2021 IEEE 4th International Conference on Soft Robotics (RoboSoft)
April 12-16, 2021, Yale University, USA

Shape Matching: Evolving Fiber Constraints on a Pneumatic Bilayer

2021 IEEE 4th International Conference on Soft Robotics (RoboSoft) | 978-1-7281-7713-7/20/$31.00 ©2021 IEEE | DOI: 10.1109/RoboSoft51838.2021.9479212

Bilige Yang1 *, Joshua Powers2 *, Atoosa Parsa2 , Josh Bongard2 , and Rebecca Kramer-Bottiglio1

pneumatic bilayer platform achieves uniform in-plane stretch
when inflated, with normal expansion limited by evenly
distributed thick pillars. Curvatures are attained by attachingstrain-limiting fibers to the surface of the bilayer sheet.
The strain-limiting fibers we employ here were previously
described by Kim, et al. [10]. The fibers are coupled to asilicone adhesive such that they can be adhered, removed,
and re-adhered in new locations and orientations. When thefibers are applied to the surface of the pneumatic bilayer,
local strains are constrained by the fibers, and the strain incompatibility facilitates complex minimal energy shapes [3].

Abstract— Shape-changing robots that can adapt their physical structure to changing tasks and environments require an
ability to locally change their surface curvatures on-demand.
Thin shape-shifting sheets represent one approach, and examples using differential growth have shown stimuli-responsive
sheets that morph from one shape to another. In this paper, we
present pneumatically-actuated bilayer sheets that can morph
between shapes using surface fibers that constrain normal
expansion while biasing in-plane expansion. The fibers canadhere to the pneumatic sheet to govern its shape trajectory,
and can be easily detached and re-attached in various patterns
to reconfigure the shape trajectory. We use artificial evolution
to solve the geometric inverse problem of determining the fiber
placements to attain a target shape upon inflation in simulation.
For simple shapes, the evolved solutions converge toward asimilar fiber placement strategy. For more complex shapes,
evolution finds more diverse fiber patterns, which is desirable
since some fiber patterns are more easily fabricated than others.
We demonstrate the utility of evolving the fiber patterns insimulation by translating two shapes, a simple saddle and amonkey saddle, from simulation to reality.

Many pneumatic soft actuators embody constraints withfibers [11] or fabric [12] to limit the strain within a direction
and thus determine the inflated shape. For a simple desired
motion, such as bending along a single axis, determinationof where to place the strain-limiting component is intuitive.
Further, the design of pneumatic fiber-reinforced actuators
for trajectory matching has been demonstrated for cylindricalactuators [11]. However, in our realization we do not havecontrol over unlimited local stretch and aim to produce
complex surface curvatures. Thus, this constrained case falls
outside of known analytical solutions.

I. INTRODUCTION
Incompatible local strains on thin sheets due to in-plane
expansion creates out-of-plane buckling; This mechanism
explains the morphological change during growth for many
processes in botany, such as unfolding of flowers [1] andnatural breaking of seed pods [2]. The inverse problemof finding the local differential-growth needed to achievea desired final shape has been solved analytically in theunconstrained case [3]. Recent demonstrations of shapeshifting sheets have leveraged composite hydrogels [4] andelastomers [5] with localized differential swelling to achievecomplex shape changes in response to temperature. However,these systems have relied on suspension in water to counteract gravitational forces, offered slow response times, andare programmed at manufacture to shift to only one target
shape. Pneumatically-actuated sheets may be an appealing
alternative as they are capable of rapid actuation [6] andhigh energy density [7]. Pikul et al. [8] and Siefert et al. [9]presented flat pneumatic structures that can inflate into complex shapes. In these approaches, programmed directionality
is permanent and constrained to one fixed motion or shape,and new shapes require the full fabrication of another device.
Here, we introduce a pneumatic bilayer sheet coupledwith reconfigurable strain limiting fibers [10], which allows
for rapid shape morphing and tunable shape matching. Our

We attempt the inverse design problem of optimizingstrain-limiting fiber placements on a pneumatic bilayer sheet
to achieve a desired inflated shape by utilizing an evolutionary algorithm. Artificial evolution has previously been
applied to a wide range of inverse design problems such as
the design of flapping wing morphologies [13], biologicalrobots [14], and antennae [15]. Our approach employs acanonical genetic algorithm, which searches over the spaceof all inert fiber patterns, where each pattern is limited to a
fixed number of fibers and each fiber has a set of evolved
start and end coordinates. Further, we assume that the fibers
are straight lines—a simplification that allows us to moreeasily transfer designs from simulation to reality.
Figure 1 shows our resulting pipeline that takes target
curvatures from physical objects or abstract equations as
input, optimizes fiber placements that achieve that curvaturein simulation, and transfers evolved designs to a real bilayer
sheet with low error. When a simple shape is used as input(e.g., a saddle curve), the evolutionary algorithm discovers
a fiber pattern similar to the known solution, which verifies
that such an approach can find “good” solutions. When amore complex shape is used as input (e.g., a monkey saddle),
several distinct but equally efficacious fiber patterns are
discovered. In short, we demonstrate that artificial evolution
can find surface fiber placements on a 2D expanding sheet
given a desired shape.

