Computationally efficient design of directionally compliant metamaterials

ARTICLE

Computationally ef

cient design of directionally

compliant metamaterials

Lucas A. Shaw

, Frederick Sun

, Carlos M. Portela

, Rodolfo I. Barranco

, Julia R. Greer

& Jonathan B. Hopkins

Designing mechanical metamaterials is overwhelming for most computational approaches

because of the staggering number and complexity of

exible elements that constitute their

architecture

—

particularly if these elements don

’

t repeat in periodic patterns or collectively

occupy irregular bulk shapes. We introduce an approach, inspired by the freedom and

constraint topologies (FACT) methodology, that leverages simpli

ed assumptions to enable

the design of such materials with ~6 orders of magnitude greater computational ef

ciency

than other approaches (e.g., topology optimization). Metamaterials designed using this

approach are called directionally compliant metamaterials (DCMs) because they manifest

prescribed compliant directions while possessing high stiffness in all other directions. Since

their compliant directions are governed by both macroscale shape and microscale archi-

tecture, DCMs can be engineered with the necessary design freedom to facilitate arbitrary

form and unprecedented anisotropy. Thus, DCMs show promise as irregularly shaped

exure

bearings, compliant prosthetics, morphing structures, and soft robots.

https://doi.org/10.1038/s41467-018-08049-1

OPEN

Mechanical and Aerospace Engineering, University of California, Los Angeles, Los Angeles, CA 90095, USA.

Division of Engineering and Applied Science,

California Institute of Technology, Pasadena, CA 91125, USA. Correspondence and requests for materials should be addressed to

J.B.H. (email:

hopkins@seas.ucla.edu

)

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1234567890():,;

M

echanical metamaterials (a.k.a. architected materials) can

achieve extreme properties th

at derive primarily from their

architecture instead of their composition

. By controlling

the locations and orientations of mic

roelements (e.g., beams, blades,

and hinges) that constitute their architecture, such materials can be

engineered with super properties otherwise not achievable (e.g.,

extreme strength-to-weight ratios

,tunablenegativethermalexpan-

sion coef

fi

cients

, and large negative Poisson

’

sratios

Past research has primarily focused on in

fi

nite periodic meta-

materials that achieve their engineered properties with isotropy

because such materials consist of single symmetric cells that

repeat without bounds and are thus manageable to design despite

their numerous constituent elements. Unfortunately, such meta-

materials have limited use because most practical applications

require materials that occupy

fi

nite and often irregularly shaped

volumes and achieve anisotropic properties tailored along pre-

scribed directions. Metamaterials that meet these demands

usually require huge numbers of aperiodic (i.e., nonrepeating)

asymmetric cells that occupy volumes with complex boundaries

and are thus too computationally expensive to design.

Previous work has sought to address these challenges by uti-

lizing precomputed databases of different cell designs to generate

aperiodic and practically shaped metamaterials that achieve

desired deformations

or targeted regions of compliance

. Finite

element analysis (FEA), sparse regularization, and constraint

optimization have been employed to generate shapes consisting of

aperiodic distributions of different materials that deform in pre-

scribed ways when actuated

. Aperiodic metamaterials have also

been designed with graded properties (e.g., elasticity

and thermal

expansion

), which vary across their lattice

’

s geometry. Addi-

tionally, metamaterials that exhibit desired textures when actu-

ated have been designed using a single anisotropic cell that is

oriented in nonrepeating patterns

. Lastly, aperiodic lattices of

shearing cells have been used to generate monolithic mechanisms

that achieve desired deformations

Despite these advances, a large computational gap remains

between metamaterial research and the ability to implement that

research within most practical applications. A new approach is

necessary to bridge this gap by leveraging simpli

fi

ed assumptions

to enable the automated design of aperiodic metamaterials of

staggering complexity and achieve customized anisotropic prop-

erties while assuming any form.

Metamaterials designed using this approach are called direc-

tionally compliant metamaterials (DCMs) because they are

engineered to achieve high compliance along desired directions

while exhibiting high stiffness along other directions. In contrast

with traditional

fl

exure systems

, which are currently used to

achieve desired directions of compliance (i.e., degrees of freedom

(DOFs)), DCMs can be engineered to assume any bulk shape

while achieving unprecedented combinations of DOFs. The rea-

son is that unlike

fl

exure systems, which achieve DOFs almost

exclusively according to how they are shaped on the macroscale,

DCMs achieve their anisotropic properties both according to

their macroscale shape as well as their architecture at smaller

scales. Thus, the design space of DCMs that achieve desired DOFs

while simultaneously assuming desired bulk shapes is sig-

fi

cantly larger than the design space of

fl

exure systems that

achieve the same objectives.

An example that demonstrates these advantages is a prosthetic

elbow joint. Although

fl

exure systems (e.g., Fig.

a) could achieve

the joint

’

s desired rotational DOF with high compliance while

possessing high stiffness in all other directions, no

fl

exure system

could also assume the irregular shape of an elbow. An aperiodic

DCM (e.g., Fig.

b) could, however, achieve the desired rotational

DOF (Fig.

c) while also assuming an elbow shape. Such a joint

would avoid the need for assembly and could be additively fab-

ricated as a monolithic structure while mimicking an elbow with

greater practicality and

fi

delity.

