pnasresearcharticle Forte

# MuA-Ori: Multimodal Actuated Origami

Antonio Elia Forte  J.A. Paulson School of Engineering and Applied Sciences,
Harvard University, Cambridge, MA 02138, USA  Department of Electronics,
Information and Bioengineering, Politecnico di Milano, Milan, 20133 Italy
Department of Engineering, King’s College London, London, WC2R 2LS, UK  David
Melancon  J.A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, MA 02138, USA  Leon M. Kamp  J.A. Paulson School of
Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA
Department of Applied Physics, Eindhoven University of Technology, Postbus
513, 5600 MB, Eindhoven, The Netherlands  Benjamin Gorissen  J.A. Paulson
School of Engineering and Applied Sciences, Harvard University, Cambridge, MA
02138, USA  Katia Bertoldi  J.A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Recently, inflatable elements integrated in robotics systems have enabled
complex motions as a result of simple inputs. However, these fluidic actuators
typically exhibit unimodal deformation upon inflation. Here, we present a new
design concept for modular, fluidic actuators that can switch between
deformation modes as a response to an input threshold. Our system comprises
bistable origami modules in which snapping breaks rotational symmetry, giving
access to a bending deformation. By tuning geometry, the modules can be
designed to snap at different pressure thresholds, rotate clockwise or
counterclockwise when actuated, and bend in different planes. Due to their
ability to assume multiple deformation modes as response to a single pressure
input we call our system MuA-Ori, or Multimodal Actuated Origami. MuA-Ori
provides an ideal platform to design actuators that can switch between
different configurations, reach multiple, pre-defined targets in space, and
move along complex trajectories.

###### keywords:

Origami | Bistabiliy | Inflatables | Actuators

## Introduction

Deformable and inflatable components have increasingly been integrated into
robotic systems, as they provide complex deformations (1, 2), the ability to
morph into target shapes (3, 4, 5, 6, 7), and inherent compliance, which in
turn enables safe interactions (8, 9, 10). These, however, suffer from a
common drawback: an intrinsic one-to-one relationship between input pressure
and output deformation. Upon inflation in fact, fluidic actuators exhibit
unimodal deformation, which accentuates as pressure increases (11). In order
to overcome this and create complex functionality, multiple actuators can be
assembled and carefully sequenced (12, 13, 14, 15, 16), or multiple chambers
of a single actuator can be pressurized independently (17, 18). As an
alternative route, bidirectional bending has been achieved by harnessing
material inextensibility (19) and non-linearity (20, 21). However, a design
strategy that enables arbitrary deformation modes with a single pressure input
is still absent.

In an effort to design novel robotic systems, engineers and scientists have
recently explored origami principles to create machines that are self-foldable
(22, 23, 24) and realizable in a variety of materials using both planar and 3D
fabrication techniques (25, 26, 27, 28). The functionality of such origami
robots can be further expanded if the crease pattern supports a non-convex
energy landscape, which enables multiple stable states (29, 30, 31, 32, 33,
34, 35, 36, 37). For example, self-locking grippers (38) and energy-absorbing
components for drones (39) were designed out of multistable origami sheets
based on the tiling of the degree-four vertex; bistable mechanical bits and
logic elements were created by introducing multistability in the classic
waterbomb origami pattern (30, 40, 41) and, finally, bistable configurations
of the Kresling pattern (42, 31) were exploited to generate locomotion via
peristaltic motion (43) or differential friction (44), create flexible joints
for robotic manipulation (45), and store mechanical memory (36, 46). All
together, these examples show that multistable origami are a promising
platform to realize fluidic actuators capable of supporting arbitrary
deformation modes when actuated with a single input.

Inspired by the potential of the Kresling pattern in the design of robotic
systems, here we employ this classic origami fold as a building block to
realize multi-output, but single-input actuators. As part of our strategy, we
start with a monostable Kresling pattern and make it bistable by introducing
two additional valley creases in one of its panels (see Figs. 1a-b). During
inflation, this panel unfolds and snaps outward, breaking the rotational
symmetry of the module. Importantly, upon vacuum such asymmetry gives rise to
bending in the opposite direction to the bistable panel up until a critical
negative pressure is reached: at this point the panel snaps inward, resetting
the module to its initial state. We use a combinatorial approach to couple
multiple modules and create actuators with prescribed and complex deformation
states that can be reached by controlling a single pressure input. Given that
they output multiple deformation modes with a single input, we call our
devices Multi-modal Actuated Origami, or MuA-Ori. MuA-Ori offers new
opportunities for the design of robotic systems capable of performing complex
tasks despite the simple actuation scheme, as demonstrated by the design of a
land rowing robot.

## The MuA-Ori building blocks based on Kresling pattern

Figure 1: Bistable origami modules as building blocks for multi-output, single
input inflatable actuators. (a) Schematics of a monostable module based on the
hexagonal-base Kresling origami pattern, along with a 3D-printed prototype.
The panels of the monostable modules remain always folded inward. We refer to
this state of deformation as state $s^{0}$. (b) Bistable module with a
modified panel (highlighted in orange) made of four triangular facets
$A^{\prime}BO^{\prime}$, $AO^{\prime}B^{\prime}$, $AO^{\prime}B$, and
$A^{\prime}B^{\prime}O^{\prime}$ and characterized by a depth $\Delta$ from
vertex $O$ to $O^{\prime}$, along with a 3D-printed prototype displayed in its
two stable states: state $s^{0}$ for which all panels (including the modified
panel) are folded inward; and state $s^{1}$ for which the modified panel is
popped outward (while all other panels are still folded inward). (c) State
diagram of the pressurized origami modules. (d-e) Pressure vs. bending angle
curves for the monostable and bistable origami modules during inflation and
deflation. At the onset of deflation for the bistable unit, $\theta$ is
slightly larger than in the monostable module due to the geometrical
incompatibility that arises in the structure when the modified panel is in the
outward configuration. (f) Experimental positive and negative pressure
thresholds, $p^{+}_{\Delta}$ and $p^{-}_{\Delta}$, as a function of the
modified panel’s depth, $\Delta$.

