# Finite Element Modeling of Pneumatic
Bending Actuators for Inflated-Beam Robots

Cosima du Pasquier, Sehui Jeong, and Allison M. Okamura Manuscript received:
June, 6, 2023; Revised August, 21, 2023; Accepted September, 17, 2023.This
paper was recommended for publication by Editor Editor Yong-Lae Park upon
evaluation of the Associate Editor and Reviewers’ comments.
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The work was supported in part by the U.S. Department of Energy, National
Nuclear Security Administration, Office of Defense Nuclear Nonproliferation
Research and Development (DNN R&D) under subcontract DE-AC02-05CH11231 from
Lawrence Berkeley National Laboratory, and National Science Foundation grant
2024247.All authors are with the Department of Mechanical Engineering,
Stanford University, United States<EMAIL_ADDRESS>Object
Identifier (DOI): see top of this page.

###### Abstract

Inflated-beam soft robots, such as tip-everting vine robots, can control
curvature by contracting one beam side via pneumatic actuation. This work
develops a general finite element modeling approach to characterize their
bending. The model is validated across four pneumatic actuator types (series,
compression, embedded, and fabric pneumatic artificial muscles), and can be
extended to other designs. These actuators employ two bending mechanisms:
geometry-based contraction and material-based contraction. The model accounts
for intricate nonlinear effects of buckling and anisotropy. Experimental
validation includes three working pressures (10, 20, and 30 kPa) for each
actuator type. Geometry-based contraction yields significant deformation
(92.1% accuracy) once the buckling pattern forms, reducing slightly to 80.7%
accuracy at lower pressures due to stress singularities during buckling.
Material-based contraction achieves smaller bending angles but remains at
least 96.7% accurate. The open source models available at
http://www.vinerobots.org support designing inflated-beam robots like tip-
everting vine robots, contributing to waste reduction by optimizing designs
based on material properties and stress distribution for effective bending and
stress management.

###### Index Terms:

Inflated-Beam Robots, Pneumatic Actuation, Explicit FEA, Anisotropic Material
Model

††publicationid: pubid:  10.1109/LRA.2023.3320010 © 2023 IEEE ]

##  I Introduction

Soft robotic systems allow for safe physical interaction with complex or
delicate environments. In contrast to to their rigid counterparts, of soft
robots use embedded design features, such as actuators or pleats, linking form
to function [1]. Typically, refining soft robot designs relies on lengthy
iterative experimentation. Yet, finite element (FE) modeling streamlines
design optimization and performance prediction, reducing the material and time
cost for new soft robots while ensuring safe and reliable operation [2, 3].

Inflated beam robots (IBRs) are a class of soft robots that use pressurized
textile or plastic sleeves to achieve a range of shapes and access constrained
or cluttered environments [4, 5, 6]. With multiple degrees of freedom (DoFs),
IBRs re controlled through pleats [7], internal devices [8, 9], tendons [4,
10], and pneumatic actuators [11, 12, 13].

