i

Topology Design of Concentric Tube Manipulators Using

Optimization

by

Kevin Ai Xin Jue Luo

A thesis submitted in conformity with the requirements for the degree of Master of Applied Sciences

Institute of Biomaterials and Biomedical Engineering University of Toronto

© Copyright by Kevin Ai Xin Jue Luo 2018

ii

Topology Design of Concentric Tube Manipulators Using

Optimization

Kevin Ai Xin Jue Luo

Master of Applied Sciences

Institute of Biomaterials and Biomedical Engineering

University of Toronto

2018

Abstract

One of the major problems facing the development and road to practical usage of concentric tube

continuum robots in surgical environments is that of instability. This issue, also known as the

snapping problem, is caused by a tube having a high bending to torsional stiffness ratio (BTSR).

Past efforts have shown that by cutting patterns on the tubes, this problem can be avoided. This

paper seeks to redesign the topology of the tubes so that BTSR is decreased and the snapping

problem is resolved in a particular tube set. The generated designs were then validated through

finite element analysis as well as experimental testing to demonstrate the elimination of the

snapping problem. Using this novel tube design in concentric tube robotic systems can increase

stable workspace because it allows the usage of greater tube curvatures and/or curve lengths.

iii

Acknowledgments

This research was carried out with the tremendous help of the following people:

T. Looi, Dr. J. Drake, Dr. H. Naguib, Dr. C. Steeves, S. Sabetian.

iv

Table of Contents

Contents

Acknowledgments ..................................................................................................................... iii

Table of Contents ...................................................................................................................... iv

List of Figures .......................................................................................................................... vii

List of Tables ............................................................................................................................ xi

List of Abbreviations ................................................................................................................ xii

List of Symbols ....................................................................................................................... xiii

List of Equations ..................................................................................................................... xiv

Introduction ................................................................................................................................ 1

Concentric Tube Robots in Minimally Invasive Surgery ........................................................ 1

Mechanical Limitations of Concentric Tube Robots ............................................................... 3

SIMP Topology Optimization ................................................................................................ 4

Thesis Overview .................................................................................................................... 5

Related Literature ....................................................................................................................... 7

Physics and Analytical Modelling of the Stability Problem .................................................... 7

Avoidance and Elimination of Snapping Instability ...............................................................11

Overview of Chapters 1, 2, 3 – Tube Set A ................................................................................14

Design Generation and FEA – Tube Set A .................................................................14

1.1 Methodology – Topology Optimization and FEA ...........................................................14

1.1.1 Setup 1 ...............................................................................................................14

1.1.2 Setup 2 ...............................................................................................................16

1.1.3 Setup 3 ...............................................................................................................17

1.1.4 Finite Element Analysis Studies .........................................................................18

1.2 Results – Design Generation and Analysis .....................................................................19

v

1.2.1 Setup 1 ...............................................................................................................19

1.2.2 Setup 2 ...............................................................................................................21

1.2.3 Setup 3 ...............................................................................................................21

Experimental Validation – Tube Set A ......................................................................25

2.1 Experiment Setup and Analysis Methodology ................................................................25

2.1.1 Stiffness Testing Method ....................................................................................25

2.1.2 Concentric Tube Rotation Test ...........................................................................27

2.1.3 Workspace Impact Analysis ...............................................................................29

2.2 Experiment Results ........................................................................................................30

2.2.1 Stiffness Results .................................................................................................30

2.2.2 Tube Rotation Test Results .................................................................................31

2.2.3 Workspace Improvement Impacts of BTSR Reduction .......................................34

Discussion – Tube Set A ...........................................................................................35

3.1 Topology Optimization ..................................................................................................35

3.1.1 Setup 1 ...............................................................................................................35

3.1.2 Setup 2 ...............................................................................................................35

3.1.3 Setup 3 ...............................................................................................................36

3.2 Analysis of Experiments ................................................................................................37

Overview of Chapters 4, 5, and 6 – Tube set B ..........................................................................40

Parametric Pattern Design Study – Tube Set B ..........................................................40

4.1 Topology Optimization ..................................................................................................40

4.2 Pattern Dimensions Design Study ..................................................................................42

4.3 Pattern Optimization Results ..........................................................................................45

Experiments and Concentric Tube Rotation ...............................................................51

5.1 Torsion and Bending Tests .............................................................................................51

5.2 Concentric Tube Parameters and Simulation ..................................................................53

vi

5.3 Heat Treatment and Tube Rotation Test .........................................................................54

5.4 Testing Results ..............................................................................................................59

Tube Set B Discussion...............................................................................................68

6.1 Optimization and Pattern Design Study ..........................................................................68

6.2 Torsion and Bending Stiffness .......................................................................................70

6.3 Curved Tube Parameter Selection and Analysis .............................................................71

6.4 Improving the Shape Setting Process .............................................................................72

6.5 Tube Rotation Stability Tests .........................................................................................74

6.6 Workspace Analysis and Impact ....................................................................................78

Conclusions ...............................................................................................................................79

Bibliography .............................................................................................................................81

Appendices................................................................................................................................84

vii

List of Figures

Figure 1. A 2 dual-tube concentric tube system ......................................................................... 2

Figure 2. Buckling Analogy of the snapping problem. .............................................................. 3

Figure 3 An example of a 2D SIMP topology optimization result for a load-bearing structure ... 5

Figure 4 Energy landscape of an unstable tube system (Webster et al) [16] ............................... 9

Figure 5 Example of a multilayer helical tube structure (Weng et al) [27] ............................... 12

Figure 6 Horizontal slit pattern used by Azimian et al [24] ...................................................... 12

Figure 7 Loading scenario for Op.I of Setup 1 ......................................................................... 15

Figure 8 Loading scenario of Setup 2 ....................................................................................... 16

Figure 9 Symmetric loading setup used in the short segment tube of Setup 3 ........................... 17

Figure 10 Raw optimization result from Op.I ........................................................................... 19

Figure 11 Raw optimization result from Op.II ......................................................................... 20

Figure 12 Post-processed feasible design based on the Op.I result ........................................... 20

Figure 13 Slit pattern tube model used for comparison............................................................. 20

Figure 14 Laser cut tube using the model in Fig. 12 ................................................................. 20

Figure 15 Topology optimization result of Setup 2 showing averaged density contours ........... 21

Figure 16 Topology optimization result of Setup 2 .................................................................. 21

Figure 17 Topology Optimization Result: Density Contour (Left), Iso (Right) ......................... 22

Figure 18 Finalized section of tube model in Solidworks ......................................................... 23

Figure 19 Design and dimensions of 1 unit of the cutting pattern ............................................. 23

viii

Figure 20 FEA Contour Results ............................................................................................... 24

Figure 21 Diamond pattern cut on nitinol tube (2.03mm OD) .................................................. 25

Figure 22 Torsional stiffness testing setup ............................................................................... 26

Figure 23 Bending stiffness testing setup ................................................................................. 26

Figure 24 Concentric tube rotation testing setup ...................................................................... 28

Figure 25 Patterned tube concentrically assembled with outer tube. ......................................... 28

Figure 26 An image used in the visual measurement of tube curvatures after heat treatment .... 29

Figure 27 Resulting trends of torsion tests on the patterned and blank tubes............................. 30

Figure 28 Resulting trends of bending tests on the patterned and blank tubes ........................... 31

Figure 29 Movement of the blank tube system, showing snapping ........................................... 32

Figure 30 Smooth motion of patterned tube set, using patterned inner tube .............................. 32

Figure 31 Graphed motion of the blank concentric tube system, showing snapping motion ...... 33

Figure 32 Graphed smooth motion of the patterned concentric tube manipulator ..................... 33

Figure 33 Workspace volumes of tube systems 1, 2, and 3 displayed from left to right ............ 34

Figure 34 Topology optimization result from Setup 1 (left) and Setup 2 (right)........................ 36

Figure 35 Stress contour result of the enforced displacement analysis ...................................... 37

Figure 36 Design space mesh and setup of inner tube short segment topology optimization .... 41

Figure 37 Topology optimization of a short segment of the inner tube ..................................... 41

Figure 38 Iso view of the short tube segment topology optimization result from Tube Set A .... 42

Figure 39 Bending loading setup (left) and mesh (right) for inner tube pattern design study..... 44

ix

Figure 40 Solidworks FEA design study using 54 design points............................................... 45

Figure 41 Inner Tube Topology Optimization Result: Density Contour (Left), Iso (Right) ....... 46

Figure 42 Outer tube topology optimization result: density contour (left), iso (right) ............... 46