*These authors contributed equally.
1 Department of Mechanical Engineering Materials Science, Yale University, 10 Hillhouse Avenue, New Haven, CT 06520, USA. (email:rebecca.kramer at yale.edu)
2 Deparment of Computer Science, University of Vermont, 82 University
Pl, Burlington, VT 05405, USA.

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Fig. 1. Beginning with a target shape, we used artificial evolution to discover fiber patterns to match a target shape, then translated the evolved fiber
patterns to reality.

II. M ETHODS
A. Pneumatic Bilayer Sheet
A silicone pneumatic bladder naturally expands in all
directions. To constrain expansion in the vertical (z) direction
in a pneumatic sheet, we added “pillars” connecting the top
and bottom surfaces of the sheet, such that the bladder may
only biaxially stretch in the x and y directions.
1) Fabrication of pneumatic bilayer: The pneumatic bilayer sheets were made out of molded silicone. Molds
were fabricated using laser-cut acrylic parts (Fig. 2a), whichwere glued together with acrylic adhesive. Two molds were
fabricated: one mold for the top silicone membrane with thepillars, and another mold for the bottom silicone membrane.
We used EcoFlex 50 elastomer (Smooth-on) with equalparts A and B, and also mixed in luminescent powder forvisualization. The mixed silicone was poured into the molds(Fig. 2b). Once cured, we removed the silicone membranes
from the molds and adhered them together using pre-curesilicone as a glue (Fig. 2c). A small cut was made on theside of the membrane to insert pneumatic tubing, which wasthen sealed with silicone adhesive (Silpoxy, Smooth-On).
2) Fabrication of strain-limiting fiber lamina: The fabrication of a composite lamina with strain-limiting fibers
and adhesive backing was described in prior work [10].
Briefly, polyester threads were embedded into silicone sheets
to provide anisotropic strain-limiting behavior. The silicone
sheets were coupled to an adhesive silicone (Silbione 4645,
Elkem Silicones), which allows patches of the fibers to easily
be adhered-to and removed-from surfaces.
3) Capturing shape with active infrared stereo camera:
To capture the inflated shape of the bilayer, we used an
active infrared (IR) stereo camera (D415, Intel Realsense).
We painted 20 evenly-distributed dots on the top surface ofthe bilayer as fiducial markers (Fig. 2d), and captured thecoordinates of these points after inflation to compare withsimulated coordinates.

Fig. 2.
Fabrication of pneumatic bilayer sheets. a: Laser cut acrylic
components were glued together with acrylic adhesive. b: Silicone was
mixed with luminescent powder, poured into acrylic molds, and cured atroom temperature. c: De-molded silicone layers were adhered using uncured
silicone. Pneumatic tubing was inserted and sealed with silicone glue. d:
Pictures of a pneumatic bladder. Green dots are the twenty fiducial markers.
Scale bar = 5cm.

materials are discretized as a collection of cubic elements
denoted as voxels. Each voxel is simulated as a point massconnected to its six neighbouring voxels by Euler-Bernoulli
beams. To represent the pneumatic bilayer, we created threesimulated layers of voxels. The top and the bottom layers
were modeled as elastic passive layers (E = 68900 Pa), while
the middle layer was modeled as an elastic expanding layer(E = 160000 Pa) that increased uniformly in volume overthe duration of the simulation. Each voxel in the bilayer
had a side length of 0.0028 m, with 50 voxels along thediameter of the layer, resulting in a total diameter of 0.14 m(50 × 0.0028). Lastly, we simulated the strain-limiting
fibers by replacing voxels in the top and bottom layers

