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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 f−1

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).

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Sadati et al. A Geometry Deformation Model

ρtip =1−C(κl)/κ S(κl)/κ 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)=n−1

i=1

Ti(κi,ϕi,li)Tn(κn,ϕn,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 =l1−κroC(−ϕ)

lc2 =l1−κroC(−ϕ+2π/3)

lc3 =l1−κroC(−ϕ−2π/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.ρ(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=n−1

i=1

Ti(κi, ϕi,li)

Ry(ϕ)0

0 1n(l−Sκl)

κl

(1−Cκl)

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(I1−3)/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=xd−x, ηy=yd−y, ηz=zd−z,(9)

σxx =ηx,x+η2

x,x+η2

y,x+η2

z,x2,

σyy =ηy,y+η2

x,y+η2

y,y+η2

z,y2,

σzz =ηz,z+η2

x,z+η2

y,z+η2

z,z2,

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(λl−1)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= (1−λlc

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

cCγ2−1

Sγ2+Sγ2

λl2

c(λl2

cCγ2−1)−λl2

c+3,

wG=pλlclcπr2

c1(λl2

cS2

γ−λl2

c+1)/S2

λ,ac=π(rc12−rc22).

(14)

For one chamber, using equation (5) (δwc+δwG=0→

(wc,λl+wG,λl)δλlc=0)and assuming λlcas the only system

state, we get (Trivedi et al., 2007)

p=E(rc12−rc22)(λl2

c−1)

6λl4

crc12(λl2

cCγ2−1)3

×(λl6

c(−2Sγ6+5Sγ4−4Sγ2+1)

+λl4

c(7Sγ6−16Sγ4+11Sγ2−2)

+λl2

c(7Sγ6−4Sγ4−3Sγ2+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 γ≈π/2. We can

next deduce λ2

rc=1−λ2

lc(π/2−γ)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(2−1/λl2

c−λl2

c)/6,

wG=pλlclcπr2

c1,(16)

p=E(rc22−rc12)(λl2

c−1)(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/κ −(1/κ −rthCψ)Cθc,

yth = (1/κ −rthCψ)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))/κ +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+κ2−2B0cot(γe)C(ψ)2−κ)

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/κ +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

1−κrthd(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)+λlc2−2)+1,

A11 =−κlc2λlc4C(ψ)(C(2γ)−1)(8nthπC(ψ)2+S(4πnth )+4πnth)2

2nth2rcπ2(S(4πnth ) + 4πnth )2(λlc2C(2γ)+λlc2−2),

A10 =lc2λlc4(C(2γ)−1)(8nthπC(ψ)2+S(4πnth )+4πnth)2

4nth2rc2π2(S(4πnth )+4πnth )2(λlc2C(2γ)+λlc2−2)

−lc2λlc2(C(2γ)−1)(8nthπC(ψ)2+S(4πnth )+4πnth )2

4nth2rc2π2(S(4πnth )+4πnth )2(λlc2C(2γ) + λlc2−2)−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)−12−1)−1,

λrc(ψ)=

λl2

ccot(γld)2(κ2r2

cC(ψ)2csc(γld)2−1)

+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(rc22−rc1 2)(−λlc

2(κ2(rc22−rc1 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+2/λl, 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)= (1−κx)λl, λx(x)=λz(x)=1/λy,

I1(x)= (1−κx)2λ2

l+2

λl(1−κx).(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=Tbρ∗consists 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=C∆θx∗+ (1−C∆θ)/κ,

yd=−S∆θx∗+S∆θ/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,ρ|=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∗(1−κx∗)y∗

,ySκy∗x∗,zCκy∗

−x∗,xSκy∗(1−κx∗)y∗

,yCκy∗−x∗,zSκy∗

0 0 z∗,z

=x∗,xy∗,yz∗,z(1−κx∗) = 1.(39)

By separation of variables, we have

x∗,x(1−κx∗) = 1/(y∗

,yz∗

,z) = 1/(λ2λ3)

y∗

,yy =0→y∗=A2y+B2

z∗,zz =0→z∗=A3z+B3.

(40)

By integrating for x*, we get

x∗=1−1−2κ(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=A34λl2−2κx A33λl+A32λl4−2κx A3λl3+1

A3λl(A3λl−2κx).(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∗(1−κx∗)y∗

,ySκy∗x∗,zCκy∗

−x∗,xSκy∗(1−κx∗)y∗

,yCκy∗−x∗,zSκy∗

z∗,x0z∗,z

=y∗

,y(x∗,xz∗,z−x∗,zz∗,x)(1−κx∗).(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∗= (1−1−2κ(h2(x)+B1))/κ

y∗

,yy =0→y∗=λly

h2(x),xz∗,z=1/λl.

(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/√1−κxbased on the simple elongation–

bending model in equation (35). Then, we have

h2(x)=−2A1(1−κx)3/2

3κ,B1=2A1

3κ,(46)

where A1=1/(λlA3). For I1, we get

I1=−12B55+4A33B4λlB53−4A3B4λl3B54+A33B4κ2λlz2B5

A3B4λlB54,

(47)

where B4=4(1−κx)3

2+3A3λl−4 and B5=κx−1. 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=v−1

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(1−κroC(−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=wENHt−wh;

T:unity matrix; %initialize

for n—number of modules in a manipulator do

T=T.TCOM(n); %COM transformation matrix calculation ww=mg.ρCOM; %weight action

end

for nL—number 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).ρr(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(l−s).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(−φ−2π/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 γ≈π/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 (γ≈π/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 (γ≈π/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 γ≈π/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

Frontiers in Robotics and AI | www.frontiersin.org June 2017 | Volume 4 | Article 2212

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 γ≈π/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,

γ≈π/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/3−ψ),C(−2π/3−ψ)]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 γ≈π/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 (γ≈π/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),γ, left) and λlc(λlc,γ , 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 (γ≈π/2) shows higher sensitivity.

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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