Mathematical Model of Pneumatic Artificial
Muscle Reinforced by Straight Fibers
NORIHIKO SAGA,1,* TARO NAKAMURA2
1
AND
KENJI YAEGASHI1
Akita Prefectural University, 84-4 Tsuchiya, Honjyo, Akita, 015-0055, Japan
2
Chuo University, 1-13-27 Kasuga, Bunkyoku, Tokyo 112-8551, Japan
ABSTRACT: This article describes a mathematical model of a pneumatic artificial muscle
reinforced by straight fibers. The pneumatic artificial muscle is lightweight and high power. In
addition, it is possible for it to emit exhaust into the atmosphere because the transmission
medium of its energy is air, and it needs neither a tank nor maintenance like hydraulic
equipment. In addition, safety to a person and the environment is high because the base of the
actuator is a soft polymeric material even if damage to the artificial muscle is caused. On
the other hand, because the device is composed of a thin film cylinder, and is easily influenced
by outside power for the conversion of the volume change in rubber into physical contraction
axially, and its passive character is strong, precise positional control is difficult. However, the
living thing that has a verbose degree of freedom enables minute motion and walking, etc. by
skillfully adjusting to avoid impedance, because it understands its own muscular characteristic
beforehand. Similarly, it is thought that the precise positional control is possible by
understanding the characteristics of an artificial muscle beforehand. In this research, it is
assumed that the expansion shape of the pneumatic artificial muscle that we developed from
the pressure distribution is equivalent to the centenary curve, and its model is based on
dynamic balance. The result shows the effectiveness of control of an artificial muscle that uses
highly accurate calculations and models.
Key Words: pneumatic actuator, artificial muscle, mathematical model, contraction characteristic.
INTRODUCTION
HIS research aims to model the pneumatic artificial
muscle, which has received much attention in the
fields of biomechanism, medical care, and welfare, etc.,
based on the concept of dynamic balance. As a result,
its application to various controls becomes possible.
There are a lot of advantages for having characteristics
similar to a human muscle by the actuator of a wearable
device and a rehabilitation robot that assists a
person’s motions. In addition, the actuator should be
lightweight and have high power (Chou and Hannaford,
1994, 1996). The pneumatic artificial muscle that we
developed consists of a lot of high-strength carbon
fibers arranged axially in a rubber tube composed of
crude rubber, expanding by adding pneumatic force
to it, and converting volume change into contraction.
As for the pneumatic artificial muscle axially strengthened by the fiber, it can obtain the contraction force
and the amount within a range which allows the elastic
deformation of the tube, since the radial restraint is
T
only the elastic force of the tube, and is different from
the McKibben type (Chou and Hannaford, 1996;
Ferraresi et al., 1999; Daerden and Lefeber 2001;
Daerden and Lefeber 2002; Ferraresi et al., 2001;
Jacobsen et al.).
The developed artificial muscle has a length of 72 mm
and a diameter of 12 mm, and the tension generated is
about 200 N in the pressure of 0.11 MPa by the
contraction rate 35% or more (Chou and Hannaford,
1996; Daerden and Lefeber 2001). The carbon fiber
inserted in the rubber tube of the artificial muscle is
composed of thin fibers bunched together. As a result,
when the artificial muscle expands, the expansion of
the rubber tube can be controlled because the bunch
of fibers diffuses, too. As a result, the rubber tube
can be thinned more. In addition, we hope a high
contraction rate will use low pressure, because the
degree of elasticity of the crude rubber used for the
artificial muscle is small. In this article, we examine
the relational equation of pressure and contraction
from the dynamic balance aiming to improve the control
of the artificial muscle.
*Author to whom correspondence should be addressed.
E-mail: [email protected]
Figures 2–4 and 8 appear in color online: http://jim.sagepub.com
JOURNAL
OF INTELLIGENT
MATERIAL SYSTEMS
AND
STRUCTURES, Vol. 18—February 2007
1045-389X/07/02 0175–6 $10.00/0
DOI: 10.1177/1045389X06063462
ß 2007 SAGE Publications
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175
176
N. SAGA
ET AL.
Y
MATHEMATICAL MODELS
OF ARTIFICIAL MUSCLE
Previous Mathematical Model
A mathematical model for artificial muscle was
proposed by Ferraresi (Daerden and Lefeber 2002).
