1
Computationally Efficient Design of Directionally
Compliant Metamaterials
Shaw et al.
2
Supplementary Figures
Supplementary F
igure
1
.
Directionally compliant metamaterials (DCMs) as irregularly
shaped flexure systems
.
a
, A
flexure system that achieves a compliant translation.
b
, An
irregularly shaped DCM that achieves the same compliant translation, and
c
, a close up of its
periodic lattice of repeating cells.
a
b
c
Translation
3
Supplementary
F
igure
2
.
“
Freedom and constraint
topologies
”
(FACT) library of vector
spaces
.
All 50 freedom spaces are provided within 7 columns organized according to the number
of degrees of freedom (DOFs) that constitute the freedom spaces in each column. The freedom
spaces within the black
-
outlined
pyramid link to constraint spaces that possess enough pure
-
force
wrench vectors (PFWVs) to synthesize
parallel systems. The constraint spaces within the region
shaded yellow, called cell spaces, are the only spaces that can be used to synthesize directiona
lly
compliant metamaterial (DCM) cells.
Rotation
Translation
Screw
Constraint
Force
Type #
4
Supplementary
Figure 3
.
Additional screw directionally compliant metamaterial (DCM)
results
.
a
,
DCM fabricated using two
-
photon lithography.
b
,
DCM compression test.
c
,
Displacement of the DCM
’
s top layer versus
the r
otation of each layer.
The experimental data
points correspond to the averaged rotation of each layer for three identical samples, while the error
bars correspond to the standard deviation amongst the samples.
d
,
Stress versu
s strain plotted for
6 cyc
les (scale bar in
a
,
2
0
μ
m,
b
, 50 μ
m)
.
a
b
c
d
5
Supplementary Figure 4
.
Directionally compliant metamaterial (DCM) shape affects degrees
of freedom (DOFs)
.
a
, Three
DOFs;
b
, their
freedom space and complementary constraint space,
c
, used to synthesize each cell such that they,
d
-
f
, achieve the DOFs within,
g
, a cube
-
shaped DCM.
h
, A blade
-
shaped homogenous material achieves three different DOFs.
i
, A blade
-
shaped DCM
with the architecture of
g
achieves 6 DOFs.
j
,
T
he
freedom space
o
f a DCM
results from the sum
of
its
architecture
’
s freedom space
and
its bulk shape
’
s freedom space (colors in
d
-
f
are defined in
Fig. 3b).
Freedom Space
Constraint Space
a
b
c
d
e
f
g
h
i
3 DOF Type 2
3 DOF Type 1
6 DOF Type 1
j
Metamaterial
Bulk Shape
DOFs
DOFs
Total DOFs
6
Supplementary
Figure 5
.
Pure
-
force wrench vector (PFWV) parameters and wire element
.
The parameters necessary to mathematically define a PFWV, which can be used to model the
behavior of a simple wire element.
f
1
x
3
r
1
x
3
x
y
z
Wire
7
Supplementary
Figure 6
.
Example flexible elements
.
A variety of flexible element geometries
shown with their corresp
onding constraint spaces labeled according to the convention established
in the freedom and constraint topologies (FACT) library of Supplementary Fig. 2.
5 DOF
Type 1
3 DOF
Type 1
3 DOF
Type 7
2 DOF
Type 5
3 DOF
Type 3
2 DOF
Type 1
1 DOF
Type 1
1 DOF
Type 1
3 DOF
Type 3
1 DOF
Type 1
2 DOF
Type 3
3 DOF
Type 3
3 DOF
Type 3
2 DOF
Type 8
2 DOF
Type 2
1 DOF
Type 3
2 DOF
Type 1
3 DOF
Type 9
8
Supplementary
Figure 7
.
Rules for sele
cting independent
pure
-
force wrench vectors
(
PFWVs
)
from within the 9 basic shapes that constitute all constraint space
s
.
a
, Single line.
b
, Disk of lines.
c
, Plane of parallel lines.
d
, Plane of lines.
e
, Sphere of lines.
f
, Box of parallel
lines.
g
, Hyperbolic paraboloid of lines.
h
, Circular hyperboloid of lines.
i
, Elliptical hyperboloid
of lines.
1 line
2 unique lines
2 unique lines
3 lines that don’t all intersect
at the same point or are not all
parallel
3 lines that don’t all lie
on the same plane
3 lines that don’t all lie
on the same plane
3 unique lines
3 unique lines
3 unique lines
a
b
c
d
e
f
g
h
i
9
Supplementary
Figure 8
.