In addition to enabling directionally compliant joints, DCMs

can facilitate other shape-morphing applications. A DCM could,

for example, be shaped on the macroscale as a propeller blade but

be engineered with a microarchitecture that exhibits a screw DOF

(i.e., a translation coupled with a rotation)

about the blade

’

s axis

while achieving high stiffness in all other directions. The pitch of

the screw DOF could be tuned such that its corresponding blade

would passively recon

fi

gure its angle of attack proportionate to

the angular speed of the propeller due to centripetal forces. Other

DCM applications are discussed in Supplementary Note 1 and

shown in Supplementary Fig. 1 and Supplementary Movie 1.

Most DCMs are currently impossible to design because their

architecture typically consists of unmanageably large numbers of

nonrepeating

fl

exible elements that collectively occupy irregularly

shaped volumes. Existing computational approaches (e.g., topol-

ogy optimization

) become overwhelmed when searching the

design space of DCMs because the space is in

fi

nitely large and the

process of searching the space requires the simultaneous opti-

mization of huge numbers of parameters.

This paper introduces the theory necessary to design arbitrarily

shaped DCMs that are locally comprised of easily computed

anisotropic constituents. Inspired by the mathematics underlying

the freedom and constraint topologies (FACT) approach

–

this theory leverages simpli

fi

ed assumptions about constituent

elements to enable the automated design of three-dimensional

(3D) DCMs of immense complexity with unmatched ef

fi

ciency.

We demonstrate the theory

’

s computational superiority using our

MATLAB tool (see Supplementary Software) and introduce the

principles that govern how both macroscale form and archi-

tecture affect the DOFs of DCMs.

Results

Design approach

. The approach introduced here leverages the

vector spaces of the FACT library

–

graphically depicted in

Supplementary Fig. 2 to rapidly generate DCMs with desired

DOFs. The vector spaces of the FACT library utilize screw the-

ory

–

and collectively embody the design space of all compliant

systems. One set of spaces, called freedom spaces

–

, consist of

red rotation lines, green screw lines, and black translation arrows

and represent all the combinations of DOFs that a system could

achieve. Another set of complementary spaces, called constraint

Rotation

Fig. 1

Introduction to directionally compliant metamaterials (DCMs).

exure system that exclusively achieves the desired rotational degree of

freedom (DOF) of an elbow but fails to assume its shape and,

aperiodic DCM that can be shaped to conform to any elbow shape while

also achieving high stiffness in all directions except about,

, the desired

compliant rotational DOF

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spaces

–

, consist of blue constraint-force lines and represent

the region of space within which

fl

exible elements must be placed

to achieve the DOFs of their corresponding freedom space.

Additional FACT-library details are discussed in Methods.

Although the FACT library was originally created to facilitate

the synthesis of

fl

exure systems via a paper

–

pencil approach, this

work demonstrates that an advanced automated approach can

leverage the same library alongside computation to enable the

rapid generation of complex DCMs with even greater bene

fi

tto

the

fi

eld of metamaterials. Whereas other approaches fail to

generate DCMs because their computational cost is too high, the

approach introduced here can generate DCMs with orders of

magnitude less cost. The reason is that unlike other computa-

tional approaches that simultaneously consider the constituent

material properties, geometric parameters, locations, and orienta-

tions of every element within a DCM, the approach introduced

here simpli

fi

es the scenario signi

fi

cantly by only modeling the

locations and orientations of each element using 6 × 1 pure-force

wrench vectors (PFWVs)

–

6×1

. These vectors are depicted

as the blue constraint-force lines within the constraint spaces of

the FACT library in Supplementary Fig. 2. The mathematics

required to de

fi

ne PFWVs and to use these vectors to model

elements of any geometry are provided in Methods.

Elements modeled using PFWVs are treated as ideal elements

that are in

fi

nitely stiff along the axes of the blue constraint-force

lines that pass through the element

’

s geometry but are in

fi

nitely

compliant in all other directions. This assumption dramatically

simpli

fi

es the design process such that the locations and

orientations of hundreds to thousands of elements per second

can be determined within DCMs using a standard desktop

computer. Although the ideal-element model produces DCMs

that would theoretically exhibit in

fi

nite stiffness in all directions

except along their in

fi

nitely compliant DOFs, once geometric

parameters and material properties are assigned to their elements,

the DOFs achieved by such DCMs actually exhibit

fi

nite

compliant values that are consistently the most compliant of all

other directions.

The proposed approach

’

s steps are brie

fl

y summarized here.

