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Interaction Forces of Soft Fiber Reinforced Bending Actuators

Authors:
  • AMOLF & Eindhoven University of Technology

Abstract and Figures

Soft bending actuators are inherently compliant, compact, and lightweight. They are preferable candidates over rigid actuators for robotic applications ranging from physical human interaction to delicate object manipulation. However, characterizing and predicting their behaviors are challenging due to the material nonlinearities and the complex motions they can produce. This paper investigates a soft bending actuator design that uses a single air chamber and fiber reinforcements. Additionally, the actuator design incorporates a sensing layer to enable real-time bending angle measurement for analysis and control. In order to study the bending and force exertion characteristics when interacting with the environment, a quasistatic analytical model is developed based on the bending moments generated from the applied internal pressure and stretches of the soft materials. Comparatively, a finite-element method model is created for the same actuator design. Both the analytical model and the finite-element model are used in the fiber reinforcement analysis and the validation experiments with fabricated actuators. The experimental results demonstrate that the analytical model captures the relationships of supplied air pressure, actuator bending angle, and interaction force at the actuator tip. Moreover, it is shown that an off-the-shelf bend angle sensor integrated to the actuator in this study could provide real-time force estimation, thus eliminating the need for a force sensor.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
1
Abstract—Soft bending actuators are inherently compliant,
compact, and lightweight. They are preferable candidates over
rigid actuators for robotic applications ranging from physical
human interaction to delicate object manipulation. However,
characterizing and predicting their behaviors are challenging due
to the material nonlinearities and the complex motions they can
produce. This paper investigates a soft bending actuator design
that uses a single air chamber and fiber reinforcements.
Additionally, the actuator design incorporates a sensing layer to
enable real-time bending angle measurement for analysis and
control. In order to study the bending and force exertion
characteristics when interacting with the environment, a quasi-
static analytical model is developed based on the bending moments
generated from the applied internal pressure and stretches of the
soft materials. Comparatively, a finite-element method model is
created for the same actuator design. Both the analytical model
and the finite-element model are used in the fiber reinforcement
analysis and the validation experiments with fabricated actuators.
The experimental results demonstrate that the analytical model
captures the relationships of supplied air pressure, actuator
bending angle, and interaction force at the actuator tip. Moreover,
it is shown that an off-the-shelf bend angle sensor integrated to the
actuator in this study could provide real-time force estimation,
thus eliminating the need for a force sensor.
Index Terms—soft robot, bending, fluidic actuator, modeling,
interaction force.
I. INTRODUCTION
IGID robots with motor-driven actuators rely on sensory
feedback and control to achieve the compliance required
for physically interacting with humans and handling
unstructured or delicate objects [1, 2]. However, soft robots [3-
5] provide an alternative approach due to their inherent
compliance and back-drivability with compact and lightweight
mechanical structures [6]. The fluidic actuation media
(pneumatics or hydraulics) provides unique characteristics
comparing with electric motors [7-10]. Piezoelectric robotic
actuators could also achieve compliance, but their application
are more towards high precision micromanipulation due to their
limited range of motion and force capabilities [11-14]. The
most widely used soft actuator is the pneumatic artificial muscle
(PAM) that generates linear contraction with pressurization.
Applications of PAMs range from biocompatible devices [15],
to humanoid robots [16, 17], and compliant manipulators [18].
Modeling and control of PAMs have been investigated
extensively in the literature [19-21]. However, in order to create
bending motions, most robotic applications use PAMs to drive
mechanisms consisting of rigid links and joints [16, 18].
Therefore despite the compliance of PAMs, the compliance of
the overall robotic systems has often been limited by the rigid
components.
It has previously been shown that soft bending actuators can
generate inherent bending motion without requiring any rigid
components [22-24]. They are constructed from polymeric [24]
or a combination of elastomeric (hyper-elastic silicones) and
inextensible materials (fabrics and fibers) [22, 23] and activated
by pressurizing fluid. There are various designs for soft bending
actuators. A single-chamber design is shown in Fig. 1, where
bending is created by asymmetrically constraining the
extension of an air chamber [25, 26] with different choices for
the cross sectional shape, such as circular [27, 28], rectangular
Interaction Forces of Soft Fiber Reinforced
Bending Actuators
Zheng Wang, Senior Member, IEEE, Member, ASME, Panagiotis Polygerinos, Member, IEEE/ASME,
Johannes T. B. Overvelde, Kevin C. Galloway, Katia Bertoldi and Conor J. Walsh, Member, IEEE
R
Fig. 1.Structure and bending motion of soft bending actuator. (a) unactuated;
(b) fully pressurized; (c) top wall details; (d) bottom layer details.
Z. Wang* is with the Department of Mechanical Engineering, The
University of Hong Kong, Hong Kong, China (email: zheng.wang@ieee.org).
P. Polygerinos is with the Ira A, Fulton Schools of Engineering at Arizona
State University, ASU at the Polytechnic School, 6075 S. Innovation Way W.
Mesa, AZ 85212 (email: polygerinos@asu,edu).
K. C. Galloway is with School of Engineering at Vanderbilt University,
2400 Highland Avenue, Nashville, TN 37215, USA (email:
kevin.c.galloway@vanderbilt.edu).
J. T. B. Overvelde and. K. Bertoldi* are with the School of Applied Sciences
and Engineering & the Kavli Institute for Bionano Science and Technology,
Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA (e-mail:
overvelde@seas.harvard.edu; bertoldi@seas.harvard.edu).
C. J. Walsh* is with the School of Applied Sciences and Engineering & the
Wyss Institute at Harvard University, 60 Oxford Street, Cam
b
ridge, MA 02138,
USA (email: walsh@seas.harvard.edu).
*Corresponding author(s).
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
2
[29], or hemi-circular [23, 30]. There are also reported designs
of multi-chamber actuators for more complex motions [31-34],
or combined opposing chambers that achieve flexing and
extending motions [30,35]. To achieve higher dexterity, a multi
degree-of-freedom (DOF) actuator could consist of several
small single-chamber bending actuator segments [27, 36].
Furthermore, different actuator designs have been reported for
biomimetic systems [30-32], safe and compliant actuators [35,
37], and delicate manipulators [25, 26, 27].