To realize our nonlinear and reconfigurable MuA-Ori, we use origami building
blocks comprising one layer of the classic Kresling pattern (also known as
nejiri ori) (42). More specifically, in its initial, undeformed state, the
single module consists of two hexagonal caps with edges of length $l=30$ mm,
separated by a distance $h=24$ mm, and rotated by an angle $\alpha=30^{\circ}$
with respect to each other (see Fig. 1a). The hexagons are connected at each
side by a panel comprising a pair of triangular facets coupled by alternating
mountain (i.e. edges $A^{\prime}B$ and $AB^{\prime}$) and valley (i.e. edge
$BB^{\prime}$) creases. Since the Kresling pattern is not rigid foldable (31),
any change in its internal volume will lead to an incompatible configuration.
To accommodate the resulting geometrical frustration, we 3D-print $1$-mm thick
triangular facets out of a compliant material (TPU95A from Ultimaker with
Young’s modulus $E=26$ MPa) and reduce the thickness locally to $0.4$ mm to
create the hinges (see prototype in Fig. 1a). Further, to facilitate coupling
between different modules, we 3D-print the hexagonal caps out of a stiffer
material (PLA from Ultimaker with Young’s modulus $E=2.3$ GPa) and, to form an
inflatable cavity, we coat the origami unit with a thin layer of
polydimethylsiloxane (PDMS) (see Supplementary Materials, Section S1 for
fabrication details). Finally, it is worth noticing that the chosen values of
the parameters $(h,l,\alpha)$ yield a monostable origami module (i.e. the
Kresling pattern is only stable in its initial, undeformed state).

To investigate the response of a single module, we inflate it with water while
submerged in water (to eliminate the effects of gravity - see Supplementary
Materials, Section S2 for details). As expected (31), the Kresling unit
extends and contracts (while twisting at the same time) upon inflation and
deflation (Fig. 1c) and returns to its undeformed state as soon as the
pressure is removed. This state of deformation, in which all panels are folded
inward, is named $s^{0}$. When in this state the bending angle, $\theta$,
remains close to zero (Figs. 1d-e).

Aiming at unlocking different deformation modes with one single pressure
input, we then take inspiration from bistability in degree-four vertices (33,
41, 47) and modify one of the original Kresling panels by introducing two
additional valley creases (i.e. $AO$ and $A^{\prime}O$ with $O$ being the
midpoint of crease $BB^{\prime}$, see Fig. 1b). While this effectively creates
a degree-four vertex, it results in a monostable origami unit, as no snap-
through instability is recorded upon inflation (see Supplementary Materials,
Section S2 for details). To increase the geometric incompatibility during
deployment and achieve bistability in the unit, we then move the degree-four
vertex inward by $\Delta$ (see Fig. 1b where $\Delta$ is the norm of vector
$\overline{OO^{\prime}}$ perpendicular to vectors $\overline{AA^{\prime}}$ and
$\overline{BB^{\prime}}$). Choosing $\Delta=3$ mm, for example, we can
fabricate an origami unit that can easily transition between two stable
states: state $s^{0}$ for which all panels are folded inward, and state
$s^{1}$ for which the modified panel is popped outward (while all other panels
are still folded inward). Similar to the unit based on the classic Kresling
pattern, upon inflation this modified module simply extends with all panels
bent inward if $p<26.1\pm 0.9$ kPa. However, at $p_{3}^{+}=26.1\pm 0.9$ kPa
(where the subscript refers to $\Delta=3$ mm and the superscript refers to
positive pressure), the unit snaps from state $s^{0}$ with the modified panel
folded inward to state $s^{1}$ where it is popped outward (Fig. 1c)—a
transition which is accompanied by a discontinuity of the bending angle
$\theta$ (see zoom-in in Fig. 1d). Finally, a further increase in pressure
causes the unit to elongate until the maximum structural limit is reached.
Afterward, when the input pressure is removed, the modified panel remains
popped outward (see Fig. 1c) because of bistability. As such, when we apply
negative pressure the unit not only contracts, but also bends (see Figs. 1c
and e), exhibiting a behaviour that radically differs from that of the
Kresling module. Specifically, we find that at first $\theta$ monotonically
increases until the two hexagonal caps come into physical contact effectively
clipping the available range of bending deformation to $\theta_{max}=21.7\pm
0.3\,^{\circ}$. As previously mentioned, this bending deformation is caused by
the modified panel, which remains in the popped outward configuration (while
the other panels fold under increasing negative pressure) and breaks the
radial symmetry. Finally, when the negative pressure passes the threshold
$p_{3}^{-}=-21.2\pm 0.7$ kPa (where the superscript refers to negative
pressure), the modified panel snaps back to the inward position (see snapping
transition from state $s^{1}$ to state $s^{0}$ in Fig. 1c) and $\theta$
suddenly decreases (dashed orange arrow in Fig. 1e). If one continues to apply
negative pressure to the module, the unit folds (almost) flat, as clearly
shown by the decreasing trend of $\theta$.

Next, we investigate the effect of the depth $\Delta$ of our degree-four
vertex panel on the positive and negative pressure thresholds,
$p_{\Delta}^{+}$ and $p_{\Delta}^{-}$, as well as the deformed configurations
reached upon snapping. The experimental results reported in Fig. 1f for
$\Delta=2$, $3$, and $4$ mm indicate that the absolute value of the pressure
thresholds increases with $\Delta$ within the considered range. Differently,
we find that the bending angle, the axial displacement of the end caps and
their relative twist remain almost constant with $\Delta$. In fact, the first
reaches an upper limit when the caps get in contact with each other and the
other two are dominated by the geometric characteristics of the Kresling
pattern (see Fig. S7). Note that for $\Delta<2$ mm, the modules are found to
be monostable. This means that negligible bending is recorded upon application
of negative pressure, since the degree-four vertex panel snaps back
immediately. Differently, for $\Delta\geq 4$ mm, the positive pressure
required to snap the modified panel outward is so high that the module fails
(see Fig. S6).

## Extending the design space for complex outputs

After demonstrating that our bistable module can transition between different
stable states with distinct deformation modes (i.e. extension, twisting, and
bending), we next combine these units to form actuators able to achieve
complex outputs: the MuA-Ori. By combining $n$ modules, we can construct
$\left(3\times 2\times 6+1\times 2\right)^{n}=38^{n}$ different actuators
since for each module $k$ we can select ($i$) either a regular Kresling module
or a unit comprising a modified, degree-four vertex panel with depth
$\Delta^{k}\in\\{2,3,4\\}$ mm; ($ii$) the upper cap to be rotated clockwise or
anticlockwise with respect to the bottom one,
$c^{k}\in\\{\textrm{//},\textrm{\textbackslash\textbackslash}\\}$, and ($iii$)
the side on which the modified panel is located, $f^{k}\in\\{1,\ldots,6\\}$
(see Fig. 2a). Note that the domain of all constraints is discrete since we
choose a limited set of values for $\Delta$ and all end plates to have their
sides aligned.

Figure 2: Extending the design space. (a) For each $k$ module we define $3$
geometrical parameters: $\Delta^{k}$, $c^{k}$, and $f^{k}$. Note that, for
simplicity, we also assign $f^{1}$, but only the location of the modified
panel relative to its neighbor is important. (b) State diagram for a $2$-units
actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textrm{//}6]$:
geometrical model prediction on the left, experimental snapshots on the right.
(c) Normalized deployment $||\mathbf{d}||/h$ and angle $\theta_{act}$ for
every stable state of the $2$-units actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textrm{//}6]$.
(d) Normalized deployment $||\mathbf{d}||/h$ and angle $\theta_{act}$ for
every stable state of any possible $2$-units (triangles) and 3-units (circles)
actuator in the design space.