The DCM

’

s volume is

fi

rst divided into smaller cell volumes

within which elements are to be placed to ensure that each cell

will individually achieve the desired DOFs. A DCM consisting of

many such smaller cells could be made to assume a variety of bulk

shapes without compromising the desired DOFs because each cell

is redundant and can, therefore, be removed from the material

’

volume with minimal consequence. Once the DCM

’

s volume is

divided into constituent cell volumes, the DCM

’

s desired DOFs

are then modeled as 6 × 1 twist vectors

–

6×1

, according to

the mathematics detailed in Methods. The freedom space that

represents the combination of all the desired DOFs is then

calculated by linearly combining the twist vectors that model each

DOF. The complementary constraint space of the resulting

freedom space is then identi

fi

ed using the FACT library. If this

constraint space belongs within the region shaded yellow in the

FACT library of Supplementary Fig. 2 (i.e., 0 DOF Type 1, 1 DOF

Type 1 through 3, 2 DOF Type 3 through 9, and 3 DOF Type 2

and 3), the geometry of that constraint space can be used to

determine the appropriate kind, number, location, and orienta-

tion of

fl

exible elements within each cell volume according to the

theory in Methods. Such constraint spaces that lie within the

yellow shaded region of the FACT library are called cell spaces

because they are the only constraint spaces that can occupy any

volume of space with enough independent PFWVs to generate

cell topologies that achieve their intended DOFs. Thus, if the

desired freedom space

’

s complementary constraint space is not a

cell space, it can

’

t be used to synthesize the DCM

’

s cells. As a

result, alternating layers of cells that each achieve some of the

DOFs within the freedom space should be designed to collectively

achieve all the DOFs within the freedom space when they are

stacked together in series. To synthesize such serially-stacked

layers, intermediate freedom spaces

should be selected from

within the freedom space according to the rules in Methods. Each

intermediate freedom space selected represents the combination

of the DOFs that each serially-stacked cell layer will contribute to

the DCM

’

s freedom space. The intermediate freedom spaces

selected must link to complementary constraint spaces that are

cell spaces because these spaces must then be used to generate the

individual cell topologies within the DCM

’

s alternating cell layers.

A case study of the design approach is provided here and

animated in Supplementary Movie 2. The case study is a DCM

that achieves a single screw DOF with a desired pitch,

, as shown

by the green line in Fig.

a. The DCM volume is

fi

rst divided into

individual cell volumes as shown. The freedom space of the

desired screw DOF is then identi

fi

ed as the freedom space labeled

1 DOF Type 2 in Supplementary Fig. 2 (Fig.

b). Its constraint

space consists of nested circular hyperboloids

fi

lled with PFWVs

that satisfy

=

ˑ

tan(

) according to the geometric parameters,

and

, labeled in Fig.

b. Since the constraint space is a cell space

(i.e., it belongs within the region shaded yellow in Supplementary

Fig. 2), each cell that constitutes the

fi

nal DCM design (Fig.

c) is

synthesized from within the geometry of the constraint space

according to the rules provided in Methods. The resulting DCM

consists of nine identical stacked layers (Fig.

d) constructed

using six different cell designs (Fig.

e) that each utilize

fi

ve wire

elements (i.e., slender cylindrical beams) aligned with indepen-

dent PFWVs from within the constraint space of Fig.

Although this space contains enough independent PFWVs that

pass through the volume of each cell within the DCM because the

space is a cell space, not all of the PFWVs

’

corresponding colinear

wire elements can directly join the cell

’

s rigid bodies together

without layer extensions that protrude from these bodies. Thus,

layer extensions are used within some of the cell designs (i.e., the

blue, green, yellow, and red cells in Fig.

e). If a higher cell

resolution had been speci

fi

ed such that many more cells would

have been generated, a propeller-blade shape could have been

carved out of the resulting DCM without altering its screw DOF

to enable the propeller application discussed previously.

We fabricated the DCM of Fig.

c at the microscale using two-

photon lithography, which achieved minimum feature resolutions

of ~1.5

μ

m (Fig.

a). To validate the desired screw DOF, we

performed in situ uniaxial compression experiments (Supple-

mentary Movie 2) while tracking the rotation of each rigid layer

using a scanning electron microscope (SEM). Imposing quasi-

static deformation (

_

=

−

−

) to the elastic strain limit

(

≈

8%) produced the corresponding clockwise rotation accord-

ing to the intended pitch of the desired screw DOF. This elastic

response was validated via FEA (Fig.

b), which showed the same

rotation upon compression. The details of this FEA are speci

fi

in Methods. The FACT-predicted pitch,

,of30

μ

m/rad was

closely matched by the FEA calculations, while the experiments

achieved an average pitch of 38.3

μ

m/rad, attributed to non-

negligible friction between the indenter and the top pyramid-

shaped layer as well as inherent manufacturing defects (Fig.

c).

To assess the repeatability of the screw deformation, we

performed cyclic compressions (Fig.

d) in which a constant

pitch was observed above a ~4

μ

m displacement. Minor

permanent deformation accumulated after the

fi

rst two cycles,

which prevented the material to revert to the zero-rotation state

upon unloading, but it did not affect the value of the pitch when

deformed in the linear regime. Additional plots are provided in

Supplementary Fig. 3. Note that although an alternative single-

screw-DOF metamaterial has previously been designed

prior to

this work, the theory of this paper enables the automated

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Max

Min

Experiment (black circles)

FACT prediction (blue line)

FEA (red circles and dashed line)

Rotation of top layer (rad)

0.0

0.1

0.2

0.3

0.4

Cycle 1 (black circles)

Cycle 2 (blue diamonds)

Cycle 3 (red triangles)

Rotation of top layer (rad)

0.0

0.1

0.2

0.3

0.4

Displacement of top layer (

Pitch,

Displacement of top layer (

Displacement

magnitude

Fig. 3

Validation of the screw degree-of-freedom (DOF) example.

