w
w
w
.
p
n
a
s
.
o
r
g
/
c
g
i
/
d
o
i
/
1
0
.
1
0
7
3
/
p
n
a
s
.
DRAFT
Supplementary Information:
Harnessing bistability for directional propulsion of soft, untethered
robots
Bistable Mechanism Analysis.
In order to analyze the bistable mechanism, we first fabricate a bistable mechanism-muscle
pair enclosed in a planar shell identical to that of a single stroke swimming robot (Supp. Fig.
1
.a and Movie S8). The four
bars within the bistable mechanism form two “V” shapes pointing to the right, representing one of the stable states of the
mechanism. The sample (the pair confined in the planner shell) is then immersed in
60
◦
C
water (state I), equivalent to the
glass transition temperature of the muscle material (VeroWhitePlus). As the muscle temperature rises due to water contact,
both of the deformed beams within the muscle start relaxing into their original/printed shape. At the onset of the instability,
where the truss-like bars of the bistable mechanism are vertical (state II), the muscle pushes the mechanism slightly towards its
second stable state (where the “V” shapes are pointing to the left) (state III).
In order to model the bistable behavior of the mechanism, we now consider one of its four bars (Supp. Fig.
1
.b). The bars
are printed with VeroWhitePlus with Young’s modulus E =
2
×
10
9
Pa, while its connections to the planar shell are printed
with Aguils300 with a Young’s modulus E =
2
×
10
6
Pa. Since the bar material has a stiffness that is
10
3
higher than its
connection points, we model the bar as an inclined rigid truss element (Supp. Fig.
1
.c) supported by two torsional springs with
a spring constant (
k
θ
) and a linear spring with a constant (
k
) (Supp. Fig.
1
.c). The rotational springs
k
θ
are the torsional
resistance of the compliant joints. The linear spring,
k
simulates bending of the flexible support through linear force in the
y
direction.
Fig. 1.
a. Video snapshot of the actuation of the bistable mechanism. Note that from stage II to III, the shape memory polymer does not contribute to propulsion. b. A zoom in
view of the bistable truss geometry, the black regions represent the flexible material, and the white is the rigid material. c. An idealization of the truss geometry in both the
original and the deformed state. The support is idealized by a linear spring
k
, and the joints by a torsional spring
k
θ
. d. The force displacement relationship of the bistable
mechanism as derived in equation
5
.
We use a Lagrangian equation to construct a relationship between the force
P
and the corresponding displacement
V
with
respect to the deformed geometry. The bar is assumed to be axially rigid.
L
=
1
2
kd
2
+
1
2
k
θ
∆
α
2
−
1
2
PV
[1]
Then we consider the deformed geometry with a rigid truss bar, to relate the different displacements in the model:
√
H
2
+
L
2
=
√
(
H
−
V
)
2
+ (
L
+
d
)
2
[2]
The solution of
d
is the difference between the deformed projected length and the original one,
d
=
√
2
HV
+
L
2
−
V
2
−
L
[3]
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| vol. XXX | no. XX |
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DRAFT
To simplify the representation, we denote the first term of the solution as
L
1
=
√
2
HV
+
L
2
−
V
2
.
The Lagrangian equation becomes
L
=
1
2
k
(
L
1
−
L
)
2
+
1
2
k
θ
(
arctan
H
−
V
L
1
−
arctan
H
L
)
2
−
1
2
PV
[4]
We obtain the relationship between
P
and
V
, by differentiating the system w.r.t. to
V
and setting the result to zero,
∂
L
∂V
= 0
. Such an equation provides the means to assess the impact each variable on the overall behavior of the system has, and
therefore design the bistable mechanism.
P
=
−
2
1
L
1
[
k
(
L
−
L
1
)(
H
−
V
)+
k
θ
(
arctan
(
H
−
V
L
1
)
−
arctan
(
H
L
))]
[5]
In the initial state (I), The shape memory muscle doesn’t exert any force
P
= 0
on the bistable mechanism (Supp. Fig. 1.d).
As the surrounding water heats the muscle, it starts to relax to its original/printed shape, pushing the bistable truss towards
its second stable state. Until the muscle pushes the truss to be vertical, i.e.
H
= 0
(II), beyond this point, the mechanism flips
to the second equilibrium state (III), where the muscle is physically detached from the mechanism.
Material characterization and Simulation.
