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The effects of nylon polymer threads on strain-loading hysteresis behavior of supercoiled polymer (SCP) artificial muscles

Supercoiled polymers (SCP) actuator, as a recently discovered artificial muscle, has attracted a lot of attention as a compliant and compact actuation mechanism. SCP actuators can be fabricated from nylon polymer threads, and generates up to 20%
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  Proceedings of the ASME 2019Dynamic Systems and Control ConferenceDSCC 2019October 9-11, 2019, Park City, Utah, USA DSCC2019-9169 THE EFFECTS OF NYLON POLYMER THREADS ON STRAIN-LOADINGHYSTERESIS BEHAVIOR OF SUPERCOILED POLYMER (SCP) ARTIFICIALMUSCLES Revanth Konda ∗ Smart Robotics LaboratoryDepartment of Mechanical EngineeringUniversity of Nevada, RenoReno, Nevada 89557Email: Jun Zhang Smart Robotics LaboratoryDepartment of Mechanical EngineeringUniversity of Nevada, RenoReno, Nevada 89557Email: ABSTRACT Supercoiled polymers (SCP) actuator, as a recently discov-ered 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 physics-inspired model is pre-sented 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 charac-terize and estimate the strain – loading hysteresis of SCP actua-tors fabricated with different numbers of nylon threads. A helicalspring model is adopted to obtain the stiffness of SCP actuatorswith different configurations. Experimental validation involvingtwo-ply, four-ply, and six-ply 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 tradi-tional actuators like electromagnetic motors and hydraulic actu-ators, mainly because they are more compliant and lightweight,and can be utilized to develop compliant structures without com-plex linkages [3,4]. Artificial muscles have shown strong poten-tial 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 off-the-shelf nylon polymer threadsor other fibers and filaments, like carbon nanotube yarns, fish-ing lines, and sewing monofilaments [8]. Under thermal stimuli,which is often realized by electric power-induced 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 fabrica-tion process, high power density, and the large actuation range[2,8,10,11]. For example, up to 21% tensile strain was demon-strated with the non-mandrel-coiled SCP actuators [11,12], up to49% strain was reported for mandrel-coiled SCP actuators [8]. In1 Copyright © 2019 by ASME  1 mm1 mm0.1 mm FIGURE 1 . (a). A nylon thread. (b). A super-coiled 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 SCP-actuated 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 effi-ciently estimate the performance of SCP actuators. An impor-tant reason is that SCP actuators exhibit significant and complexhysteresis behaviors that are difficult to be understood and cap-tured. As a typical type of nonlinearities, hysteresis appears invarious types of smart materials. For a hysteretic system, the out-put of the system depends on both the current input and the in-put history, even under quasi-static conditions [19,20]. The hys-teretic 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 three-dimensional hysteresis among force, strain, and electric powerwas presented [23].The existing studies failed to estimate the hysteresis prop-erties of different SCP actuators efficiently. They either adoptedsimplified linear models that were unable to capture the hystere-sis or phenomenological hysteresis models that required tediousprocedures for identification and implementation. Early studiespredominantly adopted linear models by approximating the hys-teresis as a linear damping term. For example, Yip et al. pre-sented the first study to model SCP actuators using a linear dy-namical model [12]. Force and position controllers were pro-posed in [16, 24, 25], but only linear models and proportional-integral-derivative (PID) controls were adopted. With a linearmodel, it was found that over 15% modeling error could be gen-erated [18]. The first approach to capture the strain – voltage hys-teresis in SCP actuators under different loading conditions usingthe augmented Preisach model and augmented Prandtl-Ishlinskiimodel was proposed in [18]. A Duhem differential model wasproposed to describe the strain – temperature hysteresis [26].The three-dimensional hysteresis was further modeled by em-bedding two Preisach operators [23]. However, all hysteresismodels presented for SCP actuators were phenomenology-basedthat 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 hys-teresis of a different SCP actuator. It is desirable to develop anapproach to capture the hysteresis behaviors of different SCP ac-tuators 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 hysteresiseffectsmightbeinducedbythefrictionamongadjacentcoilsdur-ing actuator operation [9,12], the hysteresis may also be relevantto the inherent hysteresis property of nylon polymers from whichSCP actuators are fabricated [27]. Secondly, phenomenologi-cal 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. Phenomenolog-ical models, such as the Preisach operator [20], Krasnoselskii-Pokrovskii model [19], Prandtl-Ishlinskii model [28], Duhemmodel [29], and Bouc-Wen model [30], are widely adopted