Modeling the behavior of soft bending actuators is
challenging due to the material nonlinearity and the large
deformations they produce. Most prior works on modeling soft
bending actuators were focused on the quasi-static behaviors,
in which the actuators undergo motions slow enough, such that
no dynamic effects (transients, visco-elasticity of soft material,
etc) were considered. Most previous works followed an
empirical approach [28, 38, 39], while some reported using the
finite-element method (FEM) [24, 30]. In particular, analysis of
their force exertion ability when interacting with the
environment was not reported previously. Regarding actuator
dynamics, recent work investigated the actuation speed and
hysteresis in a multi-chamber soft bending actuator design [34].
In the authors' previous work, a quasi-stationary analytical
model accounting for large deformations was developed for a
soft bending actuator with fiber reinforcements. The model was
successful in capturing the relationship between input pressure
and actuator angle. In addition, multiple candidates for the
actuator designs were compared, revealing that the hemi-
circular cross sectional shape required the lowest pressure to
bend to the same angle [40].
In this study, the quasi-static behavioral characteristics of a
soft bending actuator with fiber reinforcement and integrated
sensing is investigated. After presenting the actuator design in
Section II, the influence of the fiber reinforcement layer on
actuator motion is investigated for the first time in Section III,
using both an analytical model and a FEM model created for
the same actuator design, showing that with sufficiently low
pitch angle, the fiber layer could be regarded as radial constraint
instead of a variable with actuator motion. On this basis,
characterizations of the bending actuator motion and forces
were carried out in Section IV, where a bending sensor is
integrated into the actuator design, and the analytical model
previously derived for free-space bending in [40] is extended to
incorporate the bending sensor, and characterize both actuator
bending and tip force exertion during interactions. The newly
derived analytical model has low computational cost and is
capable of predicting interaction forces at the actuator tip
without using a force sensor. The performance of the proposed
models is validated in Section V, with experiments using
fabricated actuators.
II. A
CTUATOR DESIGN
This study is focused on the actuator design shown in Fig. 1,
previously proposed in our earlier works [10, 22, 40], which
consists of a single stretchable chamber with hemi-circular
cross section, a strain-limiting layer, a reinforcement fiber layer
encircling the chamber to constrain radial expansion during
pressurization, and a sensor layer below the strain-limiting layer
for bending angle measurements. The actuator chamber and the
sensor layer are fabricated using hyperelastic material, while
the strain-limiting layer and fiber reinforcement layer are made
from flexible but inextensible materials. With the actuator
motion generated by soft material stretch, such actuator design
could achieve smooth and continuous bending motion from its
natural resting position to around 360 degrees, with resolutions
subject to the supplied air pressure.
The forces and geometric parameters necessary for deriving
the force model are illustrated in Fig. 2.The actuator has a hemi-
cylindrical top wall with an inner radius a and thickness t, a flat
rectangular bottom layer of thickness b and width 2 , and
an initial length of L. The sensor layer at the bottom of the
actuator has a pocket throughout its length to accommodate a
thin flexible bend sensor that measures the bending angle θ.
When the air chamber is pressurized (
), the top wall
extends while the bottom layer is constrained by the
inextensible strain-limiting layer. Therefore the actuator bends
towards the bottom layer with a radius R and angle θ. With its
proximal tip firmly mounted, the actuator will exert a force if
the distal tip is in contact with an external object, which could
be decomposed into a normal factor perpendicular to the
contact surface, and a frictional factor  parallel to the contact
surface. The combined interaction force is perpendicular to
the bottom layer to ensure a constant bending moment arm of
with respect to the fulcrum O.
III. E
FFECT OF FIBERS
The fiber reinforcement layer in the actuator design
constrains the natural expansion of the actuator chamber during
pressurization and plays an essential role in generating the
bending motion of the actuator. Assuming fiber inextensibility,
the fiber pitch (i.e. density of winding) is an essential variable
to be considered in actuator design. For linear PAMs, change of
fiber pitch is part of the fundamental motion creation
mechanism, hence it must be considered in PAM design and
modeling. However, for bending soft actuators, there is no
analysis on how fiber pitch contributes to actuator motion, or
guideline available on how to choose the fiber pitch for actuator
design. Therefore, we first investigate the relations between
fiber pitch, actuator bending, and actuator deformation, to
Fig. 2. Actuator bending with the cross-sectional view and the torque
equilibrium of the distal tip. The proximal tip of the actuator is fixed and the
distal tip is in contact with an object.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
3
reveal that unlike PAMs, the fiber pitch is not a variable
involved in the actuator bending. Moreover, the conditions are
derived, where pitch angle can be ignored and the fiber layer
becomes a constraint to actuator motions.
A. Analytical study
First, we investigate the effect of the reinforcement fiber
analytically. As shown in Fig. 1, the fiber layer consists of a
left-handed helical fiber winding encircling the actuator body
for n times, and an opposing symmetrical right-handed winding
to balance out potential twisting effects. Therefore, for a given
actuator design (diameter, wall thickness, and length), the
arrangement of the fibers is fully determined by the pitch of the
helix. In fact, the pitch angle determines the fiber turn
number n for a given actuator length, and affects the ability of
the fiber layer to constrain the actuator geometry under
pressurization.
For simplicity the actuator is regarded as a hemi cylinder
with diameter 2, combining both the top and the
bottom layers enclosed by the fiber. Fig.3 shows a segment of
the unpressurized actuator, with all parameters needed to fully
define its geometry (note that for the sake of clarity only the
left-hand winding fiber is shown). The length of the bottom
layer of the corresponding actuator segment , is a function of
the fiber turn number n and the actuator length L:
. (3.1)
Here is an invariant term during actuator deformation,
since the inextensible bottom layer prevents the bottom layer
from any expansion. The top actuator wall, however, is not
subject to this constraint and hence could extend beyond its
original length of . In addition, the fiber segment is also
assumed to be inextensible with a uniform curvature. Therefore,
for the unpressurized actuator segment, the relationship for the
fiber length , the radius , the arc angle2, and the
pitch are:


2 ,  
 ,2
,


. (3.2)
After the actuator is pressurized, most of the above will not
hold due to the bending of the bottom layer. To simplify the
analysis, here we assume the bottom layer only bends
longitudinally with the actuator and ignore any lateral bending
or bulging, as shown in Fig. 3(c), Hence the width of the
actuator remains constant while the height of the hemi cylinder
h changes as a function of the bending angle.
When the actuator is pressurized (Fig. 3(c)), the bottom layer
length, diameter, andthe fiber length remain constant,
due to the inextensibilities of the strain-limiting layer and the
reinforcement fiber. All other variables ( ,, )
instead change as a function of the applied pressure (see Fig.
3(c)) and their values in the deformed configuration are denoted
with ( ,,). Denoting with the overall actuator
bending angle (see Fig. 2), the angle spanned by the
considered segment (Fig. 3(d)) is given by
. (3.3)
The parameters ,,and (defined in Fig. 3(d))
characterizing the deformed configuration can then be
expressed as function of ,,, and using the following
geometric relations

2,
,2
,
 

2,
,2
2
Fig. 3. Pitch angle analysis. (a) a segment of the actuator containing one tur
n
of fiber, (b) the un-
p
ressurized actuator section, (c) same section pressurized,
(d) bottom layer side view, (e) Effect of actuator bending angle on the fiber
pitch angle/ for an actuator design with ,,, 6, 2,2, 170mm, (f)
Effect of actuator bending angle on actuator height, change  /,
with the same actuator design.
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
4



. (3.4)
where denotes the radius of curvature of the pressurized
actuator (see Fig. 3(d)) and is the height of the arch formed
by the considered section (see Fig. 3(d)). Combining Eqns.
(3.2)-(3.4) we obtain


