Evolving variable stiffness fiber patterns for multi-shape robotic sheets

2023 IEEE International Conference on Soft Robotics (RoboSoft) | 979-8-3503-3222-3/23/$31.00 ©2023 IEEE | DOI: 10.1109/ROBOSOFT55895.2023.10121924

Atoosa Parsa1∗ , Medha Goyal2∗ , Maggy Lambo2 , Bilige Yang2 , Josh Bongard1 , and Rebecca Kramer-Bottiglio2

Abstract— Thin, planar sheets can be programmed to morph
into complex shapes through stretching and out-of-plane bending, with applicability to shape-shifting soft robots. One way tomake a morphing sheet is to use variable stiffness fibers thatcan modulate their tensile stiffness attached to the surface of a
volumetrically expanding sheet. Adjusting local stiffnesses via
tensile fiber jamming during sheet expansion allows control ofthe local shape tensor. However, finding the fiber placements
and jamming policies to achieve a set of desired shapes is a nontrivial inverse design problem. We present an additive inverse
design framework using an evolutionary algorithm to findoptimal jamming fiber patterns to match multiple target shapes.
We demonstrate the utility of our optimization pipeline with two
input curvature pairs: 1) cylinder and sphere curvatures and2) simple saddle and monkey saddle curvatures. Our method
is able to find a diverse set of sufficient solutions in both cases.
By incorporating hardware constraints into our optimization
pipeline, we further explore the transfer of evolved solutions
from simulation to reality.

I. I NTRODUCTION
Biological systems are able to adjust their morphology andbehavior to accommodate various tasks and environmental
conditions. Inspired by such dynamic plasticity, soft robots
are often designed to respond to changing tasks and environments through adaptation of their physical structure using
an increasing diversity of approaches [1]. One approach is
via shape-shifting stimuli-responsive sheets programmed tomorph into a target curvature upon actuation. This shapechange occurs when local strains arising from differential inplane growth cause out-of-plane buckling, as seen in flowersthat unfold [2] and seed pods that break open [3]. The inverse
design problem of finding local differential growth required
to morph any reference surface into any target curvature has
been solved analytically for the unconstrained case [4].
Unlike organisms, however, synthetic shape-shifting sheets
struggle to morph into more than a single pre-programmed
target shape. Demonstrations of shape-shifting sheets comprised of hydrogels [5], dielectric elastomers [6], and pneumatic bilayers [7], [8] result in fixed shapes that cannot bereconfigured without fabricating a new device.
Yang et al. [9] tackled this problem with strain-limiting
fibers (first described by Kim et al. [10]), which can beadhered, removed, and re-adhered on a pneumatic bilayer
sheet. Fiber arrangements to produce curvatures without∗ Authors contributed equally.
1 Deparment of Computer Science, University of Vermont, 82 University Pl, Burlington, VT 05405, USA.(email: josh.bongard at

uvm.edu)2 Department of Mechanical Engineering and Materials Science, Yale
University, 10 Hillhouse Avenue, New Haven, CT 06520, USA.

analytical solutions, such as the monkey saddle, were foundusing a genetic algorithm [9]. Once adhered, the inextensible
fibers created local strains for a bilayer inflating uniformly inplane, biasing deformation to ensure that the minimal energy
curvature is the target shape. New curvatures could only
be accessed via time-consuming manual intervention—fibers
must be removed, rearranged, and re-adhered—rendering theapproach infeasible in the field. A new calculation is required
to determine fiber placements for each new curvature.
To bypass manual reconfiguration, several approaches
have been employed to create multi-shape sheets. Some
include patterning actuating components on inert substrates[6], [11], or variable stiffness components on expanding substrates [12], [13], and selectively activating the components
to produce new curvatures. In prior works, however, thedynamic components are placed in simple pre-determined
positions, which limits their accessible curvatures.
To attain a multiplicity of accessible curvatures, we propose finding novel, non-intuitive placements of variable stiffness components that can turn their strain-limiting properties
on/off on the surface of an inflating sheet. We use dual-mode“tensile jamming fibers” [12], which can rapidly modulatetensile stiffness while maintaining a low bending stiffness inall directions, switching from inert (up to 200% tensile strain)
to strain-limiting (low tensile strain) via 1D stretchable
vacuum layer jamming. When a new curvature is desired,fibers need not be moved. Rather, a different combination ofvariable stiffness fibers required for the differential growth
pattern can be activated while the rest can be kept inert.
Yet, the inverse design problem compounds when morethan one shape is desired due to the complex relationshipbetween the sets of jammed fiber patterns and the sheet.
To address this, we present a multi-objective evolutionary
algorithm to optimize tensile jamming fiber placements on
an inflatable sheet for multiple desired inflated curvatures.
The evolutionary algorithm searches the space of all possible
fiber patterns, where each pattern can vary in fiber count,
positions, lengths, and orientations. The algorithm also incorporates fabrication and hardware constraints to facilitate
the transfer of selected designs from simulation to reality.
Fig. 1 shows our optimization pipeline that takes twotarget curvatures as inputs and finds two independent setsof fiber placements that achieve the target curvatures when
activated independently. We demonstrate the utility of ourpipeline with two target curvature pairs: 1) sphere & cylinderand 2) simple saddle & monkey saddle. Our optimization
framework is able to produce a diverse set of “good”
solutions in both cases. We transfer one shape-pair solution

