Preoperative Planning of Robotics-Assisted Minimally

Preoperative Planning of Robotics-Assisted Minimally

Invasive Cardiac Surgery Under Uncertainty

(Spine Title: Preoperative Planning of Robotics-Assisted Cardiac Surgery)

(Thesis Format: Monograph)

by

Hamidreza Azimian

Faculty of Engineering Science

Department of Mechanical and Materials Engineering

Submitted in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

School of Graduate and Postdoctoral Studies

The University of Western Ontario

London, Ontario, Canada

© Hamidreza Azimian, 2012

THE UNIVERSITY OF WESTERN ONTARIO

School of Graduate and Postdoctoral Studies

CERTIFICATE OF EXAMINATION

Joint-Supervisor Examiners

Dr. Rajni Patel Dr. George Knopf

Joint-Supervisor

Dr. Remus Tutunea-Fatan

Dr. Michael Naish

Dr. Mehrdad Kermani

Supervisory Committee

Dr. Simon DiMaio

Dr. Samuel Asokanthan

Dr. Bob Kiaii

The thesis by

Hamidreza Azimian

entitled:

Preoperative Planning of Robotics-Assisted Minimally Invasive Cardiac

Surgery Under Uncertainty

is accepted in partial fulfillment of the

requirements for the degree of

Doctor of Philosophy

Date:Chair of the Thesis Examination Board

ii

Preoperative Planning of Robotics-Assisted Minimally

Invasive Cardiac Surgery Under Uncertainty

Hamidreza Azimian

Ph.D. Thesis, 2012Department of Mechanical and Materials Engineering

The University of Western Ontario

Abstract

In this thesis, a computational framework for patient-specific preoperative planning of

Robotics-Assisted Minimally Invasive Cardiac Surgery (RAMICS) is developed. It is ex-

pected that preoperative planning of RAMICS will improve the rate of success by consid-

ering robot kinematics, patient-specific thoracic anatomy, and procedure-specific intraop-

erative conditions. Given the significant anatomical features localized in the preoperative

computed tomography images of a patient’s thorax, port locations and robot orientations

(with respect to the patient’s body coordinate frame) are determined to optimize charac-

teristics such as dexterity, reachability, tool approach angles and maneuverability. In this

thesis, two approaches for preoperative planning of RAMICS are proposed that enable con-

templation of uncertainties in preoperative data and surgical tasks. In the first approach,

the problem is formulated as a Generalized Semi-Infinite Program (GSIP) with a convex

lower-level problem to maximize the tolerable geometric uncertainty in the neighborhood

of surgical targets. It is demonstrated that with a proper formulation of the problem, the

GSIP can be replaced by a tractable constrained nonlinear program that uses a multi-criteria

objective function to balance between the nominal task performance and robustness to colli-

sions and joint limit violations. In the second approach, the proposed formulation attempts

to increase the chance of success by maximizing robustness with respect to uncertainties

at the task level. It is assumed that the surgical tasks can be represented by Gaussian

distributions, and the planner is formulated as a chance-constrained entropy maximization

problem. The efficacy of the proposed formulations is demonstrated by comparisons be-

tween the plans generated by the algorithms and those recommended by an experienced

surgeon for several case studies.

Keywords: Preoperative Planning, Port Placement, Medical Robotics

iii

Co-Authorship

Chapters 2, 3, and 4 contain contents from the following conference papers [1], [2], and [3].

The papers were written by the author, and reviewed by Dr. Rajni Patel and Dr. Michael

Naish. Dr. Bob Kiaii helped with the validation of the algorithms.

A version of Chapter 3 has been accepted subject to revision for publication in IEEE

Transactions on Information Technology in Biomedicine.

Chapters 2 and 4 are being prepared for journal submissions.

iv

Acknowledgements

I would like to express my gratitude to my supervisors Prof. Rajni V. Patel and Prof.

Michael D. Naish for their invaluable guidance, and relentless support and patience during

the course of my Ph.D. at the University of Western Ontario.

I would also like to thank Ms. Ana Luisa Trejos, Dr. Bob Kiaii, Prof. Samuel Asokan-

than, Mr. Jeremy Breezke, and my dear friends and colleagues that I had the chance to

work with at Canadian Surgical Technologies and Advanced Robotics (CSTAR) for the

inspiring discussions that we had from time to time.

v

Dedicated to:

My Dear Parents

vi

Contents

Certificate of Examination ii

Abstract iii

Co-Authorship iv

Acknowledgements v

Table of Contents vii

List of Figures xi

List of Tables xv

Nomenclature and Acronyms xvi

1 Introduction 1

1.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Constrained Manipulation in Actively-constrained RAMIS 8

2.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Analytical Kinematics in RAMIS . . . . . . . . . . . . . . . . . . . . . . . . 10

vii

CONTENTS viii

2.2.1 Intracorporeal Kinematics . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 Analytical Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Closed-loop Inverse Kinematics in RAMIS . . . . . . . . . . . . . . . . . . . 13

2.3.1 Differential Kinematics of the Trocar . . . . . . . . . . . . . . . . . . 13

2.3.2 Constrained Cartesian Control . . . . . . . . . . . . . . . . . . . . . 16

2.3.2.1 Task Priority Approach . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Numerical Validation: Constrained Cartesian Control for Positioning

Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.5 Tool Flexion Compensation . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Dexterity in Actively Constrained Surgical Manipulation . . . . . . . . . . . 29

2.4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 A Modified Measure of Dexterity . . . . . . . . . . . . . . . . . . . . 31

2.4.3 Dexterity in Positioning Tasks . . . . . . . . . . . . . . . . . . . . . 34

2.4.4 Dexterity in Manipulation Tasks . . . . . . . . . . . . . . . . . . . . 34

2.4.5 Optimal Placement of Surgical Cavity . . . . . . . . . . . . . . . . . 36

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3 A Semi-Infinite Programming Approach 41

3.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Coronary Artery Bypass Surgery . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.2 Endoscopic Coronary artery Bypass Grafting . . . . . . . . . . . . . 44

3.2.3 Patient Selection and Positioning . . . . . . . . . . . . . . . . . . . . 44

3.3 The da Vinci Surgical System . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 A Deterministic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.2 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4.3 Multi-Criteria Objective Function . . . . . . . . . . . . . . . . . . . 51

3.4.3.1 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

CONTENTS ix

3.4.3.2 Robot Performance . . . . . . . . . . . . . . . . . . . . . . 52

3.4.3.3 Approach Angles . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.3.4 Hand/Eye Alignment . . . . . . . . . . . . . . . . . . . . . 54

3.4.4 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.4.5 Collision Avoidance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.6 Coding of Patient-Specific Models . . . . . . . . . . . . . . . . . . . 61

3.4.7 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.8 System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 A Stochastic Approach 71

4.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1.1 Task Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.1.2 Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.2.3 Problem Transformation . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Planning of Endoscopic Coronary Artery Bypass Surgery . . . . . . . . . . 81

4.4 System Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5 Discussions 87

5.1 Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . 88

References 93

Appendices 100

CONTENTS x

A Inverse Kinematics 100

A.1 Intracorporeal (The da Vinci Active Section) Inverse Kinematics . . . . . . 100

A.2 Uncertainty Propagation in Intracorporeal Kinematics . . . . . . . . . . . . 101

A.3 Holder/Instrument Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . 103

B Mathematical Preliminaries 105

B.1 Pseudo-Inverse Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

B.2 Range Space and Null Space Properties . . . . . . . . . . . . . . . . . . . . 106

B.3 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

B.4 Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.5 Logarithmic Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

B.6 Active Set SQP Implementation of GSIP . . . . . . . . . . . . . . . . . . . . 108

B.7 Chance Constrained Programming . . . . . . . . . . . . . . . . . . . . . . . 111

