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

Deployable cuboctahedrons for adaptive space
infrastructure2024 IEEE 7th International Conference on Soft Robotics (RoboSoft) | 979-8-3503-8181-8/24/$31.00 ©2024 IEEE | DOI: 10.1109/ROBOSOFT60065.2024.10521988

Stephanie J. Woodman1 , Alex Moore1 , Siona Tagare1 , Kenneth Cheung2 and Rebecca Kramer-Bottiglio1

Abstract—Diverse space infrastructure is required for exploration missions to the Moon, Mars, and beyond. However, thecost of sending materials into space is high. One approachto ease this cost is the use of adaptive infrastructure, which
may leverage discrete building blocks that can be assembled,
disassembled, and reassembled into diverse mechanical structures based on the relevant environment and task demands.
Indeed, the NASA Automated Reconfigurable Mission Adaptive
System (ARMADAS) project is taking this approach. The
discrete building component selected by ARMADAS engineers
is a cuboctahedron, or more simply a “voxel,” as a volumetricpixel. The voxels are lightweight and simple, and assemble
into programmable mechanical metamaterial structures withhigh stiffness and stability. However, transportation of complete
voxels remains volume-inefficient, and fabrication of voxels insitu adds notable complexity to the system. Herein, we introduce
a cuboctahedron voxel design that compresses to 35% of its
deployed volume during transport and passively locks in its
expanded state at its destination, where a multitude of voxels
can then be assembled. Inspired by the Hoberman sphere,
the voxel is designed to deploy using a 1D force input. We
further confirm that the new deployable voxel is compatible
with existing ARMADAS assembly agents.
Index Terms—shape-changing robots, morphing robots, modular infrastructure, reconfigurable infrastructure

I. I NTRODUCTION
Modular, lightweight, and reconfigurable infrastructurethat can be easily assembled, disassembled, and reassembled
could apply to diverse sectors such as humanitarian aid,disaster relief, and space exploration. For example, areas
impacted by storms, fires, or flooding require quick deployment of infrastructure such as wireless antennas for restoring
communication, scaffolding for restoring power, or tents for
impromptu patient care. Off-planet, NASA is interested in
the capability to build infrastructure such as solar power,
communications, railway, and habitat systems in space oron extraterrestrial bodies. Key requirements include ease oftransportation (low volume and mass during transport tothe destination), adaptability (global reconfiguration), andstructural stability (an ability to withstand infrastructure-levelloads).

Fig. 1. ARMADAS platform for adaptable space infrastructure. a.
Vision of the system assembling infrastructures in an extraterrestrial environment. b. Our deployable Hobervoxel integrated into the present system,
surrounded by static voxels and with assembly agent robots.

S.J.W. was supported by a NASA NSTGRO Fellowship(80NSSC22K1188). R.K.B. was supported by NASA grant
80NSSC20M0098. A.M. was supported by the National Science Foundation(NSF) under IIS-1954591. S.T. was supported by the NSF under EFMA1830870. K.C. was supported by the NASA STMD Game Changing
Development Program (ARMADAS Project).
1 Department of Mechanical Engineering and Materials Science, Yale University, New Haven, CT, USA stephanie.woodman@yale.edu
2 Intelligent Systems Division, NASA Ames Research Center, Moffett Field,
CA, USA kenny@nasa.gov

979-8-3503-8181-8/24/$31.00 ©2024 IEEE

Inspired by the collective works on reconfigurable andshape-changing robots [1]–[4] as well as high-performancemechanical metamaterials [5]–[7], NASA’s Coded Structures
Laboratory (CSL) at the NASA Ames Research Centerhas been developing the Automated Reconfigurable Mission
Adaptive Digital Assembly Systems (ARMADAS) [8] projectto extend the re-usability and material efficiency of modular
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Fig. 2. Design overview and expansion ratio. Both the contracted and expanded versions of the Hobervoxel, with key features identified.

engineered systems to space infrastructure that is roboticallyassembled and maintained (Fig. 1a-b). The ARMADAS teamhas designed static and ultralight modules, called “voxels,”
which are made of a carbon fiber composite [9] that outlines a
cuboctahedron shape. The voxels can be assembled together
into cuboctahedral lattice materials, then disassembled andreassembled into variable large-scale structures based on theenvironment or use demands [8], [10]. The ARMADAS teamhas further developed two robots, both on the same scale as
the voxels, which autonomously complete the assembly tasks.
One type of robot carries and places voxels (“SOLL-E” [11],Fig. 1b), while another moves within the structure to secure
voxels together (“MMIC-I” [12], [13], Fig. 1b). Overall,
the ARMADAS system is reliable due to the simplicity ofthe task-specific robots, and the ultralight voxels render theassembled structures extremely mass efficient [14], [15].