Tip-everting inflated beam robots, or vine robots, are a subset of IBRs in
which the beam wall material is initially inverted inside the robot. Pressure-
driven eversion causes the vine robot to “grow”, extending its tip while the
beam wall is stationary relative to its environment. This is advantageous for
applications with delicate environments to be navigated via tortuous paths,
like surgery, search and rescue, and archaeology [11, 14]. Challenges arise in
beam bending due to sleeve and eversion limits, prompting the design of
pneumatic actuators that remain compact during eversion and then expand to
control beam shape [15].
Previous work on modeling of pneumatic actuators for IBRs focused on
kinematics and geometric approximations of volume change, virtual work, and
conservation of energy [13, 16, 17]. These models only offer a rough
approximation of the deformed center axis of the IBR. Pneumatic actuators rely
on geometry-based contraction or material-based contraction for bending.
Geometric contraction causes distributed buckling throughout the beam
influenced by material and stress distributions, not captured by kinematic
models. Material-based contraction relies on material anisotropy that is shear
stress distribution dependent, also not captured by kinematic modeling. Thus,
existing models only approximate deformation through iterative parametric
tuning based on experimental data, requiring time-intensive prototyping and
testing procedures. Each change in design or material requires a new tuning
process.
In contrast, finite element modeling (FEM) captures internal forces, strains,
and stresses, and can predict both buckling and anisotropy. FEM requires
initial material characterization and mesh refinement, but adapts to geometry
and material changes, seamlessly including external loads.
In this work, we contribute to the field of soft robotics in three distinct
ways. First, we propose the first comprehensive and open source FE framework
for IBRs (available at http://www.vinerobots.org), predicting bending
mechanisms incorporating both buckling and anisotropy. The framework can be
modified to simulate and verify bending in other pneumatic actuators for IBRs,
streamlining their design and fabrication. Then, we provide constitutive
models for quasi-isotropic and inextensible materials used for buckling and
anisotropic extensible materials used for anisotropy, integrated into the FE
software Abaqus CAE [18]. Third, by utilizing _Dynamic, Explicit_ 111Abaqus
CAE-specific functionalities are capitalized and italicized throughout the
text. FE analysis, our data-driven model predicts deformation for four main
types of actuators: series, compression, fabric, and embedded pneumatic
artificial muscles (sPAMs [11], cPAMs [19], fPAM [12], ePAMs [17]). sPAMs are
also referred to in literature as series Pouch Motors, so we refer to them
here as sPAMs/PMs [13].
This paper is organized as follows: first, we describe the material mechanical
characterization and the benchmark experimental bending protocol for all
actuators. Then, we explain the assumptions for the FE model mirroring the
experimental setup. We define comparison metrics and error calculations for FE
model validation. Next, we present experimental and simulation bending results
and discuss the validity and robustness of our approach. Finally, we discuss
the results in the broader context of pneumatic actuation of IBRs.

##  II Background

###  II-A Modeling of Deformation in Pneumatically Actuated IBRs

The complexity of controlling an IBR with pneumatic actuators, unlike other
mechanisms such as internal devices [8] or tendons [4] arises from the fact
that the relationship between actuation and shape cannot be directly
extrapolated from finite orientation or length changes. Differences in bending
mechanisms among pneumatic actuator types also impact their modeling.

Pneumatic actuators induce bending by shortening one side of an IBR,
achievable through geometry-based contraction, newtextwhere the transition
from flat to 3D inflated configurations shortens the actuator, or material
anisotropy, where the inherent material properties cause the actuators to
shorten.

The sPAM/PM, cPAM, and ePAM bending actuators all rely on geometry-based
contraction, often modeled with energy conservation and virtual work
principles [16, 17, 19, 20]. Analytical models describe geometric changes of
actuators during inflation, equating pressure-induced work to virtual
translation or contraction. They assume material inextensibility, constant
curvature, and idealized linear spring behavior. Niiyama et al. [13] and Greer
et al. [20] model sPAMs/PMs separately from the IBR body and for low
pressures. Kübler et al. [19] model sPAMs/PMs and cPAMs including curvature
and show promising results, but the accuracy varies with actuator dimensions.
Abrar et al. [17] model bending of ePAMs, but the model and the experimental
data differ significantly.

fPAMs rely on material anisotropy for bending. Naclerio and Hawkes [12] relate
pressure increase to volume increase based on McKibben muscle principles.
Ultimate tensile strength is used to determine maximum pressure and maximum
actuator contraction. The model performs very well for pure linear
displacement of isolated actuators. Kübler et al. [19] extend the model to
bending by including the reaction forces between the IBR and the actuators in
the equilibrium equations.

Analytical models for both contraction types are actuator-specific and tuned
using experimental data. They lack stress and strain fields, focusing only on
shape prediction without considering the full range of information required
for actuator design and fabrication.

###  II-B Considerations in FEA of thin-walled structures

The body of an IBR is a thin-walled cylindrical shell. The actuators on or in
the body surface generate asymmetric lateral loading, which in turn causes the
cylinder to bend through buckling. This specific scenario has not been
documented in literature for FE analysis, but there is a body of literature
concerning loading of thin-walled cylindrical shells under uniform lateral
pressure [21, 22].

There are two methods to solve dynamic buckling FE problems with ABAQUS:
_Standard_ and _Explicit_. They use two different approaches to solving the
general equilibrium equations in a structure:

$\mathbf{M}\mathbf{\ddot{u}}+\mathbf{D}\mathbf{\dot{u}}+\mathbf{K}\mathbf{u}=\mathbf{F},$
(1)

where M is the mass matrix, D is the damping matrix, K is the stiffness
matrix, F is the external load vector, and u is the nodal displacement vector.

