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Harnessing Viscous Flow to Simplify the Actuation

of Fluidic Soft Robots

Nikolaos Vasios,

1

Andrew J. Gross,

1

Scott Soifer,

1

Johannes T.B. Overvelde,

4

and Katia Bertoldi

1–3

Abstract

Soft robots powered by pressurized ﬂuid have recently enabled a variety of innovative applications in areas as

diverse as space exploration, search and rescue systems, biomimetics, medical surgery, and rehabilitation.

Although soft robots have been demonstrated to be capable of performing a number of different tasks, they

typically require independent inﬂation of their constituent actuators, resulting in multiple input lines connected

to separate pressure supplies and a complex actuation process. To circumvent this limitation, we embed the

actuation sequencing in the system by connecting ﬂuidic actuators with narrow tubes to exploit the effects of

viscous ﬂow. We developed modeling and optimization tools to identify optimal tube characteristics and we

demonstrate the inverse design of ﬂuidic soft robots capable of achieving a variety of complex target responses

when inﬂated with a single pressure input. Our study opens avenues toward the design of a new generation of

ﬂuidic soft robots with embedded actuation control, in which a single input line is sufﬁcient to achieve a wide

range of functionalities.

Keywords: inverse design, viscous ﬂow, ﬂuidic soft actuators, simple actuation

Introduction

Soft robots comprising several inﬂatable actuators made

of compliant materials have drawn signiﬁcant attention

over the past few years because of their ability to produce

complex and adaptive motions through nonlinear deforma-

tion.

1–11

The simplicity of their design, ease of fabrication,

and low cost sparked the emergence of soft robots capable of

walking,

12

crawling,

13

camouﬂaging,

14

assisting humans in

grasping,

15,16

and whose response can be further enhanced by

exploiting elastic instabilities.

17,18

However, to achieve a par-

ticular function, existing ﬂuidic soft robots typically require

multiple input lines, since each actuator must be inﬂated and

deﬂated independently according to a speciﬁc preprogrammed

sequence (Fig. 1a).

In an effort to reduce the number of input lines required for

actuation, band-pass valves have been designed, which can

address multiple actuators individually using a single modu-

lated source of pressure.

19

Another interesting avenue to reduce

the number of required input signals is the direct exploitation of

the highly nonlinear behavior of the system without the intro-

duction of additional stiff elements. To this end, it has been

shown that a segmented soft actuator reinforced locally with

optimally oriented ﬁbers can achieve complex conﬁgurations

upon inﬂation with a single input source.

20

Furthermore, the

nonlinear properties of ﬂexible two-dimensional metamaterials

have been proven effective in reducing the complexity of the

required input signal.

13,21

In this study, motivated by these opportunities for simpliﬁed

actuation through nonlinearities, we focus on a system com-

prising an array of ﬂuidic actuators interconnected through

tubes and demonstrate that viscous ﬂow in the tubes can be

harnessed to achieve a wide variety of target responses through

a single input (Fig. 1b). Although recent experiments with

poroelastic soft actuators indicate that viscous ﬂow is a prom-

ising candidate to simplify the actuation of soft robots,

22

the

highly nonlinear response of the system prohibits the identiﬁ-

cation of simple rules to guide its design. It is, therefore, crucial

to implement robust algorithms to efﬁciently identify the sys-

tem parameters resulting in the desired response.

1

John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts.

2

Wyss Institute for Biologically Inspired Engineering, Harvard University, Cambridge, Massachusetts.

3

Kavli Institute for Bionano Science and Technology, Harvard University, Cambridge, Massachusetts.

4

AMOLF, Amsterdam, The Netherlands.

SOFT ROBOTICS

Volume 00, Number 00, 2019

ªMary Ann Liebert, Inc.

DOI: 10.1089/soro.2018.0149

1

Downloaded by Harvard University FRANCIS A COUNTWAY from www.liebertpub.com at 05/14/19. For personal use only.

To this end, we ﬁrst derive a model that accurately captures

the viscous ﬂow in the tubes and then combine the model with

optimization to determine through inverse design the char-

acteristics of the tubes leading to desired responses using a

single input. The excellent agreement between experiments

and simulations for a wide range of prescribed target re-

sponses demonstrates the robustness of our strategy. Finally,

we show that our approach enables the realization of ﬂuidic

soft robots that can perform complex tasks when powered by

a single pressure input, as demonstrated through the design of

a simply actuated four-legged walker.

Fluidic Bending Actuators

Although the principles proposed in this study are appli-

cable to systems comprising any ﬂuidic soft actuator, to

demonstrate the concept, we focus on ﬂuidic bending actua-

tors with an embedded network of channels and chambers.

2

All actuators have length l¼75mm and a rectangular cross

section (wcþ4t)·(hþ3t)mm

2

, where wc¼16:5mm is the

chamber width, h¼7:5mm is the chamber height, and

t2[1:5, 4] mm corresponds to the thickness of the top layer

but also affects all other dimensions (Supplementary Table S1

and Supplementary Figures S1–S6). Moreover, the actuators

contain eight identical chambers connected through narrow

channels and are realized using two silicone rubbers with

different stiffness (Fig. 2a; Supplementary Data).

The geometry of the embedded chambers as well as the

contrasting properties of the two elastomers causes these

actuators to progressively bend upon inﬂation in quasi-static

conditions (Fig. 2b, c). Although the relationship between the

bending curvature jand the supplied volume Dtis almost

linear (Fig. 2d; Supplementary Movie S1), their pressure–

volume response is highly nonlinear and features a pressure

plateau (Fig. 2e; Supplementary Movie S1) caused by the

reduction in stiffness associated with the ballooning of the

top layer. Our results indicate that higher values of tlead to

actuators that are simultaneously stiffer and harder to bend.

Harnessing Viscous Flow in the Tubes

Having characterized the quasi-static response of the ﬂu-

idic bending actuators, we next investigate the response of the

elementary system comprising two actuators connected by a

tube (Fig. 3a).

To begin with, we consider two identical actuators with

t¼4 mm, connect one of them (Actuator 1, shown in blue in

Fig. 3a) to the pressure source using a tube with length

L1¼10 cm and internal radius R1¼0:38 mm, and then

connect Actuator 2 to Actuator 1 through a tube with length

L2¼10 cm and internal radius R2¼0:79 mm (Fig. 3c, Sup-

plementary Figures S7–S8). Upon supplying the system with

air pressurized at pinput ¼60 kPa for tinput ¼2:5 s (Fig. 3c), the

two actuators bend simultaneously, reach the same maxi-

mum bending curvature j1, max ¼j2, max^40 m

-1

at t¼2:5s

FIG. 2. Fluidic bending actuators. (a) Schematic of the

cross section of a ﬂuidic bending actuator. The two different

elastomers used to fabricate the sample, Ecoﬂex-30 (Smooth-

On, Inc.) and Elite Double 32 (Zhermack), are shown in gray

and green, respectively. (b, c) Snapshots of ﬂuidic bending

actuators characterized by (b) t¼1:5mmand(c) t¼4:0mm

at different actuation pressures. (d) Experimental curvature–

volume curves for four actuators characterized by t¼1:5, 2.1,

2.9, and 4.0 mm. (e) Experimental pressure–volume curves for

four actuators characterized by t¼1:5, 2.1, 2.9, and 4.0 mm.

Color images are available online.

FIG. 1. Simplifying the actuation of ﬂuidic soft robots. (a)

Each actuator is typically inﬂated and deﬂated independently

and individually, requiring a complex actuation process. (b) In

this study, we exploit viscous ﬂow in the tubes interconnecting

the constituent actuators to design soft robots capable of

achieving a variety of responses when inﬂated with a single

pressure input. Color images are available online.

2 VASIOS ET AL.

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(Fig. 3f; Supplementary Movie S2), and then deﬂate through

the inlet (since pinput ¼0 kPa for ttinput , converting the inlet

to an outlet for the system to reset). Note that by changing

pinput and tinput, we are able to control the maximum curvature

of the actuators. However, since in this system the tube used

to connect the two actuators does not impose signiﬁcant re-

strictions to the ﬂuid ﬂow, the two actuators will always bend

simultaneously.

In an effort to investigate how viscous effects in the tubes can

be harnessed to tune the rate of inﬂation of each actuator, we

replace the interconnecting tube with a narrower tube, charac-

terized by R2¼0:38 mm (keeping L2¼10 cm; Fig. 3d). The

experimental results shown in Figure 3d indicate that the actua-

tors now bend at different rates and achieve the maximum cur-

vature at different times (Supplementary Movie S3). However,

we also ﬁnd that j2, max is signiﬁcantly reduced due to energy

losses associated with the viscous ﬂow in the newly introduced

interconnecting narrow tube and that j1,max is increased because

of the restriction on ﬂuid ﬂow imposed by such a tube.

To compensate for the energy loss, we replace the second

actuator in our system with a more compliant actuator char-

acterized by t¼2:97 mm (Fig. 3e). In this case, the two actu-

ators still bend at different rates, but reach the same maximum

bending curvature j1, max ¼j2, max ^45 m

-1

(Fig. 3e; Supple-

mentary Movie S4). Therefore, our simple experiments indi-

cate that by carefully selecting both the ﬂuidic actuators and the

tubes, we can tune the bending rate as well as the maximum

bending curvature of the actuators. However, the highly non-

linear response of the system prohibits the direct identiﬁcation

of simple rules that relate its parameters to speciﬁc desired

responses. To design systems capable of achieving a target

response, we ﬁrst derive a model that describes their behavior

and then solves the inverse problem to determine the system

parameters that give rise to the target response.

Forward Modeling

Since our system comprises several ﬂuidic bending actuators

connected through narrow tubes, to predict its response we

need to be able to capture the behavior of the actuators and

determine the amount of ﬂuid transferred through the tubes

(Supplementary Figures S9–S11 and Supplementary Table S2).

To this end, we focus on the [i]-th tube in the system, which has

length L

i

(Fig. 4a), circular cross section with radius R

i

(with

FIG. 3. Harnessing viscous ﬂow in the tubes. (a) Schematic of the system considered in all three experiments. Tube 1

connects the input pressure to the ﬁrst actuator, whereas Tube 2 connects the two actuators. Tube 1 has length L1¼10 cm

and radius R1¼0:38 mm in all three experiments. (b) The rectangular pressure pulse used in all three experiments supplies

pinput ¼60 kPa for tinput ¼2:5 s. For t>tinput ,pinput ¼0 kPa and Tube 1 acts as an outlet for the system to reset/deﬂate. (c–e)

Schematics of the conﬁguration tested in the three different experiments (top) and corresponding curvature responses for the

two actuators (bottom). Color images are available online.

