Proceedings of the 2002 IEEE International Conference on Robotics & Automation Washington, DC • May 2002
Soft Actuator for Robotic Applications Based on Dielectric Elastomer : Quasi–static Analysis H. R. Choi, S. M. Ryew, K. M. Jung, H. M. Kim, J. W. Jeon, J. D. Nam Sungkyunkwan University, Korea,
[email protected] R. Maeda+ , K. Tanie+ AIST, Japan, {maeda–ryutaro, k.tanie } @aist.go.jp Abstract
proposed by previous researchers[3]. In these configurations, pre-stretched dielectric elastomer film is foiled on frame with compliant electrodes on both sides of the film and it deforms by the electric field generated by high voltages on the electrodes. However these are not able to actuate bidirectionally, which means a lot of things should be improve to be useful for practical applications. The proposed actuator realizes the bidirectional actuation since it is with a stretched film antagonistically configured with compliant electrodes. Also it is distinguished from the others with respect to the controllability of its compliance. Bidirectional actuation and compliance controllability are important characteristics for the musclelike actuator and the proposed one realizes these without any mechanical substitute or complicated algorithms. In this paper its basic concepts and working principles are introduced with quasi–static analysis. Also, preliminary results of experiments will be given to confirm the effectiveness of the proposed actuator. Its dynamic analysis and exemplary works for robotic actuating devices will be given in the companion paper submitted with this paper[4].
In this paper a new soft actuator based on dielectric elastomer is proposed. The actuator, called ANTagonistically-driven Linear Actuator(ANTLA) has the musclelike characteristics capable of performing the motions such as forward/backward/controllable compliance. Due to its simplicity of configuration and ease of fabrication, it has the advantages to be scale– independently implemented in meso– or micro scale robotic applications. Its basic concepts are introduced and quasi–static analysis is performed with experimental verifications.
1
Introduction
Recently, polymers are emerging as promising substitutes for existing actuators[1]. Among these the set of polymers actuated by electrical stimuli, called ElecroActive Polymer(EAP), are building up a new domain of actuators, though it includes several kinds of materials such as Ionic Polymer Metal Composites (IPMC), conducting polymer, polymer gel, dielectric elastomer, piezo electric polymer etc. Normally the EAP’s are classified into two groups such as ionic and non-ionic EAP’s, depending on the basic mechanism of actuation. In this stage the nonionic EAPs such as dielectric elastomer or piezo electric polymer are regarded very close to actual applications, though the others also have the possibility of being put into practical use in the near future. In this paper we introduce a new actuator using dielectric elastomer with a thorough quasi–static analysis. Dielectric elastomers can be easily found around us; for instance, polyurethane and silicone are dielectric elastomers, whose basic type of deformation is expansion (active)/contraction (passive). In fact, deformation of dielectric elastomers can be used in many ways to produce actuation[2, 3]. The stretched film type actuator is one of these actuator configurations, and also the rolled, and bow tie types, and others, have been
0-7803-7272-7/02/$17.00 © 2002 IEEE
2
Basic Principles
The actuator is based on the polymer dielectrics. Before coming to the topic its working principle is briefly introduced and the details can be referred to previous works[2, 3]. Fig. 1 illustrates an elastomer film with compliant electrodes. When a voltage is supplied across the compliant electrodes, the film shrinks in thickness and expands in area, as shown in Fig. 1. It is field–induced deformation and, based on simple electrostatic model, the effective pressure to generate the deformation of the film can be derived. Since the electrostatic force, called Maxwell stress, causes contraction along the field direction, the pressure σa can be derived as follows. V (1) σa = −²r ²o E 2 = −²r ²o ( )2 t
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C
o E
m
p
l i a n
l e c t r o ( A
C
o E
d
t e
)
m
h p
l i a n
l e c t r o ( B
d
E t
P
P
o
l y
m
D )
i e l e c t r i c
F
o
=
3
. 4
(
1
+ o
e
) a
2
e r
h
e r c e
p
p
p
1
Figure 1: Basic principle of operation
1 1 V σa = {−²r ²o ( )2 } Ya Ya t
=
1 Yo η
= α(1 + o εa )2
r e s t r a
i n
2
0
0
5
0
0
0
%
%
%
a
Also, let us define strain variables ∆p εa and o εp such as
(2)
∆p ε a =
δa − δp , δo
o
εp =
δp − δo δo
(5)
where δp is the thickness after the prestrain, δa represents the resultant thickness, and δ◦ is the original thickness. From Eqs. (2) and (4), equations about specified conditions can be derived.
