SOFT ROBOTICS
Volume 00, Number 00, 2020
ª Mary Ann Liebert, Inc.
DOI: 10.1089/soro.2019.0079
Novel Bending and Helical Extensile/Contractile Pneumatic
Artificial Muscles Inspired by Elephant Trunk
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Qinghua Guan,1 Jian Sun,1 Yanju Liu,2 Norman M. Wereley,3 and Jinsong Leng1
Abstract
Pneumatic artificial muscles (PAMs) are an extensively investigated type of soft actuator. However, the PAM
motions have been limited somewhat to uniaxial contraction and extension, restraining the development of
PAMs. Given the current strong interest in soft robotics, PAMs have been gaining renewed attention due to their
excellent compliance and ease of fabrication. Herein, under the inspiration of the elephant trunk, a family of
bending and helical extensile PAMs (HE-PAMs)/helical contractile PAMs (HC-PAMs) was proposed and
analyzed. Through both experiment and analysis, a model of generalized bending behavior of PAMs was built
and developed to investigate the properties of axial, bending, and helical PAMs in the same theoretical
framework. The topological equivalence and bifurcation were found in the analysis and utilized to explain the
behaviors of these different PAMs. Meanwhile, a coupled constant curvature and torsion kinematics model was
proposed to depict the motion of PAMs more accurately and conveniently. Moreover, a soft tandem manipulator consisting of bending and helical PAMs was proposed to demonstrate their attractive potential.
Keywords: bending, helical, bionic, soft robotics, pneumatic artificial muscle, kinematics
braided tubes.5 With the bionic braid structures similar to
creatures’ muscles, PAMs gain the properties of skeletal
muscle resemblance and variable stiffness,25 which are frequently utilized in bionic actuators18,26 or hydroskeletons.27
Especially, contractile PAMs have been widely applied to
biorobotics, humanoid robots, and rehabilitation devices,
which were collected by Andrikopoulos et al.28 Also, with
the renewed interest in soft robots, extensile PAMs have
shown promising potential and attracted more attention owing to their greater deformation and compliance.18,19,29–31
However, these studies were focused on axial PAMs
(either contractile or extensile), while animals such as elephants and octopus not only can adjust the tissue stiffness
and generate simple axial deformations but also can produce
diverse movements and shape change.32 The achievement
of these complex motions depends on muscle fibers delicately arranged in multiple orientations. Take the elephant
trunk as an example. These muscle fibers of the trunk are
arranged in three patterns (perpendicular, parallel, and helical or oblique)33 (Fig. 1a).
Introduction
A
s the soft robotic field flourishes, research on soft
actuators has gained increased attention from the general
field of robotics. Compared with robots built with rigid
structures and components and motors, soft actuators have
the advantages of greater power density, greater compliance,
high efficacy, acceptable efficiency, safer interaction, low
cost, and simple structures.1–4
Pneumatic artificial muscles (PAMs) are a frequently
investigated soft actuator type5 and have been explored
extensively. From the earliest McKibben actuator,6,7 a variety of pneumatic actuator designs to date have been derived with diverse structures such as braided-sleeved,8–10
fiber-reinforced,11–14 and pleated.15,16 They are able to realize multiple motion types, including contraction,17 extension,18,19 bending,12,20,21 and twist,22,23 which were reviewed
by Daerden and Lefeber5 and Greef et al.24
Currently, PAMs refer to a class of braided pneumatic
actuators consisting of one or more elastic tubes and matched
1
National Key Laboratory of Science and Technology on Advanced Composites in Special Environments, Harbin Institute of
Technology, Harbin, China.
2
Department of Astronautical Science and Mechanics, Harbin Institute of Technology (HIT), Harbin, People’s Republic of China.
3
Department of Aerospace Engineering, University of Maryland, College Park, Maryland.
1
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2
GUAN ET AL.
FIG. 1. PAMs inspired by the elephant trunk. (a) Muscle fiber distribution of elephant trunk (modified from the figure of
Smith et al.33); (b) the bionic methodology of PAMs; (c) schematics and fabrication of bending and HE-PAMs/HC-PAMs
and the soft trunk-like MDOF manipulator based on these PAMs. HC, helical contractile; HE, helical extensile; MDOF,
multi-degree-of-freedom; PAM, pneumatic artificial muscle.
Even though PAMs can be fabricated quickly, and at low
cost, from commercial materials, without resorting to fiberwinding and elastomer-casting, which enable ease of commercialization, the PAM motions have been typically limited
to the axial direction. Meanwhile, as another class of soft
actuator, fiber-reinforced actuators have been developed into
a large family that has capabilities of extending, contracting,
bending, twisting, and winding.12,23,34–38 Nevertheless, some
researchers had developed some bending joints based on the
compliance of PAMs by setting the PAM on an exterior
soft39,40 or rigid structure.41 Moreover, some novel designs of
PAMs also have been proposed by researchers, recently. The
family of PAMs was extended again by the bending extensilePAM (BE-PAM)21,42,43 and variable stiffness extensorcontractor PAM,27 which has promoted additional research in
PAMs (Fig. 1b).
Previously, much research has been done on modeling of
axial PAMs,44,45 however, there are limited models available
for soft bending or helical PAMs,42,43 thus constraining their
potential as an attractive class of soft actuators.13 Al-Fahaam
et al. built an output force mathematical model for extensor
bending PAMs considering the nonconstant braid angle
around the circumference during bending with a circular cross
section defined by a nominal radius.42,43 To the knowledge of
the authors, no existing model for helical PAMs is available
in the literature. Fortunately, there are also some analyses on
fiber-reinforced actuators.12,20,38 However, as a result of the
structural distinctions between the braided sleeves of PAMs
and the fiber families embedded in elastomeric actuators, the
larger range of relative gliding and rotating existing among
braided fibers not only causes frictional energy loss but also
brings more deformability, which implies that the analyses
of fiber-reinforced actuators cannot be directly applied to
bending or helical PAMs without substantial modification.
Apart from modeling the inner mechanism, the kinematics
of helical PAMs is also quite different from bending actuators
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BENDING AND HELICAL PAMS
due to the out-of-plane deformation in three-dimensional
(3D) space. Because of its simplicity and ease of computational techniques, constant curvature kinematics (CCK) has
been frequently utilized to depict the deformation of conventional continuum robots through a sequence of separate
bending or rotating transformation, reviewed by Jones and
Walker,46 Webster et al.,47 and Burgner-Kahrs et al.48
However, for soft robots, in most situations, the deformation
is the result of the coupling of bend and torsion49 (e.g., the
helical or spiral motion), which means that CCK leads to a
much larger number of elemental bend and torsion links to
approximate the deformation. As a consequence, a new easyto-compute kinematic expression is required urgently to be
developed for the coupled deformation.
In this study, under the inspiration of the muscle fiber arrangement of elephant trunks, a set of bending and helical
PAMs was realized by combining braided tubes and reinforced
flexible frames in different orientations, as shown in Figure 1c.
Since the soft actuators were usually reinforced by fibers,
strings, thin laminates, and other easy-buckling materials in
prior researches, bending contractile PAMs (BC-PAMs) were
hard to realize, and most researchers have focused on the
BE-PAMs. In this study, BE-PAM, BC-PAM, HE-PAM, and
HC-PAM were created with homologous structures using
parallel or helical reinforcement with a flexible frame having
sufficient stiffness to restrain extension or contraction and
resist buckling, which expands the operational envelope of
PAMs further and offers us a chance to characterize and
compare the performances of bending and HE-PAMs/HCPAMs with a homotypic structural design, for the first time.
Hereof, the modeling works were first conducted on the
BE-PAM and BC-PAM. The assumption of a nominal circular
cross section also was used to simplify the analysis, while to
gain more realistic modeling, the relationship between nominal radius and bending curvature of the actuator was given by
a function derived from the ratio of deformed circumference
to the original via the integral of hoop strain around the circumference instead of the direct average of the inside and
outside edges of the bend.43 Besides, the details of braid angle
changes around the circumference and along the longitudinal
axis can also be deduced by this method. Also, owing to the
nondimensional analysis, this function can be easily precomputed as a numerical solution embedded onto a microcontroller for real-time control, regardless of scale effect.
Moreover, to expand the models to helical PAMs, the generalized bending motion of PAMs was analyzed and modeled.
A set of maps was involved to describe the relationships of
some important parameters, including bending curvature, radius, axis length, volume, initial braid angle, and reinforced
helical angle, based on which the topological equivalence and
bifurcation in the bending behaviors were discussed.