,
∙,

,

1
1
. (3.5)
Eq. (3.5) can now be used to determine numerically the effect
of the overall actuator bending angle  on both the fiber pitch
angle and actuator height. A series of simulations were carried
out on an actuator design of ,,, 6,2,2,170mm,
consistent with the actuators fabricated and used in
experiments. In Fig. 3(e) and Fig. 3(f) we report the changes in
pitch angle ( /) and actuator height (  /) as
function of the actuator bending angle for n=10, 20 and 45.
The results reported in Fig. 3(e) show that the pitch angle
changes for all actuator turn numbers are within 1% for the full
bending range of 360 degrees. However, the change in actuator
height is significant for low fiber turn number (h’ decreased by
5.5% at full bending for n=10), and becomes negligible for
large turn numbers, for instance, n=20 (1.4%) and 45 (0.3%).
Two major conclusions could be drawn from the analytical
investigation: 1) fiber pitch is not a variable involved in actuator
bending. This is fundamentally distinctive from the motion
generation mechanism of PAM; 2) the fiber layer defines
actuator motion by constraining radial expansion. In particular,
to ensure sufficient constraining, a low pitch (or high turn
number) is desirable (for the current design, n>20 or 20
°).
B. Finite Element Model
To validate the analytical model, a 3D finite element (FE)
model of the actuator was created. It was constructed within the
nonlinear finite-element code ABAQUS/standard using the
Yeoh hyperelastic model [41] and assuming incompressibility
for all solid materials. The FE model consisted of an internal
hemi-cylinder chamber and an external hemi-cylinder coating
layer modeled using solid tetrahedral quadratic hybrid elements
(ABAQUS element type C3D10H). The reinforcement fibers
were then modeled by quadratic beam elements (ABAQUS
element type B32), which were connected to the internal
chamber and the coating layer with tie constraints. Here the
inextensible layer was not modeled separately, but was
combined with the bottom layer for better computational
efficiency and increasing simulation convergence. More details
on the FE model are provided in our previous work [40].
The FE model was used to study the influence of the fiber
reinforcement layer on the change of the actuator shape induced
by pressurization and results were compared with the analytical
results previously described. Seven different FEM models were
created with different number of fiber turns, n=5, 15, 25, 35, 45,
55, and 65, while keeping the actuator dimensions constant.
Each model was pressurized to 90 and 180 degrees in free space
and snapshots of the results obtained for n=5, 15, 45, and 65 are
shown in Fig. 4(c). For n=5 the sparse fiber results in large areas
of unconstrained actuator surface between the fiber turns.
Therefore, at 90 degrees bending the model presented large
nonlinear bulges, while at 180 degrees bending the simulation
could not converge due to excessive material deformation and
hence ballooning instability. However, as the fiber turn number
got higher, the radial expansion quickly decreased and became
negligible for n>35. In Fig. 4(d) we then report the normalized
pressure required to bend the actuator by 90 and 180 degrees as
function of n. The pressures required for n=45 actuator were
taken as the reference (100%) for both 90 (59.4 kPa) and 180
(113.4 kPa) result groups. These results clearly indicate that for
n>35, the response of the actuator is not affected by the specific
number of turns.
C. Comparison of analytical and numerical results
In Fig. 5(a) and Fig. 5(b) we report our analytical and
numerical predictions for the evolution of the actuator height h
and pitch angleφas a function of the turn number n for =0 and
=360 degrees. For other angles between 0 and 360 degrees, the
results also fell between those of the two angles, therefore they
were not shown here. The actuator height h was extracted from
the FE simulations as the averaged distance along the length
between the top of the actuator and the undeformed bottom
layer surface, thus ignoring the bottom layer bulging (as to
match the assumption of flat bottom layer in the analytical
model). Note that, since FE simulations with n<15 were not
Fig. 4. FEM model of the actuator (n=45). (a) pressurized actuator. (b) the
reinforcement fiber showing stress concentration at the bottom strain-limiting
layer. (c) FEM simulations with turn number of 5, 15, 45, and 65 at 90 and 180
degrees bending, (d): Pressure required to reach 90 and 180 degrees bending
for different n, with pressures for n=45 taken as references (100%).
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
5
able to converge due to ballooning instability, only numerical
results for 1565 are reported.
For the pitch angle, analytical and numerical results agree
very well and indicate that pressurization has a negligible effect
on , regardless of the number of turns n.
Differently, the analytical results reported in Fig. 5(b) show
that for small number of fiber turns (i.e.  10), pressurizing
the actuator resulted in a significant deformation in actuator
height. However, as the number of fiber turns increased (
20, the deformation of the cross section rapidly reduced and
became negligible. Note that for analytical results, actuator
height after bending is smaller than the initial height. This is not
contradicting to the bulging observed in FEM results in Fig.
4(c): while the analytical models of (3.5) describes the actuator
deformation resulting from the constraining fiber, the bulging
of FEM modeling resulted from the unconstrained soft material
between the adjacent fibers, which exhibited ballooning
expansion despite the constraining fiber was reducing the
actuator height. This was verified by the trend of reduced
bulging with increasing fiber turn number, such that denser
fiber resulted in less surface of unconstrained soft material,
hence less bulging. The FE results confirmed that for n>20 the
deformation of the actuator height was not affected by the
number of fiber turn, but showed that h converged to 6.5mm
instead of the 6mm undeformed radius. This discrepancy was
probably due to the assumption of constant actuator width,
while in reality the actuator bottom layer would bulge with
pressurization and therefore increase in height, which would
affect the actuator height as well as the structural stiffness.
In summary, both analytical and FE results clearly indicate
that fiber pitch is not a variable of actuator bending, and that the
fiber layer design does affect the actuator geometry change and
hence the bending performance. However, with sufficiently low
pitch and high turn number (20
°,n20), the fiber layer
could simply be regarded as a constraint to actuator radial
expansion. Therefore, a reinforcement fiber layer with 45
(9
°) was used for the rest of this study.
This conclusion is substantially different from those on the
PAM actuators, where the fiber pattern is a defining factor for
the actuator performance [20, 21]. This can be justified by the
fundamental difference in motion creation between the linear
PAM actuator and the bending actuator in this study. For the
PAMs, the fibers transform radial expansion into linear
contraction and generate actuator movement, therefore the
pattern of fiber winding plays a major role in the geometrical
transitions during the actuation procedure, hence affecting
actuator performance. On the other hand, for the bending
actuator in this study, expansion asymmetry is created by the
inextensible layer attached to the actuator bottom, not the fiber
winding. Instead, a sufficiently dense (n=45, 9
° for the
actuator design in this study) fiber layer only serves as a
constraint to radial expansion, does not have a critical influence
on actuator behavior. The analytical model provides a
convenient method to verify this condition for a different
actuator design: equation (3.5) could be used based on the
actuator geometrical information to decide whether a new fiber
turn number is sufficient for negligible actuator height change
(
1%) within the desired actuator bending range .
IV. ANALYTICAL MODEL TO PREDICT THE FORCE EXERTED BY
THE ACTUATOR
Modeling assumptions. The following assumptions are made
regarding the actuation process: i) the radial expansion of the
actuator is negligible because of the constraint provided by the
fiber reinforcement layer; ii) the supplied air has sufficient flow
and the air pressure changes slowly, such that the actuator
always reaches steady states; iii) the materials used to fabricate
the actuator are incompressible; iv) force interaction, if
applicable, only occurs at the actuator tip, and the bending
curvature of the actuator is uniform for all bending angles; v)
effect of gravity is not considered; vi) the hyper elastic material
of the actuator only experiences elastic deformation.
Hyper-elastic material model. Besides the actuator geometry
and the actuation process, the stress-strain properties of the
hyperelastic materials also need to be considered. The Neo-
Hookean (N-H) hyperelastic model [42] is used in this study to
capture the response of the hyperelastic material used to
fabricate the actuator. Assuming material incompressibility, the
strain energy density W is defined as:

3, (4.1)
where is the initial shear modulus of the material, is the
first invariant of the three (axial, horizontal, and vertical)
principal extension ratios , , and ,



. (4.2)
The principal nominal stresses s can be obtained as [40]:


, (4.3)
where is a common Lagrange multiplier for i=1, 2, 3. Note
that in our model different material properties ,,and
Fig. 5.Fiber analysis comparison with analytical model (Analytical) and Finite
Element Analysis (FE). Upper (a): change of fiber pitch versus turn
numbern, for 0 and 360 degrees bending; Lower (b) change of actuator heigh
versus n, for 0 and 360 degrees bending. (For turn number n=5 the FEA
simulation did not converge at full bending of 360 degrees, therefore the results
are not shown here)
1083-4435 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
6
are used for the top layer, the bottom layer, and the bend sensor,
respectively. We also want to highlight the fact that
t and
b
describe the shear modulus of the entire actuator layer (either
top or bottom) consisting of hyperelastic materials,
reinforcement fibers, and the strain-limiting layer. The values
of ,,and
are determined empirically through a
calibration process as discussed in Section 5.1.
To derive an analytical model that relates the input air
pressure and the force exerted by the actuator, we start by noting
that the radial expansion of the actuator is constrained by the
fiber reinforcement, resulting from the analysis on the fiber
layer in Section III, so that the circumferential stretch is
approximated to be equal to unity (i.e.1). Following a
similar derivation procedure as presented in [40], equation (4.3)
could be simplified using a unified extension ratio , and the
axial nominal stress becomes:



. (4.4)
Basic moment equilibrium. Based on the discussions of free-
space motion in [40], in this work we extend the moment
equilibrium to consider interaction forces at the actuator tip. At
each bending configuration, there are four bending moments
involved in the bending of the actuator (Fig. 2): is the
pressure-induced bending moment generated by the internal air
pressure acting on the actuator tip around the pivot point, O;
and are the material stretch moments of the top and bottom
layers, respectively (note that also incorporates the moment
required to bend the sensor layer); and is the tip force
bending moment, exerted by the interaction force applied to the
actuator tip if in contact with an external object. Gravity was
ignored, as the actuator weight (around 20g overall: 10g for the
proximal end connection mechanism, 10g for the rest of the
actuator, that is 0.1N gravity for the part actually involved in
bending) was significantly smaller than the actuating forces (6N
at 100kPa, 15N at 250kPa). While acts clockwise around
the fulcrum O, ,, and all act counter-clockwise to O.
Therefore the following moment equilibrium condition needs to
be satisfied: M‐MMM. (4.5)
Actuator moment components. To obtain an analytical
expression linking the applied pressure to the force exerted by
the actuator, we next proceed to obtain explicit relationships for
each moment.
1) For the hemi-circular actuator tip with radius, Ma (see
[40]) can be calculated as

 
. (4.6)
where  

, and
43. (4.7)
2) For the bottom layer, we note that the principal stretch
is a function of the vertical position. In particular, introducing
the coordinate
and the actuator bending radius R as shown in
Fig. 2, it can be written as [40]:


1. (4.8)
It follows that
2
2
,
(4.9)
where



,  (4.10)
and̅. Here we assume the moment required to
bend the sensor to be, a linear function of the actuator
bending angle, as guided from our experimental results. The
value of the sensor stiffness is determined from calibration.
3) Finally, for the top layer,
depends on the coordinates
and

defined in Fig. 2 as [40]:

, (4.11)
so that
2
.
(4.12)
Approximation. Different from the polynomial solution of
(4.9) for the bottom layer, the integral of (4.12) can only be
solved numerically, as shown in [40], which would restrict the
application of the model in analysis and real-time calculations.
In this work, we obtain an explicit expression for Mt by
replacing the hemi cylinder top wall, with a flat top wall located
at a distance from the bottom of the actuator,
denoting a coefficient to be determined. With this
approximation, in (4.11) is simplified and becomes
independent from

. (4.13)
Substitution of (4.13) into (4.12), yields
2
∙∙, (4.14)
where


, (4.15)
and


. (4.16)
The numerical value of can be determined empirically by
comparing the values of for bending angles
from 0 to 360
degrees using the exact (4.12) and approximate (4.14)
expressions for different values. For an actuator
with ,,,6,2,2,170mm , we found that 
0.65yields <5% approximation error for the entire bending
range of 0 to 360 degrees. Therefore, (4.14) with  0.65is
used in the rest of this study as a valid approximation of (4.12).
Force model. Finally, assuming that an external object is in
contact with the actuator tip as illustrated in Fig. 2, the force F
at the tip of the actuator results in an exerted moment,
∙, (4.17)
where denotes the actuator tip length. Note that we implicitly
assume that the force interaction happens at the end of the
actuator and we do not account for deformation of the actuator
tip due to force exertion.
We are now in a position to predict both the bending of the
actuator in free space and its force exertion during interaction
with an object at the tip. In particular, substituting (4.6), (4.9)
and (4.14) into the equilibrium condition (4.5) and assuming
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Mf=0, the relation between  and for an actuator bending in
free space is obtained as
P χ
, (4.18)
where  