Figure 43 Inner patterned tube stress concentration study ........................................................ 47

Figure 44 Inner patterned tube BTSR design study .................................................................. 48

Figure 45 Outer patterned tube stress concentration study ........................................................ 49

Figure 46 Outer patterned tube BTSR design study.................................................................. 50

Figure 47 Close up view of tube bending fixture ...................................................................... 51

Figure 48 Torsional stiffness testing setup using a stepper motor ............................................. 52

Figure 49 Coupler .................................................................................................................... 53

Figure 50 Laser cut pattern on the inner nitinol tube (1.8542mm OD) ...................................... 54

Figure 51 Laser cut pattern on the outer nitinol tube (2.3114mm OD) ...................................... 54

Figure 52 New aluminum mould with a patterned tube fitted inside prior to heat treatment ...... 55

Figure 53 Tip of a nitinol tube being warped/deformed at the end of a groove ........................ 56

Figure 54 Inner patterned tube fitted into an aluminum mould ................................................. 57

Figure 55 Curved patterned inner tube after shape setting ........................................................ 57

Figure 56 Concentrically assembled patterned tube system viewed under microscope ............. 58

Figure 57 Outer and inner blank tubes concentrically assembled in the rotation setup .............. 58

Figure 58 Outer Tube Bending Tests ....................................................................................... 59

Figure 59 Inner Tube Bending Tests ....................................................................................... 60

x

Figure 60 Outer Tube Torsion Tests ........................................................................................ 60

Figure 61 Inner Tube Torsion Tests ......................................................................................... 60

Figure 62 Blank tube system twist angle graph. ....................................................................... 61

Figure 63 Patterned tube system twist angle graph. .................................................................. 61

Figure 64 Maximum workspace volumes of the tube systems .................................................. 62

Figure 65 Motion of the blank tube system with 10mm curvature overlap ................................ 63

Figure 66 Polar motion graph of the blank tube system with 10 mm overlap ............................ 64

Figure 67 Clockwise motion of the blank tube system at 15mm curvature overlap. .................. 64

Figure 68 Polar motion graph of the blank tube rotation at 15mm curvature overlap ................ 65

Figure 69 Smooth rotational motion of the patterned tube system at 15mm curvature overlap .. 65

Figure 70 Polar motion graph of the patterned tube system at 15mm curvature overlap............ 66

Figure 71 Smooth motion of the patterned tube system at full (60mm) curvature overlap......... 66

Figure 72 Polar motion graph of the patterned tube system at full curvature overlap ................ 67

Figure 73 Complete failure of the tube after rotation at full curvature overlap .......................... 74

Figure 74 Cracked failure of the inner tube after rotation at 15mm curvature overlap............... 75

Figure 75 Example model of brain lateral ventricles ................................................................ 78

xi

List of Tables

TABLE 1 CONSTRAINT VALUES USED FOR SETUP 3 .................................................................... 22

TABLE 2 PATTERN PARAMETERS FOR FINAL DESIGN ................................................................. 23

TABLE 3 TUBE CURVATURE PARAMETERS ................................................................................. 27

TABLE 4 TUBE CURVATURE PARAMETERS USED IN WORKSPACE ANALYSIS .................................. 29

TABLE 5 PHYSICAL TESTING RESULTS - TUBE SET A ................................................................. 31

TABLE 6 WORKSPACE DIMENSIONS COMPARED BETWEEN 3 TUBE SYSTEMS ............................... 34

TABLE 7 NORMALIZED STRESS FROM THE INNER PATTERNED TUBE DESIGN STUDY ...................... 47

TABLE 8 NORMALIZED BTSR DESIGN STUDY RESULTS............................................................... 48

TABLE 9 NORMALIZED STRESS CONCENTRATION RESULTS FOR INNER TUBE............................... 49

TABLE 10 NORMALIZED BTSR RESULTS FOR OUTER TUBE .......................................................... 50

TABLE 11 SELECTED PATTERN DIMENSIONS FOR INNER AND OUTER TUBES............................... 50

TABLE 12 CURVATURE DIMENSIONS FOR TUBE SET B EXPERIMENTS ......................................... 55

TABLE 13 STIFFNESS PROPERTY TESTING RESULTS FOR TUBE SET B .......................................... 59

TABLE 14 WORKSPACE VOLUME COMPARISON ......................................................................... 62

TABLE 15 TUBE STIFFNESS PROPERTIES FROM PRIOR RESEARCH PERFORMED BY LEE ET AL [35]. . 76

TABLE 16 TUBE STIFFNESS PROPERTIES FOR TUBE SETS A AND B ............................................. 76

xii

List of Abbreviations

MIS Minimally Invasive Surgery

OD Outer Diameter

ID Inner Diameter

BTSR Bending to Torsional Stiffness Ratio

SIMP Solid Isotropic Material with Penalization

FEA Finite Element Analysis

FEM Finite Element Method

MLH Tube Multilayer Helical Tube

xiii

List of Symbols

𝜌 Element density

𝐶 Compliance

𝛿 Deflection (tip displacement)

𝑘 Stiffness

𝑝 Solid Isotropic Material with Penalization

𝑣𝑜𝑙𝑓𝑟𝑎𝑐 Volume Fraction Constraint

𝑐 Constant: (1 + 𝑣)‖𝑢1‖‖𝑢2‖

𝐿 Length of overlap between the curved sections of 2 concentric tubes.

𝑣 Poisson’s Ratio

𝑅 Radius of Curvature

𝑢𝑖 Curvature: 1 𝑅⁄

𝑘𝑥𝑖 Bending Stiffness

𝑘𝑧𝑖 Torsional Stiffness

𝐸 Young’s Modulus

𝐺 Shear Modulus

𝐼 2nd Moment of Area

𝐽 Polar Moment of Area

𝑠 Position along the arc length of the manipulator

𝛼(𝑠) Relative twist angle between tube at an arc length position 𝑠

𝑑𝑛(𝑢 | 𝑚) Jacobi Elliptic Function

𝜑𝑖 Torsional Deflection of the ith tube

𝑃 Pushing Load

𝑇 Torsional Load

𝑙 Cantilever distance

𝑤 Distance Between Cutouts

𝑓 Diamond Side Length

xiv

List of Equations

Equation I………………………………………………………………………………… 5

Equation 1………………………………………………………………………………… 7

Equation 2………………………………………………………………………………… 7

Equation 3………………………………………………………………………………… 7

Equation 4………………………………………………………………………………… 8

Equation 5………………………………………………………………………………… 8

Equation 6………………………………………………………………………………… 8

Equation 7………………………………………………………………………………… 8

Equation 8………………………………………………………………………………… 8

Equation 9…………………………………….…………………………………………… 10

Equation 10………..………………………….…………………………………………… 10

Equation 11…………………..……………….…………………………………………… 10

Equation 12…………………………………...…………………………………………… 10

Equation 13…………………………………...…………………………………………… 14

Equation 14…………………………………...…………………………………………… 14

Equation 15…………………………...…………………………………………………… 14

Equation 16…………………………...…………………………………………………… 17

Equation 17………..……………………………….……………………………………… 17

Equation 18…………..…………………………….……………………………………… 42

Equation 19………………...……………………………………………………………… 42

Equation 20…………………...…………………………………………………………… 71

Equation 21……..………………………………………………………………………… 71

1

Introduction

Concentric Tube Robots in Minimally Invasive Surgery

Over the past two decades, research and development in minimally invasive surgery (MIS) has

been extensive [1]. MIS procedures have replaced conventional procedures in a variety of areas,

including heart surgery, thoracic surgery, neurosurgery, orthopaedic surgery, urologic surgery,

etc. [2]. Some of the common advantages of MIS include better safety brought by smaller

incisions and reduced blood loss, a more speedy recovery, and typically a shorter length of

hospital stay [3], [4]. One method of performing MIS that is currently spreading in both clinical

use and research is robotic surgery. This usually consists of minimally invasive tools used in

conjunction with an endoscope to look inside a patient’s body, also called endoscopy. The

surgeon is able to control the robotic arm precisely using a joystick while looking at a high-

definition image of the operating site[4], [5].

While MIS has revolutionized a large breadth of surgical interventions, its application and

adaptation in areas such as heart surgery and neurosurgery still faces many challenges. To

expand the use of robotic MIS, there needs to be smaller and more dexterous robotic tools that

can navigate restricted and convoluted anatomy while doing minimal damage to normal tissue

structures. Currently, the smallest commonly used interventional instruments are steerable

needles and catheters, which are known to be as small as 0.8 mm in diameter [6]. However, these

instruments are typically only used for drug delivery and/or ablation of tissue. Their use is very

limited in surgical procedures requiring incisions, grasping and direct physical manipulation of

tissue [7] due to their lack of structural rigidity.