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Fig. 1. Pipeline overview. (a): The pipeline begins with two target curvatures and constraints imposed by the hardware setup. Constraints are too many
intersecting fibers (a-i), overly short fibers (a-ii), and too many fibers (a-iii). (b): A population of random fiber patterns is generated (Gen 0). Each pattern iscomposed of four fiber sets: two on the [T]op and two on the [B]ottom. Activation of the red fibers is an attempt to match the first target shape; Activationof the yellow fibers is an attempt to match the second target shape. An evolutionary multi-objective optimization algorithm finds a set of non-dominating
solutions that satisfy the objectives to different degrees (Gen n). (c): Each solution is a pair of fiber patterns that, when activated independently, cause theinflated sheet to form the corresponding desired curvature. (d): We choose solutions from the Pareto front to fabricate and evaluate in hardware.

for the sphere/cylinder targets to reality and demonstrate that,
because the hardware constraints have been incorporated into
the optimization process, the designs can be transferred toreality with some initial success.
II. M ETHODS
A. Simulation Setup
To simulate the sheet, we used a GPU-accelerated voxelbased physics engine called Voxcraft [14]. This physics
engine was developed based on a 3D simulator that quantitatively models the statics, dynamics, and nonlinear deformation of heterogeneous soft bodies using a Finite Element
Method (FEM) [15]. The inflatable sheet is modeled as
three layers of soft voxels placed in a circular grid. The
top and bottom layers are made of passive voxels, while
the middle layer is filled with active voxels that expand
during the simulation time. The fiber patterns are simulated
by replacing the corresponding passive voxels on the top
or bottom layers of the bladder with passive voxels of ahigher modulus that resist tensile deformation (if the fiber isjammed) or passive voxels of a lower modulus (if the fiber isunjammed). Table I includes the simulation parameters usedin all experiments, as measured from the physical hardware.
A number of additional simulator parameters control the
expansion of the active voxels, which were set by calibrating
the simulation to match a real cylinder curvature with threeparallel fibers on the top face of the sheet.
B. Optimization
Evolutionary algorithms are a class of population-based
gradient-free optimization methods that are inspired by natural evolution. They generally start with a random populationof candidate solutions and at each step of the optimization,
the fittest solutions based on problem objectives are selected
to reproduce and survive to the next generation. In this paper,

TABLE ISIMULATION PARAMETERS .

Parameter
Grid Size
DiameterDensity
Moduli of passive layers
Modulus of jammed fiber
Modulus of unjammed fiber
Modulus of expanding layer

Value
28 × 28 voxels
14 (cm)938.33 (kg/m3 )82700 (P a)1330688 (P a)68900 (P a)
160000 (P a)

we used a multi-objective evolutionary algorithm called AgeFitness Pareto Optimization (AFPO) [16]. This approach wasdeveloped to address the premature convergence problem byallowing newly added solutions to survive in the population
before being dominated by the more fitted ones, thereby
increasing the overall diversity of the population. In the lastgeneration of the optimization, the algorithm will converge
to a population of non-dominated Pareto optimal solutions.
The evolutionary algorithm has multiple components toencode the solutions (genome representation), evaluate thembased on the objectives (fitness definition), select the solutions to reproduce (selection operator), and produce the next
generation of solutions (crossover and mutation operators).
The following sections provide the details of these components in our optimization framework.
1) Genome Representation: In our optimization problem,
each solution is composed of two independent pairs offiber patterns, each associated with one of the two target
curvatures. Fiber patterns are modeled as straight line segments with four numbers indicating their starting and ending
coordinates on the sheet. We used a variable-length direct
encoding scheme for the genome representation. To map
the genotype to fiber patterns, we used a variation of theBresenham algorithm [17] as the rasterization process thatdraws the fiber segments on the simulated discrete grid.