To navigate the vast design space systematically and efficiently, we develop a
simple algorithm that predicts all stable states and snapping transitions for
actuators comprising $n$ modules upon inflation and deflation. First, we
extract key geometric features from the experiments conducted on single units,
i.e. height, twisting, and bending angle of the top cap associated to all
stable states and snapping transitions (see Figs. S7-S8). When assuming
pressure continuity, these data allow the prediction of all configurations for
the stable states and snapping transitions of any $n$-units actuator (see
Supplementary Materials, Section S3 for details on the algorithm). Note that
we also assume perfect coupling between units, so that the pressure
thresholds, $p_{\Delta}^{+/-}$, found in the experimental characterization of
Fig. 1f, remain unchanged and identical for units with the same geometrical
parameters.

As an example, in Fig. 2a, we consider an actuator comprising $n=2$ modules
characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textrm{//}6]$,
where we assume the first unit to be the one at the bottom. As expected, this
MuA-Ori has four stable states at atmospheric pressure, $s^{ij}$ (where the
subscripts $i,j=0,\,1$, refer to the state of the modified panels, with
$\Delta=2$ and 4 mm, respectively), and six snapping transitions that can be
triggered by varying the internal pressure (Fig 2b). Whereas the stable states
$s^{10}$ and $s^{11}$ can be readily obtained by simply increasing pressure, a
more complex pressure path is required to achieve state $s^{01}$, as one has
to ($i$) increase pressure above $p^{+}_{4}$ and then ($ii$) decrease it below
$p_{2}^{-}$. To validate the model, we then conduct experiments on units
connected via 3D-printed screws (see Fig. S2 for details) and find that they
closely match the predictions (see Fig 2b and Movie S3).

Since our final goal is to build actuators with programmed deformation modes,
we further quantify the deployment of our actuator by recording the vector
connecting the caps’ centroids, $\mathbf{d}$, at each stable state and
snapping transition. In Fig. 2c we report the norm of $\mathbf{d}$ normalized
by the height of a single module, $||\mathbf{d}||/h$, and the angle between
each $\mathbf{d}$ and the $z$-axis, $\theta_{act}$, for all $12$
configurations supported by the actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textrm{//}6]$.
We find that in the positive pressure regime the achievable deformation is
limited to simple extension, resulting in configurations all clustered in a
portion of the design space characterized by $\theta_{act}\sim 5^{\circ}$ and
$||\mathbf{d}||/h\sim 2$ (gray and red triangles). By contrast, behaviors
characterized by large bending angles ($\theta_{act}\sim 20^{\circ}$) unfold
under negative pressure (blue triangles). Importantly, such bending provides
opportunities for the design of actuators that can reach a wider range of
configurations. This becomes even clearer when expanding the range of possible
configurations by varying the parameters $\Delta^{k}$, $c^{k}$, and $f^{k}$
and considering not only $n=2$ but also $n=3$ building blocks (triangles and
circles in Fig. 2d, respectively). Once again, we notice that all
configurations under positive and atmospheric pressure cluster on low values
of bending angle and a normalized deployment that is roughly equal to the
number of building blocks considered. Differently, in the negative pressure
regime a rich set of deformations unfolds, characterized by a wide range of
achievable $\theta_{act}$ (the maximum values of $\theta_{act}$ increases from
$40^{\circ}$ to $50^{\circ}$ from $n=2$ to $n=3$) at near-constant normalized
deployment ($||\mathbf{d}||/h$ increases slightly from about $1.5$ to $2$ from
$n=2$ to $n=3$). These results indicate that by stacking our origami modules
to form arrays, we can realize actuators capable of supporting a variety of
deformation modes, which can be selected by varying the level of applied
pressure.

## Inverse design to reach multiple targets

Motivated by the results in Fig. 2, we then investigate the behavior of arrays
with a larger number of units. Our goal is to build actuators capable of
switching between target deformation modes when inflated with a single
pressure source. However, since the use of $n$ modules leads to $38^{n}$
possible actuator designs (i.e. $54,872$ for $n=3$ and $2,085,136$ for $n=4$),
it is crucial to use a robust algorithm to efficiently scan the range of
responses that can be achieved and identify

Figure 3: Inverse design to reach multiple targets. We employ the greedy
algorithm to inverse design actuators able to reach a set of targets with a
single pressure source. (a) Two distinct set of targets (black and red), top
and 3D view. (b)Targets error $\Psi$ as a function of total number of units:
the optimal actuators produced by the algorithm for both set of targets are
reported as ($*$) and ($**$), along with the respective parameters for each
module. (c) State diagram for the 3-units actuator ($*$) with targets $T1$ and
$T2$ highlighted. (d) Top and 3D view of the model and the experimental
prototype for the 3-units actuator reaching the black targets. (e) State
diagram for the 12-units actuator ($**$) with targets $T1$, $T2$ and $T3$
highlighted. (f) Top and 3D view of the model and the experimental prototype
for the 12-units actuator reaching the red targets.

configurations leading to the target deformation modes. Toward this end, given
the discrete nature of our design variables, we use a greedy algorithm based
on the best-first search method (48, 49)—a progressive local search algorithm
that, at each iteration, minimizes the cost function by looking at a set of
available solutions. Although there exists many algorithms to solve this type
of discrete optimization problems (50, 51), we find that the greedy algorithm
provides the best trade-off between accuracy and computational cost (see
Supplementary Information, Section S4 for details and comparison of the
different algorithms). Specifically, our greedy algorithm identifies MuA-Ori
comprising $n$ units built out of $n_{s}$ super-cells each with $n_{u}$
modules (so that $n=n_{u}\cdot n_{s}$) whose tip can reach a desired set of
targets arbitrarily positioned in the surrounding space. At the first
iteration, the algorithm starts by selecting the actuator super-cell design
that minimizes

$\Psi=\frac{1}{n_{targets}\cdot
h}\sum_{m=1}^{n_{targets}}\min||\mathbf{d}-\mathbf{T}_{m}||,$ (1)

where $n_{targets}$ is the number of targets, and $\mathbf{T}_{m}$ is the
vector connecting the $m$-th target with the origin. Once the first super-cell
is chosen, the algorithm stores it in memory and starts a second iteration.
This comes to an end when the algorithm identifies a second super-cell that,
connected to the first, one minimizes Eq. (3). The first two super-cells are
then stored in memory and the algorithm advances to the next one. Note that in
this study, to balance the number of available designs and computational cost,
we set the greedy algorithm to consider super-cells made of $3$ units (i.e.
$n_{u}=3$) (see Figs. S15 and S18 for a comparison across super-cells made
with different $n_{u}$) and, to avoid fabricating excessively long actuators
whose response could be affected by gravity, terminate the algorithm after
stacking $5$ super-cells.

