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Continuum manipulators have gained significant attention in the robotic community due to their high dexterity, deformability, and reachability. Modeling of such manipulators has been shown to be very complex and challenging. Despite many research attempts, a general and comprehensive modeling method is yet to be established. In this paper, for the first time, we introduce the bending effect in the model of a braided extensile pneumatic actuator with both stiff and bendable threads. Then, the effect of the manipulator cross-section deformation on the constant curvature and variable curvature models is investigated using simple analytical results from a novel geometry deformation method and is compared to experimental results. We achieve 38% mean reference error simulation accuracy using our constant curvature model for a braided continuum manipulator in presence of body load and 10% using our variable curvature model in presence of extensive external loads. With proper model assumptions and taking to account the cross-section deformation, a 7–13% increase in the simulation mean error accuracy is achieved compared to a fixed cross-section model. The presented models can be used for the exact modeling and design optimization of compound continuum manipulators by providing an analytical tool for the sensitivity analysis of the manipulator performance. Our main aim is the application in minimal invasive manipulation with limited workspaces and manipulators with regional tunable stiffness in their cross section.
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ORIGINAL RESEARCH
published: 30 June 2017
doi: 10.3389/frobt.2017.00022
Edited by:
Helmut Hauser,
University of Bristol, United Kingdom
Reviewed by:
Andrew T. Conn,
University of Bristol, United Kingdom
Martin F. Stoelen,
Plymouth University, United Kingdom
*Correspondence:
S. M. Hadi Sadati
seyedmohammadhadi.sadati@
kcl.ac.uk
Specialty section:
This article was submitted to
Soft Robotics, a section of the
journal Frontiers in Robotics and AI
Received: 02 February 2017
Accepted: 12 May 2017
Published: 30 June 2017
Citation:
Sadati SMH, Naghibi SE, Shiva A,
Noh Y, Gupta A, Walker ID,
Althoefer K and Nanayakkara T (2017)
A Geometry Deformation Model for
Braided Continuum Manipulators.
Front. Robot. AI 4:22.
doi: 10.3389/frobt.2017.00022
A Geometry Deformation Model for
Braided Continuum Manipulators
S. M. Hadi Sadati1*, S. Elnaz Naghibi2, Ali Shiva1, Yohan Noh1, Aditya Gupta1,
Ian D. Walker3, Kaspar Althoefer 2and Thrishantha Nanayakkara4
1Center for Robotics Research (CoRe), Department of Informatics, King’s College London, London, United Kingdom,
2School of Engineering and Material Science, Queen Mary University of London, London, United Kingdom, 3Department of
Electrical and Computer Engineering, Clemson University, Clemson, SC, United States, 4Dyson School of Design
Engineering, Imperial College London, London, United Kingdom
Continuum manipulators have gained significant attention in the robotic community due
to their high dexterity, deformability, and reachability. Modeling of such manipulators has
been shown to be very complex and challenging. Despite many research attempts, a
general and comprehensive modeling method is yet to be established. In this paper,
for the first time, we introduce the bending effect in the model of a braided exten-
sile pneumatic actuator with both stiff and bendable threads. Then, the effect of the
manipulator cross-section deformation on the constant curvature and variable curvature
models is investigated using simple analytical results from a novel geometry deformation
method and is compared to experimental results. We achieve 38% mean reference
error simulation accuracy using our constant curvature model for a braided continuum
manipulator in presence of body load and 10% using our variable curvature model in
presence of extensive external loads. With proper model assumptions and taking to
account the cross-section deformation, a 7–13% increase in the simulation mean error
accuracy is achieved compared to a fixed cross-section model. The presented models
can be used for the exact modeling and design optimization of compound continuum
manipulators by providing an analytical tool for the sensitivity analysis of the manipulator
performance. Our main aim is the application in minimal invasive manipulation with limited
workspaces and manipulators with regional tunable stiffness in their cross section.
Keywords: geometry deformation, comprehensive model, compound structure, continuum manipulator, extensile
braided actuator, artificial muscle, Cosserat rod method, variable curvature
1. INTRODUCTION
Traditional limitations posed by conventional rigid linked robots, such as vast occupied space,
rigidity, and relatively low dexterity, have resulted in an emerging trend during recent years
for scientists to show increasing interest in the concept of continuum robots (Hirose and Mori,
2004). Taking inspiration from biological examples such as the octopus arms, chameleon tongues,
and elephant trunks, researchers are looking into the possibility of replicating similar maneuver-
ability and grasping characteristics by harnessing the corresponding hyper-redundancy demon-
strated in nature (Trivedi et al., 2008). This class of continuum robots promises considerable
performance improvements in different areas, which currently witness the presence of traditional
robots, such as surgical applications, underwater manipulation, repair, etc. (Jones and Walker,
2006;Mehling et al., 2006;Webster and Jones, 2010;Cianchetti et al., 2013, 2014;Maghooa
et al., 2015;Rus and Tolley, 2015). As a natural by-product of this trend, kinematic and static
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Sadati et al. A Geometry Deformation Model
modeling and analysis of these types of robots have gained recent
attention within the research community. However, the inherent
nature of continuum robots being highly deformable has posed
new challenges in this regard (Webster and Jones, 2010).
One of the most common assumptions in continuum robotic
research is the Constant Curvature (CC) model, which has been
discussed extensively in the research literature. The constant cur-
vature model simplifies the kinematics of a continuum manip-
ulator by assuming that the backbone kinematics in a planar
deformed state can be expressed by a constant curvature pro-
file. Webster and Jones (2010) deliver a thorough discussion on
this subject in their review paper. Based on the assumption of
constant curvature, the authors reviewed several methods for
kinematic modeling of continuum robots using two separate sub-
mappings: a robot-specific and a general one. The general robot-
independent mapping can suffer from singularity, as discussed in
investigations such as Godage et al. (2011a), where a new shape
function approach suggested by Godage handles this limitation.
Although being commonly used as a simplifying assumption,
the constant curvature assumption is usually not valid in the
presence of external forces. The exact dynamic models intro-
duced in the literature, which provide better modeling accu-
racy, can be categorized into four groups: (1) lumped models,
using Lagrangian dynamics, which includes a number of rigid-
link pieces combined with springs and dampers (Godage et al.,
2015). Continuum manipulators can be modeled by extending
the lumped model as in Tatlicioglu et al. (2007), where the total
kinetic energy is computed by considering an infinite number of
rigid sections and replacing the summation over the Lagrangian
terms with an integral over the backbone axis; (2) the Principle of
Virtual Work (PVW) (Trivedi et al., 2007;Sadati et al., 2016) or
simple Euler–Bernoulli beam model (Shapiro et al., 2011) using
CC for kinematic maps, where the CC kinematic map parameters
are considered to be the model states; (3) Cosserat rod models
resulting in a boundary value problem (BVP) that can be solved,
e.g., by using numerical methods for solving systems of non-linear
equations as in Trivedi et al. (2007) and Godage et al. (2016), or by
using a weak-form series solution in a discretized finite element
domain as in Tunay (2013); (4) approximate models to identify
the system behavior, which construct a setup-specific model using
very simple solutions such as a polynomial function (Chen et al.,
2009), or more complex solutions such as using series-based shape
functions (Godage et al., 2011a).
Each of these methods can be used as a basis for a numeri-
cal finite element solution, which is not discussed here (Duriez,
2013;Duriez and Bieze, 2017). The approximate identification-
based models, appropriate for real-time control purposes, are
more precise in predicting the identified system output and are
computationally efficient, but they are only valid for the con-
ditions, input type and input values they are trained for and
do not account for the structural characteristics. For example,
Godage et al. (2011a) used a horizontally fixed orientation to
train the coefficient matrices in their series solution-based model
for the kinematics of their setup but did not consider the effect
of external loads. Their model is singularity free and provides
95% accuracy for the conditions and inputs they trained their
solution for, and the final solution is computationally efficient
and faster than lumped model and Cosserat rod models; however,
it cannot guarantee accurate results for different orientation and
loading conditions than the training assumptions. Simple but less
accurate predictions can be made by models based on constant
curvature assumptions and be used as a reference for model-based
learning, control, and observation of continuum manipulators to
enhance the accuracy, generality, and identification time, espe-
cially in surgery applications, where observations are limited and
less reliable due to limited sensory equipment in the confined
space of surgery, lack of accessibility, and general uncertainties
related to sensing of a soft tissue (Khadem et al., 2016).
On the other hand, the lumped models and Cosserat rod mod-
els suffer from intensive calculations despite being more suitable
for design and optimization purposes. The majority of methods,
numerical inaccuracy, and singularities in deriving the inverse
kinematics are inevitable especially in the case of lumped system
models. Additionally, force estimation and control, which are an
essential part in aerospace, medical, and human–robot interac-
tion applications, are often hard to implement using the current
methods because of their limitations in modeling continuum
manipulators’ compound structures. The texture and flexibility
of soft robots match well with biological properties. Different
mechanisms to control soft robot stiffness for safe interaction and
minimally invasive applications are gaining increasing interest
recently. To this end, stiffness-tuneable structures by granular
jamming (Steltz et al., 2010;Jiang et al., 2012;Ranzani et al., 2015)
and low-melting point alloys (Cheng et al., 2014;Alambeigi et al.,
2016), morphing structures (Kuder et al., 2013), stiffness control-
lable interfaces by granular (Follmer et al., 2012;Stanley et al.,
2016), layer (Kim et al., 2013), and scale jamming (Hadi Sadati
et al., 2015;Santiago et al., 2016) are recently investigated. The
new interest in the continuum manipulators with stiffness varying
and inhomogeneous compound structures indicates the need for
further investigation of their modeling and control problems.
In order to fill the gap between approximate and finite solutions
to achieve comprehensive accuracy, as well as computational effi-
ciency, and to constitute a base to model compound and tunable
stiffness structure manipulators, we introduced a new geometry
deformation-based approach in our previous publication (Sadati
et al., 2016). In our previous work, we present an approximate
analytical model for compound continuum manipulators with
pneumatic braided extensor actuators in the presence of external
forces, utilizing experimental observation of the deformed sys-
tem to model the deformation energy of the continuum media,
using the principle of virtual work to account for the behavior
of compound structures, and the constant curvature assumption
for the deformation of the backbone as an initial but not essen-
tial assumption. There, we provided a new way to model pure
elongation of pneumatic braided extensile pneumatic actuators
using geometry deformation approach. The presented geometry
deformation model is based on a famous work by Rivlin (1949) on
“the problem of flexure” where he presents a geometrical approach
to derive the strain energy function for an incompressible highly
elastic cube under pure bending with certain geometrical assump-
tions about the deformed and initial states.