In _ABAQUS/Standard_ , the Newton-Raphson method solves the equations
implicitly for stability. The stiffness matrix (K) is updated for the deformed
structure at each step. _ABAQUS/Explicit_ uses the central difference formula.
While less stable, accuracy can be improved by reducing time step size.
Despite _ABAQUS/Explicit_ requiring more time steps, in complex problems like
ours _ABAQUS/Standard_ has a high iteration cost. ABAQUS/Explicit,
acknowledged for efficiency in scenarios with frequent self-contact [18], was
chosen for our model. Also, _ABAQUS/Explicit_ offers the _*Fabric_ material
model capturing unique traits of woven fabrics. This is beneficial for our
study, given our fPAM’s anisotropic fabric, challenging to represent with
hyperelastic models.

_ABAQUS/Explicit_ offers two methods to define our time step: _Dynamic_ and
_Quasi-Static_. The _Quasi-Static_ approach omits inertial forces. This method
can be effected if calibrated for loading rate and mass scaling to limit
inertial forces while keeping simulations relatively quick. Thus, preliminary
simulation batches are necessary to estimate these values, adding complexity
to FE simulation setup.

Previous studies confirm the suitability of the _Explicit, Dynamic_ method for
simulating thin-walled inflatable structures. Nguyen & Zhang [23] and Baines
et al. [24] used _Explicit-Dynamic_ for fabric soft pneumatic actuators and
inextensible material inflatables. Thus, we use _Dynamic, Explicit_ to model
bending actuators in IBRs. Valid solutions must maintain kinetic energy below
10% of total internal energy [18].

##  III Methods

This section introduces the FE modeling approach and parameters for IBRs,
based prior thin-walled structures research. We detail actuator materials,
data acquisition methods for constitutive material models, actuator designs,
assembly methods, and performance assessment protocol.

###  III-A FE Model for IBRs

Using Abaqus CAE _Explicit_ , we developed an FE model to predict deformations
and reaction forces in IBRs and their actuators. We used a _Dynamic, Explicit_
step was. To optimize computation, IBR and actuator models were defined as
shells with thicknesses of 200 $\upmu$m and 50 $\upmu$m for TPU-coated and
silicone-coated fabrics, respectively. For efficiency, both IBR and actuator
meshes were set at 4.0 mm and 1.0 mm respectively, roughly 1% of the IBR
length and 1.7% of a single actuator length. These sizes, determined through a
preliminary study spanning mesh sizes from 0.5 mm to 4.0 mm, were chosen for
their convergence accuracy and computational efficiency.

The cPAM and ePAM actuators’ model employed _Tie_ constraints on all welded
surfaces. Similarly, sPAMs/PMs utilized Tie constraints for both welded and
glued surfaces. To model the fPAM, given distinct elastic properties of the
material and glue, the glued surface was represented as a _Composite_ surface,
combining silicone-coated Nylon material properties with those of the glue.
_General Contact_ was implemented to prevent collisions, and a 0.01 mm gap
separated IBR and actuator layers, ensuring a collision-free model
initialization.

A maximum pressure of 2 kPa was applied to the IBR, and a maximum pressure of
10, 20, and 30 kPa was applied to all actuator interiors. In a single 1.1
second time step, IBR pressure was applied in the first 0.1 seconds, then held
(A = 0 at t = 0, A = 1 at t = 0.1, A = 1 at t = 1.1), then actuator pressures
were applied for the remaining 1 second (A = 0 at t = 0, A = 0 at t = 0.1, A =
1 at t = 1.1) using two distinct _Smooth Amplitudes_ , predefined 5th-order
polynomial curves. The choice of 1.1 seconds balances computational effort
while keeping model kinetic energy under 1% of total energy. To avoid inertial
effect and to ensure that steady-state deformation has been reached, each
maximum pressure is simulated separately.

Both _Pinned_ and _Encastered_ boundary conditions (BCs) were tested. The
change did not affect the results, so pinned BCs were applied to the final
simulations.