(3)
3
where Yo denotes the elastic modulus in the original state, and η is the effective strain coefficient. η is introduced as a compensation factor. Since the elastomer has many nonlinear properties under the large strain, constants such as elastic modulus, and dielectric constant, vary during actuation, which has influence on the effective stress and strain. To effectively integrate the nonlinear property of the elastomer in the equation a compensation factor η is introduced. For instance, in the case of VHB 4905 made by 3M, η on the compression can be represented from the experimental data as follows. η
e
1
1
Figure 3: Schematic diagram ANTLA
where ∆εa is the actuated strain, Ya denotes effective elastic modulus along a direction. The effective elastic modulus Ya can be expressed such as Ya =
o
i n
i n
Figure 2: Effective strain coefficient of VHB4905
where E is the electric field, t is the final thickness, V is the applied voltage, and ²o , ²r are the permittivity of the free space and the relative permittivity of the polymer, respectively. The strain caused by the the pressure σa is approximately proportional to the square of the applied field. The main source of the deformation has been reported to be generally the electrostatic force (which should be investigated more rigorously afterwards), and it is quite significant in soft dielectric elastomers. For an isotropic solid without mechanical constraints, the strain ε along the thickness-direction is derived as ∆εa
+
r e s t r a
r e s t r a
Proposed Actuator
The proposed actuator, called ANTagonistically– driven Linear Actuator(ANTLA), is based on the principle conceptually illustrated in Fig.3. In this configuration, a pre-stretched polymer film is foiled on the frame, and compliant electrodes are coated on both sides of the elastomer film. In Fig. 3, the output terminal divides the electrode pattern between electrode A and B on the upper surface, and the common electrode C is coated on the bottom. In the proposed design the actuation is realized by using two compliantly electroded films which are engaged with uniform pretension. This mechanism makes bidirectional actuation possible in an antagonistic fashion. Figs. 4 shows the prototype of the proposed actuator. The design is based on the conceptual design and composed of two electroded polymer films (in our study polyurethane and VHB 4905 are used). The compliance of the electrode is necessary to keep the conductivity during deformation, and in this research conductive polymer
(4)
where α is scaling factor of material (in the case of VHB 4905 α is calculated as 1.0 as shown in Fig. 2) and o εa represents the thickness strain of the film on the compression. It has negative values, and it is essential for large prestrain. In the case of small prestrain, the effective strain coefficient η is negligible.
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boundary conditions, it is almost impossible to derive the generalized model for the actuator, and the model for the analysis should be simplified as much as possible. Nevertheless, the complexity of the analysis can be reduced when the actuation occurs under the plane strain condition.
4.1
Let’s consider half of the proposed actuator shown in Fig. 5 (it is expanded into a full model in next section). As illustrated in Fig. 5, half model of the actuator can be divided into a number of elements with infinitesimal width. For the element, because the elas-
Figure 4: Prototype of ANTLA
electrodes and conductive greases are used. Assuming uniform pretensions are engaged along longitudinal directions in the proposed design, the tension on both films are initially balanced along the deflecting direction of the polymer film, and the output terminal is actuated from the unbalanced force caused by the deformation of the dielectric elastomer. Thus the maximum actuation force is equal to the pretension of the elastomer film. The electrode C is common and the electrode A and B are active ones. For example, if a positive voltage is given on the electrode A, a negative one on B, and a negative on C, then the output terminal moves toward the electrode B and vice versa. Also, the compliance of the output terminal can be actively adjusted by controlling the input voltages. If positive voltages are given on electrode A and B simultaneously while keeping a negative voltage on C, the output terminal changes to a highly compliant state. Also, it becomes highly stiff when all the voltages are the same. Accordingly, the proposed actuator achieves four states desired for artificial muscle actuators, that is forward, backward, highly compliant and highly stiff. Table.1 represents the typical states of the actuator according to the given voltages.
