RESEARCH ARTICLE
www.advmat.de

Programming 3D Curves with Discretely Constrained
Cylindrical Inﬂatables
Robert Baines,* Sree Kalyan Patiballa, Benjamin Gorissen, Katia Bertoldi,
and Rebecca Kramer-Bottiglio*
and solving for the design parameters, thatis geometric and/or material parameters,
required to achieve this deformation.[4–9] Inverse design can reduce prototyping time,
material waste, and results in more performant soft systems. Many inverse inﬂatable models have focused on systems composed of materials with a high elastic modulus (making them quasi-inextensible at operating pressure). This hardware choice reduces the modeling challenge yet limits thescope of achievable deformations.[10–13]
Inﬂatables composed of hyperelastic
material can accommodate large strains,disposing them to applications requiringradical shape changes or large forces.[14–19]
Hyperelastic inﬂatables commonly incorporate stiﬀer material strain limiters,
including mesh, directional ﬁbers, slottedshells, or varying-modulus rubber on
their surface to direct deformations.[20–23] Simpler deformations
like contraction or extension are well-studied and can be designed
a priori with reduced force or energy-based models.[24–26] Models
like these assume homogeneous deformations throughout thesystem, induced by a single strain limiter on an inﬂatable undergoing typically small (<40%), planar, and axisymmetric deformations. Similarly, models for speciﬁcation of curvatures are capableof programming only planar curves and have been demonstrated
with unidirectionally extending cylinders.[27]Deformable systems that can inﬂate into non-homogeneous
3D forms open a more expansive application space. In thesemore complex emerging systems, the energetic interplay between locally stretched surface regions may give rise to topography with applications in mechanical camouﬂage and precision haptic devices,[28,29] or sinuous 3D curves with applications
in continuum manipulation.[30] Localized strain-limiting entails
boundary eﬀects arising from material compliance mismatchthat raise the required ﬁdelity of a model to capture the deforming inﬂatable’s mechanical response.
Finite element (FE) analysis has been shown to reliably predict
the behavior of 3D inﬂating continua outﬁtted in discrete strain
limiters.[31] FE simulation has recently been used for inverse design using neural networks to produce surfaces that inﬂate toprogrammed 3D topography.[32] Design strategies reliant entirely
on machine learning however suﬀer from high computational
expense, which can preclude their unaccompanied implementation in large-scale inverse design problems or transferability tosystems composed of diﬀerent materials.

Programming inﬂatable systems to deform to desired 3D shapes opens up
multifarious applications in robotics, morphing architecture, andinterventional medicine. This work elicits complex deformations by attaching
discrete strain limiters to cylindrical hyperelastic inﬂatables. Using thissystem, a method is presented to solve the inverse problem of programming
myriad 3D centerline curves upon inﬂation. The method entails two steps:
ﬁrst, a reduced-order model generates a conceptual solution giving coarse
indications of strain limiter placement on the undeformed cylindricalinﬂatable. This low-ﬁdelity solution then seeds a ﬁnite element simulation
nested within an optimization loop to further tune strain limiterparameters. We leverage this framework to achieve functionality through apriori programmed deformations of cylindrical inﬂatables, including 3D curve
matching, self-tying knotting, and manipulation. The results hold broad
signiﬁcance for the emerging computational design of inﬂatable systems.

1. Introduction
The potential of inﬂatable systems to assume complex shapes
and interact adaptively with their environment has propelledthem to the forefront of next-generation soft robotics, deployable structures, and medical devices.[1–3] Critical interest
has thus emerged regarding the inverse design of inﬂatables
and other shape-morphing systems—specifying a deformation,
R. Baines, S. K. Patiballa, R. Kramer-Bottiglio
School of Engineering and Applied Sciences
Yale University
CT 06511New Haven, USA
E-mail: robert.baines@yale.edu; rebecca.kramer@yale.edu
S. K. Patiballa
Department of Mechanical Engineering
The University of Alabama
AL 35487Tuscaloosa, USA
B. Gorissen, K. Bertoldi
J. A. Paulson School of Engineering and Applied Sciences
Harvard University
MA 02138Cambridge, USA
B. Gorissen
Department of Mechanical Engineering
KU Leuven
Celestijnenlaan 300, Leuven3001, Belgium
The ORCID identiﬁcation number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/adma.202300535