To demonstrate our approach, we select two distinct sets of targets within the
reachable space (see black and red circular markers in Fig. 3a and
Supplementary Materials Figs. S15-S19 for additional targets). For the first
set of black targets $(T1,T2)$ with small height and radius of action, we find
that the objective function $\Psi$ is minimized for an actuator with $n_{s}=1$
and
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2};\Delta^{3}c^{3}f^{3}]=[4\textbackslash\textbackslash
2;3//6;4\textbackslash\textbackslash 2]$ (see Fig. 3b). As this $3$-unit
actuator only comprises modules with $\Delta=3$ and $4$ mm, its state diagram
(shown in Fig. 3d) is qualitatively similar to the one reported in Fig. 2b. To
reach the target $T1$, the internal pressure must first be raised above
$p^{+}_{4}$ and then lowered to $p_{3}^{-}$ (see shaded gray line in Fig. 3c).
Successively, the target $T2$ is reached by further lowering the pressure to
$p_{4}^{-}$. It is worth noticing that, in this example, increasing the
pressure above $p_{4}^{+}$, decreasing it below $p_{4}-$ and finally bringing
it back to zero results in a triangular path for the centroid of the top
plate, which is spanned in a counterclockwise direction (Fig. 3d).
Importantly, when we apply this pressure history to a physical sample, the
actuator’s tip follows the predicted trajectory and approaches the two targets
very closely (see Fig. 3d), confirming the validity of our design strategy. As
predicted by our model, the actuator starts straight, bends toward $T1$, bends
toward $T2$ and finally returns to the straight configuration $O$ after reset.

For the second set of red targets $(T1,T2,T3)$, the larger height and radius
of action of the targets lead to an optimal actuator comprising $n_{s}=4$
super-cells (note that the convex shape of $\Psi$ is due to a correlation
between the optimal number of units and the average distance of the targets
from the origin—see Supplementary Materials Fig. S19). As shown in Fig. 3b,
this 12-unit actuator comprises the classic Kresling module as well as
bistable units with $\Delta=2$, $3$, and $4$ mm. This results in $8$ stable
states, $14$ snapping transitions, and a more complex state diagram in which
not all targets are reached consecutively by continuously decreasing pressure
(Fig. 3e). More specifically, to move from $T1$ to $T2$, this actuator has to
be reset by decreasing the pressure below $p_{3}^{-}$ before increasing above
$p_{4}^{+}$ and then lowering it to $p_{3}^{-}$. As such, in this case the
centroid of the top plate of the actuator passes through the straight
configuration $O$ when moving from $T1$ to $T2$ and its trajectory comprises
two disconnected loops, $O-T1$ and $O-T2-T3$ (Fig. 3f). Note that we can add
additional constraints to our greedy algorithm to make sure the targets fall
within the same closed loop on the state diagram. This obviously leads to a
different design and may increase the targets error (see Supplementary
Materials Fig. S17 for details). However, the ability to fully control the
actuator’s sequential trajectory makes the platform ideal for robotic
applications.

## MuA-Ori for robotic applications

To show the potential of the MuA-Ori for robotic applications we take
inspiration from the closed trajectory of a rowing stroke, and use the closed
triangular loop of the $3$-unit actuator in Fig. 3e to propel a land rowing
robot (Figs. 4a). The robot comprises two symmetric arms (corresponding to the
$3$-unit MuA-Ori in Fig. 3) connected through a single fluidic line and
mounted on a wheeled chassis. Further, to harnesses the cyclic motion of the
actuator and generate positive mechanical work with the ground, we connect two
rigid rods

Figure 4: Land rowing robot. (a) Schematics of the robot. (b) Robot prototype.
(c)-(d) Trajectory followed by the robot’s arms upon inflation and deflation,
as predicted by the model (c) and observed in experiments (d). (e)
Experimental snapshots of the robot in the initial configuration and after 10
and 20 cycles.

to the outer caps that serve as stroke amplifiers and attach silicon patches
at their ends to increase friction with the ground. When increasing the
pressure above $p_{4}^{+}$ and then lowering it to $p_{3}^{-}$ the two
actuators reach $T1$. At this point the rigid rods touch the ground. Then,
when we further decrease the pressure, the rigid rods move backward to T2
creating a forward thrust and eventually lift off from the ground, completing
the stroke. Finally, lowering the pressure below $p_{4}^{-}$ resets the
locomotion cycle (see Figs. 4c and d). As shown in the experimental snapshots
in Fig. 4e we can harness this particular trajectory instructed by the model
to create locomotion: the robot advances of about $16$ cm in $20$ cycles. Note
that, differently from other robotic platforms with similar performance but
requiring one or more actuators per leg with dedicated pressure sources (12,
16), our robots operate with a single pressure input, which largely simplifies
its control.

## Conclusion

To summarize, in this work we have presented a platform to design fluidic
origami robots that can switch between a range of different configurations
using only one pressurized input: the MuA-Ori. The key characteristics of our
platform is an origami building block with a degree-four vertex panel, which
can be geometrically programmed to snap at a certain input threshold, inducing
bending. This, together with the position of the modified panel in the origami
module and their direction of rotation, constitute the parameters of a rich
design space that we can efficiently scan with a custom greedy algorithm.
Therefore, instructed by this model, we are able to automatically design and
build optimal robotic arms that can reach a set of defined targets, and
actuators capable of tracing spacial trajectories in a programmed sequence,
enabling mechanical tasks and functions. While in this study we have used a
simple geometric model to identify optimal designs, a fully mechanical model
may be needed in order to extend the proposed approach to larger designs for
which the effect of gravity is not negligible. In addition, the current design
space could be further expanded through investigating the effect of other
geometrical parameters (e.g. $l$, $h$, and $\alpha$) on the resulting
deformation of the modules, as well as expanding the range of the considered
values of $\Delta$. While this could lead to more complex deformation modes
and enhanced functionality, a drawback is a more complex state diagram. This
means that a given actuator might have to go through a longer loading history
to reach some prescribed targets, increasing the operational time-span. A
potential solution to this is to measure the volume at which the module snaps
inward and outward, assume constant flow rate, and derive the time associated
to each snapping transition. This time span could then be included as variable
in the optimization algorithm, in order to find a design that reaches the
target in the shortest possible time. To conclude, we envisage that the MuA-
Ori concept hereby presented will serve as future strategy for the design of
fluidic origami robots able to overcome the unimodal deformation constraint,
despite the simple actuation.

## Materials and Methods

Details of the design, materials, and fabrication methods are summarized in
Supplementary Materials, Section S1. The experimental procedure of the
inflation with water to measure the pressure-volume curve is described in
Supplementary Materials, Section S2, along with additional experimental
data.experimental data. Details on the numerical mode are provided in
Supplementary Materials, Section S3. Finally, the optimization algorithms used
in this study are described in detail in Supplementary Materials, Section S4,
along with additional data.

