Proceedings of the ASME 2019Dynamic Systems and Control ConferenceDSCC 2019October 911, 2019, Park City, Utah, USA
DSCC20199169
THE EFFECTS OF NYLON POLYMER THREADS ON STRAINLOADINGHYSTERESIS BEHAVIOR OF SUPERCOILED POLYMER (SCP) ARTIFICIALMUSCLES
Revanth Konda
∗
Smart Robotics LaboratoryDepartment of Mechanical EngineeringUniversity of Nevada, RenoReno, Nevada 89557Email: rkonda@nevada.unr.edu
Jun Zhang
Smart Robotics LaboratoryDepartment of Mechanical EngineeringUniversity of Nevada, RenoReno, Nevada 89557Email: jun@unr.edu
ABSTRACT
Supercoiled polymers (SCP) actuator, as a recently discovered artiﬁcial muscle, has attracted a lot of attention as a com pliant and compact actuation mechanism. SCP actuators can be fabricated from nylon polymer threads, and generates up to 20%strain under thermal activation. A common challenge, however,is to accurately and efﬁciently estimate the performance of SCPactuators considering their signiﬁcant hysteresis among loading,strain, and power input. Previous studies adopted either linear models that failed to capture the hysteresis or phenomenologicalmodels that required tedious procedures for identiﬁcation and implementation. In this paper, a physicsinspired model is presented to efﬁciently capture and estimate SCP actuators’ strain – loading hysteresis by analyzing the properties of nylon threads from which they are fabricated. The strains of SCP actuators are found to be linear to that of the nylon threads under the sameloading conditions. An efﬁcient approach is proposed to characterize and estimate the strain – loading hysteresis of SCP actuators fabricated with different numbers of nylon threads. A helicalspring model is adopted to obtain the stiffness of SCP actuatorswith different conﬁgurations. Experimental validation involvingtwoply, fourply, and sixply nylon threads and SCP actuatorsare provided to conﬁrm the effectiveness of the proposed model.
∗
Address all correspondence to this author.
INTRODUCTION
Artiﬁcial muscles are materials that can change their shapesunder external stimuli and produce biologically inspired motionsto make systems move [1,2]. They are advantageous over traditional actuators like electromagnetic motors and hydraulic actuators, mainly because they are more compliant and lightweight,and can be utilized to develop compliant structures without complex linkages [3,4]. Artiﬁcial muscles have shown strong potential as the moving mechanism for robotic applications, such asbiomimetic robots, robotic prosthetics, exoskeletons, soft robots,and medical robots [5–7].Supercoiled polymers (SCP) actuators, or referred as twistedand coiled polymer actuators, belong to an emerging artiﬁcialmuscle technology [8]. SCP actuators, as shown in Figure 1,can be fabricated by coiling offtheshelf nylon polymer threadsor other ﬁbers and ﬁlaments, like carbon nanotube yarns, ﬁshing lines, and sewing monoﬁlaments [8]. Under thermal stimuli,which is often realized by electric powerinduced Joule heating,SCP actuators can contract signiﬁcantly [8–10]. While beinga new technology ﬁrst created in 2014 [8], the SCP actuationhas attracted a lot of attention in both the materials and roboticscommunities mainly because of their highly economic fabrication process, high power density, and the large actuation range[2,8,10,11]. For example, up to 21% tensile strain was demonstrated with the nonmandrelcoiled SCP actuators [11,12], up to49% strain was reported for mandrelcoiled SCP actuators [8]. In1 Copyright © 2019 by ASME
1 mm1 mm0.1 mm
FIGURE 1
. (a). A nylon thread. (b). A supercoiled polymer (SCP)actuator heat treated in an oven. (c). An SCP actuator fabricated usingprocedures in [12,18] and heat treated by applying electrical pulses.