The materials utilized in this study (Agilus30, FLX9895, VeroWhitePlus or RGD835,
high temperature resistant material or RGD525) are characterized using Differential scanning calorimetry (DSC) to determine
their glass transition temperature. VeroWhitePlus is used as the SMP muscle for all presented robots. FLX9895 is used in
conjunction with VeroWhitePlus in the Cargo Deployment example. For both of these SMPs, two further sets of material data
are obtained, 1) the fully relaxed modulus, and 2) frequency sweeps of the storage modulus at different temperatures. The
fully relaxed modulus is obtained using an Instron E3000 dynamic testing machine with a
3 kN
load cell and a temperature
insulation chamber. The experiments are performed at
80
◦
C
with a rate of
0
.
05 mm s
−
1
. Cylindrical specimens with a diameter
of
7 mm
and height of
9 mm
are used. The storage modulus is obtained using a dynamic mechanical analysis (DMA) (Mettler
Toledo DMA 861) machine. Specimens (
3
.
00 mm
×
4
.
30 mm
×
2
.
05 mm
) are tested under oscillatory shear (frequency:
0
.
01
to
100 Hz
, amplitude:
0
.
5
μ
m
). The test is conducted for temperatures
20
,
40
,
50
,
60
,
80
, and
100
◦
C
for VeroWhitePlus and
−
40
,
−
17
,
2
,
12
,
21
,
30
,
40
,
49
◦
C
for FLX9895.
We assume a time-temperature superposition principle, where the behavior of the material at a given temperature can be
modeled as multiplication of the relaxation times at a reference temperature
T
ref
and the shift factors at that temperature
(Eq. 6).
τ
i
(
T
) =
a
T
τ
i
(
T
ref
)
[6]
The Williams-Landel-Ferry (WLF) equation (Eq. 7) is then used to approximate these shift factors
a
T
to construct a
continuous storage modulus master curve at a reference temperature (
275 K
for FLX9895 and
313
.
15 K
for VeroWhitePlus) (
1
).
log
10
a
T
= log
10
τ
(
T
)
τ
(
T
ref
)
=
−
C
1
(
T
−
T
ref
)
C
2
+ (
T
−
T
ref
)
, T > T
ref
[7]
Fourier transform is then used to translate the storage modulus to a relaxation modulus which is needed as an input for
Abaqus (Eq. 8) (1).
G
(
t
) =
G
∞
+
2
π
∫
∞
0
[
G
′
−
G
∞
ω
]
sin
ωtdω
[8]
The Prony series is constructed using an approximation of equation 8 with 11 fitted coefficients (
2
). The constructed
Prony series has 19 and 29 non-equilibrium branches for VeroWhitePlus and FLX9895 respectively. The Prony series and the
temperature-time shift (TRS) coefficients are listed for both materials in Supplementary table 1 and 2.
G
(
t
) =
G
∞
+
N
∑
i
=1
G
i
e
−
t/τ
i
[9]
The resulting FEM simulations of the SMP muscle of the heating, programming, cooling, and reheating cycles are presented in
(Supp. Fig. 2). First, in a heated condition i.e., prescribed temperature (
T
≥
T
g
), with a ramping in the prescribed displacement
(Region A). The calculated corresponding force increases to its maximum. Second, in Region B, the prescribed displacement
is kept constant while the prescribed temperature decreases to below
T
g
. Third, only the constraint on displacement is
removed, i.e., no prescribed displacement. A slight rebound in displacement is observed (Region C) both in simulations and in
experiments. Finally, in Region D, the prescribed temperature is increased to
T
g
and the SMP muscle relaxes to its original
shape. The maximum force observed during the simulation is recorded for each SMP muscle, and plotted in Figure 2 against
the measured maximum force from the experiments.
2
|
DRAFT
Table 1. Prony series of VeroWhitePlus and FLX9895 as implemented in Abaqus
VeroWhitePlus
FLX9895
τ
i
G
i
τ
i
G
i
2.84E-06
0.047085
2E-09
0.037155
3.01E-05
0.047839
1E-08
0.036325
0.000215
0.021854
5E-08
0.019428
0.000214
0.039146
1E-07
0.036963
0.006686
0.046696
5E-07
0.025824
0.002274
0.060725
0.000001
0.048254
0.030904
0.078175
0.000005
0.029066
0.17528
0.079955
0.00001
0.061186
0.63842
0.05944
0.00005
0.047911
2.7039
0.11531
0.0001
0.048443
17.23
0.095187
0.0005
0.046758
58.742
0.039878
0.001
0.081033
134.64
0.097911
0.005
0.088524
996
0.10801
0.01
0.069187
10000
0.043986
0.05
0.11156
100000
0.012053
0.1
0.043557
1000000
0.001282
0.5
0.079339
10000000
0.000943
1
0.023205
50000000
0.000521
5
0.038059
10
0.004878
50
0.015653
100
5.27E-14
500
0.003613
1000
0.000806
5000
0.001064
10000
0.000212
50000
0.00029
100000
0.000218
1000000
0.000239
Table 2. TRS coefficients as input in Abaqus for both VeroWhitePlus and FLX9895
VeroWhitePlus
FLX9895
T
ref
313.15
275.00
C
1
21.5657
19.987
C
2
97.4544
109.53
PNAS |
May 4, 2018
| vol. XXX | no. XX |
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DRAFT
-1
-0 .8
-0 .6
-0 .4
-0 .2
0
T<Tg
T≥Tg
0
10
20
30
40
50
60
70
80
90
100
Displaceme
nt
Force [N/N]
Temperat
ure
Time
0
0.2
0.4
0.6
0.8
1
Region A
Region B
Region C
Region D
Fig. 2.