In this paper, we continue our previous work by present-
ing an exact yet simpler model for the actuator chamber braid
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Sadati et al. A Geometry Deformation Model
in elongation and bending with two types of braids, a highly
deformable and a stiffer braid. A shorter and simpler derivation
method is discussed for deformation energy of the continuum
media in the planar deformation case compared to the model pre-
sented in Sadati et al. (2016). Then, a more realistic solution with-
out the planar deformation restriction is introduced. Finally, two
comprehensive models for compound manipulators are discussed;
first, by using the principle of virtual energy, Neo-Hookean evalu-
ation of deformation energy, and constant curvature assumption
for the backbone deformation; and second, by employing Variable
Curvature (VC) kinematics, Hooks linear stress–strain relation,
and the Cosserat rod method for general bending of an externally
loaded continuum manipulator.
Our approach benefits from implementation of compound
structure complexities in the proposed model, i.e., braided cham-
ber and continuum media exact behavior modeling, and accurate
estimation of the cross section and backbone deformation by
combining geometry deformation and Cosserat rod methods in
a simple to derive and efficient to simulate procedure. In compar-
ison, the characteristic parameters of the chambers are not con-
sidered in most of the lumped system (Giri and Walker, 2011) and
Cosserat rod approaches despite some effort for modeling of the
braids in the pure elongation case (Trivedi et al., 2007). An exact
model for the cross-section deformation has three main benefits;
it increases the modeling accuracy for general design and control
applications (discussed in this paper), provides the necessary tool
for the trending research on the design of continuum manipula-
tors with tunable regional stiffness (Manti et al., 2016), and enables
exact planning for minimal invasive and safe robot–environment
interaction applications where the working space is limited, i.e.,
continuum manipulators in surgery (Cianchetti and Menciassi,
2017) and space applications (Cohen et al., 2016). The need for
comprehensive modeling of cross-section deformation of a con-
tinuum manipulator has been suggested in Shapiro et al. (2011)
too. Our comprehensive modeling tool for compound structures
provides a better insight in design, optimization, and control of
this class of mechanisms in a simpler, more efficient, and accurate
way, which is based on and in agreement with experimental
observations. Besides, comparing the accuracy and sensitivity of
the models help to understand what level of modeling complexity
is needed to incorporate effects of certain structural parameters
and achieve a certain accuracy in different applications.
We evaluate our simulation results against experiments on a
continuum manipulator with one STIFF-FLOP (STIFFness con-
trollable Flexible and Learn-able manipulator for surgical OPera-
tions) pneumatic actuator module (Cianchetti et al., 2013) shown
in Figure 1A having the structural properties presented in Table 1.
All the simulations are carried out based on the conditions and
inputs of the experiments.
In the following sections, first our experimental method and
the setup design are expressed in Section 2, followed by the general
description of our modeling framework in section 3. We start with
the constant curvature kinematics in section 3.1, the principle of
virtual work in section 3.2, and unit deformation energy in section
3.3 as the basis for our first modeling framework. The models
for a braided extensor actuator in simple elongation, general
elongation, and bending cases are discussed in section 3.4. The
geometry deformation method with planar and general assump-
tions for the cross-section deformationis discussed in section 3.5.
Our approach to model compound continuum structures using
the constant curvature assumption using the principle of virtual
work and the extension of the modeling tool to variable curvature
FIGURE 1 |A continuum manipulator with two STIFF-FLOP pneumatic actuator modules. Experiments on one module under body weight load of the next module
(A), module structure (B), experiments on one module under extensive external loads (C), a STIFF-FLOP module bending due to pressurization of one pneumatic
chamber (D), module cross section deforms from a perfect circular shape when bent (E), a STIFF-FLOP braided extensor actuator (F), the actuator bends instead of
pure elongation due to inhomogeneity in the braiding and tube molding (G), the tube cross section decreases and braid folds locally as the chamber elongates
because the thread cannot slide on the tube (H).
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Sadati et al. A Geometry Deformation Model
TABLE 1 |STIFF-FLOP parameters for the experiments.
Symbol [unit] ValueaValuebSymbol [unit] ValueaValueb
rc1 [mm] 2.5 2.5 rc2 [mm] 3.0 3.0
γ[°] 89.38 89 ro[mm] 18.0 17.0
r2[mm] 12.5 12.5 r1[mm] 4.5 4.5
l[mm] 44.0 44.0 lfs [mm] 16.6 10.0
E[kPa] 100 100 g [m/s2] 9.81 9.81
m[g] 24.0 24.5 mfs [g] 1.0 11.0
aBody loads.
bExtensive external loads.
The module parameters are slightly different due to fabrication inaccuracy.
kinematics using Cosserat rod method is explained in section
4. The simulation results and comparison between the models’
accuracy and sensitivity to structural parameters are discussed in
section 5. Finally, related discussion and conclusions of this work
are presented in sections 6 and 7. While analytic solutions are
presented throughout the paper, trapezoidal numerical methods
are used for evaluation of the integration as needed.
2. ROBOT HARDWARE DESIGN AND
EXPERIMENTAL PROCEDURE
We use STIFF-FLOP soft actuator modules to validate our model-
ing approach in this research. A STIFF-FLOP soft actuator module
is a three degree of freedom (DOF) pneumatic continuum actua-
tor with three braided extensor actuator chambers placed with an
offset from the module central axis and enclosed in a soft body
shell shown in Figure 1B (Cianchetti et al., 2013). Soft silicon
structures are molded using Ecoflex 50 (tensile strength 2.17MPa
and 100% modulus (E100%) 82.73 kPa from www.smooth-on.com)
(Cianchetti et al., 2013). Structural parameters are presented in
Table 1. Two different modules have been used for the tests in this
paper. They have almost identical dimensions but different active
length due to differences in their molding process. Synchronized
actuation of the pneumatic chambers causes the module to elon-
gate while asynchronized actuation causes it to bend laterally. The
module cross section deforms from a perfect circular shape shown
in Figures 1D,E. This occurs because of asymmetric actuation of
the pneumatic chambers, inhomogeneity of the module due to
molding imperfections, the flexibility of the body shell despite the
stiffening of the pressurized pneumatic chambers, and bending
of the module. STIFF-FLOP pneumatic actuator chambers are
McKibben like braided highly elastic extensile pneumatic artificial
muscles threaded helically with an ordinary sewing thread shown
in Figure 1F. The thread helix converts radial expansion of the
pressurized tube to axial deformation. An extensile chamber, with
a braid helix lead angle of more than 54.7°, elongates while a
contractile chamber, with a braid helix lead angle of less than
54.7°, shrinks as they are pressurized (Liu and Rahn, 2003;Pills-
bury et al., 2016). The thread constraints the radial and axial
deformation of the tube while it should be free to slide tangentially.
Braids in the well-known “OCT-Arm” series of continuum robots
introduced by Pritts and Rahn (2004) are to some extent free to
slide tangentially (Trivedi et al., 2007) while they are implanted
in the tube silicon body for the STIFF-FLOP. The threads may
fold locally if they are constrained tangentially to the body as
in the STIFF-FLOP case. A body shell is required to constraint
unwanted deformations of the chambers since the inhomogene-
ity of molding and braiding causes a single chamber to bend
randomly instead of pure elongation depicted in Figures 1G,H.
The STIFF-FLOP modules are highly flexible and sensitive to
changes in the input pressure. Our tests show good repeatability in
their actuation and fast linear response to the input pressure. The
molding process guarantees a robust module that can operate for
a long time; however, their long-term repeatability has not been
investigated yet. The handmade fabrication of the modules results
in structural and performance differences. Different volume ratios
of the silicone components, air bubbles trapped in the molding
process, imperfections in the radial and angular positioning of
the twin chambers, and small differences in the active length and
active surface area of the chambers due to excessive use of glue to
support the caps in some cases results in differences in structural
and performance characteristics of the modules. We carried out
our tests in this research using two different models of the STIFF-
FLOP manipulator that were different in the active length and
silicon composition (see Table 1).
STIFF-FLOP module pneumatic actuators are driven by a set
of ITV0030-3BS-Q compact pressure regulators (SMC Pneumatic
Ltd., Noblesville, IN, USA) connected to a BAMBI MD Range,
Model 150/500 pneumatic compressor (Bambi Air Compressors
Ltd., Birmingham, UK). A LabView program is designed to feed-
back control the pressure regulators through USB connection and
a USB-6211 DAQmx (National Instrument Ltd., TX, USA) data
acquisition board. We used an NDI Aurora (Northern Digital
Inc, ON, Canada) tracking system to record the movement of
each module tip. We herein present a new modeling approach to
facilitate design optimization and re al-time simul ation of this kind
of actuator in the following sections. Two sets of experiments are
carried out to validate the simulations and the manipulator tip
movement data are recorded.
One STIFF-FLOP module is randomly pressurized to investi-
gate planar deformation (Figure 1A).
One module is pressurized in two different steps while exces-
sive external loads at the tip cause large planar deformations
(Figure 1C).
3. MODELING FRAMEWORK
Our approach to modeling of a continuum manipulator consists
of a dynamic map (fD), solving the strain translational (ξ) and
rotational (ζ) rates based on the internal and external loads, and
a kinematic map (fK) finding the manipulator geometry based
on the strain rates. The forward and inverse dynamic maps can
be found based on various alteration methods; however, some
methods are better to evaluate the forward map, while others
are more suitable for the inverse map. The continuum form of
Lagrange EOM is better for the forward and inverse dynamic
maps, the Cosserat rod model is appropriate for the forward static
map, and it is more straightforward to derive the inverse static
map for a continuum manipulator using the principle of virtual
energy (Sadati et al., 2016). Static and quasi-static models are
investigated here as special cases of the dynamic model, where
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Sadati et al. A Geometry Deformation Model
the inertial forces are neglected assuming static equilibrium and
slow transitions in the system states. This is a reasonable assump-
tion since the application of continuum manipulator in precise
manipulation tasks, e.g., minimal invasive surgery as the main
application of the outcomes from this research involve less rapid
dynamic movements and more quasi-static transition of the states
in the task space (Cianchetti and Menciassi, 2017). Hooke’s law
of linear stress–strain relation is widely used in the dynamic map
(fD), while using Neo-Hookean assumptions and the principle of
virtual energy result in a more accurate static map (Gent, 2012).
A constant curvature assumption for simplicity and variable
curvature assumption for precise modeling are used to find the
kinematic map (fK). The manipulator backbone deformation has
been assumed to play the dominant role in the modeling of contin-
uum manipulators, and the cross-section deformation has usually
been neglected in the literature. However, as the manipulator
becomes softer and the cross-section diameter to the module
length ratio increases, the cross-section deformation becomes
more important. This cannot be neglected in case of emerging
studies on embedding regional stiffness-tunable structures in con-
tinuum manipulators (Alambeigi et al., 2016;Meerbeek et al.,
2016;Shiva et al., 2016).