the past 5 years, there has been a rapid increase of efforts devotedto SCP actuators and SCPactuated systems. The most popularapplications are robotic ﬁngers, hands, and arms, soft robots, andassitive robots [12–17].With the tremendous interests and preliminary successes,however, it remains to be challenging to accurately and efﬁciently estimate the performance of SCP actuators. An important reason is that SCP actuators exhibit signiﬁcant and complexhysteresis behaviors that are difﬁcult to be understood and captured. As a typical type of nonlinearities, hysteresis appears invarious types of smart materials. For a hysteretic system, the output of the system depends on both the current input and the input history, even under quasistatic conditions [19,20]. The hysteretic relationships of SCP actuators between force and strain[12,21], strain and temperature [8], and under varying loads andtemperatures [22] have been reported. Furthermore, the threedimensional hysteresis among force, strain, and electric powerwas presented [23].The existing studies failed to estimate the hysteresis properties of different SCP actuators efﬁciently. They either adoptedsimpliﬁed linear models that were unable to capture the hysteresis or phenomenological hysteresis models that required tediousprocedures for identiﬁcation and implementation. Early studiespredominantly adopted linear models by approximating the hysteresis as a linear damping term. For example, Yip et al. presented the ﬁrst study to model SCP actuators using a linear dynamical model [12]. Force and position controllers were proposed in [16, 24, 25], but only linear models and proportionalintegralderivative (PID) controls were adopted. With a linearmodel, it was found that over 15% modeling error could be generated [18]. The ﬁrst approach to capture the strain – voltage hysteresis in SCP actuators under different loading conditions usingthe augmented Preisach model and augmented PrandtlIshlinskiimodel was proposed in [18]. A Duhem differential model wasproposed to describe the strain – temperature hysteresis [26].The threedimensional hysteresis was further modeled by embedding two Preisach operators [23]. However, all hysteresismodels presented for SCP actuators were phenomenologybasedthat not only required tedious procedures for identiﬁcation andimplementation, but also only worked for a given SCP actuator.Repetitive and tedious procedures are needed to model the hysteresis of a different SCP actuator. It is desirable to develop anapproach to capture the hysteresis behaviors of different SCP actuators efﬁciently by exploring their physical properties.Modeling the hysteresis of SCP actuators directly usingphysics is a very challenging task. Firstly, the current studieson SCP actuators haven’t reached an agreement on the cause of hysteresis. Although a few studies suggested that the hysteresiseffectsmightbeinducedbythefrictionamongadjacentcoilsduring actuator operation [9,12], the hysteresis may also be relevantto the inherent hysteresis property of nylon polymers from whichSCP actuators are fabricated [27]. Secondly, phenomenological models are more popular than physical approaches to capturehysteresis in smart materials, mainly due to their effectivenessand efﬁciency. That being said, the process of developing thistype of models is tedious as mentioned before. Phenomenological models, such as the Preisach operator [20], KrasnoselskiiPokrovskii model [19], PrandtlIshlinskii model [28], Duhemmodel [29], and BoucWen model [30], are widely adopted tosuccessfully capture hysteresis in various materials and structures with standardized implementation procedures. Very fewphysical models have been presented for SCP actuators [31,32],and they couldn’t capture the hysteresis.This paper presents the
ﬁrst
study on physicsinspired modeling to capture and estimate SCP actuators’ strain – loading hysteresis by analyzing the properties of nylon threads from whichthey are fabricated. The presence of hysteresis in nylon ﬁbershas been established previously [27]. Unlike the claim by existing studies that the hysteresis of SCP actuators is induced by thefriction between adjacent coils, we found that the strain of SCPactuators is linearly correlated to that of the nylon thread underthe same loading conditions. This suggests that the strain – loading hysteresis of SCP actuators is determined by the propertiesof nylon. The developed model combines the Preisach operator and helical spring model, and can be used in a general andefﬁcient fashion to predict hysteresis in SCP actuators with different thicknesses unlike previously developed models that aremore restricted. A helical spring model is adopted to obtain thestiffness of SCP actuators. Experimental validations involvingtwoply, fourply, and sixply nylon threads and SCP actuatorsare provided to show the effectiveness of the proposed approach.The remainder of the paper is organized as follows. First,preliminary experimental illustration is provided to show the correlation between the hysteresis in SCP actuators and nylon polymer threads. The Preisach model and helical spring model arethenbrieﬂyreviewed, andtheproposedmodelispresented. Afterwhich the effectiveness of the proposed approach is experimentally veriﬁed. Finally, concluding remarks and brief discussionson future work are provided.2 Copyright © 2019 by ASME
0 200 400
Loading (g)
01234
S t r a i n ( % )
(a)
0 200 400
Loading (g)
0510152025
S t r a i n ( % )
(b)
01234
Strain of Nylon (%)
0102030
S t r a i n o f S C P A c t u a t o r ( % )
(c)
FIGURE 2
. The hysteresis between the loading and the strain of (a) a 4ply nylon thread and (b) an SCP actuator fabricated from the 4ply nylonthread. (c) The correlation of the nylon thread strain and the SCP actuator strain under the same loading condition.