y y x
z
H
a
l f
m
d
o
d
d
e l
s
y p
d
x p
P z
o
l y m
z v
z p
e r
y x
L
L
y p
z p
d
x p
=
L
x p
2
L
x p
Figure 5: Half model of proposed actuator tomer is incompressible, the strain has the following equation. (1 + o εxi + ∆i εxj )(1 + o εyi + ∆i εyj ) (1 + o εzi + ∆i εzj ) = 1
(6) (7)
where the prefix and suffix of strain variables, o, p and f denote the original state, the prestrained one and the resultant one, respectively. Also the strain variables ∆i εzj and i εzj are defined as ∆i εzj =
Table 1: States of ANTLA according to the given voltages State Electrode (ABC) Stiff state ª ª ª or ⊕ ⊕⊕ More compliant ⊕ ⊕ ª or ª ª⊕ Action toward B ⊕ ª ª or ª ⊕⊕ Action toward A ª ⊕ ª or ⊕ ª⊕
4
Half model
δzj − δzi , δzo
i
εzj =
δzj − δzi δzi
(8)
where δzi is the z–directional length of the element at i state, and i has state values defined as i = 0, p, f . The infinitesimal length extended by the prestrain, is δzp
= δz0
1 (1 +
o ε )(1 xp
+ o εyp )
(9)
After the prestrain is applied, the elastomer will be stretched as shown in Fig. 6. The resultant length δxf can be written as
Quasi–static Analysis
The dielectric elastomer is actuated with large prestrain, and the elastomer at large strain may be nonlinear. Moreover, since the strain depends on the
δxf = δxp + x
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(10)
H
a
l f
m
o
d
e l
o
f
p
o
l y m
e r
a
c t u
a
t o
r
P O
r i g
i n
a
l
l e n
g
r e s t r a
i n
1
t h
d P
l e n
g
x o
t h
d
x p
d F
i n
a
l
e x t e n
d
e d
l e n
g
b
i r t u y
a
l
e x t e n
e l e c t r o
s i o
n
s t r i c t i o
l e n
A
t h
d V
g
to the original state. When the elastomer is isotropic, only the x–directional virtual strain exists, and the y– directional virtual strain is eliminated because of the reaction force. Thus, the virtual strain along the x– direction is expressed by
x
P
x f
f
f 2
t h
n
d
εxv
=
1 1+
δzo δzf
∆f εzv
−1
(16)
x v
and we have P C
o
n
s e r v a
t i v e
f o
1
P
r c e P 1
+
P
2
σxv
= Yx
2
where x is displacement of the length between the prestrained and the actuated polymer. Thus, the resultant strain along x–direction is represented as the summation of the strain by pre–stretch and by actuation. εxf
=
o
x εxp + δx0
=
4
−dAf Yx εxf =f1 (x)
P
4.2
= δzp (1 + p εzf ) = δzp
δxp δxp + x
(12)
V 2 ) δzf
V 2 1 ²r ²o ( ) Yz δzf
(18)
P1 + P2 = f1 (x) + f2 (x, V )
(19)
Full model
= =
fL1 (x) + fL2 (x, VL ) fR1 (−x) + fR2 (−x, VR )
(20)
where VL , VR are the supply voltage of left side and right side, respectively. In a full model, the final displacement is determined at the equivalent point between the force(PL ) of the left half model and the force(PR ) of the right half model as shown in Fig. 7. Thus we get Z Z PL = PR (21) Z Z {fL1 (x) + fL2 (x, VL )} = {fR1 (−x) + fR2 (−x, VR )}
(13)
(14)
Thus, the total output force is derived as Z F = (PR − PL )dAf (22) Z = {fR1 (−x) − fL1 (x)}dδyp (23) Z + {fR2 (−x, VR ) − fL2 (x, VL )}dδyp
where the final thickness t is the length δzf . Thus, Eq. (2) becomes ∆f εzv = −
=
PL PR
where p εxf is the x-directional strain about prestrained state. In the second, let us derive the force by Maxwell stress. In this case, it is assumed that the virtual strain is induced when the actuation voltage V is applied to the elastomer. In Eq. (1), the pressure by the electrostatic force becomes σzv = −²r ²o (
= dAf σxv =f2 (x, V )
The equations obtained from the half model can be easily expanded to the full model due to its symmetry. Because δx0 , δy0 , δz0 , εxp , and εyp are constant variables, the force on the output terminal by the left elastomer and the right one, PL , PR , can be expressed by the function of final displacements and supply voltages as follows.
