Journal of Energy and Power Engineering 6 (2012) 1442-1452
A Survey on Pneumatic Muscle Actuators Modeling
Eleni Kelasidi
1
, George Andrikopoulos
1
, George Nikolakopoulos
2
and Stamatis Manesis
1
1. Department of Electrical and Computer Engineering, University of Patras, Rio 26500, Achaia, Greece
2. Department of Computer, Electrical and Space Engineering, Luleå University of Technology, Luleå 97187, Sweden
Received: October 18, 2011 / Accepted: January 05, 2012 / Published: September 30, 2012.
Abstract: The aim of this article is to provide a survey on the most popular modeling approaches for PMAs (pneumatic muscle
actuators). PMAs are highly non-linear pneumatic actuators where their elongation is proportional to the interval pressure. During the
last decade, there has been an increase in the industrial and scientific utilization of PMAs, due to their advantages such as high strength
and small weight, while various types of PMAs with different technical characteristics have appeared in the literature. This article will:
(a) analyse the PMA’s operation from a mathematical modeling perspective; (b) present their merits and drawbacks of the most
common PMAs; and (c) establish the fundamental basis for developing industrial applications and conducting research in this field.
Key words: Survey, modeling, PMA (pneumatic muscle actuator), pneumatic artificial muscle, fluidic muscle.
1. Introduction
Over the years, researchers have used various types
of actuators on their industrial, robotic and automation
applications. Hydraulic, electric, magnetic and
pneumatic actuators are some of the commonly utilized
types, with respect to the application’s characteristics,
function and limitations. During the last decade, there
has been an increase in the use of pneumatic actuators
in the industrial and medical areas, mainly due to their
advantages such as low power to weight ratio, high
strength and small weight [1].
Pneumatic muscle actuator [2], also known as the
McKibben PAM (pneumatic artificial muscle) [3-6],
fluidic muscle [7] or a biomimetic actuator [8], was
first invented in 1950s by the physician, Joseph L.
McKibben and was used as an orthotic appliance for
polio patients [3]. PMAs are well suited for the
implementation of positive load feedback, which is
known to be used by animals. They present smooth,
accurate and fast response and also produce a
significant force when fully stretched. Moreover,
Corresponding author: Stamatis Manesis, professor, research
fields: automatic control systems, industrial automation,
mechatronics. E-mail: stam.manesis@ece.upatras.gr.
PMAs are lightweight, which is a particularly useful
feature when working with applications that place
restrictions on the weight of the equipment e.g., mobile
robotic applications [9].
Until now PMAs were commonly used in areas like
those of medical, industrial and entertainment robotics.
Specifically, PMAs have been widely used in
rehabilitation engineering as an actuator in orthotic
exoskeleton appliances [10-12]. Artificial muscles
have also been used in biomimetic robotics [13, 14] and
for the development of artificial fine-motion limbs [13,
15]. PMAs can also be found in manufacturing,
laboratory and aerospace applications [16].
Typical manufacturing of a PMA can be found as a
long synthetic or natural rubber tube, wrapped inside
man-made netting, such as Kevlar, at predetermined
angle. Protective rubber coating surrounds the fibber
wrapping and appropriate metal fittings are attached at
each end. PMA converts pneumatic power to pulling
force and has many advantages over conventional
pneumatic cylinders such as high force to weight ratio,
variable installation possibilities, no mechanical parts,
lower compressed air consumption and low cost [17].
When compressed air is applied to the interior of the
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A Survey on Pneumatic Muscle Actuators Modeling
1443
rubber tube, it contracts in length and expands radially.
As the air exits the tube, the inner netting acts as a
spring that restores the tube in its original form. This
actuation reminds the operation of a single acting
pneumatic cylinder with a spring return, while this
reversible physical deformation during the contraction
and expansion of the muscle results in linear motion.