In recent years, development on concentric tube robots for use in minimally invasive robotic

surgery has become prevalent. Effective models for the mechanics and construction of concentric

tubes robots were first presented in 2006 by the Webster et al [8] and Dupont et al [9]. The

manipulator arm of a concentric tube robot is most commonly composed of 2 or 3 concentrically

assembled pre-curved tubes. The cross section sizes of the tubes are comparable to that of

steerable needles and catheters; the arm can as small as 0.8mm in outer diameter (OD) at its most

distal end and 3mm or smaller throughout the rest of the arm’s entire length [7], [10]. Concentric

tube robots have rigid manipulators, can be more dexterous than existing solutions (e.g. manual

2

laparoscopic tools), and do not require the attachment of actuation mechanisms to steer distal

sections of the tool which enter a patient’s anatomy [7], [11]. Rather, the shape of the

manipulator and position of the distal tip are controlled by rotational and translational actuators

located at the proximal end of each concentric tube. As the pre-curved tubes are actuated relative

each other, their concentric alignment forces them to conform to a mutual resultant curvature.

The length and radius of this curvature is determined primarily by the stiffness properties and

curvatures of each individual tube. This allows for the manipulator to independently navigate

narrow and winding portions of anatomy without causing unnecessary damage to surrounding

tissue, in addition to performing surgical procedures at the target site. The tubes used in

concentric tube robots are made from nitinol due to its superelastic properties. An alloy

comprising of mostly nickel and titanium, nitinol tubes can an exhibit recoverable strain limits of

up to 6-8% [10], [12], [13], a very high value for metals. The superelastic property of nitinol

allows tubes to have very tight curvatures that can be straightened out under loading and still

retain its curve upon unloading. Dupont et al demonstrated in 2011 that via simulations that the

use of concentric tube robot can be feasible for intraventricular neurosurgery [14]. Using an

algorithmic approach, parameters for the design of a concentric tube manipulator was created for

choroid plexus ablation. This was demonstrated by simulating a concentric tube robotic system

in a 3D magnetic resonance image of the lateral ventricles of a brain and identifying the choroid

plexus [14]. In 2012, a successful percutaneous beating-heart patent foramen ovale closure was

performed on a porcine model by Dupont et al [15]. Concentric tube robotic system was used to

incise and remove tissue in addition to irrigating and aspirating the region. Various types of end

effectors or endoscopes can be fitted at the tip of the innermost tube, while actuation cables and

fibre optic cables can be fed through its lumen. The potential of concentric tube robots have been

demonstrated in animal trials and simulations of surgical interventions by Dupont et al in the past

[7], [14], [16].

Figure 1. A 2 dual-tube concentric tube system, where each tube has a rotational and translational degree of motion

3

Mechanical Limitations of Concentric Tube Robots

The choice of tube parameters to be used in concentric tube manipulators are subject to certain

mechanical limitations, the one that this work will address is the problem of instability. This

instability is also referred to as the “snapping” phenomenon, where the tip can suddenly “snap”

from one position to another. This occurs when the the curved sections of the tubes have an

overlap and are rotated 180° apart (i.e. when the relative twist angle between them is too large)

[10]. The sudden motion that occurs during snapping is an unstable motion that is very difficult

to predict and uncontrollable, thus it can not be allowed to occur in surgical procedures. In

general, snapping instability occurs in tube systems which include a combination of tubes with

very high curvatures (small curve radii) or long curve lengths. The goal of this research was to

use optimization methods to find a method of laser-cutting patterns onto nitinol tubes in order to

alleviate the snapping problem in concentric tube systems. It has been found by Dupont et al that

this phenomenon can be avoided by reducing the bending to torsional stiffness ratio in the

individual nitinol tubes that make up the manipulator. Achieving this goal will allow concentric

tube robots to use tubes with longer curvatures and smaller curve radii, thus improving the

dexterity and increasing the reachable workspace size of concentric tube manipulators.

Expanding the freedom of tube parameter choice will lead to easier application and design of

concentric tube robots for challenging surgical procedures in the future. The Related Literature

chapter provides further explanation of the snapping phenomenon.

Figure 2. A slender beam under compression is unstable when it is upright, since it can buckle in either direction.

When a tube is moved from a bent stable position to the middle upright position, it will suddenly snap to a new bent

configuration.

4

To help understand the problem and motion of the snapping, Webster et al used an elastic

buckling beam analogy [17]. The snapping motion in a concentric tube system is similar to a

beam loaded in compression depicted in Fig.2. The beam is unstable when upright because it can

buckle in either direction, this can be compared to when 2 concentric tubes are rotated opposite

to one another. Further, as the beam is rotated about the joint attached to ground, there will be a

point when it will suddenly “snap through” to the other side. The factors which determine

whether the beam buckles are its stiffness, length, and the amount of load applied. Similarly, the

factors which affect snapping in a concentric tube manipulator are bending stiffness, torsional

stiffness, tube curvatures, and curve length. This research used a topology optimization method

as well as a parametric design study to generate a novel cutting pattern design to be used in

concentric tube systems. The topology optimization method used was the Solid Isotropic

Material with Penalization (SIMP) algorithm, an established algorithm first conceptualized by

Bendsoe (1988) [18], [19]. The effectiveness of the designs was evaluated using simulations as

well as experimental testing and compared to prior work.

SIMP Topology Optimization

The Solid Isotropic Material with Penalization (SIMP) algorithm is a gradient-based topology

optimization method that involves the use of finite element analysis (FEA). The objective in an

SIMP algorithm is a function of the densities of all individual elements throughout a meshed

design space [18]–[20]. This method has since been widely used in the automotive and aerospace

industries [21], [22] to design structural components. In SIMP, the user defines a meshed 2D or

3D design space in which the density of each element 𝜌𝑖 ranges between 0 and 1, where 0 is fully

void and 1 is fully dense. Each iteration of the algorithm adjusts the density of every single

element based on sensitivities as well as the surrounding elements. Common uses of SIMP

include designing structures with minimal (or maximal) compliance, where is objective can be

written as

5

{

𝑚𝑖𝑛: 𝐶(𝜌) = ∑𝜌𝑒

𝑝 𝛿𝑒𝑇𝑘𝑒𝛿𝑒

𝑁

𝑒=1

𝑤ℎ𝑒𝑛: 𝑣𝑜𝑙𝑓𝑟𝑎𝑐 = 𝑉𝐾∆= 𝐹

0 < 𝜌𝑚𝑖𝑛 ≤ 𝜌 ≤ 1

𝐼

Where 𝐶(𝜌) is compliance as a function of element density. 𝛿𝑒 and 𝑘𝑒are the displacement

vector and stiffness matrix of a single element, N is the total number of elements, and 𝑝 is a

density penalisation exponent designated as 3. Naturally, the design with the highest possible

stiffness occurs when the entire design space is solid, however this is not viable. The term

𝑣𝑜𝑙𝑓𝑟𝑎𝑐 constrains how much of the design space remains solid and how much is void. For the

purposes of this research, the formulation of the objective function will be similar, but the goal

will be to minimize the bending to torsional stiffness ratio (BTSR).

Figure 3 An example of a 2D SIMP topology optimization result for a load-bearing structure

Thesis Overview

The purpose of this research was to design nitinol tube cutting patterns which decrease the

overall bending to torsional stiffness ratio (BTSR) with the aid of topology optimization, ergo

making it possible to incorporate tubes with longer curve lengths and greater curvatures (smaller

curve radii) to be used in concentric tube robots. The tubes used in this research were split into

Tube Set A and Tube Set B; each tube set had its own method of design, experimentation and

analysis. The numeration of the chapters is continuous through the two sections, but are split up

into Chapters 1-3 which describe Tube Set A and Chapters 4-6 which focus on Tube Set B. To

achieve the goal of this research, the BTSR reduction of each cutting design was studied using

finite element analysis, and then validated by testing the difference in torsional and bending

stiffness properties between patterned tubes and blank tubes. The designs were then validated by

6

observing the elimination of snapping instability when a patterned tube combination is used

instead of blank tubes when both are tested using the same parameters.

Chapter 1 includes the methodology of how the cutting design for Tube Set A was generated and

simulated using modelling and finite element methods (FEM). This chapter includes the setup of

the SIMP topology optimization algorithm using the software Altair Hyperworks. The

methodology for initial design generation is very similar between Tube Sets A and B. The

chapter also presents the raw results of topology optimization and the post-processed final

design. The analyses using FEA are also included here and tabulated.