Here, we use the principle of virtual energy to model the cross-
section deformation caused by constant curvature bending of
one manipulator module under general gravitational loads. The
use of Neo-Hookean and Hooke’s linear stress–strain relations
has been utilized in this case. This method makes modeling
of continuum manipulators with compound structures possible
and solves the problem directly for the inverse static map f1
K,
independent of the methods based on Newtonian dynamics such
as Cosserat rod theory. Subsequently, we combine our models
for the cross-section geometry deformation with the variable
curvature kinematics and Cosserat rod theory to find a more
accurate forward static map. This model improves the modeling
accuracy in the case of having large external forces. For this com-
bined model, we only use the linear Hooke’s law for stress–strain
relation.
3.1. Constant Curvature Kinematics
The kinematics of a continuum manipulator used herein is a
geometric map for n-modules (fK(n)) between the system state
parameters and the manipulator spatial orientation, usually in
Cartesian coordinates.
We start with the constant curvature assumption shown in
Figure 2A where the manipulator backbone geometry in Carte-
sian coordinates is given based on the curvature parameters ([κ,
ϕ,l], κ=1/rb) as the system states. This map is expressed in terms
of a set of transformations given by Ry(ϕ)ρtip Rz(κl)Ry(ϕ)
(Webster and Jones, 2010),
Ry(ϕ)=
C(ϕ)0 S(ϕ)
0 1 0
S(ϕ)0 C(ϕ)
,Rz(κl)=
C(κl)S(κl)0
S(κl)C(κl)0
0 0 1
,
(1)
FIGURE 2 |Parameters for elongation and constant curvature bending of a pneumatic actuator and body shell (A), parameters for a bent helix (B), planar and
general assumptions for cross-section deformation in elongation–bending based on geometry deformation method (C), and the parameters for the variable curvature
kinematics ([ξ,ζ]) and the curvilinear frames (di) expressed by a set of smooth continuous infinitesimal constant curvature segments (D).
Frontiers in Robotics and AI | www.frontiersin.org June 2017 | Volume 4 | Article 225
Sadati et al. A Geometry Deformation Model
ρtip =1C(κl)S(κl)0T.(2)
The transformation matrix for one module is
T(κ,ϕ,l)=Ry(ϕ)0
0 1.Rz(κl)ρtip
0 1 .Ry(ϕ)0
0 1,(3)
where Ry(ϕ)is added to correct the final cross-section orien-
tation because the module does not actually twist but bends
in the direction specified by ϕ. Then, the transformation vec-
tor of a point on the backbone with axial location siof the
nth module in a manipulator consisting of several modules is
T(sn)=n1
i=1
Ti(κii,li)Tn(κnn,sn), from which the position
vector (ρ(si))and orientation matrix (R(si))can be found. For
the backbone curve length of the three pneumatic chambers,
we have,
lc1 =l1κroC(ϕ)
lc2 =l1κroC(ϕ+2π/3)
lc3 =l1κroC(ϕ2π/3)
.(4)
We present a variable curvature method as a more accurate
solution in section 4.3.
3.2. Principle of Virtual Work
Among all possible changes in the states of a system, the system
follows the one set that minimizes the system action (w). This is
known as the principle of virtual work or the principle of least
action, which can be used to derive the system’s equations of
motion (EOM). The summation of all the virtual work in the
system maintains an equilibrium described as 0 =iδwifor
rigid body systems and 0 =vu.dvfor continuum systems. In
the case of STIFF-FLOP manipulator, we have
δwL+δww+δwE+δWc+δwG=0,(5)
where δwL=fLδρLis for the point forces (i.e., external loads,
body loads, and inertial forces) and δw=δUis for 3D distributed
energy fields (i.e., body deformation energy and air pressure work)
(Zienkiewicz et al., 1977).
Body loads are distributed forces and moments on the body unit
volume, such as the weight. The body load action can be found by
integrating the unit action over the volume (ww=
v
bg(s)dv).
It can also be calculated based on the virtual displacement of the
load’s center of distribution (center of mass (COM) in the case
of the weight). Neglecting the deformation of the manipulator
cross section and considering the constant curvature assumption,
the COM position vector of the nth module is from the post-
multiplication of the traversing modules transformation matrices
(Ti(κi,ϕi,li)), a rotation mapping to the bending plane (Ry(ϕ)),
and local position vector of the curve COM in the bending
plane as
ρCOMn=n1
i=1
Ti(κi, ϕi,li)
Ry(ϕ)0
0 1n(lSκl)
κl
(1Cκl)
κl0 1T
n.
(6)
Then, for the action, we have wW=
i
gbliρCOMi. Action
for external load (fL) is wL=
i
fLδρL. For pneumatic pressure
action, we have wG=pvG, where vGis the volume in the deformed
state and vGis equal to the inner volume of the pneumatic cham-
bers. The pneumatic chamber deformation is constrained with the
helical braids and the incompressibility of their shell.
Next, we need to derive deformation action due to the elasticity
of the module body and actuator chamber shells.
3.3. Unit Deformation Energy
Elastic deformation action of the continuum media (wE) can be
derived with good accuracy based on the Neo-Hookean relation
for large deformation (Gent, 2012) using the unit deformation
energy as
wE=v
uEdv,uE=E(I13)/6.(7)
The unit deformation energy in an orthogonal frame can be
derived based on the Cauchy–Green stretch tensor first invariant
I1=
3
i=1
λ2
i,(8)
in the case that stretch values along the principle axis are known
(Gent, 2012), or by having the general deformation map (η) in one
coordinate system, we have Rivlin (1949),
ηx=xdx, ηy=ydy, ηz=zdz,(9)
σxx =ηx,x+η2
x,x+η2
y,x+η2
z,x2,
σyy =ηy,y+η2
x,y+η2
y,y+η2
z,y2,
σzz =ηz,z+η2
x,z+η2
y,z+η2
z,z2,
I1=3+2(σxx +σyy +σzz).(10)
Alternatively, (wE) can be found based on Hooke’s linear
stress–strain relation law similar to an Euler–Bernoulli beam
(Gent, 2012). This is valid for small deformations; however, it
can be used in the case of infinitesimal elements along the body,
similar to the infinitesimal constant curvature elements in the
variable curvature kinematics discussed in section 4.3. Here, the
simple Euler–Bernoulli beam stress–strain relations are used by
replacing the cross-section moment of area in the local frame
(diag[Jxd,Jyd,Jzd]) based on the deformed geometry as
wE=1
2E(Jzdκ2+a(λl1)2)ld,(11)
where Jzdand ldare found from the geometry deformation maps.
We call this as the Hooke’s law-based model for the body defor-
mation action.
3.4. Braided Extensile Actuator
Braided extensor actuators are threaded continuum chambers
similar to McKibben actuators(Tondu and Lopez, 2000) except for
the fact that they elongate when pressurized (Liu and Rahn, 2003).
The thread constraints the curvilinear axial and radial deforma-
tion of the chamber shell but slips tangentially as discussed in
section 2.
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Sadati et al. A Geometry Deformation Model
3.4.1. Simple Elongation
The thread constraint for simple axial deformation is derived from
the geometry, assuming that the thread length (sth) and the helix
total twist angle (ψ) are fixed (Liu and Rahn, 2003) (Figure 2A).
For the unit length of the thread (dsth which remains constant)
in deformed state, we have dl2
cd+ (rthddϕ)2=ds2
th. Substituting
dlcd=λlcdlc,rthd=λrcrth, dlc=dsth Cγ, and rthdϕ=dsth Sγ,
where λrc=rthd/rth and λlc=lcd/lc, we get
λrc
2S2
γ+λlc
2C2
γ=1,(12)
A simplified assumption to derive pneumatic pressure action
for a STIFF-FLOP module is neglecting the pneumatic chamber
thin shell and assuming rG=rthd=λrcrth. Then, we have
wG=plcπr2
thλlcλ2
rc, λ2
rc= (1λlc
2C2
γ)/S2
γ.(13)
This is not valid for manipulators with a thick chamber shell
and without an encapsulating body shell, i.e., OCT-Arm (Trivedi
et al., 2007). Note that, a single actuator chamber cannot be
modeled by having the pneumatic pressure action only.
The typically adopted model in the literature (Liu and Rahn,
2003;Trivedi et al., 2007) assumes that the chamber shell volume
is fully constrained to the thread and all the body volume points
follow helical radial and axial deformations. We call this method
as the constrained volume model. The shell elastic deformation
action can be derived based on the incompressibility criteria
(λψc=1/(λlcλrc)),rG=rc1,rth =rc, equations (7) and (8)
and [λlc, λrcd, λψc]as the known stretches along the cylindrical
coordinate principle axes (Liu and Rahn, 2003;Trivedi et al.,
2007). Then, for the actions, we have
wc=Elcac
6ll2
cCγ21
Sγ2+Sγ2
λl2
c(λl2
cCγ21)λl2
c+3,
wG=pλlclcπr2
c1(λl2
cS2
γλl2
c+1)/S2
λ,ac=π(rc12rc22).
(14)
For one chamber, using equation (5) (δwc+δwG=0
(wcl+wGl)δλlc=0)and assuming λlcas the only system
state, we get (Trivedi et al., 2007)
p=E(rc12rc22)(λl2
c1)
6λl4
crc12(λl2
cCγ21)3
×(λl6
c(2Sγ6+5Sγ44Sγ2+1)
+λl4
c(7Sγ616Sγ4+11Sγ22)
+λl2
c(7Sγ64Sγ43Sγ2+1) + 3Sγ4).(15)
This assumption constraints the radial and axial deformation of
the chamber to the helix, and the chamber tangential deformation
is free and can be found based on the incompressibility crite-
ria. A small modeling error is observed because of non-perfect
slip between the thread and the chamber surface as indicated
by Trivedi et al. (2007). An alternative derivation for a braided
chamber with simple elongation is given in the previous work of
the authors using geometry deformation method, which results
in a more complex solution but with similar simulation results
(Sadati et al., 2016). It has been shown that γis negligible for
a dense braid in the case of STIFF-FLOP modules (Sadati et al.,
2016) and a simpler result is possible assuming γπ/2. We can
next deduce λ2
rc=1λ2
lc(π/2γ)2from equation (12) and can
further simplify the problem by assuming a fixed radius case for
γ=π/2 and λrc=1, which suggests that the tube radius does
not change and the tube only twists. In the case of fixed radius for
equations (14) and (15), we get
wc=Elcac(21l2
cλl2
c)/6,
wG=pλlclcπr2
c1,(16)
p=E(rc22rc12)(λl2
c1)(λl2
c+3)
6λl4
crc12.(17)
3.4.2. Elongation and Bending
The same constraint relation can be derived for a bending helix
assuming no twist as shown in Figure 2B to improve the models’
accuracy. The geometrical model for a bent helix is
xth =1(1rthCψ)Cθc,
yth = (1rthCψ)Sθc,
zth =rthSψ,(18)
where θc=κsc(ψ). The helix deforms due to the thread cross-
section pure torsion (α). We consider any constant curvature
elongation–bending deformation as a separate uniform elonga-
tion followed by a constant curvature pure bending. The uniform
elongation changes γuniformly, hence from equation (12), we get
cot(γe)=δsce/(rthdδψ),(19)
where δsce=λlcδscis the variation of the axial length due to
elongation, rthd=λrcrth is the deformed radius because of pure
elongation, γe=acos(lcd/lth)is the helix lead angle after pure
elongation, lth =lc/C(γ)is the thread length, which is fixed, and
lcd=λlclc.