SIMPLIFIED FLUIDIC ACTUATION 3

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Li@Ri), and assume that (a) the tube is rigid and not deformed

by the ﬂow; (b) the head losses due to friction at the con-

nections between the tube and the actuators can be captured

by adjusting its length to Li,eq

23

;(c) the ﬂow is incompressible

and laminar; and (d) the ﬂuid velocity has the form

u¼ 2

pR2

i

d~

vi

dt

r

Ri

2

1

"#

ez, (1)

where ~

vi¼Rt

0RRi

0uez2prdrdt denotes the amount of ﬂuid

exchanged through the [i]-th tube up to time t, and e

z

iden-

tiﬁes the tangent vector to the tube (Supplementary Data).

Under these assumptions, integration of the Navier–Stokes

equations over the volume of the tube yields

Li,eq

d2~

vi(t)

dt2¼pR2

i

q(pipi1)8lLi,eq

R2

i

d~

vi(t)

dt , (2)

where p

i

is the pressure inside the [i]-th actuator and lis the

dynamic viscosity of the ﬂuid. Since for narrow tubes with

Li@Ri, as those considered in this study, the inertia term is

negligible (Supplementary Data), Equation (2) can be re-

written in dimensionless form as

d~

Vi(t)

dT þniPiPi1

ðÞ¼0, (3)

with

ni¼pGR4

itmax

8lv0Li,eq

, (4)

where ~

Vi¼~

vi=v0,Pi¼pi=G, and T¼t=tmax are the normal-

ized ﬂuid volume exchanged, pressure, and time, respectively

(v

0

,G, and tmax denoting the volume of the smallest actuator

in the system, the shear modulus of the material used to

fabricate the actuators, and the response time of the system,

respectively). Finally, since the normalized change in volume

for the [i]-th actuator, DVi¼Dvi=v0, can be expressed in

terms of the volumetric ﬂows exchanged through the two

tubes connected to it as

DVi¼~

Vi~

Viþ1, (5)

Equation (3) can be rewritten as

dDVi(t)

dt þni(PiPi1)niþ1(Piþ1Pi)¼0, (6)

where the pressure inside the [i]-th actuator, P

i

, is a function of

DVi. For a system comprising Nﬂuidic actuators interconnected

through narrow tubes, Equation (6) deﬁnes a system of Ncou-

pled differential equations, which, given a pressure–volume

relationship for the actuators, can be solved numerically to

determine the normalized change in volume for the [i]-th

actuator as a function of time (Supplementary Data). Once

the volume history for all actuators is known, their bending

curvature is determined using the corresponding curvature–

volume relationship.

To verify the validity of our model, we numerically integrate

Equation (6) using the pressure–volume and curvature–volume

relations of Figure 2d and e to simulate the experiments re-

ported in Figure 3. We ﬁnd that our numerical model (solid

lines) can successfully reproduce the responses observed in

experiments (dashed lines) for all three systems considered in

Figure 3. The capability of the numerical model to accurately

capture the response of the system in conﬁgurations involving

different tubes and actuators ensures that the model can be

used to identify optimal conﬁgurations.

Inverse Design

Although Equation (6) can be used to predict the temporal

response of arbitrary arrays of ﬂuidic actuators connected

through narrow tubes, in this study, we are mostly interested

in the inverse problem of designing a system capable of

achieving particular target responses (Supplementary Figures

S12–S16).

Speciﬁcally, we focus on systems consisting of four ﬂuidic

bending actuators characterized by t¼4:0, 2:9, 2:1, and 1:5

mm connected through narrow tubes (Fig. 4b) and want the

[i]-th actuator in the array to attain a maximum bending

FIG. 4. Forward and inverse modeling. (a) Schematic of

the system. The [i]-th tube is connected to the [i1]-th and

[i]-th actuators. (b) Schematic of the conﬁguration consid-

ered in the inverse problem, consisting of four ﬂuidic

bending actuators with thickness t¼4:0, 2:9, 2:1, and 1:5

mm connected through narrow tubes and inﬂated by a

rectangular pressure pulse. (c) The target response requires

the [i]-th actuator in the array to attain a maximum bending

curvature of Ki,max at a predeﬁned time Ti,max and then to

completely deﬂate. (d) Parameters introduced to construct

the objective function. Color images are available online.

4 VASIOS ET AL.

curvature Ki,max ¼ji,max=jref (jref ¼p=l¼41:88 m

-1

being

the curvature of a semicircle with arc length equal to the

initial length lof the actuators) at a predeﬁned time

Ti,max ¼ti,max=tmax and then to completely deﬂate (Fig. 4c).

Specifying a rectangular pulse for the input pressure

(Fig. 4b), the parameters to bedetermined to achieve the target

response are (a) the dimensionless tube parameters ni(with

i¼1, 2, 3, 4) that uniquely deﬁne the tube geometry, (b)the

magnitude of the input pressure Pinput ¼pinput=G,and(c)the

pressurization time Tinput ¼tinput=tmax. To identify a set of such

parameters resulting in the desired response, we minimize

Z¼+4

i¼1(diþ0:25si), (7)

where sidenotes the amount of time that the [i]-th actuator

spends above a threshold curvature Ki,thres ¼0:05Ki,max and

is introduced to ensure that the actuators quickly deﬂate after

approaching the target point of maximum curvature. More-

over, d

i

is the ‘‘distance’’ in the KTspace between the

target and actual points of maximum curvature for the [i]-th

bending actuator (Fig. 4d),

di¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

DK2

iþDT2

i

q, (8)

with

DKi¼Ki,max max

TKi(T), (9)

DTi¼Ti,max argmax

T

Ki(T):(10)

Ki¼ji=jref being the normalized curvature of the [i]-th

actuator.

Finally, we input our model Equation (6), the actuators’

behavior (Fig. 2d, e), and the objective function Equation (7)

into a Python implementation of the covariance matrix ad-

aptation evolution strategy algorithm

24

and solve the inverse

problem (i.e., determine the parameters ni,Pinput, and Tinput

resulting in the target response) using a population size of 50,

an initial standard deviation of 0.4, and a starting point that is

randomly drawn from a standard normal distribution (Sup-

plementary Data).

In Figure 5, we report results for two different target re-

sponses. First, we optimize the system so that all bending ac-

tuators achieve the same bending curvature Ki,max ¼1:0at

Ti,max ¼0:1þ(i1) 0:2 (with i¼1, 2, 3, 4; Fig. 5a), target-

ing a bending sequence. The optimization algorithm converges

to the optimal solution after 80 iterations (Fig. 5b) and indicates

that, if we choose the response time to be tmax ¼25 s, the system

most closely approaches the prescribed target when the tubes

have length (L1,L2,L3,L4)¼(78:6, 10:0, 43:7, 122:4) cm

and the input supplies pinput ¼102:7 kPa for tinput ¼3:4s.As

shown in Figure 5c, for this set of parameters, both the nu-

merical model (solid lines) and the experimental observations

(dashed lines) closely follow the target response, that is, the

four actuators reach the speciﬁed maximum bending curvatures

at the desired times (markers) and then deﬂate (Fig. 5d; Sup-

plementary Movie S5).

Second, we look for a system in which Ki,max ¼0:6þ

0:2(i1) and Ti,max ¼0:15 þ0:1(i1) (with i¼1, 2, 3, 4),

so that the actuators sequentially bend with progressively in-

creasing curvature (Fig. 5e). Our optimization algorithm con-

verges to the optimal solution after 60 iterations (Fig. 5f ) and

ﬁnds that this response can be achieved for (L1,L2,L3,L4)¼

(3:5, 3:0, 14:8, 43:0) cm with pinput ¼23:3 kPa and tinput ¼

5:8 s. Remarkably, for this case we again ﬁnd that both our

experiments and simulations closely match the target response

(Fig. 5f, g; Supplementary Movie S6).

We emphasize that both target responses shown in Fig-

ure 5 would require an independently controlled input line

associated with each actuator in the array, if they were to be

achieved without harnessing viscous effects in the ﬂuidic

network. Therefore, by carefully selecting the narrow tubes

connecting the ﬂuidic actuators as well as the input pressure

and pressurization time, the target response for the system

can be naturally embedded in its design, allowing for a

substantial simpliﬁcation in system actuation. Note that

even though in Figure 5 we focus on two responses, our

strategy is robust and can be used to achieve a wide variety

of responses (Supplementary Data).

Finally, it is important to note that in cases where the

careful selection of the narrow tubes, input pressure, and

pressurization time through optimization lead to system re-

sponses that do not closely approach the objective, the so-

lution space can be further enriched by further optimizing the

geometry of the ﬂuidic actuators (Supplementary Data).

However, from a practical point of view, optimizing the ge-

ometry of the ﬂuidic actuators is not always desirable, since it

requires the fabrication of new actuators.

Multiobjective Optimization

The results of Figure 5 demonstrate the robustness of our

approach in identifying systems capable of achieving a

desired target response. However, in many cases, soft ro-

bots need to be able to achieve multiple different responses

and easily switch from one to another. To this end, we

investigate whether varying the magnitude of input pres-

sure Pinput and pressurization time Tinput is sufﬁcient to

enable a single system to achieve more than one target re-

sponses. Performing a brute force search for the range of

responses that a system optimized for a speciﬁc sequence

can achieve just by varying Pinput and Tinput,weﬁndthatthe

inﬂation parameters have very little effect in changing the

initial response for which the system was optimized (Sup-

plementary Data and Supplementary Figure S17).

Therefore, to effectively identify a system capable of

switching from one desired response (Target 1) to another

(Target 2) just by varying the inﬂation parameters, we formu-

late a multiobjective optimization problem. The dimensionless

tube parameters ni(with i¼1, 2, 3, 4) and the inﬂation pa-

rameters associated with the two target responses [i.e., (P(1)

input ,

T(1)

input)and(P(2)

input ,T(2)

input )] are obtained by minimizing

Z¼aZ(1) þ(1 a)Z(2), (11)

where Z(1) and Z(2) are the objective functions corresponding to

Targets1and2anda2[0, 1] is a scalar weighing the relative

importance of each objective.