Meanwhile, the helical PAM deformation kinematics was
studied, and a coupled constant curvature and torsion kinematics (CCCTK) was proposed to depict the motion of helical PAM more accurately and conveniently by replacing
the arc links with helix links. Finally, a trunk-like tandem
multi-degree-of-freedom (MDOF) manipulator was proposed and designed according to the different characteristics
of these PAMs proposed in this study. The manipulator was
built and tested to investigate the potential applications of
bending and helical PAMs in soft robotics by verifying its
gripping and posing capabilities.
3
Bending Extensile and Contractile PAMs
BE-PAM and BC-PAM were developed based on the extensile
PAM and contractile PAM. Each BE-PAM/BC-PAM mainly consists of end fittings, an elastic tube, a braided tube, and an embedded
flexible frame (Fig. 2a, d), and these components are all fastened
together at both ends. According to the literatures for extensile
PAMs19 and BE-PAMs, the braid angle a of BE-PAMs (Fig. 2a–i)
must be larger than 54.74. While via kinematic analysis, the BCPAMs were proposed and built by adjusting the braid angle to be
below 54.74 (Fig. 2a–ii).
The difference from the axial driving PAMs is that a
flexible frame was placed concentrically between the bladder
and braided tubes to limit the extension or contraction and to
generate a bending deformation toward or against the reinforced side when inflated, as shown in Figure 2a.
Kinematic analysis of bending PAMs
To some degree, the bending PAMs, extensile PAMs, and
contractile PAMs were all developed from the McKibben actuator,5 the working mechanisms of these actuators have
commonalities. Nevertheless, for bending PAMs, the distribution of deformation at the cross section is uneven, whereas the
deformation can reasonably be assumed to be uniform for both
contractile and extensile PAMs actuated axially, which generates a mechanical distinction between bending and axial PAMs.
This distinction motivates the need to analyze bending PAMs.
Through the analysis about the constraints of the braid, the
relationship between axial strain e1 and circular strain e2
could be written as below, with a0 as the original braid angle
of the actuator at rest. k1 and k2 are stretch ratios of the braid
along the axial and circular directions (Fig. 2b).
½cosa0 ð1 þ e1 Þ2 þ ½sina0 ð1 þ e2 Þ2
¼ cos2 a0 k1 2 þ sin2 a0 k2 2 ¼ 1
(1)
During inflation, the longitudinal axis of bending PAM
bends toward or against its reinforced side under the constrain
of the comparatively inextensible and incompressible frame
(comparing with the bladder tube and the braid sleeve) with
the length of L0, and its longitudinal axis elongates or contracts to the length of L (Fig. 2c). k is the bending curvature of
the actuator axis, and the bending curvature of the flexible
frame k’ was introduced to simplify the analysis. Thus, the
strain, e1 , can be written as a function of the bending curvature of the flexible frame k’ (Figs. 2c and 3a) and the distance
to the frame h that is defined by the cylindrical coordinate F
at the cross section (Fig. 3b).
Here, it is defined that the bending curvature k’ is positive
when the actuator longitudinal axis bends toward the reinforced side. This assumes that after inflation and bending,
the actuator still maintains its circular cross section, and the
length of the flexible frame is constant.
k ¼ 1=
1
þ R ¼ k¢=ð1 þ k¢RÞ
k¢
e1 ¼ k1 1 k¢h ¼ k¢ð1 cosFÞR
(2)
(3)
The circular strain e2 can be derived from the Equations (1)
and (3) as Equation (4), shown in Figure 3b. Then the braid
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4
GUAN ET AL.
FIG. 2. The bending deformation and geometry of
BE- and BC-PAMs. (a) the
deflated and inflated states of
the BE-PAM with the inner
pressure of 0 and 0.4 MPa (i);
the deflated and inflated states
of the BE-PAM with the inner
pressure of 0 and 0.4 MPa
(ii); the curves in the figures
represent the position of the
frame. (b) The definition of
braid angle a: direction 1 is
the axial direction, and direction 2 is the circular direction, (c) main parameters
of the BE-PAM are the
bending angle h, the initial
active length L0, and the deformed axial length L. Here,
X(s) is the position of a point
with the arc length of s following the curved reinforced
frame. (d) Construction (i),
materials (ii), and fabrication
(iii–viii) of bending PAM.
(iii) Mount the end fitting
with air channel on one end of
the bladder tube. (iv) Slide the
braid sleeve over the bladder
tube and insert the frames
between the bladder tube and
braid sleeve in parallel with
the bladder tube axis. (v)
Slide the hoops over the braid
sleeve and mount the other
end fitting on the bladder. (vi)
Slide the hoops over the end
fitting/bladder/frame/sleeve,
and fix them together with
steel clamps. (vii) Mount the
pneumatic quick plug connector on the end fitting with air
channel. BC, bending contractile; BE, bending extensile.
angle around the circumference can also be obtained from
Equation (4), as depicted in Figure 3b.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 ½cosa0 ð1 þ e1 Þ2
sinaðFÞ
e2 ¼
1¼
1 ¼ k2 1 (4)
sina0
sina0
The circular ratio of the radius of actuator to the original
radius, g, can be written as a function of the bending curvature of the flexible frame k’ and the original braid angle a0 , as
shown below.
g¼
¼
R
1
¼ 2p · R 2p
1
R0
0 k2 dF
2p
1
· R 2p
1
sin a0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
dF
0
1 [ cos a0 k1 ]2
(5)
g¼
R
¼ R ¼ n1 k¢, R a0 ¼ g1 k¢, a0 ¼ w1 k, a0 ,
R0
k¢ ¼ k¢R0 , k ¼ kR0 , R ¼ R=R0
(6)
where k¢, k, and R are the nondimensionalized bending
curvatures and actuator radius, respectively. Here, g1 k¢, a0 is a
dimensionless function related to the nondimensional bending curvature k¢, and original braid angle a0 , and its numerical
solution can be obtained from the function of n1 k¢R, a0 .
Moreover, the nondimensional actuator radius can also be
written as a function w1 of k and a0 .
Nondimensional ideal model
With the principle of virtual work, the air actuation moment Mair , generated at a certain point X(s) on the frame
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BENDING AND HELICAL PAMS
5
FIG. 3. Kinematics and ideal model of BE- and BC-PAMs. Inhomogeneous distribution of axial strain e1 (a), circular
strain e2, and braid angle distribution (b) at a typical cross section, when the initial braid angle a = 70 and the axial
elongation ratio equals 1. Here, / is the cylindrical coordinate defined in the plane of the cross section. Relationships
between the nondimensional air driving moment Mair and the parameters of a0 , k of BE-PAMs with the braid angle larger
than the 54.74 (c) and BC-PAMs with the braid angle less than 54.74 (d). (The block moment is the driving moment at the
blocked state of k ¼ 0, The free-bending curvature is the curvature of actuator at the free state under no external load.)
(Fig. 2d) and considering the applied inner pressure and braid
kinematics, can be derived. Arc length, s, is along the axis of the
flexible frame. The bending actuation moment is described as a
positive value in this model, when its direction toward the side
is reinforced. The definition of Mair is similar to the Gaylord
force of axial contractile and extensile PAMs, as below:
dV dk
dV 1
d½pR2 ð1 þ k¢RÞ
¼P
¼P
dk¢
dk¢ dl
dh
dk¢
2 dR
3
þR
¼ pP 2R þ 3k¢R
dk¢
Mair ¼ P
(7)
By substituting Equation (6) into Equation (7), the expression of Mair can be expressed as a function of the initial
braid angle a0 and the nondimensionalized bending curvature, k, as shown below.
dR
3
þR
dk¢
¼ PR0 3 M air ¼ PR0 3 / a0 , k¢
Mair ¼ PR0 3 p
2R þ 3k¢R2
(8)
According to Equation (8), the actuation moment Mair can
be written as the product of two parts. The first part is proportional to the internal pressure P and the third power of R0,
and the second part is a nondimensional function that only
relates to the constraint parameter a0 and the deformation
parameter k¢, which is defined as the nondimensional air
actuation moment M air excluding the effect of pressure value
and size scale, as shown in Figure 3c and d. According to the
Equation (7), for BC-PAMs, the bending curvature k¢ and air
actuation moment Mair are negative, and so, as their nondimensional ones, but to facilitate the analysis, they are described as positive in Figure 3d.
