ORIGINAL ARTICLE

Fabrication and Dynamic Modeling of BidirectionalBending Soft Actuator Integrated with OpticalWaveguide Curvature Sensor

Wenbin Chen,1 Caihua Xiong,1 Chenlong Liu,1 Peimin Li,1 and Yonghua Chen2

Abstract

Soft robots exhibit many exciting properties due to their softness and body compliance. However, to interactwith the environment safely and to perform a task effectively, a soft robot faces a series of challenges such asdexterous motion, proprioceptive sensing, and robust control of its deformable bodies. To address these issues,this article presents a method for fabrication and dynamic modeling of a novel bidirectional bending softpneumatic actuator that embeds a curvature proprioceptive sensor. The bidirectional bending deformation wasgenerated by two similar chambers with a sinusoidal shape for reducing the internal dampness during bendingdeformation. An optical waveguide made from flexible poly (methyl methacrylate) material that is immune tothe inlet pressure was embedded into the actuator body to measure its bending angle. A dynamic modelingframework based on step response and parameter fitting was proposed to establish a simple differential equationthat can describe the nonlinear behavior of the soft actuator. Hence, a sliding mode controller is designed basedon this differential equation and the Taylor expansion. The proposed dynamical model and the sliding modecontroller were validated by trajectory tracking experiments. The performance of the bidirectional bending softactuator, such as the linear output of the curvature sensor in different inflating patterns, the proprioceptivesensitiveness to the external environment, the output force, and large bending range under relatively smallpressure, was evaluated by relevant experimental paradigms. Prototypes from the novel design and fabricationprocess demonstrated the soft actuator’s potential applications in industrial grasping and hand rehabilitation.

Keywords: soft actuator, bidirectional bending deformation, optical waveguide, trajectory tracking, sliding modecontrol, proprioceptive

Introduction

The compliant nature endows the pneumatically actu-ated soft robots with significant advantages over the

traditional rigid body robots in flexibility-required applica-tions such as service robots that can safely interact with hu-mans, rescue robots that conduct tasks in an unstructuredenvironment, and medical robots for surgery, rehabilitative,or prosthetic purpose.1–6 The most studied soft robots are the

soft actuators that have been designed to perform translationor rotation by axial or radial deformation.5,7–9 In many ap-plications of soft actuator, the actuator is expected to havecompact size and low hardness, capable of bidirectionalbending with large bending angle and appropriate stiffnessfor object holding and manipulation. Furthermore, if a softactuator can sense its deformation, it can then provide fun-damental information for closed-loop feedback control. Ty-pical tasks involving bidirectional bending, which have been

1Institute of Robotics Research, State Key Laboratory of Digital Manufacturing Equipment and Technology, School of MechanicalScience and Engineering, Huazhong University of Science and Technology, Wuhan, China.

2Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China.

� Wenbin Chen et al. 2019; Published by Mary Ann Liebert, Inc. This Open Access article is distributed under the terms of the CreativeCommons License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

SOFT ROBOTICSVolume 6, Number 4, 2019Mary Ann Liebert, Inc.DOI: 10.1089/soro.2018.0061

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reported in the literature, include autonomous slithering onthe ground,10,11 the escape maneuvers in water,12 and thecomplex motion of human thumb.8

It has been shown previously that soft-bending actuatorsare constructed from polymeric13 or a combination of elas-tomeric (hyperelastic silicones) and inextensible materials(fabrics and fibers),7,8 and activated by pressurizing fluidmedia (liquid or gas). There are various designs of soft-bending actuators by varying the number of chambers and thecross-section shape. When considering a single chamber, thecross-section shape can be circular,10,14 rectangular,15 orsemicircular5,9,16 where uniform bending is produced byasymmetrically constraining the extension of different layersof the air chamber.17 Nonuniform bending can be created byregulating the fabric distribution and winding of fibers alongthe actuators.8 More complex motions can be implementedby using multichambers.4,16,18–21

To implement bidirectional bending, some methods havebeen proposed by fabricating two22,23 or three4 individual ac-tuators together, or casting several chambers together.11,24–26

The fluid pressurizes the partial actuators of the group en-forcing them to bend toward the rest. Similarly, if the fluidonly pressurizes the rest of the chambers, the group will bendtoward another direction. Some research studies proposedmore compact designs by integrating two similarly designedchambers into one actuator to replicate the symmetric ago-nistic and antagonistic motion under pressure without fiber orshell reinforcement.12,27 Considering the possible rupture orperforation in their walls under such design, a soft pneumaticactuator enveloped by a Yoshimura patterned origami shellthat acts as an additional protection layer while providingspecific bending resilience throughout the actuator’s range ofmotion was reported.28 In another study, a negative airpressure was used to implement opposite bending corre-sponding to the deformation under positive air pressure.17