shape-adaptive mobile robot [22]. However, the advantagesof a Hoberman-type approach have not been investigated inweight- and space-limited scenarios where the Hoberman
design would need to perform similarly to a static, nondeployable system once expanded.
We applied the Hoberman mechanism to the ARMADAS
voxels to create a deployable voxel, which we call a Hobervoxel. The Hobervoxel compresses to 35% of its initial
volume (inversely, increasing in volume by 286% between
the contracted and expanded states) and passively locks into
a rigid cuboctahedron when expanded. The expanded cuboctahedron further matches the geometry of the static voxels
and is compatible with the locking [23] and gripping [13]mechanisms on the ARMADAS assembly robots. Our Hobervoxels provide a pathway to programmable, adaptive space
infrastructure by adding a layer of volumetric shape change
to the system.

Though the ARMADAS system is light, strong, and adaptable, the voxels take up a large fixed volume and thus are very
costly to transport. Therefore, we introduce an augmented
voxel design that can be initially collapsed and then deployedwhen needed. We aim to create voxels that take up a lower
storage volume during transport, and then irreversibly expand
into the original voxel building block once in the target
environment. Inspired by the global contraction/expansionof a Hoberman sphere based on a single degree of freedom (DoF) input, we applied Hoberman linkages to the
cuboctahedron voxels. The Hoberman sphere achieves its
contraction and expansion due to its rings of scissor pairs,
each composed of two struts, connected at pivot points [16].
Analyses have explored the Hoberman sphere’s kinematics [17], symmetry [18], and generality [19]. In simulation
and a close-to-ideal case, the Hoberman sphere can contract
to just 12.5% of its initial volume [20]. The Hoberman
mechanism has previously been applied to several robots,
including a snake-inspired swallowing robot using sequential
Hoberman rings [21], a soft jet propulsion system that rapidly
compresses to expel water and generate thrust [20], and a

II. D ESIGN AND FABRICATION
Our Hobervoxel design was determined by 1) the ARMADAS system’s operational constraints (e.g., voxel assembly and attachment mechanisms), 2) 1 DoF force input for
deployment, and 3) the need for structural strength, rivalingthat of the static voxel in the expanded form (Fig 2).
A. Operational constraints
The current assembly agent robots developed by NASA’s
ARMADAS project create two design constraints. First,
the SOLL-E robot, which carries and places voxels as it
crawls over the lattice structures, is tailored to the specificcuboctahedron voxel geometry (Fig. 3a), with 203 mm edge
lengths, in order to grip and traverse the voxels. Second,
the MMIC-I robot, which fastens voxels together using an
androgynous fastening mechanism [23], requires a particulargeometry of the voxel corners (Fig. 3b). This geometry has
circular holes to hold the fasteners and gripping bars forboth robots, resulting in a rigid pyramid structure at eachcorner. Our Hobervoxel design retains the specific shape,
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In a scissor mechanism that produces the desired trajectories, the orange and pink points in Fig. 4c must closely follow
the blue and purple ideal trajectories (shown in Fig. 4c,d),respectively. The blue line refers to the axial trajectory of
the corner and base, while the horizontal purple line refers
to the constant orientation of the vertices. To characterize
the performance of a mechanism with specific L and ϕ,
we calculated the optimal position of the orange and pink
points at each increment of folding in the simulated scissor
mechanism. Specifically, we ensured that, at each incrementof folding, the tracked points were as close as they could beto the ideal trajectory. We recorded these positions and usedthem to plot the mechanism’s optimal simulated trajectory,
as in Fig. 4d.
To find functional and spatially efficient solutions, we
swept L and ϕ values and calculated a quality metric foreach pair in order to compare their performance. We took the
distance between the ideal and calculated trajectories at every
point along the lines and set the quality metric as the standard
deviation of these distances, which could be expressed as

Fig. 3. Static voxel form factor. a. The complete static voxel with acharacteristic length of the system. b. The corner of the static voxel withrobot and fastener interface features. c. The corner of the Hobervoxel ismade to replicate the corner of the static voxel.