DOI: 10.1002/adma.202300535

Adv. Mater. 2023, 2300535

2300535 (1 of 6)© 2023 Wiley-VCH GmbH

www.advmat.deFigure 1. Adhering discrete strain-limiting patches to the surface of hyperelastic inﬂatable cylinders gives rise to complex and functional forms.
A) Several inﬂatables fabricated using this technique are shown. B) Imageof prototype inﬂatable with patches that inﬂuence curvature 𝜅 and torsion
𝜏 in extremely non-linear ways based on their dimensions, orientation, andplacement along the continuum surface. The zoomed inset shows howeven a localized 1 mm-wide, 4 cm-long patch can produce curvature that
substantially changes the global morphology of an inﬂatable at the meterscale (Inset scale bar: 2 cm; Image scale bar: 15 cm). We propose an inverse design pipeline to yield strain limiter design parameters for an uninﬂated cylinder such that when inﬂated, its centerline matches a user-input
space curve.

Despite recent progress, there is still a dearth of inverse design models for 3D discretely constrained inﬂatables, which acts
as a bottleneck to adopting multifunctional soft systems in critical application spaces. This fact motivates a framework capableof capturing nuanced mechanics for solution accuracy yet able togenerate conceptual solutions in reduced time. Here, we present
such a framework and evaluate it by focusing on a useful, albeit
relatively unexplored case of elongated inﬂated cylinders clad in
discrete strain-limiting “patches.” The number, dimensions, locations, and orientations of patches placed on the cylinder may betuned to coerce myriad space curves (Figure 1, Video 1, Supporting Information), with applications ranging from shape matching, manipulation, navigation, to self-tying knots. Our inverse
method solves for patch design parameters to be placed on an
uninﬂated cylinder such that the centerline of the inﬂated cylinder closely approximates a user-input space curve.
Starting with a user-input space curve ( ∈ ℝ3 ), we use kinematics to generate a conceptual solution for strain limiter placement and heights on the uninﬂated geometry. Next, the conceptual solution seeds an FE simulation nested within an optimization structure that tunes strain limiter placement and dimensions. The conceptual solution emerges from a reduced-ordermodel that provides intuition about the mechanics that govern
elongate hyperelastic inﬂatables clad in discrete strain limiters;
furthermore, seeding the FE optimization loop with this conceptual solution reduces the design search space, facilitating fasterconvergence on valid parameters than using FE alone.

2. Results and Discussion
Rectangular strain-limiting patches i = 1 … n with dimensions Hi
× Wi are placed on an undeformed cylindrical inﬂatable of length
L at positions [Xi ,Yi ] (on the developable surface), and angles withrespect to the longitudinal axis Θi (Figure 2). Upon inﬂation, theinﬂatable undergoes large extensional and radial stretches while

Adv. Mater. 2023, 2300535

Figure 2. Visual depiction of inverse design pipeline, starting with a target
centerline going into kinematic segmentation algorithm, the solution ofwhich is in turn used to seed a FE simulation nested within an optimization
loop. The output of this pipeline are strain limiter design parameters tomatch a target centerline.