In the case of pure bending, we assume B0as a uniform polar
rotational deformation rate on the thread cross section due to
external bending moment (τ) on the bent helix (similar to a bend-
ing moment effect τ/(J2th Gth), where τis a bending moment, sth is
the thread length, and J2th is the polar second moment of area for
the thread cross section), the change in the cross-section torsion
angle is δα(ψ)=B0C(ψ)S(γ(ψ))δsth, where δsth =rthdδψ/S(γ(ψ))
is the thread length element in axial direction and rthd=λrcrth is
the deformed helix radius due to pure elongation. Here, λlcand
λrcare the stretch ratios caused by pure elongation. Then, the
variation of the bending angle (δθ(sc(ψ)))along the axis becomes
δθc(sc(ψ))=κδscb(ψ)=δαC(ψ)S(γ(ψ))=B0C2
(ψ)S(γ(ψ))rthdδψ,
(20)
where δscb(ψ)is the variation of the axial length due to bending.
Using the geometry of the local lead angle (γ(ψ)) and equations
(19) and (20), we have
δsc(ψ)=δsce+δscb(ψ)=cot(γ(ψ))rthdδψ,
= (B0C2
(ψ)S(γ(ψ))+cot(γe))rthdδψ, (21)
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Sadati et al. A Geometry Deformation Model
which results in a biquadratic quartic form equation for S2
(γ(ψ))
that can be solved as (Rees, 1922)
S2
(γ(ψ))=
κ(4B2
0C(ψ)4+4cot(γe)B0κC(ψ)2+κ22B0cot(γe)C(ψ)2κ)
2B2
0C(ψ)4,
(22)
where γe=acos(lcd/lth)is used to substitute for cot(γe). This is
considered as the exact solution for γ(ψ). This exact model gives
complex solutions for B0and γ(ψ)in the following steps. There-
fore, we assume γ(ψ)π/2 [only in equation (20)], which results
in a simpler solution as δθc(sc(ψ))=κδscb(ψ)=B0C2
(ψ)rthdδψ.
Integrating this w.r.t. ψ, we get
θc(sc(ψ))=rthdB0(ψ+S(2ψ)/2)/2,(23)
where B0is found from θc(lcd)=λlclcκand lcd=λlclc=sc(2nthπ)
as
B0=2λlclcκ/(λrcrth(2nth π+S(4nth π)/2)),(24)
where nth is the number of full turns. Using equations (21), (23),
and (24) and substituting for cot(γe), the simplified solution for
γ(ψ) becomes
cot(γ(ψ))=B0C2
ψ+cot(γe).(25)
In initial state, when the helix is straight, the local (dlc(ψ)) and
backbone (dlc) axial unit lengths are equal. In the deformed bent
state shown in Figure 2B, we have
dlcd
dlc(ψ)d
=λlc
λlc(ψ)
=1
1κrthd(ψ)Cψ
.(26)
Similar to equation (12), in the local curvilinear frame, we have
λ2
rc(ψ)S2
γ(ψ)+λ2
lc(ψ)C2
γ(ψ)=1.(27)
By substituting rthd(ψ)=λrc(ψ)rc, equations (25) and (26) in
equation (27), we obtain a quadratic function for λrc(ψ)
0=A12λ2
r(ψ)+A11λr(ψ)+A10 ,
A12 =
κ2lc2λlc4C(ψ)2(C(2γ)1)(8nthπC(ψ)2+S(4πnth )
+4πnth)2
4nth2π2(S(4πnth )+4πnth )2(λlc2C(2γ)+λlc22)+1,
A11 =κlc2λlc4C(ψ)(C(2γ)1)(8nthπC(ψ)2+S(4πnth )+4πnth)2
2nth2rcπ2(S(4πnth ) + 4πnth )2(λlc2C(2γ)+λlc22),
A10 =lc2λlc4(C(2γ)1)(8nthπC(ψ)2+S(4πnth )+4πnth)2
4nth2rc2π2(S(4πnth )+4πnth )2(λlc2C(2γ)+λlc22)
lc2λlc2(C(2γ)1)(8nthπC(ψ)2+S(4πnth )+4πnth )2
4nth2rc2π2(S(4πnth )+4πnth )2(λlc2C(2γ) + λlc22)1.
(28)
The final result is presented in Appendix A.1 due to space
limitation. We call this as the exact helix model for λrc(ψ).
The exact helix model is valid for stiff threads and springs;
however, the softer braids may bend as well as rotate. A sim-
pler result is possible if we assume that the lead angle with the
curvilinear axis is constant and equal to the lead angle after pure
elongation (γ(ψ)=γe). This model assumes that the thread tends
to retain its lead angle and follows the shell deformation without
any limitation other than the thread length. From equations (26)
and (27), we have
0=λrc(ψ)2+cot(γld)2(λl2
cκλrc(ψ)rcC(ψ)121)1,
λrc(ψ)=
λl2
ccot(γld)2(κ2r2
cC(ψ)2csc(γld)21)
+csc2
(γlrmd )
+κrcC(ψ)cot(γld)2λl2
c
cot(γld)2κ2λl2
cr2
cC(ψ)2+1,
(29)
which we call the constant lead angle bending model.
As in the case of pure elongation, actions can be found based on
incompressibility criteria λψc(ψ)=1/(λrc(ψ)λlc(ψ)),rG=rc1,
rth =rc2, equations (7) and (8) as
wc=lcdrc2
rc1 2π
0
ucdaG,wG=plcdrc1
02π
0
daG,(30)
daG=rcλrc(ψ)dψdrc,uc=E(λ2
ψc(ψ)+λ2
rc(ψ)+λ2
lc(ψ)3)/6,
(31)
where the integrals are dealt with numerically in the simulations.
The actions depend on κ, hence a standalone relation for one
chamber is not derived here. A similar result to equation (13) in
the case that we neglect the chamber shell can be derived by sub-
stituting λrcwith λrc(ψ). Assuming a dense thread (γ(ψ)π/2)
does not reduce the complexity of the final solutions in equa-
tions (28) and (30) significantly; however, the fixed radius case
(γ=π/2), λrc(ψ)=λrc=1 results in a simpler solution for the
actions as
wc=
πElc(κ2λl2
c(rc22rc1 2)(λlc
2(κ2(rc22rc1 2)
+4) + 8) + 16)
24κ2λl2
c
,(32)
wG=pπlcλlcrc12.(33)
An alternative derivation for a bent braided pneumatic actuator
is presented based on the deformation geometry approach in the
section 3.5.2.
3.5. Geometry Deformation Method
The unit deformation energy for simple pure elongation of a
symmetric cylinder is presented in Gent (2012) based on incom-
pressibility criteria and principal stretch ratios. We use equations
(7) and (8) and
I1=λ2
l+2l, λr|ψ=1/λl,(34)
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Sadati et al. A Geometry Deformation Model
to derive the unit deformation energy for the pure symmet-
ric elongation case, where λiare the stretch ratios along other
principle directors. This can be extended to a simple model for
elongation–bending of a cylinder based on equation (26) as
λy(x)= (1κx)λl, λx(x)=λz(x)=1/λy,
I1(x)= (1κx)2λ2
l+2
λl(1κx).(35)
This model accounts for the inhomogeneous elongation of the
volume elements but does not consider their bending. We call this
as the simple bending model.
We use a similar approach as the solution by Rivlin (1949) for
“The problem of flexure” to find a simple but exact analytical solu-
tion for the general deformation map of a bent hollow cylinder.
We assume that the continuum bodies as an incompressible media
and the cross-section plane remains planar under deformation
as in the Euler–Bernoulli beam assumption. Therefore, the final
orientation of the deformed cross-section plan is determined by
the deformation of the backbone. The deformed state for each
point ρd= [xd,yd,zd]T=Tbρconsists of the deformation
of the backbone in the form of a transformation matrix (Tb)
and a vector function presenting the planar deformation of the
cross section ρ= [x,y,z]T. Assuming constant curvature,
neglecting the pneumatic chamber holes, and dealing with the
deformation of each module separately, we have
Tb=Rz(κl)ρtip
0 1 .(36)
The deformation map becomes,
xd=Cθx+ (1Cθ)/κ,
yd=Sθx+Sθ/κ,
zd=z,(37)
where θ=y*κ. Incompressibility criteria hold if the determi-
nant of the deformation map Jacobian w.r.t. the initial states
(ρ=[x,y,z]) becomes one (Rivlin, 1948),
|ρd|=1.(38)
The above relation is derived and then solved for ρ*
following some assumptions on the shape function variable
dependency and boundary conditions. Our general assumptions
are having a fixed backbone on xz-plane (ρ*(0) =0), symme-
try w.r.t. yz-plane z
(z=0)=0and having the neutral plane
along the main axis and perpendicular to the curvature radius
x
(x=0)=0 and z
(x=0)=z; however, we drop some of these
assumptions in different models. We discuss the solution based
on two main assumptions.
3.5.1. Planar Deformation
Here, the deformation only occurs in the xy-plane. Assuming no
deformation in zdirection (z*=z) similar to Rivlin (1949), we can
separate the variables as x*(x,z), y*(y), and z*(z). From equation
(38), we get
x,xCκy(1κx)y
,ySκyx,zCκy
x,xSκy(1κx)y
,yCκyx,zSκy
0 0 z,z
=x,xy,yz,z(1κx) = 1.(39)
By separation of variables, we have
x,x(1κx) = 1/(y
,yz
,z) = 1/(λ2λ3)
y
,yy =0y=A2y+B2
z,zz =0z=A3z+B3.