In situ nanomechanical compression experiment on a screw directionally compliant

metamaterial (DCM) fabricated using two-photon lithography, during which the corners of the rigid layers were tracked (red circles) and compared wi

the undeformed con

guration (yellow circles).

Finite element analysis (FEA) assuming fully linear behavior depicting the clockwise rotation observed in

experiments.

Pitch comparison between freedom and constraint topologies (FACT) prediction and FEA (30

μ

m/rad), and experiments (38.3

μ

m/rad).

Cyclic compression of the screw-DOF DCM. The experimental data points in

and

correspond to the averaged top-layer rotation for a minimum of two

identical samples, while the error bars correspond to the standard deviation amongst the samples (scale bar in

,50

μ

Screw

Freedom

space

Constraint

space

Screw

Cell

volume

FACT

Layer

extension

Fig. 2

Single degree-of-freedom (DOF) screw example.

The available volume is divided into individual cell volumes and the desired screw DOF is

speci

ed.

The screw DOF

’

s freedom space and its complementary constraint space shown with parameters that relate its geometry to the pitch,

p

, of the

screw,

the resulting aperiodic directionally compliant metamaterial (DCM) design consisting of,

nine identical layers each made of,

six unique cell

designs

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synthesis of metamaterials that achieve any combination of DOFs

(i.e., screws, translations, and rotations) located and oriented any

way desired.

Single-DOF case study

. Suppose a cube-shaped 5×5×5-cell DCM

is desired that is stiff in all directions except about a single

rotational axis through its center as shown in Fig.

a. The free-

dom space that embodies the desired rotational DOF is depicted

as the red line, labeled 1 DOF Type 1 in Supplementary Fig. 2. Its

constraint space consists of the intersecting blue planes shown in

Fig.

b. Since this constraint space is a cell space, the portion of

the space that

fi

lls each cell volume can be used to synthesize their

respective topologies. Two blade elements per cell can, for

instance, be selected such that each blade

’

s plane corresponds

with a plane from the constraint space as shown in Fig.

c, d to

ensure that each cell individually achieves the desired DOF. Recall

that the rules for determining the number and way

fl

exible ele-

ments should be selected from within constraint spaces to achieve

the desired DOFs embodied by their freedom spaces are provided

in Methods. The remaining cells can be similarly synthesized to

generate the aperiodic DCM of Fig.

e. Note from the view shown

in Fig.

f that the planes of the blade elements all intersect the

rotational axis. Modal analysis demonstrates that regardless of

what constituent material the resulting design is assigned, the

fi

rst

mode shape corresponds with the desired compliant rotation

(Fig.

g) for a variety of DCM bulk shapes, e.g., a hollowed-out

cube (Fig.

h) or a halved cube (Fig.

i). Many more irregular

shapes (e.g., the elbow shape of Fig.

b,c) could be carved out of

the cube-shaped DCM without compromising its desired rota-

tional DOF if a higher cell resolution is applied. The process for

designing this case study is animated in Supplementary Movie 3

and details regarding its FEA veri

fi

cation are provided in

Methods.

Multi-DOF case study

. It is not always obvious which freedom

space maps to a given set of DOFs when more than one DOF is

desired. Suppose a cube-shaped 4×4×4-cell DCM is desired that

achieves the three rotational DOFs shown in Fig.

a. The freedom

space that represents the combination of these intersecting rota-

tions, labeled 3 DOF Type 3 in Supplementary Fig. 2, is the sphere

of all red rotation lines that intersect a common point as shown in

Fig.

b. To determine this freedom space, the desired DOFs were

modeled using twist vectors according to the theory in Methods

and were linearly combined to generate all the other twist vectors

within the resulting freedom space. The freedom space

’

s com-

plementary constraint space is a sphere of PFWVs that intersect

the same point as the rotation lines within the freedom space.

Since this constraint space is a cell space, the DCM of Fig.

could be synthesized by aligning the axes of three wire elements

in each cell with three independent PFWVs from within the

constraint space of Fig.

b according to the rules detailed in

Methods. Note that many of the resulting cell designs require

layer extensions. Regardless of constituent material properties, the

fi

nal DCM

’

fi

rst three mode-shapes correspond with the three

abc

def

ghi

Fig. 4

Example with a rotational degree-of-freedom (DOF).

A rotational freedom space and,

its constraint space of intersecting planes can be used,

, to synthesize individual cells that individually achieve the desired rotation so,

, an aperiodic directionally compliant metamaterial (DCM) with,

intersecting blade elements can be generated to achieve,

–

, the desired rotation for a variety of bulk shapes (colors in

–

are de

ned in Fig.

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desired rotations as shown in Fig.