σ(A) f ordimensions, and vertex geometry as the original static voxels
to maintain compatibility with the assembly agents (Fig. 3c).

A = {T calc y (a) − T ideal y (a) |a=1:240 }where T calc y (a) is the y-coordinate of the calculated trajectory at increment a along the trajectory, T ideal y (a) is they-coordinate of the ideal trajectory at increment a along the
trajectory, and the trajectory is divided into 240 increments.
We then plotted the quality metric over the space of all possible mechanisms defined by L and ϕ, where smaller valuesof the quality metric described mechanisms whose calculated
trajectory was closer to the ideal path (Fig. 4e). The simulated
results did not account for the fact that some solutions are
physically impossible, like values of L approaching 60 mm.
Thus, the chosen design was not simply the mechanism withthe smallest quality metric, especially because many different{L, ϕ} pairs have a sufficiently small quality metric, but
are non-functional in practice. Ultimately, the chosen design,
shown as the red star in Fig. 4e, set L = 40 mm and
ϕ = 40◦ . This was a middle-ground between a minimized
quality metric and maximized mechanical feasibility. This
design achieved a compression ratio of 35% (Fig. 2), thetheoretical ideal given the large size of our corners.

B. Hobervoxel deployment
Working within the operational constraints, we aim to
reduce the volume of the voxel for transport using a simple
compression mechanism.
Our approach is inspired by the compression and expansionof the Hoberman sphere, which actuates using a single,
uniaxial force input. Each vertex in a Hoberman sphere has
an internal and external component, which are the “base” and“corner,” respectively, in our design (Fig. 2). The vertices arejoined by a scissor mechanism, comprising two pairs of two
struts, Strut 1 and Strut 2 (Fig. 4a,b), where Strut 1 connects
to the base, and Strut 2 connects to the corner (Fig. 2). Holistically, the Hoberman sphere works by preserving the relative
geometry and orientation of the vertices during compression.
Using our terminology, this means that the corners and bases
follow fixed relative trajectories, as dictated by the geometryof the scissor mechanism. The corner and base pieces moveaxially away from the center of the cuboctahedron during
expansion while maintaining their orientation relative to theother vertices.
We developed a simulation to find and optimize a scissor
linkage geometry that would meet the above constraints formotion. The simulation is based on a closed-form geometricmodel that uses a strut length of 60 mm (Fig. 4a,b) and a
Strut 1 height of 5.5 mm (Fig. 4a) based on the operational
constraints detailed in § II.A. The equations comprising
this model are detailed in § VI for completeness. The freevariables of the model are length L and angle ϕ of Struts
1 and 2 (Fig. 4a,b). These variables characterize the widthand height of the isosceles triangle formed when the scissor
mechanism expands (Fig. 4c).

C. Hobervoxel structural strength
After designing the deployment mechanism, we sought tomake the Hobervoxel robust enough to act as a permanent
structural component once deployed. We printed all parts on a
Formlabs Form 3+ resin printer, due to the need for precision
at small scales, and assembled the parts with 0-80 hardware.
Without any locking mechanisms, the Hobervoxel tends tocontract toward its low-volume state. To solve this problem,
we chose to use two locking mechanisms based on rivets andbuckles.
To lock the base and corner pieces together, we integrated
split-shank rivets (Fig. 5b) onto the corners and corresponding receiving holes into the bases. As the Hobervoxel
deploys, the rivets move toward their receiving holes and77

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Fig. 4. Identifying an optimal design using simulation. a. Strut 1 of a 2-strut scissor pair with constraints called out, and the variable L that was
optimized in the simulation. b. Strut 2 of a 2-strut scissor pair with constraints called out, and the variables L and ϕ that were optimized in the simulation.
c. Two scissor pairs that make up an edge of the compressible voxel showing three points along the trajectory of (d), and the expanded and compressededge lengths. Strut 1 highlighted in orange, Strut 2 highlighted in pink, and the ideal trajectories in purple and blue. d. The calculated trajectory resultsof the simulated L, ϕ pair and how they compare to the ideal trajectories. e. Varied L and ϕ plotted against their quality metric, with the chosen design
pointed out.

each other. The locking mechanism requires a strong andelastic material that allows the male buckles and rivet heads toflex temporarily before snapping into the expanded position,
which led us to choose Formlabs’ Tough 1500 resin for our
Hobervoxel prototypes.