C The Heart Anatomy 112

D Permissions 115

Vita 118

List of Figures

2.1 Kinematics of the intracorporeal section with a roll-pitch-yaw wrist (the dis-

tance between frames 5 and 6 is denoted by a6) . . . . . . . . . . . . . . . . 11

2.2 The convention utilized for solving the inverse kinematics under the trocar

constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3 A disjunctive approach to solving the inverse kinematics problem for actively-

constrained manipulation in RAMIS . . . . . . . . . . . . . . . . . . . . . . 14

2.4 A snapshot of the tool action subject to a trocar constraint at a given time t 15

2.5 Block diagram of the closed-loop inverse kinematics scheme . . . . . . . . . 19

2.6 A helical trajectory inside a box resembling the surgical cavity accessed

through a port on the top face . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Results of trajectory tracking under a trocar constraint: the desired trajec-

tory (dashed line) is compared with the actual wrist trajectory (solid line) . 23

2.8 Lateral deflection at the RCM . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9 Admissible rectangle within the workspace of the robot . . . . . . . . . . . . 24

2.10 Topography of accuracy in positioning tasks normalized within the admissible

rectangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.11 2-norm of the position error for two different trajectories at different box

locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.12 Flexible tool operating under a remote center of motion . . . . . . . . . . . 28

xi

LIST OF FIGURES xii

2.13 Different manipulability ellipsoids to examine the effectiveness of the pro-

posed measure of dexterity: four manipulability ellipsoids, denoted by, E1,

E2, E3, and E4 whose singular values are given as {2, 5, 6}, {2, 3, 6}, {3, 3, 4},

and {1.5, 1.5, 2}, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.14 Topography of normalized dexterity in the admissible rectangle for position-

ing tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.15 The overlaid accuracy and dexterity topographies and the Pareto optimal

location of the box inside the admissible region (marked by a circle) . . . . 37

2.16 The reach dexterity Dr and orient dexterity Do calculated inside the box for

various wrist orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.17 The accuracy A of manipulation tasks calculated inside the box for various

wrist orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.18 The Pareto-optimal port locations for different manipulation tasks: The rect-

angles highlight the regions where the Pareto-optimal port locations can be

found . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 A hierarchical diagram of the proposed framework for robust preoperative

planning of RAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Left: The location of coronary arteries on the heart (source: [4])). Right:

The left internal mammary artery (LIMA) located on the chest wall close to

the sternum, extended from the first rib to the sixth rib . . . . . . . . . . . 44

3.3 The da Vinci surgical system . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.4 A schematic illustration of the surgical workspace and wrist uncertainty with

p = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 A conceptual illustration of design centering . . . . . . . . . . . . . . . . . . 51

3.6 0 ≤ φ0 ≤ 90° is the optimum approach angle of the tool at the surgical site 54

3.7 Hand/eye alignment in terms of the elevation and manipulation angles . . . 55

3.8 The da Vincir instrument arm (left), and the endoscope arm (middle) mod-

eled by geometric primitives: capsules and spheres . . . . . . . . . . . . . . 59

3.9 The da Vinci hollow robot composed of geometric primitives . . . . . . . . 59

LIST OF FIGURES xiii

3.10 Wireframe representation of the da Vinci active arm . . . . . . . . . . . . . 60

3.11 Surgical fixtures (composed of target positions and normal vectors) extracted

from a patient’s 3D model reconstructed from the patient’s CT images. The

fixtures are determined based on the procedure and are identified by a sur-

geon or a radiologist. The surface normal vector at each target location is

determined by measuring three adjacent points surrounding the target and

fitting a plane to those points. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.12 Left: Port locations recommended by the planner and the computed wrist

uncertainty volume as a union of octahedra. Right: Port locations recom-

mended by the planner based on the triad recommended by the surgeon, and

the computed wrist uncertainty volume as a union of octahedra . . . . . . . 64

3.13 A 3D view (left) and the top view (right) of the positioning of the rib cage

and the arms pertaining to the best plan recommended by the algorithm . . 64

3.14 The da Vincir arms in action: a collision-free trajectory of the active sections

as they reach the individual target sites inside the thorax . . . . . . . . . . 65

3.15 Comparison of the overall scores of the plans pertaining to ARP, SRP and

SRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.16 Comparison of the achieved robustness in the plans pertaining to ARP, SRP

and SRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.17 Comparison of the achieved dexterity in the plans pertaining to ARP, SRP

and SRT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.18 Actual reliability in terms of the constraint satisfaction rate based on the

plans recommended by the algorithm for four different patients: each bar

represents the constraint satisfaction rate at one target fixture k = 1, · · ·N . 69

4.1 Maneuverability of the tool confined by reachability constraints at the trocar

and target(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2 An illustrative comparison of the unscented transform with linearization and

probabilistic sampling: red dots represent the samples taken from the original

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

LIST OF FIGURES xiv

4.3 The da Vincir arms in action: a collision-free trajectory of the active sections

as they reach the individual target sites inside the thorax (for k = 1, 2, 3, 4) 83

4.4 The position of the admissible task fixtures generated in a Monte Carlo sim-

ulation for a LIMA harvesting task . . . . . . . . . . . . . . . . . . . . . . . 84

4.5 Comparison of the overall scores pertaining to the SRP, SRT and ARP plans 85

4.6 Comparison of the (normalized) Cartesian task entropy pertaining to the

SRP, SRT and ARP plans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Comparison of the (normalized) joint space/Cartesian space cross entropy

pertaining to the SRP, SRT and ARP plans . . . . . . . . . . . . . . . . . . 86

A.1 Frame attachment in intracorporeal kinematics (the distance between frames

5 and 6 is denoted by a6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

A.2 Target pose uncertainty propagation in intracorporeal kinematics. . . . . . 102

B.1 Log-smoothing technique: While the convexity of z = g(x) in the neighbor-

hood of x1 is much smaller than that of y = f(x) within the same neigh-

borhood, the convexity of the function in the vicinity of x3 is not affected as

much. This lowers the chance that gradient-based minimization techniques

getting stuck at x1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

C.1 The blood flow through the heart atria and ventricles: 1. right atrium 2.

right ventricle 3. pulmonary trunk 5. pulmonary veins 6. left atrium 7. left

ventricle 8. aortic artery 10. Venae Cavae (source: [4]) . . . . . . . . . . . . 113

C.2 The heart valves (source: [4]) . . . . . . . . . . . . . . . . . . . . . . . . . . 114

List of Tables

2.1 DH parameters of the intracorporeal mechanism with roll-pitch-yaw wrist

(left) and roll-pitch-roll wrist (right) . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Comparison of the accuracy measures and measured error norms for two

different trajectories at different box locations . . . . . . . . . . . . . . . . . 25

2.3 Significant robot manipulator performance measures . . . . . . . . . . . . . 31

2.4 Comparison of different measures in quantification of different manipulability

ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1 The criteria contributing to the objective function . . . . . . . . . . . . . . 55

