integration of delayedrelays, called hysterons. The output of the hysteron, γβ ,α , iswritten as

γβ ,α [i(·);ζ0(β ,α)] =

+1 if i > α

−1 if i < β

ζ0(β ,α) if β ≤ i≤ α

, (1)

where α and β determine the thresholds, i(·) denotes theinput such that i(τ) ∈ [imin, imax], imin and imax are theminimum and maximum values of the inputs, respectively,0≤ τ ≤ t, ζ0(β ,α) ∈ {−1,1} is the initial condition.

Through the integration of all hysterons, the output of a

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α

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𝛿 #$! !

𝛿#!

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𝛿&(#$!)𝛿&&

…

…

… … …

imax

imin

Fig. 2: Discretization of the Preisach density function.

Preisach operator, Γ, is expressed as

u(t) = Γ[i;ζ0](t) =∫

P0

µ(β ,α)γβ ,α [i;ζ0(β ,α)](t)dβ dα,

(2)where µ is the density function and P0 = {(β ,α) : imin ≤β ≤ α ≤ imax} is the Preisach plane. The Preisach planedefines the region of integration. Based on the history of theapplied input i(τ), 0≤ τ ≤ t, the outputs of all hysterons arecomputed using Eq. (1).

For efficient computation and model implementation, thefollowing discretization procedure is employed: the densityfunction, µ(β ,α), is discretized to a piecewise constantfunction – the density value is constant within each latticecell but varies from cell to cell [28]. Fig. 2 shows an exampleof discretization of the density function into L levels andL(L+1)

2 cells, each cell is associated with a constant densityvalue, {µi j}. With this discretization scheme, the modeloutput at time instant n is written as

u[n] = uc +L

∑i=1

L+1−i

∑j=1

µi jsi j[n], (3)

where uc is a constant bias, si j[n] is the signed area of thecell (i, j), namely, its area occupied by hysterons with output+1 minus that occupied by hysterons with output −1. Theboundary of the two regions is the memory curve, which isdetermined by the input history. As an example, the memorycurve shown in Fig. 2 is denoted as ψ0. The outputs of thehysterons within the shaded area are +1 (activated) and theremaining are −1 (inactivated). Readers are referred to [11],[29] for more details about the Preisach operator.

III. PROPOSED APPROACH

A. Proposed Model

Consider imin, imax, and iext being the physically feasibleminimum and maximum values, and the physically infeasibleminimum value of an input sequence, as shown in Fig.3(a). When the input sequence in Fig. 3(a) is applied toa system with lonely stroke and hysteresis, the obtainedinput-output correlation is shown in Fig. 3(b). Please note

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t

Index

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Physically infeasible

(a)

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Outpu

t

Physically infeasible

(b)

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!!"#

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Physically Infeasible

imax

imin

iext

1 6 19

(c)

Fig. 3: (a) Input sequence expanded to physically infeasible region. (b) The input-output relationship of the proposed modelunder the input sequence in (a). (c) Preisach plane with the proposed approach.

that the input never goes below imin. In the conventionalPreisach operator formulation [29], the Preisach plane shownin Fig. 2 is constructed with imin and imax. When the inputis imin, all hysterons will be −1 and the output is a constant,regardless of input history. With the lonely stroke, the outputof the system under the input imin also depends on theinput history, as shown in Fig. 3(b). Hence, the traditionalmodeling approaches needs to be modified.

The proposed method involves expanding the input rangeto physically infeasible region (Figs. 3(a)-(b)). From a math-ematical point of view, the proposed process offers therequired flexibility to capture the lonely stroke by expandingthe input region of a Preisach operator to [iext, imax]. From aphysics point of view, residual stress, defined as the stressremaining in a material after the actual cause of stress isremoved [26], could be a reason. In that case, after an inputcycle, when the actual physical load is removed from theartificial muscle, the residual stress acts as a virtual inputto the artificial muscle, thereby resulting in a non-constantoutput value. Hence, by expanding the input range, the virtualinput caused due to residual stress is compensated.