PRELIMINARY EXPERIMENTAL ILLUSTRATION
Preliminary experiments were conducted to ﬁnd the correlation of the hysteresis properties in SCP actuators and the corresponding nylon threads that the SCP actuators were fabricatedfrom. For a 4ply nylon thread, different loading conditions wereapplied within the range of [40, 500] g. Each loading condition was held for 3 minutes to ensure steadystate conditions arereached. In the end of each step, the steadystate elongation andthe corresponding strain value were recorded.The relationship between the loading and strain of a 4plynylon thread is shown in Figure 2(a). As it illustrates, a signiﬁcant hysteresis was observed. The same loading conditions weretested for the SCP actuator fabricated from 4ply nylon thread,and the hysteresis between the loading and strain of the SCP actuator is shown in Figure 2(b). Since the tested loading conditions were the same for the nylon thread and the SCP actuator,the corresponding strain values of the nylon thread and SCP actuator were correlated in Figure 2(c). It’s found that the straincorrelation can be approximated as a linear term. It can be inferred that there exists a linear mapping between the strains of the nylon thread and the SCP actuator. The loaddisplacementhysteresis of the SCP actuators is thus induced by the inherentproperty of nylon thread [27]. Based on the correlation, a modelwill be proposed to predict the hysteresis of an SCP actuatormade out of nylon thread with any given ply number. A moredetailed discussion will be presented in Sections
ExperimentalSetup
,
Experimental Results
, and
Proposed Model
.
REVIEW OF THE PREISACH OPERATOR AND HELICAL SPRING MODELPreisach Operator
The Preisach operator is one of the most popular and accurate models to capture hysteresis [19,20], and is thus adopted inthis study. The Preisach operator can be expressed as a weightedintegration of delayed relays, called Preisach
hysterons
. The output of the Preisach hysteron,
γ
β
,
α
, can be expressed as
γ
β
,
α
[
v
(
·
)
;
ζ
0
(
β
,
α
)] =
+
1 if
v
>
α
−
1 if
v
<
β ζ
0
(
β
,
α
)
if
β
≤
v
≤
α
,
(1)where
α
and
β
determine the thresholds,
v
(
·
)
denotes the inputhistory
v
(
τ
)
, 0
≤
τ
≤
t
,
ζ
0
(
β
,
α
)
∈ {−
1
,
1
}
is the initial condition.By integrating different Preisach hysterons, the output of aPreisach operator,
Γ
, can be expressed as
u
(
t
) =
Γ
[
v
;
ζ
0
](
t
) =
P
0
µ
(
β
,
α
)
γ
β
,
α
[
v
;
ζ
0
(
β
,
α
)](
t
)
d
β
d
α
,
(2)where
µ
is the density function and
P
=
{
(
β
,
α
)
:
v
min
≤
β
≤
α
≤
v
max
}
is the
Preisach plane
. The Preisach plane deﬁnes theregion of integration. Given the input history
v
(
τ
)
, 0
≤
τ
≤
t
, theoutputs of all hysterons can be computed using Eq. (1).To identify the density function
µ
(
β
,
α
)
efﬁciently, the following discretization procedure is often employed: the densityfunction is discretized to a piecewise constant function – the density value is constant within each lattice cell but could vary fromcell to cell [33]. Figure 3 shows an example of discretizationof the density function into L levels and
L
(
L
+
1
)
/
2 cells, eachcell is associated with a constant density value,
µ
ij
. With thisdiscretization scheme, the model output at time
n
is written as:3 Copyright © 2019 by ASME
L
m
L
L
m
m L m
L
L
L
L
FIGURE 3
. Discretization of the Preisach density function.