(40)
By integrating for x*, we get
x=112κ(A1x+B1)
κ+h1(z),A1=1
A2A3
,(41)
where considering the fixed backbone root, yz-plane symmetry,
and neutral plane, we have B1=B2=B3=h1(z)= 0. Then, for I1and
wEfrom equations (7), (9), and (10), we have
I1=A34λl22κx A33λl+A32λl42κx A3λl3+1
A3λl(A3λl2κx).(42)
I1is not a function of y, and the integration for wEis dealt with
numerically. From the neutral plane assumption (z
(x=0)=z)
and for the planar elongation–bending deformation case, we get
A3=1; however, in the case of planar pure bending after a pure
elongation, from equation (34), we have z
(x=0)=z/λl, hence
A3=1/λl. A weak approximate analytical solution is presented
in Sadati et al. (2016). The elastic deformation action for the body
shell as a hollow cylinder is
wE=wE(r2)wE(r1).(43)
3.5.2. General Deformation
A more realistic model can be derived based on a more general
assumption for z
(x,z). Then, similar to equations (39)–(41), we
have
1=
x,xCκy(1κx)y
,ySκyx,zCκy
x,xSκy(1κx)y
,yCκyx,zSκy
z,x0z,z
=y
,y(x,xz,zx,zz,x)(1κx).(44)
By separation of variables and from the result for x* in equation
(40), the fixed backbone root, yz-plane symmetry, and neutral
plan assumptions, we have
x= (112κ(h2(x)+B1))
y
,yy =0y=λly
h2(x),xz,z=1l.
(45)
To satisfy xy-plane symmetry neutral plane assumptions and
reach a realistic cross-section deformation similar to Figure 2C,
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Sadati et al. A Geometry Deformation Model
we propose z=A3z/1κxbased on the simple elongation–
bending model in equation (35). Then, we have
h2(x)=2A1(1κx)3/2
3κ,B1=2A1
3κ,(46)
where A1=1/(λlA3). For I1, we get
I1=12B55+4A33B4λlB534A3B4λl3B54+A33B4κ2λlz2B5
A3B4λlB54,
(47)
where B4=4(1κx)3
2+3A3λl4 and B5=κx1. Similar
to the planar pure bending case in the last section, we assume that
A1=A3=1/λl. A more general solution for the braided case
is hard to achieve and has not been considered here.
4. CONTINUUM MANIPULATOR MODELS
4.1. Constant Curve Model Implementation
In the next step, we use different combinations of the discussed
methods to model the pneumatic chamber elastic deformation,
gas pressure, and a body shell elastic deformation actions for
a STIFF-FLOP module. The inverse map that results from the
principle of virtual energy is used to predict the required pressure
for any constant curvature geometry. We substitute rod=ro/λ
for roin equation (4) to approximate the deformation of the
placing radius for the actuators too. The algorithm to imple-
ment the method is presented in Algorithm 1. The results are
compared to the experimental measurements, and the advantage
of the geometry deformation approach in increasing the model
prediction accuracy is presented in Table 2.
4.2. Extension to Variable Curvature Model
The solutions provided in the previous sections improve the mod-
eling of the cross-section deformation with constant curvature
assumption. This solution can be extended to the variable cur-
vature backbone model, where the backbone consists of a series
of infinitesimal constant curvature curves (Trivedi et al., 2007;
Burgner-Kahrs et al., 2015;Neumann and Burgner-Kahrs, 2016).
The variable curvature assumptions improve the accuracy of the
backbone deformation modeling, and the presented methods in
this research can improve the associated cross-section deforma-
tion models.
4.3. Variable Curvature Kinematics
Variable curvature kinematics presents the relation between local
curvilinear frames with normal (d1), tangential (d2), and binormal
(d3=d2×d1) unit vectors along the backbone of a continuum
media (Figure 2D). The backbone is considered to be consisting
of a series of infinitesimal length constant curvature curves along
the main axis. The translational (ξ) and rotational (ζ) strain rates
w.r.t. the unit length along the manipulator backbone curve (s) in
the local curvilinear frames are found from a dynamic map based
on Cosserat rod model or Euler–Bernoulli rod model as a special
case with infinite shear modulus.
ALGORITHM 1 |The pseudo code for a Matlab program to derive the inverse static relation for a continuum manipulator using the principle of virtual energy and constant
curvature assumptions.
Data: par—structural parameters, mthd,mthdc—modeling method for body and chamber
Result: inverse model for a continuum manipulator based on the principle of virtual work
initialization;
[vG,wT]= jacobian(w_func(par,mthd)); %numerical jacobian
return p=v1
G.wT; %inverse map
Function [vG,wT]= w_func(par,mthd) {
Data: par—structural parameters, mthd—modeling method, ρL—external load position vector
Result: actions and actuator volume
ψc= [0 2π/3 2π/3]; %angular position for pneumatic chambers
For i—number of pneumatic chambers do
lc=l(1κroC(ψ+ψc));
[vGt,wENH t,wEEBt]= action_fun(lc,par,mthdc); %wfor pneumatic chambers
vG(i)=wG; %gas and chamber shell actions
wc+ = wcENHt;
[vGt,wcENH t,wcEEBt]= action_fun(lc,par,mthd); %wfor the chamber holes in the body shell
wh+ = wcENHt;
end
[vGt,wENH t,wEEBt]= action_fun(l,par,mthd); %wfor the body shell
wE=wENHtwh;
T:unity matrix; %initialize
for n—number of modules in a manipulator do
T=T.TCOM(n); %COM transformation matrix calculation ww=mgCOM; %weight action
end
for nLnumber of external loads do
wL= +fLρL; %external load action
end
wT=wE+Wc+wL+ww; %total action
return [vG,wT]}
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Sadati et al. A Geometry Deformation Model
TABLE 2 |Performance comparison between different models for random pressurization of one STIFF-FLOP module under body weight loads vs. 43 experimental data
points, and under an extensive external load vs. 33 experimental data points.
Body shell Pneumatic chamber Abs. error Error (%) Ref. error (%)
Model Model CC [kPa]–VC [mm] CC–VC CC–VC
Elong. CG simple sym. Fixed radius 60–7.6 74–10 60–12
Elong. CG simple sym. Elong. braided 111–9 179–12 113–14
Bend. Eu–Be fix section Fixed radius 61–13 75–17 61–20
Bend. CG simple model Elong. braided 95–9 118–12 95–14
Bend. GD planar Bend. const.γ40–10 65–14 40–16
Bend. GD general Elong. braided 53–14 93–19 53–22
Bend. GD general Bend. exact helix 38–7 62–10 38–11
Bend. GD general Bend. const. γ38–6.6 61–9 38–10
Error percentage is used for comparison between the CC and VC models, and reference percentage error is used for comparison with previous research as in Trivedi et al. (2007) and
Godage et al. (2011a).
A set of rotation matrices (R(s)) gives the relation between
the local transnational strains ρr(s),s=ξ+ [0,1,0]Tand the
deformation rates of the spatial geometry of the manipulator
ρ(s),sas
ρ(s),s=R(s)r(s),s=R(s).(ξ+ [0,1,0]T),(48)
where ξis the strain rates in the local curvilinear frame, which the
rotation matrix R(s)rotates to be expressed in the reference Carte-
sian frame. This rotated representation is equal to the variation of
the position vector from one local frame to the next one (ρ(s),s)
along the backbone unit length (s).
R(S)is found from the local rotational strain rates (ζ). The
rotation matrix variation along the backbone unit length (R(s),s)
can be found as
R(s),sζ]×
.R(s),(49)
where []×is an operator to create a skew symmetric matrix. Equa-
tion (49) presents the kinematic relation between the rotation
matrix variation (R(s),s) and the rotational strain rates expressed
in the local frame (ζ) along the manipulator backbone curve
(Neumann and Burgner-Kahrs, 2016). Integrating equation (48)
for ρ(s)and equation (49) for R(s)presents the variable curvature
kinematics. The next step is finding the local strain rates based on
the system loads from the dynamic map.
4.4. Beam Theory (Cosserat) Model
A Cosserat rod model can be used to derive the dynamic map.
Cosserat rod models derive the equilibrium between the forces
on each infinitesimal element of a continuum media based on
Newtonian approach and a free force body diagram of each ele-
ment (Trivedi et al., 2007). Then, the inter-element load effects are
replaced by the resultant stresses. Hooke’s law is usually used to
relate these stresses to the strain rates. Rearranging the derivation
to find the strain rates based on the loads, results in a boundary
value problem with partial differential equations (PDE) of order
two that needs to be integrated over volume and time. This
derivation becomes simpler for the static planar case (Trivedi et al.,
2007).
The general derivation of the Cosserat rode method can be
found in the literature (Trivedi et al., 2007;Tunay, 2013;Neu-
mann and Burgner-Kahrs, 2016) and not presented here. The
Euler–Bernoulli beam model is an alternative approach to derive
the same final relation in the form of a PDE of order one, where
the resultant stresses are found from the total load exerted on each
cross section from one side of the beam. The loads are usually
calculated based on the loads from the free end side of the beam.
The Cosserat rod model reduces to the Euler–Bernoulli beam
model in the case of infinite shear modulus. The Euler–Bernoulli
method results in the following static map for a planar continuum
manipulator in the static case,
ξ=
Gad0 0
0Ead0
0 0 Gad
1
.RT.(fL+fb) + fpd,
ζ=
EJ1d0 0
0GJ2d0
0 0 EJ3d
1
.RT.(τbd+τLd) + τpd(50)
where fb(s)=ba(ls).gis the body weight force, fpd=pacd.d2
is the pneumatic pressure force, g= [0, g, 0]Tis the gravity
acceleration vector (upward in the simulations), τLd= (ρLd
ρd(s))×fLis the external load moment, τbd=ba l
s(ρd(s)
ρd(ε))dε×gis the body weight moment, τpd=apdrodp.
[C(φ)C(φ+2π/3)C(φ2π/3)]T.d3is the pneumatic pressure
moment exerting at the manipulator tip, and rod=ro/λlis
the deformed placement radius for the pneumatic chambers, all
in deformed state. The deformed values in each element along
the backbone curve are found by substituting λt=ξ2+1 and
κ=ζ3/(ξ2+1)in equations from section 3.4 and 3.5. Substituting
equation (50) in equations (48) and (49) and integrating the result
give the full model for the manipulator. We use the backbone area
and second moment of inertia for the deformed body to account
for the cross-section deformation.
5. SIMULATION AND COMPARISON
We investigate the accuracy of the simulations from different
models in predicting the experimental results and the sensitivity
of each model to structural and kinematic parameters. The com-
parison between the models helps to understand how the model-
ing accuracy improves by increasing the mathematical derivation
complexity and to find the proper model to observe certain behav-
iors, incorporate effects of important structural parameters, and
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=[
Sadati et al. A Geometry Deformation Model
achieve a desired modeling accuracy. Modeling absolute error is
defined as the positive value of the difference between the output
vector (pressure (p) for the inverse model (section 5.4) and tip
position vector (ρtip) for the forward model (section 5.5)) from
the model simulation and the experiments. Error percentage is
found by dividing the absolute error by the vector value from
the experiments as the reference value and used as a means
to compare constant curvature and variable curvature modeling
results despite their differences. To make comparison with similar
research easier, a reference error percentage is defined where the
absolute error is divided by a structural parameteras the reference
value (Efor the inverse model (section 5.4) and module length
(l+2lfs) for the forward model (section 5.5) (Trivedi et al., 2007)).
We use error percentage to compare different models in this
research; however, the reference error percentage should be used
for comparison to results from other similar research.