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2) Mutation Operator: To produce the next generation in
the evolutionary algorithm, we implemented two mutation
operators. The first operator (applied with a probability of25%) changes the number of fibers by deleting one randomly
chosen fiber or adding a randomly generated one. The second
mutation operator (applied with a probability of 75%) causes
a smoother variation in the configuration space by changing
one endpoint of a randomly chosen fiber to a new position
drawn from a uniform random distribution within the grid.
3) Optimization Constraints: We include fabrication constraints in the optimization process to ensure the selected
fiber patterns can be transferred to reality. Fibers shorter
than six cm are challenging to fabricate, thus providing
the first constraint. Because the fibers are bulky, we cannot
overlay more than two without losing their coupling to theunderlying sheet (i.e., a fiber laid on top of others would
not make full contact with the sheet in the vicinity of theoverlap). Thus, a second constraint limits the number offibers at any given intersection to two. The last constraint
is on the total number of fibers in a pattern: we limit thisto a maximum of six fibers on each face of the sheet
for the sphere/cylinder curvatures. However, this number
is increased to 10 for the saddle/monkey-saddle curvaturesdue to their complexity. All constraints were imposed in
the initialization step and checked during the optimization
process, to ensure the optimal fiber patterns found by thealgorithm were realizable in hardware.
4) Fitness Definition: With evolved fiber patterns, we
simulated the sheet with both fiber patterns (i.e., spherefibers, cylinder-fibers) on the top and bottom faces. We used
the heights of specific and distributed points to determine
how much the inflated sheet matches target curvatures when
the corresponding set of fibers is jammed. The error between
the target and the simulated solution (es ) is defined as:
Pnv|hi − ti |(1)
es = i=1nvwhere hi is the height of voxel i at the end of the simulation
time, nv is the number of voxels on the top layer of theinflated sheet, and ti is the height of the corresponding pointfrom the target curvature.
The es metric represents the average error per voxel,
measured in mm. In each of the case studies presented inthis paper, we have two target curvatures. For every candidate
solution, we calculated two errors associated with matchingeach of the two targets. These two errors are used as thefitness values that will be minimized by our multi-objective
optimization algorithm. In the results section, we also report
the error after normalizing the values by the minimum andmaximum attainable heights in the simulated targets, to givean unbiased indication of matching the target shapes.
5) Parameters and Setup: We ran the evolutionary algorithm with a population size of p = 100 for g = 300
generations for the sphere/cylinder and g = 350 for thesaddle/monkey-saddle. One experimental run takes about
15 hours to finish on a Penguin Relion XE4118GTS GPU

node. We conducted three independent runs for each of thetwo case studies. All optimizations1 were run on the GPU
cluster at Vermont’s advanced computing core.
C. Hardware Setup
The pneumatic bilayer sheet was fabricated following themethod detailed in Yang et al. [9]. The sheet contains pillars
connecting the top and bottom layers, thus constraining
expansion to be in-plane. In prior work, strain-limiting fibers
were adhered to a sheet’s surfaces to induce curvatures upon
inflation, but the output was immutable. Here, we introduce
variable tensile stiffness fibers to this platform to adjust local
stiffness during volumetric expansion of the sheet, allowing
control of the local shape tensor. The fibers were fabricatedfollowing the procedure described in Yang et al. [12]. Thisprior work included a demonstration of the tensile jamming
fibers arranged in a grid on a pneumatic sheet. In this paper,
we open the design space to allow the fibers to vary inlength and placement. We adhere the fibers to the sheet using
Silpoxy™ (Smooth-On).
Fiber patterns for specific target sheet curvatures obtained
via evolutionary algorithm were transferred to hardware.
Variable stiffness fibers were fabricated with lengths andplacements on the sheet provided by the optimization. Selectfibers were jammed with vacuum to activate a target shape.
The sheet was inflated and held at 3.9±0.1 psi (3.6±0.1 psi
for the neat sphere due to occlusions at higher psi) andheights along the inflated curvature were measured at 16
representative points using a commercial motion capturesystem (Phasespace, Inc.) and averaged over five trials.
D. Design of Experiments
In our prior work, we searched for a set of inert fiber
placements, each of which enabled a sheet to obtain a desired
curvature [9]. Here we find patterns that leverage the fibers’
stiffness-switching ability to achieve shape change. The
simplest approach is to run two evolutionary algorithms—
The first evolves a specialized pattern Pi that produces
shape Si ; The second evolves a specialized pattern Pj that
produces shape Sj . The best pattern from each algorithm canthen be extracted and layered on to the same sheet. Shape
Si can be achieved with the merged pattern by stiffening
Pi and slackening Pj , and shape Sj can be achieved bystiffening Pj and slackening Pi . However, we found that thisapproach of simply layering optimal fiber patterns could nottranslate well to the physical hardware, as it resulted in manyfiber intersections (3+ overlapping fibers) and otherwise
ineffective combined pattern signatures.
We sought to resolve this problem by evolving patterns Pi
and Pj in parallel and with imposed constraints that would
increase the likelihood of successful transferal to reality.
With the evolved fiber pattern pairs, we asked two questions:
1) How well do the evolved simulated shapes match thetarget shapes (simulation error es ), and how well do the1 https://github.com/AtoosaParsa/robosoft-2023 contains the source codenecessary for reproducing our results.