˜
u
(
n
) =
u
c
+
L
∑
i
=
1
L
+
1
−
i
∑
j
=
1
µ
ij
s
ij
[
n
]
,
(3)where
u
c
is a constant bias,
s
ij
[
n
]
is the
signed
area of the cell
(
i
,
j
)
, namely, its area occupied by hysterons with output
+
1 minus that occupied by hysterons with output
−
1. The boundary of the two regions is the memory curve. As an example, the memory curve shown in Figure 3 is denoted as
ψ
0
. The outputs of the hysterons within the shaded area are 1 and the remaining are
−
1. Readers are referred to [34–36] for more details about thePreisach operator.
Helical Spring Model
The helical spring model predicts the stiffness coefﬁcient of a spring based on its geometry. It can be employed to computethe stiffness of the SCP actuators manufactured with different plynumbers of nylon thread. The stiffness coefﬁcient
K
of a helicalspring is determined using the following equation:
K
=
d
4
G
8
nD
3
,
(4)where
d
is the wire diameter,
D
is the outer diameter,
G
is therigidity modulus, and
n
is the number of coils. More details onthe helical spring model can be found from chapter 10 in [37].The equation presented here is used to calculate the theoreticalvalues of stiffness of SCP actuators.
PROPOSED MODEL
A mathematical model is presented that correlates the hysteresis in SCP actuator with the hysteresis in the correspondingnylon thread from which the SCP actuators are fabricated. Thisimplies that given the strain – loading hysteresis of the nylonthread, the strain – loading hysteresis of the SCP actuator canbe predicted. Furthermore, the correlation between the strains of nylon threads with different ply numbers is intuitively explained.The correlation between the strains of different SCP actuators isobtained using the helical spring model.Without loss of generality, the derivation is presented tocompute the hysteresis of SCP actuators manufactured from a
q
ply nylon thread based on the knowledge of a
p
ply nylon thread.Assume the stiffness of a
p
ply nylon thread is
K
nylon
,
p
. Thestrain – loading hysteresis of a
p
ply nylon thread can be expressed as
S
nylon
,
p
=
Γ
[
F
load
;
ζ
0
](
t
)
,
(5)where
S
nylon
,
p
denotes the strain of the
p
ply nylon thread, and
F
load
is the applied loads.For the SCP actuator fabricated by the
p
ply nylon thread,the strain – loading hysteresis can be described as
S
SCP
,
p
=
K
nylon
,
p
K
SCP
,
p
Γ
[
F
load
;
ζ
0
](
t
)
,
(6)where
K
SCP
,
p
is the stiffness of the SCP actuator manufacturedfrom the
p
ply nylon thread, which can be obtained using Eq.(4).For a
q
ply nylon thread, the strain – loading hysteresis relationship can be written as
S
nylon
,
q
=
Γ
[
q p
·
F
load
;
ζ
0
](
t
) =
pq
·
Γ
[
F
load
;
ζ
0
](
t
)
.