(11)
where dAf is the cross–sectional area of the element, and f1 (x) represents the force due to the prestrain with displacement x. When only x-directional strain(p εxf ) exists, the final thickness δzf is derived from Eqs. (6) and (9) such as δzf
(17)
Thus, the resultant force on the half model is obtained as
Thus, in Eq. (11) the restoration force P1 is written by P1
−1
4
P2
δxf − δx0 = δx0
Yz δzo V 2 δzf ²r ²o ( δzf )
In Eq. (17), the electrostatic force is
Figure 6: Forces in arbitrary element of half model
o
δxf δxo Yz −
(15)
where Yz denotes the z–directional effective elastic modulus, and ∆εzv is the virtual strain with respect
⇒
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gk · K(x) − ge · E(x, VL , VR )
(24)
- 0
. 1
2
- 0
. 1
4
- 0
. 1
6
- 0
. 1
8
L
F 2
. 2 L
e f t m
. 2
2
- 0
. 2
4
- 0
. 2
6
k
E V
q
u
i v a
l e n
t
p
o
i n
o
r c e
s e n
s o
r
o
l y m a
t
a
s e r
d
i s p
l a
s e n
s o
c e m
e n
t
r
e r
c t u
a
t o
r
o d e l
0
k
V
S
l i d
e
O
u
t p
u
t
F
o
r c e ,
( N
)
- 0
- 0
. 5
P
- 0
. 2
P
- 0
8
0
. 3
k
R
V
h i g
t m
L
- P R
e l o d
G B M
- 0
. 3
o
v e
e q
u
i v a
l e n
t
p
o
i n
a
e a &
l l s c r e w
r e d E
n
m
o
c o
d
t o
r
e r
t
2 - 1
- 0
. 8
- 0
. 6
- 0
. 4
- 0 D
. 2 i s p
0 l a c e m
e n
0 t ,
( m
m
. 2
0
. 4
0
. 6
0
. 8
1
)
Figure 8: Experimental setup
Figure 7: Simulation results for full model: compare with each model (o εxp = 2.0, o εyp = 2.0) 1
0
. 9
0
. 8
0
. 7
0
. 6
0
. 5
0
. 4
0
. 3
0
. 2
0
. 1
x
p
e r i m
e n
t
( m
m
)
E
K(x) = Yx Lyp {o εxp (δLzf − δRzf ) x(1 + o εxp ) (δLzf + δRzf )} + δxp
(25)
E(x, VL , VR ) = Yx Lyp × δLxf Yz [δLzf − 1} { δLxo Yz − δδLzo ²r ²o ( δVL )2 Lzf Lzf
(26)
− δRzf
δRxf { δRxo Yz −
Yz δRzo VR 2 δRzf ²r ²o ( δRzf )
D
i s p
l a c e m
e n
t ,
where K(x) and E(x, VL , VR ) represent the force by prestrain and by electrostatic force, respectively such as
0
5
0
0
1
0
0
0
1 V
o
5
l t a g
0
0 e ,
2 V L
( V
0
0
i m
0
u
l a
t i o
2
n
5
0
0
3
0
0
0
)
Figure 9: Experimental and simulated results of displacement; without load, 0.01Hz, 200% prestrain
The actuator is actuated using a driving circuit that generated sine waves of high voltages with variable frequency. Parameter values for the simulation are as shown in Table 2. VHB 4905 acrylic dielectric elastomer film by 3M was used as the elastomer for the actuator. The actuator was fabricated with a single layered film(prestrain 200%) and a compliant electrode was coated as shown in Fig. 3. In the first ex-
− 1}]
where δLjf and δRjf are the j–directional final length of the left side and the right side, and δLjo and δRjo are the j–directional original length of the left side and the right side such as j = x, z, respectively. gk and ge are the effective restoration and electrostatic coefficients, respectively. These are influenced by geometric structure such as the ratio of the frame size, the thickness of output terminal, etc. Based on these equations we can control the displacement and stiffness of the actuator independently.
5
0
S
Table 2: Parameters in simulation notation value units notation value o Lxp 10 mm εyp 200 Lyp 20 mm gk 2.0 Lzo 0.5 mm ge 0.5 o εxp 200 % α 3.4
Simulations and Experiments
In this experiments, several aspects of the actuator were studied to confirm the validity of the proposed actuator model. Simulations and experiments were conducted and the results were compared. In the first, the force and displacement characteristics of the proposed actuators were determined. Also, the variation of stiffness according to input voltages was investigated. As shown in Fig. 8, an experimental setup consisting of a moving slide table, a laser displacement sensor, and a force sensor. Positions and forces can be simultaneously recorded using a Pentinum PC.