B. Simulation
The pneumatic bilayer was simulated with the multimaterial physics simulator Voxelyze [16]. In the simulator,
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ti is the height of voxel i on the target shape, and ui is theheight of voxel i on the uninflated baseline.
The transfer error is computed similarly but rather basedon the 20 fiducial markers mentioned. Each marker corresponds to 4 voxels on the simulated designs. Thus, thetransfer error can be computed as:Pm
j=0 |sj − ri |et = Pm(2)
j=0 |rj − uj |with stiff/inelastic voxels (E = 9000000 Pa). Our simulationneglected contact with the ground and gravity. We note that
we calibrated the simulator to these physical parameters byhand until we observed visual matching between simulation
and reality.
C. Experimental Design
To assess the fidelity of our inverse design approach, we
conducted experiments on two representative desired shapes.
The first is the well-known and well-understood saddle shape(Fig. 1) with negative Gaussian curvature [17]. The second
shape, the “monkey saddle” (Fig. 1), was chosen for its
slightly more complex mathematical formulation [18].
1) Simple case—saddle: For the simple saddle, we started
with the known solution of limiting strain in orthogonal
directions on the top and bottom of the sheet (i.e., limiting
x-direction stretch on the top surface and limiting y-direction
stretch on the bottom surface). We tested this design in reality
and achieved the desired saddle shape. We then translated thefiber placements to simulation to validate that the simulated
sheet produced the same inflated shape as the physicalsheet (the forward problem). Finally, we used the generated
shape in simulation as the target shape for the evolutionary
algorithm to match using evolved sets of fiber placements(the inverse problem).
2) Complex case—monkey saddle: For the monkey saddle
shape, we used the known mathematical function, f (x, y) =x3 − 3xy 2 , to generate the input target shape, then usedthe evolutionary algorithm to discover fiber placements thatshould generate the target shape in the inflated pneumatic
bilayer (the inverse problem only).
3) Sim2Real validation and error categorization: We used
two error metrics in our study: simulation error and transfer
error. Simulation error represents the error between a target
shape and a computer generated design, during the optimization process with the evolutionary algorithm. Transfer
error represents the error between the physical realizationand computer generated design (calculated as the percentage
error between heights of the 20 fiducial markers in reality
and in simulation). Simulation error measures the abilityof the algorithm to match a target shape, while transfer
error validates that the computer generated design can be
replicated in reality.
As described in §II.B., we employed a soft-body finiteelement simulation in which the elements take the form ofsmall cubical volumes: voxels. We used the height of thevoxels on the sheet to compare different shapes. The uninflated sheet provided the baseline heights of all voxels, andthe inflated sheet deviated from the baseline by some height
difference. Our optimization function aimed to minimize theerror between the target and evolved voxel heights, given as:Pn|si − ti |es = Pni=0(1)
i=0 |ti − ui |where et is the reality transfer error, m is the number ofmarkers, sj is the average height of the four marked voxels
on the evolved (simulated) shape corresponding to marker jin reality, rj is the height of marker j on the real bilayer,
and uj is the average height of the four marked voxels on
the uninflated baseline that correspond to marker j in reality.
D. The Evolutionary Algorithm
Evolutionary algorithms are a class of population-based,gradient-free optimization methods [19]. In most such algorithms, at each generation, individuals in a population areevaluated based on some fitness measure and individuals
with better fitness values are selected to reproduce in thenext generation.
We used a canonical genetic algorithm [20] to evolve fiber
patterns on the expanding sheet. We used a direct encoding
scheme for the genome, where the genotype indicates thestarting and ending points of the fibers on the bottom and top
surfaces of the sheet. The genetic algorithm was implemented
with elitism, with an elite size of one: the individual withthe highest fitness in each generation survives to the next,
without any changes to its genome. The fitness function was
to reduce error as described in Eq. 1.
To produce the next generation, we did not implement
crossover but instead utilized a two-way tournament selection(with replacement), followed by a mutation on the selectedindividual. The mutation randomly selects one of the fiber
endpoints and changes its position on the surface by drawing
a new (x, y) end position from a uniform distribution thesize of the total sheet. In other words, the new endpoint
position can be anywhere on the sheet’s surface. To fill in
the fiber locations between the starting and ending points on
the simulated sheet we employed Bresenham’s digital line
plotting algorithm [21].
For each of the two target shapes, we ran the evolutionary
algorithm with a population size p = 20, for g = 120
generations (i.e., 120 generations is one independent trial).
Population size and number of generations were restricted
to these low numbers, since the bladder was simulated in a
CPU-based soft body simulator (described in section §II.B.)
and evaluating each individual takes considerable time (≈ 22
seconds on a Quad-Core Intel Core i7 Macbook Pro). Despite
this low search effort, significant decreases in error wereachieved, as demonstrated in the next section.
We conducted three independent trials each for the normal
saddle and for the monkey saddle. Only a small number oftrials was possible, as each trial ran on average a little over
14 hours. Lastly, we limited the number of fibers generated towhere es is the error in simulation, n is the number of voxels,
si is the height of voxel i on the evolved (simulated) shape,
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TABLE IRESULTS SUMMARY FOR THE SADDLE CURVATURE .