5.1. Helix Lead Angle Models
Simulation results for γ(ψ)in terms of different values of γ,κ,
and λlcshow considerable changes in the local lead angle, w.r.t.
the initial lead angle (γ) and the uniform changes in the lead
angle after pure elongation (γe), even for γπ/2 (Figure 3A).
The variations from the base lead angle (γ) increase significantly
as the γitself decreases and as λlcincreases. However, it decreases
slightly when κincreases and there is no notable variation w.r.t.
θ=κlc.γproduces the dominant effect and enhances the effect
of other parameters as it decreases. Results from the exact model
[equation (22)] and the simplified model [equation (25)] are
almost identical while the simplified model predicts slightly larger
variations in the local lead angle; Hence, we continue with the
simplified model. The small lead angle assumption (γ(ψ)π/2)
in all the derivations causes a larger error w.r.t. the exact model
and predicts more local change in γ(ψ). With local changes in γ(ψ),
the errors of the three models remain small for γe>80°, and they
have identical results for γ=π/2 as expected.
5.2. Comparison of Deformation Models
for a Braided Actuator
Simulation results for dif ferent derived models of λrc(ψ)and cross-
section shapes of a STIFF-FLOP pneumatic chamber in terms
of γ,κ,θ, and λlcare presented in Figures 4A,B. The exact
helix model as in equation (28) predicts the largest variation
from the circular shape such that the cross-section radius will
shrink more in both the inner (concave) and outer (convex)
sides of the curve in the bending plane and w.r.t. the tangen-
tial plane. The deformation of the radius on the inner side is
smaller than the deformation of the outer side, and this difference
increases as either θor κincreases, in addition the deformation
magnifies when γdecreases and/or λlcincreases. However, it
decreases when κis increased for a constant θand increases
when κis increased for a constant λlc. The most significant
parameter on the cross-section asymmetric deformation is the
bending angle (θ), and its effect is enhanced by the braid helix
lead angle (γ) decreasing as well as the axial stretch ratio (λlc)
increasing. The observed effects intensify for bigger radii. The
small lead angle case (γπ/2) for the exact helix model predicts
a larger uniform decrease in the shell radius, which we found
unsuitable for modeling purposes and is not presented in the
graphs.
Despite the exact helix model, the constant lead angle assump-
tion [equation (29)] predicts an almost uniform shift in the cross
section toward the bending axis. The structural parameters have
almost the same effects on the deformation predicted by this
model as discussed for the exact helix model; however, their
effects are not significant. The constant lead angle model in the
general case corrects the pure elongation general case, and the
small lead angle case (γπ/2) corrects the small angle case for
the pure elongation. The small lead angle models predict a slightly
smaller radius compared to the general cases; however, they are
not significantly different from the fixed radius models (γ=π/2)
and from each other in terms of the prediction of the shell radius
deformation. Assuming γπ/2 results in slightly larger predic-
tions for radius and thread local lead angle. The results are in
good agreement with the exact models for γe>80°; however,
the simplification does not reduce the complexity of derivations
significantly.
The exact helix model assumes the braid helix deforms due
to pure torsion of the thread cross section, similar to a stiff
spring, without any bending. This assumption for the constant
lead angle model is not valid, and this model’s prediction is similar
to the result from the pure elongation of a helix [equation (12)].
We suggest using the exact helix model for the chambers with
stiff braids, and using the constant lead angle model for more
deformable braids, i.e., sewing threads, where the deformation of
the chamber shell is more dominant. The deformation of the cross
section and the difference between the models become noticeable
when γ > 1.309 rad or λlc>1.8. However, to understand the
significance of this difference on the static model of a cham-
ber, we investigate the action predicted by any of the models
later.
Note the different predicted profile for a module in nearly pure
bending with a big radius (λlc=1.1187 and θ=1.0482 [rad]) in
Figure 4B compared to amo dule with smaller radius in Figure 4C.
Different predicted deformations based on different models for
the outer and the inner radius of a braided pneumatic chamber
with inner and outer diameter of, respectively, 12 and 18 mm and
γ=1.48354 rad are presented in Figures 3B,C. Results for a body
shell without braids are presented for comparison purposes too.
The deformation predicted by all the braided models other than
the bending model with the exact helix γassumption is almost
identical to the prediction of the constant radius model for this
γ. A closer look shows a small decrease in the radius in case
of pure elongation of a helix. The radius changes somewhat for
the bending model with the constant γassumption and slightly
expands inward. The bending model with the exact γassumption
shows a helix that does not allow lateral deformation other than
the uniform reduction due to the elongation of the axis. It causes a
change in the twist angle at the back side of the bent where two
consecutive rounds of the helix have different local lead angles
and create a heart shape. While a tube inside a helical spring
behaves similar to the exact γassumption model, the thread tends
to follow the deformation of the body shell for the softer braids
and to maintain γ; therefore, it behaves similarly to the constant γ
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Sadati et al. A Geometry Deformation Model
A
B
C
FIGURE 3 |The change in the local lead angle (γ(ψ)) of a pneumatic chamber’s helical thread in terms of different values of γ,κ, and θ=κlcdwhere λlc=1.2418
[(A), left] and γ=1.4835 rad [(A), right]. The model with dense thread (small γπ/2) tends to predict more change in γ(ψ)(A). The change in the outer radius
stretch ratio (λrc(ψ))[(B), left] and shell cross-section deformation [(B), right] of a pneumatic chamber for rc2 = 12 mm, λlc=1.8368, κ=27.8036 rad,
θ=1.0482 rad, and γ=1.4835 rad (B). The change in the inner radius stretch ratio (λrc(ψ))[(C), left] and cross-section deformation [(C), right] of a pneumatic
chamber where rc1 = 8 mm, λlc=1.5549, κ=41.6922 rad, θ=1.0482 rad, and γ=1.4835 rad (C). Note how the cross-section shape changes based on different
model assumptions.
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Sadati et al. A Geometry Deformation Model
0 2
0.9
0.95
1
rc( ) ( = 0.001,
= 0.026525) [rad]
lc
= 1.1187 [rad]
0 2
0.6
0.7
0.8
0.9
1
( = 0.026525)
lc
= 1.5549 [rad]
0 2
0.5
0.6
0.7
0.8
0.9
( = 13.9151)
lc
= 1.8368 [rad]
0 2
0.6
0.7
0.8
0.9
1
( = 13.9151)
0 2
0.8
0.9
1
rc( ) ( = 0.5246,
= 13.9151) [rad]
0 2
[rad]
0.8
0.9
1
1.1
1.2
rc( ) ( = 1.0482,
= 27.8036) [rad]
0 2
[rad]
0.6
0.7
0.8
0.9
1
( = 27.8036)
0 2
[rad]
0.5
0.6
0.7
0.8
0.9
( = 27.8036)
0 2
0.8
0.85
0.9
0.95
1
rc( ) ( = 0.001,
= 0.026525) [rad]
= 1.4835 [rad]
0 2
0.8
0.85
0.9
0.95
1
= 1.3963 [rad]
0 2
0.7
0.8
0.9
1
= 1.309 [rad]
0 2
0.8
0.85
0.9
0.95
1
rc( ) ( = 0.5246,
= 13.9151) [rad]
0 2
0.8
0.85
0.9
0.95
1
0 2
0.6
0.7
0.8
0.9
1
0 2
[rad]
0.8
0.85
0.9
0.95
1
rc( ) ( = 1.0482,
= 27.8036) [rad]
0 2
[rad]
0.8
0.85
0.9
0.95
1
0 2
[rad]
0.5
0.6
0.7
0.8
0.9
Const. r ( = /2)
Pure Elong
Pure Elong- /2
Bend- CG exact
Bend- CG Const.
Bend- CG /2
Bend- GD planar
Bend- GD planar fix z
Bend- GD general
0 2
0.8
0.85
0.9
0.95
1
rc( ) ( = 0.001,
= 0.026525) [rad]
= 1.4835 [rad]
0 2
0.8
0.85
0.9
0.95
1
= 1.3963 [rad]
0 2
0.7
0.8
0.9
1
= 1.309 [rad]
0 2
0.8
0.85
0.9
0.95
1
rc( ) ( = 0.5246,
= 13.9151) [rad]
0 2
0.8
0.85
0.9
0.95
1
0 2
0.6
0.7
0.8
0.9
1
0 2
[rad]
0.8
0.85
0.9
0.95
1
rc( ) ( = 1.0482,
= 27.8036) [rad]
0 2
[rad]
0.8
0.85
0.9
0.95
1
0 2
[rad]
0.5
0.6
0.7
0.8
0.9 Const. r ( = /2)
Pure Elong.
Pure Elong.- /2
Bend- CG exact
Bend- CG Const.
Bend- CG /2
Bend- GD planar
Bend- GD plan. fix z
Bend- GD general
0 2
0.9
0.95
1
rc( ) ( = 0.001,
= 0.026525) [rad]
lc
= 1.1187
0 2
0.6
0.7
0.8
0.9
1
( = 0.026525)
lc
= 1.5549
0 2
0.5
0.6
0.7
0.8
0.9
( = 0.026525)
lc
= 1.8368
0 2
[rad]
0.5
0.6
0.7
0.8
0.9
( = 13.9151)
0 2
0.6
0.7
0.8
0.9
1
( = 13.9151)
0 2
0.9
0.95
1
rc( ) ( = 0.5246,
= 13.9151) [rad]
0 2
[rad]
0.85
0.9
0.95
1
rc( ) ( = 1.0482,
= 27.8036) [rad]
0 2
[rad]
0.6
0.7
0.8
0.9
1
( = 27.8036)
A
B
FIGURE 4 |The change in the outer radius stretch ratio (λrc(ψ))(left) and shell cross-section deformation (right) of a pneumatic chamber in terms of different values
of γ,λlc,κ, and θ=κlcwhere γ=1.4835 rad (left) and λlc=1.2418 (right). For a module with large radius (rc2 = 12mm) for better presentation of the model
assumption effects on the cross-section deformation (A), and the STIFF-FLOP pneumatic chamber (rc2 = 3 mm) (B). Different scales for the y-axis are used for better
visibility. Note how the different assumptions change the cross-section shape by changing the λrc(ψ)profile; and the different predicted profile for a module in nearly
pure bending with a large bending radius (λlc=1.1187 and θ=1.0482 rad) in panel (B) compared to a module with bigger radius in panel (A).
assumption model. It is clear that the helix tends to maintain the
radius, and the braided chamber does not shrink in radius as does
the simple cylinder without braids.