, ,,andare material properties
for the bottom layer, top wall, and the bend
sensor,,,χ,andare given in (4.7), (4.10), (4.15)
and (4.16), respectively.
Moreover, if the tip of the actuator is constrained, the
following relationship between input pressure, bending angle,
and bending force can be obtained from Eq. (4.5)

 22,(4.19)
If both the pressure Pin and the bending angle can be
measured, (4.18) can be used to calculate the interaction force
of the actuator both under isotonic (constant pressure) and
isometric (constant angle) conditions. Interestingly, since Eq.
(4.18) consists of polynomial functions of  and , its right
hand side can be calculated easily and it is therefore suitable for
real-time applications.
V. EXPERIMENTAL VALIDATION
A. Experimental setup and calibration
The actuators were fabricated using a multi-step molding
process with 3D printed molds [22, 40]. A hemi-circular steel
rod was used to create the air chamber in the first rubber layer
(Elastosil M4601 A/B Wacker Chemie AG). Woven fiberglass
(S2-6522 plain weave) was attached to the flat bottom surface
using silicone adhesive as the strain limiting layer. Kevlar fiber
of 0.38 mm diameter was then hand wound in a double helix
pattern around the length of the actuator body. Fiber
reinforcements were further secured by placing the entire
assembly into another mold to encapsulate the actuator body in
a 1.0mm thick silicone layer (Ecoflex-0030 silicone, Smooth-
on Inc.). The actuator body was then removed from the steel rod
and capped at both ends with silicone (Elastosil M4601 A/B).
A vented screw (10-32) was fed through the silicone cap and
became the connection for the pneumatic tubes. A sensor layer
of 12x170x2mm (Ecoflex-0030, Smooth-on Inc.) was molded
separately and attached to the actuator bottom layer with
flexible silicone adhesive. The sensor layer had a 6x170mm
central pocket for a flexible bend sensor (Spectra Symbol Flex
Sensor FS-L-0095-103-ST) of 5x100x1mm. The sensor
exhibited very good linearity (0.99) between angle and
resistance when calibrated with a goniometer. For this study we
fabricated actuators of design ,,, 6,2,2,170mm.
An experimental setup was developed (Fig. 6(b)) with
integrated air pressure regulation and force/torque sensing
(Nano17, ATI Industrial Automation), which was previously
reported in [10, 22]. The proximal tip of the actuator was
mounted firmly to the mounting base and connected to the air
regulator. As the distal end bends downwards, the overall
contact force F on the distal tip (as shown in Fig. 2, F is always
perpendicular to the actuator bottom surface) will vary in
direction with respect to the horizontal surface. To adapt to the
variation of contact direction, an extension bar was attached to
the force sensor, the 6-DoF force/torque measurements were
combined using the dimensions of the extension bar to obtain
the overall force at the actuator tip regardless of its direction.
The material properties ,,and for the analytical
model were determined by calibration. Three actuators were
used in this study, with the same actuator design and the same
material for their top and bottom walls, therefore we used
̅. The same calibration procedure was followed to obtain
the averaged material property ̅ as presented previously in
[40]. Despite using a different actuator design of ,,,
6,2,2,170m, the calibration resulted in a very similar ̅ 
0.313MPa with those of [40], mainly due to the same wall
thickness and material combination used in both designs. The
same calibration procedure was used again to estimate the
sensor stiffness0.103Nm/rad, using the same group of
actuators bending in free space with/without the bend sensor.
The estimated material properties were used in all experiments.
B. Free-space bending test
In the free-space bending test, the pressure-angle relationship
of (4.18) is validated. The effect of integrating a bend sensor
into the actuator is also accessed by comparing the results of
(4.18) with or without sensor stiffness. In the experimental
Fig. 7. Free space bending results of measured air pressures (Exp), predicted
air pressures before and after considering sensor stiffness.
Fig. 6.Experimental setup. (a) sensorized actuator (b) evaluation platform with
actuator mounting base and force/torque sensor.
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IEEE/ASME Transactions on Mechatronics
8
setup, an actuator was mounted horizontally on the platform,
with the distal tip bending downwards in the vertical plane
without obstacles. Without rigid structural support, the mounted
actuator exhibited a 60-degree downwards natural bending
angle due to gravity.
Three trials were conducted, and for each trial the actuator
was pressurized from its natural resting position with air
pressure increased from 0 to 250 kPa, and actuator bending
angles were measured by the integrated bend sensor.
To validate the analytical model of (4.18), the measured
bending angles were used to calculate air pressure, and compare
with the actual supplied air pressure values. To illustrate the
influence of the integrated bend sensor, two pressure
estimations were made by setting 0and0.103Nm/
rad, respectively. They are compared in Fig. 7 against the
measured air pressure (Exp). The analytical model was derived
based on internal material stretch, which should be zero at the
natural resting condition. Therefore, the 60 degree initial
bending angle was subtracted from sensor measurements before
applying the analytical model. Similar procedures were
conducted in other tests where the analytical model was used.
In the results shown in Fig. 7, the estimated pressure values
match those obtained experimentally throughout the actuator
bending range. Moreover, considering the sensor stiffness
results in a noticeably better match to the experimental
measurements with a standard deviation of =12.5kPa, or 5.0%
of the full pressure range of 250kPa, comparing with
=18.9kPa (7.6%) when sensor stiffness is not considered. The
calibration result of 0.103Nm/rad was therefore
justified and used in the rest of this study.
Actuator hysteresis is a well-known issue for linear PAM
actuators, which is generally accepted to be caused by the
friction of fiber braiding and between different layers of
materials within the actuator [6]. However, although our