–

f. If a higher cell resolution

had been used, the resulting DCM could have been formed to

mimic the DOFs and shapes of natural wrist, shoulder, or hip

joints for various prosthetic or soft-robot applications. The pro-

cess for designing this case study is animated in Supplementary

Movie 3 and details regarding its FEA veri

fi

cation are provided in

Methods.

Case study with a freedom space not linked to a cell space

.Not

every constraint space can be used to generate the layers of a

DCM. Suppose, for instance, a DCM is desired that achieves

two intersecting rotations on its top surface as shown in Fig.

The freedom space that represents the combination of those

DOFs, labeled 2 DOF Type 1 in Supplementary Fig. 2, is a

planar disk of red rotation lines that intersect at the same point

(Fig.

b). Its constraint space consists of a plane of PFWVs that

is coplanar with the disk of rotations and a sphere of PFWVs

that intersect at the same point where the rotations intersect

(Fig.

c). Since the PFWVs on the plane of the constraint space

don

’

t pass through the cells in the DCM, there are not enough

independent PFWVs in the rest of the constraint space (i.e., the

sphere) to synthesize the cells with

fl

exible elements that

directly connect the layers together. Thus, alternating layers of

cells that each achieve some of the DOFs within the freedom

space should be designed to collectively achieve all the DOFs

within the freedom space when they are stacked together. To

synthesize such cell layers, intermediate freedom space should

be selected from within the freedom space according to the

rules discussed in Methods. Th

e intermediate freedom spaces

should also link to complementary constraint spaces that are

cell spaces since those are the only spaces that can occupy any

volume of space with enough independent PFWVs to generate

correct cell topologies located anywhere. Note that the freedom

spaces of all previous examples link to constraint spaces that are

cell spaces but the freedom space of Fig.

b does not link to a

cell space, which is why intermediate freedom spaces that do

link to cell spaces are required. Suppose, for this example, the

two rotations shown in Fig.

a were each selected as the

intermediate freedom spaces from within the space of Fig.

The intersecting planes of the

fi

rst intermediate freedom space

’

complementary constraint space (Fig.

b) can be used to syn-

thesize the

fl

exible elements of each cell (Fig.

d) in the

fi

rst

layer (Fig.

e) such that the cells in that layer individually and

collectively achieve the rotation of their intermediate freedom

space. The second intermediate freedom space

’

scom-

plementary constraint space (Fig.

b) can then be used to

synthesize the

fl

exible elements of each cell (Fig.

f) in the

second layer (Fig.

g) such that the cells in that layer indivi-

dually and collectively achieve their differently oriented inter-

mediate freedom space. If this process continues for each

successive alternating layer, the resulting aperiodic DCM

(Fig.

h) will achieve all the DOFs within the full freedom space

of Fig.

basshowninFig.

i, j. The

fi

nal design can then

be additively fabricated and shaped as desired (Fig.

k). The

process for designing this case study is animated in Supple-

mentary Movie 3 and details regarding its FEA veri

fi

cation

are provided in Methods.

Automated design tool

. A MATLAB tool (provided in Supple-

mentary Software) was created to automate the design of DCMs.

The tool

fi

rst prompts users to specify cell size and resolution. In

the example of Fig.

, a cell size of 2.54 cm and a resolution of

4×4×4 cells was chosen. The tool then prompts users to specify

the desired DOFs and to identify their corresponding freedom

space. In the example of Fig.

a, two orthogonal translational

DOFs and two orthogonal rotational DOFs were chosen on the

top surface of the DCM, which combine to produce the freedom

space, labeled 4 DOF Type 8 in Supplementary Fig. 2. This

freedom space contains a disk of translations and an in

fi

nite

number of stacked disks

fi

lled with rotations and screws (Fig.

b).

If the freedom space selected links to a constraint space that is a

cell space, this constraint space is used by the tool to generate all

the cells within the DCM using the mathematics detailed in

Methods. If, however, the freedom space does not link to a cell

space, the tool then requires the user to identify intermediate

freedom spaces that link to constraint spaces that are cell spaces

and combine to produce the freedom space. Since the freedom

space in Fig.

b does not link to a cell space, the freedom space

Freedom

Space

Constraint

Space

Fig. 5

Example with three intersecting rotational degrees of freedom (DOFs).

Three desired intersecting rotational DOFs and,

the freedom and

constraint spaces used to synthesize,

a directionally compliant metamaterial (DCM) that,

–

, achieves the desired compliant rotations (colors in

–

are

ned in Fig.

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labeled 2 DOF Type 8 in Supplementary Fig. 2 was chosen twice

and oriented as shown in Fig.

c. Note that these spaces do link to

cell spaces and they combine to form the freedom space of

Fig.

b. The tool then uses the constraint spaces (Fig.

d) of these

intermediate freedom spaces to generate the appropriate number,

location, and orientation of wire elements within each cell of the

DCM (Fig.

e, f). The tool also automatically generates layer

extensions when necessary. Note from Fig.

e, f that the wires

within the alternating layers, labeled

and

, lie within the

parallel disks of their respective constraint spaces and some of the

cells required layer extensions. The tool then generates an.stl

fi

of the resulting design (Fig.

g), which can be uploaded to 3D

printers (Fig.

h). The tool also uses a custom-developed modal-

analysis approach, which is provided in Supplementary Software

and discussed in Methods, to generate animated .gif

fi

les of the

DCM

’

s DOFs (Fig.

i). A demo of the tool is provided in Sup-

plementary Movie 4. The computational times required by a

standard desktop computer to generate

DCM designs that

achieve the DOFs of Fig.

a, i are plotted in Fig.