###  III-B Material Characterization

The sPAMs/PMs, cPAMs, and ePAMs all rely on geometric shape change for
contraction, and thus require a sturdy inextensible material. We chose a 70D
TPU-coated Ripstop Nylon because it easily bonds to itself when the coating is
heated, which we achieve through ultrasonic welding, and it does not rip at
the current working pressures. The fPAM relies on material anisotropy for
contraction. We chose a silicone-coated Ripstop Nylon because of its proven
anisotropic behavior in previous work [12]. The material properties needed for
FEA were measured using uniaxial testing to break on an Instron 5565 under
ASTM D882 measurement standard and rectangular test specimens (20
mm$\times$350 mm, 250 mm gauge length). Five specimens were tested per
material and per relevant orientation. The measured elastic moduli for the
Nylon and the silicone fabrics are shown in Table I for each orientation.

In the experiments, the sPAMs/PMs, cPAMs, and ePAMs and their respective IBRs
are all cut along the main axes of the textile (0∘ and 90∘). The _Material
Evaluation_ module on Abaqus CAE was used to fit the 0∘ and 90∘ uniaxial test
data of the TPU-coated Nylon to the _Reduced Polynomial_ hyperelastic
constitutive model, also known as Neo-Hookean model, with $N=1$, where
$U=C_{10}(\bar{I}-3)$, resulting in the following material parameters: C10 =
50.3, D10 = 0.

TABLE I: Elastic moduli for both materials used to fabricate four types of IBR actuators.  Material | $E_{0}$ | $E_{45}$ | $E_{90}$
---|---|---|---
| TPU-coated Nylon,
---
Quest Outfitters
215 MPa | - | 201 MPa
| Silicone-coated Nylon,
---
Rockywoods SCA
96.5 MPa | 4.05 MPa | 120 MPa
Figure 1: Top: uniaxial loading orientations; bottom: stress-strain curve for
Silicone-coated TPU; the simulation data in black matches the test data until
a validity threshold of roughly 24% strain.

The main axis of the fPAM is cut at 45∘, because the actuation mechanism
relies on anisotropy for the fPAM to contract and cause the IBR to bend. Given
that hyperelastic material models commonly used in FEA of soft robotics
actuators (second order polynomial [2], Yeoh [25], Ogden [26, 27]) do not
account for anisotropy [23], the material properties of the silicone-coated
Nylon were modeled using the built-in _*Fabric_ material model in
ABAQUS/Explicit. _*Fabric_ is a data-based model that uses uniaxial test data
along the fill and warp directions of a fabric, and its shear response. The
experimental data acquired as described above was thus implemented directly
into Abaqus for the fill and warp directions (0∘ and 90∘). The shear stress-
strain curve was calculated by using the 45∘ uniaxial test data as a bias-
extension test data (a common alternative to the picture-frame test) [28]. At
the center of the test specimens, the material is in pure shear and the shear
angle is directly related to $d$, the displacement during the uniaxial test:

$\gamma=\frac{\pi}{2}-2\arccos\left(\frac{L_{0}+d}{\sqrt{2L_{0}}}\right),$ (2)

where $\gamma$ is the shear angle and $L_{0}$ is the initial specimen length.
The shear force can be fit to uniaxial test data by relating it to the shear
angle and $d$ using the following equation:

Figure 2: Inflated beam robot (IBR) fabrication and dimensions overview; top
to bottom: schematic top, front, and side view (actuated) and main dimensions
for A. series pneumatic artificial muscle/pouch motor (sPAM/PM), sides welded
over two layers and taped onto the IBR, B. compression pneumatic artificial
muscle (cPAM), sides welded over four layers and directly onto the IBR, C.
embedded pneumatic artificial muscle (ePAM), two cylinders welded in a pattern
to integrate the actuator in the IBR surface, D. fabric pneumatic artifical
muscle (fPAM), glued using Silpoxy; the bold red lines indicate a _Pinned_
boundary condition in the model; all portions described as ‘Transfer Tape’ and
‘Weld Lines’ are defined in the model with _Tie_ constraints; the areas
denoted as ‘Silpoxy’ are modeled as a _Composite_ surface.