Chapter 2 encompasses the methodology and results of all physical experimentation on the

patterned tube design. This chapter also describes an analysis on the change in maximum

possible workspace of a concentric tube robot due to the employment of the laser-cut pattern.

This includes images and tabulations of the equipment used, the resulting data, and visualizations

of manipulator workspace. Chapter 3 provides the discussion of results from the previous 2

chapters, as well as further analyses, lessons learned, comparisons, explanations of what was

observed, and the rationale for further work to be done using Tube Set B.

Chapter 4 is the first of three chapters focusing on Tube Set B, and outlines the methodology and

results of the design generation, design study and parameter choices made for this tube set. The

topology optimization setup here was very much shared with the methodology described in

Chapter 1, so this chapter elaborates more on the design study of cutting pattern dimensions.

The fifth chapter describes the methodology and results of material property testing, design

validation, and analytical analyses regarding Tube Set B. It expounds on the improvements and

differences in procedure and results in comparison to the content in Chapter 2.

Chapter 6 analyses and discusses the results obtained from Chapters 4 and 5. There are also

comparisons made to the results for Tube Set A, as well as similar work performed by other

groups in the past. The lessons learned, impacts, and potentials of the results and process for this

research is also stated here. Finishing this thesis is a conclusion that provides a brief summary of

the research which addresses all 6 Chapters.

7

Related Literature

Physics and Analytical Modelling of the Stability Problem

As for the snapping problem in concentric tube robot design, prior work by Dupont et al [11] has

demonstrated that the factors determining whether instability will occur are the curve length,

curve radius, and the Poisson’s ratio of each individual tube. So long as a tube set satisfies the

criterion given in Eq.1, the manipulator will not exhibit snapping instability [11]. The constant 𝑐

is expressed in Eq.2.

L√c <π

21

𝑐 = (1 + 𝑣)‖𝑢1‖‖𝑢2‖ 2

1 + 𝑣 =𝑘𝑥𝑖𝑘𝑧𝑖

=𝐸𝐼

𝐺𝐽3

Here, 𝐿 represents the length of overlap between the curved sections of two tubes. The constant 𝑐

is a function of the Poisson’s ratio (𝑣) and the curvatures of tubes 1 and 2, represented by 𝑢𝑖

(reciprocal the radius of curvature 𝑟𝑖). Meanwhile the expression (1 + 𝑣) equates to the ratio

between the bending stiffness (𝑘𝑥𝑖) and torsional stiffness (𝑘𝑧𝑖) of an individual tube, also called

the bending to torsional stiffness ratio (BTSR). This stability criteria was derived separately by

the Dupont group [11] in 2009 as well as the Webster group [17] in 2016 using two different

approaches. The former used a torsional model which predicted the location and twist angle of

the distal tip of the concentric tube system based on the actuation input angle, while the latter

used a minimum energy model to predict the configuration of the tube system. Eqs.1-3 point out

that we can potentially satisfy the stability criteria without having to compromise the curvature

parameters of the tubes. Rather, in order to lessen then value of the left side of Eq.1, one can do

so by reducing the BTSR.

Torsional Model Approach

The torsional model was developed to create the forward and inverse kinematics for concentric

tube robots. The relative twist angle between the two tubes at any point along their arc length is

expressed as a function of the relative actuated rotation angle at the proximal end (input twist

angle) for a specified length of curvature overlap. Eq.4 is a differential equation derived by

8

Dupont et al [11] which describes relative twist angle 𝛼 as a function of the location on the tube

system along its arc length 𝑠.

�̈�(𝑠) = 𝑐 sin 𝛼(𝑠) 4

Here, �̈�(𝑠) is the second derivative of the relative twist angle with respect to arc length 𝑠. The

system is defined such that 𝑠 = 0 at the actuation point and 𝑠 = 𝐿 at the most distal tip of the

system and 𝑠 ∈ [0, 𝐿]. Integrating Eq.4 gives us

𝑠 =±1

√2𝑐∫

𝑑𝛼

√cos(𝛼𝐿) − cos(𝑎)

𝛼(𝑠)

𝛼(0)

5

The expression in Eq.5 is identified as an elliptic integral of the first kind [11] [23], which can be

written in terms of Jacobi elliptic functions, thus Eq.6 relates the relative twist angle 𝛼(𝑠) at any

point along the tube and the twist angle 𝛼(𝐿) at the distal end.

sin (𝛼(𝑠)

2) =

sin (𝛼(𝐿)2 )

𝑑𝑛 ((𝐿 − 𝑠)√𝑐 | 𝑐𝑜𝑠2 (𝛼(𝐿)2 ))

6

sin (𝛼(0)

2) =

sin (𝛼(𝐿)2 )

𝑑𝑛 (𝐿√𝑐 | 𝑐𝑜𝑠2 (𝛼(𝐿)2 ))

7

sin (𝛼(𝐿)

2) = 𝑑𝑛 (𝐿√𝑐 | 𝑐𝑜𝑠2 (

𝛼(𝐿)

2)) 8

Here, 𝑑𝑛(𝑢 | 𝑚) is a Jacobi elliptic function [23]. Setting 𝑠 = 0, we can relate the input twist

angle 𝛼(0) and the distal twist angle 𝛼(𝐿), obtaining Eq.7. Since snapping occurs when the tubes

are rotated directly opposite of one another, we are only interested when 𝛼(0) = 𝜋 and hence

sin (𝛼(0)

2) = 1, which yields Eq.8. Using a variety of identities, Dupont et al concludes that

multiple solutions to Eq.8 exists when 𝐿√𝑐 > 𝜋/2 [11]. This is effectively a singularity in the

workspace of the robot, and the manipulator will exhibit sudden snapping when one tube is

continuously rotated against the other.

9

Energy Model Approach

The snapping motion can also be viewed as sudden release of torsional strain energy. As the two

tubes rotate against one another, they build up in torsional strain energy which, in a tube set that

does not satisfy Eq.1, will be released, moving the system to a new configuration [17], [24]. The

most stable configuration between any two pre-curved tubes is when their input relative twist

angle is 0, which means that their planes of curvature are exactly aligned. Tube systems aligned

this way have the lowest possible amount of a stored strain energy. If the curvatures and curve

lengths of the tubes are the exact same, the system will theoretically have zero stored energy.

When two tubes have a relative twist angle, they will each be twisted individually by a certain

angle 𝜑𝑖 and experience static torsion. A tube system will always naturally seek to obtain the

configuration that results in the lowest possible energy, and for a globally stable tube set, this

corresponds to a specific 𝜑𝑖 (torsional deflection of the ith tube) for both the inner tube and outer

tube. Thus, a set of all the possible combinations of 𝜑1and 𝜑2 for a specific relative input twist

angle 𝛼(0) between the two tubes gives us an entire energy landscape[10], [17]. This is shown in

Fig.4 .

Figure 4 Energy landscape of an unstable tube system at different input relative rotation angles (Webster et al) [17]

For a tube set in which local instability exists, there will be more than one local minimum

when 𝛼(0) = 𝜋, this is called a bifurcation in the energy landscape. As 𝛼(0) is steadily increased

from 0 to 180° in a system where snapping occurs, a new local minimum will develop in the

10

energy landscape in addition to the global minimum to which the tube set is configured. As 𝛼(0)

gets closer to 180°, the value of secondary minimum approaches that of the global minimum

until they reach the same value when 𝛼(0) = 180°. It is at this point where the local minimum

becomes the new global minimum and the tube snaps through to a new configuration. The new

configuration has a new set of 𝜑1 and 𝜑2 twist angles and thus a new end effector position and

resultant curvature plane. Using the energy approach, the Webster group developed the same

stability criteria independent from the torsional model approach [17]. They express it as

𝜆 <𝜋2

4, 𝑤ℎ𝑒𝑟𝑒 𝜆 = 𝐿2‖𝑢1‖‖𝑢2‖

𝑘𝑧1−1 + 𝑘𝑧2

−1

𝑘𝑥1−1 + 𝑘𝑥2

−1 9

This takes into account the difference in stiffnesses and Poisson’s ratios between the inner and

outer tubes, and is the same criteria described by Dupont et al. Combining Eqs.1-3, we re-write

the stability criteria as Eq.10.