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hardware-realized shapes match the evolved simulated
shapes (transfer error et )?
2) When the fiber patterns are combined on the sheet,does the order of the layering matter? That is, does itmatter if Pi is layered over Pj , or vice versa?
We designed our experiments to address these questions.
First, we focused on simple curvatures with known fiber
placement solutions: sphere/cylinder. We translated evolved
fiber patterns for the sphere shape Ss and the cylindricalshape Sc to reality, and quantified differences between thesimulated and hardware curvatures. Then, we layered thetwo patterns (cylinder on sphere; sphere on cylinder), andmeasured the shape fidelity when the patterns are combined
and in different orders. We then extended the evolutionary algorithms to the more complex saddle/monkey-saddle shapes.

A. Spherical and Cylindrical Curvatures
Spherical and cylindrical curvatures are two Gaussian
curvatures (zero, cylinder; positive, sphere) that can be madeusing only six jamming fibers placed in a perpendicular grid
pattern on the top face of the sheet [10]. We used this casestudy as a “sanity check” to verify our sim2real pipeline.
1) Simulation: Target curvatures are made by simulating
the known fiber pattern solutions shown in Fig. 3a(i-ii). The
best evolved designs—limited to six fibers per top/bottom
face—from the three independent trials are shown in Fig. 3bd. The low simulation errors (es ) indicate that the evolutionary algorithm discovers new solutions that are different
from the known optimal solution but use the same numberof fibers. Normalizing es according to the maximum andminimum heights of the target curvatures, we find the percent
errors range between 10% and 30%, as shown in Table II.

E. Calculation of Transfer Error
We quantified the error between the simulation output
curvatures from evolved fiber patterns and the real hardware
curvatures, et , as follows: Heights (measured via motion
capture) for several points along the inflated sheets werenormalized by the greatest height difference between the
sheet when inflated and when uninflated. Normalization was
also performed in simulation to ensure a fair comparison.
The transfer error et is defined as:
Pnp|ri − si |(2)et = i=1
np
where ri is the normalized height of the sheet at point i
in hardware, si is the normalized height in simulation, andnp is the number of points measured. Normalization was toensure a direct comparison between curvature in simulationand hardware without considering scaling differences, since
the heights achieved in hardware were highly dependent on
the inflation pressure.
III. R ESULTS
We demonstrate the performance of our optimization
pipeline with two target shape-pairs: 1) sphere and cylinder
curvatures, and 2) saddle and monkey-saddle curvatures.
Fig. 2a,c depicts the optimization progress in terms of theaverage population fitnesses. As expected, the average error
between the target shapes and evolved simulated shapes, es ,decreases during the optimization. Fluctuations in the average error are due to the multi-objective nature of the problem
and the underlying principles of the optimization algorithm.
Fig. 2b,d shows the Pareto-optimal solutions found by one
multi-objective optimization run. These solutions each satisfy
the problem objectives to different extents, and since we areinterested in minimizing error values for two curvature inputs
in parallel, we break the trade-off by devising a selection
metric based on distance from the average error of randomly
produced solutions.

TABLE IIERROR BETWEEN TARGET SHAPES & EVOLVED SIMULATED SHAPES .