the patches inﬂuence curvature (𝜅) and torsion (𝜏) of the centerline parameterized by arclength t (see Figure S1, Supporting
Information, for fabrication of inﬂatables and a typical pressure–volume curve). Herein, we strive to model the relationship between patch placement and 𝜅(t) and 𝜏(t) to systematically program curves.
The inﬂation of patch-clad cylindrical inﬂatables is characterized by multiple snap-through instabilities. The ﬁrst of these instabilities is associated with the inﬂatable reaching a critical pressure and developing an initial bulge.[33] Subsequent snap-troughs
arise from overcoming local stiﬀness discontinuities due to thepresence of patches.[34] Once the air bulge has propagated to thecylinder’s distal tip, the inﬂatable assumes its fully inﬂated shape.
This state of static equilibrium during the ﬁrst inﬂation cycle, inwhich curvature caused by patches is fully “activated,” is the concern in this work (see Note S1, Supporting Information, for morediscussion on inﬂation mechanics of patch-clad inﬂatables).
2.1. Kinematic Segmentation Algorithm
The ﬁrst part of our inverse design pipeline is a conceptual solution, enabled by a reduced-order model, which we dub "kinematic segmentation algorithm" (KSA). Given a target curve ,
KSA quickly generates coarse estimates of n, ([Xi ,Yi ]), Hi , andΘi along an uninﬂated cylinder, overlooking complexities arisingfrom material properties. The advantage of KSA is twofold: it provides insight into the mechanics of the deforming cylinder and an
initial guess for the FE optimization loop to reduce convergencetime due to the highly non-convex search space.
In KSA, ﬁrst, a discrete Frenet–Serret triad consisting of thē normal N,̄ and binormal B̄ triad, is calculated for .
tangent T,
̄We then identify where curvature (𝜅 = ||dT∕dl||)
exceeds a predeﬁned threshold 𝜅 tol (Note S2, Supporting Information). If apoint exceeds 𝜅 tol , we ascribe that point along the arclength to2300535 (2 of 6)© 2023 Wiley-VCH GmbH

15214095, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300535 by Yale University, Wiley Online Library on [27/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com

www.advmat.dehave a patch. Close-by segments within a speciﬁed tolerance are
merged, and the axial midpoints of the resulting sections deemed
to have a patch are calculated, indicating the axial midpoint ofa patch in the deformed conﬁguration, yi . Knowing the initiallength of an uninﬂated cylinder and the length of a target centerline (Lf ), we calculate the axial stretch 𝜆 = Lf /L and then ﬁnd
the patch axial midpoint in the reference conﬁguration Yi = yi /𝜆.
Hi is estimated as the length of the segments that exceed 𝜅 tol .
We estimate Xi using the angle of twist of B̄ along the curve up
to a midpoint of a segment. Similarly, we estimate Θi by calculating the angle of twist of B̄ through a segment and assuming
the patch angle in the undeformed conﬁguration maps directly
to the centerline twist it exerts in the deformed conﬁguration. Although it rapidly yields a conceptual solution, KSA alone does
not furnish a complete solution because it does not estimate thewidth of the patches Wi . The inﬂuence of patch parameters on 𝜅and 𝜏 depends on the patch’s material properties, and the boundary eﬀects between the patch and the inﬂated cylinder. Both ofthese considerations require a model beyond the simpliﬁed kinematic representation.
2.2. Finite Element Model
An FE simulation nested within an optimization loop constitutes
the second part of the inverse design pipeline. The FE simulations (which were conducted using the commercial package
ABAQUS 2020/Explicit) account for the mechanical boundaryeﬀects and material properties omitted from KSA through nonlinear elasticity; they model the cylindrical inﬂatable and patches
as isotropic membrane elements governed by hyperelastic constitutive laws (Note S3, Supporting Information). We performed
forward validations to ensure the accuracy of the FE model (NoteFigure S2 and S4, Supporting Information).
In the inverse FE problem, a shape-matching objective is formulated as a minimization of the squared diﬀerence between target and simulation torsion and curvature:min

Xi ,Yi ,Hi ,Wi ,Θi()2
𝜅(t) − 𝜅(t)∗ + (𝜏(t) − 𝜏(t)∗ )2()
s.t : Lbi ≤ Xi , Yi , Hi , Wi , √i ≤ Ubi

(1)
(2)

Here, * denotes the target values. An optimal solution is computed by searching the design space while satisfying the constraints (bounds on reasonable sizes of patches, denoted as Lbi
through Ubi ; see Note S5, Supporting Information). Due to thenon-convex search space of the optimization problem and thenumber of free variables, we chose an evolutionary algorithm
known as the "co-variance matrix adaptation evolution strategy"(CMA-ES), which is well-suited to such problems.[35]2.3. Assessment of Inverse Design Pipeline
We evaluated the inverse design pipeline with four input curves(Figure 3A–D): Hilbert’s space-ﬁlling curve [36] (which is not traditionally diﬀerentiable, but we use a smooth approximation),
two bespoke 3D curves, and an equation-driven curve describing the contour of a hyperbolic paraboloid (Note S6, SupportingAdv. Mater. 2023, 2300535

