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Automatic design of fiber-reinforced soft actuatorsfor trajectory matchingFionnuala Connollya, Conor J. Walsha,b,1, and Katia Bertoldia,c,1

aHarvard John A. Paulson School of Engineering and Applied Sciences, Cambridge, MA 02138; bWyss Institute for Biologically Inspired Engineering, HarvardUniversity, Cambridge, MA 02138; and cKavli Institute, Harvard University, Cambridge, MA 02138

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved November 21, 2016 (received for review September 12, 2016)

Soft actuators are the components responsible for producingmotion in soft robots. Although soft actuators have allowed fora variety of innovative applications, there is a need for designtools that can help to efficiently and systematically design actua-tors for particular functions. Mathematical modeling of soft actu-ators is an area that is still in its infancy but has the potential toprovide quantitative insights into the response of the actuators.These insights can be used to guide actuator design, thus accel-erating the design process. Here, we study fluid-powered fiber-reinforced actuators, because these have previously been shownto be capable of producing a wide range of motions. We presenta design strategy that takes a kinematic trajectory as its inputand uses analytical modeling based on nonlinear elasticity andoptimization to identify the optimal design parameters for anactuator that will follow this trajectory upon pressurization. Weexperimentally verify our modeling approach, and finally wedemonstrate how the strategy works, by designing actuators thatreplicate the motion of the index finger and thumb.

soft robotics | fiber-reinforced actuators | customized actuators

In the field of robotics, it is essential to understand how todesign a robot such that it can perform a particular motion for

a target application. For example, this robot could be a robot armthat moves along a certain path or a wearable robot that assistswith motion of a limb. For conventional hard robots, methodshave been developed to describe the forward kinematics (i.e., forgiven actuator inputs, what will the configuration of the robotbe) and inverse kinematics (i.e., for a desired configuration ofthe robot, what should the actuator inputs be) (1–4).

Recently, there has been significant progress in the field ofsoft robotics, with the development of many soft grippers (5, 6),locomotion robots (7, 8), and assistive devices (9). Althoughtheir inherent compliance, easy fabrication, and ability to achievecomplex output motions from simple inputs have made softrobots very popular (10, 11), there is growing recognition thatthe development of methods for efficiently designing actuatorsfor particular functions is essential to the advancement of thefield. To this end, some research groups have begun focus-ing their efforts on modeling and characterizing soft actua-tors (12–20). In particular, significant progress has been madeon solving the forward kinematics problem (16–19) and evenon using dynamic modeling to perform motion planning (14).However, the practical problem of designing a soft actuatorto achieve a particular motion remains an issue. Finite ele-ment (FE) analysis has previously been used as a design toolto find the optimal geometric parameters for a soft pneumaticactuator, given some design criteria (15). Although this pro-cedure yields some nice results, only basic motions (linear orbending) were studied, because the method is computationallyintensive. An alternative approach is to use analytical modelingcombined with optimization to determine the properties of a softactuator that will achieve a particular motion for some targetapplication.

Here, we focus on fiber-reinforced actuators (17–21), andgiven a particular trajectory, we find the optimal design param-

eters for an actuator that will replicate that trajectory uponpressurization. To achieve this goal, we first use a nonlin-ear elasticity approach to derive analytical models that pro-vide a relationship between the actuator design parameters(geometry and material properties) and the actuator deforma-tion as a function of pressure for each motion type of inter-est (extending, expanding, twisting, and bending). Then, weuse optimization to determine properties for actuators thatmatch the desired trajectory (Fig. 1). Whereas similar actua-tors were previously designed empirically (22, 23), here, we pro-pose a robust and efficient strategy to streamline the designprocess. Furthermore, this strategy is not limited to the spe-cific cases presented here (namely the trajectories of the indexfinger and thumb) but, rather, could be applied to producerequired trajectories in a variety of soft robotic systems, suchas locomotion robots, assembly line robots, or devices for pipeinspection.