(7)This can be explained considering the stiffness of the
q
ply nylonthread is
q p
K
nylon
,
1
. Another way of explanation is that load
q
·
F
load
applied to a
q
ply nylon thread and load
p
·
F
load
applied toa
p
ply thread will generate the same amount of strain.Finally, for an SCP actuator fabricated from a
q
ply nylonthread, the strain – loading hysteresis can be expressed as
S
SCP
,
q
=
K
SCP
,
q
K
SCP
,
p
·
S
SCP
,
p
(8)4 Copyright © 2019 by ASME
Through the proposed model, the hysteresis in SCP actuators can be estimated using the hysteresis in nylon threads andmodelled using the Preisach operator (Eq. (3)) and the helicalspring model (Eq. (4)).
EXPERIMENTAL SETUPSCP Actuator Fabrication
Following our previous study [18], the V Technical TextilesConductive Yarns (110
/
34 dtex, Denier:110
/
34f) were used tomanufacture SCP actuators. They can consistently generate 1015% strains. The manufacturing process consists of two steps,namely, coiling and heat treatment. Coiling was realized by amotor, and heat treatment was realized either using an oven orapplying electrical pulses. Samples of fabricated SCP actuatorsunder different heat treatment conditions were shown in Figure1. After heat treatment, the length of the coiled thread convergesand the SCP actuator is created. In this study, different SCP actuators were fabricated from 2ply, 4ply and 6ply nylon threads,and their resting lengths were all approximately 100 mm. Thenumber of turns for coiling, the temperature or peak voltage applied for heat treatment, and the number of cycles during heattreatment varied with the ply number and were usually determined experimentally. More details about the manufacturing of SCP actuators can be found in [12] and [18].
Experimental Setup
The experimental setup is shown in Figure 4. A rigid beamwas used to suspend the nylon thread/SCP actuator. To measurethestrain, alineardistancesensor(SPSL035LATS,Honeywell)which has a resolution of 0.04 mm was used. It measured thedisplacement based on the position of a magnet. The magnetwas attached to the free end of the nylon thread/SCP actuator.The additional weight due to the magnet was around 20 g. Theoutput voltage from the sensor was recorded using a multimeter.These voltage readings were later converted into displacementand strain values. The free end was used for loading.
EXPERIMENTAL RESULTS
The proposed model can capture the hysteresis of SCP actuators by utilizing the hysteresis of a nylon thread that the actuatoris fabricated from. The linear correlation of strains is brieﬂy presented in Section
Preliminary Experimental Illustration
andwill be discussed in detail in this section. To evaluate the performance of the modeling performance, the average absolute errorand standard deviation were utilized in this study:
SCP ActuatorDistance SensorLoadMultimeterPower Supply
FIGURE 4
. The experimental setup for strain measurements.
E
average
=
M
∑
i
=
1

e
i

M
,
(9)
σ
=
M
∑
i
=
1
(

e
i
−
E
average
)
2
M
−
1
,
(10)where
M
is the number of datapoints to be evaluated and
e
i
denotes the error of the
i
th
point.
Model Identiﬁcation2PlyNylonThreadandSCPactuators
Experimentswere conducted to measure the hysteresis of a 2ply nylon threadand the corresponding SCP actuator. A sequence of loading conditions ranged from [20, 250] g were applied, as shown in Figure5(a). This sequence followed a damped oscillations proﬁle to satisfy the sufﬁcient excitation condition required to fully identifythe Preisach operator [38].The model was identiﬁed efﬁciently as a linear leastsquaresoptimization problem [20]. The level of discretization of thePreisach operator,
L
, was chosen to be 10, and the constant bias,
u
c
, was found to be 1.73%. Figure 5(b) shows the hysteresismeasurement and modeling results of the loading – strain hysteresis of 2ply nylon thread. The modeling error was 0.02
±
0.02%, as presented in Figure 5(c)). In this paper, the unit of theerror, %, is the error of the strain output, not the error percentage. The term “index” refers to the numbering of the quasistaticstrain values. Considering that the strain range was over 3%, the5 Copyright © 2019 by ASME