Focusing on a system capable of switching between the

two responses deﬁned by the anchor points shown in

Figure 6a and b, our optimization algorithm ﬁnds that both

objectives are best approached for a¼0:5 (Fig. 6c) when

SIMPLIFIED FLUIDIC ACTUATION 5

FIG. 5. Solution of the inverse problem. (a) The ﬁrst target response requires all actuators in the system to achieve the

same maximum bending curvature Ki,max ¼1:0 but at different times Ti,max ¼0:1þ0:2(i1) (with i¼1, 2, 3, and 4). (b)

Evolution of the objective function during CMA-ES iterations. (c) Curvature response for the optimal system, as determined

from the numerical model (solid lines) and experiments (dashed lines). (d) Snapshots of the four actuators at

T¼0:13, 0:27, 0:5, 0:74, corresponding to the times at which each actuator achieves its maximum curvature during the

experiment. (e) The second target response requires all actuators in the system to achieve the maximum bending curvature

Ki,max ¼0:6þ0:2(i1) at Ti,max ¼0:15 þ0:1(i1) (with i=1, 2, 3, and 4). (f) Evolution of the objective function during

CMA-ES iterations. (g) Curvature response for the optimal system, as determined from the numerical model (solid lines)

and experiments (dashed lines). (h) Snapshots of the four actuators at T¼0:23, 0:27, 0:33, 0:43, corresponding to the times

at which each actuator achieves its maximum curvature during the experiment. CMA-ES, covariance matrix adaptation

evolution strategy. Color images are available online.

6

(L1,L2,L3,L4)¼(16:5,10:0,48:0, 124:0) cm, p(1)

input ¼39:6 kPa,

t(1)

input ¼6:32 s, p(2)

input ¼58:5 kPa, and t(2)

input ¼3:33 s. The cor-

responding numerical (solid lines) and experimental (dashed

lines) responses are again in excellent agreement for both

system responses and come sufﬁciently close to both objec-

tives (Fig. 6d, e; Supplementary Movie S7). Consequently,

our multiobjective optimization approach can be used to

successfully design systems that can achieve different target

responses just by varying the input pressure magnitude Pinput

and duration Tinput (Supplementary Figure S18).

Conclusions

In summary, using a combination of optimization tools and

experiments, we have shown that viscous ﬂow in the tubes

interconnecting ﬂuidic actuators can be exploited to design

soft robots that, although inﬂated through a single input, are

capable of achieving a wide range of target responses.

Throughout our study, we have found an excellent agree-

ment between the numerical predictions and experimental

ﬁndings—a clear indication of the predictive power and ro-

bustness of our framework. Even though in this work we

focused on systems in which the actuators inﬂate according

to a target sequence, we believe that our strategy can be

directly applied to design a wide range of ﬂuid-actuated soft

robots capable of performing multiple different tasks using

a single input.

To demonstrate how actuation sequencing through viscous

ﬂow can simplify the actuation of ﬂuidic soft robots, we

design a soft robot that comprises the four bending actuators

considered throughout this study (with top layer thicknesses

t¼4:0, 2:9, 2:1 and 1.5 mm), connect Actuator 1 (t¼4:0 mm)

to the pressure input through a tube with L1¼78:6cmand

R1¼0:38 mm, and supply pinput ¼102:7kPafortinput ¼3:4s.

If the four actuators are interconnected using tubes that do not

impose signiﬁcant restrictions to ﬂuid ﬂow (i.e., Ri¼0:79 mm

for i¼2, 3, 4), only the most compliant actuator inﬂates and no

FIG. 6. Multiobjective optimi-

zation. (a, b) The curvature anchor

points deﬁning Target 1 (a) and

Target 2 (b), respectively. (c) Par-

eto front. The black line connects

members of the Pareto set. The

color of the markers corresponds to

value of aused in the optimization.

The overall optimal solution that

most closely approaches both ob-

jectives is found for a¼0:5. (d, e)

Numerical (solid lines) and exper-

imental (dashed lines) curvature

responses for the optimal solution

of the multiobjective inverse

problem for Target 1 (d) and Tar-

get 2 (e), respectively. (f, g) Input

pressures required to achieve Target

1(f) and Target 2 (g), as determined

from the solution of the multi-

objective inverse problem (solid

lines) and as provided in experi-

ments (dashed lines). Color im-

ages are available online.

SIMPLIFIED FLUIDIC ACTUATION 7

functionality is achieved (Supplementary Movie S8). In con-

trast, if the actuators are connected using the tube lengths that

correspond to the optimal solution of Figure 5a and b (Fig. 7a),

the soft robot walks in a consistent and predictable manner

covering a distance of ^15 cm for 10 inﬂation cycles (Fig. 7b;

Supplementary Movie S8).

Finally, although in this study we only considered ob-

jectives for which a single curvature–time point was sufﬁ-

cient to describe the desired response of each actuator, one

could differently focus on the smooth control of ﬂuidic

actuators and deﬁne an objective function in terms of mul-

tiple target points in the curvature–time space for each ac-

tuator. We expect that very few modiﬁcations would be

necessary to achieve a smoother response for every actuator,

since viscous ﬂow is inherently a ‘‘smoothing’’ process.

Acknowledgment

This research was supported by the NSF under grant

number DMR-1420570.

Authors’ Contributions

N.V., J.T.B.O., and K.B. designed research; N.V. per-

formed research; N.V., J.T.B.O., and K.B. analyzed data;

N.V., A.J.G., and S.S. performed experiments; A.J.G. helped

design experiments; and N.V. and K.B. wrote the article.

Author Disclosure Statement

The authors declare no conﬂict of interest.

Supplementary Material

Supplementary Data

Supplementary Movie S1

Supplementary Movie S2

Supplementary Movie S3

Supplementary Movie S4

Supplementary Movie S5

Supplementary Movie S6

Supplementary Movie S7

Supplementary Movie S8

References

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logical inspiration, state of the art, and future research.

Appl Bionics Biomech 2008;5:99–117.

2. Ilievski F, Mazzeo AD, Shepherd RF, et al. Soft robotics

for chemists. Angew Chem Int Ed Engl 2011;50:1890–

1895.

3. Shepherd RF, Ilievski F, Choi W, et al. Multigait soft robot.

Proc Natl Acad Sci U S A 2011;108:20400–20403.

4. Kim S, Laschi C, Trimmer B. Soft robotics: a bioinspired

evolution in robotics. Trends Biotechnol 2013;31:287–294.

FIG. 7. Actuating a four-legged ﬂu-

idic soft robot using a single pressure

input. (a) The four actuators considered

throughout this study are interconnected

using narrow tubes with length (L

1

,L

2

,

L

3

,L

4

)=(78.6, 10.0, 43.7, 122.4) cm

and radius R¼0:38 mm. Tube 1 is

connected to a pressure input supplying

the robot with pinput ¼102:7 kPa for

tinput ¼3:4 s every 40 s. (b) Snapshots

of the position of the soft robot after 5

and 10 inﬂation cycles, demonstrating

its ability to walk in a predictable and

consistent manner (Supplementary Mo-

vie S8). Color images are available

online.

8 VASIOS ET AL.

5. Carmel M. Soft robotics: A perspective—Current trends

and prospects for the future. Soft Robot 2014;1:5–11.

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robots. Nature 2015;521:467–475.

7. Laschi C, Mazzolai B, Cianchetti M. Soft robotics: Tech-

nologies and systems pushing the boundaries of robot

abilities. Sci Robot 2016;1:eaah3690.

8. Zhao H, O’Brien K, Li S, et al. Optoelectronically inner-

vated soft prosthetic hand via stretchable optical wave-

guides. Sci Robot 2016;1:eaai7529.

9. Dian Y, Verma MS, So JH, et al. Buckling pneumatic linear

actuators inspired by muscle. Adv Mater Technol 2016;1:

1600055.

10. Polygerinos P, Correll N, Morin SA, et al. Soft robotics:

Review of ﬂuid-driven intrinsically soft devices; manufactur-

ing, sensing, control, and applications in human-robot inter-

action. Adv Eng Mater 2017;19:1700016.

11. Krishnan G, Bishop-Moser J, Kim C, et al. Kinematics of a

generalized class of pneumatic artiﬁcial muscles. J Mech

Robot 2015;7:041014.

12. Tolley MT, Shepherd RF, Mosadegh B, et al. A resilient,

untethered soft robot. Soft Robot 2014;1:213–223.

13. Rafsanjani A, Zhang Y, Liu B, et al. Kirigami skins make a

simple soft actuator crawl. Sci Robot 2018;3:eaar7555.

14. Morin SA, Shepherd RF, Kwok SW, et al. Camouﬂage and

display for soft machines. Science 2012;337:828–832.

15. Polygerinos P, Wang Z, Galloway KC, et al. Soft robotic

glove for combined assistance and at-home rehabilitation.

Robot Auton Syst 2015;73:135–143.

16. Paoletti P, Jones GW, Mahadevan L. Grasping with a soft

glove: Intrinsic impedance control in pneumatic actuators.

J Royal Soc Interface 2017. [Epub ahead of print]; DOI:

10.1098/rsif.2016.0867.

17. Overvelde JTB, Kloek T, D’haen JJA, et al. Amplifying the

response of soft actuators by harnessing snap-through in-

stabilities. Proc Natl Acad Sci U S A 2015;112:10863–

10868.

18. Rothemund P, Ainla A, Belding L, et al. A soft, bistable

valve for autonomous control of soft actuators. Sci Robot

2018;3:eaar7986.

19. Napp N, Araki B, Tolley MT, et al. Simple passive valves

for addressable pneumatic actuation. 2014 IEEE Interna-

tional Conference on Robotics and Automation (ICRA),

Hong Kong, China, 2014, pp. 1440–1445.

20. Connolly F, Walsh CJ, Bertoldi K. Automatic design of

ﬁber-reinforced soft actuators for trajectory matching. Proc

Natl Acad Sci U S A 2017;114:51–56.

21. Yang D, Mosadegh B, Ainla A, et al. Buckling of elasto-

meric beams enables actuation of soft machines. Adv Mater

2015;27:6323–6327.

22. Futran CC, Ceron S, Murray BM, et al. Leveraging ﬂuid

resistance in soft robots. In: 2018 IEEE International

Conference on Soft Robotics (RoboSoft). Livorno, Italy:

IEEE, April 2018:473–478.

23. Menon E. Piping Calculations Manual. New York: McGraw-

Hill Education, 2005.

24. Hansen N, Mu

¨ller SD, Koumoutsakos P. Reducing the time

complexity of the derandomized evolution strategy with

Covariance Matrix Adaptation (CMA-ES). Evol Comput

2003;11:1–18.