Analytical Modeling of Actuator SegmentsOur approach is based on assuming a desired actuator consists ofmultiple segments (mimicking the links and joints of the biologi-cal digit), where each different segment undergoes some combi-nation of axial extension, radial expansion, twisting about its axis,and bending upon pressurization. To realize actuators capableof replicating complex motions, we use segments consisting of acylindrical elastomeric tube surrounded by fibers arranged in ahelical pattern at a characteristic fiber angle α (Fig. 2A) (23, 24),because it has been shown that by varying the fiber angle andmaterials used, these segments can be easily tuned to achieve awide range of motions (17–21). When the elastomeric tube is ofuniform stiffness, the segment undergoes some combination of

Significance

Fluid-powered elastomeric soft robots have been shown tobe able to generate complex output motion using a simplecontrol input such as pressurization of a working fluid. Thiscapability, which mimics similar functions often found in biol-ogy, results from variations in mechanical properties of thesoft robotic body that cause it to strain to different degreeswhen stress is applied with the fluid. In this work, we outlinea mechanics- and optimization-based approach that enablesthe automatic selection of mechanical properties of a fiber-reinforced soft actuator to match the kinematic trajectoryof the fingers or thumb during a grasping operation. Thismethodology can be readily extended to other applicationsthat require mimicking or assisting biological motions.

Author contributions: F.C., C.J.W., and K.B. designed research; F.C. performed research;F.C. and K.B. analyzed data; and F.C., C.J.W., and K.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence may be addressed. Email: walsh@seas.harvard.edu orbertoldi@seas.harvard.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1615140114/-/DCSupplemental.

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extension

twist

α1,L1

α2,L2

α3,L3

α4,L4

α6,L6

α5,L5

single pressure

inlet

input 2: complex kinematics

output: input 1: actuator models

A B

C

expansion

bending

Fig. 1. Designing an actuator that replicates a complex input motion.(A) Analytical models of actuator segments that can extend, expand, twist, orbend are the first input to the design tool. (B) The second input to the designtool is the kinematics of the desired motion. (C) The design tool outputs theoptimal segment lengths and fiber angles for replicating the input motion.

axial extension, radial expansion, and twisting about its axis uponpressurization (17–19). In contrast, when the tube is composedof two elastomers of different stiffness, pressurization producesa bending motion (25, 26).

Previous work has explored the design space of fiber-rein-forced actuators capable of extending, expanding, and twistingusing FE analysis (20) and kinematics and kinetostatics model-ing (17–19). Although these existing analytical models providegreat insight into the behavior of fiber-reinforced actuators, theyare restricted to exactly two sets of fibers (a set of fibers beingfibers arranged at the same angle).

Model 2: bending

isotropic core

thin anisotropic outer layer

Model 1: extension, expansion and twist

isotropic elastomer

reinforcement

simplify to

A B

SdZ

Ri

Rm

Ro

ri

rm

ro

θ

dz

Ro

dψ

dφ

Apply pressure

C

α

Rm

Ri

Fig. 2. Modeling a fiber-reinforced actuator. (A) The actuator consists of an elastomeric tube surrounded by an arrangement of fibers. In the model, thissystem is simplified to an isotropic tube, with an outer layer of anisotropic (but homogeneous) material. (B) Parameters for the analytical model of anextending, expanding, twisting actuator. (C) Parameters for the analytical model of a bending actuator.

Here, we use a nonlinear elasticity approach, which facilitatesmodeling actuators with an arbitrary number of sets of fibers.Rather than modeling the tube and the fibers individually, wetreat them as a homogeneous anisotropic material (27–29). Morespecifically, because the fibers are located on the outside of thetube and not dispersed throughout its thickness, we model theactuator as a hollow cylinder of isotropic incompressible hypere-lastic material (corresponding to the elastomer), surrounded bya thin layer of anisotropic material (corresponding to the fiberreinforcement), and impose continuity of deformation betweenthe two layers (Fig. 2A). The isotropic core has initial innerradius Ri and outer radius Rm , and the outer anisotropic layerhas initial outer radius Ro . The anisotropic material has a pre-ferred direction that is determined by the initial fiber orienta-tion S = (0, cosα, sinα). We define a deformation gradient F,from which we calculate the left Cauchy–Green deformation ten-sor B = FFT , the current fiber orientation s = FS, and the tensorinvariants I1 = tr(B) and I4 = s.s.

