2024 7th International Conference on Soft Robotics (RoboSoft)San Diego, CA, USA. April 14-17, 2024

Merging variable stiffness fiber patterns on multi-shape robotic sheets

2024 IEEE 7th International Conference on Soft Robotics (RoboSoft) | 979-8-3503-8181-8/24/$31.00 ©2024 IEEE | DOI: 10.1109/ROBOSOFT60065.2024.10521946

Medha Goyal*1 , Maggy Lambo*1 , Josh Bongard2 , and Rebecca Kramer-Bottiglio1

in systems where actuating components are patterned onto
inert substrates [4], [8], [9] or variable stiffness components
are patterned on expanding substrates [10], [11]. However,in these sheets that can achieve multiple curvatures, theconstraints are placed in intuitive positions, limiting theiraccess to complex shapes.
The inverse design problem of patterning sheets withconstraints able to produce multiple complex, non-intuitive
shapes was tackled by Yang et al. [12]. Therein, inextensible
patches (introduced by Kim et. al [13]) were patterned ona radially inflating pneumatic bilayer according to solutions
found using a genetic algorithm, initiating pre-determined
out-of-plane deformations to produce target shapes. While
the patches could be adhered, removed, and re-adhered toswitch between patterns and produce multiple shapes on
the same sheet, they worked passively and could not be
selectively activated. Therefore, the sheet could not produce
different shapes without manual reconfiguration of the inextensible fiber patches.
To allow for reconfiguration without manual intervention,
Yang et. al replaced the inert patches with variable stiffness fibers [10]. These fibers work via 1D layer jammingand contain strips of silicone with inextensible polyester
inclusions that jam under negative pressure (vacuum). Whenjammed, the fibers increase in tensile stiffness by over 20×,allowing them to constrain bilayer expansion and produce
out-of-plane curvatures. The jamming can be reversed byremoving the vacuum, resulting in passive fibers that nolonger constrain local growth. Yang et. al [10] used thevariable stiffness fibers on the surface of an inflatablesheet to produce basic curvatures. We expanded on thisby introducing a generalized simulation-to-reality (sim2real)pipeline that accepts two input shapes {Si , Sj } and generates
two fiber patterns {Pi , Pj }, one for each shape, using a
multi-objective evolutionary algorithm [14]. The two fiber
patterns were overlaid to create a multi-shape sheet: shape
Si was accessed by stiffening Pi and slackening Pj , andshape Sj by slackening Pi and stiffening Pj . Designs fora sheet that could switch between cylindrical and sphericalcurvatures were successfully transferred into hardware in[14], demonstrating the feasibility of this pipeline.
The optimization algorithm used in our prior work returned designs consisting of fiber patterns for two target
shapes on a single sheet. Each pattern was compared against
its own corresponding target shape and errors for eachshape were reported separately. Each combined design was
regarded as a whole only when being checked for violationsof hardware constraints, such as the presence of too manyfibers or fibers that were too short to feasibly manufacture. In

Abstract— Shape morphing can be achieved using thin, planar sheets patterned with strain-limiting constraints to direct
differential growth into desired shapes. Our previous work
produced such shape-shifting sheets using an inflatable sheetwith variable stiffness fibers placed in patterns optimized by amulti-objective evolutionary algorithm. With our pipeline, we
generated two specialized fiber patterns {Pi , Pj } to produce
shapes {Si , Sj }, which we then layered onto the same sheet.
However, this layering approach had translation inefficiencies in
hardware, such as fiber redundancies. To reduce fiber crowding
and increase shape fidelity, we propose that fibers identified assimilar in both Pi and Pj can be implemented as a single fiber
belonging to both patterns. Extending our previous work, herein
we implement a post-optimization fiber merging protocol. Applying the protocol to sheets patterned with fibers targeting two
shape-pairs, cylinder/sphere and simple saddle/monkey saddle,
we demonstrate that fiber merging reduces the total numberof fibers on each sheet, thus reducing sheet bulk and weight.
We further measure the error between the target shapes andactual hardware shapes: For the cylinder/sphere shape-pair,the error increases with merging implemented when comparedto the original, unmerged fiber patterns. For the monkey
saddle/simple saddle shape-pair, the error decreases for one
or both shapes for all fiber mergers implemented. The results
indicate that fiber merging is increasingly useful with increasing
fiber pattern complexity as measured by fiber count. Therefore,
fiber merging is a potentially useful strategy to simplify thecomplexity of fiber designs required for shape-matching.