𝐿√𝑘𝑥𝑘𝑧‖𝑢1‖‖𝑢2‖ <

𝜋

210

1

𝑘𝑥=

1

𝑘𝑥1+

1

𝑘𝑥2 ,

1

𝑘𝑧=

1

𝑘𝑧1+

1

𝑘𝑧211

𝐿√𝑘𝑧1

−1 + 𝑘𝑧2−1

𝑘𝑥1−1 + 𝑘𝑥2

−1‖𝑢1‖‖𝑢2‖ <

𝜋

212

To account for the different stiffnesses of tubes 1 and 2, we recognize the tubes as being separate

sets of 2 bending springs and 2 torsional springs in series and use the relation Eq.11 to obtain

Eq.12, which equivalent to Eq.9. For the purposes of this research, we will use the stability

criteria expressed as Eq.12.

Once again, the goal of this research was to decrease the bending to torsional stiffness ratio of

nitinol tubes to be used in concentric tube systems. Two past groups Azimian et al [25] and Lee

et al [26], [27] have demonstrated that this could be achieved by cutting anisotropic patterns onto

the surface of straight nitinol tubes before shaping them into the desired pre-curvatures. These

cut patterns are largely similar in design and are primarily comprised of very small horizontal

11

slits patterned throughout the curved length of the nitinol tubes. More information regarding

related prior work is covered in the next section of this chapter. These past designs however,

were mostly based off the educated intuition of the researcher, and no attempt has been made to

use optimization methods to find a cutting pattern from the ground up.

In 2016, Webster et al had attempted to use the sudden motion of the snapping phenomenon for

benefit [28]. A high amount of energy is released when the snapping motion occurs, and this was

used to generate the force needed for soft tissue suturing. Using model of the stored energy of a

concentric tube system, the occurrence of snapping and the amount of energy released could be

predicted. This application was tested by the Webster group on synthetic skin, and the concentric

tube system was able to successfully able to drive the need through the simulated tissue [28].

However, a specific application had not been specified where it would be beneficial to employ a

concentric tube system to perform a suture operation. As a proof-of-concept, this showed that

there are other potential uses for concentric tube robot and that perhaps instead of avoiding the

snapping motion, it could be harnessed.

Avoidance and Elimination of Snapping Instability

Two groups, Lee et al [26], [27] and Azimian et al [25], have been successful in reducing tube

BTSR by cutting patterns. They were also successful in alleviating the snapping problem in

concentric tube systems using their designs. The research by Azimian et al was performed at

CIGITI laboratory at the Hospital for Sick Children. In said research, a multilayer helical (MLH)

tube (Fig.5) solution to the snapping problem was considered and tested [25]. However, the

nonlinear nature of MLH tubes meant that their incorporation in concentric tube systems would

be difficult and conventional beam-based models would lead to significant inaccuracies. In

addition, despite using 3 layers of helical tubes, the torsional stiffness of the MLH tube in one

direction turned out to be much greater than that of the other. This difference resulted in the

motion of the tested concentric tube system being smooth in one direction while still exhibiting

snapping in the other. Hence, this solution was rejected in favour of a laser etched cellular

pattern which consisted of small horizontal slits repeated throughout the curved length of the

tube (Fig.6). The formulation of this pattern was a trial and error process based on models of the

material properties of auxetic cellular tubes; by controlling its pattern geometry, one can

decrease the BTSR of a tube. After both numerical and experimental testing, they found that the

12

horizontal slit design yielded good results and was able to improve the stability margin

(maximum length of curve overlap) by up to 40%.

Figure 5 Example of a multilayer helical tube structure (Weng et al) [29]

Figure 6 The horizontal slit pattern used by Azimian et al [25]; dimensions are in millimetres

In 2015, the Lee group published a more in-depth study [26] in which an analytical model of the

slit tube design was made and several tube sets with differently dimensioned varying patterns

were tested. The analytical model was created so that one could predict the effect of pattern

parameters on the stiffness properties of the tube. Although assumptions in the model resulted in

inaccuracies, it made the choice of parameters easier without having to go through a blind trial

and error process. Six patterned tubes of different dimensions were tested and compared to their

13

corresponding blank tubes. The results showed that the BTSR of a patterned tube could be

reduced to 26.9% of that of a blank tube. Employing this tube design resulted in smooth motion

in a concentric tube system whereas a set of blank tubes would have exhibited snapping should

the same parameters be used. The slit pattern however, has two significant drawbacks. The first

one is that the cut-out pattern on the tubes would interlock with one another and become

“completely stuck together” [26]. A temporary fix was found by applying a PTFE sleeve

between the tubes to physically separate the patterned tubes. Another drawback mentioned was

that the bending stiffness of the patterned tube could be as low as 15% of that of the blank tube.

This makes for a very flimsy manipulator and such a system could see the same limitations in

surgical applications as the steerable needles and catheters mentioned in the Introduction.

14

Overview of Chapters 1, 2, 3 – Tube Set A

In the following 3 Chapters, the presented information encompasses all design, setup,

experimentation, and analysis performed using Tube Set A. This covers the first

experimentations using SIMP topology optimization to optimize for tube design and the

subsequent refinements of the optimization setup. The pattern was generated and finalized for the

first time, and validation was performed to observe its effects. These chapters showed that using

topology optimization as a starting point to design tube cutting patterns was a viable method and

showed great potential.

Design Generation and FEA – Tube Set A

1.1 Methodology – Topology Optimization and FEA

Altair Hyperworks Optistruct was used to perform the optimization using the SIMP algorithm.

The setups were split into two parts: Setup 1, Setup 2, and Setup 3, with the latter being a

progression of the former. Setup 1 was also split into Operations I and II to observe how

different object functions affected the optimization result.

1.1.1 Setup 1

Setup 1 used a design which consists of a tube of 3 mm OD, 2.5 mm ID, and 20 mm in length.

Single point constraints, which constrain all 6 DOFs, are applied to all nodes on the proximal end

face of the tube. A variety of loading scenarios are applied to the distal end of the tube. Eight

bending loads are applied in eight different directions (Fig.7). Two torsional moments of

opposite directions are applied along the Z axis (longitudinal). Two different objective functions

were used in separate optimization operations. Operation I (Op.I) used an objective function

(Eq.13) that was based on compliance.

15

Figure 7 Loading scenario for Op.I of Setup 1

𝑚𝑖𝑛: 𝑓(𝜌) =∑ 𝐶𝑧𝑖𝑁𝑖=1

∑ 𝐶𝑥𝑗𝑀𝑗=1

𝑠. 𝑡. 𝑣𝑜𝑙𝑓𝑟𝑎𝑐 ≤ 𝑣 13

𝑘𝑥𝑘𝑧=𝐶𝑧𝐶𝑥

14

Here, 𝜌 represents the density of individual elements in the design space, 𝐶𝑧𝑖 and 𝐶𝑥𝑗 represent

the compliances of the tube under each torsional and bending load scenario respectively. An

heuristic volume fraction constraint “𝑣𝑜𝑙𝑓𝑟𝑎𝑐” was set to control the amount of material

remaining in the design space. Since compliance is the reciprocal of stiffness for both bending

and torsion, Eq.14 holds true, and thus Eq.13 could be used as the objective function. Operation

II (Op.II) was based on static deflection of the distal tip of the tube instead of the compliance of

the entire tube. Since static deflection is inversely proportional to stiffness, an objective function

similar to that in Op. I was used (Eq.15). Here, 𝛿 represents static deflection.

𝑚𝑖𝑛: 𝑓(𝜌) =∑ 𝛿𝑡𝑖𝑁𝑖=1

∑ 𝛿𝑏𝑗𝑀𝑗=1

𝑠. 𝑡. 𝑣𝑜𝑙𝑓𝑟𝑎𝑐 ≤ 𝑣 15

Different volume fraction (𝑣𝑜𝑙𝑓𝑟𝑎𝑐) constraints were experimented with in the process for both

operations. Multiple optimization runs were performed in order to obtain a variety of topologies,

16

so that the most feasible result could be chosen for post-processing and analysis further. The

chosen result from each operation was exported and remodelled in Solidworks, where fillets

were added in order to smooth out high stress concentrations.

Once it was recognized that this method of using topology optimization was useful in reducing

the BTSR, a version of the tube design was cut using a CNC laser machine. Material property

testing was performed on the tube to evaluate its bending stiffness, torsional stiffness, as well as

strength under strain. Based on results seen from the actual behaviour of the sample, additional

constraints to subsequent optimization setups were made.

1.1.2 Setup 2

For the 2nd optimization setup, the same tube length was used with a simplified loading scenario.

Instead of employing 8 different bending moments distributed in all directions around the centre

node (Fig.7), only 4 were used so that the centre node was surrounded only on one side, shown

in Fig.8.

Figure 8 Loading scenario of Setup 2, taking into account the symmetry of stiffness in a tube structure.