Information). We crafted inﬂatables according to the output of
the inverse design pipeline by laser cutting patches and adhering them to the surface of the inﬂatable. Then, after inﬂation, we
took a 3D scan of the inﬂatable from which the centerline was
extracted (Note S7 and Figure S3, Supporting Information).
Results testify that the inverse design pipeline can accurately
generate valid design parameters to approximate a variety ofspace curves with diverse torsion and curvature proﬁles. Due tothe diﬀerent lengths of target and experimental centerlines, weused dynamic time warping [37] to quantitatively evaluate similarity (an explanation of dynamic time warping, as well as its value
for each curve’s torsion and curvature as a function of arclength
are tabulated in Note S6, Supporting Information). Calculations
yielded a value averaged over all curves of 0.5887 for curvature,and 1.4899 for torsion.
The patch-clad inﬂatables’ successful replication of certain target curve features, while diﬃculty in replicating others, elucidates general design considerations for elongate hyperelastic systems clad in discrete, distributed strain limiters. From the similarity metric provided by dynamic time warping, we remark thattorsion approximation was less accurate than curvature approximation. Additionally, sharp peaks in curvature or torsion overrelatively small distances in arclength prove diﬃcult, owing to
the inability of the inﬂatable to produce true inﬂection points ortwist very rapidly. This challenge is embodied in the Hilbert curveof Figure 3A, which does not consistently reach the full extent
of target curvature and torsion. Another apparent challenge ismaintaining constant 𝜅 or 𝜏 over a sizable span of arclength. For
instance, the inﬂatable centerline in Figure 3B does not maintain
constant 𝜅 at each peak. Likewise, Figure 3C has small ﬂuctuations in 𝜅 that prevent it from tracking the smooth dip and riseof the target curve.
We ascertained the performance of the FE simulation alone
and when it was seeded with KSA in generating designs for
Figure 3C. Results show that KSA generates nearby solution
guesses for the patch parameters, limiting the FE search space inthe otherwise rugged energy landscape, and reducing the number of required iterations to converge on designs which minimize
the objective in Equations (1) and (2) (Note S8 and Figure S4, Supporting Information).
2.4. Applications
Equipped with the inverse design framework, we showcase
how an inﬂated shape can be harnessed for robotic functionality beyond mere curve matching. First, we created a self-tying
knot (Figure 4A, Video S2, Supporting Information), an application necessitating both highly curved deformations and nonhomogeneous inﬂation, due to the intersections that are present
when mapping the undeformed to the deformed deformation.
Due to a base-to-tip inﬂation,[38] the inﬂatable experiences no selfintersection until it is deﬂated and the retraction force tightens
the knot. We envision programmable self-tying knots with a single input volume that could be used in medicinal applications, tostaunch blood ﬂow, or in deployable structures to rapidly createarchitectural ﬁxtures.
Additionally, we demonstrated programmable grasping ofvarious objects across diﬀerent length scales using unusual

2300535 (3 of 6)© 2023 Wiley-VCH GmbH

15214095, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300535 by Yale University, Wiley Online Library on [27/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com

www.advmat.deFigure 3. Benchmark target centerlines used in assessing the KSA-FE optimization pipeline. The curves span a variety of diﬀerent curvature and torsion
proﬁles as a function of arclength. For each we show the resulting inﬂatable and a 3D pointcloud of the inﬂatable with a juxtaposed centerline indicating
magnitude of curvature. A) Smooth version of Hilbert’s space-ﬁlling curve. Scale bar: 8 cm. B,C) Hand-drawn curves. Scale bars: 11.5 and 7.5 cm,
respectively. D) Equation that describes the contour of a hyperbolic paraboloid surface. Scale bar: 10 cm.

continuum grasp modes (Figure 4B–E, Video S3, Supporting
Information). Programmed helix diameters or hooking curvatures also enabled the inﬂatable to manipulate multiple objects at
once. For instance, tight helices were able to pinch and lift smalldiameter objects, like the nozzle of a soap dispenser (Figure 4B),
whereas looser helices served as a power grasp to extract and subsequently return objects of greater diameter, like plastic pipes,
in a recycling bin (Figure 4C). Multi-object grasping was accomplished by programming distributed functional torsion and curvature across the inﬂatable length. As examples, with an inﬂatable consisting of a single volume, we ensnared and lifted two
buckets (Figure 4D), or encoded helices and hooks to retrieve disparately shaped objects—a pipe and a metallic ﬁxture—from a
recycling bin (Figure 4E). Ultimately, multiple dissimilar object
grasping is viable with the elongate inﬂatables and holds promisefor robotic manipulation applications.