The inner and outer layers require different strain energyexpressions, so let W (in) be the strain energy for the isotropiccore and W (out) be the strain energy for the anisotropic outerlayer. For the isotropic core, we choose a simple incompress-ible neo-Hookean model, so that W (in) =µ/2(I1 − 3), with µdenoting the initial shear modulus. For the anisotropic layer,let W (out) be the sum of two components, W (out) = c1W

(iso) +

c2W(aniso), where W (iso) =µ/2(I1− 3) is the contribution from

the isotropic elastomeric matrix, W (aniso) is the contributionfrom the fibers, and c1 and c2 are the corresponding volume frac-tions. To derive a suitable expression for W (aniso), we consider asmall section of the helical fiber and model it as a rod subject toan axial load (SI Appendix, Fig. S5). It is trivial to show that thestrain energy density of the rod is (SI Appendix)

W (aniso) =E(√I4 − 1)

2

2, [1]

where E is its Young’s modulus. By slightly modifying the strainenergy, the above equations can easily be extended to accountfor more than one set of fibers. For example, to achieve a pureextending actuator, we might require two sets of fibers, with fiberorientations s1 and s2. In this case, the strain energy density is

W (aniso) =E(√I4 − 1)

2

2+

E(√I6 − 1)

2

2, [2]

where I4 = s1.s1 and I6 = s2.s2.

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A

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F

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I

H

E

B

Fig. 3. Analytical predictions and experimental results for extending, twist-ing, and bending actuators. (A) Heat map illustrating the axial extension (λz)as a function of fiber angle and input pressure. (B) Front and bottom viewsof an extending actuator. (C) Comparison between analytical prediction andexperimental results. (D) Heat map illustrating the twist per unit length (τ )as a function of fiber angle and input pressure. (E) Front and bottom viewsof a twisting actuator. (F) Comparison between analytical prediction and

We can then use the strain energies to calculate the Cauchystresses, which take the form

σ(in) = 2W(in)1 B− pI [3]

σ(out) = 2W(out)1 B + 2W

(out)4 s1 ⊗ s1 + 2W

(out)6 s2 ⊗ s2 − pI,

[4]

where Wi = ∂W∂Ii

, I is the identity matrix, and p is a hydrostaticpressure (30).

To further simplify the analytical modeling, we decouple bend-ing from the other motions. In the following, we first introducean analytical model describing an extending, expanding, twistingactuator and then a model for a bending actuator.

Modeling Extension, Expansion, and Twist. When the elastomericpart of the actuator is of uniform stiffness, we assume that thetube retains its cylindrical shape upon pressurization, and theradii become ri , rm , and ro in the pressurized configuration(Fig. 2B). The possible extension, expansion, and twisting defor-mations are then described by (SI Appendix)

F =

Rrλz

0 00 r

Rrτλz

0 0 λz

, [5]

where R,Φ,Z and r , φ, z are the radial, circumferential, andlongitudinal coordinates in the reference and current configura-tions, respectively (28, 31). Moreover, λz and τ denote the axialstretch and the twist per unit length, respectively. To determinethe current actuator configuration, we first apply the Cauchyequilibrium equations, obtaining

P =

∫ rm

ri

σ(in)φφ − σ

(in)rr

rdr +

∫ ro

rm

σ(out)φφ − σ(out)

rr

rdr , [6]

where P is the applied pressure. Assuming there are no externalaxial forces or external axial moments applied to the tube, theaxial load, N , and axial moment, M , are given by

N = 2π

∫ rm

ri

σ(in)zz rdr + 2π

∫ ro

rm

σ(out)zz rdr = Pπr2i [7]

and

M = 2π

∫ rm

ri

σ(in)φz r2dr + 2π

∫ ro

rm

σ(out)φz r2dr = 0. [8]

Eqs. 6–8 with the Cauchy stress σ as given in Eqs. 3 and 4 andthe deformation gradient of Eq. 5 are then solved to find λz , ri ,and τ as functions of P (SI Appendix).

Modeling Bending. Because the exact solution for the finite bend-ing of an elastic body is only possible under the assump-tion that the cross-sections of the cylinder remain planar uponpressurization—a condition that is severely violated by ouractuator—we assume (i) that the radial expansion can beneglected (i.e., r/R = 1) and (ii) vanishing stress in the radialdirection (i.e., σrr = 0). Note that these conditions are closelyapproximated when the actuator has fiber angle less than 30◦ andthe actuator walls are thin (16). Furthermore, because the actu-ators have a symmetric arrangement of fibers, no twisting takesplace, so the deformation gradient reduces to (SI Appendix)

F =

λz (φ)−1 0 00 1 00 0 λz (φ)

. [9]

experimental results. (G) Heat map illustrating the bend per unit length (ω)as a function of fiber angle and input pressure. (H) Front view of a bend-ing actuator. (I) Comparison between analytical prediction and experimen-tal results.