Although experienced engineers may be able to intuit some of

the DCM designs provided previously, the automated tool of this

work can rapidly generate designs that are too complex for most

humans to visualize. Two such examples, which were generated

by the tool, are provided in Fig.

. The design of Fig.

a achieves

ab c

de f

gh i

Fig. 6

Example with two intersecting rotational degrees of freedom (DOFs).

Two desired intersecting rotational DOFs and,

their freedom and,

constraint spaces. Since the plane of the constraint space doesn

’

ll all space, two intermediate freedom spaces must be used.

The constraint space of

the

rst intermediate freedom space is used to synthesize the unit cells of,

the

rst layer.

The constraint space of the second intermediate freedom

space is used to synthesize the unit cells of,

the second layer.

This process is repeated for alternating layers until the

nal directionally compliant

metamaterial (DCM) is synthesized that,

achieves the desired DOFs.

The DCM can be additively fabricated and shaped as desired (scale bar in

10 cm, and colors in

and

are de

ned in Fig.

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7

the 2 DOF Type 4 freedom space in Supplementary Fig. 2. This

freedom space consists of a disk of intersecting screws of the same

pitch. The tool

’

s modal analysis shows that the two independent

screw DOFs (Fig.

b, c) that combine to generate the desired

freedom space are successfully achieved by the design generated.

The design of Fig.

d achieves the 3 DOF Type 6 freedom space in

Supplementary Fig. 2. This freedom space consists of two parallel

planes of parallel rotation lines oriented in orthogonal directions

with respect to each other and a translation arrow that is

perpendicular to these planes. The freedom space also possesses

other screw lines that are not shown in Fig.

d to avoid visual

clutter. The tool

’

s modal analysis shows that the two desired

independent rotational DOFs and the one desired independent

translational DOF (Fig.

–

g) that combine to generate the

desired freedom space are successfully achieved by the design

generated. Animated .gif

fi

les that show how the designs of Fig.

deform are provided in Supplementary Movie 4.

Effect of bulk shape on DOFs

. The DOFs achieved by a DCM

are similarly affected by its bulk shape and architecture. Thus,

the freedom space of a DCM is determined by linearly com-

bining the twist vectors that constitute the freedom space of the

DCM

’

s architecture and the freedom space of the DCM

’

sbulk

shape if it were

fi

lled with a homogenous material. We

experimentally demonstrate this principle using the example of

Fig.

(see Supplementary Movie 5). The freedom space of a

homogenous material shaped like the system shown in Fig.

ais

a single translation arrow (i.e., 1 DOF Type 3 in Supplementary

Fig. 2). If the same shape rotated 90

is used as the system

’

2 DOF type 8

4 DOF type 8

2 DOF type 8

Intermediate freedom spaces

FACT

Translation DOFs

Rotation DOFs

Constraint space

Time to generate design (sec)

02040

Number of unit cells,

, in a

lattice

100

0.10

0.06

0.02

–0.02

0.10

0.06

(m)

0.02

–0.02

–0.04

–0.08

–0.012

(m)

Fig. 7

Automated design tool.

Users specify the desired degrees of freedom (DOFs) and,

the freedom space that results from the combination of those

DOFs. If that freedom space does not link to a constraint space that is a cell space,

, intermediate freedom spaces must be selected from within the

freedom space that do link to cell spaces. The tool then generates the directionally compliant metamaterial (DCM) topology using,

the constraint spaces

of the intermediate freedom spaces.

The elements in the alternating layers lie within their corresponding constraint spaces.

The tool generates an.stl

le that,

can be used to additively fabricate the design.

Animated .gif

les of the DCM

’

s DOFs are also generated.

Plot of the computational time

required by the tool to generate a

u

u

u

DCM that achieves the four DOFs speci

ed (scale bar in

, 5 cm, and colors in

are de

ned in Fig.

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periodic architecture as shown in Fig.

b, the freedom space of

the resulting DCM, as predicted by the principle discussed

previously, is the disk of translations (2 DOF Type 10) shown in

Fig.

c. This freedom space results from the linear combination

of the translational DOF of the DCM

’

s bulk shape and the

translational DOF of the DCM

’

s architecture. The DCM was 3D

printed (Fig.

d) and loaded along the two directions of the

DCM

’

s translational DOFs (Fig.

e, f). The plot of Fig.

demonstrates that the compliance along these directions (i.e.,

and

-axes) are similar. Another example that demonstrates

this section

’

s principle is provided in Supplementary Note 2

and shown in Supplementary Fig. 4 and in Supplementary

Movie 5.