Cross-section deformation of a pneumatic chamber for the
STIFF-FLOP module is presented in Figure 5A for different
bending angles. It is clear that the change in the shape of the
cross section in the Figure 5A is less obvious than in Figure 4C
with bigger inner and outer radius. The inner cross-section shape
deformation is more obvious. The chamber tends to remain cir-
cular in all cases except for the bending with exact γmodel,
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Sadati et al. A Geometry Deformation Model
B
A
FIGURE 5 |The cross-section deformation model for a STIFF-FLOP pneumatic chamber where λlc=1.5549 and γ=1.4835 rad (A). The cross-section model for
deformation of STIFF-FLOP body shell where λlc=1.5549 and γ=1.4835 rad [(B), left] and comparison with experiment [(B), right]. The area changes to an egg
shape and shifts toward the inner of the bent. The GD- planar model predicts less material shift compared to the general model. The GD- general method provides a
better prediction of the cross-section deformation (B).
where the tube shell tends to bend inward and become more like
a bean. This bean shape is consistent with our observation for the
STIFF-FLOP pneumatic chamber, despite the thread that is used
in making the chamber bends easily.
The predicted deformation by different models for a STIFF-
FLOP body shell is presented in Figure 5B, where deformation
due to a pure symmetric elongation is presented too for compar-
ison. The simple bending model based on Cauchy–Green stretch
ratios (CG- simple bending model) and the planar bending model
from the geometry deformation model (GD- planar model) pre-
dict more shift in the cross section toward the bending side. The
CG- simple bending model predicts an almost uniform shift in
the cross section toward the bending axis. The planar bending
model predicts an egg shape cross section with the sharp side
toward the bend center. The general geometry deformation model
(GD- general bending model) predicts a smaller shift toward
the bending axis, and a small lateral shrink at the back of the
bending side and a small lateral expansion at the inward. Based
on the shift and the overall shape, we conclude that the GD-
general bending model predicts the most realistic deformation.
A comparison between the GD- general bending model and the
actual cross-section deformation of the body shell is presented in
Figure 5B too.
5.3. Comparison of Pressure and Action
Models for a Braided Actuator
The pressure required for one pneumatic chamber to reach a
certain λtis found by exploiting the principle of virtual energy
for one pneumatic chamber. The result from different braided
chamber models is compared against experiments with a STIFF-
FLOP chamber in Figure 6. The variation w.r.t. the bent curvature
is neglected for the bending models; however, as the standalone
actuator chamber bends when it pressurized shown in Figure 1G,
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Sadati et al. A Geometry Deformation Model
FIGURE 6 |Required gas pressure for a certain deformation of a STIFF-FLOP pneumatic chamber, experimental data vs. model simulations (top), sensitivity of the
predicted required pressure to the structural parameter γby plotting p,γfor different values of θ,γ,λlc, and E (down). The CG- exact helix model shows better
prediction in terms of mean error and profile shape, however, predicts high non-linear sensitivity.
we simulate the results for different λlcin θ=1.482 rad. It is
observed that the constant γmodel and the exact helix model
give the best prediction based on the actual measured values. The
results are very sensitive to the verified parameters. For example,
in Figure 6, by choosing a slightly different value for Eand γ
the bending with constant helix model can predict the pressure
values precisely while this model is less sensitive to a change in
Ecompared to the other models. Therefore, we choose the model
that predicts the behavior trends in the experimental data and then
we identify the unknown structural parameters to fit the model to
the experimental readings. The trend in the experimental values
and small sensitivity to the γvalue show that the constant γ
model has the best prediction for this chamber. This result was
predictable due to easily deformable threads having been used in
making this chamber. There is no significant change in the model
predictions for different values of θand γ, other than the pressure
value predicted by the GD- general bending model, which is
shown to be very sensitive to the value of γ, where the pressure
increases significantly as γreduces. This very sensitive behavior
needs more experiments for verification when using chambers
with variable thread angles.
The elastic deformation from the Neo-Hookean (NH) and
Euler–Bernoulli (EB) methods, and gas pressure actions w.r.t.
different values of θ,λlc, and γpredicted by different models are
presented in Figure 7A for a STIFF-FLOP pneumatic chamber.
The gas pressure action does not change for different values of θ
and γ, except for the GD- exact helix model that demonstrates a
significant increase as γincreases. The most important parameter
is λlc, which causes a significant increase in the actions. For the
CG- bending model with fixed radius (the case of dense braids,
γπ/2), a decrease in the Neo-Hookean elastic deformation
action is observed, which suggests this assumption cannot be
used with the Neo-Hookean method. The graphs show some
models predict the actions up to 3 times more compared to
other models, which show the importance of a proper choice
of modeling method. The predictions from GD- exact γand
the planar deformation assumptions are quite similar. The
Euler–Bernoulli method predicts significantly higher values than
the Neo-Hookean method. For the models with fixed radius,
dense tread, or neglecting the pneumatic chamber shells, larger
values for the actions are calculated. In general, any simplifying
assumption causes an overestimation of the actions. While
the action values are representations of the energies stored in the
system, the actions variation w.r.t. system states determines the
final static map for the system.
5.4. Constant Curvature Model
We attempted to identify the unknown values such as γ,E, and
sensor frame initial register by fitting the model simulations
to the actual experiment information for eight data points and
subsequently verifying the result against 43 data points. All the
remaining parameters were measured manually. It was observed
that a combination of the geometry deformation method with
general assumptions for the body shell and the bending model
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Sadati et al. A Geometry Deformation Model
A
B
FIGURE 7 |The gas pressure action (wG), body deformation energy using Neo-Hookean (wENH )and Hooke’s (wEEB )methods vs. change in the helix and curvature
parameters (A). Sensitivity analysis of the action values w.r.t. γby plotting w,γfor different values of θ,γ,λlc, and E (B). High complexity models, i.e., GD- exact
helix models, predict high non-linear sensitivity, and the simplified models, i.e., CG- braided fix radius, predict higher and, in some cases, incorrect sensitivity values
and profile.
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Sadati et al. A Geometry Deformation Model
0 5 10 15 20 25 30 35 40 45
-5
0
5
10
p [Pa]
10 4P2
0 5 10 15 20 25 30 35 40 45
-5
0
5
10
p [Pa]
10 4P1
-0.04 -0.02 0 0.02 0.04
x [m]
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
y [m]
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
34
35
36
37 38
40
33
39
0
-0.04-0.04
0.02
z [m]
0.04
-0.02 -0.02
y [m] x [m]
0 0
0.020.02 0.04 0.04 0 5 10 15 20 25 30 35 40 45
trials
-5
0
5
10
p [Pa]
10 4P3
Fix section Exp. Bend- GD general & Bend- CG const.
FIGURE 8 |The simulation vs. experimental results from 43 data points for inverse static relation between the input pressures and the tip position (p=f(ρtip))for a
STIFF-FLOP pneumatic chamber, using principle of virtual work. A combination of the GD- general method and the constant γmodel shows 13% lower mean error
percentage and 23% lower mean reference error percentage compared to the Euler–Bernoulli rod model without any cross-section deformation.
with the exact helix lead angle assumption provides the best
modeling accuracy. This might have been predicted based on
the previous observations about the body shell and pneumatic
chamber. This method increases the accuracy of the model by
up to 13% mean error compared to a simple CG- symmetric
elongating body shell with fixed radius chambers and 14% com-
pared to Euler–Bernoulli bending model with fixed cross section.
A better accuracy is possible if a CG- pure elongating braided
helix model is used instead of the helix fixed radius model. Using
the simplified models such as the CG- simple bending or fixed
radius assumption increases the modeling error by up to 57%
compared to the best possible solution. The general deforma-
tion derivation [equation (45)] provides a better solution for the
problem of flexure compared to the planar deformation method
[equation (40)] based on Rivlin’s method (Rivlin, 1949). The
Euler–Bernoulli model with fixed radius assumption for the actu-
ator chambers results in better accuracy compared to the Neo-
Hookean models in this case; however, the correct choice of
the combination of the methods is still important, i.e., in incor-
porating the structural parameters’ effect. It is clear that more
exact modeling of the deformation of the cross section increases
the modeling accuracy significantly, even if this prediction has
been made based on a simple symmetric pure elongation model.
However, a carefully chosen combination of the models based on
the nature of the module is important to achieve the best result.
The overall mean error of the model is high (61% mean error
percentage and 38% mean reference error percentage) because
of the inaccuracy in constant curvature assumption, especially
for near straight configurations, as observed in similar previous
research (Trivedi et al., 2007;Godage et al., 2011a). A comparison
between the best and the worst model prediction is presented in
Figure 8 showing the importance of considering the cross-section
deformation.
5.5. Variable Curvature Model
Equation (50) needs to be solved for the deformed geometry in
the equilibrium state. We start with a guess for the deformed
geometry, i.e., straight configuration, and solve for the manipu-
lator geometry where the result should be identical to the first
guess. This leads to a system of non-linear equations that can be
solved with numerical methods such as Powell dogleg (Powell,
1970) used in Matlab “fsolve” function. Starting from the straight
undeformed state shows good convergence, and the algorithm
usually needs two trials to find the equilibrium configuration
for a single-curve formation and three trials for a double-curve
formation (Figure 1C).
We investigate the Cosserat rod model accuracy to simulate the
planar deformation of one STIFF-FLOP module with extensive
external loading shown in Figure 1C. A Matlab program algo-
rithm is presented in Algorithm 2. The overall accuracy of the
Cosserat rod model is about 52–100% better than the constant
curvature model for any combination of the models. Approxi-
mately the same trend in the accuracy of the models is observed
here compared to the constant curvature model. The combination
of the bending GD general deformation or planar assumption and
the exact or constant γhelix model results in a small 9–10% error.
The error for the combination of the CG symmetric elongation
model and the helix pure elongation results in 12% error, which is
slightly less than the accuracy of the models based on the general
deformation method. A comparison between the combination of
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Sadati et al. A Geometry Deformation Model
ALGORITHM 2 |A pseudo code for a Matlab program to solve the forward static relation for a continuum manipulator using the Cosserat rod method.
Data: par—structural parameters, mthd,mthdc—modeling method for body and chamber
Result: inverse model for a continuum manipulator based on Cosserat rod model
initialization;
first guess for the equilibrium geometry (q0)—undeformed state;
q= fsolve(@(q0)(ode113(@bvp(l,q0,par)),q0); %to solve the boundary value problem
return q; %equilibrium geometry
Function bvp(s,q0,par) {
Data: par—structural parameters, mthd—modeling method, q0—initial geometry, s—axial position
Result: ODE derivatives
[x0,y0]= interp1(q0,s); %interpolate the geometry of the current axial position
[λl,s, κ,s]=...; %the element differential kinematics based on the guessed geometry from section 4.4.