3. Conclusion
The burgeoning ﬁeld of soft robotics is rife with soft actuation
technologies that expand the bounds of what is possible in robotic
functionality. Applying strain limiters to the surface of soft volumetrically expanding bodies makes it possible to elicit intricate
3D shapes. Despite the ability to easily create them, there is adistinct lack of principled design methods for these highly nonlinear soft deforming bodies.

Adv. Mater. 2023, 2300535

With elongated cylindrical systems clad in discrete strain limiters as a model system, we present an inverse design methodthat elicits strain limiter parameters required to match target 3D
curves. Programming myriad 3D curves is enabled by the proposed modeling pipeline (akin to initial-guess generation techniques [39,40] ) that seeds a high-ﬁdelity FE model with a conceptual solution resulting in a decrease in iterations required toreach desired levels of objective ﬁtness. Moreover, in inverse design, FE can often act as a black box, obfuscating intuition of
the design solution,[41,42] which motivates a reduced-order model
to provide intuition to the designer. We quantitatively evaluatedthe eﬃcacy of shape matching with several benchmark curves ofvarying curvature and torsion proﬁles, and subsequently demonstrated how inﬂation to programmed 3D curves can be harnessed
to interact dexterously in the environment with a single pressurized volume.
There are many exciting avenues related to this study to explore. For instance, a rich design space is available when patch
directional moduli are left as a free parameter in the optimization. Novel shapes could be achievable when the patch geometry
is not constrained to a rectangle but left to vary. Another remaining challenge is to study how trajectories change under dynamic
inﬂation rather than in quasi-static equilibrium manifolds. Lastly,
considering self-contact in the model would expand utility to applications with speciﬁed force requirements. In the interest of reducing prototyping time, material waste, and stepping toward optimal performance, we anticipate such an inverse design method

2300535 (4 of 6)© 2023 Wiley-VCH GmbH

15214095, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300535 by Yale University, Wiley Online Library on [27/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com

www.advmat.deFigure 4. Applications. A single volumetrically inﬂating cylinder clad in discrete strain-limiting patches can accomplish highly dexterous functions.
A) Self-tying knot. B) Grasping and lifting a bottle by a 10 mm diameter nozzle using a tight helix. C) Grasping and lifting a PVC pipe of diameter 40 mm
using a helix with larger radius. D) Simultaneous grasping and lifting of two buckets by exploiting a loop knot. E) Grasping multiple diﬀerently shaped
objects using a combination of enveloping and hinge grasps.

could be applied to various other soft systems where modelinghas proven a challenge: soft robotics, smart structures, or moregenerally, any ﬁeld in which inverse design of 3D inﬂatables may
be applicable (Note S9, Supporting Information).

Supporting Information

4. Experimental Section
To make strain-limiting patches, Spandex (Polyester Lycra/Spandexfour-way stretch fabric LY 902, Paylessfabric) was dredged in Liquid Latex(Kangaroo Monster Liquid Latex) and allowed to cure for 24 h. Patch dimensions were drawn as a DXF ﬁle and cut from the spandex composite
with a laser cutter (Universal Laser Systems, VLS 2.3). Then, rubber cement (BestTest White Paper Cement, Bestine) served to adhere the patch
to the cylindrical inﬂatable (Qualtex 260d balloons), essentially solvent

Adv. Mater. 2023, 2300535

welding the latex-infused spandex patch to the latex surface. To ensurecomplete cure and maximum adhesion waiting was done for 24 h before
inﬂating. Please see Supporting Information for more details about experiments and simulations.