Recently PMAs have found significant medical
applications [18-20] as PMA actuator resembles the
function of a human muscle, and thus has been given
the name pneumatic artificial muscle [4, 21]. It should
be noted that the most significant advantage of utilizing
PMA in control applications, is that for their position
control, only one analog variable needs to be controlled,
while for the same operation with a pneumatic cylinder,
two analog variables need to be controlled (one for
each chamber). As a result in the case of a pneumatic
cylinder, it is more difficult to find an equilibrium
between the two gauge pressures in the chambers, that
it is for the case of PMAs.
As this type of actuators is becoming increasingly
popular, many alternative types of PMAs have
appeared in the scientific literature that differs in their
mechanical construction and in the mathematical
model describing their principal operation. Until now,
except from the Festo AG Company’s PMAs [4,
22-24], there has not been standardized and
commercial versions of PMAs and in applications
containing such actuators, is of paramount importance
to: (1) choose PMA that fits the application, and (2)
utilize the correct mathematical PMA model. The high
non-linearities due to: (1) the existence of the
pressurized air; (2) the elastic-viscous material; and (3)
the geometric features of PMA, are the first problem
that a control engineer will have to deal with in order to
derive and utilize a proper PMA mathematical model.
PMAs can be addressed in the literature with
different names, like fluidic muscle [25-27], air muscle
[28, 29], pneumatic muscle actuator [29], fluid actuator
[24], fluid-driven tension actuator [4], axially
contractible actuator [30, 31], tension actuator [32, 33],
PPAM (pleated pneumatic artificial muscle) [34, 35],
biomimetic actuator [34], BPA (braided pneumatic
actuator) [36-40], paynter hyperboloid muscle [27, 41,
42], ROMAC (robotic muscle-like actuator) muscle
[34, 42], and yarlott netted muscle [7, 42, 43]. Typical
types of PAMs and the corresponding naming are
depicted in Fig. 1.
The aim of this article is to provide an up-to-date
literature review of the various manufacturing types of
PMAs and their operation, while focusing on analysing
the most popular and functional mathematical models
of PMAs that can be found in the scientific literature.
The survey, in this promising technology of PMAs,
will also provide a wide set of the most important
references in this field that could be further
investigated by the interested readers.
This article is organized as follows. In Section 2, the
basic PMA’s principles of operation are presented. The
most significant modeling approaches for PMAs are
presented in Section 3. Finally, some conclusions are
drawn in Section 4.
2. Principles of Operation
The basic principles of the PMA’s operation can be
categorized in two cases: (1) under a constant load and
with varying gauge pressure, and (2) with a constant
gauge pressure and a varying load. To illustrate this
operation, a PMA of an arbitrary type is considered
[34]. In the first case (Fig. 2), PMA is fixed at one end
and has a constant mass load hanging from the other
side. The pressure difference across the membrane, i.e.,
its gauge pressure, can be increased from an initial
value of zero. At zero gauge pressure the volume
enclosed by the membrane is minimal
min
V
and its
length maximal
max
l
. If the actuator is pressurized to
some gauge pressure
1
P
, it will start to bulge and at the
same time develop a pulling force that will lift the mass
until the equilibrium point where the generated force
will equal to the mass weight
M
g
.
At this point the PMA’s volume will have been
increased to
1
V
and its length contracted to
1
l
.
A Survey on Pneumatic Muscle Actuators Modeling
1444
Fig. 1 Various types of PMAs: (a) McKibben
muscle/braided muscle; (b) pleated muscle; (c) yarlott
netted muscle; (d) ROMAC muscle; and (e) paynter
hyperboloid muscle.
Fig. 2 PMA operation at constant load.
Increasing the pressure further to
2
P
will continue this
process, until the gauge pressure reaches the maximum
allowable value
max
P
. During this type of operation
PMA: (1) will shorten its length by increasing its
enclosed volume, and (2) will contract against a
constant load if the pneumatic pressure is increased.