A.1 D-H Parameters of the Mitsubishi PA10-7C mounted with a roll-pitch-yaw

instrument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

xv

Nomenclature and Acronyms

Nomenclature

p Position vector

ptroc Position vector of trocar

pelb Position vector of the elbow on the holder mechanism

pw Position vector of instrument wrist

prcm Position vector of the remote-center-of-motion on the tool

J Jacobian matrix of the tool tip

Jv Translational submatrix of Jacobian matrix

Jω Rotational submatrix of Jacobian matrix

J1 Primary task Jacobian matrix

J2 Secondary task Jacobian matrix

J r Reach Jacobian matrix

Jo Orientation Jacobian matrix

J Constrained Jacobian matrix

Jelb Elbow Jacobian matrix

Jw Wrist Jacobian matrix

Jc Jacobian matrix of the trocar constraint

xd Vector of the desired task in Cartesian space

x1 Vector of the primary task

x2 Vector of the secondary task

xvi

NOMENCLATURE AND ACRONYMS xvii

xt Vector of the tool tip task

n Unit normal vector

l Unit vector

m Unit Vector

rtool Tool approach unit vector

K1 Primary task gain matrix

K2 Secondary task gain matrix

q Joint vector

qic Intracorporeal joint vector

qins instrument joint vector

qhldr Holder joint vector

g Transformation matrix

gt Target frame

gtroc Trocar frame

ghldr Holder frame

gic Intracorporeal transformation matrix

ercm RCM error

et Tip error

Et(s) Tip error Laplace transform

Xd(s) Desired task Laplace Transform

S(s) Sensitivity function Laplace transform

σ Largest singular value

P Pressure

I Inertia

E Young’s Modulus

w deflection

NOMENCLATURE AND ACRONYMS xviii

λi Eigenvalue

κ(·) Condition number operator

Dp Positioning dexterity

Dr Reach dexterity

Do Orientation dexterity

A Accuracy

p Position vector

p Augmented position vector

n Unit normal vector

P Optimization parameters (plan)

B Tolerable uncertainty

B Tolerable uncertainty in dual wrist space

C Uncluttered subset of the dual wrist space

r Uncertainty ball radius

φ Approach angle

ψ Elevation angle

η Manipulation angle

J Jacobian matrix

W Scalar weight

U Optimality criterion

R Rotation matrix

c(u) Parametric spatial curve

c Augmented vector of the ICS curve triad

Q Joint space

κ(·) The condition number operator

d The minimum distance

NOMENCLATURE AND ACRONYMS xix

f Scaling function

I Set of intercostal curves

gi Semi-infinite constraint functions

hi Ordinary constraint functions

c ICS curve triad

X Surgical procedure task sequence

xR Right instrument task frame

xL Left instrument task frame

µRx Right instrument task frame mean value

µLx Left instrument task frame mean value

QRx Right instrument task frame covariance matrix

QLx Left instrument task frame covariance matrix

µRq Right instrument joint mean value

µLq Left instrument joint frame mean value

QRq Right instrument joint frame covariance matrix

QLq Left instrument joint frame covariance matrix

N Gaussian distribution

Ent Shannon entropy

P (.) Probability operator

γ1 Approach angle at trocar

γ2 Approach angle at target

H Hessian matrix

L Lagrangian

NOMENCLATURE AND ACRONYMS xx

Acronyms

OCS Open Cardiac Surgery

MIS Minimally Invasive Surgery

MICS Minimally Invasive Cardiac Surgery

RAMIS Robotics-Assisted Minimally Invasive Surgery

RAMICS Robotics-Assisted Minimally Invasive Cardiac Surgery

RCM Remote Center of Motion

ICS Intercostal Space

GSIP Generalized Semi-Infinite Programming

SQP Sequential Quadratic Programming

SRP Surgeon’s Recommended Ports

SRT Surgeon’s Recommended Triad

ARP Algorithm’s Recommended Ports

DH Denavit-Hartenberg

Chapter 1

Introduction

1.1 Synopsis

According to World Health Organization [6] and Health Canada [7] cardiovascular disease

is the number one killer and one of the most costly diseases in Canada and around the

world. Open cardiac surgery (OCS), the conventional treatment of cardiovascular disease;

however, has proved to be intensely invasive and is usually followed by a long recovery

period. Minimally invasive cardiac surgery (MICS) was proposed to reduce the invasiveness

of OCS by using laparoscopic surgical tools and an endoscope that enter the chest cavity

through small incisions in the chest wall. Lower morbidity, shorter hospital stay and less

trauma are among the benefits of MICS, compared to OCS. Nevertheless, the popularity of

MICS has remained limited due to technical difficulties such as reduced dexterity and lack

of direct vision. In fact, compensation for the degrees of freedom lost at the entry ports,

was the main motivation for introduction of Robotics-Assisted Minimally Invasive Cardiac

Surgery (RAMICS). Hand motion scaling and tremor filtering are among other features

of RAMICS. Despite their advantages, drawbacks such as collisions and singularities of

robotic manipulators can severely reduce the success rate of robotics-assisted minimally

invasive procedures. These difficulties are further exacerbated for thoracic interventions by

the complexity of the anatomy and the restricted workspace.

1

1.2 Motivation 2

1.2 Motivation

It is expected that preoperative planning will improve the performance of RAMICS by

contemplating the robot kinematics, the patient’s thoracic anatomy and the requirements

of the surgical procedure. The location of the ports (two for the instruments and one

for the endoscope) on the ribcage, as well as the configuration of the robotic arms with

respect to the patient’s thoracic anatomy, are the main parameters of the planning. In

fact, an improper port/robot placement may make a robotic MICS task infeasible due to

extracorporeal collisions, joint limits, an inability to reach surgical targets or a lack of

dexterity. This may require port repositioning or conversion to open surgery, leading to

higher morbidity.

Currently, the surgeon selects the position of the port triplet based on his/her experience

and by looking at preoperative chest CT images. In current techniques, port positions are

selected based on heuristic guidelines and mainly to enhance surgical target reachability

without explicit consideration of the performance of the robot, while tool manipulability,

collisions, and target reachability are strongly dependent on the configuration of the robot.

This provides the motivation for developing a planning strategy based on patient-specific

anatomical models. It is expected that this strategy will improve surgical outcomes and

eliminate additional risks such as bleeding, trauma and infection. The planning procedure

will start by quantifying the patient’s eligibility for RAMICS via a RAMICS feasibility

measure; once the patient qualifies for RAMICS, the strategy will address aspects such as

optimal port and robot placement.

1.3 Literature Review

In general, work related to the topic of this thesis can be categorized into four groups:

Clinical approach: The problem of patient preoperative preparation and port place-

ment for RAMIS has been widely visited from a clinical perspective (e.g. see [8, 9] for

cardiac surgery, [10] for endoscopic knot tying, and [11, 12] for urological surgery). The

highlight of this approach is that it is based on statistical inference, surgeons experience

1.3 Literature Review 3

and heuristics, and lacks more objective in-depth analytical reasoning. However, due to its

innate simplicity, it has been well-received among clinicians.

Visualization of Surgical Workspace: The second common approach has been

mainly proposed by medical imaging scientists. In this technique, an interactive 3D model

of the surgical workspace is constructed. The surgeon can choose a suitable port location

simply by navigating in the surgical workspace. Similar to the clinical approach, this ap-

proach lacks objective analytical reasoning. As such, a system for interactive visualization

of intra-thoracic models constructed from preoperative Computed Tomography (CT) and

Magnetic Resonance Imaging (MRI) was reported in [13]. This system assists cardiac sur-

geons in preoperative examination of the angle of view through the candidate port locations

in the chest wall. The system is also capable of calculating some optimal port locations

based on the criteria proposed in [8]. A similar visualization system for preoperative plan-

ning of RAMICS for the da Vinci surgical system was reported in [14]. The system also

has built-in systems for collision checking and spatial registration of the anatomy.

Workspace optimization: The third group of papers have treated preoperative plan-

ning of RAMIS as objective optimization of various criteria such as dexterity and reacha-

bility usually inside a geometric volume resembling the surgical workspace without utilizing

any patient-specific information. While this approach is analytically more rigorous com-

pared to the above-mentioned approaches, it lacks essential context in terms of taking into

account patient- and procedure-specific requirements. As such in [15, 16], optimal port

placement for the da Vinci surgical system based on various robot dexterity measures was

studied. In [17], optimal pose planning for collision-free operation of a new surgical robotic

system was addressed. Optimal placement of the Raven IV with respect to the patient’s

body for maximum reachable workspace was studied in [18].