Supporting Information is available from the Wiley Online Library or fromthe author.

Acknowledgements
The authors thank Dr. Gaurav Singh for thought-provoking discussions,and labmates for accommodating guerilla balloon art installations on the2300535 (5 of 6)© 2023 Wiley-VCH GmbH

15214095, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300535 by Yale University, Wiley Online Library on [27/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com

www.advmat.debenchtops for several months. This project was sponsored by the Oﬃce ofNaval Research under award N00014-21-1-2417. Any opinions, ﬁndings,
and conclusions or recommendations expressed in this material were
those of the authors and do not necessarily reﬂect the views of the Ofﬁce of Naval Research. R.B. was supported by the NSF Graduate Research
Fellowship Program (DGE-1752134)

Conﬂict of Interest
The authors declare no conﬂict of interest.

Data Availability Statement
All data needed to evaluate the conclusions in the paper are present
in the paper and/or the Supporting Information. Additional data related to this paper are available from the authors upon request. Design pipeline scripts can be downloaded at https://github.com/thefaboratory/3d_discrete_inﬂatables.

Keywords
inﬂatables, inverse design, programmable matter, soft actuators, soft
robotics
Received: January 17, 2023
Revised: March 17, 2023
Published online:[1] E. Siéfert, M. Warner, Proc. R. Soc. A 2020, 476, 20200047.
[2] E. W. Hawkes, L. H. Blumenschein, J. D. Greer, A. M. Okamura, Sci.
Rob. 2017, 2, eaan3028.
[3] Y.-J. Kim, S. Cheng, S. Kim, K. Iagnemma, IEEE Trans. Rob. 2013, 29,1031.
[4] E. Siéfert, E. Reyssat, J. Bico, B. Roman, Nat. Mater. 2019, 18, 24.
[5] D. Melancon, B. Gorissen, C. J. García-Mora, C. Hoberman, K.
Bertoldi, Nature 2021, 592, 545.
[6] A. S. Gladman, E. A. Matsumoto, R. G. Nuzzo, L. Mahadevan, J. A.
Lewis, Nat. Mater. 2016, 15, 413.
[7] T. J. Jones, E. Jambon-Puillet, J. Marthelot, P.-T. Brun, Nature 2021,
599, 229.
[8] D. Jiao, Q. L. Zhu, C. Y. Li, Q. Zheng, Z. L. Wu, Acc. Chem. Res. 2022,
55, 1533.
[9] H. Aharoni, Y. Xia, X. Zhang, R. D. Kamien, S. Yang, Proc. Natl. Acad.
Sci. USA 2018, 115, 7206.
[10] E. Siéfert, E. Reyssat, J. Bico, B. Roman, Proc. Natl. Acad. Sci. USA
2019, 116, 16692.
[11] M. Skouras, B. Thomaszewski, P. Kaufmann, A. Garg, B. Bickel, E.
Grinspun, M. Gross, ACM Trans. Graphics 2014, 33, 1.
[12] J. Panetta, F. Isvoranu, T. Chen, E. Siéfert, B. Roman, M. Pauly, ACM
Trans. Graphics 2021, 40, 1.