Regarding pose sensing of soft actuators, some researchwork employs the side-polished optical waveguide in whichthe intensity of transmitted light is related to flexion angle todesign the proprioceptive curvature sensing element. Thefabrication strategies include the use of macrobend stretchoptical fibers29 and stretchable optical fibers30–32 for re-sponding to multiple modes of deformation, the thin-film-designed waveguide applied in robot hands,33 engraved plas-tic optical fiber for bend measurement,34–36 and joint angledetection.37–40 The stiff magnetic sensors measure curvatureof a bending segment by mapping the position of the magnetwith respect to the stiff Hall element.11,25 Some other re-searchers integrate resistive sensors that basically measure achange in the current flow through conductive materials, suchas the conductive ink,41,42 conductive polymers,43 eutecticgalliumindium,44 and hydrogels.45 The capacitive sensingelements are also built to reveal a soft body bending angle bycomparing the two sensory responses around the concave andthe convex side of the bent body,46 or by detecting elonga-tional strains.47 Most of the developed proprioceptive sensorsdetecting bending curvature, for example, the optical wave-guide sensors made from elastomer material, are usuallyplaced on the surface of soft object. They can only work wellwhen none of the radial deformations are overlapped on theaxial deformation because radial deformations will possiblydistort the linear relationship between output voltage and theaxial deformation corresponding to the bending angle.

Several issues still exist for the fabrication of bidirectionalbending soft actuators integrated with sensing ability. First,the bidirectional soft actuator that works in constrained en-vironment often requires a compact size to meet the taskconditions such as the hand rehabilitative glove for strokepatients. Consequently, the external reinforced technique forprotecting the actuator from perforation, such as the externalshell reinforcement, will not be suitable for use in such con-strained environments. Second, ordinary three-dimensional(3D) printing techniques have more difficulties in integratingthe flexible sensor or interaction force sensor into the softstructure compared with the multistep molding techniques.Furthermore, the available materials for 3D printing aregenerally with greater hardness than that of the popular softelastomers, hence it will consume more fluid pressure toovercome the inner stress of the material deformation. Third,most of the developed sensing elements integrated in the softactuator for curvature detection are susceptible to the inflatedpressure. Therefore, the calibrated relationship between theoutput voltage and curvature tends to be affected if multi-chambers are simultaneously inflated. This significantly af-fects performance in practical applications where closed-loopfeedback control is required.

In this article, we present a new compact design of a softpneumatic actuator with curvature proprioceptive ability. Theactuator consists of two similar chambers with sinusoidalsection shape, and each chamber is fiber reinforced by onlywinding around the wave peaks. The soft actuator is able tobend bidirectionally with large angle and exerts large tip forceunder a relatively small pressure. The optimized geometricalparameter is explored through finite element analysis (FEA)simulation. The customized optical waveguide with an in-tendedly roughened surface, made from poly (methyl meth-acrylate) (PMMA) material, is integrated into soft actuator asthe proprioceptive curvature sensor. The optical waveguide isfree of radial deformation and can provide steady linear outputunder air pressure. Because of the nonlinear behavior of thesoft actuator, the modeling of dynamics and precise control ofactuator’s behavior are two fundamental issues of soft robot-ics. Therefore, a simplified dynamic modeling frameworkbased on step response experiment is proposed. Consideringthat the dynamic model includes pressure-dependent param-eters, the variable structure control strategy is studied and acompetent sliding mode controller is developed. The proposedsystem was validated by a trajectory tracking experiment.

The main contributions of this article are as follows: (1)Differing from the popular reinforced fiber uniformly acrossthe whole chamber, which results in relatively high materialstress in pressure-induced bending deformation, the wavechamber with sinusoidal section shape proposed in this re-search is fiber reinforced only around the wave peak. Thisdesign enhances the bidirectional bending ability of the softactuator in lower air pressure. (2) Through step responseexperiment, the dynamic modeling framework consideringthe pressure-dependent parameters is developed toward re-producing the nonlinear behavior of the soft actuator. Thetechnique of Taylor expansion is used to locally linearize thecontrol variable and further establish the model-based vari-able structure controller. By embedding the plastic opticalfiber curvature sensor, the feedback control is implemented.

This article is organized as follows: the objective of thiswork is given in the Objectives section. In the Materials and

496 CHEN ET AL.

Methods section, the fabrication of the soft actuator and theintegration of optical waveguide are first provided. Then,the influence and mechanism of the section parameter of thechamber on bending deformability are studied. Third, thecurvature description and the measuring principle by usingthe optical waveguide are presented. Finally, the dynamicmodeling strategy of the soft actuator is discussed, and thevariable structure sliding mode controller is designed forclosed-loop feedback control. In the Results section, weverify the performance of the soft actuator in bidirectionalbending, sensor sensitivity, output force, and angular trajec-tory control by different experimental paradigms.