I. INTRODUCTION
Soft robots that adapt their form (shape) on-demand would
be able to function across multiple environments or task
contexts [1]. For thin sheets, incompatible local strains due
to in-plane expansion creates out-of-plane buckling, yielding
local curvature changes that can collectively translate to
global shape change. The inverse problem of designing local
differential growth patterns on a thin sheet to achieve a target
shape has been solved analytically [2]. Several works havedemonstrated shape-shifting sheets using a variety of materials including cellulose-embedded hydrogels [3], dielectric
elastomers [4], pneumatic silicone bilayers, [5] [6], liquidinfused ribbed sheets [7], and rubber sheet kirigami [8].
While some of these systems are reversible, most achieve
only a fixed, singular shape-change per device. Sheets thatcan morph into multiple shapes have been demonstrated*Authors contributed equally.
1 Department of Mechanical Engineering & Materials Science,
Yale University, New Haven CT, USA. email:{medha.goyal,

maggy.lambo, rebecca.kramer}@yale.edu
2 Department of Computer Science, University of Vermont, Burlington
VT, USA. email: jbongard@uvm.edu
This material is based upon work supported by the National Science
Foundation (NSF) under the Emerging Frontiers in Research and Innovation(EFRI) program (EFMA-1830870) and the Designing Materials to Revolutionize and Engineer our Future (DMREF) program (CMMI-2118988).

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Fig. 1. Pipeline overview. (a): Two target curvatures (for example, simple saddle on the left and monkey saddle on the right) and constraints imposed by
the hardware. Constraints are too many intersecting fibers (a-i), overly short fibers (a-ii), and too many fibers (a-iii). (b): An evolutionary multi-objective
optimization algorithm finds a set of solutions that satisfy the objectives to varying degrees. Each solution set has a pattern of fibers (shown as red andyellow) on the top and bottom of the sheet. Red fibers correlate to the first target shape; yellow fibers correlate to the second target shape. Simulated
inflations with each shape-pattern activated (now shown as bright and dark red) yield shape-matching to each target shape. (c): The fiber merging protocol
is applied post-optimization. Candidate fibers to merge must be in close proximity, defined as less than half of the sheet diameter (c-i); have a similar
orientation, defined as being less than 10◦ off parallel (c-ii); and be similar in length, with less than 2 cm variation (c-iii). The fiber pair must also compriseone fiber from each pattern associated with the target shapes. Merging is performed on the chosen fiber patterns to simplify the designs. (d): The original
and merged designs are transferred to hardware.

shape. For a shape-pair {Si , Sj }, we layer the unique fiber
patterns {Pi , Pj } generated by evolutionary optimizationand implement a post-optimization fiber merging protocol(Fig. 1). We identify fiber pairs, where one fiber is from Pi
and one is from Pj , and choose pairs that appear to constrain
the sheet in similar ways when producing the target shapes.

disregarding the presence and impact of one fiber pattern on
the other—Pi still passively exists while inactive, and therefore affects the shape fidelity of Sj when Pj is active— theprior sim2real pipeline misses the opportunity to consider the
shape similarity between the two target shapes programmed
onto the same sheet and leverage this similarity to simplify
the resulting overlaid fiber patterns.
Shape similarity between two target shapes may imply
fiber redundancies, or two fibers that belong to the uniquepatterns Pi and Pj but are close together and “crowded.” To
exploit shape similarity, reduce fiber crowding, and potentially increase shape fidelity, we propose that fibers identified
as similar in both Pi and Pj can be implemented as a single
fiber belonging to both patterns. Merging a pair of fibers
reduces the total fiber count in the set {Pi , Pj } by one.
A lower fiber count is desirable for a few reasons. First, thesimulation models fibers as being embedded into the sheet
instead of sitting atop the sheet, as is the case in reality.
Therefore, each additional fiber adds weight and bulk notaccounted for in the model, introducing a source of error.
Second, reducing the number of fibers reduces the labor
required for fabrication, speeding up the sim2real transfer.
Third, using fewer fibers frees up space on the sheet, which
could allow the transfer of more complex target shapes or
potentially even a third curvature.
Our work tackles this problem by exploring how fibers
associated with one target shape might aid in producing theother shape. We consider designs for the multi-shape shifting
sheets found in simulation in our prior work [14], looking forsimilarities between the fiber patterns associated with each