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Because the actuator bends due to the moment created bythe internal pressure acting on the actuator caps (SI Appendix,Fig. S9), we equate this moment

Mcap = 2PR2i

∫ π

0

sin2 φabs(Ri cos φ̄− Ri cosφ

)dφ, [10]

with the opposing moment due to the stress in the material

Mmat =

∫ ∫λ−1z σzz (Ri + τ)

(Ri cos φ̄− (Ri + τ) cosφ

)dφdτ,

[11]

where φ̄ denotes the location of the neutral bending axis, dτ isthe differential wall thickness element, and dφ is the circumfer-ential angle element (Fig. 2C).

Now solving Mmat =Mcap yields the relationship betweeninput pressure and output bend angle:

P =

∫∫λ−1z σzz (Ri + τ)

(Ro cos φ̄− (Ri + τ) cosφ

)dφdτ

2R2i

∫ π0

sin2 φabs(Ri cos φ̄− Ri cosφ)dφ,

[12]

where σzz can be obtained by substituting F into Eqs. 3 and 4(SI Appendix).

Comparing Analytical and Experimental Results. To fabricate theextending, expanding, twisting actuators, we used the elastomerSmooth-Sil 950 (µ2 = 680 kPa; Smooth-On), and for the bend-ing actuators, we used both Smooth-Sil 950 and Dragon Skin 10(µ2 = 85 kPa; Smooth-On). The fiber reinforcement was Kevlar,with a Young’s modulus E = 31,067 MPa and radius r =0.0889 mm. Each actuator had an inner radius of 6.35 mm, wallthickness of 2 mm, and length of 160 mm. The effective thicknessof the fiber layer (8.89× 10-4 mm) is a fitting parameter here andwas identified using the results in SI Appendix, Fig. S7.

For the bending model, we used an FE simulation (SIAppendix, Fig. S10) to determine the location of the neutral axis(φ̄= 35◦). Using FE analysis, we determined that our bendingmodel was accurate for thin-walled actuators (SI Appendix, Fig.S11). However, for thicker-walled actuators, the model yieldedlower than expected bend angles at any given pressure. Tosolve this problem, we used one FE simulation (with fiber angleα= ±5◦) to determine an effective shear modulus µ̄ (78 kPa)for the actuator (rather than using the shear moduli µ1 andµ2). We found that using this fitting parameter, we could accu-rately predict the response for actuators with other fiber angles(SI Appendix). Note that because FE analysis generally providesmore accurate results than our analytical bending model, analternative solution would be to use FE simulations to build adatabase of simulation results for actuators with a range of dif-ferent fiber angles. However, this option would be more compu-tationally expensive, and so, although not ideal, it is preferablein our case to use just one FE simulation to identify the fittingparameters for the analytical bending model, rather than relyingsolely on FE.

We first consider extending actuators (with two sets of fibers,arranged symmetrically). Fig. 3A shows how the amount of exten-sion undergone (illustrated by the color) depends on the fiberangle of the actuator (y axis) and the current actuation pressure(x axis). We see that an actuator with fiber angleα= 0◦ yields themost extension, whereas, in contrast, actuators with larger fiberangles undergo contraction. We fabricated and tested an actua-tor with fiber angles α= ±3◦ (highlighted in red in Fig. 3A), andthe results (Fig. 3 B and C) show good agreement between themodel and the experiment. Second, we consider twisting actu-ators, which have only one set of fibers. From Fig. 3D, we cansee that an actuator with fiber angle around 30◦ produces themaximum amount of twist per unit length. Fig. 3 E and F shows

that the analytical model accurately represents the twist per unitlength undergone by an actuator with fiber angle α= 3◦.

Finally, Fig. 3G illustrates the bend angle per unit length as afunction of fiber angle and actuation pressure. At any given pres-sure, for larger fiber angles, we see less bend per unit length.Comparing analytical and experimental results for a bendingactuator with fiber angles α= ±5◦ (Fig. 3 H and I), we see agood match between the model and the experiment.

Replicating Complex MotionsWe have presented two analytical models, which describe extend-ing, expanding, twisting, and bending actuator motions. In addi-tion to using these models to explore the actuator design space,we can use them for more complex operations, such as design-ing a single-input, multisegment actuator that follows a spe-cific trajectory. In the following sections, we will demonstratethis methodology by determining the properties of multisegmentactuators that can replicate finger and thumb motion.