Through the nondimensional analysis, the scale effect
of the BE-PAMs can be drawn such that the actuation
moment is proportional to the third power of R0, but has no
relation to the actuator length. Moreover, as mentioned
previously, the model demonstrates that the critical initial
braid angle is equal to 54.74. Above which the bending
PAM will be a BE-PAM and below which it will be a BCPAM. Also, the initial braid angle has significant effects on
the output force, moment, or deformation. Therefore, the
next section of experimental characterization focuses on
6
the relationships between the actuation moment or force
and the parameters of a0 , k, and P.
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Experimental characterizations
Two BE-PAMs (BE-1 and BE-2) and three BC-PAMs
(BC-1, BC-2, and BC-3) with different braid angles were
fabricated and tested to analyze the effect of the braid angle
on their mechanical properties, as shown in Figure 4a. The
free-bending curvature and block force were both measured
with these two BE-PAMs and three BC-PAMs under a range
of working pressures to characterize their deformation and
actuation capabilities.
The PAMs were built with polyethylene terephthalatebraided sleeve tube, latex bladder tube, glass fiber-reinforced
epoxy flexible frame, 3D-printed polyacrylic end fittings, and
some fasteners (Fig. 2d—i). The geometry and material
property parameters of these BE/BC-PAMs are summarized
in Table 1.
Free-bending curvature. The free-bending curvature is
the bending curvature of the flexible frame at a certain pressure under zero external force or moment load. The curvature
was indirectly measured by capturing the positions of several
marked points on the actuator, as shown in Supplementary
Figure S1.
The bending angles of BE-1 and BE-2 are 60.1 and 69.0,
and that of BC-1, BC-2, and BC-3 are 32.5, 20.7, and 18.6,
respectively. It is a little different from the axial contractile
PAMs that a contractile PAM with a small braid angle usually
has more contraction stroke, while the BC-PAM with a too
small braid angle seems to generate less bending motion,
which may be caused by the synthetic effects of the end effect
of nontorus deformation,44 pressure deadband, and increasing
distance to the comparatively inextensible frame (Fig. 4d). As
shown in Figure 4c and Supplementary Figure S1a and b, the
BE-PAM with a larger braid angle has larger bending angles
at the same working pressure. As shown in Figure 4e and
Supplementary Figure S2, under the same bending curvature
(or pressure), the BC-PAM with a smaller braid angle has the
larger circular expansion ratio, which also makes it more
prone to failure due to larger bending curvature of the frame at
the two ends caused by end effect (Fig. 4i—i). The frames of
BC-2 and BC-3 broke at the pressure around 0.225 MPa. The
test data relating circular ratio g to the nondimensional
bending curvature k’ are compared with the analysis in
Figure 4e. As seen in Figure 4e, the kinematic analysis predicts bending PAM expansion convincingly, which also
verifies the validation of the analysis.
Figure 4c and d depicts nonlinear bending curvature during
inflation (pressurization) and deflation (depressurization).
The bending curvature exhibited hysteresis at the working
pressures, which is mainly caused by the internal friction of
the braid and the viscoelasticity of the bladder. For BCPAMs, the hysteresis of bending curvatures is not that obvious during inflation and deflation, which is mainly caused
by lower radial interaction forces between the braided tube
and the bladder tube, and the consequential less internal
friction of the braid. Furthermore, it was found that there was
a pressure deadband both in BE-PAMs and BC-PAMs before
the actuator starts to generate a bending motion.
GUAN ET AL.
Block force. Block force is the force that the bending
PAMs generate when one end is clamped on the test frame
and the other is in contact with the force sensor (digital
weight scale) (Fig. 4b).
Block force was tested for both BE-1 and BE-2 with a
cycle of pressures ranging from 0 to 0.30 MPa (Fig. 4e). The
BC-1 PAM was first tested with the pressures from 0 to
0.25 MPa, and its frame suddenly fractured around 0.24 MPa.
So after being repaired, it was tested with cyclic pressures in
the range between 0 and 0.20 MPa. The BC-2 and BC-3 were
test in the range from 0 MPa to the pressure of 0.150 and
0.155 MPa around which their elastic frame buckled but not
fractured and the output forces reduced with increasing
pressures (Fig. 4g and i). Nevertheless, when the pressure is
decreased to a certain level, the BC-2 and BC-3 PAMs returned to the prior forward pressuring curves (Fig. 4g), indicating that this process is invertible, unlike the sudden
fracture of BC-1 PAM. According to the buckling behaviors
of BC-PAMs (Fig. 4b and g), the larger the braid angle is, the
less the flexible frame of BC-PAM buckles under the same
pressure, which is due to the decreasing axial compressive
force from the contraction trend and increasing lateral constraint force from the braid sleeve and bladder tube with braid
angle. However, when the braid angle is large enough, the
frame would break suddenly under increasing load/pressure
instead of reversible buckling due to higher lateral constraint
force to axial compressive force ratio.
Moreover, the friction generated during deformation is
much less, and the hysteresis effect is not as obvious as that in
the free-bending test. Nevertheless, there is still a pressure
deadband before the actuator starts to output force. This
pressure deadband has been mentioned in prior studies and is
usually attributed to the need for the bladder to inflate to fill
the initial gap between the bladder and the braid so that the
bladder is in contact with the braid.50
Figure 4f and g also implies a linear relationship between
the block force and the test pressure for both BE-PAMs and
BC-PAMs. BE-2 PAM (braid angle of 69.0) generated almost double the force of BE-1 PAM (braid angle of 60.1)
with the lower braid angle (Fig. 4f). The linearly increasing
rate of BE-2 PAM block force with pressure is also about
twice that of BE-1 in the linear interval. Similar patterns were
found among BC-PAMs. Even though the generated block
forces of BC-2 and BC-3 PAMs were lower than BC-1 PAM
at same pressures due to the pressure deadband and buckling
of flexible frames, their increasing rate is much larger than
BC-1 PAM. We also observe that the smaller the braid angle
with which the BC-PAM is built, the greater is its rate of
increase of block force to pressure (Fig. 4g).
As indicated by Equation (7) in the Nondimensional Ideal
Model section, the block moment generated at the block state
is proportional to the working pressure. The larger the braid
angle of the BE-PAM is, the higher is the generated block
moment, which is contrary to the behavior for BC-PAMs.
Meanwhile, due to the block state, the actuation moment
generated at the end fixed on the test frame is the product of
the block force and the distance from the tip. Thus, the
analysis in the Nondimensional Ideal Model section successfully predicts the behavior of the block forces to increasing pressures. The increasing rates of block force to
pressure at different braid angles were depicted and compared with the analysis, shown in Figure 4h. As shown in
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FIG. 4. Experimental characterization of BE- and BC-PAMs. (a) Two BE-PAMs with different braid angles, BE-1
(a0 = 60.1) and BE-2 (a0 = 69.0), and three BC-PAMs with different braid angles, BC-1 (a0 = 32.5), BC-2 (a0 = 20.7), and
BC-3 (a0 = 18.6). (b) Buckling behaviors of BC-PAM during block force tests. Free bending test results of BE-PAMs (c)
and BC-PAMs (d). (e) The test and ideal model results of radial expansion ratio of BC-PAMs to inner pressures. Block force
test results of BE-PAMs (f) and BC-PAMs (g). (h) Block force test and ideal model results of nondimensional air actuation
moments Mair to initial braid angle a0. (i) The break of the actuator frame. (i) The inflated BC-3 PAM before and after the
frame break, (ii) broken frame of BC-3 PAM at free bending test, and (iii) broken frame of BC-1 PAM at block force test.
7
8
GUAN ET AL.
Table 1. Geometry and Material Property Parameters of Bending Extensile Pneumatic Artificial Muscles
and Bending Contractile Pneumatic Artificial Muscles
BE-1
Initial braid angle
Outer diameter R0, mm
Active initial length L0
Outer radius Rb0 of the bladder tube
Thickness t0 of the bladder tube
Sizes of the elastic frame
Tensile modulus of the flexible frame
BE-2
BC-1
BC-2
BC-3
69.0
60.1
32.5
10.2
10.5
9.2
13.5 cm
8.5 mm
2.5 mm
0.5 mm (thickness) · 5 mm (width)
6.796 GPa
20.7
9.0
18.6
9.5
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BC, bending contractile; BE, bending extensile.
Figure 4h, BC-PAMs generated much higher block moment
at the same conditions, which is verified by both analysis and
test. Moreover, the test results of BC-PAMs were overestimated by the analysis, which is mainly caused by the end
effect and large radius expansion of the bladder under experimental conditions.
According to the free bending curvature and block force tests,
it can be concluded that BE-PAMs can generate larger deformations, but produce lower output force, than the contractile
PAM, which has been demonstrated by the idealized analysis.