Whereas fiber patterns found previously contained fibersthat would only be active (jammed) for one shape or theother, by implementing fiber merging, we replace some pairsof fibers with a single fiber that is active (jammed) for
both shapes. This fiber merging strategy reduces the total
number of fibers required on the sheet for multi-shape access.
We show visually and by sampling inflation values acrossrepresentative points that for simple shapes requiring fewer
fibers (cylinder/sphere curvatures), reducing the number offibers causes the shape to diverge from the target shapes.
For more complex target shape-pairs (simple saddle/monkey
saddle) consisting of more total fibers, implementing one
fiber merger has a negligible effect on shape fidelity, andimplementing a second fiber merger slightly improves themonkey saddle (while slightly worsening the simple saddle).
Overall, the results herein demonstrate that fiber patterns
including fiber mergers have the potential to match at least
two target shapes with a similar target-to-real error as theoriginal patterns (without fiber mergers) while requiring
fewer fibers. The results thus suggest that the introductionof post-optimization fiber merging is a useful strategy todecrease the fiber count while increasing the shape fidelityof some shapes, and the strategy’s efficacy is dependent onfiber count and similarities between the target shapes.
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II. METHODS

B. Fiber Designs: Requirements for Merging
Merged patterns were created by identifying pairs of fibers
across designs that appeared to serve the same function as
one another. Although line clustering algorithms exist [15],the current work aims to investigate nonlinear interactions
when fibers are close to or pass over one another, which
are unique challenges. Therefore, in this work, fiber mergers
were achieved by ensuring each fiber pair satisfied a set of
conditions described as follows: a) each fiber pair contains
one fiber from subset Pi and one fiber from subset Pj ; b)
the fibers have a similar orientation, defined as being lessthan 10◦ off parallel; c) the fibers are similar in lengthwith less than 2 cm variation; and d) the fibers are in close
proximity, defined as less than half of the sheet diameter.
These conditions were chosen because they describe thesmallest separation between any two fiber pairs as observedby the eye for the designs explored in this paper. Each pairthat satisfied these conditions was merged by being replacedwith a single fiber placed at their midpoint in the sameorientation and with a length equal to their average, describedin more detail in Section II-D.

A. Fiber Designs: Original
Two fiber patterns from our prior work [14] were transferred into hardware: one for a sheet that switches between
cylindrical and spherical curvatures, the other for a sheetthat switches between monkey saddle and simple saddle
curvatures. Among the cylinder/sphere designs presented in[14], all had the same number of fibers—six in total—and
we selected “Design 3” from [14] (Fig. 2a,b) for its relatively low sim2real errors. Among the monkey saddle/simple
saddle designs, we again selected “Design 3” from [14](Fig. 2c,d) for its lower number of fibers—14 in total—
relative to the other designs.