Objectives

In this work, we address two main issues: (1) develop a softactuator with bidirectional bending ability and propriocep-tive curvature sensing function that is robust in large airpressure; (2) formulate a general framework for dynamicmodeling of the soft actuator and implement precise curva-

ture control with only proprioceptive information. Aroundthese two objectives, the proposed design strategy will bevalidated through experiment. The related geometrical pa-rameters of the chamber for desired bending deformation willbe investigated. The performance of the optical waveguidebased on PMMA material in measuring the general curvature,such as the robustness and output linearity under free de-formation, will be studied. The dynamic model and thefeedback control through the proprioceptive curvature will bedeveloped.

Materials and Methods

Fabrication of the soft actuator

The structure and main component of the soft-bendingactuator are shown in Figure 1. The elastomeric body wasfabricated by the silicone composite (ELASTOSIL� RT622 A). The soft body comprises two identical chambers,which were cast together through the middle layer that was

FIG. 1. Typical fabrication steps of the soft actuator, structure of the optical waveguide and freely continue bendingdeformation of the soft actuator. (A) Fabricating the open chamber with a sinusoidal shape. (B) Closing the open side of thechamber with an elastomer sheet. (C) Attaching the strain limited layer and winding the reinforced fiber. (D) Encapsulatingthe two fiber-reinforced chambers and flexible sensor together. (E) The final soft actuator with bidirectional bending ability.(F) The inner structure of the soft actuator: (I) the wall of the chamber corresponding to different fabrication steps, (II) thestrain limited layer (woven fiberglass), and (III) the optical waveguide. The two main geometrical parameters for thechamber with the sinusoidal wave: pitches s and amplitude h. (G) The customized optical waveguide with transmit probe(Tx) and receive detector (Rx). (H) Bending deformation of the neural axis (optical waveguide) of the soft actuator. Colorimages are available online.

FABRICATION AND DYNAMIC MODELING OF SOFT ACTUATOR 497

integrated with nonstretchable fiberglass fabric. The shape ofthe chamber was a sinusoidal wave with multiperiod, whichwas radially reinforced by the Kevlar fibers incorporated intothe elastomeric material (Fig. 1C). The fibers were onlywound around the peak, and then crossed through the bottomin the normal and inverse direction. Such fabrication strategyconstrains the radial bulging around the wave peak but letgroove region between peaks freely bulge, which signifi-cantly improves the natural axial elongation ability of theelastomeric body. Furthermore, the fabrication decreases theinner squeezing stress from material stacking during com-pression. When inflating one of the two chambers, the elon-gation of stretching side chamber and the shortening of thecompressed side chamber are then transformed to the lateralbending deformation under the constraint of middle strainlimited layer.

The integration of curvature proprioceptive function

To measure the general curvature of the soft actuator underinflated pressure, the optical fiber made from transparentPMMA material was fabricated to be intentionally lost. Aslight propagates through it, some laterally refract the envi-ronment, and some repeatedly reflect the end port. The more itis deformed, the more the light is lost. Thus the bending de-formation can be indicated by measuring the light power loss.The relationship between curvature and power loss can becalibrated by individual experiment. Considering bendablebut not stretchable feature of the PMMA, the opticalwaveguide was embedded into the middle layer whoselength keeps constant due to the limitation of the glass fiber(Fig. 1D). It must be noted that the inflated chambers willprovide pressure force to the optical waveguide, which willintroduce unexpected radial deformation of the waveguideif it is fabricated by the transparent elastomeric silicone.30

Such radial deformation will bring additional light loss, andfurther affects the original relationship between curvatureand light loss. Hence the PMMA-based optical waveguidewas chosen as the curvature sensor, for its incompressiblefeature under air pressure.

To fabricate the sensory waveguide, the thermoplasticsheath was used as the jacket to steadily hold the probesrelative to the ends of waveguide (Fig. 1G). As the highlyabsorptive composite, the thermoplastic sheath insulates theoptical waveguide from the nonuniform local deformationintroduced by the tensioned Kevlar fibers in inflation.

Curvature-based configuration description

For the traditional rigid link-based manipulator, the con-figuration of the mechanism is determined by the joint angleor prismatic displacement. For the soft actuator, the curvatureis the independent variable. The chosen variable is appro-priate because the input pressure to the soft segment has adirect effect on the curvature. In this work, the curvature ismeasured on the neural axis of the soft actuator whose lengthapproximately keeps constant and is equal to the initial lengthof the actuator. The free bending of the neural axis is a uni-form arc whose curvature can be measured by the integratedsensor. Thus, the free continual bending deformation of thesoft actuator can be seen as the time-varying arc whosecurvature continually varies from the great value to the smallvalue. If the level length and the curvature of the neural axis

of the soft actuator are denoted by l0 and k(t), respectively,then the subtended angle h(t) at the instantaneous center ofthe circle corresponding to the arc is

h(t)¼ l0k(t): (1)

Because the curvature is proportional to the subtendedangle, it is reasonable to model the configuration of the softactuator by the subtended angle instead of the curvature. Thusthe generalized coordinate to describe the system configu-ration is the subtended angle. As an example of this point, thecontinual deformation of neural axis of a soft actuator withconstant length 6 mm, that is, continual variation of curvaturek, is shown in Figure 1H by varying the subtended angle hfrom 0� to 330�.