actuator design utilizes a fiber layer, there is minimum sliding
between the fiber and the actuator body, and hence the friction
loss is negligible during actuation. Therefore the dominating
internal interaction is material stretch, which does not lead to
hysteresis, as observed in previous work on soft bending
actuators made from the same soft material [34]. In our previous
experiments with the same actuator design, no significant
hysterical effect was observed [40], therefore hysteresis is not
investigated in this study.
C. Isometric test
In an isometric test, the actuator was constrained at constant
bending angles while input pressure increased from 0 to
250kPa. In each state the contact forces were measured by the
force sensor. The experiment consisted of three trials, with three
actuator positions considered for each trial: 90, 135, and 180
degrees (without subtracting the natural resting angle), as
shown in Fig. 8(a)-Fig. 8(c).The analytical model was used to
estimate bending forces using the measured bending angles.
Comparatively, FEM simulations were conducted following the
same experimental procedure and bending angles.
The calculated forces from the analytical and FEM models
are compared with the experimental force measurements in Fig.
8(d). The analytical results form parallel lines for each bending
angle. This trend is matched by the FEM and experimental
results, with FEM results exhibiting increasing deviations at
larger bending angles, possibly caused by large material strains
and hence FEM element deformations. The experimental
measurements are closely matched by the analytical results with
a standard deviation of =0.26 N (6.0% of a maximum force of
4.4 N) for 90 degrees, =0.13 N (4.6% of 2.83 N) for 135
degrees, and =0.12 N (5.6% of 2.1 N) for 180 degrees.
D. Isotonic test
In the isotonic test, the input pressure was kept at a range of
constant values, and interaction forces were measured at
different bending angles. Similar with the previous
experiments, force predictions from the analytical model were
compared with measured forces.
The experimental results are shown in Fig. 9, where each trial
forms an isotonic line that intersects with both axes. The
intersections with the horizontal axis indicate the maximum
bending angle for the particular input pressure where force
drops to zero. The intersections with the vertical axis indicate
the maximum achievable force for that pressure, which occurs
at zero bending. Between the two intersections, the force-angle
relationship is nonlinear as described in (4.18). This
nonlinearity is introduced by the hyperelastic material property
and therefore captured very well by the analytical model. The
comparison results for each pressure configuration are listed in
TABLE I. The analytical model provides a better estimation for
larger pressures, with 2.7% average error for 241kPa,
comparing with 16.1% for 60kPa. This is mainly due to the
Fig. 8. Isometric test results. (a)-(c) actuator bending for 90, 135, 180 degrees,
(d) comparison of analytical model, FEM, and experimental (Exp) results.
Fig. 9.Isotonic test results. Comparison of force predictions from the analytical
force model (Analytical) and experimental force measurements (Exp).
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IEEE/ASME Transactions on Mechatronics
9
complex actuation procedure the actuator underwent not fully
considered in the analytical model, such as interactions between
different material layers, nonlinear bulging and deformations
on radial and circumferential directions. The above factors were
more significant for lower pressures, where the actuator started
deforming from its original state. Although an averaged
material shear modulus was used to incorporate the above
factors, the calibration process for the shear modulus resulted
in an optimal value for the entire pressure range from 0 to
250kPa. Therefore, the resulting analytical model provided
better predictions for higher pressures. The isotonic test results
also illustrated the compliant feature of soft bending actuators,
where each isotonic line in Fig. 9 gave a quantitative measure
of how much the actuator would deform with an exerted force.
VI. CONCLUSIONS AND FUTURE WORK
This work demonstrated that the bending and tip-force
capabilities of the soft fiber-reinforced bending actuators under
quasi-static conditions could be characterized, modeled, and
controlled. Moreover it was shown that integrating an off-the-
shelf bend angle sensor to the actuator could provide force
estimation without requiring a dedicated force sensor. This is a
critical step to demonstrate the capability of soft bending
actuators for robotic applications. Although the study assumed
quasi-static states with slow motions, its conclusions are
applicable to most application scenarios where soft actuators
have advantages, ranging from manipulating unmodeled/fragile
objects to interacting with unstructured environments, where
compliance, adaptability and safety are more critical concerns
than actuation speed and high bandwidth.
An analytical model was developed to describe the force
generation mechanism and quantify the relationship between
the input pressure, bending angle and tip bending force. The
resulting force model consists only of polynomial functions,
therefore it can be easily embedded onto a micro controller for
real-time calculation and control in robotic applications. The
effect of the reinforcement fiber layer was also investigated. A
geometrical study was conducted for the actuator on the
influence of different fiber pitch angles on actuator geometry.
To validate the analytical models, a FEM model of the actuator
was developed closely resembling the real actuator. In the
comparison results, the FEM simulation could provide a closer
match to the reality, while compromising real-time capability.
In addition, the FEM model also highlighted the critical regimes
of stress concentration in the internal structures.
Actuators were fabricated and tested to validate the analytical
and FEM models in free-space, isometric, and isotonic
experiments. The results demonstrated that the analytical model
was able to capture the relationship between input pressure,
bending angle, and the output bending force at the actuator tip
with good accuracy, despite the approximations and