C. Transferring Designs to Hardware
Each shape-shifting sheet consists of a flat, circular, pneumatically driven bladder made of Ecoflex 50, containing
a uniform grid of pillars that limit expansion in the zdirection, patterned with variable stiffness fibers as describedin prior works [10], [14]. Each fiber contains strips ofEcoflex-50 (Smooth-On) with polyester thread inclusions.
The strips slide past one another at atmospheric pressure
but are jammed into contact under vacuum, creating a force
chain that amplifies the fiber’s tensile strain by over 20× andconstrains the sheet’s expansion along its length, producing
out-of-plane deformation when the sheet expands. Fibers
were adhered to the sheet using Silpoxy™ (Smooth-On).
Variable stiffness fibers were fabricated with lengths andplacements according to designs chosen from the multiobjective algorithm in our previous work [14], as describedin Section II-B. The fibers were jammed with vacuum toactivate a target shape. The sheets were inflated and held
at 5.0 ± 0.1 psi for both cylinder/sphere sheets and at
4.0 ± 0.1 psi for all saddle/monkey saddle sheets. The
inflation pressures were chosen to ensure that the curvature
was visible by the eye but not so high that the sheets werestrained close to the point of rupture.
D. Design of Experiments
In our prior work, we searched for two variable stiffness
fiber patterns Pi and Pj , which were overlaid onto one sheet,enabling us to switch between two shapes Si and Sj . In thiswork, we find pairs of fibers such that one fiber is part of Pi
and the other fiber is part of Pj . We measure the distance
between each pair of fibers d, the length of each fiber li andlj , and the angle of one fiber with respect to the other, α.
For all α < 10°, d less than half the sheet diameter, and|li − lj | < 2, the pair of fibers is replaced with a single
representative fiber of length (li + lj )/2 at an angle of α/2

Fig. 2. Target shapes and simulated designs chosen for hardware
implementation. (a, c) The target shapes for (a) the cylinder and sphereshape-pair, and (c) the monkey saddle and simple saddle shape-pair. (b,d) The fiber pattern designs for (b) cylinder and sphere, and (d) monkey
saddle and simple saddle, found by the multi-objective algorithm [14] andchosen for our study. Adapted with permission [14]. Copyright 2023, IEEE.

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with respect to one of the original fibers, and at the midpoint
between the positions of the two original fibers (a distanceof d/2 between the two original placements).
Following these thresholds and guidelines for fiber merging, we ask: Does merging two fibers—one each from Pi
and Pj , into a single fiber belonging to both {Pi , Pj }—affectthe shape fidelity of Si and Sj ? We answer this question bymeasuring the error between the target shapes and the real
inflated shapes Si and Sj . We did this for two sets of shapepairs: {Pi , Pj } = {cylinder, sphere} and {monkey saddle,
simple saddle}.

curvature with the following equation:Pnp|ri − si |e = i=1np(1)where e is the target-to-hardware error, ri is the z-coordinate(height) of the sheet at point i in hardware (normalized by thegreatest height difference), si is the normalized height of thetarget curvature, and np is the number of points measured.
III. R ESULTS & D ISCUSSION
Applying the merging rules listed in Section II-D to thefiber patterns chosen from [14] yielded one candidate fiber
pair for merging in the cylinder/sphere design (Fig. 3a)and two candidate fiber pairs (which we will call fiber
pairs A and B) in the simple saddle/monkey saddle design(Fig. 3b,c). The following six fiber patterns were subsequently transferred into hardware: (1) the original design
without merging for cylinder/sphere, (2) a design with one
merge in cylinder/sphere, (3) the original design without
merging for simple saddle/monkey saddle, (4) a design withfiber pair A merged in simple saddle/monkey saddle, (5) adesign with fiber pair B merged in simple saddle/monkey
saddle, and (6) a design with both fiber pairs A and B mergedin simple saddle/monkey saddle (Fig. 3d).
We assessed the error between the original designs andmerged designs by comparing the curvature produced in
hardware for each shape to the target curvature (definedmathematically). Images of hardware inflation of the sphereand cylinder shapes are shown in Fig. 4, and the simple saddle and monkey saddle are shown in Fig. 5. The quantitative
error results are given in Tables I and II.
The results indicate that for the cylinder/sphere design,
merging a pair of fibers into a single fiber that belongs toboth shape patterns yields inflated shapes that are less faithful
to the target shapes, compared to those produced by the

E. Error Calculation
To compare the error between hardware curvatures andtarget curvatures, we measured the z coordinates of 15-16
representative points and compared them to the z coordinates at the corresponding x- and y- positions of the target