The second type of PMA’s operation, which is the
case of operation under constant gauge pressure, is
depicted in Fig. 3. In this case, the gauge pressure is
kept at a constant value
P, while the load is decreasing,
driving the PMA to inflate and decrease its initial
length
3
l
. If the load is completely removed, the
swelling goes to its full extent, at which point the
volume will reach its maximum value
max
V
, the length
of its minimal value
min
l
, and the pulling force will
drop to zero. PMA will not be able to contract beyond
Fig. 3 PMA operation at constant pressure.
this point and it will operate as a bellows at shorter
lengths, generating a pushing instead of pulling force.
During this type of operation PMA: (1) will decrease its
length at a constant pressure if its load is decreased, and
(2) its contraction has an upper limit at which it
develops no force and its enclosed volume is maximal.
For these two principal operations, it should be
highlighted that for each pair of pressure and load,
PMA has an equilibrium length. This characteristic is
in a total contrast to the operation of a pneumatic
cylinder where the developed actuation force only
depends on the pressure and the piston surface area so
that at a constant pressure, it will be constant regardless
of the displacement [1].
3. Modeling Approaches
In recent years, there has been quite a lot of activity
regarding the mathematical modeling of PMAs. The
aim of such a model is to relate the pressure and length
of the pneumatic actuator to the force it exerts along its
entire axis. In the process of deriving a proper PMA
model, variables such as pulling force, actuator’s
length, air pressure, diameter and material properties,
play a major role in the PMA dynamical behavior and
this is why the mathematical models aim to describe
the relationships between these factors. Understanding
those relationships is of paramount importance in every
application that consists of PMAs and especially if the
A Survey on Pneumatic Muscle Actuators Modeling
1445
main goal is to control its overall function (mainly the
length of actuation). Unfortunately, PMAs evidence
strong non-linear force-length characteristics that make
it more difficult to control them and obtain the
demanded performance features [33, 44]. In the
following subsections an analysis to the most common
and valuable PMA’s models will be presented.
3.1 Geometrical Model of PMA
The original method of modeling was based on the
geometry of PMA, an approach that has not been very
useful for predicting the dynamical characteristics of
PMAs because their parameters are not easily
measured during actuation. Thus, various different
geometrical models were proposed to describe the
behavior of PMA. Among these models, the Chou and
Hannaford model [45] and the Tondu and Lopez model
[46] have been widely used.
The Chou and Hannaford model is the simplest
geometrical model for a static performance of a PMA.
The proposed model is valid under the following
assumptions: (1) the actuator is cylindrical in shape; (2)
the threads in the sheath are inextensible and always in
contact with the outside diameter of the latex bladder;
(3) frictional forces between the tubing and the sheath
and between the fibers of the sheath are negligible; and
(4) latex tubing forces are negligible.
With this approach the PMA actuator can be
modeled as a cylinder, depicted in Fig. 4, with a length
L, thread length b, diameter D, and number of thread
turns n. The angle ș is defined as the angle of the
threads with the longitudinal axis [47].
When the PMA actuator inflates, D and L change,
while n and b remain constant, while the expressions
for the PMA’s length and diameter can be formulated
as:
sin
cos ,Lb Db
n
T
T
S
(1)
By combining Eq. (1) the thread length can be
calculated as:
1/2
2222
bLDn
S
(2)
Fig. 4 Simplified geometrical model of PMA.
Eq. (2) is referred in the literature as the geometric
relationship for PMA, while its volume is provided by:
32
2
cos sin
4
b
V
n
T
T
S
(3)
Utilizing the energy conservation principle, PMA
simple geometric force
g
F
can be calculated as the
gauge pressure multiplied by the change in volume
with respect to length (this model can also be found in
[47]):
2
2
2
2
31
4
g
L
pb
b
F
n
S
§·
¨¸
©¹
(4)
Another simple and widely-used geometrical model
of PMA is that of Tondu and Lopez [46]. Based on this
approach and by: (1) utilizing similar geometric
description of the muscle [45]; (2) assuming
inextensibility of the mesh material; and (3) angle
changes during the alteration of PMA’s length, the
following expression of the contraction force F
generated by the muscle, as a function of the control
pressure P and the contraction ratio
H
, based on the
theorem of virtual work , is deduced [46, 48, 49]:
22
0
(, ) (1 )FP rp
S
TE
ªº
u
¬¼
(5)
where:
0
22
000
131
,0
tan s
,
in
max
l
l
TE
T
T
dd
(6)
In Eqs. (5) and (6),
0
r
is the nominal inner radius, l
is the length of the muscle, l
0
is the initial nominal
length,
P
is the pressure and ș
0
is the initial angle
between the membrane fibres and the muscle axis,
while this model can also be found in Ref. [21].