Essentially, the Preisach plane is constructed with iextand imax, with iext < imin. It is demonstrated in Fig. 3,where the physically infeasible inputs and correspondingoutputs are highlighted. Consequently, this ensures that theexpanded densities, the first few expanded columns of cellswith densities {ρi j} and {σi j} in the Preisach plane (Fig.3(c)) are active when the input takes on the value of imin atthe end of the first input cycle (Fig. 3(a)).

B. Model Identification

The model identification procedure comprises of the fol-lowing steps:• Firstly, the Preisach plane is constructed using the

proposed approach, as depicted in Fig. 3(c);• Secondly, based on the complexity of the input/output

correlations data, the level of discretization L is deter-mined empirically;

• Finally, the model is identified as a linear least-squaresoptimization problem, similar as [11].

The densities of the first few expanded columns, {σi j}and {ρi j}, are primarily responsible for capturing the lonelystroke, while the densities {µi j} are responsible for modelingthe remaining output cycles of hysteresis. Let the number ofexpanded columns be denoted by m. Based on the notationin Fig. 3(c), when m is a non-negative integer, the output ofthe proposed model can be written as

u[n] =L

∑i=L+1−m

L+1−i

∑j=1

σi jsi j[n]+L−m

∑i=1

m

∑j=1

ρi jsi j[n]

+L−m

∑i=1

L+1−i

∑j=m+1

µi jsi j[n]

(4)

For example, Fig. 3(c) shows that the value of L is 19 withm = 6. With the proposed approach, the system always startswith the cells corresponding to the densities {σi j} activated(all σi j being +1). Due to this, it is no longer needed toinclude the constant bias uc like Eq. (3). This is due to thefact that the densities {σi j} play the role of a constant bias.The pattern of {σi j} is thus expected to be different fromthe rest of the densities that are to model the lonely strokeand hysteresis nonlinearities.

An ideal way to determine an appropriate value of L isto first determine the level of discretization L0 required tocapture the repeatable hysteresis. Then the final level ofdiscretization L would be L0+1, since one column for lonelystroke(m = 1) ensures that it is accurately modeled. This isdue to the following considerations: When an input cycleshown in Fig. 3(a) is applied, the cells corresponding todensities {σi j} will remain active, while the densities {ρi j}will remain active (+1) after the first cycle – when the inputincreases during the first input cycle, the memory curve ofthe expanded Preisach operator will monotonically move up,converting each row of densities {ρi j} from −1 to +1. Inthat regard, increasing the number of expanded columns willnot improve the accuracy of the proposed model. This canalso be inferred as follows:

• when m> 1 with a fixed L, this is equivalent to decreas-ing the discretization level L, for the repetitive hysteresis

and may generate less accurate results, assuming L isconstant.

• When m< 1 with a fixed L, then the first column will bedivided into two portions vertically, with the left portionresponsible for the lonely stroke and the right portion forthe repetitive hysteresis. Ideally, different densities mustbe assigned for left and right portions of the first columnto properly capture the lonely stroke and hysteresis,respectively, which is impossible in practice. Hence, italso leads to a higher error.

C. Comparison Models

A polynomial model and a conventional Preisach operatorwere selected for performance comparison purposes. Detailsabout the Preisach operator have been provided in SectionII. A second-order polynomial model was employed in thiswork, as presented in Eq. (5):

O(I) = p2I2 + p1I + p0 (5)

where O denotes the actuator output, p2, p1, and p0 areconstants, and I is the input to the actuator.

D. Inverse Compensation

Since the input-output relation is monotonous, an inversionalgorithm based on methods presented in [29] and [11] canbe adopted. The inversion problem is, given a desired outputvalue ud and the memory curve ψ0 which is determinedby the input history, to find a new input v, such that ud =Γ[v;ψ0]. Let the input and output corresponding to the initialmemory curve ψ0 be v and u respectively. The algorithm tofind v is presented as Algorithm 1.