Since it is known that the bending stiffness of a beam is the same if the loading direction is

reversed, halving the number of loads has no effect on the outcome. Based on the results from

the first setup, it was judged that using compliance as the argument in the objective function was

17

the better choice. Thus, Eq.13 was used in this setup as well in addition to three constraints: a

minimum member size constraint, an upper bound constraint on stress, and an upper bound

constraint of variance of stress throughout all bending directions. The minimum member size

constraint simply specified the minimum thickness of the tube design at any location. This was

employed to make the design more feasible to manufacture. The stress constraint was employed

so that structural integrity would already be taken into consideration even before post-

processing. Lastly, the constraint on the variance of stress in all 4 loading directions worked to

improve uniformity and continuity of the topology. The optimization results from this setup

helped adjust the limits used for the three constraints mentioned earlier. These constraints were

also used in the following Setup 3.

1.1.3 Setup 3

The 3rd setup was performed in an effort to obtain better patterning, standardization, strength,

and feasibility. This setup was designed to be a unit cell pattern optimization, meaning that the

tube was symmetrically loaded instead of using boundary conditions. Such a method creates a

short, but more continuous pattern/topology throughout the length of the short tube. To account

for variation in topology all around the tube, 8 bending loads were used this time instead of 4,

thus doubling the density of bending directions (see Fig.9).

Figure 9 Symmetric loading setup used in the short segment tube of Setup 3

18

The hypothesized outcome of the unit cell optimization setup is so that a repeatable pattern can

be generated from the 3rd setup. The design space consisted of a tube that was 2.3mm in diameter

and 3mm in length. Drawing lessons from the 2nd setup, the same constraints on member size,

element stress, and stress variance were used in this 3rd setup. After several adjustments to the

algorithm parameters, one resultant topology was chosen to be post-processed using Solidworks.

This procedure was meant to generate a cell pattern that is repeated both laterally around and

longitudinally along the tube. The average dimensions of the pattern were obtained by making

measurements within HyperView. These dimensions were used to generate a repeating pattern

on a tube using Solidworks.

1.1.4 Finite Element Analysis Studies

The resultant 3D models were then meshed for FEA. This analysis applied bending loads and

torsional moments to the distal tip of the post-processed tubes. Eq.16 and Eq.17 were used to

determine the bending and torsional stiffness values.

𝐾𝑏 = 𝐸𝐼 =𝑃𝑙3

3𝛿16

𝐾𝑡 = 𝐺𝐽 =𝑇𝑙

3𝛿17

The ratio 𝑲𝒃

𝑲𝒕 was taken as the BTSR for the specific bending direction. The mean BTSR value for

the X and Y bending directions was compared to the BTSR for a blank tube of same length with

no cuts. The reductions in BTSR for Op.I and Op.II were then normalized with respect to the

BTSR of a blank tube. In order to compare these results to previously existing cut patterns, the

same FEA was performed on a slit pattern tube. To have a controlled comparison, the volume of

the slit tube was roughly equal to either the Op.I or Op.II model, depending on which had a

lower BTSR. The BTSR of the slit tube obtained from FEA was then compared to the best-

performing model out of the two optimization operations. The material used for all the analyses

was nitinol. An accurate parameter for the elastic and shear moduli could not obtained because

the material properties of nitinol can vary drastically between batches of nitinol; this is

something that depends greatly on the heat settings and material composition at the time of

manufacture.

19

In addition, a analysis on stress concentration of the post-processed design from Setup 3 was

performed by applying an enforced displacement. This approach in studying stress was chosen

because simply comparing the maximum element stresses between the blank tube and the cut

tube (model generated from the 3rd setup) would not be realistic. When employed separately in a

concentric tube robot, the blank tube and the patterned tube would not experience the same

magnitudes of loading. Thus, comparing stresses by controlling the loading magnitude and

direction would not be a valid study. Hence the two tubes were set up to be displaced by the

same amount at the tip. The tube lengths used throughout this analysis were all 11.9mm. An

extremely drastic displacement of 2.3507mm was applied to both analyses. This value was based

on a tube radius of curvature of 30mm, which is also a very tight bend. For the cut tube, this

displacement was also applied in 8 different directions. The results from this FEA comparison

would include the BTSR reduction from a regular tube, as well as the increase in stress caused by

the cut topology.

1.2 Results – Design Generation and Analysis

1.2.1 Setup 1

Fig.10 and Fig.11 represents the optimized results from Op.I and Op.II respectively as they

appear in Altair Hyperworks, the solid areas of the design are colored red while the void spaces

are in blue. Fig.12 shows the remodelled and post-processed versions. These are designated as

model 1 and model 2. The upper bound volfrac used for both results was 0.7. The FEA results

showed that operation 1 exhibited the greater BTSR reduction of 61%.

Figure 10 Raw optimization result from Op.I, using the compliant-based objective function

20

Figure 11 Raw optimization result from Op.II, using the displacement-based objective function

Figure 12 Post-processed feasible design based on the Op.I result

Figure 13 Slit pattern tube model used for comparison

The total volume of the model resulting from Op.I was 40.9 mm3. The volume of the slit tube

model (Fig.13) made for comparison was 39.84 mm3. FEA analysis showed that the BTSR of the

slit tube model had a 49% reduction from the original blank tube. The bending stiffness in the

X+ and Y+ directions were 1.31Nm2 and 0.662Nm2 respectively, which has a variance of at least

20.99%. Fig.14 shows the topology cut on a tube. This cut topology was made using a CNC laser

cutter. This cut tube however, proved to be too weak under bending stress and exhibited

plasticity very easily.

Figure 14 Laser cut tube using the model in Fig. 12

21

1.2.2 Setup 2

Fig.15 and Fig.16 show one set of results from the 2nd optimization setup. The values for

maximum Von Mises stress, stress variance, and minimum member size constraints were

determined in a trial-and-error process to come up with a feasible result. Additionally, further

adjustments of the volfrac parameter showed that the volume fraction constraint was unnecessary

and redundant; the objective function used does not force the entire workspace to become either

void or completely solid. Therefore the volfrac constraint was omitted from this point onwards.

Fig.11 shows the result with the void spaces taken out, called an isolated view. This was done by

only displaying elements with 0.5 density and above. The results of the 2nd setup demonstrated

the effectiveness of the new constraints added. It also showed that the omission of the volfrac

constraint did not lower the quality of the optimization results. Decreasing the stress constraint

mostly caused the cut-out features to be smaller, and more rounded. Reducing variance was very

effective in producing a more distinctive recurring pattern in the results. This was a problem in

Setup 1. The incorporation of the minimum member size constraint made sure that there were no

open edges in the optimization result.

Figure 15 Topology optimization result of Setup 2 showing averaged density contours

Figure 16 Topology optimization result of Setup 2 using averaged element densities, showing volumes with density above 0.5

1.2.3 Setup 3

The results for the 3rd setup (unit cell setup) were generated and are shown in Fig.17. Once again,

the left image shows the density contour map, and the right displays all portions of the result

22

where the density is 0.5 or above (iso view). During the post-processing of the result shown in

Fig.17, the average size of the rhombus-shaped orifices and the average distances in between

were used to make a diagonal grid pattern on the tube. The constraint parameters to obtain the

results shown in Fig.17 are listed in Table 1. This pattern is modelled onto a tube that is 11.9mm

long, as shown in Fig.18. The post-processing procedure involved taking two important

dimensions from the rhombus pattern: the side lengths and the distances between them. In the

end, these dimensions do not correspond strictly to the direct optimization result because stress

concentration has to be taken into consideration. Hence the addition of fillets and additional

member thickness. The dimensions chosen are listed in Table 2.

TABLE 1 CONSTRAINT VALUES USED FOR SETUP 3

Constraint Value

Maximum Stress 15 MPa

Variance 5 MPa

Min. Member Size 0.2 mm

Figure 17 Topology Optimization Result: Density Contour (Left), Iso (Right)

23

Figure 18 Finalized section of tube model in Solidworks

TABLE 2 PATTERN PARAMETERS FOR FINAL DESIGN

Dimension Measurement

Distance between cutouts (𝑤) 0.201mm

Diamond side length 0.15mm

Diamond radius (𝑓) 0.05mm

Instances around circumference 10

Figure 19 Design and dimensions of 1 unit of the cutting pattern

24

This model was compared to a blank tube model that was also 11.9mm long. The FEA, carried

out using Hyperworks, showed that there was BTSR reduction of 24%. Under the same set

displacement, the maximum element von Mises stress increased by a factor of 1.63. Fig.20 below

shows parts of the FEA result for the pattern cut model. The types of loads applied are described

in the Materials and Methods section. Stress Variance was lso improved. In the Setup 3 results,

the average bending stiffness per unit length throughout 8 different loading directions was

27.5Nm2 and never varied by more than 0.1Nm2. The pattern shown in Fig.18 was the design to

be used in real tube experiments in the following Chapter 2.