## Acknowledgments

We thank Eder Medina for his assistance in the development of the fabrication
methodology. Author contributions: A.E.F., D.M., B.G. and K.B. proposed and
developed the research idea. A.E.F designed and fabricated the samples.
A.E.F., D.M., B.G. and L.K. performed the experiments. A.E.F. and D.M.
designed the optimization. D.M. conducted the numerical calculations. A.E.F.,
D.M., and K.B. wrote the paper. K.B. supervised the research. Funding:
Research was supported by the NSF grants DMR-2011754, DMR-1922321 and
EFRI-1741685. A.E.F. acknowledges that this project has received funding from
the European Union’s Horizon 2020 research and innovation program under the
Marie Skłodowska-Curie grant agreement No 798244. Competing interests: The
authors declare no conflict of interest. Data and materials availability: All
data needed to evaluate the conclusions of this study are present in the paper
or the Supplementary Materials.

## Supplementary materials

Section S1. Fabrication
Section S2. Testing
Section S3. Model
Section S4. Optimization
Figure S1. 3D-printed origami modules
Figure S2. Multi-unit actuator fabrication and assembly
Figure S3. Single pressure input origami robot
Figure S4. Experimental setup of the inflation with water
Figure S5. Experimentally measured response of the $\Delta=3$ mm module
Figure S6. Experimental pressure, displacement, and bending angle curves of
our origami modules
Figure S7. Effects of $\Delta$
Figure S8. Modeling the stable states and snapping transitions of our origami
modules
Figure S9. Actuator’s parameters
Figure S10. State diagrams
Figure S11. A $2$-units actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textbackslash\textbackslash
3]$.
Figure S12. Design space for actuators with $n\in\\{1,2,3,4\\}$
Figure S13. Greedy algorithm
Figure S14. Comparison between optimization algorithms with integer
constraints
Figure S15. Greedy algorithm results
Figure S16. A $6$-units actuator reaching three targets
Figure S17. The $12$-units actuator with additional constraint
Figure S18. Random targets error
Figure S19. Optimal number of units as a function of the target radius
Movie S1. Fabrication
Movie S2. 2-units actuators
Movie S3. 12-units actuator, triple target objective
Movie S4. Robotic application

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## Supplementary Information

## S1. Fabrication

The actuators tested in this study are constructed by connecting 3D-printed
origami modules. This section gives details on the design and fabrication of
the 3D-printed modules as well as their assembly to create multi-unit
actuators and robots.

### Design of the 3D-printed origami units

Each origami module is 3D-printed using a commercially available multi-
material printer (Ultimaker $3$). To account for geometric incompatibility
during inflation, we print the $1$-mm thick triangular facets out of compliant
thermoplastic polyurethane (Ultimaker TPU 95A with tensile modulus $E=26$
MPa). The thickness decreases to $0.4$ mm at the junction of the triangular
facets (hinges), to allow more compliance. This value is the lowest possible
thickness our printer is able to print with a $0.4$ mm print core. Further, to
enable rigid connection of different units and increase bistability, we print
the end caps as well as the four triangular facets of the bistable cell out of
stiff polyactic acid (Ultimaker PLA with tensile modulus $E=2.3$ GPa). As
shown in Fig 5, the single module consists of two hexagonal caps with edges of
length $l=30$ mm, separated by a distance $h=24$ mm, and rotated by an angle
$\alpha=30^{\circ}$ with respect to each other. To enable coupling of
different units, we print a screw and a threaded hole on the top and bottom
surfaces with length $w=6$ mm an thread size $d_{T}=24$ mm.

Figure 5: 3D-printed origami modules. Isometric and projected views of the
origami module.

### Assembly of an inflatable multi-unit actuator

Below are the eight steps needed to fabricate and assemble an inflatable
origami actuator sample made of multiple modules (see Fig. 6 and Movie S1):

  * •

Step 1: we 3D-print (Ultimaker $3$) each origami unit out of polyactic acid
(Ultimaker PLA) and thermoplastic polyurethane (Ultimaker TPU 95A), using
$0.4$ mm print cores with the fine default setting.

  * •

Step 2: we cut the 3D-printed adhesion skirt with scissors.

  * •

Step 3: we remove the 3D-printed support material inside of the origami unit
with pliers.

  * •

Step 4: we insert a toric joint on the connection screw to make the unit
airtight (see Steps 5-8).

  * •

Step 5: we assemble multiple units together through the connection screws
ensuring a tight assembly through the toric joints inserted in Step 4.

  * •

Step 6: we coat the sample with a $0.5$ mm layer of polydimethylsiloxane
(PDMS) and let it cure for $24$ hours.

  * •

Step 7: we fit end caps, making sure to have a tight assembly through the
toric joints inserted in Step 4, to create an airtight cavity. Note that one
of the end caps has an inlet for fluidic actuation.

  * •

Step 8: we test the origami actuator by connecting it to a fluidic supply.

Figure 6: Multi-unit actuator fabrication and assembly. Snapshots of the eight
steps required to fabricate and assemble inflatable multi-unit origami
actuators.

### Design of the single pressure input origami robot

The origami robot is constructed by mounting two multi-unit actuators (each
one acting as a crawling arm) on a wheeled chassis made of LEGO bricks (see
Figs. 7a and b). To create a single airtight cavity, we connect the two multi-
unit actuators through a fluidic line (1/16" soft plastic tubing and luer lock
connectors) and a coupler (1/16" acrylic plate fixed on the chassis). Finally,
we glue $10$ cm-long amplifiers (1/4" acrylic plate) on the top caps of the
actuators to accentuate the stroke during inflation and deflation. Note that
we add a patch of silicon rubber on the amplifiers’ tip (ecoflex 5 from
Smooth-on) to increase the friction coefficient when in contact with the
ground.

Figure 7: Single pressure input origami robot. (a) The two identical actuators
forming the arms of the robot. (b) Schematics of the origami robot. (c)
Components of the origami robot. (d) Assembled origami robot.

## S2. Testing

To characterize the experimental response of the fabricated origami building
blocks, we inflate them with water— to eliminate the influence of air
compressibility— and measure their internal pressure while tracking their
height and bending angle. As shown in Fig. 8, we use a syringe pump (Pump
33DS, Harvard Apparatus) to displace water into the origami unit at $10$
mL/min, measure the pressure using a pressure sensor (ASDXRRX015PDAA5 with a
measurement range of $\pm 15$ psi by Honeywell), and track the height, bending
angle, and twist angle of the upper cap using two digital cameras (front and
top view with two SONY RX100 V). Note that we submerge the entire unit in a
water tank to eliminate the influence of gravity while eliminating air from
all supply tubes and calibrate the pressure sensor to atmospheric pressure
before each measurement cycle.