Algorithm 1 Inverse Compensation Algorithm [29]

v(0) := v, u(0) = u, ψ(0) = ψ0, n = 0Choose any d such that 0 < d < imax−iext

Lif u > u then

sign = 1else

sign = -1Step 1: Calculate a1

(n) and a2(n) such that Γ[v(n)+ sign ·

d;ψ(n)]−Γ[v(n);ψ(n)] = a1(n)d2 +a2

(n)dCalculate d0 such that ud−Γ[v(n);ψ(n)] = a1

(n)d20 +a2

(n)d0Step 2: n = n+1, v(n) = v(n−1)+ sign ·d,u(n) = Γ[v(n)+ sign ·d;ψ(n)]Choose δ1 and δ2 such that 0 < δ1 < 1,0 < δ2 < 1if ||d0−d||< δ1 or ||ud−Γ[v(n)+d;ψ(n)]||< δ2 then

v = v(n)

elseGo back to Step 1

IV. EXPERIMENTAL SETUP

A. Fabrication Procedure

Based on our previous study [12], the V Technical Tex-tiles Conductive Yarns (110/34 dtex, Denier:110/34f) wereused to fabricate the SCP actuators that can generate 10-15%

Rectangular Frame

Power Supply

Arduino UNO

Distance Sensor

Load

SCP

Magnet

Tangential Fan

Fig. 4: Experimental setup.

strains. The process comprises of two steps, namely, coilingand heat treatment. Coiling was performed by twisting thenylon fibers using a motor, and heat treatment was realizedthrough Joule heating by applying electrical pulses. Afterheat treatment, the length of the coiled fibers convergesand the SCP actuator is created. More details about themanufacturing of SCP actuators can be found in [7] and[12].

B. Experimental Setup

The experimental setup consists of a rectangular framefrom which the SCP actuator is suspended. On one side ofthe frame, a distance sensor (SPS-L225-HALS, Honeywell)is used to measure the vertical displacement of the actuator.On the other side, a tangential fan is attached. A magnet andmechanical loads are suspended from the free end of theactuator. The tangential fan is adopted to promote adequateair flow and ensures consistent displacement measurements[7], [12]. A power supply is used to apply the voltage inputsto the actuator. Finally, an Arduino UNO microcontrollerboard is used to operate the power supply and acceptmeasurements from the displacement sensor. The setup isillustrated in Fig. 4.

V. EXPERIMENTAL RESULTS

A. Model Identification and Validation

The model identification and validation results of thequasi-static voltage-strain relationship of an SCP actuator arepresented, when the load is switched from a higher value toa lower value. A voltage input sequence ranging from 0 Vto 11.5 V was applied with a step size of 0.23 V, as shownin Fig. 5(a). The wait time for each applied voltage was 10seconds. This value ensured that a quasi-static strain valuewas obtained. The voltage input sequence was applied afterwaiting for approximately 60 seconds once the load wasswitched. Note that this wait time is for mechanical loadinput, in contrast to the 10 seconds wait time for the voltage

0 1000 2000 3000 4000

Time (sec)

0

5

10

Voltage (

V)

(a)

0 5 10

Voltage (V)

-15

-10

-5

0

Str

ain

(%

)

Actual

Estimated

(b)

0 5 10

Voltage (V)

-15

-10

-5

0

Str

ain

(%

)

Actual

Preisach

Polynomial

(c)

0 5 10

Voltage (V)

-15

-10

-5

0

Str

ain

(%

)

Actual

Estimated

(d)

0 2 4 6 8

Voltage (V)

-10

-5

0

Str

ain

(%

)

Actual

Estimated

(e) (f)

0 50 100 150 200

Index

-15

-10

-5

0

Str

ain

(%

)

Desired

Actual

(g)