Figure 20 FEA Contour Results under bending, torsional and displacement loading conditions

25

Experimental Validation – Tube Set A

The ultimate goal of this project was to show that cutting the diamond shape pattern generated

from Setup 2 would help in avoiding the snapping problem when implemented on a real tube.

Whereas a system of blank tubes would exhibit instability (snapping) when the same curvature,

material, and curvature overlap parameters are used. To test this, this aforementioned pattern

(Fig.18) was laser cut onto a nitinol tube of 2.03mm OD, 1.33 mm ID, and over a length of 77mm.

A magnified image of the cut tube is shown in Fig.21. The both the blank tube and patterned tube

system were tested for their bending stiffness, torsional stiffness, and employed in a concentric

tube assembly to observe the effects of the diamond shape design.

Figure 21 Diamond pattern cut on nitinol tube (2.03mm OD)

2.1 Experiment Setup and Analysis Methodology

2.1.1 Stiffness Testing Method

To test the reduction in BTSR, a torsion stiffness test fixture and bending stiffness test fixture

were set up separately. The torsion fixture (shown in Fig.22) consisted of a Transducer Techniques

TRT-50 torsion sensor, an ALPINE potentiometer encoder, and a 3 Nm Maxon DC motor to turn

the tube. Data was transferred directly to a computer where analysis takes place.

26

Figure 22 Torsional stiffness testing setup

The bending test setup (Fig.23) consisted of an ATI force sensor mounted onto a DENSO VP-

G 6-axis robotic arm. A pushing anvil tool was manufactured (by 3D printing and milling) and

mounted onto the ATI Gamma Transducer force sensor in order to push down on the tube in a

cantilever style test. The cantilever is achieved by fixing the tool at one end using a chuck and

collet assembly. Due to the fact that the pushing anvil slips on the tube during a cantilever test, we

compensated by using an adjusted torque arm length to calculate the bending stiffness for every

step down that the anvil pushes. To estimate the torsional and bending stiffness, we use the slope

generated by a line of best fit based on the data for 2 tests for each tube.

Figure 23 Bending stiffness testing setup

27

2.1.2 Concentric Tube Rotation Test

In order to test for elimination of snapping, we deployed the blank and patterned tube in separate

concentric tube assemblies each with a larger outer tube. Thus, the blank and patterned tubes would

serve as the inner tubes in their respective assemblies. All the tubes were then pre-curved by heating

them to 565˚ Celsius for 10 minutes in a Vulcan 3-1750 convection oven while being set inside a

mold. The dimensions of the tubes are listed in Table 3. The choice of these dimensions was based

on the stability criterion (Eq.12) and the bending to torsional stiffness ratio of the tubes. At a

curvature overlap of 77mm, the blank tube system should experience snapping while the patterned

tubes should not.

TABLE 3 TUBE CURVATURE PARAMETERS

Tube ID (mm) OD (mm) Curve Radius (mm) Curve Length (mm)

Patterned Tube 1.33 2.03 47.5 77

Blank Tube 1.33 2.03 47.5 77

Outer Tube 2.13 2.33 50.25 77

The patterned tube and blank tube were both separately fitted concentrically into the outer tube

(Fig. 25) with their curved sections overlapping by a length of L=44mm. This system of tubes was

then mounted into a concentric tube rotation setup (Fig.24). This setup consisted of a 3 Nm Maxon

DC Motor clamped onto the straight section of the inner tube, an encoder attached to the inner tube

to track the input rotation angle, and a clamp to fix the outer tube in place. The movement of the

curved section and the tip of the tube assembly was observed and recorded by a video camera

located to the left of Fig.24, pointing at the tip of the tube. The tip position was tracked using the

video analysis and tracking software Xcitex ProAnalyst. The tube rotation was recorded at every

6 frames (0.5 second) and graphed by comparing the rotation angle of the tip (from its centre of

rotation) and the input rotation from the motor which was measured by the encoder.

28

Figure 24 Concentric tube rotation testing setup

Figure 25 Patterned tube concentrically assembled with outer tube.

One of the biggest challenges faced in the experiment portion was the shape setting of the nitinol

tubes. A common way to shape set nitinol place it into a mould to be held in a oven for a specified

amount of time [30]. To constrain the nitinol tube to a desired curvature, a block of aluminum was

milled out so that the tubes can be individually fitted into grooves. The fitting process subjects the

tubes to great amounts of strain. The heating procedure was refined using an iterative procedure;

the two parameters adjusted were heating temperature and heating time. In failed heat treatment

procedures, the tubes either would have a large a mount of springback or could not withstand the

29

strain of being bent back to a straight tube. The final chosen procedure used in this test resulted in

an average curve radius springback of 13.68% for the inner tube and 14.10% for the outer tube.

This was measured by taking photo of the curved tube placed on a page of 5mm grid paper and

evaluating the curve length and radius based on the images (see Fig.26).

Figure 26 An image used in the visual measurement of tube curvatures after heat treatment

2.1.3 Workspace Impact Analysis

Based on results from the experiments, we can observe the differences in workspace between

patterned tubes and blank tubes. This will provide a visual and quantitative identification of how

diamond-shape patterning of tubes can improve the usability of concentric tube robots by

allowing a greater stable operational work space. In order to demonstrate this improvement, we

chose nitinol tube combinations that satisfied the stability condition described by Eq.12. For

consistency, the tube sizes used in this analysis are kept the same as those used in Table 3. The

tube curvatures and curve lengths used for the workspace analysis are documented in Table4.

TABLE 4 TUBE CURVATURE PARAMETERS USED IN WORKSPACE ANALYSIS

Tube System # Tube Curve Radius Curve Length

1 Blank Inner Tube 47.5 mm

77 mm Blank Outer Tube 61.68 mm

2 Patterned Inner Tube 47.5 mm

77 mm Blank Outer Tube 52.84 mm

3 Patterned Inner Tube 47.5 mm

77 mm Patterned Outer Tube 47.15 mm

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The curvature of the inner tube in all analyses was kept at 47.5mm, while the outer tube

curvature was chosen so that the robot would be stable at all points throughout its workspace. A

third Tube system was also used in this analysis where the outer tube also assumed the same

percentage decrease in stiffness and BTSR. To create a representation of the workspace for a

single set of tubes, a Matlab concentric tube forward kinematics model was used to generate a

point cloud where every point was a possible tip position of the robot. The point cloud was

generated by incrementally translating and rotating the inner tube as well as rotating the outer

tube, thus the longitudinal position of the outer was fixed. A 3D volume was then created using

fitted curves around the outer surfaces of this point cloud. The size and dimensions of the

workspace volume were then compared between different tube sets.

2.2 Experiment Results

2.2.1 Stiffness Results

The graphed results of the torsion and bending tests on the patterned and blank tubes are shown in

Fig.27 and Fig.28. The results of the tests are tabulated in Table 5 and compared to the FEA

predictions.

Figure 27 Resulting trends of torsion tests on the patterned and blank tubes

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Figure 28 Resulting trends of bending tests on the patterned and blank tubes

TABLE 5 PHYSICAL TESTING RESULTS - TUBE SET A

Tube Test Torsional Stiffness Bending Stiffness BTSR

Patterned Tube Experiment 0.0127 N∙ m2 0.0119 N∙m2 0.94

Simulation 0.0143 N∙ m2 0.0136 N∙m2 0.95

Blank Tube Experiment 0.0236 N∙ m2 0.0290 N∙m2 1.23

Simulation 0.0234 N∙m2 0.0293 N∙m2 1.25

From the experimental results, we see that the BTSR was decreased from 1.23 to 0.937, a

reduction of 23.6%. The results from FEA simulation shows that the BTSR decreased from 1.25 to

0.95, a reduction of 24.0%. Reasons for this small discrepancy may be caused by the high gripping

force on the tubes needed for the torsion test, this force may cause localized deformation in the

tubes. In addition, the Poisson’s ratio used to predict the torsional stiffness of the tube (v=0.3) was

provided by the manufacturer and may not be the most accurate. Nonetheless, this test did prove

that the patterned tube exhibited a substantially reduced BTSR.