Address correspondence to:

Johannes T.B. Overvelde

AMOLF

Science Park 104

Amsterdam 1098XG

The Netherlands

E-mail: overvelde@amolf.nl

Katia Bertoldi

John A. Paulson School of Engineering

and Applied Sciences

Harvard University

Cambridge, MA 02138

E-mail: bertoldi@seas.harvard.edu

SIMPLIFIED FLUIDIC ACTUATION 9

Supporting Information

Nikolaos Vasios1, Andrew J. Gross1,

Scott Soifer1, Johannes T. B. Overvelde4,∗, Katia Bertoldi1,2,3,∗

1J. A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138, USA

2Wyss Institute for Biologically Inspired Engineering

Harvard University, Cambridge, MA 02138, USA

3Kavli Institute for Bionano Science and Technology

Harvard University, Cambridge, MA 02138, USA

4AMOLF, Science Park 104, 1098XG Amsterdam, The Netherlands

∗To whom correspondence should be addressed;

E-mail: bertoldi@seas.harvard.edu or overvelde@amolf.nl

S1 Fluidic Bending Actuators

In this section we provide details on the design, fabrication, testing and modeling of the

individual ﬂuidic bending actuators considered in this study.

Design

Although the principles proposed in this study can be applied to any type of ﬂuidic soft

actuator, to demonstrate the idea we focus on ﬂuidic bending actuators that consist of a

network of channels and chambers embedded in an elastomer (PneuNets).2Speciﬁcally,

we consider an actuator with initial length l= 75 mm and a rectangular cross section

with width wand height htotal (see Fig. S1). Note that, since in this study the response

1

of the actuators is tuned by varying the thickness of the upper layer t(with t∈[1.5,4]

mm), to make sure that all of them bend upon inﬂation both wand htotal depend on t

with w= 33 + 4tand htotal = 7.5+3t(if the width and height of the actuator remain

unchanged when tchanges, the actuators corresponding to larger values of twill just

expand and not bend, upon inﬂation).

To achieve bending upon inﬂation,

(i) we embed eight chambers within the actuator, each with length lc= 4 mm and

width wc= 3 mm connected via narrow channels with length dc= 3 mm , width

wt= 3 mm and height ht= 1.875 mm (see Fig. S1);

(ii) we use two diﬀerent elastomers to fabricate the actuator: a more compliant one for

the top part (shown in gray in Fig. S1) and a stiﬀer one for the bottom part (shown

in green in Fig. S1).

The values for all geometric parameters of the actuators modeled and fabricated in

this study are summarized in Table S1. Finally, we point out that our design is fully-

parameterized, so that the response of the actuators is unaﬀected (i.e. the normalized

pressure-volume and normalized curvature-volume curves remain the same) if all the di-

mensions are scaled by the same factor.

Fabrication

The actuators tested in this study are made of silicone rubbers. Speciﬁcally, we used

Ecoﬂex 00-30 (Smooth-On, Inc.) for the top (shown in gray in Fig. S1) and Elite Double

32 (Zhermack) for the bottom (shown in green in Fig. S1). The two layers were casted

and joined together using the 3-part mold shown in Fig. S2. The mold was designed in

Solidworks and 3d printed in Vero-blue using an Objet Connex 500 printer (Stratasys).

2

t

h

tb

l

t

h

tb

l

lc

dc

lc

ht

wc

w

AA’

B’

B

wt

a) b)

c)

d)

tw+dc

tw

tw

tw+dc

dc

dcdc

dc/2

A

A’

B’

B

C ’

Cl

w

htotal

x

y

z

xy

z

xy

zx

y

z

htotal

C’

C

dc/2

Figure S1: 3D model of the bending actuator considered in this study. (a) Isometric view.

Note that the gray and green regions correspond to EcoFlex-30 (Smooth-On, Inc.) and

Elite Double 32 (Zhermack) respectively. (b) Side view highlighting the top and bottom

layer thickness. (c) Side cross–section highlighting the details of the inner chambers

and channels. (d) Top cross–section highlighting the details of the inner chambers and

channels.

Table S1: Geometric parameters of the actuators considered in this study

Geometric parameter Value

Actuator Length, l75 mm

Chamber Height, h7.5 mm

Chamber Width, wc16.5 mm

Number of Chambers, n8 mm

Chamber Distance, dc3.0 mm

Channel Height, ht=h/4 1.875 mm

Channel Width, wt=wc/5.5 3.0 mm

Top Layer Thickness, t1.5 −4.0 mm

Bottom Layer Thickness, tb= 2t3.0 −8.0 mm

Wall Thickness, tw= 2t3−8.0 mm

Actuator Width, w= 2wc+ 2tw36.0 −41.0 mm

Chamber Length, lc= [l−2tw−(n+ 1)dc]/n 5.25 −4.0 mm

3

Parts I and II slide into one another and were used to cast the top portion of the actuator

made of EcoFlex-30. Part III was used to cast the bottom layer of the actuator made of

Elite Double 32.

Part I

Part II

Part III

Figure S2: A 3D render of Parts I, II and III of the mold used to cast our ﬂuidic bending

actuators.

Our actuators can be fabricated using the following 12 steps (see Fig. S3):

Step 1: expose all inner surfaces of the mold to Ease Release 200 spray (Mann Release

Technologies) to facilitate the process of removing the cured elastomer later on;

Step 2: prepare EcoFlex-30 by (a) dispensing equal amount of part A and B in a clean

container, (b) mixing thoroughly and (c) vacuum degassing for about 10 minutes.

Step 3: pour the Ecoﬂex mixture inside part II of the mold.

Step 4: slowly place part I of the mold on top of part II, while allowing for any excess

silicone to ﬂow out of the mold.

Step 5: cure the EcoFlex for about 4 to 5 hours at room temperature, while securing a

tight seal between mold parts I and II.

Step 6: (a) remove the cured EcoFlex from the mold and (b) trim any protruding edges

(if necessary).

4

Step 1 Step 2a Step 2b Step 2c Step 3 Step 4

Step 5 Step 6a Step 6b Step 7a Step 7b Step 7c

Step 8 Step 9 Step 10a Step 10b Step 11 Step 12

Figure S3: Snapshots of the 12 steps required for the fabrication of our ﬂuidic bending

actuators

Step 7: prepare Elite Double 32 by (a) dispensing an equal amount of base and catalyst

in a clean container, (b) mixing thoroughly and (c) vacuum degassing for about 3 minutes.

Step 8: pour the Elite Double 32 mixture inside part III of the mold.

Step 9: carefully place the EcoFlex-30 part of the actuator on top of the liquid Elite

Double 32 and allow the latter to cure for about 25 minutes and bond to the EcoFlex.

Step 10: remove the cured actuator from part III of the mold and trim any protruding

edges if necessary.

Step 11: insert a tube in one end of the actuator

Step 12: test the fabricated actuator for any leaks by inﬂating with a syringe pump. If

leaks are present, patch them using the appropriate silicone rubber.

Testing

In order to fully characterize the quasi-static response of the fabricated bending actua-

tors we conducted experiments to determine their pressure-volume and curvature-volume

relationships. Note that as a part of this study we fabricated and tested actuators with

5

four diﬀerent values of the top layer thickness, namely t= 1.5,2.1,2.9 and 4.0 mm. All

the actuators were tested using a syringe pump (Standard Infuse/Withdraw PHD Ultra;

Harvard Apparatus) equipped with two 50-mL syringes (1000 series, Hamilton Company)

with an accuracy of ±0.1%.

Pressure-Volume

For the pressure-volume measurements the actuators were inﬂated using water (to avoid

eﬀects of air compressibility) at a rate of 50ml/min, ensuring quasi-static conditions. The

pressure inside the actuators was measured during inﬂation using a MPX5050DP (NXP

USA Inc.) pressure sensor, connected to an Arduino Nano. The Arduino was able to log

the pressure in a text ﬁle with the use of a Python script and the serial module.

In Fig. S4 we report the evolution of the pressure pas a function of the volume

change ∆vinside the actuator for all actuators tested in this study1. The results of

Fig. S4 show that the pressure-volume curves for all actuators are nonlinear and feature

a pressure plateau. The plateau indicates the maximum pressure that the given actuator

can withstand and can be tuned by varying t(i.e. it monotonically increases with t).

Curvature-Volume

For the curvature measurements all actuators were inﬂated with air (the use of water for

inﬂation was avoided to eliminate the inﬂuence of gravitational eﬀects on the curvature

of the actuators). Upon inﬂation, we recorded videos of the deformation of each actuator,

which we processed to extract the curvature. Speciﬁcally, for each recorded frame we

identiﬁed the bottom edge of the actuator (highlighted in red in Fig. S5b) using a Python

image processing script. We then determined the radius Rof the circle that best ﬁts

1Note that the curves reported in Fig. S4 were determined by averaging the pressure volume curves

from 4 inﬂation cycles per actuator

6

t=1.5mm

t=2.1mm

t=2.9mm

t=4.0mm

0 10 20 30 40 50 60

Volume ∆v[ml]

0

3

6

9

12

15

18

Pressure p[kPa]

Figure S4: Experimental pressure-volume curves for four actuators characterized by t=

1.5, 2.1, 2.9 and 4.0 mm.

(minimizing the squared distance – least squares solution) the bent shape of the edge

(Fig. S5c) and calculated the average curvature as κ= 1/R.

As for the volume inside the actuator’s cavity corresponding to each curvature mea-

surement, it is important to note that we had to account for air compressibility. Since the

syringe pump, the tubes and the actuator form a closed system, application of Boyle’s

law yields,

p0vsys

0=pvsys (S1)

where p0is the initial pressure, pis the current pressure, vsys

0is the total volume of the

system at pressure p0and vsys is the total volume of the system at pressure p. Note that

the total initial volume of the system vsys

0can be written as,

vsys

0=v0+vsyringe

0+vtube

0(S2)

where v0is the initial volume inside the actuator, vsyringe

0is the initial volume inside the

syringe pump and vtube

0is the initial volume inside the tubes used to connect the actuator

7

a) b)

Edge Trace

Fitted Circle

R=1/κ

c)

Edge Trace

κref=π/l

d)

lπR=l

0 10 20 30 40 50 60

Volume ∆v[ml]

0

10

20

30

40

50

Curvature κ[1/m]

t=1.5mm

t=2.1mm

t=2.9mm

t=4.0mm

e)

Figure S5: Determining the curvature of ﬂuidic bending actuators upon inﬂation. (a) A

snapshot of a bending actuator during inﬂation. (b) An image processing code identiﬁes

the bottom edge of the actuator, highlighted here in red. (c) The curvature of the actuator

is calculated as the inverse radius of the circle that best ﬁts the shape of the bottom edge.