Section parameter of chamber on bendingdeformability

As for section shape with the sinusoidal wave, there areseveral shape parameters that determine the dampness andaffect the bending ability for the given inlet pressure, such asthe lateral wall thickness, middle layer thickness, pitch, and theamplitude of the sinusoidal wave. For real application, it isdesired that the actuator can bend bidirectionally under smallpressure while produces acceptable output force. To reduce theanalysis complexity and find the general influence, we mainlyconsider the following two significant parameters (Fig. 1F):pitch s and the amplitude h. The FEA simulation was used tostudy such two parameters on the bending performance.

The thickness and longitudinal length of the chamber werefixed for all simulations; that is, 3 and 115 mm. For thespecified pitch and amplitude of the wave, the soft actuatorwas pressurized by 0.1 MPa, and the final subtended anglewas recorded. This simulation was repeated under differentpitches and amplitudes. In the simulation experiment, theamplitude varied from 1.5 to 3.5 mm at step 0.5 mm. For eachamplitude, the pitch was varied to let the number of waves ofthe chamber vary from 6 to 15. The simulation result is givenin Figure 2.

FIG. 2. The subtended angle of the soft actuator for dif-ferent pitches under the specified amplitude of the sinusoidalwave. Color images are available online.

498 CHEN ET AL.

For the same air pressure, the greater subtended anglerepresents the relatively low inner dampness and higherbending ability. As shown in Figure 2, the pitch of the sinu-soidal wave is approximately monotonically decreasing withthe subtended angle. Basically, the smaller pitch of the si-nusoidal wave brings greater bending deformation under thesame pressure. Meanwhile, for the specified pitch, the greateramplitude generally brings better bending ability. The softactuators with greater amplitude are basically easy to bendcompared with the soft actuator with smaller amplitudes. Inthe specified range of the parameters, when one chamber wasinflated with 0.1 MPa pressure, the greatest subtended anglereached 280� under pitch 8.5 mm and amplitude 3.5 mm, theleast subtended angle was just 50� under pitch 8.5 mm andamplitude 1.5 mm. For great pitches, the variation of ampli-tudes shows a relatively small influence on the bendingability. However, the bending ability is more sensitive to theamplitude when fabricating the soft actuator with a smallpitch. The illustrative steady bending deformation of the softactuator with pitch 8.5 mm and amplitude 3.5 mm is shown inFigure 3 when individually inflating the two chambers.

Although the geometrical parameters are discretized in avery limited range, the general tendency of the bending underthe different geometrical parameters is effectively revealed.The greater amplitude and smaller pitch endow the soft ac-tuator better bending ability.

Metric of the curvature by the optical waveguide

If the output power of the waveguide under no deformationis defined as the baseline power V0, the power loss of thewaveguide is defined as the following logarithm function30:

I¼ 10log10(V0=V), (2)

where V is the output power. From this definition, the outputpower loss I is zero under no deformation, I>0 with in-creasing deformation. In this work, the voltage of the pho-todiode was used to indicate the output power.

The linearity of the waveguide under different inflationpatterns is essential for the usability as the curvature sensor.To specify such property of the waveguides, three inflationpatterns are studied: (I) inflating only one chamber, (II) in-

flating both the chambers, and (III) bending without inflation.For the bending under inflation (patterns I and II), the camerawas used to capture the steady image, which was then used tomeasure the subtended angle. Three markers prestuck on theneural axis, that is, one at the middle and two at the end, wereused to find the arc center, which was further used to form thesubtended angle. The measuring principle is shown in Fig-ure 4. For the bending without inflation (pattern III), the 3Dprinted molds with a curved groove to contain the soft ac-tuator were used to shape it to the desired angle. The voltagefrom the photodiode was simultaneously recorded. The cor-responding relationship between the subtended angle and theoutput power loss is shown in Figure 5.

The linear relationship between the subtended angle of thesoft actuator and the output power loss is highly significant inthe cases of inflating only one individual chamber or both thechambers. The radial incompressibility of the PMMA mate-rial eliminates the complex local deformation introduced bythe inflated air pressure from the optical waveguide and helpsmaintain the linear relationship. This characteristic assignsthe PMMA waveguide with superiority over the flexiblesensor based on conductive ink and the optical waveguidemade of elastomeric silicone.

Benefiting from the high linearity between subtended an-gle and the output power loss, the power loss can be utilizedas an indication of the bending angle of the soft actuatorthrough calibration using the experimental data shown inFigure 5.

Dynamics response modeling

The exact model of the soft actuator is a challenging issuedue to the high nonlinearity of the interaction between theelastomer and pressured fluid. However, we can explore thefundamental principle of the system by observing the stepresponse under different pressure. Because of the completelysymmetrical structure of the soft actuator, we only considerthe dynamic response in the case of inflating one chamber.The illustrative dynamic responses under different pressureare given in Figure 6.