Discussion

We created an approach that leverages the vector spaces of the

FACT library to enable the automated synthesis of metamaterials

(i.e., DCMs) that achieve desired combinations of compliant

DOFs while assuming any form. The reason such materials can

achieve these properties is that their DOFs are independently

determined by both the DOFs of their architecture and the DOFs

2 DOF type 4

3 DOF Type 6

ab c

defg

Screw

Rotation

Translation

Fig. 8

Less intuitive designs generated by the automated tool.

A design that achieves,

two intersecting screw degrees of freedom (DOFs) with the

same pitch.

A different design that achieves,

two orthogonally skew rotational DOFs and,

an orthogonal translational DOF

Metamaterial

DOF

Bulk shape

DOF

Total DOFs

1 DOF

type 3

1 DOF

type 3

2 DOF type 10

160

140

120

100

0.1

0.2

0.3

Direction (blue line)

Direction (red dashed line)

0.4

0.5

Force (N)

Displacement (mm)

Fig. 9

Directionally compliant metamaterial (DCM) shape affects degrees of freedom (DOFs).

A bulk shape that achieves a translational DOF but would,

achieve two translational DOFs if the shape were rotated 90

and used inside of itself.

The freedom space of a DCM results from the sum of its

architecture

’

s freedom space and its bulk shape

’

s freedom space.

DCM was additively fabricated and,

tested in two different directions.

The

stiffness in both directions are the same (scale bar in

, 5 cm)

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9

of their bulk shape. To maintain this independence for ensuring

high shape versatility, the cell resolution of DCMs should be

suf

fi

ciently high (i.e., cell size should be orders of magnitude

smaller than the characteristic size of the DCM) such that enough

redundant cells exist in the architecture to span any cross-section

of the material

’

s shape.

Compared to other computational approaches (e.g., topology

optimization) that typically require 10 s of hours to generate a

single two-dimensional (2D) cell design within a periodic meta-

material, the approach proposed here requires only 10 s of sec-

onds to generate thousands of different 3D cell designs within

aperiodic DCMs (i.e., ~6 orders of magnitude more cells

per second can be generated). It also does not require pre-

computed databases of cell designs

, which typically demand

signi

fi

cant time to populate and large amounts of memory to

store. Rather, the approach rapidly searches the most promising

branches of the mathematically complete design tree to generate

DCM solutions, which enable irregularly shaped

fl

exure bearings,

compliant prosthetics, morphing structures, and soft-robots that

are too complex to synthesize using alternative approaches. The

theory introduced also paves the way for enabling the synthesis of

general metamaterial con

fi

gurations beyond the stacked-layer

serial designs of this work.

Methods

FACT library

. Supplementary Fig. 2 contains the mathematically complete library

of all 50 freedom spaces. The chart organizes the freedom spaces into seven dif-

ferent columns according to the number of DOFs that combine to generate them.

Each freedom space is labeled with a type number at its upper left corner. The

freedom spaces that lie outside the black-outlined pyramid of Supplementary Fig. 2

are not shown with their complementary constraint spaces because such spaces do

not possess enough independent PFWVs to synthesize parallel

fl

exure systems

–

(i.e., systems that directly join two bodies together using parallel elements like the

layered cells within the DCMs of this paper). Additionally, only freedom spaces

that link to constraint spaces that are cell spaces (i.e., spaces that lie within the

region shaded yellow in Supplementary Fig. 2) can

fi

ll any volume of space with

enough independent PFWVs to enable the synthesis of cells that successfully

achieve their intended DOFs regardless of where they are located in a DCM.

Although others have mathematically categorized screw systems similar to the

vector spaces of FACT for other applications

–

, the library of Supplementary

Fig. 2 has been organized to facilitate the design of DCMs. Exploded views of each

freedom and constraint space type in the library are provided and described in

detail with the equations that de

fi

ne their geometry in prior publications

Mathematically de

fi

ning PFWVs

. PFWVs

–

6×1

, are graphically depicted as

the blue constraint-force lines (Supplementary Fig. 5) that constitute the constraint

spaces of the FACT library. These vectors are de

fi

ned according to

́

́

́

́

́

where

1×3

is a 1×3 force vector that points in the direction of the blue constraint-

force line

’

s axis, and

1×3

is a 1×3 location vector that points from the coordinate

system to any location along that line

’

s axis. Physically speaking, blue constraint-

force lines represent the axis about which a force can be imparted.

Modeling general

fl

exible element geometries

. There are three categories of

fl

exible elements

—

parallel

, serial

, and hybrid

. Since parallel elements are

suf

fi

cient for generating DCM examples that achieve any desired combination of

DOFs and require the least amount of computation to model them compared to

serial or hybrid elements, parallel elements are used exclusively to generate the

DCMs of this work. If, however, a future work desires serial or hybrid elements, the

theory to model them exists

An element is parallel if blue constraint-force lines can

fi

ll the element

’

s entire

geometry without exiting the geometry at any point and directly connect the two

rigid bodies that the element joins together. A parallel element is modeled using the

constraint space that graphically depicts the linear combination of the constraint-

force lines

’

corresponding PFWVs that satisfy the previous two conditions. Thus,

the constraint space of an element represents the forces that the element is capable

of resisting (i.e., the element

’

s directions of highest stiffness). As an example,

consider the parallel wire element shown in Supplementary Fig. 5. The constraint

space that models this element is the single blue constraint-force line that satis

fi

the two conditions speci

fi

ed above. This model treats the wire element as if it is

fi

nitely stiff along its axis but is in

fi

nitely compliant in all other directions since

the constraint-force line can only impart forces along its axis. Additionally, note

that the constraint space models only the location and orientation of the element

and does not consider its material properties or its geometric parameters (i.e., its

diameter or length).