linearization during the modeling process. In addition, the
proposed analytical model provided a direct computational link
between an actuator design and its performance, hence the
parameters of an actuator design could be calculated from a
given performance criteria, simplifying the design iteration and
hence reducing the time and cost. Potential applications include
glove-type wearable robotic devices where interaction forces
are mostly encountered at the actuator tip [10, 23], and new
actuator designs featuring the same topology.
In future work, analytical modeling of nonlinear actuator
behaviors, such as non-uniform bending and radial bulging will
be tackled. Viscoelastic effect, air compressibility and heat
dissipation arising from fast actuation will be investigated to
analyze the dynamic actuator behaviors. Distribution of
interaction forces along the actuator body will be investigated,
together with engineering aspects such as actuator fatigue and
failure modes. The modeling approach could be generalizable
to other actuator designs following the same fundamental
structure of a single air-chamber and fiber reinforcements.
Applications of soft bending actuators will be carried out
exploring the integrated bending sensor, in scenarios such as
contact detection and force control. Through modeling analysis
and experimental validations, the static and dynamic properties
of the actuators could be demonstrated to the robotics research
community. With its unique features and the tools for analysis,
design, and control, soft bending actuator will be a capable
competitor of rigid-bodied actuators for robotic applications.
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Comparison between Analytical Force Estimations and Experimental
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Pressure (kPa) (N) (N) /
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IEEE/ASME Transactions on Mechatronics
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Zheng Wang (M’10, SM’16) received the BSc from
Tsinghua University (China), MSc with distinction from
Imperial College (UK), and PhD with merit from
Technische Universität München (Germany). He was a
postdoctoral research fellow in Nanyang Technological
University (Singapore) between 2010 and 2013, and a
postdoctoral fellow with the School of Engineering and
Applied Sciences and the Wyss Institute of Bioinspired
Engineering at Harvard University between 2013 and 2014. Since July 2014 he
has been an Assistant Professor in the Department of Mechanical Engineering
at the University of Hong Kong. His research interest include: soft robotics,
underwater robots, medical robots, and human-robot interaction.
Panagiotis Polygerinos (M’11) received the B.Eng.
degree (top of his class) in mechanical engineering from the
Technological Educational Institute of Crete, Heraklion,
Greece in 2006, the M.Sc. (with distinction) degree in
mechatronics and Ph.D. in mechanical engin eering/medical
robotics from King’s College London, London, U.K., in
2007 and 2011, respectively. From 2012 until 2015, he was
a postdoctoral fellow of technology development with the
Harvard Biodesign Lab and the Wyss Institute for Biologically Inspired
Engineering at Harvard University. He is currently an Assistant Professor with
the Ira A. Fulton Schools of Engineering at Arizona State University, USA. His
research interests focus on the realization of tasks that are essential to the
design, implementation and integration of novel robotic systems and
mechatronic devices that improve patient care and human activity.
Johannes T. B. Overvelde received his BSc and MSc (both
with distinction) in Mechanical Engineering from Delft
University of Technology, the Netherlands in 2012.
Currently, he is an Applied Mathematics PhD candidate in
Katia Bertoldi’s group at the School of Engineering and
Applied Sciences at Harvard University. His research
interests are in the field of nonlinear structural optimization
and computational mechanics.
1083-4435 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMECH.2016.2638468,
IEEE/ASME Transactions on Mechatronics
11
Kevin C. Galloway received his B.S.E. and Ph.D. in
mechanical engineering from the University of
Pennsylvania and is currently a research assistant professor
in mechanical engineering and the Director of Making at
Vanderbilt University. His current research interests
include new approaches to design, rapid prototyping from
the micro- to the macro-scale, active soft materials, and the
manufacture and control of wearable robotic devices.
Katia Bertoldi received her master’s degrees from Trento
University (Italy) in 2002 and from Chalmers University of
Technology (Sweden) in 2003, majoring in Structural
Engineering Mechanics. Upon earning a Ph.D. degree in
Mechanics of Materials and Structures from Trento
University, in 2006, Katia joined as a PostDoc the group of
Mary Boyce at MIT. In 2008 she moved to the University
of Twente (the Netherlands) where she was an Assistant Professor in the faculty
of Engineering Technology. In January 2010 Katia joined as Associate
Professor of Applied Mechanics the School of Engineering and Applied
Sciences at Harvard University and established a group studying the mechanics
of materials and structures.
Conor J. Walsh received his B.A.I and B.A. degrees in
Mechanical and Manufacturing engineering from Trinity
College Dublin, Ireland in 2003 and an M.S. degree and
Ph.D. in Mechanical Engineering from the Massachusetts
Institute of Technology in 2006 and 2010 with a minor in
entrepreneurship through the Sloan School of Management
and also a Certificate in Medical Science through the
Harvard-MIT Division of Health Sciences and Technology. During his time at
MIT, he had received the Whitaker Health Sciences Fund Fellowship as well as
numerous other design, entrepreneurship and mentoring awards. He is an
Associate Professor in the Harvard School of Engineering and Applied Sciences
and a core faculty member at the Wyss Institute for Biologically Inspired
Engineering at Harvard University. His research efforts are at the intersection
of science, engineering and medicine with a focus on developing smart medical
devices for diagnostic, therapeutic, and assistive applications.
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