Figure 8: Experimental setup of the inflation with water. Schematic of the
test setup used to characterize the bending angle vs./ pressure and height
vs./ pressure curves of the origami units with (1) syringe pump, (2) pressure
sensor, (3) water tank, (4) origami unit, and (5) digital cameras.

We report in Fig. 9 the experimentally measured pressure, $p$, as a function
of the end caps displacement, $u_{z}$, and the bending angle, $\theta$, for
the module with $\Delta=3$mm. Further, we highlight on the curves the pressure
thresholds, $p_{3}^{+}$ and $p_{3}^{-}$, as well as the maximum displacement,
$u_{z}^{max,\pm}$, and bending angle, $\theta_{max,\pm}$, at snap-through for
both the inflation and deflation regime.

Figure 9: Experimentally measured response of the $\Delta=3$ mm module. (a)
State diagram of the pressurized origami module. (b) Experimentally measured
pressure, $p$, as a function both end caps’ displacement, $u_{z}$, and bending
angle, $\theta$ for the origami module with $\Delta=3$ mm. In the plot, we
highlight the pressure thresholds, $p_{3}^{+}$ and $p_{3}^{-}$, as well as the
maximum displacement, $u_{z}^{max,\pm}$, and bending angle,
$\theta_{max,\pm}$, at snap-through for both inflation and deflation.

We summarize all experimental results in Fig. 10 and plot the pressure,
displacement, and bending angle of each design tested in this study as a
function of normalized time, $T$, for both the inflation and deflation regime.
To validate repeatability, we test for each design three specimens and report
the mean (solid lines) and standard deviation (shaded region). We find that
the classic Kresling module as well as the modified-panel module with
$\Delta=0$ mm do not show any snap-through instability. Modules with
$\Delta=1$ mm show a snapping transition during inflation, but bend marginally
when later deflated. Modules with $\Delta=2$, $3$, $4$ mm are bistable as they
exhibit discontinuity in their $p-T$, $u_{z}-T$, and $\theta-T$ curves, and
show substantial bending when deflated from the snapped configuration. Modules
with $\Delta=5$ mm break during the inflation before the panel snaps outward.

Figure 10: Experimental pressure, displacement, and bending angle curves of
our origami modules. . Experimentally measured pressure (left vertical axis),
end caps displacement (right vertical axis), and bending angle (right vertical
axis) of each design tested in this study as a function of normalized time,
$T=t/t_{end}$ (where $t$ is real time and $t_{end}$ the duration of the test),
for both (a) inflation and (b) deflation. To validate repeatability, we test
for each design three specimens and report the mean (solid lines) and standard
deviation (shaded region). We find that the classic Kresling module as well as
the modified-panel module with $\Delta=0$ mm do not show any snap-through
instability. Modules with $\Delta=1$ mm show a snapping transition during
inflation, but bend marginally when later deflated. Modules with $\Delta=2$,
$3$, $4$ mm are bistable as they exhibit discontinuity in their $p-T$,
$u_{z}-T$, and $\theta-T$ curves, and show substantial bending when deflated
from the snapped configuration. Modules with $\Delta=5$ mm break during the
inflation before the panel snaps outward.

From the experimental curves, we can extract the pressure thresholds at which
the modified panel snaps outward during inflation, $p_{\Delta}^{+}$ and inward
during deflation, $p_{\Delta}^{-}$. We summarize the mean and standard
deviation of these pressure thresholds in Fig. 11 for the bistable modules
with $\Delta=2$, $3$, and $4$ mm. We also report in Fig. 11, the displacement
and the bending angle at snap-through, $u_{z}^{max,\pm}$ and
$\theta_{max,\pm}$ for both the inflation and deflation regime.

Figure 11: Effects of $\Delta$. Bar chart of the experimentally measured
pressure thresholds, $p_{\Delta}^{\pm}$, end caps displacement,
$u_{z}^{max,\pm}$, bending angle, $\theta^{max,\pm}$, and twist angle,
$\phi^{max,\pm}$, at snap-through during both inflation (a) and deflation (b)
The values for $p_{\Delta}^{\pm}$, $u_{z}^{max,\pm}$, and $\theta^{max,\pm}$
are extracted from the curves shown in Fig. 10. The values for
$\phi^{max,\pm}$ were obtained measuring the twist angle at each stable
configuration and snapping transition during the experiments.

## S3. Model

### Origami module

We start by using the geometric quantities that we track in our experiments
(shown in Fig. 11) to reconstruct the configuration of the origami module just
before and just after each snapping transition (see Fig. 12).

Figure 12: Modeling the stable states and snapping transitions of our origami
modules. (a) Typical state diagram for a origami module with a modified panel
of depth $\Delta$. The unit can transition between two stable states: state
$s^{0}$ when the modified panel is folded inward, and state $s^{1}$ when the
panel is popped outward and stays in that position even when the input
pressure is removed (b-g) Reconstructed geometry of the origami module at each
stable state and before and after each snapping transition.

### Actuator comprising $n$ modules

We can create actuator made of $n$ units by simply combining the different
stable states and snapping transitions found for the single unit model
described in Fig. 12. Note that we impose that any $n$-units actuator forms a
closed, inflatable cavity (i.e. they are all subjected to the same internal
pressure). By combining $n$ modules, we can construct $\left(3\times 2\times
6+1\times 2\right)^{n}$ different actuators since for each module $k$ we can
select ($i$) either a regular Kresling module or a unit comprising a modified
panel with depth $\Delta^{k}\in\\{2,3,4\\}$ mm; ($ii$) the upper cap to be
rotated clockwise or anticlockwise with respect to the bottom one,
$c^{k}\in\\{\textrm{//},\textrm{\textbackslash\textbackslash}\\}$, and ($iii$)
the side on which the modified panel is located, $f^{k}\in\\{1,\ldots,6\\}$
(see Fig. 13).

Figure 13: Actuator’s parameters. Our reconfigurable actuators are defined by
the modified panel’s depth, $\Delta^{k}$, the initial rotation of the upper
cap, $c^{k}$, and the position of the modified panel, $f^{k}$, of each their
$k$ unit.

For an actuator made of $n$ units, the number of stable states is equal to
$2^{n_{\Delta}}$, where $n_{\Delta}$ is the number of unique modified panel
depths $\Delta$. Note that we assume all units with the same $\Delta$ snap
synchronously at the pressure thresholds, $p_{\Delta}^{+}$ and
$p_{\Delta}^{-}$. Since in this study, we consider only the discrete set
$\Delta\in\\{2,3,4\\}$ mm, all our actuators have either $n_{\Delta}=0$, $1$,
$2$, or $3$. For each different $n_{\Delta}$, we report the corresponding
state diagram in Fig. 14.

Figure 14: State diagrams. (a) State diagram for any $n$-units actuator made
only of Kresling modules. (b-d) State diagram for any $n$-units actuator made
with $1$, $2$, and $3$ unique modified panel’s depth, i.e. $n_{\Delta}=1$,
$2$, and $3$, respectively.