2.2.2 Tube Rotation Test Results

The tip position for both the blank tube and the patterned tube setups are shown as either blue or

red dots in Fig.29 and Fig.30. The snapping motion occurred in the blank tube system when the

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tubes curvatures were overlapped by 44mm. Each dot represents the tip position for every single

frame of video, which was recorded at 24 frames per second. Snapping motion was observed in

the blank tube setup; the tip suddenly jumped around 70˚ while rotating clockwise in the bottom

right quadrant. The patterned tube setup exhibited smooth motion, although the tip path was

more angled and different than that of the blank tube. This is attributed to the fact that the tip of

the patterned tube setup traveled at an angled plane relative to the view of the camera, as well as

the fact that the blank tube had a higher bending stiffness, which made for differing orientations

at all input angles. The graph for the motion of the blank tube is shown in Fig.31, a clear jump

can be seen in the middle of the motion, and this is attributed to snapping. Fig.32 shows the

motion for the patterned tube, where the tip exhibits much smoother motion in the same region.

Figure 29 Movement of the blank tube system, showing snapping

Figure 30 Smooth motion of patterned tube set, using patterned inner tube

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Figure 31 Graphed motion of the blank concentric tube system, showing snapping motion

Figure 32 Graphed smooth motion of the patterned concentric tube manipulator

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2.2.3 Workspace Improvement Impacts of BTSR Reduction

Figure 33 Workspace volumes of tube systems 1, 2, and 3 displayed from left to right

The workspace volumes for tube systems 1, 2, and 3 are displayed in Fig.33 together with 3 tube

configuration examples to show the reach of a concentric tube robot. Tube system 2 showed a

workspace increase of 21.4%, and tube set 3 had a workspace increase of 20.5%. The maximum

width of the workspace for tube system 2, where a patterned tube was paired with a blank outer

tube, was the largest (186.75mm), had greater loss of central distal workspace. The sizes and

dimensions of the three workspaces are presented in Table 6.

TABLE 6 WORKSPACE DIMENSIONS COMPARED BETWEEN 3 TUBE SYSTEMS

Tube System #

Workspace Dimension

Volume (mm3) Max. Width (mm3) Height (mm3)

1 8.33x105 161.35 82.39

2 1.06x106 186.75 82.39

3 1.05x106 175.46 87.45

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Discussion – Tube Set A

3.1 Topology Optimization

3.1.1 Setup 1

The results from this setup resulted in a 61% BTSR reduction, this was the result of using

compliance in the objective function. One significant observation from both Op.I and Op.II

results is the clear diagonal cut pattern, angled approximately 45 ̊ from the Z axis of the tube.

This can be correlated to the fact that in a unit stress element, the maximum shear stress occurs

along the 45 ̊ angle [31]. Since the goal is to increase torsional stiffness with respect to bending,

then having more material aligned in the direction of maximum shear stress would be beneficial.

Setup 1 demonstrated that a drastic reduction in BTSR can be achieved by using topology

optimization as the basis tube cutting pattern generation. Indeed, heuristic input is always

necessary when using SIMP optimization methods, thus this topology is far from a definitive

solution to this optimization problem. One significant aspect of this result is that the topology is

vastly different than the slit cutting patterns used in prior studies [25]–[27]. Furthermore, it

achieved a lower BTSR than a slit tube model which removed a similar amount of material from

the original tube. This showed us that using SIMP topology optimization to design concentric

tube manipulators has its merits and was worthwhile pursuing.

3.1.2 Setup 2

Here, once again the 45 ̊ angle of the topology is clear. Furthermore, this characteristic could be

seen when using most reasonable values for the new constraints added (values that lead to

realistic optimization results). Comparing these results to the 1st setup, the patterns generated by

the compliance-based algorithm were spread more evenly along the tube. Such was the reason

that compliance was chosen to be used in the objective function in the 2nd setup.

Traditionally, in SIMP optimization, a volume fraction (volfrac) upper or lower constraint is

necessary to develop a useful result. This is because SIMP is normally used to optimize for either

strictly stiff or compliant structures where the element densities would all either converge to 1 or

0 respectively. However, the BTSR objective function needs to balance between bending

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stiffness (EI) and torsional stiffness (GJ) and thus the element densities do not tend to either

extreme. This is the reason why the volfrac constraint is not necessary for this type of

optimization. The constraint parameters used for maximum stress, directional stress variance,

and minimum member size were chosen through an iterative process of trial and error.

Decreasing the stress constraint mostly caused the cut-out features to be smaller, and more

rounded. Reducing variance was very effective in producing a more distinctive recurring pattern,

which was a problem in Setup 1. Furthermore, the incorporation of the minimum member size

constraint was effective in increasing the average thickness of material throughout the design

space. While this constraint did not fully solve the issue of disconnected members, we did not

see the isolated “islands” of solid material from the Setup 1 results. See Fig.34 for comparisons.

Figure 34 Topology optimization result from Setup 1 (left) without constraints, and results from Setup 2 (right) with

stress, stress variance, and member size constraints added

3.1.3 Setup 3

In this unit cell optimization setup, the most distinctive diamond/rhombus pattern was observed,

compared to the previous 2 setup results, this result had the most consistent pattern. The

improved result was most likely caused by the both the shortened tube segment and the

symmetric loading. Based on the FEA, the model resulting from Setup 3 was less effective in

reducing the BTSR than that from Setup 1, in that the analysis showed a 24% decrease. When

the same amount of displacement was applied both the blank tube and the patterned tube, the

patterned tube also exhibited a maximum stress that was 1.63 times that of the blank tube. The

stress contour graph (Fig.35) of the model tube under enforced displacement helps to identify the

areas of stress concentration. As seen in Fig.35, the locations where stress is highest are the

fillets on the rhombus orifices. This is to be expected, because orifices and corners should be the

areas of highest stress concentration. Methods to reduce the amount of stress concentration may

include increasing the radius of the fillets or rearranging the dimensions of the rhombus pattern.

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As mentioned in Chapter 1.2.3, the variance of stiffness along different bending directions was

also reduced to 0.0727% from the 21.29% variance analysed in Setup 1.

Figure 35 Stress contour result of the enforced displacement analysis showing areas of stress concentration (red)

3.2 Analysis of Experiments

Chapter 2.2 has shown that this cellular design based on topology optimization was effective in

reducing the BTSR of a nitinol tube, as well as avoiding the snapping problem in a concentric tube

assembly. The sparseness in data points observed in Fig.32 is different than the empty gap

observed in Fig.31. This sparseness can be mostly attributed to the fact that the output twist angle

was calculated based on the centre of rotation on the tip, the path of which was an elliptical shape

instead of a circle. Thus the data points would be sparser for the portion of the path that is on the

flat section of the ellipse. The other reason that the paths for the two tubes were so different was

because of the patterned tube had lower bending stiffness, which means the stiffness of the outer

tube was more dominant. This problem can be potentially avoided by doing away with the 2.33mm

OD outer tube and instead use a smaller tube that fits inside the 2.03mm OD tube, thus re-

designating the 2.03mm as the outer tube.

This diamond-shape design is also distinct from the horizontal slit designs demonstrated in prior

work by Azimian et al [25] and Lee et al [26], [27]. A major drawback of the horizontal slit design

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is the drastic decrease in bending stiffness of the tube. This can inhibit the ability of a concentric

tube robot to perform load-bearing operations. In research conducted by Lee et al [26], the bending

stiffness for a 2mm tube (similar to the tube size used in this study) decreased from 0.015898

N*m^2 to 0.002385 N*m^2, a reduction of 85.0%. The diamond patterned tube used in this study

had a lower 59.0% bending stiffness reduction compared to a blank tube. However, BTSR

reduction in the horizontal slit tube demonstrated by Lee et al was 59.2%, while the diamond

patterned tube in this study only reduced BTSR by 24%. This points to a trade-off between not

only the two designs but also between bending stiffness and BTSR reduction. Further geometric

optimization and other analysis of the diamond pattern was completed for Tube Set B (Section B).

Two areas for improvement in the methodology presented in chapter 2 are the torsional stiffness

testing setup and the shape set procedure. As seen in Fig.27, the torsional data graph has a large

amount of noise compared to the graph for bending data. This is because the bending data was

generated in steps of deflection; positional and load data would be recorded for 1s-3s for each step

of deflection. The data for the torsion testing was recorded continuously as the DC motor spun,

thus we end up with only 1 data point per step. The shape set procedure also needed improvement

so that there would be less springback in the resultant curvature. The drastic change in coloration

could also point to a change in material properties. This could explain the discrepancy in the

prediction of the snapping problem elimination results in Chapter 2.2.2. According to the bending

and torsional stiffness val