Equations (1), (3), (4), and (6) are still valid in this case.
However, to be applied to the general bending motion,
Equations (2) and (5) are rewritten as Equations (11) and (13)
by submitting the new expression of k1 and replacing the
curvature of the reinforced frame with that of reference line.
k1 ðhÞ ¼ kRefl
Helical Extensile and Contractile PAMs
In this section, HE-PAMs or HC-PAMs were developed by
reinforcing the extensile or contractile PAMs helically
around the axis. As bending PAMs, each helical PAM also is
made up of end fittings, an elastic tube, a braided tube, and an
embedded flexible frame (Fig. 5b–d), and with the same
materials of bending PAMs. When the HE-PAM or HC-PAM
inflated, the helical reinforced frame will constrain its extension or contraction and generate the bending and rotating
motions at the same time, which cause the helical deformation of the actuator like the helical or oblique muscle fibers of
an elephant trunk (Fig. 5c and d).
Kinematic analysis of generalized bending behaviors
As given previously, the reinforced side of BC-PAMs and
BE-PAMs is undeformed, while during general bending motions of PAMs, the deformation can occur around the entire
circumference, for example, when it comes to the helical PAMs.
Here, a reference line was introduced as the intersection
line of the infinitesimal unit actuator segment surface and the
osculating plane to build the relationship among bending,
axial, and radial deformations (Fig. 5a). When the reference
line is on the inner side surface of the bend, the axial bending
curvature of the PAM is positive, as shown in Figure 5a. In
contrast, the bending curvature is negative, when the reference line is on the outer side surface. For BE-PAMs and BCPAMs, the reference line is the line of the reinforced frame
and the reference line deformation ratio kRefl = 1. From
Figure 5a, Equations (9) and (10) are given as below.
¼ k¢R ¼ k¢R
k ¼ kaxis , k¢ ¼ kRefl , k¢R ¼ k¢R
kaxis ¼ kRefl
qaxial
¼ kRefl 1 þ k¢R
qRefl
qðhÞ
1 ¼ kRefl ð1 þ k¢Rð1 cosFÞÞ ¼ e1 þ 1
qRefl
(11)
(9)
(10)
aðFÞ ¼ Arctan
k1 cosða0 Þ
k2 sinða0 Þ
R
¼ n2 k¢R, kRefl , a0 ¼ g2 k¢, kRefl , a0
R0
¼ w2 k, kRefl , a0 ¼ f2 k, kaxis , a0
g¼
¼
V
¼ kaxis g2
V0
(12)
(13)
(14)
The range of braid angle is limited by the gap and width
between the threads. So here, the braid angle ranges from 85
to 5. The relationship of the circular ratio g with nondimensional axis bending curvature of axis k or reference line k¢
and reference line ratio kRefl or axis ratio kaxis was depicted
by several sets of surfaces corresponding to the initial braid
angles from 85 to 5, shown in Figure 6a–c. Also, the relationship of the volume ratio with k and axis ratio kaxis is
shown in Figure 6d. As shown in Figure 6 and Equations (13)
and (14), besides kRefl or kaxis , the circular ratio g and the
volume ratio also vary with k or k¢.
The topological equivalence is found in these maps of
circular ratio g2 , w2 , f2 or volume ratio at different initial
braid angles (Fig. 6 and Supplementary Fig. S3). The circular
ratio g becomes not that sensitive to the axis length, when the
elongation ratio kaxis reduced to a much low level, and vice
versa. Keep the axis length constant and the bending motions
will decrease the circular ratio g, whatever the initial braid
angle is. Meanwhile, the differences also exist among these
maps of different initial braid angles. The map with a lower
initial braid angle has a much narrower range of axis or reference line bending curvature, and higher and broader range
of circular ratio g. Also, the lower the braid angle is, the more
sensitive the circular ratio is to the bending motion.
BENDING AND HELICAL PAMS
9
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FIG. 5. Kinematic analysis of helical
PAMs. (a) Kinematic analysis of generalized bending behaviors. kaxis is the axis ratio,
kRefl is the reference line ratio, qaxis and qRefl
are bending radii of axis and reference line
separately, and R is the radius of the actuator; (b) kinematic parameters of helical
PAMs, b is the helix angle of the frame, d is
the winding angle of the frame around the
actuator. Uninflated and inflated states of
HE-PAM (c) and HC-PAM (d), the curves in
the figures represent the position of the
frame. (e) Materials (i) and constructions
(ii–v) of helicial PAM. (ii) Mount the end
fittings on the bladder, (iii) slide the braid
sleeve over the bladder tube and insert the
frames between the bladder tube and braid
sleeve helically with a desired helix angle.
(iv) Fix the end fittings, braid sleeves,
frames, and bladder tube together with cable
ties. (v) Mount the air hose quick connectors
on the end fitting with air channel.
According to Figure 6d and Supplementary Figure S3, bifurcation happens when the initial braid angle a0 ¼ 54:74 . As
is well known, PAMs have the tendency to increase in the
volume under inner pressure. When a0 > 54.74, the PAM will
tend to extend, and on the contrary, when a0 < 54.74 the PAM
will be prone to contraction. Moreover, as seen in Figure 6d,
the surface convexity increasing with braid angles impacts
that the axial stiffness and bending stiffness of PAMs increase
with the braid angle at a certain pressure. For the PAM with an
initial braid angle higher than 54.74, the extensile PAM, the
convexity of the volume ratio to the k is much lower than that
of the PAM with a lower initial braid angle than 54.74, the
contractile PAM. It means that the volume ratio of extensile
PAMs is not as sensitive to the bending motion as is a contractile PAM. In other words, extensile PAMs are more flexible and contractile PAMs are stiffer.
Kinematics of generalized helical PAMs
According to Figure 5b, due to the restriction of helical
reinforced frame, Equation (16) is given as below. d, b0 are
the winding angle around the actuator and the initial helical
angle of reinforced frame, respectively. Due to the helical
angle, the reference line changes with the position along the
reinforced frame, but Equation (10) is still valid considering
the infinitesimal length of unit actuator segment, as shown
in Figure 5b—iii. Also, Equations (15) and (16) are obtained
as below, Bishop-Moser et al. introduced a similar expression for spiral fiber-reinforced actuator.23,38 d0 is the initial
winding angle.
l0
laxial
b0 ¼ arctan
¼ arctan
d0 R 0
dR
(15)
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10
GUAN ET AL.
FIG. 6. A set of surfaces to depict relationships among the generalized bending deformation parameters of PAMs with
initial braid angles from 85 to 5. (a) The relationship of the circular ratio g with nondimensional reference line bending
curvature k¢ and reference line ratio kRefl ; (b) the relationship of g with nondimensional axis bending curvature k and kRefl ;
(c) the relationship of g with k and axis ratio kaxis ; (d) the relationship of the volume ratio with k and axis ratio kaxis . Color
images are available online.
kRefl 2 sin2 b0 þ
R
R0
2
cos2 b0 ¼ 1
(16)
Then, the relationship between the curvature of axis or
reference line and circular ratio could be draw as a function,
Equation (17), derived from Equations (13) and (16). a0 , b0
are constant terms related to the braided sleeve and reinforced
frame. c is the helical angle of actuator axis, and s is the
torsion. Rhelical is the radius of the helical axis.
g¼
c ¼ arctan
d
klaxial
R
¼ C k, a0 , b0
R0
d
s
¼ arctan
¼ arctan
kl
k
1
Rhelical ¼ h
2 i
k 1 þ kld
(17)
(18)
(19)
Figure 7a presents the relationship of nondimensional axis
bending curvature k with circular ratio g and reference line
elongation ratio kRefl , at a0 ¼ 65 and b0 = 5 – 85. Figure 7b
presents the relationships of nondimensional axis bending
curvature k with circular ratio g and kRefl , when a0 ¼ 65 or
35 and the reinforced frame helical angle b0 = 80. Figure 7c
presents the relationships of volume ratio with k and axis
ratio kaxis , in the same conditions as above. From Figure 7a, it
can be deduced that when b0 closes to 90, the kRefl tends to 1,
which means that the reinforced side of helical PAM is
almost inextensible and its bending behavior is close to bending PAMs. Meanwhile, according to Figure 7b and c, with the
same helical angle, when a0 ¼ 65 , the PAM will extend and
bend with positive curvature. In contrast, when a0 ¼ 35 , the
PAM will contrast in axis, expand in radius, and bend with
negative curvature.
Constant curvature kinematics. To obtain the whole
configuration of the actuator, the CCK and CCCTK based on
Frenet–Serret frames are introduced, as shown in Supplementary Figure S4.