Design1
23

Cylinder Curvature
es [mm] % error
0.93
16.94%
0.74
9.97%
0.90
11.21%

Sphere Curvature
es [mm]% error
1.19
19.28%
1.29
29.72%
1.00
14.00%

2) Transfer to Reality: Design-pairs produced in simulation with the lowest es and most pronounced curvatures
were chosen as possible solutions, and of these, Design 1
was transferred to hardware (Fig. 4). We then assessed theerror between the evolved simulated shapes and the realized
hardware shapes, et , as given in Table III. The results indicatethat sheets patterned only with fibers for a single target
curvature have lower errors than sheets with both sets offibers (i.e. the transfer errors et are lowest when only thesphere pattern or only the cylinder pattern exist for sphereand cylinder curvatures, respectively). We surmise that in
sheets patterned with both sets of fibers, the presence of
unjammed fibers (e.g., fibers for the cylinder design thatremain unjammed when only the sphere design is activated)
impacts the overall stiffness of the sheet, producing some
general curvature not associated with the activated design.
Interestingly, the error for the neat cylinder is much lower
than for the neat sphere. Yet, the errors for each curvatureare similar for the combined sheets.
TABLE IIIERROR BETWEEN EVOLVED SIMULATED SHAPES & HARDWARE SHAPES

Fiber Placements
Cylinder Neat
Sphere Neat
Cylinder on Sphere
Sphere on Cylinder

Cylinder et
17.1% ± 0.5%—
25.3% ± 0.4%
27.7% ± 0.6%

Sphere et—28.6% ± 0.9%
31.1% ± 1.1%
33.9% ± 1.0%

We predicted that the order in which fibers are attachedto the sheet would impact the fidelity of each curvature.
Fibers placed on top are further from the surface of the
sheet at points of overlap, which reduces their ability totranslate tensile strain during jamming into constraining the

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Fig. 2. Optimization summary. Optimization progress and the non-dominated Pareto front of the last generation for (a, b) cylinder/sphere curvaturesand (c, d) saddle/monkey saddle curvatures. The evolutionary search plots (a, c) show the average errors of the population, averaged over 3 independent
trials. The shaded area depicts the standard deviation of the error in the population. Each point in the scatter plots (b, d) represents one solution-pair (twoindependent fiber patterns) from the last generation of our optimization process. Colormap indicates the distance of the solution points from the average
errors of 100 randomly generated configurations (L1 norm). As we move away from the vertical and horizontal red dotted lines (average random errors),
the solutions become better at matching both targets at the same time. The selected solution from the Pareto front set is marked in orange.

bladder during inflation. Therefore, we suspected that thepattern closest to the bladder would produce a curvature withless error compared to the curvature produced by the pattern
on top. However, the data show that there are negligible
differences between the shape fidelity of each curvature,
regardless of the pattern overlay order, suggesting that there
is no need to prioritize one accessible state over another.
As noted above, the transfer error et for the neat cylinder
is lower than for the neat sphere. The neat spherical curvature
possesses areas of unwanted curvature, as indicated by the∼ 29% transfer error (compared the ∼ 17% transfer error forthe neat cylinder). We speculate that these areas of unwanted
curvature could be redirected by the presence of additional
fibers, which could be incorporated by relaxing our hardware
constraints on the number, lengths, and proximity of fibers
in the constrained multi-objective optimization.
We further note that simulated jammed fibers are represented as stiffer voxels embedded within the top or bottom
layers of the three-layer simulated sheet (as described in§IIA), while in hardware the variable stiffness fibers areattached to the sheet’s surface. This discrepancy between

simulated and hardware instantiations is another source oftransfer error, which could be mitigated by simulating thefibers as separate voxels that are unique from the top andbottom faces of the simulated sheet.
Finally, we found that the sheet curvature was dependent
on inflation pressure. The transfer error for the neat sphereat 2.6 psi was 30.5% ± 1.5%, at 3.0 psi was 34.5% ± 1.4%,
and at 3.5 psi it was 28.6% ± 0.9%. Additionally, weobserved time-dependent relaxation effects in the sheets, andwe needed to wait a sufficient time between the inflation ofunique shapes. The relaxation time was taken into consideration when switching between shapes during data collection
and could be quantified more formally in future work.
B. Simple Saddle and Monkey Saddle Curvatures
We next investigated a more complex target shape-pair:saddle and monkey saddle. While there is a known fiber
pattern solution for the simple saddle, the monkey saddle
represents an unknown solution space. The results of three independent optimization runs are given in Fig. 5 and Table IV.
We see that optimization discovers a diverse set of solutions
with each random initialization. Inspecting Table IV, we see