Patient- and procedure-specific planning: The most rigorous treatments to pre-

operative planning of RAMIS to date, have been reported in [19–24]. These papers have

approached preoperative planning of RAMIS as optimization of certain criteria while consid-

ering patient-specific geometry. Optimal port placement for Totally Endoscopic Coronary

Artery Bypass (TECAB) grafting using preoperative patient data was reported in [20]. The

1.4 Problem Statement 4

authors reported up to 43% improvement in dissection time. A planning strategy for the

da Vincir surgical system (from Intuitive Surgical Inc.) was reported in [19] that included

optimal port/robot placement for collision-free operation with enhanced dexterity and vis-

ibility. Preoperative planning of the da Vinci instrument arms with application to totally

endoscopic coronary artery bypass was addressed in [22]. In this work, the port locations

and robot orientation are selected such that dexterity and clearance are optimized. However,

this work did not take the endoscope arm into account for planning. Finally, a system for

intraoperative planning and control of RAMICS was reported in [23]. The system consists

of a preoperative planning stage as well as an intraoperative planning and control stage. To

the best of our knowledge, the first major effort on preoperative planning of RAMIS under

uncertainty was addressed in [2].

Finding the optimal insertion location for percutaneous interventions, e.g. needle steer-

ing, has been studied in several papers including [25–27]. Planning under uncertainty has

been well-addressed in these papers. However, needle steering is generally a much easier

problem to solve due to the lower degrees-of-freedom that are involved when compared to

more sophisticated surgical manipulation tasks.

1.4 Problem Statement

The ultimate objective of this research is to develop a framework for preoperative planning

of RAMICS, based on patient-specific preoperative images. The planning will attempt to

optimize several criteria including but not limited to

• Robot dexterity

• Surgical target reachability

• Collision avoidance

• Hand/eye coordination

It is expected that by employing such a planning strategy the success rate of RAMIS will be

improved. This can be implicitly quantified by reduction in surgical time, port-repositioning

1.4 Problem Statement 5

rate, robot-repositioning rate, and collision rate as well as reduction in the rate of conversion

to open cardiac surgery.

Preoperative plans can only be inferred from patients’ preoperative data. Obviously,

these plans are naturally unable to contemplate variability of actual intraoperative condi-

tions with respect to preoperative models, and, as a result, their reliability needs to be

justified. To the best of our knowledge, so far no study has addressed the robustness of

preoperative plans with respect to geometric uncertainties of preoperative data with appli-

cation to RAMIS. As a matter of fact, deviations of intraoperative conditions from preop-

erative models could severely undermine the effectiveness and practicality of the methods

and algorithms developed thus far for preoperative planning of robotics-assisted minimally

invasive surgery. Patient-specific preoperative planning without taking into account the

mismatches between preoperative data and intraoperative conditions may lead to results

that are generally too unreliable to be useful in clinical practice. Moreover, teleoperated

surgical interventions are highly prone to spatio-temporal uncertainties pertaining to tissue

deformation, patient placement error, and physiological motions (e.g., heart and respira-

tory motion). Variability in the surgeon’s hand gestures is another major source of uncer-

tainty. This suggests that in order to address intraoperative uncertainty, patient-specific

modeling and characterization of uncertainty sources including tool/tissue interaction and

heart/respiratory motion are essential; however, these models may vary from patient to

patient, and a great deal of time and effort is required to develop patient-specific models

of uncertainty. Such models, if exist, should be rendered as nonlinear partial differential

equations that are naturally hard to solve.

The above rationale motivates further research on developing novel, efficient, and yet

relatively-accurate planning techniques that not only take into account the patient, proce-

dure and robot requirements, but also attempt to improve the robustness of preoperative

plans with respect to uncertainties in preoperative data. In this thesis, we propose a novel

formalism for preoperative planning of robotics-assisted minimally invasive cardiac surgery

that is focused on this goal. While the main target of this formalism is cardiac surgery,

every attempt has been made to ensure that the formalism is general and can be applied

1.5 Thesis Outline 6

to different surgical procedures with few modifications; however, the application of the

formalism to other procedures remains unexplored.

1.5 Thesis Outline

The thesis is composed of five chapters. In addition to the first and the last chapters

that provide the introductory material and the concluding remarks, respectively, the main

contributions of the thesis are presented in Chapters 2, 3, and 4. The following is an

overview of these chapters:

Chapter 2: In this chapter, various aspects of the kinematics of RAMIS are studied.

This chapter establishes the foundations upon which further control and planning algorithms

are built. The subsequent chapters employ some of the results of this chapter to formulate

preoperative planning of RAMICS. In this chapter, analytical solutions for forward and

inverse kinematics under the trocar constraint are derived. Moreover, compliant motion

control under the trocar constraint is exposed as a closed-loop inverse kinematics scheme,

and the accuracy of the proposed scheme is investigated. Finally, a novel dexterity measure

is proposed, and it is demonstrated that by optimal placement of the surgical cavity within

the robot workspace, dexterity as well as accuracy can be significantly improved. This

chapter show how the trocar constraints can affect various measures (particularly in active

RCM realization). Moreover, through a number of illustrative examples some preliminary

justifications for preoperative planning of RAMIS procedures are presented.

Chapter 3: In this chapter, a deterministic formalism for preoperative planning of

RAMICS is presented. The principle contribution of the proposed formalism that distin-

guishes the work presented in this chapter from related work for preoperative planning of

RAMIS is the recognition of geometric uncertainty in preoperative data. Furthermore, the

chapter establishes a framework for incorporating patient- and procedure-specific informa-

tion into the planning. The planning problem is rendered as a robust optimization problem,

and an efficient solution method is proposed.

Chapter 4: As an alternative formalism, a stochastic approach for preoperative plan-

ning under task uncertainty is presented in this chapter. The approach attempts to regain

1.5 Thesis Outline 7

some of the qualities that have been compromised in the previous approach such as consid-

eration of both position and orientation uncertainties. The formalism is based on analyzing

the propagation of the task-level uncertainty in multi-arm scenarios in RAMICS. The plan-

ning is formulated as a chance-constrained entropy maximization problem, and an efficient

solution method is proposed.

Chapter 2

Constrained Manipulation in

Actively-constrained RAMIS

2.1 Synopsis

Minimally Invasive Surgery (MIS) was originally introduced to reduce postoperative trauma

and recovery time. Nevertheless, its popularity remained limited due to drawbacks such as

impaired hand/eye coordination, hand/tool motion reversal and reduced dexterity. Hence,

robotic manipulators were adopted to overcome these problems, mainly due to motion con-

straints imposed on the surgical tool by the trocar placed within the entry port. The trocar

constraint creates a Remote Center of Motion (RCM) by constraining the lateral motion of

the laparoscopic tool at the port of entry. This constraint is essential to ensure minimally

invasive access. An RCM can be either realized by a built-in compliant mechanism, referred

to as passive compliant motion control, or enforced by the robot controller, referred to as

active compliant motion control. The da Vincir [28] robotic system is an example that

utilizes a compliant mechanism to enforce the RCM. The DLR MIRO [29] robot, on the

other hand, employs active compliant motion control. While the former is superior from

the safety point of view, to the knowledge of the authors no reliable comparative study on

these two solutions has been reported.

The literature on the kinematics of robotic manipulators is very mature due to significant

8

2.1 Synopsis 9

research activity over the last few decades. However, with the emergence of new applications

for robotic manipulators that involve new operating conditions, the area deserves further

consideration.

As a matter of fact, the number of papers addressing the kinematics of robotic ma-

nipulators with application to RAMIS is quite limited. Of those that have been reported,

the results are usually restricted to a special type of manipulator and are ad hoc in their

approach. A thorough kinematic analysis should entail all aspects that concern Cartesian

control, planning and analysis of RAMIS.