(d) To normalize the curvature measurements we use κref =π/l, which is the curvature

of a semi-circle with arc length equal to the initial length of the actuator’s bottom edge

(l). (e) Experimental curvature-volume curves for four actuators characterized by t= 1.5,

2.1, 2.9 and 4.0 mm.

to the syringe pump. Moreover, the total volume of the system vsys at pressure pcan be

similarly expressed as,

vsys =vsys

0+ ∆v−∆vsyringe (S3)

where ∆vis the change in volume inside the actuator and ∆vsyringe is the volume dispensed

by the syringe pump. By combining equations Eq. S1, Eq. S2 and Eq. S3 and solving

with respect to ∆vwe obtain,

∆v= ∆vsyringe −p−p0

pvsys

0(S4)

which we use to determine the volume change in the actuator given the volume dispensed

8

by the syringe ∆vsyringe and the pressure pmeasured by the pressure sensor.

In Fig. S5e we report the evolution of the bending curvature κas a function of the

volume change inside the actuator, ∆v, for all actuators tested in this study. The re-

sults show that the curvature increases almost linearly with the volume change and that

actuators with larger top layer thickness need a larger ∆vto achieve the same bending

curvature, indicating that an increase in tleads to stiﬀer actuator.

Finite Element modeling

In an eﬀort to better understand the eﬀect of the actuator’s thickness ton its response, we

performed a series of Finite Element simulations. All numerical analyses were carried out

using the commercial non-linear Finite Element software Abaqus (SIMULIA, Providence,

RI). EcoFlex-30 was modeled as an incompressible Gent solid23 (via a UHYPER user

deﬁned subroutine) with a shear modulus of G= 19.43kP a and an extension limit of

Jm= 37.54. Elite Double 32 was modeled as a nearly incompressible neo-Hookean solid24

with shear modulus G= 0.375M P a and Poisson’s ratio ν= 0.4998. All components

of the actuator were meshed using 8-node fully integrated hybrid linear bricks (C3D8H)

and perfect bonding was assumed between the two silicone rubbers. To determine the

pressure-volume and curvature-volume relationships of each actuator, quasi-static non-

linear simulations were performed using Abaqus/Standard. One end of the actuator

was held completely ﬁxed in space while the actuator was inﬂated using volume control

through the ﬂuid ﬁlled cavity interaction. While the pressure and volume inside the ﬂuid

cavity are both provided as post-processing variables, the curvature was determined by

following the exact same procedure we used to post-process the experimental results.

To verify the validity of our numerical simulations, we ﬁrst compared the numerical

and experimental results for the pressure-volume and curvature-volume response of the

9

four tested actuators. The results shown in Fig. S6a and Fig. S6b compare the normalized

pressure-volume and curvature-volume curves between FEA and experiments where pres-

sure is normalized with the initial shear modulus of EcoFlex-30 (G= 19.43kP a), volume

is normalized with the initial volume inside the actuator with top layer thickness t= 4mm

(vref

0= 4078.125mm3) and curvature is normalized using κref =π/l = 41.88 ×10−3mm−1,

which is the curvature of a semi-circle with arc length equal to the initial length of the

actuator’s bottom edge l(see Fig. S5d). The results indicate that the FEA simulations

accurately capture the pressure-volume and curvature-volume responses of the ﬂuidic

bending actuators for all four considered values of t.

Having veriﬁed the accuracy of our numerical analyses, we next used our simulations to

investigate the eﬀect of ton both the pressure-volume and the curvature-volume responses.

Speciﬁcally, we simulated the response of 22 actuators with diﬀerent values of top layers

thickness t(t∈[1.5,4.0] mm). The results shown in Figs. S6c and S6d indicate that

for any given change in volume ∆v, both the pressure and the curvature of the actuators

vary almost linearly with t. As such, by linearly interpolating between these 22 curves,

we built a response library from which we can determine the behavior of actuators with

arbitrary top layer thickness within the range t∈[1.5,4.0] mm (Figs. S6e and S6f).

S2 Arrays of Interconnected Fluidic Actuators

In this section we ﬁrst describe the setup we built to inﬂate an array of ﬂuidic actuators

connected via narrow tubes and then derive the governing equations that describe the

response of such a system.

10

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pressure p/G

t=1.5mm

t=2.1mm

t=2.9mm

t=4.0mm

a)

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Curvature κ/κref

FEA

EXP

b)

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pressure p/G

e)

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Curvature κ/κref

f)

1.50

2.00

2.50

3.00

3.50

4.00

Top Layer Thickness (mm)

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Pressure p/G

c)

0 2 4 6 8 10 12 14 16

Volume ∆v/v0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Curvature κ/κref

d)

1.50

2.00

2.50

3.00

3.50

4.00

Top Layer Thickness (mm)

Figure S6: Finite Element simulations (a) Comparison between the experimental (mark-

ers) and numerical (continuous lines) pressure-volume curves for four actuators character-

ized by t= 1.5, 2.1, 2.9 and 4.0 mm. (b) Comparison between the experimental (markers)

and numerical (continuous lines) curvature-volume curves for four actuators characterized

by t= 1.5, 2.1, 2.9 and 4.0 mm. (c) Numerical pressure-volume curves for 22 actuators

with t∈[1.5,4.0] mm. (d) Numerical curvature-volume curves for 22 actuators with

t∈[1.5,4.0] mm. (e) Evolution of the pressure-volume response of the actuators as a

function of t. (f) Evolution of the curvature-volume response of the actuators as a function

of t.

Experimental Setup

The setup used to test our array of connected actuators is shown in Fig. S7a and consists

of the actuators, narrow tubes to connect them, two pressure regulators, two solenoid

valves, an LCD screen, a camera and an Arduino.

As shown in Fig. S7a, our system comprises four ﬂuidic bending actuators connected to

each other via narrow tubes (Extreme-Pressure PEEK Tubes -McMaster Carr 51085K41-

51085K48). Speciﬁcally, we used tubes with an inner diameter of 0.005”, 0.007”, 0.01”,

0.02” and 0.03” (depending on the experiment) and an outer diameter of 1/16”. Note

11

Soft Actuators

LCD Screen

Outlet Solenoid Valve

Inlet

Solenoid Valve

Camera

Air Inlet

High

Pressure

Inlet

Actuator Inlets/Outlets

Pressure

Regulators

LCD Screen

Solenoid Valves

Pressure Sensors

Arduino

Power

Supply

b)

a)

Figure S7: Experimental setup for testing arrays of interconnected ﬂuidic actuators. (a)

A photo of the setup highlighting the solenoid valves, the pressure regulators, the ﬂuidic

actuators, the LCD screen and the camera. (b) A schematic of the wiring for the Arduino

with the solenoid valves, the pressure sensors and the LCD screen

that, since the tubes are rated for pressures up to 1000psi (6.98MPa), we expect the tube

walls to not deform during our experiments and thus we consider them as rigid in our

model (see Section Modeling).

To ensure a leak-free connection between the narrow tubes and the actuators, during

fabrication we equipped all ﬂuidic actuators with two PVC tubes with an inner diameter of

1/16” and outer diameter of 1/8” (McMaster Carr 9446K11) (see Fig. S8b). By choosing

the inner diameter of the PVC tubes (1/16”) to be the same as the outer diameter of

the narrow tubes (1/16”) we minimize the leaks occurring in the connections. Finally,

we note that the last actuator in the sequence is only connected by a single tube (see

Fig. S8a).

All narrow tubes except for the ﬁrst connect two neighbouring actuators. As for

the ﬁrst tube, it connects the ﬁrst actuator in the sequence to the pressure source (see

Fig. S8a). In order to inﬂate the connected actuators at pressures in the order of a few

kPa, we decreased the pressure from the wall air-outlet (∼200 psi) using two pressure

regulators (1/4 NPT 15CFM by Wilkerson and 1/4 NPT 9CFM by Coilhose Pneumatics)

connected in series. The ﬁrst pressure regulator reduces the inlet pressure from ∼200

12

PneuNet

Outlet Tube

Outlet Tube

Narrow Tube 1

Narrow Tube 2

b)

System Inlet/Outlet

Tube 3

Tube 4

Tube 2

Tube 1

a)

Figure S8: Array of interconnected ﬂuidic actuators. (a) A 3D render of a system com-

prising 4 ﬂuidic bending actuators interconnected via narrow tubes. The inlet of the ﬁrst

actuator serves as the inlet/outlet of the whole system. The last actuator in the array is

only connected to a single narrow tube. (b) A 3D render of an individual ﬂuidic bending

actuator highlighting the inlet and outlet tubes. The narrow tubes are inserted into the

outlet tubes which in turn are connected to the actuator

psi to about ∼40 psi, while the second one accurately controls the pressure in the range

[0,40] psi. Further, to turn on and oﬀ the input pressure we used two standard two-way

solenoid valves (SC8256A002V - ASCO). One of the valves was used to switch on/oﬀ the

input pressure, while the second one was used to switch on/oﬀ the outlet of the system to

the atmosphere, so that the actuators could deﬂate and return to their initial state (i.e.

when the input solenoid was open, the outlet solenoid was closed and vice-versa). Both

solenoid valves were powered through an external power supply using ∼9Vand ∼0.1A

and controlled via an NPN transistor (IRF520 - Vishay Siliconix).

To monitor and record both the input pressure and the pressure inside each actuator

we equipped the setup with 5 pressure sensors all connected to the Arduino. To measure

the input pressure we used a sensor rated for use up to ∼400 kPa (MPX5100DP-ND -

NXP USA Inc.), while to measure the pressure inside the actuators we used sensors rated

for use up to ∼200 kPa (MPX5050DP-ND - NXP USA Inc.).

13

Finally, to capture the deformation experienced by the actuators upon pressurization

we used a digital camera (Sony RX100 IV). Note that we used the LCD screen, which

displayed whether the input pressure was on or oﬀ, to sync the recorded videos with the

pressure readings.

Modeling

The system considered in this study comprises several ﬂuidic bending actuators connected

via narrow tubes. To capture its response we not only need to be able to capture the

behavior of the actuators, but also to determine the amount of ﬂuid exchanged through

the tubes. To this end, we focus on the [i]-th tube in the system, which has length Li,

circular cross section with radius Ri(with LiRi) and is connected to actuators [i−1]

and [i], as shown in Fig. S9a. We further introduce an orthonormal local direction basis,

where ezis identiﬁed as the unit vector along the tube’s length and erand eθlie in the

plane of the cross-section (see Fig. S9a). To determine the amount of ﬂuid exchanged

through the tube up to time t, ˜vi(t), we assume that

(i) the tube walls are rigid and not deformed by the ﬂow. Note that this assumption

is motivated by the fact that the tubes used in our experiments are rated for much

higher pressures (∼103psi) than the ones developed in our actuators (∼3psi).