Design1
2
3Simulation Error
3.37%
2.79%
3.19%

Transfer Error
10.31%
11.11%
12.93%

saddle with three valleys and three mountains—as shownin Figure 4. The shape was generated with the equation:f (x, y) = x3 − 3xy 2 . The output of this equation wasthen normalized within the range [a, b], where a was theminimum voxel height attained from the simple saddle andb the maximum voxel height. Using this target shape input,
the algorithm was able to achieve solutions with an averagesimulation error of ≈ 11.13%, with the lowest at 10.16%,
and an average transfer error of ≈ 15.84%, with the lowest
at 14.67%. It is worth noting that both the simulation andtransfer errors for the monkey saddle are higher than theerrors for the simple saddle, possibly due to the increased
complexity of the target monkey saddle shape.
TABLE IIRESULTS SUMMARY FOR THE MONKEY SADDLE CURVATURE .

Fig. 3. Inflated saddle shapes with different strain-limiting fiber placements.
Left column shows top-down and bottom-up views of fiber placements on
the top and bottom of the sheet in simulation. Illuminated sheets are physicalhardware instantiations. a: Hand-designed fiber placement. Three fibers on
top and three fibers on bottom making perpendicular angles. Scale bar = 5
cm. b-e: Evolved fiber placements (designs 1-3). All placements were able
to generate the simple saddle shape with a moderate transfer error (≈ 10%).

Design1
2
3Simulation Error
11.57%
11.66%
10.16%

Transfer Error
16.67%
14.67%
16.19%

C. Error minimization and solution diversity
We show the simulation error during evolution for boththe saddle and monkey saddle designs in Figure 5. The
simulation error for the saddle continues to drop until theend of the 120 generations, while the simulation error for themonkey saddle decreases faster at first but plateaus after the60th cycle. This discrepancy could potentially be mitigatedby employing an evolutionary algorithm based on AgeFitness Pareto Optimization (AFPO) [22], a multi-objective
optimization method for avoiding premature convergence.
We note here that while the generated curvatures from each
evolved solution in Figures 3 and 4 appear globally similar,
the fiber placements converge more in the simple saddle case.
For the simple saddle, the fiber placements from each trial(design) are visually similar, and this convergence suggests
the existence of one underlying solution for the saddle shape.
For the monkey saddle, the evolved fiber placements are
more visually different, which could mean that increasing
shape complexity corresponds to a wider variety of possible
fiber pattern solutions.
To measure the degree of convergence for the evolved
solutions quantitatively, we compared the different designs
using the Hausdorff distance metric, which has previously
been used to compare the similarity of images [23]. We
have adapted this distance metric for use here by representing
an evolved fiber design on the upper or lower surface as abinary 50 × 50 matrix in which unity values represent areas

be five fibers per face (top and bottom) for the simple saddle
case to simplify future transfer to reality. We increased thislimit to seven fibers per face for the monkey saddle case,due to its more complicated final curvatures.
III. R ESULTS
A. Simple saddle
The best evolved designs from the three independent trials
for the saddle, shown in Fig. 3b-d, embodied a set offiber placements similar to the hand-designed set of fiber
placements (the known solution, shown in Fig. 3a). This
result indicates that the evolutionary algorithm “rediscovers”
known optimal designs. On average, the evolved designs
achieved a small simulation error—the error between thetarget shape and evolved shape in simulation—of ≈ 3.12%,
with the lowest simulation error of 2.79% (see Table I). On
average, the physical realization achieved a moderate transfer
error—the error between the physical bladder and simulated(evolved) bladder—of ≈ 11.45%, with the lowest transfer
error of 10.31%. We note that the simulation errors are
similar across designs, as are transfer errors, demonstrating
the reliability and reproducibility of our pipeline.
B. Monkey saddle
To test our pipeline on a more complex shape with no
known solutions, we chose the monkey saddle shape—a633