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Fig. 4. Sphere/cylinder curvatures in hardware: Design 1. Inflated sheet
in sphere/cylinder shapes, with neat and combined evolved fiber placements.

their visual similarities.
IV. C ONCLUSION AND F UTURE W ORK

Fig. 3. Target and evolved simulated solutions for sphere/cylinder
curvatures. Target sphere/cylinder curvatures in (a) and the three best
designs for fiber placements from three independent evolutionary runs in(b-d). Fiber patterns are shown for the top (left) and bottom (right) faces ofthe sheet. Each line on the circular sheet represents a fiber that is jammed(light red) or unjammed (dark red). The blue voxels represent passive voxels
on the top and bottom faces of the sheet. Green voxels in the middle layer
are active voxels expanding during the simulation. The side view in eachpanel shows the inflated sheet at the end of the simulation time.

that the simple saddle generally has a lower simulation error(es ∼ 8-10%) than the monkey saddle (es ∼ 20-36%).
TABLE IVERROR BETWEEN TARGET SHAPES & EVOLVED SIMULATED SHAPES

Design1
2
3

Monkey Saddle Curvature
es [mm]%error4.54
25.28%
4.31
19.31%
4.58
36.51%

Saddle Curvature
es [mm]%error4.80
10.68%
4.46
8.72%
4.74
8.40%

It is worth noting that although the errors seem to be high(es ∼ 5 mm), the optimized fiber patterns are substantiallybetter than a randomly generated fiber pattern (see Fig. 2b).
Plotting the simulated output curvatures from evolved fiber
patterns (Fig. 6a,c) and target (Fig. 6b,d) curvatures shows

We presented a multi-objective evolutionary algorithm tooptimize variable stiffness fiber patterns on the surface ofan inflatable sheet for multi-shape sheets. Leveraging thefibers’ stiffness-switching ability, we evolved fiber patterns
that enabled the sheet to access two unique shapes bystiffening/slackening the subsets of fibers corresponding toeach shape. We demonstrated the utility of the optimization
for two target shape-pairs: 1) sphere/cylinder curvatures, and2) saddle/monkey-saddle curvatures. We showed that theevolutionary algorithm successfully produced a diverse setof designs for each case. We also transferred one designfrom the sphere/cylinder case study to reality and concluded
that, because the optimization algorithm takes the hardware
constraints into account, the transfer of evolved solutions
from the Pareto front to hardware is feasible. Furthermore,
we found that when combining fiber patterns on the sheet’s
surface to attain multi-shape access, the order of patternoverlay is not an important consideration (i.e., combined fiber
patterns display no dominant shape, regardless of attachment
order). Future work involving the transfer of additional
designs from target to simulation to hardware would deepen
our understanding-of and confidence-in our design pipeline.
Because our optimization pipeline is additive—it produces
independent sets of jamming fiber patterns, which are thenoverlayed on top of each other on the inflatable sheet—webelieve it is scalable to an increasing number of target shapes
and could produce patterns matching highly complex target
curvatures. In the future, a post-processing step could further
simplify the patterns by merging the most similar fibers (viathresholding distance, length, and orientation). A merged

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Fig. 6. Optimized and target complex curvatures. Optimized (red) andtarget (blue) monkey saddle shape (a-b) and simple saddle shape (c-d) for
the first design from the first evolutionary trial.

Fig. 5. Target and evolved simulated solutions for saddle/monkeysaddle curvatures. Target saddle/monkey-saddle curvatures in (a) and thethree best designs for fiber placements from three independent evolutionary
runs in panels (b-d). The targets are generated using the equations andnormalized in the desired range. Fiber patterns are shown for the top andbottom faces of the sheet. Each line on the circular sheet represents a fiberthat is jammed (light red) or unjammed (dark red). The side view in eachpanel shows the inflated sheet at the end of the simulation time.

fiber could belong to two or more fiber patterns, rather thankeeping the fiber patterns mutually exclusive. Such a merging
approach could reduce the total number of fibers, stay within
our hardware constraints, and open access to a multiplicityof sheet curvatures that transfer from target to simulation toreality with low simulation and transfer errors.
ACKNOWLEDGMENT
This material is based upon work supported by the National Science Foundation under the Emerging Frontiers in
Research and Innovation (EFRI) program (EFMA-1830870).
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