Equation (A) and Equation (B) are shown as follows.
R1
High pressure
Low pressure
Moving
ðBÞ
New Mathematical Model
MUSCLE SHAPE
As for the pneumatic artificial muscle, the internal
pressure that causes the expansion balances the tension
of the fiber, the elastic deformation of rubber, the load,
etc. [4,8]. The pneumatic artificial muscle in this study
is composed of a tube of crude rubber, and allows large
contractions with little change in the internal pressure.
Therefore, it is necessary to strictly select all the physical
quantities related to the artificial muscle in order
to make a highly accurate model (Daerden and
Lefeber 2001; Daerden and Lefeber 2002). We made
a model Figure 1 assuming the expansion shape to be
equivalent to the catenary curve of Figure 2 because
of uniform pressure inside the artificial muscle. When a
is assumed to be a constant, the equation concerning
the surface of the expansion of the artificial muscle is
as follows:
x
x
a x
fðxÞ ¼ a cosh
¼ exp
þ exp : ð1Þ
a
2
a
a
A curve equation that shows the expansion configuration of the artificial muscle is approximated, with
consideration of the initial diameter d0 of the boundary
and the artificial muscle, as follows:
fðxÞ ¼ 2
x
l
d0
þ þ :
2a 8a 2
R2
ðAÞ
However, because the shape of the artificial muscle
is assumed to be circular when it is contracted by air
pressure, it is necessary to measure the radius and the
angle of the circle. Therefore, it is difficult to design
a feed-forward controller in order to compensate
for the nonlinear characteristics of the muscle. In this
article, it is possible to create a feed-forward control by
modelling the shape on catenary’s curve.
2
R1 < R2
ð2Þ
Center of the circle
Figure 1. A mathematical model for this type of artificial muscle has
already been proposed.
0.03
Amount of expansion (m)
2ER2 s
r0 ri
ðsin cos Þ þ Elf s
ri
ri
(
þN=4
X
2
n
þ T sin cos
N
n¼N=4
R2
2
p R þ 2Rr0 sin sin 2 ¼ 0
2
2
F þ p ðr0 þ Rð1 cos Þ NT ¼ 0:
Z
−0.04
at rest
0%
5%
10%
15%
20%
25%
0.025
0.02
0.015
0.01
0.005
−0.03
−0.02
−0.01
0
0
0.01
0.02
0.03
0.04
Length (m)
Figure 2. Catenary curve.
Here, l is the contraction length and
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
l3
:
a¼
24ðl0 l Þ
ð3Þ
STATIC MODEL OF MUSCLES
An explanation of the mathematical model for the
artificial muscle follows. The mathematical model
concerning the contraction of the artificial muscle is
derived from various power relationships acting on the
artificial muscle. Figure 3 is a model of the artificial
muscle that uses the Equation (4). The tension vector
of the fiber is an arbitrary point and the angle of the
shaft is obtained as follows:
x
’ ¼ arctan
:
ð4Þ
a
x-Axial Direction:
The next concern is axial balance. Power that
works axially can be shown by the pressure Px that
acts on the artificial muscle, the load Fx, and the
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177
Mathematical Model of Pneumatic Artificial Muscle
1
σy
ϕ
σx
σt
Stress (MPa)
0.8
dPy
0.6
0.4
0.2
dx
d0
0
0
0.5
1
1.5
2
2.5
3
Strain (mm)
Figure 4. Stress–strain diagram of rubber material.
Figure 3. Pressure impressed to the rubber.
tension Sx of the fiber. The balance equation concerning
these is as follows:
Px þ Fw Sx ¼ 0:
ð5Þ
Internal Axial Pressure
Pressure Px applied to the edge of the artificial muscle
is illustrated in the following equation:
P¼
d02
P:
4
ð6Þ
Here, P is pressure.
Tension of the Fibers
Considering the number of the fibers, tension of the
fiber Sx is obtained as follows:
l
Sx ¼ mx ¼ mt cos arctan
:
ð7Þ
a
Therefore, the force balance of the x-axial direction is
obtained as follows:
t ¼
d20 P=4 þ FW
:
m cos½arctanðl=aÞ
ð8Þ
y-radial Direction:
We next discuss the force balance for the radial
direction. The force balance is given as follows:
Ry þ Sy Py ¼ 0:
ð9Þ
Here, Py is pressure in the y-axial direction, Sx is the
tension of the fiber, and Ry is the contraction forces
caused by the elasticity force of the rubber.