units %
periment, the displacement and force characteristics of the actuator were investigated. During these experiments the voltage VL applied to the actuator was sinusoidally varied from 0 to 2.7kV at quasi-static frequency 0.01Hz. The force of the actuator was obtained as shown in Fig. 10 in which the actuator was kept at fixed position (x = 0 mm). The result of displacement is depicted in Fig. 9. As shown in Figs. 9 and 10, a larger hysteresis is observed in the displacement than in the force, which may be considered due to the visco–elastic property of the polymer. Though there
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0
. 0
0
. 0
6
0
. 0
4
0
. 0
2
4
0
0
. 0
0
3
. 0
V
2
. 0
3
k
3
2
5
V
k
3
V
. 5
k
V
. 0
2
0
F
o
o
0
r c e ,
r c e ,
( N
( N
)
)
0
k
5
0
1
t
n
u
t i o
5
. 0
. 0
l a
u
. 0
0
u
E
x
p
e r i m
e n
t
O
0
i m
t p
F
S
0
0
5
0
0
1
0
0
0
1
V
o
l t a g
5
e ,
0
0
V
2
( V L
0
0
0
2
5
0
0
3
0
0
0
0
. 0
6
0
. 0
5
. 0
4
- 0
. 0
6
q
u
i v a
l e n
t
- 1
0
. 0
4
. 0
3
0
. 0
2
0
. 0
1
p
o
i n
t
p
e r i m
e n
/ m
0
0
1
0
0
0
1
5 V
0 o
0 l t a g
2 e
V L
,
V
0 R
0
0
- 0
. 6
- 0
. 4
- 0
i m
u
l a
t i o
( V
)
2
5
0
0
n
3
0
0
0
3
5
0
. 2 i s p
0 l a c e m
e n
0 t ,
( m
m
. 2
0
. 4
0
. 6
0
. 8
1
)
analysis which has been performed with experimental verifications. The proposed actuator realizes bidirectional actuation as well as controllable actuation. Bidirectional actuation and compliance controllability are important characteristics for the musclelike actuator and the proposed one realizes without any mechanical substitute or complicated algorithms. Its dynamic analysis and the example of robotic actuating devices will be given in the companion paper submitted with this paper[4].
t
)
x
( N t i f f n
5
. 8
Figure 12: Simulation results of stiffness about displacement
S
S
0
- 0
D
E
e s s
- 0
)
Figure 10: Experimental and simulated results of force; displacement x is 0, 0.01Hz, 200% prestrain
0
2
5
0
0
. 0
E
- 0
1
0
Figure 11: Experimental and simulated results of stiffness; displacement x is 1mm, 0.01Hz, 200% prestrain
Acknowledgments This work was performed under the management of the Intelligent Microsystem Center at the 21C Frontier R&D Program sponsored by Korea Ministry of Science and Technology.
exist hysteresis effects, simulated results are quite similar to the experimental results. In the second experiment, the stiffness according to the input voltage was studied and the results are as shown in Fig. 11. The stiffness of the proposed actuator exponentially decrease according to the input voltages VL and VR . The voltage applied to the actuator was sinusoidally varied from 0 to 3.5kV, where the actuator was kept at fixed strain(p εxf = 10%). From the results of simulations and experiments, though a larger hysteresis is still observed in the displacement that the force, they are quite similar to each other. Also, it should be noted that the stiffness of the actuator varies exponentially according to input voltages. Thus, it is expected that the stiffness of the actuator can be controlled by modulating the supply voltages VL and VR . The stiffness about the displacement has nearly linear values within the limits of a controllable strain, as depicted in 12. The experimental results show that the proposed actuator man realize desired stiffness without any mechanical substitute or complicated control algorithms.
6
References [1] Y. Bar-Cohen, Electroactive Polymer(EAP) Actuators as Artificial Muscles. SPIE press, 2001. [2] R. Pelrine, R. Kornbluh, and J. Joseph., “Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation,” Sensor and Actuators A: Physical 64, pp. 77–85, 1998. [3] R. Kornbluh, R. Pelrine, J. Eckerle, and J. Joseph, “Electrostrictive polymer artificial actuators,” Proc. Int. Conf. on Robotics and Automation, pp. 2147–2154, 1998. [4] H. R. Choi, S. M. Ryew, K. M. Jung, H. M. Kim, J. W. Jeon, J. D. Nam, R. Maeda, and K. Tanie, “Soft actuator for robot applications based on dielectric elastomer : Dynamic analysis and applications,” to be presented in IEEE Int. Conf. on Robotics and Automation, 2002.
Conclusions
In this paper a new soft actuator based on dielectric elastomer was proposed. Its basic concepts and working principles were introduced with quasi–static
3217