$\displaystyle F_{sh}(\gamma)$
$\displaystyle=\frac{1}{(2H-3W)\cos{\gamma}}\left(\left(\frac{H}{W}-1\right)\cdot
F\cdot\right.$ (3)
$\displaystyle\quad\left(\cos{\frac{\gamma}{2}}-\sin{\frac{\gamma}{2}}\right)-W\cdot
F_{sh}\cdot\left(\frac{\gamma}{2}\right)\cdot\cos{\frac{\gamma}{2}}\left.\vphantom{\frac{H}{W}-1}\right),$

where $H$ and $W$ are dimensions of the sample along and across loading-
direction respectively, and $F$ is the external load applied during the
uniaxial test. The fitting results are shown in Fig. 1. The model matches the
45∘ test data well up to roughly 24% strain; values beyond that differ due to
the assumption of thread inextensibility. Preliminary tests on the isolated
fPAM muscle (not attached to the inflated beam) showed that the maximum
contraction under the maximum pressure (30 kPa) were under the threshold,
ascertaining the validity of the model in these conditions.

###  III-C Actuator Designs and Mechanisms

The actuator dimensions relate to bending force and actuator footprint.
Considering $Pressure=\frac{Force}{Area}$, same area actuators have comparable
bending forces. Second, IBRs often use eversion actuation, where larger
foodprints mean higher eversion pressure (e.g., a wider actuator is harder to
evert). Comparing actuators of same footprint equals comparing actuators of
same bending efficiency.

The dimensions are given in Table II, and the design, bonding method,
actuation method, and physical prototypes are shown in Fig. 2. The fabrication
steps followed previous work (sPAMs/Pouch Motors [13], cPAMs [15], ePAMs [17],
fPAM [12]). For the sPAMs/PMs and the fPAM, the actuators are fabricated
separately from the IBR body and then attached using adhesive transfer tape
(3M, Saint Paul, Minnesota, United States) and Silpoxy adhesive (Reynolds
Advanced Materials, Broadview, Illinois, United States) respectively. For the
cPAMs and the ePAMs, the actuators are integrated on the surface of the IBR.
The cPAMs are three layers of fabric selectively welded to create a form of
origami pouch that unfolds under pressure. The ePAMs are created by
selectively welding two parallel cylindrical IBR bodies into an actuation
pattern. All four mechanisms are shown in the Front view of Fig. 2.

TABLE II: Dimensions for the sPAM/PM, cPAM, ePAM, and fPAM designs. | sPAMs/PMs | cPAMs | ePAMs | fPAM
---|---|---|---|---
$\text{L}_{\text{IBR}}$ | 360 mm | 360 mm | 360 mm | 360 mm
$\text{D}_{\text{IBR}}$ | 80 mm | 80 mm | 80 mm | 80 mm
$\text{L}_{\text{actuator}}$ | 60 mm | 60 mm | 60 mm | 300 mm
W | 60 mm | 60 mm | 60 mm | 60 mm
f | - | 24 mm | - | -

###  III-D Experimental Evaluation

Figure 3: Experimental setup: A. schematic pressure control system; B. IBR
displacement measurement setup, including both air inlets (left), camera
(middle front), and checkerboard (right) for measurement calibration; white
dots show the six markers used for displacement measurement.

The experimental setup used to capture bending of the IBR and actuators is
shown in Fig. 3. The control system, in Fig. 3A, is attached to a wall
pressurized air inlet regulated manually to 100 kPa. The air is then split
into two QB3 solenoid valves that regulate pressure through digital sensors
placed close to the IBR and the actuators. A microcontroller (Arduino Uno)
records the pressure measurements to the valves and controlled the valves
based on a predetermined test sequence. An Intel Realsense D145 camera
captures RGB and depth images, using a checkerboard for calibration. Six
markers placed along the IBR and at the height of the middle of each actuator
pouch are used to measure displacement. The data acquired by the camera is
post-processed using Python and plotted using Matlab.

Previous work showed that gravity has only a negligible effect on the
measurements due to the light weight of the materials used in these
experiments [19]. The experimental and simulation setup are otherwise built to
be as physically similar as possible.

Each actuator type was tested using the protocol below:

  1. 1.

IBR pressurized at 2 kPa

  2. 2.

Actuator sequentially pressurized to 10, 20, and 30 kPa consecutively for 5
seconds each

  3. 3.

Actuator depressurized for 5 seconds

The 5-second duration for the inflation and deflation allow for the pressure
in the actuators to achieve steady-state, which provides a more accurate
reading of the pressure/displacement relationship. It is also close to what
the FE procedure simulates, where each pressure is simulated and held to a
threshold to avoid inertial effects.