0 50 100 150 200

Index

-1

-0.5

0

0.5

1

Str

ain

Err

or

(%)

(h)

Fig. 5: (a) Input sequence for model identification. Model identification results with (b) the proposed approach, (c) aconventional Preisach operator, and a second-order polynomial model. Validation results under (d) input sequence #1. (e) inputsequence #2. (f) Preisach plane with densities. (g) Open-loop control performance for the presented inverse compensationalgorithm. (h) Strain control error.

input. This was done to ensure that the SCP actuator settledat a quasi-static value as a result of the change in loads. Thestep size was selected to obtain smooth output curves. Theload was switched from 400 g to 200 g.

The density function of the proposed model is identifiedsuch that all the densities are less than or equal to 0,since strain output decreases monotonically with voltageinput. The L value was chosen to be 13, and the Preisachplane was constructed with iext and imax as −0.95 and 11.5,respectively. This choice of iext ensured that m = 1. Theresults obtained through the proposed model are comparedwith a conventional Preisach operator and a second-orderpolynomial model. The model identification results are pre-sented in Figs. 5(b)-(c). As it can be seen, only the proposedapproach captures the lonely stroke phenomenon with highdegree of accuracy. The modeling error with the proposedapproach, the traditional approach and the polynomial modelare 0.09%, 0.59%, and 1.36% respectively.

The proposed model is further validated by applying twodifferent voltage sequences to the SCP actuator. The strainestimates for the two verification correlations are presentedin Figs. 5(d)-(e). The model verification errors were slightly

higher than the model identification error. The validation er-ror with the proposed approach for the correlations presentedin Figs. 5(d) and (e) are 0.29% and 0.20%, respectively.The errors with traditional approach are 0.71% and 1.18%while with the polynomial model are 1.53% and 2.09%respectively. Also, as suggested in Section III.B, the densityof the first cell in the Preisach plane is not comparable tothe remaining densities, and is thus not plotted in Fig. 5(f).The value of the density of the first cell was −9.93V−1.The model validation further confirms the effectiveness ofthe proposed modeling approach.

B. Inverse Compensation

The performance of the presented inversion algorithm wasexamined through an open-loop strain-tracking experiment.The values of δ1 and δ2 were selected to be 0.04 and 0.12,respectively. These values were determined experimentallyand ensured maximum accuracy while not consuming largecomputational times. A randomly chosen sequence of the de-sired steady-state strains is generated, as shown in Fig. 5(g).Fig. 5(g) shows the experimental strain output measurementsobtained under the proposed inverse compensation scheme.

Fig. 5(h) shows the corresponding control error between thedesired output and actual output. The inverse compensationscheme is proven to be effective, with the mean error of0.34% strain.

VI. CONCLUSION AND FUTURE WORK

In this paper, a novel modeling approach is proposed tocapture the lonely stroke nonlinearity coupled with hystere-sis. A detailed analysis of the proposed modeling approachis presented where the effects of the offset and the levelof discretization of the density function on the modelingerror are presented. The proposed approach is applied tocapture the quasi-static voltage-strain correlation of an SCPactuator. The proposed approach exhibits high accuracy inmodeling the lonely stroke and hysteresis. The approach isfurther validated by estimating the outputs for two moreinput sequences. Lastly, an inverse compensation algorithmis presented and its advantages in control performance areconfirmed in experiment.

As a part of the future work, we plan to include the mod-eling of lonely stroke behavior in the load-strain and voltage-strain relationship when the mechanical load is switchedfrom a lower value to a higher value. The shape of the lonelystroke in the voltage-strain relationship when the mechanicalload is switched from a lower load to higher load is differentfrom the case presented here [22]. In the case presentedin this work, the repeatable region lies above the lonelystroke. However, for the other case, the repeatable regionlies below the lonely stroke. Hence, the proposed approachhas to be modified to capture this phenomenon. In this case,the densities in the expanded columns will also take negativevalues.

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