In Fig. 15a we show the state diagram of an actuators comprising two modules
characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textbackslash\textbackslash
3]$ and compare the outputs from the geometric model to the experiments done
on a physical prototype. Further, for each of the stable states and snapping
transitions in the state diagram, we record the vector connecting the caps’
centroids, $\mathbf{d}$. In Fig. 15b, we report the normalized deployment,
$||\mathbf{d}||/h$, and angle, $\theta_{act}$ for each stable state and
snapping transition of the actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textbackslash\textbackslash
3]$. Further, in Fig. 16 we report $||\mathbf{d}||/h$ and $\theta_{act}$ for
each stable state and snapping transition of every actuator design with
$n\in{1,2,3,4}$.

Figure 15: A $2$-units actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textbackslash\textbackslash
3]$. (a) State diagram for a $2$-units actuator characterized by
$[\Delta^{1}c^{1}f^{1};\Delta^{2}c^{2}f^{2}]=[2\textrm{//}3;4\textbackslash\textbackslash
3]$: geometrical model prediction on the left, experimental snapshots on the
right. (b) Normalized deployment, $||\mathbf{d}||/h$, as a function of the
bending angle $\theta_{act}$, for every stable state and snapping transition.

Figure 16: Design space for actuators with $n\in\\{1,2,3,4\\}$. Normalized
deployment, $||\mathbf{d}||/h$, and bending angle, $\theta_{act}$, for all
possible $1$-unit (a), $2$-unit (b), $3$-unit (c), and $4$-unit (d)
constructed by combining our library of modules.

## S4. Optimization

To identify actuators capable of achieving target deformation modes, we solve
the following discrete optimization problem

$\displaystyle\min_{\Delta^{k},c^{k},f^{k}}$
$\displaystyle\Psi\left(\Delta^{k},c^{k},f^{k};n\right)$ (2) s.t.
$\displaystyle\Delta^{k}\in\\{2,3,4\\}\textrm{mm}$ $\displaystyle
c^{k}\in\\{\textrm{//},\textrm{\textbackslash\textbackslash}\\}$
$\displaystyle f^{k}\in\\{1,2,3,4,5,6\\}$ $\displaystyle
k\in\\{1,2,\ldots,n\\}$ $\displaystyle n\in\mathcal{Z}^{+},$

where $\Psi$ is the cost function, $n$ is the number of units making the
actuator and $\Delta^{k}$, $c^{k}$, and $f^{k}$ are are the modified panel
depth, the orientation of the upper cap with respect to the bottom one, and
the side on which the modified panel is located for the $k$-unit in the array.
Note that all design variables (i.e. $\Delta^{k}$, $c^{k}$, and $f^{k}$) are
constrained to be integer value and, for the sake of simplicity, we solve the
optimization problem multiple times for fixed number of units $n\in[1,15]$.

In the main text we use the optimization algorithm to identify actuators whose
tip can approach a desired set of target points and therefore define the cost
function as

$\Psi=\frac{1}{n_{targets}\cdot
h}\sum_{m=1}^{n_{targets}}\min||\mathbf{d}-\mathbf{T}_{m}||,$ (3)

where $n_{targets}$ is the number of targets, and $\mathbf{T}_{m}$ is the
vector connecting the $m$-th target with the origin. Here, we also consider
two additional optimization problems and use the algorithm to identify
actuators that

  * •

maximize the angle $\theta_{act}$ between the vector connecting the cap’s
centroids, $\mathbf{d}$, and the $z$-axis. For this case we define the cost
function as

$\Psi\left(\Delta^{k},c^{k},f^{k};n\right)=-\theta_{act}.$ (4)

  * •

maximize the deployment height $||\mathbf{d}||/h$. For this case we define the
cost function as

$\Psi\left(\Delta^{k},c^{k},f^{k};n\right)=-||\mathbf{d}||/h,$ (5)

### Optimization algorithms

There are many algorithms able to solve an optimization problem with integer
constraints such as the one presented in Eq. [2]. In this study, we used three
classic algorithms: ($i$) the genetic algorithm with integer constraints;
($ii$) the integer optimization via a surrogate model; and ($iii$) the greedy
algorithm based on best-first search. Note that given the high-dimensionality
and complexity of this optimization problem, there is no guarantee that these
algorithms will lead to a unique global minimum.

#### Genetic algorithm with integer constraints

We started by using the genetic algorithm with integer constraints, which
attempts to minimize a penalty function that depends on the fitness (value of
the cost function $\Psi$) and feasibility (design variables are integer) of an
individual. For this study, we used the Matlab implementation of the algorithm
(Matlab function ga) and imposed the constraint that all design variables,
i.e. $\Delta^{k}$, $c^{k}$, and $f^{k}$ must have integer values with upper
and lower bounds reported in Eq. [2]. We ran the function ga multiple times,
each time considering a fixed value of $n\in[1,15]$, using a population size
of $200$, a max stall generations (i.e. the consecutive number of generations
with no change to the cost function value) of $500$ and a maximum number of
generations of $1000$.

#### Integer optimization via a surrogate model

Next, we used the surrogate model optimization, which is a derivative-free
method that replaces the complex and non-smooth objective function by a
surrogate (i.e. an approximation of that function), which is created by
sampling the objective function. For this study we used the Matlab
implementation of the algorithm (Matlab function surrogateopt) and imposed the
constraint that all design variables, i.e. $\Delta^{k}$, $c^{k}$, and $f^{k}$,
must have integer values with upper and lower bounds reported in Eq. [2]. We
ran the algorithm for fixed values of $n$ with a maximum number of function
evaluations of $20,000$.

#### Greedy algorithm based on best-first search

For the greedy algorithm, we developed and in-house Matlab code based on the
best-first search method that creates an actuator with $n$ units out of
$n_{s}$ super-cells, each super-cell made of $n_{u}$ modules (so that
$n=n_{u}\cdot n_{s}$). At the first iteration, the algorithm selects the
super-cell design that minimizes $\Psi$ and stores it in memory. Then, in the
second iteration, we identify a second super-cell that, when connected to the
first one, minimizes $\Psi$. The first two super-cells are then stored in
memory and the algorithm advances to the next iteration (see Fig. 17 and
Algorithm 1 below).