(ii) the head losses due to friction at the connections between the tubes and the actuators

can be captured by the equivalent length method, adjusting the tube’s length to

Li,eq.

(iii) the radial (ur) and angular (uθ) components of the ﬂuid velocity ﬁeld uare zero

(since LiRi), so that

u(r, θ, z, t) = uz(r, z, t)ez,(S5)

14

S1

where we also dropped the dependency on θbecause of cylindrical symmetry.

(iv) the ﬂow is incompressible, so that

∇ · u= 0 ⇒∂ur

∂r +1

r

∂uθ

∂θ +∂uz

∂z = 0 ⇒∂uz

∂z = 0,(S6)

indicating that uzdoes not depend on z(i.e. uz(r, t))

(v) the ﬂuid ﬂow is laminar and governed by the Navier–Stokes equations, which in

light of Eqs. S5 and S6 reduces to 2

∂uz

∂t =−1

ρ

∂p

∂z +µ

r

∂

∂r r∂uz

∂r ,(S8)

where ρis the ﬂuid density, µis the dynamic viscosity of the ﬂuid and pis the

pressure in the tube.

(vi) the ﬂuid velocity proﬁle has the form,

uz(r, t) = Fi(t)"r

Ri2

−1#,(S9)

where Fi(t) is an unknown function that, since the volumetric ﬂow rate through the

[i]-th tube (d˜vi(t)/dt) is given by

d˜vi(t)

dt =ZRi

0

uz(r, t)2πrdr =−1

2πR2

iFi(t),(S10)

2Note that a closed-form analytical solution to Eq. S8 exists for steady state conditions and is given

by ,

uz(r) = ∆p R2

i

4µLi"r

Ri2

−1#,(S7)

where ∆pis the (constant) diﬀerence in pressure between the two ends of the tube. However, this solution

is unable to capture the response of our system, since ∆pis determined by the pressure inside the two

actuators connected to the tube and continuously changes with time as the ﬂuid ﬂows. The velocity ﬁeld

proposed in Eq. S9 does not satisfy the Navier–Stokes equations in a point-wise manner, but meets the

no-slip boundary conditions at the tube walls and provides a very good approximation for the volumetric

ﬂow, which is the main interest of this study. Analytical solutions to Eq. S8 under transient conditions

only exist for the starting ﬂow in a tube with a constant pressure gradient and are very complicated.

The highly non-linear and time-varying pressure gradient experienced by the ﬂuid in our system prohibits

analytical solutions to Eq. S8

15

S 2

S 3

can be expressed as

Fi(t) = −2

πR2

i

d˜vi(t)

dt .(S11)

Tube N

Actuator N

∆VN

PN(∆VN)

c)

VN

˜

Tube i+1

Tube i–1Tube i

Actuator i–1Actuator i

∆Vi

Pi(∆Vi)

ez

∆Vi–1

Pi–1(∆Vi–1)

a)

Vi

˜Vi+1

˜

Vi–1

˜

er

Tube 1

Actuator 1

∆V1

P1(∆V1)

V1

˜

Pressure Input

b)

Pinput , Tinput

Normalized

Pressure P

Normalized

Time T

Figure S9: Array of interconnected ﬂuidic actuators. (a) A schematic of the system. Note

that the [i]-th tube is connected to the [i−1]-th and [i]-th actuators. (b) A schematic of

the ﬁrst tube in the array, which has the left end connected to the pressure source and

the right one connected to the [1]-st actuator in the array. (c) A schematic of the last

tube in the array, which is connected to the [N−1]-th and [N]-th actuators

Next, we substitute Eqs. S9 and S11 into Eq. S8 and integrate along the volume of

the tube to obtain,

Li,eq

d2˜vi(t)

dt2=−πR2

i

ρ(pi−pi−1)−8µLi,eq

R2

i

d˜vi(t)

dt ,(S12)

where piis the pressure inside the [i]-th actuator. Upon the introduction of the normalized

volumetric ﬂow ˜

Vi= ˜vi/v0(v0denoting the volume of the smallest actuator in the system),

time T=t/tmax (tmax denoting the response time of the system) and pressure Pi=pi/G

(with Gbeing the shear modulus of the material used to fabricate the actuators), Eq.

(S12) can be rewritten in dimensionless form as

εi

d2˜

Vi(T)

dT 2+d˜

Vi(T)

dT +ξi(Pi−Pi−1) = 0,(S13)

with

εi=R2

iρ

8µtmax

,and ξi=πGR4

itmax

8µLi,eqv0

.(S14)

16

It is important to point out that for narrow tubes with Li,eq Ri, as those considered

in this study, the product εiξiis typically very small (εiξi1). Using the representative

values presented in Table S2 we ﬁnd that for air εair = 5.1×10−6and ξair = 268.27

and for water εwater = 7.25 ×10−5and ξwater = 4.92, resulting in εair ξair = 0.001368

and εwaterξwater = 0.00035. Consequently, for the tubes in our systems, in which viscous

forces dominate inertia, Eq. S13 can be simpliﬁed to (see Section “On the Simpliﬁcation

of Equation S13”).

d˜

Vi(T)

dT +ξi(Pi−Pi−1) = 0.(S15)

Eq. S15, which describes the volumetric ﬂow in a narrow tube connected to ﬂuidic ac-

tuators, is frequently considered the Ohm’s law analog for electrical circuits. In this

context, 1/ξiexpresses the equivalent resistance that the tube imposes to ﬂuid ﬂow; when

ξiis large, high ﬂow rates dVi/dT are achieved for relatively low pressure diﬀerences,

whereas when ξiis small, the opposite is true. Finally, since the normalized change in

volume for the [i]-th actuator, ∆Vi= ∆vi/v0, can be expressed in terms of the volumetric

ﬂows exchanged through the two tubes connected to it as

∆Vi=˜

Vi−˜

Vi+1,(S16)

Eq. (S15) can be rewritten as

d∆Vi(T)

dT +ξi(Pi−Pi−1)−ξi+1 (Pi+1 −Pi) = 0.(S17)

For a system comprising Nﬂuidic actuators interconnected via narrow tubes Eq. S17

result in a system of Ncoupled diﬀerential equations, which given a pressure-volume

relationship for the actuators that can be numerically solved to determine the normalized

change in volume for the [i]-th actuator as a function of time. Once the volume history for

all actuators is known, their bending curvature is then determined using the corresponding

17

S 4

curvature-volume relationship. Finally, we note that for the ﬁrst and last tube in the array

Eq. S17 needs to be modiﬁed to

d∆V1(T)

dT +ξ1(P1−Pinput(T)) −ξ2(P2−P1)=0,(S18)

d∆VN(T)

dT +ξN(PN−PN−1)=0,(S19)

to account for the pressure input (see Fig. S9b) and the end of the array (see Fig. S9c).

b)

Actuator 1

Inward Projecting Re-entrant

K1

Outlet Tube 1

2R

˜

c)

Square Reduction

K2

Flow Direction

Narrow Tube

Outlet Tube 1

2Ri

2R

˜

Tube Exit

e)

K4

Actuator 2

Outlet Tube

2

2R

˜

d) Square Expansion

K

Flow Direction

Narrow Tube

Outlet Tube

2

2R

˜

a)

Actuator 1

Flow Direction

Outlet Tube 1 Outlet Tube 2

Narrow Tube

2R2Ri

˜

2R

˜

Actuator

2Flow Direction

f)

Actuator 1

Actuator 2

Tube curvature

Figure S10: Minor Losses in the tubes. (a) Schematic of the tube connections between

two actuators, highlighting the sudden radii transitions from one tube to another. (b)

Schematic of the inward projecting re-entrant transition associated with K1(b) Schematic

of the square reduction transition associated with K2. (c) Schematic of the square-

expansion transition associated with K3. (d) Schematic of the tube exit transition, asso-

ciated with K4. (e) Schematic of the tube curvature associated with K5

As for the adjusted length Li, eq , according to the equivalent length method it can be

written as

Li, eq =Li+Ri

fi, D

Ni, t

X

α

Ki, α,(S20)

where Ni, t are the number of minor losses associated with the [i]-th tube and fi, D is the

Darcy friction factor, which for laminar ﬂow is deﬁned as

fi, D =64

Rei

,(S21)

18

S1

S 2

Rei= (2 ρ v0)/(µ π Ritmax )dVi/dT being the Reynolds number for the [i]-th tube. More-

over, Ki, α is referred to as the K-value for the α-th minor loss in the tube and is either

read from tables or has particular formulas depending on the type of geometric transitions

in the tube. Speciﬁcally, for all tubes in our system we have Ni, t = 5 (see Fig. S10) and

•(α=1) minor loss associated with the transition from the actuator chamber to the

PVC outlet tube inserted into the actuator (see Fig. S10b), for which

Ki, 1= 0.78,∀i= 1, . . . , N (S22)

•(α=2) minor loss associated with the transition from the outlet PVC tube with

radius ˜

Rinserted into the actuator to the narrow tube with radius Ri(see Fig. S10c),

for which

Ki, 2=1.2 + 160

Re

˜

R

Ri!4

−1

,∀i= 1, . . . , N (S23)

•(α=3) minor loss associated with the transition from the narrow tube with radius

Rito the outlet tube with radius ˜

Rinserted into the actuator (see Fig. S10c), for

which

Ki, 3= 2 "1−Ri

˜

R4#,∀i= 1, . . . , N (S24)

•(α=4) minor loss associated with the transition from the outlet tube inserted into

the actuator to the actuator chamber (see Fig. S10e), for which

Ki, 4= 1.0,∀i= 1, . . . , N (S25)

•(α=5) minor loss due to the curvature of each tube for which

Ki, 5= 1.5,∀i= 1, . . . , N (S26)

19

S1

S1

S1

S1

S1

Table S2: Representative parameter values for this study

Parameter Value

Tube Radius, R0.381 mm

Tube Length, L20 cm

Stiﬀness, G19.43 kPa

Initial Actuator Volume, v04078.125 mm3

System Response Time, tmax 25 sec

In this study we consider systems comprising either 2 or 4 interconnected bending

ﬂuidic actuators (i.e. N= 2 or 4), so that Eq. S17 becomes a system of either 2 or 4

coupled ODEs. To determine the pressure Piinside the [i]-th actuator (characterized by

the geometric parameter ti) for a given change in volume ∆Vi, we use the numerically

determined pressure volume response library shown in Fig. S6e. For the pressure input

supplied to the system, we consider a rectangular pulse (see Fig. S9b)

Pinput(T) = Pinput ,forT≤Tinput

0,forT > Tinput

(S27)