###  III-E Metrics

The finite element model and experimental displacement data are compared based
on the XY-displacement of the markers. First, using a _Path_ along the center
line of the body of the IBR, the XY displacement coordinates vs. pressure are
extracted from the output database (ODB) of the simulations. They are then
plotted against the measured marker displacements at P = 10, 20, 30 kPa, using
the pressures measured at the inlet for the actuators and the final simulation
pressure. The accuracy is calculated as one minus the least square error
between the marker positions ($y_{\text{EXP}}$) and the simulation
displacement curve ($y_{\text{FEA}}$):

$a=1-\frac{1}{N_{\text{markers}}}\sum_{n=1}^{N_{\text{markers}}}\left(\frac{y_{\text{FEA}}-y_{\text{EXP}}}{y_{\text{FEA}}}\right)^{2}$
(4)

##  IV Results and Discussion

Figure 4: Results and comparison of experimental and FEA results for the four
actuator types; in each actuator section there is a photograph of the actuator
at 30 kPa (top, left), a capture of the von Mises Stress in FEA at 30 kPa from
the side and a cross-section (center), and a graph comparing the displacements
of the experiments (mean) and FEA in length and height at 10, 20, and 30 kPa
(right); the experimental error bar represents the variation from the mean;
the stress scale at the bottom is the same for all FEA results and capped at
20 MPa.  TABLE III: Accuracy of FE model and maximum experimental vertical
displacement for the
sPAM/PM, cPAM, ePAM, and fPAM actuators

| 10 kPa | 20 kPa | 30 kPa | $\text{d}_{\text{max}}$
---|---|---|---|---
sPAMs/PMs | 96.4% | 95.8% | 94.3% | 67.5 mm
cPAMs | 87.1% | 97.0% | 97.3% | 159.7 mm
ePAMs | 80.7% | 92.1% | 94.9% | 119.4 mm
fPAM | 97.6% | 97.3% | 96.7% | 67.2 mm

In this section, we first compare experimental and FEA results for individual
actuators, then compare the actuators to each other. We discuss differences
bending mechanisms, bending range, and their effect on the accuracy of the
simulation. We discuss the FEA strategy and its implications for the results.
Finally, we analyze how the FEA presented in this work contributes to the
fields of IBRs and Soft Robotics in terms of design and control.

###  IV-A sPAMs

The results for each actuator are shown in Fig. 4, and FEA accuracy values and
maximum displacements (at 30 kPa) are given in Table III. Starting with the
sPAMs (Fig. 4.A), the experimental and simulation data correspond very well,
with a minimum accuracy of 94.3%. The actuators are external to the IBR body
so their axis of contraction is further away from the IBR central axis, which
in turn limits the bending radius. The actuator location also implies that the
IBR cross-section is affected very little by bending. The stress peaks, very
similar in location and magnitude to those of the ePAMs, are at the weld
seams. The buckling pattern is irregular compared to the cPAMs and ePAMs,
which indicates that the effect of external actuators is less evenly
distributed and limits the bending range.

###  IV-B cPAMs

The cPAMs (Fig. 4.B) display the most substantial displacement and pronounced
curvature. Their maximum vertical displacement is 1.34 times that of ePAMs and
2.37 times that of sPAMs/fPAM. Embedded at the surface like the ePAMs, the
cPAMs are closer to the center axis of the IBR than the sPAMs and fPAM. The
added fabric folds lead to a 2.60-fold increase in their inner area compared
to ePAMs. These folds contribute to a dual buckling pattern: one between the
actuators (similar to ePAMs) and another at the actuator itself. Stress peaks
primarily emerge at weld lines, especially where four fabric layers connect.
The FEA accuracy of the cPAMs exceeds 97% for pressures over 20 kPa but drops
by 10% for lower pressures due to buckling-induced stress singularities. This
inherently unstable mechanism induces sudden and substantial bending changes.
At 10kPa, when the buckling pattern is only partially formed, deformation is
unstable and very sensitive to slight pressure variations. To support this
hypothesis, we show the cPAMs deformation at 11, 12, and 13 kPa in Fig. 5. A 3
kPa pressure difference increases maximum vertical displacement by nearly 30%.
Given that the experimental setup includes a margin of error for pressure
control, this explains the lower accuracy at 10 kPa.

###  IV-C ePAMs

Figure 5: FEA simulation values for the cPAM at 11, 12, and 13 kPa to test the
hypothesis that the FEA has lower accuracy at partially buckled states.