tosuccessfully capture hysteresis in various materials and struc-tures 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 physics-inspired mod-eling to capture and estimate SCP actuators’ strain – loading hys-teresis 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 exist-ing 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 – load-ing hysteresis of SCP actuators is determined by the propertiesof nylon. The developed model combines the Preisach opera-tor and helical spring model, and can be used in a general andefficient fashion to predict hysteresis in SCP actuators with dif-ferent thicknesses unlike previously developed models that aremore restricted. A helical spring model is adopted to obtain thestiffness of SCP actuators. Experimental validations involvingtwo-ply, four-ply, and six-ply 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 cor-relation between the hysteresis in SCP actuators and nylon poly-mer threads. The Preisach model and helical spring model arethenbrieflyreviewed, andtheproposedmodelispresented. Afterwhich the effectiveness of the proposed approach is experimen-tally 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 4-ply nylon thread and (b) an SCP actuator fabricated from the 4-ply 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 corre-lation of the hysteresis properties in SCP actuators and the cor-responding nylon threads that the SCP actuators were fabricatedfrom. For a 4-ply nylon thread, different loading conditions wereapplied within the range of [40, 500] g. Each loading condi-tion was held for 3 minutes to ensure steady-state conditions arereached. In the end of each step, the steady-state elongation andthe corresponding strain value were recorded.The relationship between the loading and strain of a 4-plynylon thread is shown in Figure 2(a). As it illustrates, a signifi-cant hysteresis was observed. The same loading conditions weretested for the SCP actuator fabricated from 4-ply nylon thread,and the hysteresis between the loading and strain of the SCP ac-tuator is shown in Figure 2(b). Since the tested loading condi-tions were the same for the nylon thread and the SCP actuator,the corresponding strain values of the nylon thread and SCP ac-tuator were correlated in Figure 2(c). It’s found that the straincorrelation can be approximated as a linear term. It can be in-ferred that there exists a linear mapping between the strains of the nylon thread and the SCP actuator. The load-displacementhysteresis 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 HELI-CAL SPRING MODELPreisach Operator The Preisach operator is one of the most popular and accu-rate 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 out-put 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 condi-tion.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 fol-lowing discretization procedure is often employed: the densityfunction is discretized to a piecewise constant function – the den-sity 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 mi-nus that occupied by hysterons with output  − 1. The boundary of the two regions is the memory curve. As an example, the mem-ory 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 hys-teresis 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 ex-pressed 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 re-lationship 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 actua-tors 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 10-15% 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 actu-ators were fabricated from 2-ply, 4-ply and 6-ply nylon threads,and their resting lengths were all approximately 100 mm. Thenumber of turns for coiling, the temperature or peak voltage ap-plied for heat treatment, and the number of cycles during heattreatment varied with the ply number and were usually deter-mined 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(SPS-L035-LATS,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 actu-ators by utilizing the hysteresis of a nylon thread that the actuatoris fabricated from. The linear correlation of strains is briefly pre-sented in Section  Preliminary Experimental Illustration  andwill be discussed in detail in this section. To evaluate the perfor-mance 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 data-points to be evaluated and  e i  de-notes the error of the  i th  point. Model Identification2-PlyNylonThreadandSCPactuators  Experimentswere conducted to measure the hysteresis of a 2-ply nylon threadand the corresponding SCP actuator. A sequence of loading con-ditions ranged from [20, 250] g were applied, as shown in Figure5(a). This sequence followed a damped oscillations profile to sat-isfy the sufficient excitation condition required to fully identifythe Preisach operator [38].The model was identified efficiently as a linear least-squaresoptimization 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 hys-teresis of 2-ply 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 percent-age. The term “index” refers to the numbering of the quasi-staticstrain values. Considering that the strain range was over 3%, the5 Copyright © 2019 by ASME
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