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BENDING AND HELICAL PAMS
11
FIG. 7. A set of surfaces to depict the relationships among the deformation parameters of helical PAMs. (a) The
relationships of the circular ratio g with nondimensional axis bending curvature k and reference line deformation ratio kRefl
with initial braid angles a0 = 65 and the reinforced frame helical angle b0 from 5 to 85; (b) the relationship of g with k
and kRefl with a0 = 65 or 35 and the reinforced frame helical angle b0 = 80; (c) the relationship of the volume ratio with
k and kRefl with a0 = 65 or 35 and b0 = 80. Color images are available online.
CCK has been widely used in continuum robots due to the
facilitation in the analysis of static/differential kinematics
and real-time control. For CCK, variable curvature elastic
structures can be considered a finite number of curved links,
which can be described by a finite set of arc parameters and
converted into analytical frame transformations.47
For continuum robots without torsion, the frame transformation of each link is usually described by a bending translation matrix defined with bending curvature and bending
direction angle and arc length. However, the local frame related to the reinforced frame rotates along the length of the
helical actuator. Considering the torsion of the local frame,
the translation matrix is defined by two independent transformations of bending and rotation. It should be noted that the
change of bending direction is different from the torsion
deformation, especially for the body of soft robots. For soft
robotic structures with bending and rotating deformation, the
transformation of each link needs two independent transformations with an additional parameter of torsion angle to
describe the torsion instead of a dependent translation matrix.
Moreover, due to the coupling of bending and torsional deformation, a larger number of arc links are needed to obtain
sufficient precision.
Here, U0 is the matrix describing the relative position and
orientation of the actuator initial end. Ui ðsi Þ is the transformation matrix at a point of initial arc length si from the initial
end. The actuator was divided into N links, each of which is
described by the translation matrix Me. le, he , de are deformed
arc length, bending angle, and torsion angle of each link. Te is
the torsion translation matrix, and Be is the bending translation matrix. Here, the bending direction angle w is constant
and equals to 0, which means the bending direction in local
coordination is fixed.
Ui ðsi Þ ¼ U0 Me i ¼ U0 ðTe Be Þi , i ¼ 0, 1, 2 N
le ¼
l0 kaxis
ile
, he ¼ kaxis le , de ¼ sle
, si ¼
N
kaxis
(20)
(21)
12
GUAN ET AL.
2
cosðde Þ
6 sinðde Þ
Te ¼ T ðde Þ ¼ 6
4 0
0
sinðde Þ
cosðde Þ
0
0
3
0
07
7
05
1
0
0
1
0
2
(22)
~
n¼
1
d
6
~
t ¼ C3 · 3 4
kð1 þ qÞ ds
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
k2 þ s2
cos ð1 þ qÞ k2 þ s2 s
k
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
2
2
k kþ s sin ð1 þ qÞ k2 þ s2 s 5
0
(30)
82
cosðhe Þ
>
>
>
6
>
>
0
6
>
>
4 sinðh Þ
>
>
e
>
<
0
2
Be ¼
>
>
>
>
6
>
>
6
>
>
4
>
>
:
0
1
0
0
1
0
0
0
sinðhe Þ
0
cosðhe Þ
0
0 0 0
1 0 0
0 1 le
0 0 1
1 cosðhe Þ
kaxis
0
sinðhe Þ
kaxis
3
3
7
7 , kaxis 6¼ 0
5
1
k
s
7
7, kaxis ¼ 0
5
(31)
(23)
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pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
ks sin ð1 þ qÞ k2 þ s2 s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
1
d
k
6
~
~
nþ ~
t ¼ C3 · 3 4 ks cos ð1 þ qÞ k2 þ s2 s 5
b¼
sð1 þ qÞ ds
s
2
Coupled constant curvature and torsion kinematics. The
torsion exists ubiquitously in the deformation of soft robotic
body due to asymmetric actuation or loads. However, as
mentioned above, with independent bending and rotation
transformation matrix,47 CCK needs large numbers of element
arc links to obtain high accuracy approximation, which brings
much burden of calculation. Here, we first introduced a coupled
constant curvature and torsion transformation base on Frenet–
Serret frames to get higher order precision. For example, to
helical actuator with constant curvature and torsion, precise
solution can be obtained with only one coupled transformation
instead of a finite set of bending and rotating translations.
d
~
t ¼ kð1 þ qÞ~
n
ds
(24)
d
~
n ¼ kð1 þ qÞ~
t þ sð1 þ qÞ~
b
ds
(25)
d~
b ¼ sð1 þ qÞ~
n
ds
d
~
u ¼ ð1 þ qÞ~
t
ds
~
u¼
ð1 þ qÞ~
tds
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
cos ð1 þ qÞ k2 þ s2 s þ C1¢
6
7
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1
¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C3 · 3 6
sin ð1 þ qÞ k2 þ s2 s þ C2¢ 7
4
5
2
2
k þs
(32)
The variable v was introduced to simplify the expression, as
Equation (33).
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
v ¼ ð1 þ q Þ k 2 þ s2 s
(33)
According
toi the initial boundary conditions: S = 0,
h
! ! = E , U! = 0 .
!
3·3
0
3·1
n0 b0 t0
"
! !
!
n0 b0 t0
0 0 0
¼
C3 · 3
01 · 3
(26)
2
!
u0
1
#
2
1
6 cosc
6
03 · 1 6 0
6
1 6
6
4 0
0
0
sinc
cosc
0
1
cosc
sinc
1
0
0
3
1 0 0 0
60 1 0 07
6
7
¼6
7,
40 0 1 05
0 0 0 1
k
s
sin c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi , cos c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
2
k þs
k þ s2
(27)
(28)
2
ð1 þ qÞs þ C3¢
The general solution of ~
t, ~
n, ~
b, ~
u can be written as Equations
(29–32).
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
2
2
2
2
~
t ¼ C3 eið1 þ qÞ k þ s s þ C2 e ið1 þ qÞ k þ s s þ C1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3
2 sin (1 þ q) k2 þ s2 s
(29)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7
6 ¼ C3 · 3 4 cos (1 þ q) k2 þ s2 s 5
1
l0
0
~
t, ~
n, and ~
b are tangent, normal, and binormal unit vectors of
the actuator axis, respectively. ~
u is the vector from the origin
to a point of the actuator axis. q is the strain of the axis. Since
the curvature k and torsion s are constant, from the Equations
(24) to (27), Equation (28) is derived as follows:
d
d3
~
~
t¼0
t þ ð1 þ qÞ2 k2 þ s2
3
ds
ds
Z
3
1 þ C1¢
pffiffiffiffiffiffiffiffiffiffi
k2 þ s2 7
7
C2¢
7
pffiffiffiffiffiffiffiffiffiffi
2
2
k þs 7
7
7
C3¢
pffiffiffiffiffiffiffiffiffiffi
5
k2 þ s2
1
(34)
Equations (35) and (36) are obtained by solving Equation
(34), and then, the CCCTK transformation matrix can be
derived as Equation (37) (for more detailed derivation, see
Supplementary Data).
C3 · 3
01 · 3
2
cosc
03 · 1
¼4 0
1
0
C1¢ ¼ 1,
0
sinccosc
cos2 c
C2¢ , C3¢ ¼ 0
3
0
sinccosc 5 (35)
sin2 c
(36)
BENDING AND HELICAL PAMS
2
cos (v)
6
6 sin c sin (v)
~
~
~
~
n
b
t
u
MC F ¼
¼6
6
0 0 0 1
4 cos c sin (v)
0
sin c sin (v)
sin2 c cos (v) þ cos2 c
sin c cos c(1 cos (v))
0
3
cos (v)
cos c 1pffiffiffiffiffiffiffiffiffiffi
k 2 þ s2
sin (v) 7
7
sin c cos c(1 cos (v)) sin c cos c vpffiffiffiffiffiffiffiffiffiffi
k2 þ s2 7 (37)
2
2
7
cos c sin
(v) þ sin cv
pffiffiffiffiffiffiffiffiffiffi
5
sin2 c þ cos2 c cos (v)
k2 þ s2
0
1
cos c sin (v)
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13
FIG. 8. Simulations and validations of generalized helical PAM kinematics. (a—i) A set of calculated configurations of
piecewise CCK for a helical curve with different link numbers, and (a—ii) the calculated tip end position of CCK and
CCCTK to the elemental link number. The simulation results of deformed configurations and braid angle distributions about
HE-PAM (b—i, ii) and HC-PAM (c—i, ii) under inflated states. (d) The volumes ratio of HC-PAM and HE-PAM to
bending curvature. (e, f) The modeling and test results of unit vector trajectories of three end axial directions during
inflation (0–280 kPa) and deflation (280–0 kPa). CCCTK, coupled constant curvature and torsion kinematics; CCK, constant
curvature kinematics. Color images are available online.