Adv. Mater. 2023, 2300535[13] R. Baines, S. Patiballa, R. Kramer-Bottiglio, in 2021 IEEE 4th Int. Conf.
on Soft Robotics (RoboSoft), IEEE, Piscataway, NJ, USA 2021, pp. 29–
34.
[14] N. El-Atab, R. B. Mishra, F. Al-Modaf, L. Joharji, A. A. Alsharif, H.
Alamoudi, M. Diaz, N. Qaiser, M. M. Hussain, Adv. Intell. Syst. 2020,
2, 2000128.
[15] J. C. Case, E. L. White, R. K. Kramer, Soft Rob. 2015, 2, 80.
[16] A. Miriyev, K. Stack, H. Lipson, Nat. Commun. 2017, 8, 596.
[17] S. A. Morin, R. F. Shepherd, S. W. Kwok, A. A. Stokes, A. Nemiroski,
G. M. Whitesides, Science 2012, 337, 828.
[18] N. G. Kim, J.-H. Ryu, Adv. Intell. Syst. 2022, 5, 2200264.
[19] K. Althoefer, Nat. Rev. Mater. 2018, 3, 76.
[20] C.-P. Chou, B. Hannaford, IEEE Trans. Rob. Autom. 1996, 12, 90.
[21] F. Connolly, C. J. Walsh, K. Bertoldi, Proc. Natl. Acad. Sci. USA 2017,
114, 51.
[22] L. Belding, B. Baytekin, H. T. Baytekin, P. Rothemund, M. S. Verma,
A. Nemiroski, D. Sameoto, B. A. Grzybowski, G. M. Whitesides, Adv.
Mater. 2018, 30, 1704446.
[23] R. V. Martinez, J. L. Branch, C. R. Fish, L. Jin, R. F. Shepherd, R. M.
Nunes, Z. Suo, G. M. Whitesides, Adv. Mater. 2013, 25, 205.
[24] J. Bishop-Moser, S. Kota, IEEE Trans. Rob. 2015, 31, 536.
[25] G. Singh, G. Krishnan, Smart Mater. Struct. 2017, 26, 065024.
[26] A. Sedal, D. Bruder, J. Bishop-Moser, R. Vasudevan, S. Kota, J. Mech.
Rob. 2018, 10, 024501.
[27] G. Singh, G. Krishnan, Soft Rob. 2019, 7, 109.
[28] J. Pikul, S. Li, H. Bai, R. Hanlon, I. Cohen, R. Shepherd, Science 2017,
358, 210.
[29] W. M. van Rees, E. A. Matsumoto, A. S. Gladman, J. A. Lewis, L. Mahadevan, Soft Matter 2018, 14, 8771.
[30] B. Yang, R. Baines, D. Shah, S. Patiballa, E. Thomas, M. Venkadesan,
R. Kramer-Bottiglio, Sci. Adv. 2021, 7, eabh2073.
[31] S. Y. Kim, R. Baines, J. Booth, N. Vasios, K. Bertoldi, R. KramerBottiglio, Nat. Commun. 2019, 10, 1.
[32] A. E. Forte, P. Z. Hanakata, L. Jin, E. Zari, A. Zareei, M. C. Fernandes, L. Sumner, J. Alvarez, K. Bertoldi, Adv. Funct. Mater. 2022, 32,
2111610.
[33] S. Kyriakides, C. Yu-Chung, Int. J. Solids Struct. 1991, 27, 1085.
[34] J. T. B. Overvelde, T. Kloek, J. J. A. D’haen, K. Bertoldi, Proc. Natl. Acad.
Sci. USA 2015, 112, 10863.
[35] N. Vasios, A. J. Gross, S. Soifer, J. T. Overvelde, K. Bertoldi, Soft Rob.
2020, 7, 1.
[36] H. Sagan, Space-Filling Curves, Springer, Berlin, Germany, 1994.
[37] D. J. Berndt, J. Cliﬀord, in AAAIWS’94: Proc. 3rd Int. Conf. on Knowledge Discovery and Data Mining, ACM, New York 1994, pp. 359–370.
[38] I. Müller, P. Strehlow, Rubber and Rubber Balloons: Paradigms of Thermodynamics, Lecture Notes in Physics, vol. 637, Springer, Berlin, Germany, 2004.
[39] L. Bayón, J. M. Grau, M. M. Ruiz, P. M. Suárez, J. Math. Chem. 2010,
48, 28.
[40] P. Tsiotras, E. Bakolas, Y. Zhao, in AIAA Guidance, Navigation, andControl Conference, American Institute of Aeronautics and Astronautics, Reston, VA, USA 2011, https://doi.org/10.2514/6.2011-6689.
[41] S. K. Patiballa, G. Krishnan, Mech. Based Des. Struct. Mach. 2020, 142,052310.
[42] Y. Bar-Sinai, G. Librandi, K. Bertoldi, M. Moshe, Proc. Natl. Acad. Sci.
USA 2020, 117, 10195.

2300535 (6 of 6)© 2023 Wiley-VCH GmbH

15214095, 0, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/adma.202300535 by Yale University, Wiley Online Library on [27/06/2023]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License

www.advancedsciencenews.com