Currently, the most popular robotic manipulator that has been approved for robotic pro-

cedures is the da Vincir that employs passive RCM control. Instances of actively-controlled

RCM use can be found in research set-ups such as in [30] and [29]. Herein, the most notable

work addressing kinematic problems in RAMIS including kinematic control, planning and

performance analysis are reviewed. Solving the inverse kinematics of RAMIS under a tro-

car constraint with two different manipulators and using geometric methods was addressed

in [31] and [24]. Closed-loop kinematic control of robots in surgical applications has been

addressed in [32–38]. In [32] a Cartesian control algorithm was developed for the AESOP.

In the proposed scheme, differential kinematics of the trocar were formulated in terms of

the active joints and an observer was designed for estimation of the active joint velocities.

In [33], Cartesian control of a Mitsubishi PA10-7C for RCM realization was implemented

using the extended Jacobian approach; however, no generic explicit expressions for the tro-

car differential kinematics were given. As an alternative approach to kinematic control,

real-time constrained Cartesian control using constrained optimization techniques was re-

ported in [34–38]. Although this approach allows strict consideration of task constraints, it

requires solving a constrained optimization problem at every time step.

Proposing new measures or applying existing kinematic measures to RAMIS is another

research direction. A measure of accuracy in terms of encoder accuracy was developed

in [39]. The authors also defined manipulability in terms of the maximum joint velocity at-

tained for the desired minimal instrument tip velocity. This is particularly significant when

the permissible joint velocity range is limited; however, this measure does not reflect the

2.2 Analytical Kinematics in RAMIS 10

conditioning of the manipulability ellipsoid. Finally, [40] and [41] studied the performance

of kinematic control in minimally invasive surgery subject to instrument flexion and error

in port locations.

This chapter studies the kinematics of active compliant motion control in RAMIS and

extends the results given in [1], [24] and [30]. Firstly, an algorithm for solving the inverse

kinematics problem of trocar-constrained manipulation for general nonredundant manip-

ulators is given. While it is not common to use analytic inverse kinematics for real-time

control, it is obvious that such an algorithm is invaluable for analysis and planning purposes.

As a principal contribution of this chapter, a closed-loop inverse kinematics (CLIK) scheme

based on differential kinematics of the trocar and the concept of task priority is formulated.

The proposed CLIK scheme achieves accurate RCM realization, but at the expense of the

accuracy of the pose of the instrument tip. Moreover, the chapter revisits the concept of

kinematic dexterity in actively constrained surgical manipulation and proposes novel well-

defined dexterity measures. Finally, through an illustrative example, it is demonstrated

that the accuracy and dexterity can be significantly improved by proper placement of the

surgical cavity within the workspace of the robot.

This chapter is organized as follows: In Section 2.2 an analytical approach to solving

the forward and inverse kinematics problems in RAMIS is presented. A closed-loop inverse

kinematics scheme is formulated in Section 2.3, and an accuracy analysis is presented for

the proposed scheme and numerical validation is also provided. In Section 2.4 the concept

of dexterity with application to RAMIS is discussed.

2.2 Analytical Kinematics in RAMIS

2.2.1 Intracorporeal Kinematics

In this section, a framework for the analysis of actively constrained surgical manipulation

is established by a disjunctive treatment of the intra- and extracorporeal sections. The

kinematics of the instrument tip with respect to the coordinate frame attached to the trocar

are referred to as the intracorporeal kinematics. In other words, these are the kinematics


θ1

.

z0, z1, x2, x3

.

x0, x1

.

z2θ2

.

z5

θ5

θ3

.

z3, z4, x5

.

x4

.

x6

θ6

.

z6

.

.

d4

.

Figure 2.1: Kinematics of the intracorporeal section with a roll-pitch-yaw wrist (the distancebetween frames 5 and 6 is denoted by a6)

observed from within the body cavity accessed through the port. On the other hand,

the extracorporeal section consists of the mechanism situated between the base and the

trocar coordinate frames. The kinematics of the intracorporeal section can be modeled

by a spherical joint composed of two revolute joints θ1 and θ2 at the RCM, a prismatic

joint d4, and a wrist mechanism composed of revolute joints θ3, θ5 and θ6. Roll-pitch-yaw

and roll-pitch-roll are the most common wrist mechanisms used in surgical instruments.

Comparative studies between these two wrist types can be found in [42] and [43]. Fig. 2.1

illustrates the frame assignment of the intracorporeal section with a roll-pitch-yaw wrist.

The Denavit-Hartenberg parameters of the intracorporeal kinematics for both wrist types

are provided in Table 2.1

While the kinematics of the intracorporeal section can be generically formulated, a

variety of different mechanisms can be used for the extracorporeal section. A study of

different extracorporeal mechanisms can be found in [44].


Table 2.1: DH parameters of the intracorporeal mechanism with roll-pitch-yaw wrist (left)and roll-pitch-roll wrist (right)

Joint θ d a α

1 θ1 0 0 0

2 θ2 0 0 −π2

3 θ3 0 0 π2

4 0 d4 0 0

5 θ5 0 0 −π2

6 θ6 0 a6 −π2

Joint θ d a α

1 θ1 0 0 0

2 θ2 0 0 −π2

3 θ3 0 0 π2

4 0 d4 0 0

5 θ5 0 0 −π2

6 θ6 a6 0 π2

2.2.2 Analytical Inverse Kinematics

An algorithmic approach to solving the inverse kinematics problem for manipulators oper-

ating under a trocar constraint is proposed in this section. This algorithm is based on a

disjunctive inverse kinematics solution of the extra- and intracorporeal sections. The entire

mechanism (see Fig. 2.2) consists of an instrument holder and an instrument that provide 8

degrees of freedom for the successful accomplishment of any given manipulation task subject

to a trocar constraint.

For a given trocar coordinate frame, denoted by gtroc ∈ SE(3), and a desired task,

denoted by gt ∈ SE(3), the intracorporeal inverse kinematics, gic = g−1trocgt, is first solved

for the intracorporeal joint values, qic ∈ R6. Given the intracorporeal kinematics, attributes

of the instrument holder can be determined. This includes the position of the elbow on the

holder mechanism, pelb ∈ R3, and the approach of the tool, rtool ∈ R3. These attributes

are sufficient to solve the inverse kinematics of the holder mechanism, ghldr ∈ SE(3), to

obtain the holder joint values, qhldr ∈ R5. Finally, by solving gins = g−1hldrgtrocgic, the

instrument joint values, denoted by qins ∈ R3, can be obtained and the vector of aggregate

joint values may be rendered as q =

qhldr

qins

. Algorithm 2.1 and Fig. 2.3 provide an

overview of the procedure for solving the inverse kinematics problem. The function GetQic

returns the intracorporeal joint values, GetElbPos returns the pose of the holder elbow,

and the functions GetQhldr and GetQins return the holder and the instrument joint values,

respectively. In Appendix A, closed-form solutions are given for the case that a roll-pitch-

2.3 Closed-loop Inverse Kinematics in RAMIS 13

gbase

gtroc

gtg6

Intracorporealkinematics

Instrumentkinematics

Holderkinematics

Figure 2.2: The convention utilized for solving the inverse kinematics under the trocar con-straints

yaw instrument is mounted on a Mitsubishi PA10-7C with a locked shoulder joint1.