14
GUAN ET AL.
Table 2. Geometry Parameters of the Helical Extensile-Pneumatic Artificial Muscle
and Helical Contractile-Pneumatic Artificial Muscle for Experimental Validations
Outer
Active
Initial
diameter
initial
braid
R0
length L0
angle a0
HE
HC
68.5
35.0
12.5 mm
9 mm
Initial
radius
RRefL0
of the
frame
Outer
radius
Rb0 of the
bladder
tube
16.0 cm
17.0 cm
8.5 mm
8.5 mm
Thickness
t0 of the
bladder
tube
10.5 mm 2.5 mm
8.75 mm
Sizes of the
elastic frame
The initial
The winding reinforcing
helical
angle d
of the frame angle b0
0.5 mm (thickness)
· 5 mm (width)
200
150
77.7
82.5
HC, helical contractile; HE, helical extensile.
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According to Equation (37)
When s = 0, c = 0 and v ¼ ð1 þ qÞjkjs:
2
cosðvÞ
6
0
6
M1 ¼ 4
sinðvÞ
0
0
1
0
0
sinðvÞ
0
cosðvÞ
0
1 cosðvÞ
k
0
sinðvÞ
k
3
7
7
5
(38)
1
When k = 0, c = p/2 and v ¼ ð1 þ qÞjsjs.
2
1
6 sinðvÞ
M2 ¼ 6
4 0
0
sinðvÞ
cosðvÞ
0
0
0
0
1
0
3
0
07
7
v 5
jsj
1
(39)
Equations (38) and (39) correspond to the bending and
rotating portion of constant curvature transformation, which
verifies the validity of CCCTK.
Figure 8a—i shows the calculated configurations of CCK for
a helical curve with length of 160 mm, curvature of 0.025 mm-1,
and torsion of p/160 mm-1 with different link numbers from 1 to
1000. Figure 8a—ii shows that the calculated tip end position
varies with the elemental link number, which tends to be the
result of CCCK when the number is larger enough. In this case,
to keep the error lower than 0.01%, the number of links has to be
larger than *900 (Supplementary Fig. S5).
Kinematic simulation and experimental validation
An HE-PAM and an HC-PAM were fabricated, tested, and
simulated. The HE-PAM was built with the initial braid angle
a0 of 68.5, the initial reinforcing helical angle b0 of 77.7,
and initial length L0 of 160 mm, and the HC-PAM was built
with a0 of 35.0, b0 of 82.5, and L0 of 170 mm (for more
detailed geometric information, see Table 2). The helical
PAMs were also built with the same materials as bending
PAMs mentioned above.
According to the general bending and kinematic analysis,
the deformations and braid angle changes during inflation can
be simulated and are shown in Figure 8b and c. The plot of the
volume ratio to axis bending curvature is presented in
Figure 8d. As shown in Figure 8b and c, when inflated, a
spiral region appeared around the reinforced frame both in
HE-PAM and HC-PAM and sustains the original braid angle.
Braid angle changes around the circumference and spirally
along the axis, consistent with the test results (Supplementary
Movie S1). Also, as predicted by the analysis and simulation,
with the initial braid angle above the bifurcation angle of
54.74, the bending curvature of HE-PAM is positive, and, on
the contrary, the bending curvature is negative for HC-PAM.
As a result, in the opposite side of reinforced frame, the braid
angle of HE-PAM decreases with the helical deformation and
the braid angle of HC-PAM increases with the deformation.
Whereas no matter the bending curvature is either positive or
negative, the chirality of the helical PAM is always consistent
with the helical direction of bending frame.
Moreover, during inflation, the HE-PAM volume increases with axial elongation and the bending curvature, and
the HC-PAM volume increases with axial contraction and
the negative bending curvature (Fig. 8d). The reason is that the
volume of HE-PAM is dominated by axis length and the volume of HC-PAM is dominated by the circular radius.
To present and analyze the complex out-plane transformation of helical PAMs, test results of end axial direction
trajectories during inflation and deflation are compared with
the analysis in Figure 8e and f. Directional data were obtained
via an inertial measurement unit (IMU; MPU-9250; InvenSense). Due to the installation deflection angle, the axes
of IMU were transferred into the coordinate system of the
kinematic model. The simulated trajectories of projected
IMU axes in modeling coordinates are shown and compared
with the data collected. The normalized standard deviations
and correlation coefficients are shown in Table 3. According
to Figure 8e and f and Table 3, the test results meet very well
Table 3. Standard Deviations and Correlation Coefficients of Axes Direction Trajectories
Axis
Normalized standard deviation (HE-PAM)
Correlation coefficient (HE-PAM)
Normalized standard deviation (HC-PAM)
Correlation coefficient (HC-PAM)
PAM, pneumatic artificial muscle.
X
Y
Z
Average
9.34%
0.9834
2.00%
0.9989
3.51%
0.9894
3.97%
0.9926
8.65%
0.9926
3.92%
0.9901
7.17%
0.9885
3.30%
0.9939
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BENDING AND HELICAL PAMS
with the modeling results. The average normalized standard
deviations of HE-PAM and HC-PAM are 7.17% and 3.30%,
respectively, and the average correlation coefficients of
three axes are 0.9885 and 0.9939, which validated the
general bending behavior analysis and kinematic models.
Due to gravity and braid to bladder friction of the PAM, the
relatively larger deviation and hysteresis occurred for the
HE-PAM. In contrast, as a result of larger load capability
and less braid angle change, the test results of the HC-PAM
agreed much better with analysis and there was much less
hysteresis.
It should be pointed out that this study is only focused on
the constraining effects of the braid sleeve and reinforced
frame, which dominates the most significant behaviors of
PAM and is the basis of further accurate mechanical models.
Also, the end axial direction trajectories were chosen as the
reference parameters to verify the model, as they were mainly
defined by the constraining effects of the braid sleeve and
reinforced frame instead of the elastic deformation of the
bladder tube. Nevertheless, the bladder still has great effects
on the behaviors of bending and helical PAMs, especially on
the relationship of the displacement with inner pressures and
external loads. Other factors such as the end effects, friction,
and viscoelasticity also have more or less influence on the
actuator behavior, whereas they are not involved in this study
since they do not have much effect on the kinematics of the
braid sleeve and reinforced frame of tested helical PAMs.
However, it should be also noted that the effect of the bladder
and other factors still need to be considered for further accurate mechanical models.
Trunk-Like Manipulator
In this section, potential applications of bending and helical PAMs in soft robotics are explored by demonstrating a
trunk-like MDOF manipulator (Fig. 9). As mentioned above,
BC-PAMs own much output or load capability, and BEPAMs can generate more deformation. The first segment was
a two-directional bending PAM that comprised one bladder
and two pieces of braided sleeves with a braid angle lower
than 54.74 (contractile part) and the other one higher than
54.74 (extensile part) (Fig. 9a–i). The end of the contractile
part was fixed to the base to bear the largest load, and the
extensile part was used to contribute a larger range of
movement. Meanwhile, the HE-PAM was mounted in tandem on the end to grasp objects such as elephant trunk
(Fig. 9a and Supplementary Movie S2). The pressures of
PAMs were controlled by electropneumatic regulators
(ITV2030-212; SMC Corporation). The active lengths of
contractile bending and extensile bending parts are 10.5 and
9 cm, respectively, and the length of the helical part is
*28 cm. The total length and maximum diameter of the
manipulator are 65.5 and 2.5 cm. As a result, the length/diameter ratio of the soft manipulator reaches up to 26.2, even
larger than the real elephant trunk.
Operation space
The bending directions of bending contractile and extensile parts are orthogonal to enable the transformations in
different planes and broaden the workspace (as presented in
Fig. 9a and Supplementary Fig. S6). Also, a long helical part
15
was fixed at the end to extend the operation range and facilitate reaching and gripping target objects.