Fig. 3. Fiber pairs selected for merging. Original, unmerged designs
found in simulation [14] are shown on the left. Pairs of fibers identified as
candidates for merging are circled in yellow. The subsequent merged designs
are shown on the right. (a) Original (left) and merged (right) fiber designs
for cylinder/sphere. The subset of fibers shown in bright red corresponds tothe cylinder curvature; dark red fibers correspond to the sphere curvature.
(b) The original design for simple saddle/monkey saddle with merge pair A
circled (left), and Merge A implemented (right). Bright red fibers correspondto the monkey saddle curvature; dark red fibers correspond to the simple
saddle curvature. (c) The original design for simple saddle/monkey saddle
with merge pair B circled (left), and Merge B implemented (right). (d) The
original design for simple saddle/monkey saddle with merge pairs A and B
circled (left), and both merges implemented (right).

Fig. 4. Cylinder and sphere sheet inflations in hardware. Original
and merged designs were transferred to hardware. Shapes were attained by
activating the subset of fibers corresponding to each target shape during
sheet inflation. The merged fiber patterns visually contain one less fiber
than their unmerged fiber pattern counterparts. Sheet diameter = 14 cm.

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reduced to five for the merged pattern. With fewer total fibers,
each fiber on average contributes more to the system’s shape.
Therefore, the absence of even one fiber could substantially
impact shape fidelity.
In contrast, the errors for the sheets patterned for simple
saddle/monkey saddle shapes were not substantially impacted by a single pair of fibers merging, regardless of whichpair was merged. The monkey saddle error notably decreased
after two fiber pair merges. We speculate that because thesimple saddle/monkey saddle design has a relatively greaternumber of fibers (14 total fibers), the presence of more
fibers adds weight and bulk to the sheet. With two fibers
eliminated via merging (12 total fibers when both pairs
A and B are merged), some weight is removed allowing
the sheet to inflate in a manner more similar to the target
shape. Our visual inspections support this suggestion: an
asymmetry was observed during inflation for the simple
saddle/monkey saddle sheets, where the side of the sheetwith a greater concentration of fibers (the left in Fig. 5), hada less pronounced inflation than the other side.
IV. C ONCLUSION AND F UTURE W ORK
Fig. 5. Simple saddle and monkey saddle sheet inflations in hardware.
Original and merged designs were transferred to hardware. The resulting
inflations are shown for each sheet (original, Merge A, Merge B, and bothmerged), where the subset of fibers corresponding to each target shape wasselectively activated, producing either a simple saddle or monkey saddle.
Sheet diameter = 14 cm.

Our prior works introduced variable stiffness fibers thatcan be patterned on inflatable sheets to selectively constrain
inflation and produce designed inflated shapes. We proposeda sim2real pipeline to generate (using a multi-objective evolutionary algorithm) unique fiber patterns for target shapes,
which could be overlaid to achieve a multi-shape sheet [14].
Activating one unique pattern would produce one shape,
and activating the other unique pattern would produce theother shape. However, we found that overlaying uniquepatterns caused fiber crowding on the sheet—–a problem
that would become increasingly severe with an increasing
number of target sheet shapes. To alleviate this problem, we
implemented a strategy to merge redundant fibers.
In general, fiber patterns may be more easily merged ifthey are similar, where a fiber identified as similar in both
fiber patterns Pi and Pj can be implemented as a single fiber
belonging to both patterns to reduce crowding. We proposeda set of thresholds to identify pairs of candidate fibers, where
each fiber of the pair belongs to a different fiber pattern (andis associated with a different target shape). For the mergeablefiber pairs, we replaced each pair with a single representative
fiber of averaged length at the midpoint of their separation
by both distance and angle.
We explored the viability of this merging strategy as a
post-processing step for further simplifying designs foundin simulation on two target shape-pairs: 1) cylinder/sphereand 2) simple saddle/monkey-saddle. We selected one design
produced in simulation for each case, applied merging rules
to the designs, replicated the original designs and merged
designs in hardware, and compared the shapes produced
before and after merging. We demonstrated that all fiber
patterns, original and merged, successfully produced thetarget shapes with reasonable fidelity and errors comparable
to those previously reported.