The target motion of the actuator was determined using elec-tromagnetic trackers that were placed on the hand at the wrist,at each joint along the finger, and at the fingertip (SI Appendix,Fig. S13) (22). Time series data of the coordinates of each sensorin 3D space were recorded as the hand was opened and closed.Using these data, the configuration of the fingers and thumb dur-ing a grasping motion can be obtained. Adjacent sensors areconnected by links, and we use the data to calculate the lengthof each link and the angles between the links at each time. Wesmooth the data by applying a Savitzky–Golay filter. Since it willnot be possible to produce an actuator that will match the finger

extending segmentsbending segments

input

initialize design

variables

calculate required segment

update design

variables

fabricate actuator

objective function (Eq 13)

minimized?

yes

no

pressure P(j)

length lext,i twist angle θi

pressure P(j)

pressure P(j)

bend angle ψi

use analytical models

input:

output:

extending twisting bending

compare to:

Ŷ Ŷ Ŷ(j)

(j)

lext,i( j) θi

(j) P(j)

(j)

ltw,1

llink,1(j)Ψ1

Ψ2

θ

llink,2llink,3

(j)

( j)

lbend,1

(j)

lbend,2(j)

lbend,3(j)

lext,1(j)

lext,2

lext,3(j)

( j)

( j)

( j)

( j)Ψ3

twisting segments

(j)

αi αi αilext,i(0) ltw,i

(0) lbend,i(0)

ŶŶ Ŷ

Ŷ

Ŷ

Ŷ

ŶŶ

Ŷ

Ŷ

Fig. 4. Optimization algorithm for actuator design.

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Fig. 5. (A and B) Finger actuator: qualita-tive comparison of actuator motion and fingermotion. (C) Comparison of input finger trajec-tory and output actuator trajectory. (D) Illustra-tion of segment lengths and fiber angles. (E)Quantitative comparison of desired, expected,and achieved link lengths (Left) and bend angles(Right). (F and G) Thumb actuator: qualita-tive comparison of actuator motion and thumbmotion. (H) Comparison of input thumb trajec-tory and output actuator trajectory. (I) Illustra-tion of segment lengths and fiber angles. (J)Quantitative comparison of desired, expected,and achieved twist angles (Left), link lengths(Center), and bend angles (Right).

trajectory exactly at every point, we choose just four configura-tions to match. (These configurations are equally spaced alongthe trajectory; SI Appendix, Figs. S14 and S15.) Because the inputdata represent the motion of the top of the finger, the top of theactuator we design should mimic the input motion.

The actuator will consist of multiple segments, each with a dif-ferent length and fiber angle. We prescribe the number of seg-ments and the type of each segment. (For replicating finger andthumb motion, expanding segments are not required, so eachsegment type is extend, bend, or twist.) The radius, wall thick-ness, and material parameters are the same as in the previoussection. (Here, we use Dragon Skin 10 for the extending andtwisting segments.) We set the maximum allowed pressure to80 kPa. (At higher pressures, the Matlab solver fsolve is unable tosolve Eqs. 6–8.) Furthermore, to simplify the fabrication proce-dure, we impose a minimum fiber angle of 5◦. Also, we prescribea maximum allowed fiber angle of 50◦, because above this angle,the radial expansion of the actuators becomes significant (20).Note that for a finger, bending occurs at discrete joints, but forthe actuator, it will of course take place over some finite length.To approximate the motion of the finger as closely as possible, wewant this length to be as short as possible, so we impose a max-

imum allowed bending segment length of 30 mm (SI Appendix).We then input all of this information, together with the mod-els we developed in the previous sections, into a nonlinear least-squares optimization algorithm in Matlab (lsqnonlin) (Fig. 4). Tofind the design parameters for an actuator that will move throughthe given configurations (combinations of link lengths and bendangles) as it is pressurized, we minimize

f = c1∑N

j=1

∑ntw

i=1

∣∣∣θ(j)i − θ̂(j)i

∣∣∣2 +∑N

j=1

∑next

i=1

∣∣∣l (j)ext,i − l̂(j)ext,i

∣∣∣2+ c2

∑N

j=1

∑nbend

i=1

∣∣∣P (j)i − P̂ (j)