Figure 9f shows the unit vector trajectories of three end axial
directions of the two-direction segment during inflation and
deflation. The pressure of extensile bending part was tuned in
the range of 0–250 kPa continuously with a cyclic ramp input
function, while the pressure of contractile bending part was set
at a set of pressures (0 kPa and pressures from 50 to 250 kPa
with equal intervals of 40 kPa). As shown in Figure 9a and f,
the first bending contractile part bends in the plane perpendicular to Y axis with the pitch angle from 0 to 46.2, and the
second part bends in the plane vertical to the X axis with the
roll angle from 0 to 99.6. Figure 9g shows the complex unit
vector trajectories of X, Y, and Z axial directions of helical part
under the pressures of 0–250 kPa (for three orthographic
views, see Supplementary Fig. S7), with the inner helical diameter of *50 mm at 280 kPa (Fig. 9a–iii). Combine the two
orthogonal bending motions, and the two-direction segment
can cover most of a quarter of a hemisphere (Fig. 9f). The trails
of helical part end in different orientations are also shown in
Figure 9h. As shown in Figure 9a and h, the large range of
configuration transformation and end workspace can enable
the helical part to warp or hook the object easily.
Gripping and poses of the manipulator
In this part, the gripping and posing capabilities of the
trunk-like manipulator are presented. The proposed soft
manipulator can grip objects of different shapes, weights, and
sizes by winding or hooking with a single helical actuator.
Lifting of objects and adjusting of the orientations can also be
achieved by commanding different poses of the manipulator
with the bending contractile and extensile actuators, as shown
in Figure 9b–d.
Figure 9b shows that the soft manipulator grips and lifts
different objects of various shapes, weights, and sizes.
Figure 9c shows that the manipulator wraps a bottle of water
with the highly flexible helical part and pours the water into
the beaker steadily by the out-of-plane bending of the contractile and extensile bending parts (Supplementary Movie
S2). Besides, the manipulator can hook the object by
threading the tip though a hole or a handle and clasping the
object with the helical deformation, as presented in Figure 9d.
With the length/diameter ratio of 24, the soft manipulator still
can grip and lift an object 10 times its bodyweight: 528 g
(Fig. 9b and Supplementary Movie S2). However, with high
payload, the operation space of the manipulator reduced
significantly. For example, when the manipulator grips a full
bottle of water of 528 g, the maximum pitch angle of the first
bending contractile part under 250 kPa is *21, only half of
that without payloads. Also, the maximum roll angle of the
second bending extensile part under 250 kPa is *34, only
one-third of that without payloads. That is understandable
considering its intrinsic property of soft and the high length/
diameter ratio of 24 (Supplementary Movie S2).
According to the above, the proposed soft manipulator is
verified to be adaptable, capable, and flexible for gripping
and lifting diverse objects of different shapes, weights, and
sizes, and adjusts to the orientation desired. Compared with
other manipulators with multifinger grippers and multiactuator joints, our soft tandem manipulator shows more
structural simplicity, compactness, and dexterity.
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FIG. 9. The soft tandem trunk-like MODF manipulator. (a) Deformed configurations of the manipulator composed of three
parts (contractile bending part, extensile bending part, and helical part, P1–P3 are pressures of these three parts in sequence),
under a set of conditions. (b) Gripping objects of different shapes, sizes, and weights: (i) Gripping a full bottle of water
(528 g, F64 mm), (ii) griping a screwdriver head can (121 g, F50 mm), (iii) gripping hot melt glue gun by holding the head
(248 g, length 200 mm), (iv) gripping universal meter (443 g, width 90 mm, thickness 45 mm). (c) Griping and pouring the
half bottle of water into the beaker. (d) Hooking a roll of tape by threading the tip through its center and clasping the object
with the helical deformation. (e) The definition of X, Y, and Z axes. (f) Unit vector trajectories of three end axial directions of
two-direction segment during inflation and deflation (P1: 0–250 kPa, P2: 0–250 kPa). (g) Unit vector trajectories of three
axial directions of the helical segment (P3: 0–250 kPa). (h) Workspaces and trails of the three segmental ends. (i) Materials
(i) and fabrication (ii–vii) of two-directional bending PAM. (ii) Mount the air hose and quick connectors on the end fitting 1
(with air channels) and the middle cork. (iii) Push the middle cork into the bladder tube at a desired position and mount the
end fittings on the bladder. (iv) Slide the braid sleeve 2 over the right part of the bladder tube and insert the frames 1 and 2
between the bladder tube and braid sleeve in parallel at different positions on the left and right part. (v) Slide the braid sleeve
1 over the left part of the bladder tube. (vi) Fix the end fittings, middle cork, braid sleeves, frames, and bladder tube together
with cable ties. (vii) Mount the air hose quick connectors on the end fitting 1. Color images are available online.
16
BENDING AND HELICAL PAMS
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Conclusions
Through bionics, bending and HE-PAMs/HC-PAMs can
be realized by reinforcing extensile or contractile PAMs in
parallel or helical direction, without fiber-winding and
elastomer-casting, which enriches the categories of PAMs
and provides a methodology to develop new soft actuators.
The most significant contribution of this article is in proposing a method to study the properties of axial, bending, and
helical PAMs to the deformation parameters in one theory
scheme through the analysis of the generalized bending behavior of PAMs, which can provide a map for designers to
find a methodology to build an actuator meeting their requirement. The analysis revealed how the generalized
bending motion effects on the braid angle distribution, radius
change, and inner volume, and described the relationships
among the bending, axial, and radial deformations of PAMs.
Moreover, the topological equivalence and bifurcation were
found in the behaviors of PAMs with different initial braid
angles, which imply the similarity of different PAMs and the
existence of extensile and contractile bifurcation behavior in
all of the axial, bending, and helical PAMs. Besides, according to analysis and experiments, no matter the PAM is
axial, bending, or helical, the contractile PAM can generate
relatively higher output forces or moments, but presents
lower deformation capability than the extensile one.
Also, a CCCTK analysis was proposed to replace the CCK
and depict the motion of helical PAM more accurately and
conveniently. It can also be utilized in nonconstant curvature
and torsion kinematics with piecewise links to get higher
order precision than piecewise CCK, especially for these soft
robots with coupled bending and torsion deformation under
actuation or load.
Finally, based on all the above, a soft tandem manipulator
composed of bending and helical PAMs was proposed. Also,
its adaptation, capability, and flexibility were demonstrated
by a sequence of operational or functional tests, indicating
that bending and helical PAMs can be applied to a wide variety of situations, including soft sorting manipulators, search
robots, biorobots, assistant exosuits and force feedback
wearable devices.
In the future, since the elastic forces of its own structures and
load effects of weight and interaction forces were not included
in this study, a further model involving these factors will be
investigated to better depict the behaviors of these PAMs. What
is more, the dynamic behaviors surrounding material viscoelasticity, air compressibility, and kinetic friction will also be
investigated to acquire higher control accuracy. Besides, we
will attempt to propose more interesting applications of bending and helical PAMs to explore more possibilities with them.
Acknowledgment
The first author extends his sincere gratitude to the graduate student, Tianpeng Wang, for his assistance in the
pneumatic control systems and experiments.
Author Disclosure Statement
No competing financial interests exist.
Funding Information
This work was supported by the National Natural Science
Foundation of China (Grant No. 1102076), and the Young
17
Elite Scientists Sponsorship Program by CAST (Grant No.
2018QNRC001).
Supplementary Material
Supplementary Data
Supplementary Figure S1
Supplementary Figure S2
Supplementary Figure S3
Supplementary Figure S4
Supplementary Figure S5
Supplementary Figure S6
Supplementary Figure S7
Supplementary Movie S1
Supplementary Movie S2
References
1. Marchese AD, Katzschmann RK, Rus D. A recipe for soft
fluidic elastomer robots. Soft Robot 2015;2:7–25.
2. Laschi C, Mazzolai B, Cianchetti M. Soft robotics: technologies and systems pushing the boundaries of robot
abilities. Sci Robot 2016;1:eaah3690.
3. Gorissen B, Reynaerts D, Konishi S, et al. Elastic inflatable
actuators for soft robotic applications. Adv Mater 2017;29:
1604977.
4. Hines L, Petersen K, Lum GZ, et al. Soft actuators for
small-scale robotics. Adv Mater 2017;29:1603483.
5. Daerden F, Lefeber D. Pneumatic artificial muscles: actuators for robotics and automation. Eur J Mech Environ Eng
2002;47:10–21.
6. Gaylord RH. Fluid actuated motor system and stroking
device. US2844126, 1958.
7. Chou CP, Hannaford B. Measurement and modeling of
McKibben pneumatic artificial muscles. IEEE Trans Rob
Autom 1996;12:90–102.