The ePAMs (Fig. 4.C) combine bending elements from the sPAMs and cPAMs. The
deformed actuator shape is the same as the sPAMs/PMs, and, as for the cPAMs,
the actuators are embedded directly on the IBR and rely on a regular buckling
pattern to deform. The displacement range of the ePAMs thus lies between the
two other actuator types. Because the bending mechanism is the same as the
cPAMs, the same stress singularities are found at low pressures ($<$20 kPa)
and partially buckled states. Although the ePAMs are slightly less accurate
than the cPAMs in our experiments, the fabrication of ePAMs is more
straightforward because it only involves welding two surfaces (no extra
folds), and they can easily be combined on the IBR surface for 3D control.

###  IV-D fPAMs

Finally, the fPAM (Fig. 4.D) has the smallest bending range of the four, given
that the bending mechanism relies solely on material contraction and that the
actuator axis is furthest away from the IBR center axis. However, the bending
mechanism is the most stable, and the FEA corresponds to the experiments with
over 96.7% accuracy for all pressures. The fabric model in Abaqus causes
oscillations in the results that can be mitigated by controlling the
simulation time step. The deformation amplitude is not affected, but the
computational cost of the FEA increases with increasing time step. A judicious
choice of time step and mass scaling will affect the run time and the
oscillation frequency. We find that the oscillations always have the same
amplitude, which corresponds to the experimental deformed state.

###  IV-E General Considerations

The mesh size significantly affects the accuracy of Finite Element Analysis
(FEA), especially for cPAMs and ePAMs that rely on buckling for bending. We
tested mesh sizes from 0.5 to 4.0 mm. While ePAMs and fPAMs were not
sensitive, accuracy for cPAMs and ePAMs increased by almost 20%. Finer mesh
led to deeper buckling patterns, increasing bending radius. However, finer
mesh also exponentially increased run times, so our chosen mesh balances
accuracy and efficiency. Users can adjust mesh size based on needing
qualitative insights or quantitative data for designing purposes.

Overall, the FEA results confirm the accuracy and potential of using FEA in
IBR design and deployment. Actuators differentiate based on bending mechanism
(geometry-based or material-based contraction), external/embedded system, and
inner surface area. Combining geometry and embedded actuation with a large
surface area provides the broadest deformation range but also shows more
buckling and inaccuracies at lower pressures. Material-based external
actuation is highly accurate but with a limited range. These insights help
roboticists pick optimal designs. For intricate path navigation without much
environment interaction, cPAMs excel. For minor obstacle navigation and tip
steering, fPAMs are simpler.

FEA is valuable for aiding IBR fabrication in multiple aspects. For instance,
the von Mises stress distribution can highlight potential rupture zones in a
design. If weld strength is known, ruptures can be predicted and managed via
design tweaks, material adjustments, or refining weld parameters
computationally. This accelerates the process, saving time and resources. Our
model stands out by offering this kind of support, motivating us to share it
openly online for others to use and improve, even introducing new actuator
types.

An additional advantage of using FEA modeling in IBR design is that very long
IBR models can be easily built and tested virtually. While other models tend
to mimic isolated IBR deformation, FEA allows interactions with the
environment through additional steps and timed external loads. By simulating
an environment virtually, the actuator choice and design can be tailored for
specific applications or experimental challenges an IBR might face.

##  V Conclusion

This work validates a new approach of modeling pneumatically-actuated inflated
beam soft robots, or IBRs using FEA. The _Dynamic, Explicit_ formulation of
FEA accurately captures complex buckling patterns and anisotropic material
models, suitable both geometry-based and material-based bending mechanisms.
The model converges accurately for the four primary types of pneumatic
actuators used with IBRs.

Combining buckling-based and embedded actuator design yields in the highest
bending curve and deformation range. However, stress singularities during
buckling at lower pressures affect model accuracy. Material-based deformation
retains over 96.7% accuracy at all pressures but has limited deformation
potential.

The FEA approach proposed here can be adapted to achieve qualitative and
quantitative results that streamline different stages of the IBR design
process. This tool is crucial for enhancing IBR efficiency, avoiding wasteful
ruptures or failures, and by tailoring actuator use and positioning to
specific applications.

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## Acknowledgment

The authors thank Alexander Kübler for the experimental procedure and
measurements performed with the sPAM/PM, cPAM, and fPAM.