Algorithm 2.1 DisjunctiveConstrainedInverseKinematics(gt, gtroc)

1 qic ← GetQic(gtroc, gt)2 rtool,pelb ← GetElbPos(qic, gtroc)3 qhldr ← GetQhldr(rtool,pelb)4 qins ← GetQins(qhldr, qic, gtroc)

5 q ←(

qhldr

qins

)

6 return q

2.3 Closed-loop Inverse Kinematics in RAMIS

2.3.1 Differential Kinematics of the Trocar

While the inverse kinematics solution proposed in the preceding section is beneficial for

offline applications such as planning, a closed-loop inverse kinematics scheme may be pre-

1See a simulation video for RCM realization using the analytical inverse kinematics at www.youtube.com/

watch?v=0Fef5sIj3HI


Get trocar co-

ordinate frame

Get desired co-

ordinate frame

Solve intracorporeal

inverse kinematics

Compute the instru-

ment holder’s pose

Solve instrument holder

inverse kinematics

Solve instrument

inverse kinematics

Figure 2.3: A disjunctive approach to solving the inverse kinematics problem for actively-constrained manipulation in RAMIS

ferred for implementing real-time Cartesian control. In this section, the trocar constraint

that impacts the operation of a surgical instrument in RAMIS is formulated in the form of

generic expressions. Intuitively, these expressions should depend upon the geometry of the

trocar as well as the kinematics of the manipulator.

Let us define the tangent plane (on the patient’s skin surface) Γ at the trocar port as:

Γ = {p ∈ R3|nT (p− ptroc) = 0}, (2.1)

where n ∈ R3 is a unit vector normal to the plane, p ∈ R3 is a position vector, and

ptroc ∈ R3 is the position of the trocar.

To reach inside the cavity, the tool passes through the trocar creating an RCM. At a

given time t, the position of the RCM on the tool can be given as:

prcm(t) = νrcmpw(t) + (1− νrcm)pelb(t), for 0 ≤ νrcm ≤ 1, (2.2)


.

m.

n

.

l

.

.

.

.

prcm

(t) = ptroc

pw(t)

.

pelb

(t)

.

rtool(t)

Figure 2.4: A snapshot of the tool action subject to a trocar constraint at a given time t

where pw ∈ R3 is the wrist position. Since the RCM must coincide with the trocar, νrcm is

calculated by:

νrcm =nT (pelb(t)− ptroc)

nT

(

pelb(t)− pw(t)

) . (2.3)

The instantaneous velocity of the RCM point is obtained as:

prcm = νrcmpw + (1− νrcm)pelb, for 0 ≤ νrcm ≤ 1. (2.4)

It should be noted that in general, (2.2) gives the coordinates of a point on the tool that

coincides with ptroc at time t, and at time t + δt, the RCM occurs at a different point on

the tool that must be calculated from (2.2) for a different νrcm (Fig. 2.4).

To ensure that the tool does not have any lateral deflection in the plane at the RCM,

the following equality constraint must be satisfied:

lT

mT

prcm = 0, (2.5)


where l ∈ R3 and m ∈ R3 are orthonormal vectors that span the plane Γ, and {l, m, n} is

an orthonormal basis for R3.

Equation (2.5) can be formulated in terms of the joint velocities of the mechanism, i.e.,

Jcq = 0, (2.6)

where Jc ∈ R2×n is the Jacobian matrix of the RCM point on the tool that should operate

subject to the constraint and q ∈ Rn is the joint vector. Jc can be expressed in terms of

the Jacobian matrices of the elbow and the wrist, given by Jelb ∈ R3×n and Jw ∈ R3×n

respectively; i.e.,

Jc =

lT

mT

(νrcmJw + (1− νrcm)Jelb) . (2.7)

Equations (2.6) and (2.7) express the trocar differential kinematics in terms of the differen-

tial kinematics of the wrist and the elbow.

2.3.2 Constrained Cartesian Control

2.3.2.1 Task Priority Approach

In this section, the concept of task priority [45] and the notion of a restricted Jacobian [46]

are utilized to formulate a constrained Cartesian control scheme for actively-constrained

surgical manipulation.

Assume that x1 and x2 are two individual velocity-level tasks recognized based on their

priorities as the primary and the secondary tasks respectively, which must be accomplished

with the same order of priority:

x1 = J1q,

x2 = J2q,

(2.8)

where J i ∈ Rmi×n and xi ∈ Rmi for i = 1, 2, are the Jacobian matrices and the vector of


twists, respectively. A least-squares solution for (2.8) is given as [45]:

q = J+1 x1 + (I − J+

1 J1)(J+2 (x2 − J2J+

1 x1) + (I − J+2 J2)z), (2.9)

where B+ , BT (BBT )−1 is the pseudo-inverse of B ∈ Rm×n for m < n, J2 , J2(I −

J+1 J1) is the restricted version of J2 (projection of J2 into the null space of J1), and z ∈ Rn

is an arbitrary vector.

Now consider constrained surgical manipulation where the task at the instrument tip

can be given by a velocity-level equation as:

Jq = xd. (2.10)

Since the motion constraint at the RCM must be satisfied prior to the instrument tip task,

(2.6) and (2.10) are recognized as the primary and the secondary tasks, respectively. By

substituting (2.6) and (2.10) in (2.9), i.e. for J1 = Jc, J2 = J , x1 = 0 and x2 = xd, (2.9)

can be rewritten as:

q = (I − J+c Jc)J

+xd + (I − J+

c Jc)(I − J+

J)z, (2.11)

where the trocar-constrained Jacobian matrix, i.e., the restricted version of J , denoted by

J , is defined as:

J , J(I − J+c Jc). (2.12)

By using (2.12) and the definition of pseudo-inverse, (2.11) can be rewritten as:

q = (I − J+c Jc)(I − J+

c Jc)TJT (J J

T)−1xd + (I − J+

c Jc)(I − J+

J)z, (2.13)

Since the null space projector is idempotent and hermitian [47], (2.13) can be simplified as:

q = JT

(J JT

)−1xd + (I − J+c Jc)(I − J

+J)z, (2.14)


that can be further simplified as:

q = J+

xd + (I − J+c Jc)(I − J

+J)z. (2.15)

Now by multiplying both sides by J , (2.15) can be rewritten as:

J q = J J+

xd + J(I − J+c Jc)(I − J+

c Jc)(I − J+

J)z, (2.16)

using the same properties of the null space projector, (2.16) can be rewritten as:

J q = J J+

xd + J(I − J+c Jc)(I − J

+J)z, (2.17)

that can be further simplified as:

J q = J J+

xd + J(I − J+

J)z, (2.18)

and under the condition that J is full-rank (when the arm’s configuration is not at kinematic

or algorithmic singularities2), it can be shown that:

J q = xd. (2.19)

In other words, the surgical task equation of motion is derived by projecting the instrument

task into the null space of the constraint.

While the joint velocity solution obtained in (2.15) provides an invaluable closed-form

solution, in practice it might not be useful for joint control due to inaccuracies in numerical

integration and error accumulation. However, (2.15) can be modified to form a CLIK scheme

that is robust to noise and numerical drifts. Surveys on CLIK can be found in [48] and [49].

2At an algorithmically singular configuration, J2(I − J+

1J1) loses rank while J1 and J2 are full rank.


+ K2 + J+

+

+ K1

(

lT

mT

)

J+c

∫

Robot

xd(t)

ptroc

prcm(t)

-

x(t)

-

xd(t)

Figure 2.5: Block diagram of the closed-loop inverse kinematics scheme

A CLIK scheme based on task priority for trocar-constrained manipulation can be given as:

q = J+c K1

lT

mT

ercm + J+

xd + K2et − JJ+c K1

lT

mT

ercm

, (2.20)

where Ki are diagonal positive-definite matrices, ercm and et are the position errors at

the RCM and the instrument tip, respectively. Ki with larger elements result in a faster

response, smaller steady state errors, and a higher chance of instability.

As an alternative, a numerically robust variant of (2.20) was proposed in [50] that does

not suffer from algorithmic singularities:

q =J+c K1

lT

mT

ercm + (I − J+c Jc)J

+(xd + K2et). (2.21)

The block diagram of the CLIK scheme is shown in Fig. 2.5.

Although the above joint velocity solution is an approximation of (2.20), it provides

reliable compliant motion control at the RCM. Nevertheless, its accuracy for performing

surgical tasks should be further analyzed.