A Survey on Pneumatic Muscle Actuators Modeling
1446
A disadvantage of the model is that its design is
based on the hypothesis of a continuously
cylindrical-shaped muscle, whereas it takes a conic
shape at both ends when it contracts. Consequently, the
more the muscle contracts, the more its active part
decreases. This phenomenon results in the actual
maximum contraction theoretically being smaller than
that expected from Eq. (5) [50]. These models still
have limitations in predicting the behavior of PMA in
no-load conditions. However, there is a major
phenomenon which Tondu and Lopez [46] considered
to improve Eq. (5), which is the addition of an
empirical correction factor k to account for an end
deformation of PMA:
22
0
(, ) (1 )FP rP k
S
TE
ªº
u
¬¼
(7)
where again:
0
max
dd
and
max
is provided from:
max
(1 / )(1 / )k
E
T
(8)
Inserted in this way within the considered static
model, the parameter k does not modify the value of the
maximum force given at zero contraction ratio. This is
in concordance with the conducted experiment since
PMA has a cylindrical shape only when its contraction
ratio is zero. Furthermore, the parameter k allows
adapting the model maximum contraction ratio given
by Eq. (8) to the experimental data. Thus, it tunes the
“slope” of the considered static model.
In addition it has established two options for the
selection of the parameter
k : (1) a constant value
which may vary depending on the material that the
muscle is made of, and (2) the parameter k depends on
the pressure in the muscle at any given time. It has been
observed in Refs. [45, 46] that during operation there is
a force/displacement hysteresis in the muscles caused by
friction between the braid stands. Chou and Hannaford
[45] produced a model including an experimentally
obtained force offset which was added to calculate
forces during muscle contraction and subtracted during
extension. Tondu and Lopez [46] took this concept
further by attempting to quantify the offset force by
modeling the friction. Although the model produced
was more accurate than that of Chou and Hannaford, it
still relied on a degree of experimental data. In Ref.
[51], Chou and Hannaford proposed that physical
configuration and the behavior of PMA hinted the
variable stiffness similar to spring-like characteristics.
In Ref. [52], more realistic muscle geometry has
been utilized, in order to model the muscle’s true form,
by taking into consideration the irregular shape of the
end sections during inflation. The proposed geometry
includes a frustum cone that models each end section
and a cylinder to model the muscle middle section. The
middle-section geometry is governed by the
deformation of the braid and its relationship is based on
the form already shown in Fig. 4. With respect to the
geometrical model depicted in Fig. 5, L is the overall
muscle length, L
L
is the horizontal length of the cone,
L
Z
is the cone generator length, ȕ is the frustum-cone
angle of the muscle ends, L
m
is the middle-section
length, D is the middle section diameter and d is the
end-fixture diameter.
All parameters are a function of the muscle-contracted
length, except L
Z
that is determined experimentally based
on the muscle end diameter and maximum contracted
diameter. The relationships between the muscle diameter
D and the muscle middle-section length L
m
are again
provided by Eq. (1), where b, ș and n were previously
described in Fig. 4. By utilizing Eq. (1), the braid angle is
eliminated and the braid diameter is calculated by:
221/2
()
m
bL
D
n
S
(9)
During contraction, the cone base diameter expands
beyond the end-fixture diameter. The cone horizontal
length L
L
is:
Fig. 5 Geometrical model of PMA.