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of the surface covered by fibers and nullity values represent
uncovered areas. The distance metric can then be written as
H(A, B) = max(h(A, B), h(B, A))(3)

where A and B are the sets of points/pixels in two evolved
designs A and B, in either the upper or lower surface, andh(A, B) = maxa∈A minb∈B ||a − b||. Here, a and b are nonzero points in A and B. The function h(A, B) identifies
the point a ∈ A that is farthest from any point of b ∈ B
and measures the distance from a to its nearest neighbour in
B [23]. H(A, B) then indicates how far one pattern has to“expand” in order to “cover” the other pattern. H(A, B) = 0
means the two patterns are identical. The larger H(A, B) is,
the more different the two patterns are.
TABLE IIIH AUSDORFF DISTANCES BETWEEN EVOLVED FIBER PATTERN DESIGNS .
Simple saddle (Top surface)
Design 1
Design 2
Design 3
Simple Saddle (Bottom surface)
Design 1
Design 2
Design 3
Monkey Saddle (Top surface)
Design 1
Design 2
Design 3
Monkey Saddle (Bottom surface)
Design 1
Design 2
Design 3

Design 1
0
Design 1
0
Design 1
0
Design 1
0-

Design 2
8.0
0
Design 2
12.1
0
Design 2
10.2
0
Design 2
14.3
0-

Design 3
8.0
6.1
0
Design 3
7.6
7.6
0
Design 3
10.3
7.0
0
Design 3
10.8
25.1
0

Fig. 4. Inflated monkey saddle shapes with different evolved strain-limiting
fiber placements. Left column shows top-down and bottom-up views of fiber
placements on the top and bottom of the sheet in simulation. Illuminatedsheets on the right are physical hardware instantiations. a-c: Evolved fiber
placements (designs 1-3). All placements were able to generate a monkey
saddle-like shape. Scale bar = 5 cm.

We computed the Hausdorff metric H between the fiber
patterns evolved to match the simple saddle and the monkey
saddle shapes, for both the top and bottom sheet surfaces(Table III). The average distance between the designs evolved
to match the simple saddle (8.23; average of top two panels
in Table III) is lower than the average distance between
designs evolved to match the monkey saddle (12.95; averageof bottom two panels in Table III). This result confirms
the qualitative observation that evolved fiber designs forthe simple saddle are more similar to each other while
the evolved fiber designs for the monkey saddle are morediverse. We conducted two control trials in which 36 random
fiber patterns were generated—one with five fibers on boththe top and bottom surfaces, and another with seven fibers
on each surface—and computed the Hausdorff distances
between each pattern according to Equation 3. The averageof these metrics was 17.98 for the random five-fiber patterns
and 16.33 for the random seven-fiber patterns. These two
numbers are larger than the average of all of our metrics
in the table above, which was 11.08, indicating that ouralgorithm does find sets of solutions that are more similar
than random solution sets.

Fig. 5. Simulation Error between evolved shape and target shape, over 120
generations for the simple saddle and the monkey saddle shapes. The solid
curves represent the average error for all designs; Shaded regions show one
standard deviation of the averaged error.

the sheet to morph into different shapes upon inflation. We
have also demonstrated a sample pipeline for automaticallydesigning the placement of the strain-limiting fibers on thepneumatic sheet to induce a desired curvature. We haveshown that evolutionary search is a promising approachthat repeatedly converged on a known solution for a simple
target shape (saddle curvature), and produced a diversity ofconvincing results for a more complex target shape (monkey
saddle curvature). We believe this pipeline could extend to
the design of fiber patterns that produce increasingly complex
shapes in pneumatic sheets via improvements in the bilayer
construction, the use of more fibers, and/or stronger searchmethods that may enable the discovery of additional patterns.
Although formal methods for deriving strain limitation

IV. C ONCLUSION AND FUTURE WORKIn this paper, we introduced a pneumatic bilayer sheetwith reconfigurable strain-limiting surface fibers that enable634

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are possible for complex shapes, such methods assume
perfect control over local strain at all points and tend to bemathematically burdensome. Thus, empirical approaches likeevolutionary algorithms can be reasonable alternatives. Evolutionary algorithms are particularly well-suited to optimizations in which the number of parameters is not fixed [24],[25]. For the design space herein, additional objectives could
include rewarding less total fiber material, fewer numbers offibers, fewer fiber crossing points, or making fibers with nonzero curvatures. Such multi-objective optimization could be
employed as a method to reduce either the simulation error(indicating “better” solutions) or transfer error (indicating
more manufacturable solutions).
Finally, our implementation of reconfigurable fibers ona pneumatic bilayer sheet allowed us to use the same
pneumatic bilayer for all experiments herein. With continued
progress toward variable tensile stiffness materials [26], [27],our pipeline could be used to develop shape-changing sheetsthat can autonomously adapt without any human handling.
Evolving fiber placements could also augment existing evolutionary algorithms that already simultaneously optimize theshapes, material properties, and control policies of soft robots[28], [29], [30], thus paving the way toward automatedshape-changing robots that adapt to different environments
and tasks [31].

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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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