Furthermore, we consider µ= 1.568 ×10−5P a ·s(corresponding to the dynamic viscosity

of air at room temperature), tmax = 25 sec, v0= 4078.125 mm3(corresponding to the

volume of a bending actuator with t= 4 mm) and G= 19.43 kPa (corresponding to the

shear modulus of EcoFlex-30), yielding

ξi= 2.98309 ×1015 ×R4

i

Li, eq

(S28)

Finally, to integrate Eq. S17 we use a Python implementation of the Real-valued Variable-

coeﬃcient Ordinary Diﬀerential Equation solver, (LSODA) with initial conditions

∆Vi(0) = 0 ∀i= 1, . . . , N. (S29)

20

Equivalence between Eqs. S13 and S15 in systems for which εξ << 1

In this Section we demonstrate that in systems for which the product εξ is very small

Eq. S15 (i.e. the simpliﬁed form of the governing equation) is identical to Eq. S13 (i.e.

the governing equation). To this end, we ﬁrst quantify the product r=εξ in our system

comprised of narrow and slender tubes in which viscous forces dominate

r=εξ =R2ρ

8µ tmax

πGR4tmax

8µLv0

=

=1

64πG(πR2)

(µ2/ρ)

| {z }

rF

πR2L

v0

| {z }

rV

R

L2

| {z }

r2

a

=

=1

64πrFrVr2

a1 (S30)

where,

•rFis the ratio between the force associated with the pressure gradient (GπR2) and

the viscous forces in the tube (µ2/ρ)

•rVis the ratio between the volume inside the tubes (LπR2) and the volume inside

the actuators (v0)

•rais the aspect ratio of the tube deﬁned as the radius (R) divided by the length (L)

Since in our system rF,rVand raare always individually very small numbers, Eq. S30

indicates that εξ << 1.

Next, to explain the validity of Eq. S13 when εξ 1, we study the analytical solution

of a very simple system consisting of a single narrow tube connected to a single pneumatic

actuator and a pressure source (see Fig. S11), for which Eq. S13 reduces to

εd2˜

V

dT 2+d˜

V

dT +ξ(P−Pinput) = 0.(S31)

21

Assuming that (i) the input provides a constant pressure, Pinput = 1; (ii) the actuator

has a linear pressure-volume response, P(∆V) = ∆V; and (iii) the initial conditions are

˜

V(0) = 0, ˜

V0(0) = −ξ, Eq. S31 admits the analytical solution in the form

Tube

Actuator

V

Pressure Input

˜

Normalized

Pressure P

Normalized

Time T

1

Pressure P

Volume ∆V

1

ε , ξ

1

Figure S11: A simple system consisting of a single narrow tube, a single ﬂuidic actuator

and a pressure source supplying a constant pressure

˜

V(T) = 1 + A(ε, ξ) exp −T

2ε1 + p1−4εξ

−B(ε, ξ) exp −T

2ε1−p1−4εξ,(S32)

where

A(ε, ξ) = 1−2εξ −√1−4εξ

2√1−4εξ ,(S33)

B(ε, ξ) = 1−2εξ +√1−4εξ

2√1−4εξ .(S34)

By substituting ε=r/ξ, Eqs. S32-S34 can be rewritten as

˜

V(T) = 1 + A(r) exp −ξT

2r1 + √1−4r

−B(r) exp −ξT

2r1−√1−4r,(S35)

22

with

A(r) = 1−2r−√1−4r

2√1−4r,(S36)

B(r) = 1−2r+√1−4r

2√1−4r,(S37)

which for r1 can then be expressed using Taylor expansion as

˜

V(T)=1−exp(−T ξ) + Or2.(S38)

Having determined the analytical solution of Eq. S31, we now focus on the simpliﬁed

governing equation,

d˜

V

dT +ξ(P−Pinput) = 0.(S39)

Importantly, we ﬁnd that for the same system and boundary conditions Eq. S39 admits

the analytical solution

˜

V(T)=1−exp(−T ξ),(S40)

which is identical to Eq. S38 up to second order terms with respect to r. Therefore, the

analysis of the simple system justiﬁes the simpliﬁcation of the governing equation used in

this study.

Non-Dimensional Extents within which the Model Assumptions are Valid

Our numerical model, just like all models, is fundamentally based on the assumptions

stated in the Forward Modeling Section of the Main text (also in the Modeling Section

of the SI). If any of the assumptions is violated the model is not expected to maintain its

23

predictive capabilities. In the following we will address each of the assumptions on which

the model is based on and quantify the relevant non-dimensional extents within which

each assumption is valid.

i) The tube walls are rigid and not deformed by the ﬂow.

To assess the validity of this approximation one should analytically estimate

the expected change in the radius of the tube due to the internal pressure. To

this end, we assume that the maximum pressure developed due to the ﬂow

is equal to pmax and that the tube has thickness t, radius Rand is made of

a material with Young’s modulus Eand Poisson’s ratio ν. Furthermore, we

approximate the tube as a thick-walled linearly elastic pressure vessel, so that

the stresses at the inner surface of the tube are given by,

σrr =−pmax,(S41)

σθθ =pmax

(R+t)2+R2

(R+t)2−R2,(S42)

σzz =pmax

R2

(R+t)2−R2.(S43)

It follows from Eqs. S41–S43 that the circumferential strain can be expressed

as

εθθ =1

E[σθθ −ν(σrr +σzz)] = pmax

E(2 −ν)R2+ 2(ν+ 1)Rt + (ν+ 1)t2

t(2R+t),

(S44)

At this point is it important to point out that the rigid tube-walls assumption

is valid if εθθ 10−3= 0.1%. Since in this study we used tubes characterized

24

by E'3.8GPa, ν'0.38 R= 0.381mm, and t= 0.4064mm and the maximum

pressure in the tubes (due to the input pressure) was pmax = 100kPa, we ﬁnd

through Eq. (S44) that εθθ = 4.9·10−5, justifying the validity of the rigid wall

assumption made in our study.

ii) The head losses due to friction at the connections between the tubes and the

actuators can be captured by the equivalent length method, adjusting the tube’s length

to Leq.

As long as such connections exist in the system, this approximation will always

be valid provided that the ﬂow is laminar, incompressible and inviscid eﬀects

are negligible.

iii) The radial (ur) and angular (uθ) components of the ﬂuid velocity ﬁeld uare zero

(since LR), so that

u(r, θ, z, t) = uz(r, z, t)ez.

This assumption relies on the fact that the length of the tube is much larger

than its radius LRand is valid if

L

R10.

iv) The ﬂow is incompressible.

25

To assess the validity of this assumption we start from the continuity condition

which states,

Dρ

Dt = 0 ⇒∂ρ

∂t +∇·(ρu) = 0 ⇒∂ρ

∂t =−ρ∇·u−u·∇ρ(S45)

The density gradient ∇ρcan be expressed in terms of the pressure gradient

∇pby making use of the chain rule to ﬁnd

∇p=dp

dρ

|{z}

c2

∇ρ⇒∇p=c2∇ρ⇒∇ρ=1

c2∇p(S46)

where c=pdp/dρ is the local speed of sound. Therefore, by combining

Eq. S45 and Eq. S46 we ﬁnd

∂ρ

∂t =−ρ∇·u−u

c2∇p(S47)

For the ﬂow to be incompressible, the term ∂ρ/∂t needs to vanish implying

that the density of the ﬂuid does not vary as a result from the ﬂow. To this

end, the incompressibility assumption is valid if,

∇·u= 0 and u

c2∇p= 0 (S48)

Given that the radial and angular velocity components vanish ur=uθ= 0

(following from a prior assumption) the ﬁrst requirement for incompressibility

suggests that,

26

∇·u=∂ur

∂r +1

r

∂uθ

∂θ +∂uz

∂z = 0 ⇒∂uz

∂z = 0 (S49)

which is valid for long and narrow tubes for which,

L

R10 (S50)

The second requirement for incompressible ﬂow is immediately satisﬁed in the

case where the velocity magnitude of the ﬂow is much smaller than the speed

of sound since,

u

c1⇒u

c2∇p= 0 (S51)

v) The ﬂow is laminar and governed by the Navier Stokes equations which reduce to

∂uz

∂t =−1

ρ

∂p

∂z +µ

r

∂

∂r r∂uz

∂r

To assess the validity of this assumption we have to estimate the Reynolds

number for the system

Re =uρR

µ=

dv

dt ρR

πR2µ=dv

dt

ρ

πRµ ,

If Re is found to be less than 2500 then the assumption holds; otherwise the

ﬂow is not expected to be fully laminar. To this end, we express the non-

dimensional ﬂow rate d˜

V /dT in terms of the Reynolds number as,

d˜

V

dT =dv

dt

tmax

v0

=ReπtmaxRµ

ρv0

27

According to Eq. S15, the governing equation for a system comprising a single

actuator connected to the input source via a single narrow tube,

d˜

V

dT +ξ∆P= 0

For such a system, the maximum Reynolds number is expected ar T= 0 where

the pressure gradient is maximum. At T= 0 we have ∆P=−Pinput and thus,

max

d˜

V

dT

=ξPinput (S52)

Therefore, the maximum expected Reynolds number in this case is given by,

Remax =ξρPinput v0

πtmax Rµ

Substituting representative values for our study (see Table S2) we ﬁnd that

Remax ≈2400. As a result, since the maximum expected Reynolds number is

less than 2500 the ﬂow is expected to be laminar justifying this assumption.

(vi) The ﬂuid velocity proﬁle has the form

uz(r, t) = Fi(t)"r

Ri2

−1#,(S53)

where Fi(t) is an unknown function.

This assumption is motivated by the Poiselle ﬂow since the spatial component

of the velocity proﬁle is chosen so that is satisﬁes the no-slip boundary con-

ditions at the tube walls. The temporal component of the velocity ﬁeld is an

unknown function of time to be determined. This assumption is expected to

28

be valid in all scenarios where the ﬂow satisﬁes the no-slip boundary condition.

However, as noted in the Section ”Modeling” of the Supporting Information

it doesn’t satisfy the Navier-Stokes equations in a point-wise manner, but still

provides a very good approximation for the volumetric ﬂow which is the main

interest of this study.