According to the profile of the step response curve, thesystem shows both features of decaying oscillation and the

FIG. 3. The simulated bidirectional bending behaviorunder different pressure when separately inflating the twochambers. Color images are available online.

FIG. 4. The principle to calculate the subtended angle forsensor calibration. Color images are available online.

FABRICATION AND DYNAMIC MODELING OF SOFT ACTUATOR 499

exponential decaying toward the steady angle. The oscilla-tion component could be obtained from the internal reso-nance of the soft actuator. It is reasonable to infer that the softactuator is similar to a compositive system. The compositivesystem is formed by superimposing two individual systemsthat have complex conjugate pole pair and a real pole, re-spectively, on the left-half complex plane.48 The complexconjugate pole pair on the left-half complex plane corre-sponds to the decaying oscillation feature of an underdampingsystem, whereas the real pole corresponds to the exponentialdecaying feature of an overdamping system. Thus, the generalresponse of the soft actuator under step inflation can be viewedas a third-order system (three poles on the left-half complexplane).

When modeling the system, keeping the oscillation term ishelpful to improve the high-frequency motion ability. How-ever, the oscillation term will bring more parameters andincrease the model complexity. Furthermore, the estimationof modeling uncertainty will be difficult and further increase

the complexity of the controller. To reduce the modelingcomplexity while keeping the system basic feature, we keepthe real part of the complex conjugate pole pair. Then, thesystem is approximated by superimposing two overdampingsystems. The introduced error in this approximation is themodel uncertainty, which can be included in the variablestructure controller.49 Under this framework, the response ofthe system can be fitted by the following time domainequation:

h¼C1e� k1tþC2e� k2tþC3, (3)

where C1 2 R� , C2 2 R� , and C3 2 Rþ are dependent onthe initial conditions and the inlet pressure. Especially, C3 isthe steady-state angle. k1 and k2 are both positive and alsodepend on the pressure. As different step input pressurebrings a different response curve, this formula is a generalexpression. For the clusters of response curves, the least-square fitting technology can be used to establish depen-dent functions about coefficients k1, k2, and C3 to thepressure u.

The step response given in Equation (3) is actually the so-lution of a second-order differential system [Eq. (4)] in thetime domain. Although Equation (4) is similar to the work inthe literature10,22,50 from appearance, it is with a differentmeaning. The system given in Equation (4) is a nonlinear time-invariant system due to the pressure-dependent coefficients:

€hþ k1þ k2ð Þ _hþ k1k2h¼ k1k2C3, (4)

where u is the outlet pressure from the pneumatic propor-tional valve into the chamber. It must be noted that the for-mula (4) is just roughly approximation, since there is fittingerror when determining the relationship between coefficientsk1, k2, C3 and the pressure u. However, it is possible to es-timate the boundary of such model uncertainty and imple-ment precise angle control by using the variable structurecontrol strategy.

If the Dk1 and Dk2 are defined as the variation of k1 and k2,substitute k1 with k1þDk1, k2 with k2þDk2, we can obtainthe following formula Dr to represent the model uncertainty:

Dr¼ � Dk1þDk2ð Þ _hþDk1Dk2þ k1Dk2þ k2Dk1ð Þ � C3� hð Þ

: (5)

When providing system performance requirement, such asthe maximum speed and range of input pressure, the Dr canbe generally estimated.

Considering the compromise between the model simplicityand fitting accuracy, the three parameters k1, k2, and C3 canbe fitted as the function of outlet pressure u from the pneu-matic proportional valve:

k1¼ a1uþ a0

k2¼ b1uþ b0

C3¼ c1u2þ c0u,

(6)

where a0, a1, b0, b1, c0, c1 are constant coefficients dependingon the data of the step response experiment.

FIG. 5. The linear relationship between the subtendedangle and the power loss. Color images are available online.

FIG. 6. The sampled dynamic step response under dif-ferent pressure. Color images are available online.

500 CHEN ET AL.

After substituting Equation (6) into Equation (4), we ob-tain the following formulas:

€h¼ f1 uð Þ _hþ f2 uð Þhþ f3 uð Þ, (7)

where

f1(u)¼ � (a1þ b1)u� a0� b0

f2(u)¼ � a1b1u2� (a1b0þ a0b1)u� a0b0

f3(u)¼ a1b1c1u4þ (a1b0c1þ a0b1c1þ a1b1c0)u3

þ (a1b0c0þ a0b1c0þ a0b0c1)u2þ a0b0c0u:

The system [Eq. (7)] is related to the high order of thesystem control and is not common for designing the slidingmode controller. Considering that the input voltage of thevalve was always varying in very limited range, it is reasonableto reduce the order of the system control by using the Taylorexpansion. Here we use the first-order Taylor expansion toapproximate the high-order term f2(u) and f3(u), and thenEquation (7) can be approximated to the following expression:

€h¼ f1 u0ð Þþ f ¢1 u0ð Þ � u� u0ð Þð Þ _hþf2 u0ð Þþ f ¢2 u0ð Þ � u� u0ð Þð Þhþ

f3 u0ð Þþ f ¢3 u0ð Þ � u� u0ð Þ,(8)

where u0 is the reference input pressure and

f ¢1(u)¼ � (a1þ b1)

f ¢2(u)¼ � 2a1b1u� (a1b0þ a0b1)

f ¢3(u)¼ 4a1b1c1u3þ 3(a1b0c1þ a0b1c1þ a1b1c0)u2

þ 2(a1b0c0þ a0b1c0þ a0b0c1)uþ a0b0c0:

(9)

Thus, further reformulating Equation (8) to the normalform:

€h¼ g _h, h� �

þ h _h, h� �

� u� u0ð Þ, (10)

where

g _h, h� �

¼ f1 u0ð Þ _hþ f2 u0ð Þhþ f3 u0ð Þ

h _h, h� �

¼ f ¢1 u0ð Þ _hþ f ¢2 u0ð Þhþ f ¢3 u0ð Þ: (11)

Sliding mode controller design

The dynamic model of the soft actuator carries pressure-dependent parameter whose variation and error bring thenonlinear feature for the system. As a systematic approach tothe problem of maintaining stability and consistent perfor-mance in the face of modeling uncertainty, the variablestructure sliding mode control strategy is especially suitableto design the controller for the system of the soft actuator.

Let x1¼ h, x2¼ _h, and considering the system error,the differential equation of the system model [Eq. (10)] iswritten as

_x1¼ x2

_x2¼ g x1, x2ð Þþ h x1, x2ð Þ � ~uþDr, (12)

where the ~u¼ u� u0 is assigned as the system control and islimited to a small range in the controller, so that small vari-ation around the reference pressure u0 will occur for using theTaylor expansion. Dr is a manually added term to considerthe system modeling error, and its detailed expression is gi-ven in Equation (5).

The tracking problem for the system is to find a control law~u, such that the closed-loop system satisfies limt!1xd(t)�x(t)¼ 0, where xd(t) is the target trajectory. First, let

e¼ xd � x1

s¼ ceþ e, (13)

where xd is the desired trajectory, c is the positive constant,and is used to adjust for the weight difference between thebending angle and bending velocity. The Lyapunov functioncan be defined as V ¼ s2

�2. And then, the derivative of this

function is

_V ¼ s c _eþ€xd � g� h~u�Drð Þ: (14)

Thus, to _V < 0, the sliding mode controller can be de-signed as

~u¼ 1

hc _eþ€xd � gþ gsþD � sgn sð Þð Þ, (15)

where g is the positive constant which is used to control theconvergence speed of the difference between practical tra-jectory and the desired trajectory. sgn( � ) is the sign function,which is assigned to be +1 for the positive variable and -1 forthe negative variable. D is the maximum fluctuation range ofsystem model error Dr, and it is given by

Drj j � Dk1þDk2ð Þ _h�� ��þ

Dk1Dk2þ k1Dk2þ k2Dk1j j � C3� hj j ¼D: (16)

Results

Considering the compromise of easy fabrication and per-formance, soft actuator with pitch 8.5 mm and amplitude3.5 mm was fabricated to verify the proposed design andmodeling framework. The electric proportional pressurevalve with high rate of flow (VPPM-NPT, FESTO Ltd.) wasused in the following tests.

Sensor sensitivity test

To test the sensitivity of the optical waveguide under ex-ternal disturbance, we conducted the lateral scanning ex-periment to detect the contour of the object.30 To steadilycontact the surface for improving the sensitivity, one of thechambers was pressurized with 0.1 MPa to bend a probableangle and keep a constant stiffness. Three 3D printed objectswith different surface contours were placed on a movableplatform. The proximal end was placed on the fixed platform,and the distal end of the soft actuator kept contacting thesurface. When the object moved ahead in average speed0.2 cm/s, the subtended angle was recorded. The height pro-file of the contacted surface was reconstructed by the varyingsignal from the optical waveguide. The fourth-order low-pass

FABRICATION AND DYNAMIC MODELING OF SOFT ACTUATOR 501

Butterworth filter with 10 Hz cutoff frequency was used toproceed the noise from the sensor. Before the lateral scan-ning, we carried out a calibration procedure in which the softactuator moved along an inclined plane. The obtained rela-tionship between the height of the contact point and the angleof the soft actuator was used to reconstruct the contour in thelateral scanning experiment.

The experimental paradigms and the lateral scanning re-sults are shown in Figure 7. From these data, we observe thatthe soft actuator can distinguish multishapes with the con-cave surface, triangular wave, and sinusoidal wave. Suchexperiment demonstrates the ability of the PMMA-basedoptical waveguide to dynamically sense time-varying dis-turbance acting on the soft actuator.