∣∣∣2, [13]

where ntw , next , and nbend are the total numbers of twisting,extending, and bending segments, respectively, and N is thenumber of goal configurations. The first term measures the dif-ference between the required (θ̂) and achieved (θ) twist angles;the second term measures the difference between the required(̂lext ) and achieved (lext ) segment lengths, and the third termmeasures the difference between the pressures at which therequired bend angles should be achieved (P̂) and the pressuresat which the required bend angles are actually achieved with

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the current set of variables (P). The parameters c1 = 100 andc2 = 1,000 are weights that balance the relative importance ofthe twisting, bending, and extending segments. If f is not suf-ficiently small, the variables are updated and the optimizationloop repeats. When the minimum value of f is found, the opti-mization outputs (i) the fiber angle αi for each segment, (ii) theinitial length of each of the bending and twisting segments, and(iii) the pressures P̂ (j) at which the goal configurations will occur.

Index Finger Motion. The fiber angles and lengths required to imi-tate the movement of the index finger are illustrated in Fig. 5D.Segments 1, 3, and 6 have length 70 mm, 22 mm, and 15 mm,with fiber angles of ±40◦, ±50◦, and ±50◦, respectively. Thesesegments undergo axial contraction when pressurized. Segments2, 4, and 5 are bending segments of length 22 mm, 28 mm, and15 mm, with fiber angles of ±6◦, ±5◦ and ±5◦, respectively.

We fabricate the actuator as detailed in SI Appendix. To com-pare the actual performance of the actuator to the expected per-formance, we characterize it by taking pictures of the actuator atvarious different actuation pressures. We evaluate its motion byusing Matlab to track points on the actuator. We see good agree-ment between expected and actual motion (Fig. 5 C and E andMovie S1), with some discrepancies that are most likely due todefects in the actuator fabrication (for example, segment lengthsbeing up to 4 mm shorter than expected, due to the fibers beingwound around the actuator between segments; SI Appendix).

Thumb Motion. As a second example, we consider the design ofan actuator that upon pressurization, replicates the motion of athumb. The motion of the thumb is more complex than that ofthe finger, because it moves out of plane. We capture this out-of-plane motion as a twisting motion. We calculate the amountof twist by fitting a plane to the twisting links at each time andthen finding the angle between the normal to this plane and thenormal to the initial plane.

Fig. 5I illustrates the fiber angles that are needed to repro-duce the motion of the thumb, as predicted by the model. Seg-ment 1 is a twisting segment of length 25 mm, with fiber angle5◦; segments 2, 4, and 6 are bending segments of length 15 mm,

30 mm, and 22 mm, each with fiber angles ±5◦; and segments3, 5, and 7 are extending segments of length 20 mm, 8 mm, and17 mm, with fiber angles ±41◦, ±50◦, and ±47◦. To analyze themotion of this actuator, we placed two cameras at right anglesto each other and took pictures of the actuator at various differ-ent actuation pressures. The images were combined to reproducethe 3D motion of the actuator (SI Appendix). Fig. 5 H and J com-pares the input thumb kinematics and the output actuator motion(Movie S2). We see reasonable agreement between the expectedand achieved motions, with discrepancies in this case most likelydue to inaccuracies in measuring the actuator motion (for exam-ple, misalignment of cameras), as well as defects in actuator fab-rication (such as nonuniform wall thickness).

ConclusionsUsing analytical models for fiber-reinforced actuators thatextend, expand, twist, and bend, we have devised a method ofdesigning actuators customized for a particular function. Giventhe kinematics of the required motion, and the number andtype of segments required, the algorithm outputs the appro-priate length and fiber angle of each segment, thereby pro-viding a recipe for how the actuator should be made. Theprocedure is somewhat limited in its current form because itrequires the user to input the type of segments required, butfuture versions will eliminate the need for this step, thus fur-ther automating the procedure. Future work will also focus ondeveloping a model that combines bending with other motions,to increase the versatility of the algorithm. The design tool wehave presented here has immense potential to streamline andaccelerate the design of soft actuators for a particular task,eliminating much of the trial and error procedure that is cur-rently used and broadening the scope of fiber-reinforced softactuators.

ACKNOWLEDGMENTS. The authors thank Dr. J. Weaver for assistance with3D printing and Dr. P. Polygerinos and Dr. S. Sanan for helpful discussions.This work was partially supported by National Science Foundation Grant1317744, the Materials Research Science and Engineering Center underNational Science Foundation Award DMR-1420570, the Wyss Institute, andHarvard’s Paulson School of Engineering and Applied Sciences.

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