[acd,apd,Jc3]= cross_sec(λl,s, κ,s,par,mthdc); %gas cross section area, chamber cross section area and second moment of area based on the element
differential kinematics
[ad,Jc3]= cross_sec(par,mthd); %boody shell cross section area and second moment of area
rod= sqrt(ad)ro; %approximates the chamber radial position change
τp=p[cos(ψ),C(2π/3ψ),C(2π/3ψ)]Tapdrod+τL; %total planar moment due to gas pressure and the external tip moment
ρw=bag(trapz(@(x)(interp1(q0,x)-x0), s,l)); %integration for the body weight load moment
[ξ, ζ] = ...;
[ρ,s,R,s] = ...; %substitute in equations (48)–(50) to calculate the derivatives.
return [ρ,s,R,s]}
FIGURE 9 |The simulation vs. experimental result from 33 data points for planar deformation of one STIFF-FLOP module under extensive external loads (Figure 1C).
A combination of GD- general method and exact helix model shows 7% lower mean error percentage and 10% lower mean reference error percentage compared to
the Euler–Bernoulli rod model with no cross-section deformation. The VC model shows 52% lower mean reference error percentage than the CC model (Figure 8).
The iteration to solve the BVP problem for some of the simulation steps is presented (bottom).
the GD- general bending models and the constant helix angle (γ)
model with the model without any change in the cross section is
presented in Figure 9 showing 7% increase in the model accuracy
by taking into account the cross-section deformation based on
mean reference error. The model reference error percentage is
10% for the case with extensive external load and less than 5%
in the case that only the body weight is present, which is similar
to the results reported in Trivedi et al. (2007) and Godage et al.
(2011b). Model complexity provides a more accurate solution but
increases the time that the numerical non-linear solver needs to
find a feasible solution. This is due to increased computation time
and highly non-linear behavior of the model complexity.
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Sadati et al. A Geometry Deformation Model
5.6. Sensitivity Analysis
An analytical model is beneficial for design by providing an effi-
cient means for sensitivity analysis and optimization of the system
performance w.r.t. the structural parameters. Here, we investigate
the sensitivity of the predicted deformation profiles, the input
pressure, and deformation energy actions to changes in γand E, as
the two important structural parameters in the design of stiffness
controllable braided actuators, by deriving the gradient of these
values w.r.t. γand E. The resulting non-linear relations are plotted
for different values of γ,κ,λlc, and E, showing the sensitivity of the
results in any geometrical action point and for any combination of
the other structural parameters.
For γ(ψ)as in Figure 10, the exact γand the dense thread
(γ=π/2) models sensitivity to a change in γdecreases as γ
decrease and λlcincreases. Unlike γ(ψ), the sensitivity of λrc(ψ)
to the change of γincreases by the increase in λlcand κand
decrease in γshowing more sensitivity in pure elongation cases
for a module with a less dense braid. The model with simplify-
ing assumption of γπ/2 overestimates the predicted values as
observed in the previous sections.
The sensitivity of the deformation action, gas pressure action
(Figure 7B), and required pressure for a braided actuator to reach
a certain elongation (Figure6) to changes in γincreases for a more
stiff actuator (increase in E) with a less dense thread (decrease
in γ) and in a pure elongation case (decrease in θand increase in
λlc). The sensitivity of pdecreases for a high bending angle cases
while slightly increases for the deformation actions. pand ware
linear functions of E, and their sensitivity to changes in the module
stiffness is similar to the graphs for pand wwhere the values are
divided by the value of Ein our simulation (10kPa).
The CG- exact model that uses a simplifying assumption for γ,
as in equation (23), results in a highly non-linear gradient w.r.t. γ,
which is not similar to that of the exact γmodel in Figure 10. This
is predictable as the simplifying assumption only holds for small
values of γand predicts greater changes in the sensitivity, similar
to the other models with similar simplifying assumption.
6. DISCUSSION
First, cross-section deformation of a STIFF-FLOP chamber in
terms of structural and bending parameters using different meth-
ods is presented in Figure 4A,B. The exact helix model predicts
a bean like cross section with small lateral deformations, and the
constant γassumption predicts a uniform shift in the cross section
toward the bending axis. A larger uniform decrease in the cross
section is predicted for the small lead angle assumption (γπ/2).
The most effective parameter on the asymmetric deformation of
the cross section is the decrease in the bending angle (θ) and
the braid helix lead angle (γ), where the increase in the axial
stretch ratio (λlc)and initial radius enhances these effects. The
structural parameters have fewer effects on the models with the
constant γand dense braid assumptions. The dense braid assump-
tion predicts a slightly smaller deformed radius compared to the
model for simple elongation of a helix. These models are valid
for γ > 80°[deg], however, are not recommended as they do not
reduce the derivation complexity significantly.
The braided chamber cross-section deformations predicted by
any of the models are presented in Figure 3B,C. The elongation
of a helix results in a small uniform reduction of the radius
and the constant γassumptions added a small shift toward the
bending angle too. There is no lateral deformation in the exact
helix model in pure bending resulting in a heart shape deformed
cross section due to a notable different twist angle at the start
and end of each thread rounds. In general, the braided chamber
FIGURE 10 |Sensitivity of predicted values of γ(ψ)(γ(ψ), left) and λlc(λlc, right) from different models to change in γfor different values of γ,κ, and λlc. The
CG- exact model that uses simple γmodel shows highly non-linear sensitivity for different values of γ, which is not similar to that of the exact γmodel. The model
with dense thread assumption (γπ/2) shows higher sensitivity.
Frontiers in Robotics and AI | www.frontiersin.org June 2017 | Volume 4 | Article 2220
Sadati et al. A Geometry Deformation Model
radius deformation is smaller than the cases without braids. The
experimental results for a STIFF-FLOP module are presented in
Figures 5A,B.
The exact helix model assumes pure torsion of the thread cross
section, which is valid for a stiff thread similar to metallic springs.
The bendable thread, i.e., sewing threads, being use for many
micro soft manipulators tend to follow the deformation of the
chamber body; hence, the constant γassumption should work
better with them.
Next, we investigated the action predicted by the models. A
considerable shift in the cross section toward the bending axis
is observed for the CG- simple bending model (uniform shift),
the GD- planar models (egg shape), and the GD- general bending
model (trapezoidal shape). The most realistic shift and overall
deformation compared to the experiment results are predicted
by GD- general bending model (Figure 5B). As predicted, the
constant γmodel gives the best prediction of pressure vs. λcfor
one pneumatic chamber (Figure 6). Despite that the results are
very sensitive to the identified value for E, we choose the model
that shows a better agreement with the behavior trends observed
from the experiments. The results for the GD- general bending
model are shown to be very sensitive to the value of γas can be
predicted for a spring.
The elastic deformation and gas pressure actions are shown to
be very sensitive to the backbone elongation (λlc)compared to
the bending angle (θ), showing the dominance of the pure elon-
gation effect over the bending for static modeling (Figure 7A).
Simplifying assumptions (fixed radius, dense tread, neglecting the
chamber shells, and Euler–Bernoulli linear stress–strain relation)
results in the overestimation of the actions values (up to three
times).
A comparison of using different combination of the models for
the pneumatic chambers and the body shell with the experiment
results shows that a combination of GD- general bending model
for the body shell and the bending helix with constant γprovides
the best modeling accuracy as we anticipated from the previous
observations (Table 2;Figure 8). It is clear that the modeling
accuracy increases by using more realistic models for the defor-
mation of the cross section (13% less mean error percentage). The
results from the simple symmetric elongation and the fixed radius
chamber models can be more accurate if, instead of using the real
measured values, we identify the structural parameters by fitting
the simulation results to the experiment values.
Despite all the improvements from implementing the cross-
section deformation models, the overall simulation accuracy is
relatively low (up to 62% mean error and 38% mean reference
error). We showed 52–100% fewer simulation errors by using
the Cosserat rod model in the case of high external loading
where the combination of the GD- general deformation or planar
assumption and the exact or constant γhelix model results in
a small 8% error, and the combination of the CG symmetric
elongation model and the helix pure elongation shows 10% error
(Figures 2C and 9). The model error is less than 5% in the case
that only the body weight is present. The Hooke’s law for linear
stress–strain relation used in equation (50) can be substituted
with a more realistic model, i.e., Neo-Hookean, Mooney-Rivlin,
or Gent relations based on the deformation rate matrix invariants
of a highly deformable material, in the future to further improve
the modeling accuracy (Gent, 2012).
We observed that considering the cross-section deformation
increases the simulation accuracy up to 10% for the Cosserat rod
model and 13% for the constant curvature model compared to the
constant cross-section models. The combination of the geometry
deformation method and bending helix model has the best result
with up to 2% more accurate results compared to the other models
for the cross-section deformation. The reference error percentage
is used to compare the modeling accuracy with similar research
where we showed 38% error for the inverse relation in the model
with constant curvature assumption and body loads, 5% error for
the forward relation in the model with variable curvature assump-
tion and body loads, and 10% error for the case with extensive
external loads. Our results comply with the 5% error observed by
the modeling errors reported in Trivedi et al. (2007) and Godage
et al. (2011a) for the cases without extensive external loads, while
our models incorporate the information about the cross-section
deformation too. Our results suggest the importance of the cross-
section deformation in the modeling accuracy, which confirms its
evident importance for minimal invasive manipulation with small
workspaces and for the manipulators with regional controllable
stiffness in their cross section.
The sensitivity of local deformation rate of the cross-section
radius λrc(ψ)predicted pressure for a certain elongation (p), and
deformation and pressure actions (w) to changes in the braid ini-
tial lead angle (γ) increases as the module elongates and for a less
dense braiding, while the sensitivity of the local lead angle (γ(ψ))
w.r.t. γdecreases in the same conditions. γ(ψ)is not sensitive to
the actuator curvature, pbecomes less sensitive, and the sensitivity
of wslightly increases as the actuator bends. The sensitivity of
λrc(ψ)to γfor different bending angles is related to the polar
position of the points with decreasing sensitivity for the points on
the inside of the bending module and increasing sensitivity for the
ones on the outside. The model is linearly sensitive to changes in
the module stiffness (E) as all the relations are linear functions
of this parameter. Models with more complexity or simplifying
assumptions about the thread lead angle predict higher values and
non-linearity for the model sensitivity. This shows the importance
of an accurate derivation for the models with increased complexity
and avoiding simplifying assumption about the thread lead angle
to achieve robust results. The fact that the resulting predictions
for the system behavior and model sensitivity to parameters can
be different for different models show the importance of proper
choice of model assumptions and complexity level.
We believe our simple analytical model for the cross-section
deformation can play the same role as the constant curvature
assumptions in providing a simple estimation of the backbone
kinematics, for the manipulators with a variable stiffness cross
section. A suggestion for a future research is to improve the model
by considering the material hysteresis and damping effects as
well as incorporating adaptive terms in the model to take into
account for the changes in the material properties and repeatabil-
ity of experimental results as presented to some extent in Shapiro
et al. (2011). For the minimal invasive surgery applications with
limited workspace, as the main aim of our future research, we
plan to implement the tactile information from the continuum
Frontiers in Robotics and AI | www.frontiersin.org June 2017 | Volume 4 | Article 2221