Finite element simulations of the SMP muscle showing force, displacement and the corresponding ambient temperature. In Region A, the SMP muscle is programmed
using a prescribed displacement under high temperature. The corresponding force is measured. In Region B, the displacement is kept constant as the system is cooled to below
the glass transition temperature. In Region C, the prescribed displacement is removed, Which shows that the programmed shape remains unaltered at a cold temperature. In
Region D, the temperature is increased, and the SMP muscle recovers to its printed shape.
The role of bistability vs muscles in propulsion.
In order to assess the contribution of the muscle-induced force on the distance
traveled by the swimming robot, we deploy the same robot with various muscles. We systematically increase the thickness of
the beams within the muscle, therefore increasing the resultant force (Supp. Fig. 3.a). For beams with thickness
<
1.2 mm,
the robot did not move forward, as the muscle force is not strong enough to overcome the bistability energy barrier. All the
muscles with beams
>
1.2 mm overcame the energy barrier and were able to propel the robot forward. However, the robot
traveled the same distance (Supp. Fig. 3.b), regardless of the increase of the force amplitude by a factor 2. Therefore, the
distance traveled by the swimmer depends on the bistable element rather than the muscle force, as long as the muscle is strong
enough to push the mechanism to the onset of the instability.
c
d
t=1.2 mm
t=1.4 mm
t=1.6 mm
b
a
Fig. 3.
Three swimmers travel the same distance when the shape memory actuator of three different beam thicknesses are used. This shows that propulsion comes
predominantly from triggering of the bistable mechanism. The beam thickness values are
1
.
2
,
1
.
4
,
1
.
6mm
.
Movies.
SI Movie 1
Propulsion of a single stroke swimmer. The distance travelled is approximately
1
.
15
l
where
l
is the body length
of the swimmer.
4
|
DRAFT
SI Movie 2
Shape memory muscle of different thickness exhibiting different time to activation.
SI Movie 3
Propulsion of a two-stroke swimmer. The sequence of activation is controlled by the thickness of the shape
memory muscle. The distance traveled is approximately
1
.
9
l
(where
l
is the length of the single actuator swimmer).
SI Movie 4
Propulsion of a two-stroke directional swimmer. The programmed path is straight followed by a left turn. The
distance traveled is approximately
0
.
5
l
after the first stroke, and a turn of
23
.
85
°
after the second stroke.
SI Movie 5
Propulsion of a two-stroke directional swimmer. The programmed path includes a left turn followed by a right
turn. The rotation is approximately
21
.
64
°
after the first stroke, and
−
21
.
45
°
after the second stroke.
SI Movie 6
Shape memory muscle of different material exhibiting controlled activation depending on the surrounding
temperature.
SI Movie 7
Propulsion of a reversing swimmer. The programmed operation includes a forward stroke, deployment of cargo,
and a reverse stroke. The sequence of activation is controlled by the temperature of the surrounding environment. The
first stroke is activated when water reaches
T
L
, deployment occurs when water is heated to
T
H
, then the reverse stroke
occurs.
SI Movie 8
Internal mechanism of the actuator showing the shape memory muscle pushing the bistable mechanism from one
equilibrium state to the next.
1. Ferry JD (1980)
Viscoelastic Properties of Polymers
. (John Wiley & Sons, Inc., Toronto), 3rd edition, pp. 1–672.
2. Schwarzl FR (1975) Numerical calculation of stress relaxation modulus from dynamic data for linear viscoelastic materials.
Rheologica Acta
14(7):581–590.
PNAS |
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| vol. XXX | no. XX |
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