2.3.3 Accuracy

With the joint velocity solution given by (2.21), the twist associated with the instrument

tip can be expressed as:

xt =JJ+c K1ercm + J(I − J+

c Jc)J+(xd + K2et). (2.22)

where ercm =

lT

mT

ercm. Correspondingly, the error dynamics of the tip pose can be

given as:

et = −JJ+c K1ercm − J(I − J+

c Jc)J+K2et + (I − J(I − J+

c Jc)J+)xd. (2.23)

Under the assumption that the Jacobian matrices are exact and nonsingular, the error

dynamics at the RCM point can be given by

˙ercm = −K1ercm, (2.24)

which signifies that if K1 is positive definite and the initial error at the RCM is nonzero, it

will eventually tend to zero. For small displacements, the Jacobian matrices can be assumed

constant and (2.23) can be considered as a locally linear differential equation. Therefore,

the Laplace transformation of (2.23), under the assumption of constant J and Jc, can be

written as:

Et(s) =(

sI + J(I − J+c Jc)J

+K2

)−1 (

et(0)−

JJ+c K1(sI + K1)−1ercm(0) + (I − J(I − J+

c Jc)J+)sXd

)

, (2.25)

where Et(s) and Xd(s) are the Laplace transformations of et(t) and xd(t), respectively. The

first two terms on the right-hand side of (2.25), if nonzero, can be likened to nonpersistent

disturbances that are only present during the transient phase; in other words, the steady

state error will be dominated by the last term. The sensitivity function can be defined in


terms of J and Jc as:

S(s) ,(

sI + J(I − J+c Jc)J

+K2

)−1(

I − J(I − J+c Jc)J

+)

. (2.26)

For a given desired motion xd, the instrument tip error can be determined by the

sensitivity function. Therefore, it is reasonable to define a measure of accuracy in terms

of the largest singular value of the sensitivity function; i.e., σ(S). The infinity norm of

S(s), ‖S(jω)‖∞ = supω σ(S(jω)), gives the supremum of the largest singular value of the

sensitivity function over its frequency response. In order to maximize accuracy, a suitable

robot configuration should be sought by solving

minq‖S(jω)‖∞, (2.27)

that correponds to a minimax optimization problem. Solving (2.27), in general, is hard

and computationally expensive. Furthermore, it should be noted that the expression given

for S(jω) in (2.26) has been obtained based on a quasi-static assumption of surgical ma-

nipulation. While such an assumption is sufficiently realistic in surgical manipulation, to

address more general applications with higher frequency components, the dynamics of the

robot must also be taken into account and S(jω) may no longer be accurate at higher fre-

quencies. Therefore, a measure of kinematic accuracy based on (2.27) will be more reliable

at lower frequencies. In fact, at lower frequencies, i.e., ω � mini∣

∣

∣λi(

J(I − J+c Jc)J

+K2

)∣

∣

∣,

where λi(B) are the eigenvalues of matrix B , S(jω) can be approximated as3:

S(jω) '(

J(I − J+c Jc)J

+K2

)−1(


+)

, (2.28)

meaning that at lower frequencies, the sensitivity function is equivalent to a constant matrix(

J(I − J+c Jc)J

+K2

)−1(


+)

, and by decreasing a norm of this matrix,

sensitivity becomes smaller as well. This suggests that at lower frequencies (2.27) can be

3This definition is valid for nonsingular J(I − J+c Jc)J+K2.


the port location

the desired trajectory

.

z

. x.y

Figure 2.6: A helical trajectory inside a box resembling the surgical cavity accessed througha port on the top face

rewritten as:

minqA−1, (2.29)

where

A ,1

σ

(

(

J(I − J+c Jc)J

+K2

)−1(


+)

) (2.30)

is a measure of accuracy that is independent of ω. The above expression is significant due

to the fact that it can quantify the accuracy of the instrument tip in terms of the robot

configuration. It should be noted that in real world applications, desired tasks are specified

by their motions, i.e., xd, and in this case by minimizing A−1 over the configuration space

one can seek an optimal configuration at which the desired task can be accomplished with

maximum accuracy.

2.3.4 Numerical Validation: Constrained Cartesian Control for Position-

ing Tasks

To validate the expressions derived earlier in this section, an experiment is designed and

implemented numerically. In this experiment, a situation similar to RAMIS is simulated,

in which the wrist has to reach a set of target positions while maintaining an RCM. Due to


the complexities of manipulation tasks and without loss of generality, a simple positioning

task is implemented in this simulation. In fact, positioning tasks are common in noninvasive

therapeutic techniques such as in radiotherapy and palpation.

For this experiment, a Mitsubishi PA10-7C arm equipped with a surgical instrument

is considered. The Mitsubishi PA10-7C is a redundant manipulator with 7 revolute joints.

In order to avoid redundancy, the shoulder joint is locked. In this scenario, the wrist of

the instrument follows a given trajectory fixed inside a box while passing through a port

situated on the top face of the box, creating a remote center of motion (Fig. 2.6).

The results of the trajectory tracking for two different trajectories are given in Fig. 2.7,

while Fig. 2.8 shows the corresponding lateral deflections at the RCM. 4. According to

the figures, despite relatively large deviation of the wrist position from the desired tra-

jectory, the lateral deflections at the RCM are rather small. This simulation verifies that

the proposed CLIK scheme can be considered as a reliable compliant motion controller for

trocar-constrained manipulation.

0.34

0.36

0.38

0.4

−0.02

0

0.02

0.38

0.4

0.42

0.44

0.46

x (m)y (m)

z(m

)

(a) Trajectory I

0.34

0.36

0.38

0.4

−0.02

0

0.02

0.38

0.4

0.42

0.44

0.46

x (m)y (m)

z(m

)

(b) Trajectory II

Figure 2.7: Results of trajectory tracking under a trocar constraint: the desired trajectory(dashed line) is compared with the actual wrist trajectory (solid line)

To assess the accuracy of the instrument when following desired trajectories, a second

4See the simulation video at www.youtube.com/watch?v=_hjFZpO8M7Y.


0 1 2 3 4 5−1

−0.5

0

0.5

1x 10

−3

x(m

)

Time (s)

0 1 2 3 4 5−1

−0.5

0

0.5

1x 10

−3

y(m

)

Time (s)

(a) Trajectory I

0 1 2 3 4 5−1

−0.5

0

0.5

1x 10

−3

Time (s)

x(m

)

0 1 2 3 4 5−1

−0.5

0

0.5

1x 10

−3

Time (s)

y(m

)

(b) Trajectory II

Figure 2.8: Lateral deflection at the RCM

.

z

.x

AdmissibleRectangle

×

Box

.Port

Figure 2.9: Admissible rectangle within the workspace of the robot

experiment is designed. The objective of this experiment is to determine if accuracy is

related to the location of the box considered in the first experiment (Fig. 2.6) within the

workspace of the robot.

Due to the symmetry of the robot workspace, the 3D analysis of the workspace can be

reduced to a 2D analysis. A rectangular region on the plane y = 0 within the workspace

is considered as an admissible region for the location of the port (Fig. 2.9). It is assumed


Table 2.2: Comparison of the accuracy measures and measured error norms for two differenttrajectories at different box locations

x (m) z (m) Accuracy (10−2) Max. error norm I (10−3m) Max. error norm II (10−3m)

0.5 0.4 17 22 29

0.41 0.37 23 17 24

0.45 0.45 40 16 23

0.37 0.50 83 11 17

that every location inside the rectangle corresponds to one potential port location that can

be considered as a reference point on the box, i.e., each location inside the rectangle can

be uniquely associated with a box location. By utilizing the inverse kinematics scheme

presented in Section 2.2.2, the average accuracy of the wrist (over all positions) inside the

box is calculated for every location inside the rectangle. This results in a topography for the

proposed measure of accuracy inside the admissible rectangle as illustrated in Fig. 2.10. In

fact, the topography demonstrates the expected accuracy as a function of the box location

within the admissible rectangle.