Example exactly
-
constrained cells synthesized using different
flexible elements
.
a
,
A cell that achieves a single rotational degree of freedom (DOF) and con
sists
of a blade and two wire elements.
b
, The cell
’
s freedom and constraint spaces.
c
, A different view
of the same cell.
d
,
e
, Two different views of a cell that achieves a single screw DOF and consists
of a circular hyperboloid element and two wire elem
ents.
Blade
Wires
Blade
Wires
Hyperboloid
Circular
Wires
Wires
b
c
d
e
a
10
Supplementary
Figure 9
.
Parameters used within the automated design tool
.
The tool
synthesizes each cell within a directionally compliant metamaterial (DCM) with the minimum
necessary number of independent wire elements that lie within the DCM
’
s desired constraint space
and directly join the cell
’
s two rigid bodies together. The
two bodies of cell
(
a
)
are shown with
corresponding parameters.
y
x
z
x
a
y
a
z
a
t
t
s
s
s
(
a
)
R
R
11
Supplementary Note 1:
Additional
a
pplications for
directionally compliant
metamaterials (
DCMs
)
In addition to enabling
soft
-
robotic
joints
(Fig. 1b)
that achieve directionally compliant
properties
and
bodies that passively deform in controlled ways such as the propeller example
with
the screw
degree of freedom (
DOF
)
(
Fig. 2
c
)
discussed in the main text, DCMs can be used to
enable new precision flexure
systems with demanding shape
requirements
.
Flexure systems
currently achieve compliant directions according to how they are shaped on the macroscale. The
flexure system of
Supplementary
Fig.
1
a, for instance, achieves a single translational DOF with
high
compliance because its homogenous constituent material is shaped with macro
-
sized parallel
blade elements
(see Supplementary Movie 1)
. Suppose, however, a system is desired that achieves
the same translational DOF but must fit within a cylindrical shape wi
th a filleted circular hole
through its geometry. Such a system would not be possible to achieve by shaping a homogenous
material as a macroscale flexure system but would necessitate the implementation of a DCM like
the example in
Supplementary
Fig.
1
b.
Al
though the DCM of
Supplementary
Fig.
1
b consists of a
single repeating cell design (
Supplementary
Fig.
1
c), most
other
DCMs typically require aperiodic
cell designs, which differ throughout the material
’
s geometry.
Since the theory of this paper
enables th
e design of any kind of DCM
including aperiodic designs
, th
is theory will disrupt the
field of flexure
-
system design
such that flexure systems will be enabled that can conform to almost
any desired shape.
12
Supplementary Note 2:
E
xample
that demonstrates the effect of a DCM
’
s bulk
shape
This section
provides a
supplementary
example
that demonstrates
how
the DOFs achieved by
a DCM are similarly affected by both the DCM
’
s
architecture
and its bulk shape.
Supplementary
Movie 5 provides anim
ations of the example presented here.
Suppose a cube
-
shaped DCM is
desired that achieves two orthogonal translations and an orthogonal rotation as shown in
Supplementary
Fig.
4
a. These DOFs combine to generate the 3 DOF Type 2
freedom space
in
Supplementar
y
Fig.
2
. This
freedom space
consists of an infinitely large box of rotation lines that
are parallel to the axis of the rotational DOF and a disk of translation arrows that are orthogonal
to this axis
as shown in
Supplementary
Fig.
4
b
. The
freedom space
’
s
complementary
constraint
space
is an infinitely large box of
pure
-
force wrench vectors (
PFWVs
)
that are parallel to the
rotation lines of the
freedom space
. Since the
freedom space
contains
n
=3 DOFs, three wire
elements should be selected according to
equation (
4
) with axes that are colinear with
m
=3
independent PFWVs from the
constraint space
(
Supplementary
Fig.
4
c) so that the resulting cell
achieves its desired DOFs (
Supplementary
Fig.
4
d
-
f). If this process is repeated for all the cells
within the c
ube
-
shaped DCM, a periodic design can be generated that achieves the desired DOFs
(
Supplementary
Fig.
4
g). Suppose, however, that the DCM is shaped like a flat blade instead of a
cube. Whereas a cube
-
shaped homogenous material would achieve no DOFs, a blad
e
-
shaped
homogenous material would achieve the two orthogonal intersecting rotations and the orthogonal
translation shown in
Supplementary
Fig.
4
h. If the same
architecture
shown in
Supplementary
Fig.
4
g where applied to a blade
-
shaped DCM (
Supplementary
F
ig.
4
i), the resulting material would
exhibit 6
DOFs. Thus, the
freedom space
of a DCM is determined by linearly combining the twist
13
vectors that constitute (i) the
freedom space
of the DCM
’
s
architecture
and (ii) the
freedom space
of the DCM
’
s bulk shape if it were filled with a homogenous material (
Supplementary
Fig.
4
j).