All other parallel element geometries can be similarly modeled. Example

parallel elements and the constraint spaces that model their behavior are shown in

Supplementary Fig. 6. The DOF column and type numbers for each of these

constraint spaces are labeled in the

fi

gure using the convention established in

Supplementary Fig. 2.

Modeling DOFs and freedom spaces

. Just as constraint spaces are generated by

linearly combining their independent PFWVs de

fi

ned in equation (

), freedom

spaces are generated by linearly combining their DOFs. DOFs can be mathema-

tically modeled using 6 × 1 twist vectors

–

6×1

,de

fi

ned by

́

́

́

́

́

ðÞþ

́

where

1×3

is a 1 × 3 angular velocity vector that points along the twist

’

s axis,

1×3

a 1 × 3 location vector that points from the coordinate system to any location along

the twist

’

s axis, and

is the scalar pitch of the twist. If the twist

’

s pitch is zero, the

twist is a red rotation line. If the twist

’

s pitch is any other

fi

nite nonzero value, the

twist is a green screw line. If the twist

’

s pitch is in

fi

nity, the twist is a black

translation arrow and is de

fi

ned according to

́

́

́

where

1×3

is a 1 × 3 zero vector, and

1×3

is a 1 × 3 linear velocity vector that points

along the axis of the twist. Although all the compliant directions contained within a

freedom space are modeled using twist vectors, the DOFs of a freedom space are

the independent twist vectors that linearly combine to generate the other twist

vectors (i.e., compliant directions) within the freedom space.

Selecting elements within constraint spaces

. This section explains how con-

straint spaces can be used to determine the location and orientation of

fl

exible

elements from within the constraint spaces

’

geometries to ensure that the resulting

system achieves its intended DOFs. For a parallel system to successfully achieve the

DOFs of its intended freedom space,

fl

exible elements that collectively contain

independent PFWVs should be selected from within the freedom space

’

s com-

plementary constraint space where

Thus, since the freedom space of Fig.

b consists of one screw DOF (i.e.,

=

1),

each cell within the

fi

nal DCM (Fig.

c) requires

fl

exible elements that together

contain

=

=

5 independent PFWVs from within the freedom space

’

complementary constraint space. Consequently, each of the cells in the DCM of

Fig.

c consist of

fi

ve wire elements with axes that are colinear to

fi

ve independent

PFWVs from within the constraint space of Fig.

Thus, although Eq. (

) can be used to determine the correct number,

,of

independent PFWVs to select from within a constraint space, the equation does

not provide guidance on how to select the

PFWVssuchthattheyare

independent. Gaussian elimination

could be used as a mathematical approach

to con

fi

rm whether a collection of

PFWVs are independent by determining if

a matrix consisting of the PFWVs possesses a rank of

. The rules provided with

the shapes of Supplementary Fig. 7, however, offer a more intuitive approach for

selecting PFWVs from constraint spaces such that they are independent. Each

constraint space in the FACT library consists of various combinations of the

nine shapes shown in Supplementary Fig. 7a-i. The instructions above each

shape in the

fi

gure describe how many independent PFWVs lie within the shape

and how they should be selected from the shape such that they will be

independent.

Different

fl

exible elements contain different numbers of independent PFWVs

within their geometry. Whereas a wire element contains a single independent

PFWV, blade elements contain three independent PFWVs. The number of

independent PFWVs within a general

fl

exible element is the number of

independent PFWVs within the element

’

s constraint space. Thus, the number,

of independent PFWVs within each element shown in Supplementary Fig. 6 can be

determined by subtracting the labeled DOF number,

, from 6 according to Eq. (

The principles of this section can be used to synthesize the parallel topologies of

general DCM cells. Suppose, as an example, a parallel cell is desired that achieves a

single rotational DOF located on the edge of the cell

’

s two rigid bodies as shown in

Supplementary Fig. 8a. The complementary constraint space of this single-rotation

freedom space, labeled 1 DOF Type 1 in Supplementary Fig. 2 and shown larger in

Supplementary Fig. 8b, is the set of planes that intersect the rotation

’

s axis. Thus,

according to Eq. (

=

5 total independent PFWVs must be selected from within

this constraint space because its freedom space contains

=

1 DOF. Since the

constraint space consists of intersecting planes, which according to Supplementary

Fig. 6 are each the constraint space of a single blade element (i.e., 3 DOF Type 1), a

blade element could be selected from within any one of the intersecting planes.

Additionally, since the planar constraint space of a blade element contains only

three independent PFWVs according to Supplementary Fig. 7d, two more

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