Output force test

To validate the output force of the soft actuator, wecollected the tip force exerted by the actuator through acompression load cell (FC10-50N; Forsentek Co., Limited,China) (Fig. 8). The tip of the actuator was in contact with theload cell. The proximal end of the actuator was mounted onthe platform and was linked to the air source. During pres-surization, the actuator exerted force to the load cell throughflexion, which was different from the blocked force obtainedby constraining the curvature of the pressurized soft actua-tor.27 When one chamber was pressurized, the other one waskept in nonpressurized status. The pressure increased in stepsof 10 kPa and the tip force was recorded after the pressureholding for 20 s in each step to obtain the steady status of thesoft actuator. The experiment results are provided in Figure 8.The force at the inlet pressure 0.17 MPa is up to 5.5 N. Due tothe minor difference in manual fabrication, the output forceof the two pressurized chambers under the same pressure isnot the same. However, the output force from the two pres-surized chambers is ascending with the increased pressure inthe range of 0.18 MPa.

Controller-related parameter determination

Before validating the sliding mode controller of the systemunder the proprioceptive curvature sensor, there are two

groups parameters need to be predetermined. One group in-volves the system dynamics-related parameters; that is, k1,k2, and C3. Another group involves the constant parametersrelated to the controller; that is, D, c, and g.

In this work, the maximum pressure of the chamber waslimited to 0.2 MPa. The fitting procedure was performed to

FIG. 7. Contour detection ofdifferent surfaces: concave wave,triangle wave, and sinusoidal wave(left); the corresponding re-constructed contour (right). Colorimages are available online.

FIG. 8. The output force test. The upper image shows anexperimental setup and the lower figure represents the ex-erted tip force under increasing air pressure. Color imagesare available online.

502 CHEN ET AL.

find the relationship between k1, k2, C3 and the input pressureu in Equation (6) to obtain the system dynamic equation. Fora specified input pressure u, the least-squares technique wasemployed to obtain the particular value of the parametersk1,k2, and C3. The fitting procedure was repeated for differentstep responses with input pressure from 0.08 to 0.2 MPa insteps of 0.01 MPa. Then, a collection of k1, k2, and C3 underdifferent input pressure u was obtained (Figs. 9 and 10). Thus,the corresponding relationship between the three parametersand the inlet pressure can be approximately established.

Because D is the maximum fluctuation range of systemmodel error Dr, its specific value depends on the requirementof maximum speed and range of input pressure. From thegeneral performance of the system, it is supposed that _h

�� �� �400�=s, C3� hj j � 180�. As shown in Figure 10, Dk1j j � 4,Dk2j j � 0:25. In the range of 0.2 MPa pressure, the maximum

nominal values of k1 and k2 are 34 and 1.5, respectively.Then, the D approximates to 4500�

�s2.

For the c and g, these two coefficients were tuned by handto maximize the performance in step response, focusing onminimizing rise time and limiting subsequent oscillations. Inthis work, we used c¼ 10=s and g¼ 50=s.

Model validation and controller performance test

To validate the proposed dynamic model and controller byusing the proprioceptive angle sensor, the model step re-sponse and trajectory tracking experiment were furtherperformed.

First, the dynamic performance of the proposed model andcontroller for the soft actuator was verified by the step responseof different pressure inputs. The output of the model and theexperimental data from 0.11 to 0.19 MPa are compared andshown in Figure 11. The corresponding values of the param-eters k1, k2, and C3 in the model were determined by thespecified pressure through an averagely fitted formula shownin Figures 9 and 10. Through integration along the time, themodel outputs were obtained. Due to the order reduction andparameters fitting error, the model shows the overdampingfeature without any overshoot and oscillation. The model’s

rising speed to the level of steady angle is generally slowerthan the natural response of the soft actuator. However, thetime taken to reach the steady angle of the model is alwaysshorter than the time of natural response. It is inferred from theoverdamping characteristic of the model that the system willshow good performance in lower frequency and would beweak in higher frequency. Generally speaking, the modeloutput can basically match the natural response of the softactuator. The trajectory difference found in the experimentprovides the tuning space for the sliding mode controller.

Second, the sine wave and square wave were used as thedesired trajectory to test the controller performances on fre-quency tracking and amplitude response, respectively. Theresults on following the 0.5–4 Hz sine waves at an amplitude of40� are shown in Figure 12A. As shown, the controller per-forms well in a lower frequency (<1 Hz). When the frequencyfurther increases, the phase delay and amplitude decrease ap-pear great. The phase delay of 0.1 s and amplitude decrease of20� (*50%) are observed at a frequency of 4 Hz. Because theoverall performance of the used pneumatic valve on bandwidthand air flow can fully support the high-frequency motion, the

FIG. 9. The relationship between the parameter C3 andinput pressure u. Color images are available online.

FIG. 10. The relationship between parameters k1, k2, andinput pressure u. (A) k1 versus u, and (B) k2 versus u. Colorimages are available online.