Elastic Force of the Rubber
Strict conditions are needed for the mathematical
models of an artificial muscle that is composed of
a rubber tube (in which a large quantity of elastic
deformation is allowed), including air with compressibility (Nakamura, et al., Verrelst et al., 2002; Verrelst
et al., 2002; Verrelist et al., 2003; Yaegashi et al., 2003).
This chapter shows the best conditions for establishing
the relationship between Young’s modulus of crude
rubber and the expansion rate of the artificial muscle.
Because the artificial muscle that we produced has
axially installed fibers, it does not expand spherically
like a rubber balloon, and it does not become an
expansion of the biaxial tensile. Then, we did an
unconfined compression test based on the JIS standard.
It is necessary to calculate the Young’s modulus of
the crude rubber taking into consideration the beginning
of the curve in the elasticity region of the stress–strain
diagram (Figure 4). The expansion rate of the artificial
muscle was found to be about 300% in the experiment
to determine fundamental characteristics. Therefore,
because one only has to examine Young’s modulus
consisting of the same range, it can be approximated
by the following equations:
y ¼ x þ ð10Þ
The radial stress of the rubber tube is shown as
follows:
l4
x2 þ ð11Þ
¼
d0 a 4
Figure 4 shows the stress–strain diagram of the
artificial muscle actuator calculated by the experiment.
The contraction force of the rubber Ry, is given by the
following equation:
Z l=2
Ry ¼ 4t
ð12Þ
dx
0
However, ¼ 0.1263, ¼ 0.3
Tension of the Fibers
Figure 5 shows a radial model of the artificial muscle
that takes into account the number of fibers. When the
number of the fibers is assumed to be an even number
(m 4), the angle of 0 of each fiber in the range
can be defined as follows:
m
ð2n þ 1Þ n ¼ 0, 1, . . . :
ð13Þ
m
2
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178
N. SAGA
y
ET AL.
Laser position sensor
Solenoid valve
π
(2n+1)
m
Artificial muscle
z
Pressure sensor
Regulator
Figure 5. Cross-section of the artificial muscle.
Load
Therefore, the tension of the fibers in the y-axial
direction Sy is shown as follows:
l
Sy ¼ 2t sin arctan
M:
ð14Þ
a
Here,
M¼
m=2
X
n¼0
sin
h
m
i
ð2n þ 1Þ :
ð15Þ
Internal pressure in the radial direction
Next, we discuss internal pressure in the radial
direction of the artificial muscle. Figure 3 shows a
model of pressure on the surface of the artificial
muscle. In this figure, minute pressure in the y-axial
direction dPy is:
d
dPy ¼ P sin d dx:
2
ð16Þ
2
l
Py ¼ Pl
þ 2d0 :
3a
ð17Þ
Here,
The force balance of the y-radial direction is obtained
as follows:
R l=2 Pl l2 =3a þ 2d0 þ 4t 0 =d0 a l4 =4 x2 þ dx
:
t ¼
2 sin½arctanðl=aÞM
ð18Þ
Therefore, the following equation
2mt l2 þ 6ad0 þ 12d0 MFw
:
P¼
d0 2mðl2 þ 6ad0 Þ 3Md20
ð19Þ
Data logger
Trigger
Amplifier
Figure 6. Experimental setup.
or right depending on the expansion or contraction of
the actuator using an artificial muscle, so as to loosen
the self-weight of the attachments and a weight setup
on the axial where this free attachment is attached.
We measured the contraction characteristic and the
response characteristic by using the changing weight of
the internal pressure as a parameter. Also, we measured
the amount of loading contraction by changing the load
under constant internal pressure by an attached load cell
instead of the tester’s weight.