permanently lock each base and corner together once it iscompletely deployed.
The scissor mechanisms undergo a higher combined load
than the bases and corners and thus require a more robust
mechanism to rigidly lock them once deployed. We integrated
buckles to provide locking and rigidity. The bottom two struts
have the male portion of the buckle, and the top two struts
have the female part. The male part locks into the female
part as the Hobervoxel deploys (Fig. 5a). Once expanded, thescissor mechanism is impossible to fold further because its
yoke-style design would cause parts of the struts to interfere,and cannot be unfolded without the large force required tosnap the prongs in the male buckle. The buckles act as stabilizing tension elements between the top and bottom struts
when the scissor mechanisms are locked, increasing theirrigidity. Importantly, the Hobervoxel’s uniform volumetric
expansion means both the buckle and rivet mechanisms lockpassively, as all mating pieces simultaneously move toward

III. E XPERIMENTAL D ESIGN
After designing the Hobervoxel, we aimed to quantify thedifferences in mechanical performance between the staticand deployable voxels, so we fabricated them out of thesame resin. Since these voxels will ultimately make up a
cuboctahedral lattice metamaterial and be treated as a cellularsolid, we additionally compare the performance of the voxel
subunits to their constituent material—Tough 1500 resin—asa baseline to evaluate the geometric designs.
To capture the differences in mechanical performance, we
tested a cylinder of the neat Tough 1500 resin (5 samples),the static voxel fabricated with the same resin (3 samples),78

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Fig. 5. System rigidity and locking. a. The buckle that locks the scissor pairs together in the expanded state. b. The split-shank rivets that lock the cornerto the base.

voxel. As we saw failure modes only within the struts on
both voxel types, we estimate that the increased performance
metrics of the Hobervoxels are due to the increased bulk of
the struts.

and the Hobervoxel again fabricated with the same resin(3 samples), in compression according to ASTM D695-15(Fig. 6). From this data, we extracted the average compressive
strength of each sample and used this, along with the average
weights of the samples, to get the strength-to-weight ratios(σWRs) (Table I). We also calculated the stiffness as the slopeof the linear regions, the elastic modulus, and the proportional
limit (stress just before plastic deformation at the end of theelastic region).

This study was constructed using the same material forboth the static and deployable voxel prototypes, such thatthe differences in mechanical performance would generalize
across other materials for future systems. Indeed, because the
Hobervoxel includes additional hinges, locking mechanisms,
and strut bulk relative to the static voxel, we expect the slight
performance increases to generalize due to increased weight,
with the assumed reduction in volume during travel (allowing
for fewer trips with material) outweighing the weight penalty.
Nevertheless, given the ARMADAS static voxel geometry
used here is optimized for weight and mechanical performance when manufactured from a carbon fiber-reinforced
polymer material, it will be necessary to carry out the same
static vs. deployable voxel comparisons with the specializedmaterial in the future.

IV. R ESULTS
The Hobervoxel has a higher compressive strengthand strength-to-weight ratio (σwR) than the static voxel(Fig. 6b,c, Table I). An interesting property of the Hobervoxelis that we did not see the traditional decrease in stresswith increasing strain beyond the ultimate tensile strength
point during the compression testing. Before our Hobervoxel samples could fail traditionally, the internal corners of
the structure hit each other, causing the stress to increase
dramatically at the highest strain. We expected to see adecreasing stress-strain relation much sooner due to added
joints in the Hobervoxel relative to the static voxel. Instead,the Hobervoxel showed many stick-slip points that let thestress increase slowly, but continuously, even to extreme
strains. Given this, we took our compressive strength as the
maximum strength before the structure contacted itself.
The neat material showed the highest compressive strength,
σWR, stiffness, elastic modulus, and proportional limit–
orders of magnitude above both voxel types. Though the neatmaterial is impressive, it would weigh too much at larger
scales to be worth the increases in strength—the structure
would be overbuilt with solid blocks. The Hobervoxels had
higher average stiffness, elastic modulus, and proportional
limit than the static voxels, though they were on the sameorder of magnitude, showing an overall increase in mechanical performance than the static voxels.

