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Abstract and Figures

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.
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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
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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)
dt2¼pR2
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.
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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
+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),
di¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
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
aZ(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
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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
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(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
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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
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2. Ilievski F, Mazzeo AD, Shepherd RF, et al. Soft robotics
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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.
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5. Carmel M. Soft robotics: A perspective—Current trends
and prospects for the future. Soft Robot 2014;1:5–11.
6. Rus D, Tolley MT. Design, fabrication and control of soft
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.
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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.
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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.
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untethered soft robot. Soft Robot 2014;1:213–223.
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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
Downloaded by Harvard University FRANCIS A COUNTWAY from www.liebertpub.com at 05/14/19. For personal use only.
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= 2t38.0 mm
Actuator Width, w= 2wc+ 2tw36.0 41.0 mm
Chamber Length, lc= [l2tw(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.
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
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+ ∆vvsyringe (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 pp0
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 ×103mm1,
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 [i1]
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 ruz
∂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 i1Tube i
Actuator i1Actuator i
Vi
Pi(∆Vi)
ez
Vi1
Pi1(∆Vi1)
a)
Vi
˜Vi+1
˜
Vi1
˜
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 [i1]-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 [N1]-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(pipi1)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(PiPi1) = 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 = 5.1×106and ξair = 268.27
and for water εwater = 7.25 ×105and 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(PiPi1) = 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, 1iexpresses 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
dVi(T)
dT +ξi(PiPi1)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
dV1(T)
dT +ξ1(P1Pinput(T)) ξ2(P2P1)=0,(S18)
dVN(T)
dT +ξN(PNPN1)=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 "1Ri
˜
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 ,forTTinput
0,forT > Tinput
(S27)
Furthermore, we consider µ= 1.568 ×105P 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)
| {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)
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(PPinput) = 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 + p14εξ
B(ε, ξ) exp T
2ε1p14εξ,(S32)
where
A(ε, ξ) = 12εξ 14εξ
214εξ ,(S33)
B(ε, ξ) = 12εξ +14εξ
214εξ .(S34)
By substituting ε=r/ξ, Eqs. S32-S34 can be rewritten as
˜
V(T) = 1 + A(r) exp ξT
2r1 + 14r
B(r) exp ξT
2r114r,(S35)
22
with
A(r) = 12r14r
214r,(S36)
B(r) = 12r+14r
214r,(S37)
which for r1 can then be expressed using Taylor expansion as
˜
V(T)=1exp(T ξ) + Or2.(S38)
Having determined the analytical solution of Eq. S31, we now focus on the simplified
governing equation,
d˜
V
dT +ξ(PPinput) = 0.(S39)
Importantly, we find that for the same system and boundary conditions Eq. S39 admits
the analytical solution
˜
V(T)=1exp(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)2R2,(S42)
σzz =pmax
R2
(R+t)2R2.(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 103= 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 = 4.9·105, 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,
Dt = 0 ρ
∂t +·(ρu) = 0 ρ
∂t =ρ·uu·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
|{z}
c2
ρp=c2ρρ=1
c2p(S46)
where c=pdp/dρ is the local speed of sound. Therefore, by combining
Eq. S45 and Eq. S46 we find
∂ρ
∂t =ρ·uu
c2p(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
c2p= 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
c1u
c2p= 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 ruz
∂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
=Reπtmax
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 +ξP= 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
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, maxref 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 KTspace between the target and actual points of
maximum curvature for the [i]-th bending actuator,
di=qK2
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; w0
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=ξiξmin
ξmax ξmin
(S58)
ˆ
ti=titmin
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
pressurization time Tinput and the length of the four tubes in the array, Li. For each target
response we report the numerically obtained evolutions of curvature and pressure of the
optimal system as a function of time.
Pinput,Tinput that correspond to the optimal solution. Moreover, for each target response
we test the response of the system with the tubes and pressure input parameters deter-
mined by the optimization algorithm. For the six cases presented in Fig. S12 we test
the response both numerically and experimentally, while for those reported in Fig. S13
we only perform numerical simulations. In all our tests we find a very good agreement
34
A1 A2 A3 A4
pinput =53.1kPa , tinput =2.66s
Tube(1)Tube(2)Tube(3)Tube(4)
L1=84.7cm
L2=6.6cm
L3=42.5cm
L4=121.0cm 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
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 =61.3kPa , tinput =2.52s
Tube(1)Tube(2)Tube(3)Tube(4)
L1=133.7cm
L2=7.6cm
L3=104.0cm
L4=84.1cm 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
d)
e)
K1,max =0.6 , 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
K1,max =0.5 , T1,max =0.1
K2,max =1.0 , T2