TABLE IERROR BETWEEN HARDWARE & TARGET CURVATURES :CYLINDER / SPHERE

Sheet
Original
MergedCylinder e10.7% ± 1.6%
31.2% ± 1.5%

Sphere e10.0% ± 0.9%
36.7 % ± 2.2%

TABLE IIERROR BETWEEN HARDWARE & TARGET CURVATURES : SIMPLESADDLE / MONKEY SADDLE

Sheet
Original
Merge A
Merge B
Both Merged

Simple Saddle e
15.4% ± 1.4%
12.2% ± 0.5%
14.8% ± 1.5%
19.2% ± 1.0%

Monkey Saddle e
20.5% ± 0.5%
20.3% ± 0.8%
23.7 % ± 1.0%
18.6% ± 0.5%

original, unmerged fiber patterns. The increased error e isseen for both the cylinder and the sphere shapes.
For the simple saddle/monkey saddle designs, we implement two fiber pair mergers, which we denote as Merge A
and Merge B. Merge A slightly improves the shape fidelityof the simple saddle shape and has a negligible effect on themonkey saddle. Merge B has a negligible effect on the simple
saddle shape and slightly worsens the shape fidelity of themonkey saddle. Implementing both merges slightly worsens
the simple saddle while improving the monkey saddle shape.
We surmise that merging was a disadvantageous strategy
for the simpler cylinder/sphere shapes due to their low
number of fibers: six for the unmerged pattern, which was
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By comparing the shapes produced by our original andmerged designs to the target shapes, we observed thatmerging increases the target-to-hardware error in sheets withfewer fibers, but can improve or negligibly affect shape
matching for sheets with more complex features that requirea greater number of fibers. This finding suggests that fiber
merging is a useful strategy to simplify complex fiber patterns, reducing both the labor required to transfer designs tohardware and the bulk that could be weighing the sheet down
and potentially contributing to the error. Future work will
investigate the variables associated with pattern complexity(e.g., fiber count, number of fiber intersections) to elucidate
the thresholds at which merging becomes advantageous.
Since we only compared multiple inflations of a single
sample in this work, future work should consider sample
variance. Further insight into the efficacy of fiber merging forpattern simplification and shape-fidelity enhancement could
also be gleaned in simulation, where errors between target
and simulated shapes could be assessed much faster thantarget-to-hardware errors.
In this work, we studied the impacts of merging as applied
to fiber pairs that were aligned in orientation, similar inlength, and in proximity to one another relative to other fiber
pairs. Future work could explore these variables further anddetermine the extent to which merging is useful for fiber pairsof different lengths, relative angles, and separation distances,and whether additional merging steps are beneficial. Opening
up the parameter space could give us more options for
merging when fiber patterns are found via simulation andoptimization, and also a tool to empirically investigate which
fibers contribute the most to producing a target shape, toguide the choice of candidates for merging.
Herein, we focused on relatively related shape pairs, as
we aimed to exploit their shape similarities. Future work
could instead utilize pairs of dissimilar target shapes not
typically placed on the same sheet, such as cylinder andmonkey saddle. The study of multi-shape sheets capableof switching between two relatively dissimilar shapes may
elucidate quantitative metrics to evaluate shape similarity
in the context of fiber merging. Furthermore, such metrics
may reveal thresholds corresponding to the likelihood that ashape-pair will contain mergeable fibers.
As merging rules are simultaneously expanded and refined(i.e., the design space gets larger while we systematicallynarrow the candidate fiber pairs), merging could be integrated into the simulation and optimization components ofour pipeline. For instance, a neural network could learn
latent representations of fiber patterns for various shapes.
Additional training could then be performed to ensure that
decompressing linear combinations of latent representations
drawn from different shapes results in appropriately merged
fiber patterns. In the longer term, artificial intelligence (AI)models that successfully generate images from text prompts
may be adapted to generate fiber patterns from text prompts
that describe individual target shapes, resulting in a trained
generative model that generates merged fiber patterns fromtext prompts that describe a set of different target shapes.

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