Aside from material choice and design refinement, an
important next step will be to implement an actuation mechanism for deployment from the compressed state once inthe target environment. A few methods have been discussed,
including the development of a new assembly agent robotthat would move voxel to voxel with a simple inflatingor linear actuator, expanding the voxels into their locked
forms. Another approach may be a spring-loaded designthat could be transported in a constrained volume, and thenautomatically expand when released from the constraint.
In general, robotic assembly of structural modules isseen as a strong candidate solution for very large spacestructures, such as habitats, antennas, industrial facilities, andscientific instrumentation requiring large apertures. While
there are ongoing efforts to develop the ability to source
material from space, near-term operations will rely heavily
on materials launched from Earth, at an extremely high cost.
Therefore, these structures must be highly efficient—in the
low-gravity environments currently considered, this naturally
favors frameworks of modules that are composed of slender
struts, like the voxels presented herein. An ability to pack anddeploy such structural modules onsite presents an appealing
alternative to the robotic or manual assembly of modules
from stock strut and node parts. Thus, the Hobervoxel and

V. D ISCUSSION
The Hobervoxel saw an increase in mechanical performance compared to the static voxel, with benefits in σWR,
stiffness, elastic modulus, and proportional limit, while significantly improving volumetric efficiency for space transport. The results indicate that the Hoberman mechanism isa feasible approach to creating a deployable cuboctahedron
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Fig. 6. Results of compression testing. Stress vs. strain for the a neat material, b static voxel, and c Hobervoxel, with the compressive strength identified.
Error clouds represent one standard deviation.
SampleMass [g]σWR [N/kg ∗ m2 ]Stiffness [N/m]Elastic Modulus [N/m2 ]Proportional Limit [N/m2 ]Neat Material
Static Voxel
Hobervoxel

3.73
211.9
491.2

2.79e39.61e-4
16.70e-44.38e6654.3
2.16e32.71e83.87e37.15e37.27e6106.5
125.1

TABLE IM ECHANICAL PERFORMANCE METRICS OF NEAT MATERIAL , STATIC VOXEL , AND H OBERVOXEL .

ψx1 is the X coordinate of the pink point. The set of equations
that describe the system are:
5.5



d=
5.5
−1
sin tan
D−L


5.5
−1
θ = tan
D−L

ARMADAS system may be understood as a conceptual
bridge between a status quo of manually assembled spacesystems and an envisioned future of deployable and robotically assembled space systems. Such hybrid deployed andassembled systems hold promise in realizing the benefits ofboth types of systems where appropriate.
We hope that this Hoberman-inspired voxel deployment
strategy will be broadly used to achieve large volumetricshape-change across the fields of modular and soft robotics
and adaptive infrastructure. In the context of this project,
our simulation and design approach show promise for use inNASA’s ARMADAS system, the deployment of which will
enable the next phase of space exploration to the Moon, Mars,and beyond.

b=Lcos(ϕ)

1
κ = (D − L)2 + b2 − 2(D − L) ∗ b ∗ cos(π − ϕ) 2b ∗ sin(π − ϕ)−1
η = sinκ
γ =ϕ−η
1
h = L2 + b2 − 2 ∗ b ∗ l ∗ cos(π − ρ + ϕ 2
L ∗ sin(π − ρ + ϕ−1
α = 2sin =hVI. E QUATIONS FOR THE G EOMETRIC M ODELThis set of equations describes the geometry and motionof the scissor linkage mechanism during compression, which
we used to simulate the system. L, and ϕ (Fig. 4a,b) describe
the geometry of the scissor mechanism: L is the length ofthe portion of the scissor closer to the center, and ϕ is theangle of the bend in Strut 2. In our chosen case, L = 40 mm
and ϕ = 40◦ . D is a constant describing the length of eachstrut, which was 60 mm.
The variables ρ and δ control the position of the mechanism in simulation, where ρ dictates how folded the scissor
is, and δ dictates its overall angle relative to the coordinate
axes. Simulating over small increments of ρ and δ ensuressmoothness in the simulation results (Fig. 4d,e). The calculated values ψ1 and ψ2 refer to the location of the labeled pink
and orange points, respectively, from Fig. 4c. For example,1

ϵ = (2κ2 − 2κ2 cos(α + 2γ)) 21

σ = ((D − L)2 + d2 − 2d(D − L)cos(π − ρ + θ)) 2
dsin(π − ρ + θ)
−1
N = sinσπ − α − 2γ
+δ−η
ψx1 = ϵ ∗ cos2
π − α − 2γ
+δ−η
ψy1 = ϵ ∗ sin2
ψx2 = σsin(δ − N )ψy2 = σcos(δ − N )80Authorized licensed use limited to: Yale University. Downloaded on May 24,2024 at 15:35:07 UTC from IEEE Xplore. Restrictions apply.

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