8. Tuffield P, Elias H. The shadow robot mimics human actions. Ind Robot 2013;30:56–60.
9. Ball EJ, Meller MA, Chipka JB, et al. Modeling and testing
of a knitted-sleeve fluidic artificial muscle. Smart Mater
Struct 2016;25:115024.
10. Yi J, Chen X, Song C, et al. Fiber-reinforced origamic
robotic actuator. Soft Robot 2018;5:81–92.
11. Lightner S, Lincoln R. The fluidic muscle: a ‘new’ development. Int J Mod Eng 2002;2:8.
12. Bishop-Moser J, Kota S. Towards snake-like soft robots: design of fluidic fiber-reinforced elastomeric helical manipulators. In Proceedings, IEEE/RSJ International Conference on
Intelligent Robots and Systems. Tokyo, Japan: IEEE, 2013,
pp. 5021–5026.
13. Polygerinos P, Wang Z, Overvelde JTB, et al. Modeling of
soft fiber-reinforced bending actuators. IEEE Trans Robot
2015;31:778–789.
14. Deimel R, Brock O. A novel type of compliant and underactuated robotic hand for dexterous grasping. Int J Rob
Res 2016;35:161–185.
15. Daerden F, Lefeber D. The concept and design of
pleated pneumatic artificial muscles. Int J Fluid Power
2001;2:41–50.
16. Noritsugu T, Yamamoto H, Sasaki D, et al. Wearable
power assist device for hand grasping using pneumatic
artificial rubber muscle. SICE Annual Conference, Sapporo, Japan, August 4–6, 2004. IEEE, 2004;1:420–425.
17. Wirekoh J, Park YL. Design of flat pneumatic artificial
muscles. Smart Mater Struct 2017;26:035009.
Downloaded by Harbin Industrial University from www.liebertpub.com at 03/06/20. For personal use only.
18
18. Mcmahan W, Chitrakaran V, Csencsits M, et al. Field trials
and testing of the OctArm continuum manipulator. In
Proceedings, IEEE International Conference on Robotics
and Automation (ICRA). Orlando, Florida, USA. IEEE,
2006, pp. 2336–2341.
19. Pillsbury TE, Wereley NM, Guan Q. Comparison of contractile and extensile pneumatic artificial muscles. Smart
Mater Struct 2017;26:095034.
20. Wang Z, Polygerinos P, Overvelde J, et al. Interaction
forces of soft fiber reinforced bending actuators. IEEE/
ASME Trans Mechatron 2017;22:717–727.
21. Guan Q, Sun J, Liu Y, et al. A bending actuator based on
extensile PAM and its applications for soft robotics. In
Proceedings, 21st International Conference on Composite
Materials, 2017, Xi’an. Xi’an, International Committee on
Composite Materials (ICCM), 2017, pp. 20–25.
22. Kleinwachter H, Geerk J. Device with a pressurizable
variable capacity chamber for transforming a fluid pressure
into a motion. US Patent No. 3,638,536. Feb. 1, 1972.
23. Bishop-Moser J, Kota S. Design and modeling of generalized fiber-reinforced pneumatic soft actuators. IEEE Trans
Robot 2017;31:536–545.
24. Greef AD, Lambert P, Delchambre A. Towards flexible
medical instruments: review of flexible fluidic actuators.
Precis Eng 2009;33:311–321.
25. Kobayashi J, Okumura K, Watanabe Y, et al. Development
of a variable stiffness joint drive module and experimental results of joint angle control. Artif Life Robot 2010;15:
72–76.
26. Booth JW, Shah D, Case JC, et al. OmniSkins: robotic
skins that turn inanimate objects into multifunctional robots. Sci Robot 2018;3:eaat1853.
27. Al-Fahaam H, Nefti-Meziani S, Theodoridis T, et al. The
design and mathematical model of a novel variable stiffness
extensor-contractor pneumatic artificial muscle. Soft Robot
2018 Jul 24.
28. Andrikopoulos G, Nikolakopoulos G, Manesis S. A survey
on applications of pneumatic artificial muscles. In Proceedings, 19th Mediterranean Conference on Control &
Automation (MED), 2011, Corfu, Greece. IEEE, 2011, pp.
1439–1446.
29. Kim S, Laschi C, Trimmer B. Soft robotics: a bioinspired
evolution in robotics. Trends Biotechnol 2013;31:287.
30. Al-Ibadi A, Nefti-Meziani S, Davis S. Novel models for the
extension pneumatic muscle actuator performances. In
Proceedings, 23rd International Conference on Automation
and Computing (ICAC), 2017, Huddersfield, UK. IEEE,
2017, pp. 1–6.
31. Fei Y, Wang J, Pang W. A novel fabric-based versatile and
stiffness-tunable soft gripper integrating soft pneumatic
fingers and wrist. Soft Robot 2019;6:1–20.
32. Trivedi D, Rahn CD, Kier WM, et al. Soft robotics: biological inspiration, state of the art, and future research.
Appl Bionics Biomech 2014;5:99–117.
33. Smith KK, Kier WM. Trunks, tongues, and tentacles:
moving with skeletons of muscle. Am Sci 1989;77:
28–35.
34. Bishop-Moser J, Krishnan G, Kim C, Kota S. Design of soft
robotic actuators using fluid-filled fiber-reinforced elastomeric enclosures in parallel combinations. In Proceedings,
IEEE/RSJ International Conference on Intelligent Robots
and Systems, 2012, Vilamoura, Algarve, Portugal. IEEE,
2012, pp. 4264–4269.
GUAN ET AL.
35. Connolly F, Polygerinos P, Walsh CJ, Bertoldi K. Mechanical programming of soft actuators by varying fiber
angle. Soft Robot 2017;2:26–32.
36. Kurumaya S, Phillips BT, Becker KP, et al. A modular soft
robotic wrist for underwater manipulation. Soft Robot
2018;5:399–409.
37. Yan J, Zhang X, Xu B, et al. A new spiral-type inflatable
pure torsional soft actuator. Soft Robot 2018;5:527–540.
38. Uppalapati NK, Krishnan G. Towards pneumatic spiral grippers: modeling and design considerations. Soft robot 2018;5:
695–709.
39. Zhao F, Dohta S, Akagi T, Matsushita H. Development of
a bending actuator using a rubber artificial muscle and its
application to a robot hand. In Proceedings, SICE-ICASE
International Joint Conference, 2006, Bexco, Busan, Korea.
IEEE, 2006, pp. 381–384.
40. Noritsugu T, Gao L. Development of wearable waist power
assist device using curved pneumatic artificial rubber muscle. Trans Jpn Hydraulics Pneumatics Soc 2005;36:143–151.
41. Zhang JF, Yang CJ, Chen Y, et al. Modeling and control of
a curved pneumatic muscle actuator for wearable elbow
exoskeleton. Mechatronics 2008;18:448–457.
42. Hassanin AF, Steve D, Samia NM. A novel, soft, bending
actuator for use in power assist and rehabilitation exoskeletons. In Proceedings, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2017,
Vancouver, BC, Canada. IEEE, 2017, pp. 533–538.
43. Al-Fahaam H, Davis S, Nefti-Meziani S. The design and
mathematical modelling of novel extensor bending pneumatic artificial muscles (EBPAMs) for soft exoskeletons.
Robot Auton Syst 2017;9963–9974.
44. Tondu B. Modelling of the McKibben artificial muscle: a
review. J Intell Mater Syst Struct 2012;23:225–253.
45. Zhang Z, Philen M. Pressurized artificial muscles. J Intell
Mater Syst Struct 2012;23:255–268.
46. Jones BA, Walker ID. Kinematics for multisection continuum robots. IEEE Trans Robot 2006;22:43–55.
47. Webster Iii RJ, Jones BA. Design and kinematic modeling
of constant curvature continuum robots: a review. Int J
Robot Res 2010;29:1661–1683.
48. Burgner-Kahrs J, Rucker DC, Choset H. Continuum robots
for medical applications: a survey. IEEE Trans Robot 2015;
31:1261–1280.
49. Renda F, Cianchetti M, Giorelli M, et al. A 3D steady-state
model of a tendon-driven continuum soft manipulator inspired by the octopus arm. Bioinspiration Biomimetics
2012;7:025006.
50. Hocking EG, Wereley NM. Analysis of nonlinear elastic
behavior in miniature pneumatic artificial muscles. Smart
Mater Struct 2013;22:014016.
Address correspondence to:
Jinsong Leng
National Key Laboratory of Science and Technology
on Advanced Composites in Special Environments
Harbin Institute of Technology
No. 2 Yikuang Street
Harbin 150080
People’s Republic of China
E-mail: lengjs@hit.edu.cn