Figure 17: Greedy algorithm. Schematic of the greedy algorithm with $n_{u}=3$.
At each iteration, the algorithm selects the actuator super-cell design that
minimizes $\Psi$. Algorithm 1 Greedy algorithm based on best-first search

Set $n_{max}$;
Set $n_{u}$;
Set $n_{s}=0$;
While $n_{s}\cdot n_{u}\leq n_{max}$
$n_{s}=n_{s}+1$;
if $n_{s}=1$ then
$\cdot$ Calculate $\Psi$ for each actuator design with
($\Delta^{k},c^{k},f^{k}$), $k=1:n_{s}\cdot n_{u}$;
$\cdot$ Find the actuator that minimizes $\Psi$ and set its design variables
to ($\Delta^{k^{*}},c^{k^{*}},f^{k^{*}}$)
else
$\cdot$ Calculate $\Psi$ for each actuator design with
($\Delta^{k},c^{k},f^{k}$), where the set of variables from
$k=1:(n_{s}-1)\cdot n_{u}$ are coming
from the previous iteration of ($\Delta^{k^{*}},c^{k^{*}},f^{k^{*}}$) and
$k=(n_{s}-1)\cdot n_{u}+1:n_{s}\cdot n_{u}$ are free.
$\cdot$ Find the actuator that minimizes $\Psi$ and set its design variables
to ($\Delta^{k^{*}},c^{k^{*}},f^{k^{*}}$)

### Results

In the following we first compare the performance of the three algorithms and
then present additional results obtained using the greedy algorithm.

#### Comparison between the three algorithms

To test and compare the three algorithms, we considered the set of three
targets ($T_{1}$, $T_{2}$, $T_{3}$) shown in Fig. 3 of the main article and
minimized the cost function given in Eq. [3]. In Fig. 18, we report the cost
function value with respect to the number of generations/function evaluations
as obtained using the three algorithms. We find that for all considered values
of $n$ both the genetic algorithm with integer constraints and the surrogate
model stall quickly, with a minimum value of the cost function of $1.04$ and
$1.12$ reached for $n=15$, respectively. Further, we find that the greedy
algorithm with $n_{u}=3$ outperforms the genetic algorithm and the surrogate
model optimization as it identifies an actuator design that leads to
$\Psi=0.729$ for $n=12$. Note that, for $n_{u}=3$, the greedy algorithm
requires about $2.75\times 10^{5}$ evaluations of $\Psi$ to identify the
optimal design, whereas the surrogate model takes about $1\times 10^{5}$
evaluations of $\Psi$ (the genetic algorithms requires about $8\times 10^{5}$
evaluations of $\Psi$ with a population size of $200$). However, the greedy
algorithm does not require any other operation apart from a simple computation
of $\Psi$ during each iteration. Differently, the surrogate algorithm has to
update the underlying model. The simplicity of the greedy algorithm leads to a
CPU time of $850$ s (parallelized on $24$ cores) to solve the algorithm
compared to $2,500$ s and $4,000$ s for the genetic algorithm and the
surrogate model, respectively. We therefore use the greedy algorithm to
identify optimal configurations for our actuators.

Figure 18: Comparison between optimization algorithms with integer
constraints. Comparison of the (a) generic algorithm, (b) surrogate model, and
(c) greedy algorithm based on best-first search to solve the integer
optimization problem of minimizing the targets error $\Psi$.

#### Additional results generated by the greedy algorithm

In Fig. 19 we present results obtained when the greedy algorithm is programmed
to maximize $\theta_{act}$ (Fig. 19a-c) and $||\mathbf{d}||/h$ (Fig. 19d-f).
Note that we present present results for three different values of $n_{u}$
(i.e. $n_{u}=1,$ 2 and 3).

Then, in Fig. 20 we consider a set of three targets $(T1,T2,T3)$ different
from that included in the main text and present the results for the optimal
design identified by the greedy algorithm. Next, in Fig. 21 we show the
inverse design of an actuator reaching the same set of three targets
considered in Fig. 3 of the manuscript, but with the additional constraint
that the targets much be reached successively by decreasing pressure.

Finally, in Fig. 22 we show how the minimum value of $\Psi$ found by the
greedy algorithm varies with the number of targets, $n_{targets}$, and the
units forming a super-cell, $n_{u}$, and in Fig. 23 how the target radius
(i.e. the radius of the sphere fitted with the targets) influences the optimal
number of units of the actuator.

Figure 19: Greedy algorithm results. Normalized deployment,
$||\mathbf{d}||/h$, and bending angle, $\theta_{act}$, for actuators with
$n_{s}\in\\{1,2,3,4,5\\}$ and $n_{u}\in\\{1,2,3\\}$. (a-c) The greedy
algorithm is programmed to maximize $\theta_{act}$ at each increment of
$n_{s}$. (d-f) The greedy algorithm is programmed to maximize
$||\mathbf{d}||/h$ at each increment of $n_{s}$.  Figure 20: A $6$-units
actuator reaching three targets. For a specific set of three targets, we use
the greedy algorithm to find the actuator design that minimizes $\Psi$, i.e
the error between the targets and the top cap’s centroid. Note that we fix
$n_{u}=3$ and consider $n_{s}\in\\{1,2,3,4,5\\}$. (a) Targets error $\Psi$ as
a function of total number of units: the optimal actuator produced by the
greedy algorithm for the three targets is reported as (*), along with the
respective parameters for each module. The considered set of targets is shown
in the inset. (b) State diagram for the $6$-units actuator (*) with targets
$T1$, $T2$, and $T3$ highlighted. (c) Top and 3D view of the model and the
experimental prototype for the $6$-units actuator reaching the targets.
Figure 21: The $12$-units actuator with additional constraint. We focus on the
same set of three targets considered in Fig. 3 of the manuscript and further
impose that each targets much be reached successively by decreasing pressure.
(a) Targets error $\Psi$ as a function of total number of units: the optimal
actuator produced by the greedy algorithm for the three targets is reported as
(*), along with the respective parameters for each module. (b) State diagram
for the optimal $12$-units actuator (*) with targets $T1$, $T2$, and $T3$
highlighted. (c) Top and 3D view of the model for the $12$-units actuator
reaching the targets. Figure 22: Random targets error. For a random set of
$n_{targets}$, we use the greedy algorithm to find the actuator design that
minimizes the target error $\Psi$. Note that each target is bounded by a cubic
domain centered with the actuator and norm equal to $1/2(n_{u}\cdot n_{s})$ to
ensure that it is within reach. We report the average error based $1000$
simulations with $n_{targets}\in\\{1,2,3,4\\}$ and $n_{u}\in\\{1,2,3\\}$. Note
that for each set of targets, we fix $n_{u}$ and choose the $n_{s}$ (with the
constraint that $n=n_{u}\cdot n_{s}\leq 15$) that minimizes the error. Figure
23: Optimal number of units as a function of the target radius. For $100$
random sets of $n_{targets}=3$, we measure the target radius, $R_{targets}$,
i.e. the radius of the sphere fitted with the targets, and we use the greedy
algorithm to find the number of units that minimizes the error between the
target and the top cap’s centroid. We report here the target radius normalized
by the module height, $R_{targets}/h$, as a function of the optimal number of
units $n$ found by the greedy algorithm. We find that the average target
radius increases with the number of units to minimize the error $\Psi$.