FABRICATION AND DYNAMIC MODELING OF SOFT ACTUATOR 503

weak performance of the controller on high frequency mainlycomes from the overdamping characteristic of the systemmodel. The material owned stress, which damps the config-uration restoring speed, also plays a significant role. As forthe local fluctuation observed in tracking the wave with afrequency of 1 Hz, the system model error and the switch ofsliding surface in the controller are mainly responsible fortheir emergence.

Figure 12B shows the controller following the 4 s periodsquare wave at an amplitude of 40�. This frequency is lowerthan the frequencies used in the previous sine wave trackingtests, which allow the soft actuator to reach the desired angleand hold. We can observe from the result that the step re-sponse from a lower angle to the upper angle with amplitude80� occurs in *0.6 s and returns back in *0.3 s. Whenreaching the desired angle, the controller is capable of holdingconstant angles indefinitely without significant error. No ob-vious overshoot and oscillation are found along the time.

Conclusion

In this article, we have presented the design and fabricationof a bidirectional bending pneumatic soft actuator with em-bedded curvature sensor (optical waveguide made of PMMAmaterial). The main design parameters of the sinusoidal wavechamber, that is, pitch and the amplitude, which significantlyaffect the bending ability of the soft actuator, are analyzed byFEA method. The simulation result shows that the greateramplitude and smaller pitch endow the soft actuator betterbending ability. With proper selection of actuator parameters,the bending angle of the soft actuator can reach 280� underonly 0.1 MPa pressure. This shows the potential ability of theproposed actuator in obtaining impressive bending perfor-mance. What’s more, the output force experiment also indi-cates that the prototype actuator can obtain favorable momentin relatively low pressure.

The plastic optical waveguide for sensing the curvatureshows the superiority of robust sensing of curvature underdifferent pressure. It overcomes the weakness of sensorsmade of elastomer or conductive liquid that suffers from thecrimped surface deformation that likely introduces the un-expected output nonlinearity. The lateral scanning experi-ments validate the sensitivity of the plastic optical waveguideon external disturbance. The superiorities of steady and linearoutput voltage corresponding to the increased curvature pro-vide reliable information for closed-loop feedback control.

The dynamic model of the soft actuator is validated by thecomparison of the model output against the experimental dataunder step input in different pressure. Although reducing thesystem order introduces some fitting error, the model caneffectively reproduce the actuator’s response to pressure in-put with acceptable accuracy. The Taylor expansion is usedto locally linearize the control variable in designing thesliding mode controller, which requires the control variableto be tuned in small range around the initial value. However,the soft actuator can still be well controllable to track sinewave with an amplitude of 40� and frequency of 1 Hz.

FIG. 11. Comparison of model output and the naturalresponse of soft actuator by step input at different pressure.The solid lines represent the model output, while the dottedlines indicate the natural response when inflating onechamber. For the same excitation pressure, results from themodel and soft actuator are marked by the same color. Colorimages are available online.

FIG. 12. Dynamic performance of the controller in dif-ferent conditions. (A) Sinusoidal wave with a frequency of0.5, 1, 2, and 4 Hz from top row to bottom row, respectively;the amplitude is 40� for all frequencies. (B) Square wavewith a frequency of 0.25 Hz and an amplitude of 40�. Colorimages are available online.

504 CHEN ET AL.

The proposed bidirectional bending soft actuator withcurvature proprioceptive ability can be used in a variety ofgripper-related applications in industry and medical fields,such as the universal gripper for automatic fruit sorting infood production, exoskeletal glove for hand rehabilitation ofstroke survivor, and prosthetic hand for an amputee. Inmedical field, the bidirectional bending capacity is favorableto cover multisymptoms from early to middle stage of astroke survivor, such as flaccid paralysis patient whose handcannot actively flex and dystonia patient whose hand cannotactively open. The prosthetic hand with variable stiffnessfingers is another potential application of the proposed softactuator. The finger’s angle and force sensitivity to the ex-ternal environment are favorable for an amputee. Further-more, the good performance of the controller is helpful inprecise grasping and manipulation for using soft prosthetichand in daily living.

Toward these potential applications in industry and medicalfield, future work will focus on force proprioceptive ability,more robust controllers based on system model without pre-cise information, and adaptive control of force and stiffness.

Acknowledgments

This article is based upon the work supported by the NationalNatural Science Foundation of China (Grant No. 91648203),the National Key R&D Program (Grant No. 2016YFE0113600,2018YFB1307201), and the Natural Science Foundation ofHubei Province (Grant No. 2018CFB431).

Author Disclosure Statement

No competing financial interests exist.

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Address correspondence to:Caihua Xiong

Institute of Robotics ResearchState Key Laboratory of Digital

Manufacturing Equipment and TechnologySchool of Mechanical Science and Engineering

Huazhong University of Science and Technology1037 Luoyu Road, Wuhan

430074China

E-mail: chxiong@hust.edu.cn

506 CHEN ET AL.