Inverse Design & Optimization

Numerical solutions of Eq. S17 can be used to predict the temporal response of arbitrary

arrays of ﬂuidic actuators connected via narrow tubes. Here we are interested in the

inverse problem of designing a system capable of achieving a target response. Speciﬁcally,

we want the [i]-th actuator in the array to attain a speciﬁed maximum bending curvature

Ki, max =κi, max/κref at a predeﬁned time Ti, max =ti, max /tmax and then to completely

deﬂate. Since in this study we only consider systems consisting of 4 narrow tubes and 4

ﬂuidic bending actuators with diﬀerent top layer thickness tand use a rectangular pulse

as input pressure, the parameters that need to be determined to achieve such a target

response are

•the parameter tideﬁning the geometry of each actuator in the array (with i=

1,2,3,4);

•the radius to length ratio of the tubes deﬁned by the dimensionless parameter ξi

(with i= 1,2,3,4);

•the magnitude of the input pressure Pinput and the pressurization time Tinput.

To ﬁnd the set of parameters resulting in the desired response, we then minimize

Z=

4

X

i=1

(di+w τi) (S54)

29

where diis the “distance” in the K−Tspace between the target and actual points of

maximum curvature for the [i]-th bending actuator,

di=q∆K2

i+ ∆T2

i,(S55)

with

∆Ki=Ki,max −max

TKi(T),(S56)

∆Ti=Ti,max −argmax

T

Ki(T).(S57)

Moreover, τidenotes the amount of time that the [i]-th actuator spends above a threshold

curvature κ(i)

thres = 0.05κ(i)

max and is introduced to ensure that all actuators quickly deﬂate

(i.e. reach κ= 0) after reaching the target point of maximum curvature. Finally, the

factor wis a weight that sets the relative importance of the two objectives; w→0

expresses a bias towards solutions that just minimize d(i), while very large wresults in

solutions that minimize only τ(i). By trial and error, we found that for our system w= 1/4

leads to the best results.

Finally, we input all of this information together with the models we developed in the

previous section (Eq. S17) and the actuators’ response (see Figs. S6e and f) into a Python

implementation of the Covariance Matrix Adaptation Evolution Strategy (CMA-ES).

CMA-ES is an evolutionary algorithm that is used to solve optimization/inverse prob-

lems by iteratively solving several forward problems to adjust a covariance matrix of the

solution. CMA-ES is a derivative free algorithm, well suited for optimization problems of

high dimensionality. Even though CMA-ES is not as fast as gradient-based algorithms, in

our study it outperformed the latter since the objective function deﬁned by Eqs. S54–S56

is non-diﬀerentiable. To ensure that all the parameters involved in the optimization have

30

S 5

similar orders of magnitude, we renormalize them to lie in the interval [0,1]. To this end,

we deﬁne the renormalized tube resistances ˆ

ξi, actuators top layer thicknesses ˆ

ti, input

pressure ˆ

Pinput and pressurization time ˆ

Tinput as

ˆ

ξi=ξi−ξmin

ξmax −ξmin

(S58)

ˆ

ti=ti−tmin

tmax −tmin

(S59)

ˆ

Pinput =Pinput −Pinputmin

Pinputmax −Pinputmin

(S60)

ˆ

Tinput =Tinput −Tinputmin

Tinputmax −Tinputmin

(S61)

where ξmax = 62863, ξmin = 31.4, tmax = 4.0 mm, tmin = 1.5 mm, Pinputmax = 25,

Pinputmin = 0, Tinputmax = 0.25 and Tinputmin = 0. The initial values for all variables

used at the beginning of the optimization are drawn from a standard normal distribution.

Apart from the initial values of all the variables, CMA-ES also requires the initial standard

deviation to generate new candidate solutions in the ﬁrst generation of solutions. After

trial and error, we found that an initial standard deviation of σ= 0.4 was a reasonable

choice to ensure a “rich in variety” ﬁrst generation of solutions given that all variables

lie in the interval [0,1]. Note that the parameter bounds are enforced using rejection and

resampling. Whenever CMA-ES generates new parameter values that lie outside the [0,1]

interval, the values are rejected and new ones are generated until all candidate parameter

values are within the [0,1] interval.

Results

In this study we consider systems comprising four bending actuators connected via four

narrow tubes, choose the system response time to be tmax = 25 secs and use tubes with

31

A1 A2 A3 A4

pinput =102kPa , tinput =3.39s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=78.6cm

L2=10.0cm

L3=43.0cm

L4=122.0cm

Optimized System

0.0 0.5 1.0 1.5 2.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

Curvature Response

0.0 0.5 1.0 1.5 2.0

Normalized Time T

0.0

0.2

0.4

0.6

0.8

1.0

Normalized Pressure P

Pressure Response

A1 A2 A3 A4

pinput =110.0kPa , tinput =2s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=87.0cm

L2=23.0cm

L3=118.0cm

L4=400.0cm 0.0 0.5 1.0 1.5 2.0 2.5

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

A1 A2 A3 A4

pinput =48.88kPa , tinput =7.43s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=316.0cm

L2=1.91cm

L3=31.8cm

L4=94.1cm 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized Pressure P

a)

b)

c)

d)

Target

Model

Experiment

K1,max =1.0 , T1,max =0.1

K2,max =1.0 , T2,max =0.3

K3,max =1.0 , T3,max =0.5

K4,max =1.0 , T4,max =0.7

Target Response

K1,max =1.0 , T1,max =0.1

K2,max =0.8 , T2,max =0.3

K3,max =0.6 , T3,max =0.5

K4,max =0.4 , T4,max =0.7

K1,max =0.15 , T1,max =0.1

K2,max =0.4 , T2,max =0.3

K3,max =0.7 , T3,max =0.5

K4,max =1.0 , T4,max =0.7

t=1.5mm

t=2.1mm

t=2.9mm

t=4.0mm

Figure S12: Optimal solutions to the inverse problem for a system comprising four bending

actuators connected via four narrow tubes. The top layer thickness of the actuators are

ﬁxed to be t1= 4.0 mm, t2= 2.9 mm, t3= 2.1 mm and t4= 1.5 mm. The system

response time is chosen as tmax = 25 sec and the tube radii are ﬁxed to R= 0.38 mm.

The optimization algorithm determines the magnitude of the input pressure Pinput, the

pressurization time Tinput and the length of the four tubes in the array, Li. For each target

response we report the evolutions of curvature and pressure of the optimal system as a

function of time, as obtained both numerically (solid line) and experimentally (dashed

line).

radius R= 0.38 mm. To begin with, we also ﬁx the top layer thicknesses of each actuator

to be t1= 4.0 mm, t2= 2.9 mm, t3= 2.1 mm and t4= 1.5 mm for actuators 1, 2, 3

and 4, respectively. Consequently, the parameters to be determined by the optimization

algorithm are the input pressure Pinput, the pressurization time Tinput and the length of

32

A1 A2 A3 A4

pinput =93.6kPa , tinput =3.21s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=44.29cm

L2=3.5cm

L3=20.5cm

L4=65.5cm 0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.2

0.4

0.6

0.8

1.0

Normalized Pressure P

A1 A2 A3 A4

pinput =22.1kPa , tinput =4.64s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=4cm

L2=32.4cm

L3=26cm

L4=66cm 0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

A1 A2 A3 A4

pinput =23.3kPa , tinput =5.8s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=3.5cm

L2=3cm

L3=14.8cm

L4=43.0cm 0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.2 0.4 0.6 0.8 1.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

d)

e)

f)

K1,max =1.0 , T1,max =0.15

K2,max =1.0 , T2,max =0.25

K3,max =1.0 , T3,max =0.4

K4,max =1.0 , T4,max =0.55

K1,max =0.8 , T1,max =0.15

K2,max =0.6 , T2,max =0.25

K3,max =0.4 , T3,max =0.35

K4,max =0.2 , T4,max =0.45

K1,max =0.6 , T1,max =0.15

K2,max =0.8 , T2,max =0.25

K3,max =1.0 , T3,max =0.35

K4,max =1.2 , T4,max =0.45

Optimized System Curvature Response Pressure Response

Target Response

Target

Model

Experiment

t=1.5mm

t=2.1mm

t=2.9mm

t=4.0mm

Figure S12: (Contd.) Optimal solutions to the inverse problem for a system comprising

four bending actuators connected via four narrow tubes. The top layer thickness of the

actuators are ﬁxed to be t1= 4.0 mm, t2= 2.9 mm, t3= 2.1 mm and t4= 1.5 mm. The

system response time is chosen as tmax = 25 sec and the tube radii are ﬁxed to R= 0.38

mm. The optimization algorithm determines the magnitude of the input pressure Pinput,

the pressurization time Tinput and the length of the four tubes in the array, Li. For each

target response we report the evolutions of curvature and pressure of the optimal system

as a function of time, as obtained both numerically (solid line) and experimentally (dashed

line).

each tube in the array, Li(which can be determined from ξi). While in Fig. 5 of the main

text we focus on two target responses, in Figs. S12 and S13 we show the results obtained

solving the inverse problem for 11 diﬀerent target responses, in which we vary both Ki,max

and Ti,max. For each case, we report the tube lengths Liand pressure input parameters

33

A1 A2 A3 A4

pinput =33.4kPa , tinput =3.26s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=48.9cm

L2=9.1cm

L3=54.0cm

L4=400.0cm

Optimized System

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

Curvature Response

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

Pressure Response

A1 A2 A3 A4

pinput =427.4kPa , tinput =0.58s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=192.3cm

L2=46.0cm

L3=86.3cm

L4=81.7cm 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

A1 A2 A3 A4

pinput =141.9kPa , tinput =1.16s

Tube(1)Tube(2)Tube(3)Tube(4)

L1=109cm

L2=38.0cm

L3=95.4cm

L4=248.6cm 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.5

1.0

Normalized Curvature K

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Normalized Time T

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalized Pressure P

a)

b)

c)

Target

Target Response

K1,max =0.5 , T1,max =0.1

K2,max =1.0 , T2,max =0.3

K3,max =1.0 , T3,max =0.5

K4,max =0.5 , T4,max =0.7

K1,max =1.0 , T1,max =0.1

K2,max =0.5 , T2,max =0.3

K3,max =0.5 , T3,max =0.5

K4,max =1.0 , T4,max =0.7

K1,max =1.0 , T1,max =0.1

K2,max =0.6 , T2,max =0.3

K3,max =0.6 , T3,max =0.5

K4,max =0.6 , T4,max =0.7

Figure S13: Optimal solutions to the inverse problem for a system comprising four bending

actuators connected via four narrow tubes. The top layer thickness of the actuators are

ﬁxed to be t1= 4.0 mm, t2= 2.9 mm, t3= 2.1 mm and t4= 1.5 mm. The system

response time is chosen as tmax = 25 sec and the tube radii are ﬁxed to R= 0.38 mm.

The optimization algorithm determines the magnitude of the input pressure Pinput, the

pressurizatio