Fundamental Characteristics
Figure 7 is an illustration of the contraction of the
artificial muscle. A ring made of aluminum is installed
in the artificial muscle as shown in the figure, and it
helps to improve the contraction force while controlling
the expansion of the artificial muscle. An artificial
muscle that can endure high pressure can generate
a high contractile force because the contractile force
of the artificial muscle depends on the supply pressure
(Jacobsen et al., Nakamura, et al., Verrelst et al., 2002;
Verrelst et al., 2002; Verrelist et al., 2003; Yaegashi
et al., 2003). However, this does not create an
essential improvement because the contraction of the
low-pressure area is sacrificed according to Young’s
modulus, depending on the design changes in thickness,
diameter, etc. of the rubber tube. Therefore, a ring
was installed in the artificial muscle, designed with
the contraction of the low-pressure power area in
mind and the ability to endure high-pressure power
was increased.
is obtained.
The Contraction Characteristic of the Artificial Muscle
EVALUATION OF THE ARTIFICIAL MUSCLE
Experimental Setup
Figure 6 shows the experimental setup used to
measure the characteristics of the artificial muscle.
This experimental setup constitutes moving to the left
The contraction characteristics of the artificial muscle
can be shown in the relationship between the applied
pressure and the contraction rate. The distortion energy
function of the artificial muscle has large initial stress
expansion as well as expansion of the rubber balloon.
Therefore, high pressure is generally needed at the
initial stage in the contraction of the artificial muscle.
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179
Mathematical Model of Pneumatic Artificial Muscle
Contraction rate ×100 (%)
0.4
ring 0
ring 1
ring 2
ring 0:†
ring 1:†
ring 2:†
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
0 .02 0.04
0.06
0.08
0.1
0.12
0.14
0.16
Pressure (MPa) [†: Simulation result]
Figure 7. Contraction of the artificial muscle.
Figure 8. Relationship between the experimental result and the
simulation result.
Table 1. Specification of muscle.
Type of rubber
Type of fiber
Number of fiber
Length (mm)
Diameter (mm)
Thickness (mm)
Natural rubber
Carbon fiber
16
72
12
2
the simulation value and the experiment results stopped
responding to the rise of the contraction percentage.
The cause was thought to be because the thickness of
the rubber tube changed while expanding.
CONCLUSION
However, an artificial muscle is preferably one in
which a large contraction and contraction force can be
generated in a small, light, low-pressure area, and which
also takes into consideration the expanding diameter.
Therefore, we developed a small, lightweight artificial
muscle which takes into consideration the expansion of
the diameter. Moreover, the contraction characteristics
of the artificial muscle were measured by installing
a ring made of aluminum. Table 1 and Figure 8 show
the specifications and the experimental results of the
artificial muscle, respectively. Contraction starts from
low pressure. The artificial muscle without a ring has
the largest contraction percentage. Its contraction
exceeded 35% at a pressure of 0.11 MPa. Therefore,
the rubber tube can be made thinner by covering
it with a high-intensity fiber. And, it is possible to
make the artificial muscle’s contraction start from a
low pressure.
Static Pressure Relationship between Force and Length
This chapter shows the effectiveness of an artificial
muscle model based on a comparison between simulation results and the experimental data obtained from
Equation (19). Figure 8 shows the simulation and
experimental results of the contraction characteristics
of the artificial muscle. In this figure, the experimental
results are shown as a broken line, and the simulation
value as a solid line that contains the measurement
point. High accuracy was obtained in a low contraction
percentage of less than 20% though the accuracy of the
value of the simulation. We were able to derive a
mathematic model with high accuracy for the contraction of the artificial muscle. However, the correlation of
We considered the dynamic equilibrium conditions
of the contraction characteristics of a developed
artificial muscle, and formed a mathematical model.
As for the comparison between the mathematical model
calculation results and actual experimental results, the
correlation and accuracy are high. It is thought that it
could become a more complete mathematical model by
improving its identification of the thickness of rubber,
and using Young’s modulus of the artificial muscle’s
contraction characteristics.
NOMENCLATURE
E ¼ Young’s modulus of the deformable chamber
R ¼ curvature radius of the deformed actuator
S ¼ thickness of the chamber
lf ¼ length of one fiber
ri ¼ initial radius of the chamber
ro ¼ radius of the end-cap
T ¼ tension in one fiber
N ¼ number of fibers
p ¼ supply pressure
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Proceedings IEEE Transactions on Robotics and Automation,
12(1):90–102.
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N. SAGA
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