To determine how the location of the box can affect the accuracy of the CLIK scheme

proposed in this section, the same trajectory following experiment is repeated for four

different box locations inside the rectangle. The 2-norms of the resulting position errors

are shown in Fig. 2.11. According to the figure, in both cases the largest error norm is

obtained at x = 0.5 m and z = 0.4 m while the smallest error norm is associated with

the box located at x = 0.37 m and z = 0.5 m. On the other hand, the error norms

corresponding to x = 0.45 m and z = 0.45 m, and x = 0.41 m and z = 0.37 m are between

these two values and are relatively close. Interestingly, this qualitative assessment can be

verified by the values suggested by the topography. The maximum error norms and the

associated accuracy measures, read from Fig. 2.10, are compared in Table 2.2. The values

are normalized within the rectangle. According to the table, as the maximum error norm

decreases, the proposed accuracy measure increases.

This experiment demonstrates that the proposed measure can be physically associated

with the error bound of trajectory tracking using the proposed CLIK scheme. Additionally,

it demonstrates that the accuracy of surgical tasks can be improved by proper placement of


0.1

0.1

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

z(m

)

x (m)0.35 0.4 0.45 0.5 0.55

0.35

0.4

0.45

0.5

0.55

Figure 2.10: Topography of accuracy in positioning tasks normalized within the admissiblerectangle

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

2-n

orm

ofth

ew

rist

position

erro

r(m

)

0.50

0.4

0.410

0.37

0.450

0.45

0.370

0.50

(a) Trajectory I

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

2-n

orm

ofth

ew

rist

position

erro

r(m

)

(b) Trajectory II

Figure 2.11: 2-norm of the position error for two different trajectories at different box lo-cations

the cavity within the robot workspace. Note that the absolute accuracy, however, depends

on the general kinematics of the robot, and the Mitsubishi PA10-7C is not designed to

provide optimum accuracy for an application such as RAMIS, but is being utilized in this

thesis solely for proof of concept.


2.3.5 Tool Flexion Compensation

So far, it has been assumed that the surgical tool is completely rigid. However, this assump-

tion may not always be valid particularly with the current trend in developing finer and

lighter tools for more delicate interventions such as in pediatric surgery. In such situations

it is necessary to compensate for the deflections at both the RCM and the tool wrist. This

not only guarantees the accuracy in the implementation of the task, but neutralizes the

lateral forces that might be exerted at the RCM.

Effects of flexion can be abstracted as static deflection and dynamic vibrations [51].

Herein, by employing the static equations of a cantilever beam for small flexions, under

quasi-static assumption of the surgical process, the proposed CLIK can be adjusted to

compensate for the deflection. Note that employing the static equations of the deflection is

further justified for light weight instruments due to their small mass-to-length ratio. The

displacement of a flexible tool due to an end load P can be given as [52]:

w(l) =Pl2(3L− l)

6EI, (2.31)

where E is the Young’s modulus of the tool, I is the moment of inertia along the neutral

axis, L is the length of the tool, and l is the distance along the tool. Using (2.2) and (2.31),

assuming that the distance between the flange (where the tool is mounted) and pelb is L′,

the deflections at the RCM and the tool tip can be calculated by:

wrcm =Pρ2

rcmL3(3− ρrcm)

6EI, (2.32)

ww = wmax =PL3

3EI, (2.33)

where ww is the maximum deflection at the wrist, and ρrcm = L+L′

Lνrcm − L′

Lwhich renders

wrcm =wmax

2ρ2

rcm(3− ρrcm). (2.34)


.

rtool

.

s.

wrcm

.

.

wmax

.

.

wmax

flange

pelb

l = 0

l = L + L′

.

Figure 2.12: Flexible tool operating under a remote center of motion

The loci of the RCM on the deflected tool can be given as:

prcm(t) = νrcmpw(t) + (1− νrcm)pelb(t) + wrcms, for 0 ≤ νrcm ≤ 1, (2.35)

where s, the load direction, along with wmax can be measured using strain gauges (see Fig.

2.12). Note that pw and pelb are calculated from the forward kinematics of the robot under

the rigidity assumption. Since the direction of the applied end load is unknown in general,

an orthogonal pair of strain gauge Wheatstone bridges mounted along the tool’s neutral axis

can be utilized. Moreover, νrcm can be calculated by solving the following cubic equation:

nT

(

pelb(t)− pw(t)

)

νrcm − nT swrcm = nT (pelb(t)− ptroc). (2.36)

Once νrcm is calculated the rest of the procedure is similar to the rigid tool case. However,

it is worth mentioning that for general manipulation tasks the position and the orientation

of the end-effector, i.e., x will be affected by the deflection, and both have to be taken into

account.

2.4 Dexterity in Actively Constrained Surgical Manipulation 29

2.4 Dexterity in Actively Constrained Surgical Manipulation

2.4.1 Background

In the preceding section, the accuracy of the proposed CLIK scheme for RAMIS was inves-

tigated by proposing a suitable measure of accuracy. While accuracy is dependent on the

control scheme being utilized, there are other significant qualities that objectively quantify

the robot capabilities. Among them, dexterity is of the primary importance for the robotic

manipulators used in minimally invasive surgery.

Developing suitable measures to objectively quantify the capabilities of robotic manip-

ulators has been an active area of research over the past two decades. Recent surveys on

kinematic measures can be found in [1] and [53]. Many kinematic performance measures

for robotic manipulators have been proposed in the last two decades. Among them, manip-

ulability, dexterity and isotropy are the most popular robot performance measures. Ideally,

any kinematic measure has a physical interpretation and is objective in the sense of being

invariant with respect to the choice of a reference coordinate frame and length unit. Since

most of the existing dexterity measures are defined in terms of the Jacobian matrix, and

the Jacobian matrix is generally not invariant with respect to the choice of the reference

frame and length unit, the measures fail to meet the invariance requirements. Addition-

ally, for the application of actively constrained manipulation in RAMIS, one can intuitively

verify that the trocar constraint has a negative impact on dexterity and a true dexterity

measure for RAMIS should inherently reflect this impact. In this section, well-defined dex-

terity measures applicable to constrained positioning and manipulation tasks in RAMIS are

proposed.

While objectivity is essential for a kinematic metric, in certain situations, some types

of subjective kinematic criteria might be desired. This happens especially when some kine-

matic features of a task need to be emphasized. This invokes the notion of task-specific

kinematic measures that subjectively evaluate the compliance of the manipulator when

performing a given task in a certain configuration.

The manipulability Ellipsoid was proposed in [54], and its volume was assumed to be a


measure of uniformity of the mapping from joint-space to task-space. Consider the Jacobian

matrix as a linear mapping between the velocities in the joint and Cartesian spaces. In

general, this mapping projects the unit ball in the joint space into an ellipsoid in Cartesian

space whose volume can be given as√

JJT . The volume of the manipulability ellipsoid

was utilized as a measure of distance from singular configurations in [54]. In a singular

configuration, the actual degrees-of-freedom of the manipulator drops below the number of

joints. This is equivalent to the situation that at least one of the singular values of the

Jacobian matrix becomes zero. In [55], it was argued that the Jacobian determinant could

not be relied upon as a true measure of distance from singularity. The author proposed the

Jacobian condition number as a measure of isotropy. The Jacobian condition number was

originally proposed as a measure of dexterity in [56]. Riemannian geometry was employed

in [57] to construct a class of manipulability measures by associating a Riemannian metric

in task-space to another Riemannian metric in joint-space. However, the authors showed

that there was no natural choice of metrics in task-space and hence this brings arbitrariness

into the measure. In [58] and [59] a similar approach was employed for construction of

performance measures for constrained manipulators. They proposed the ratio of a task-

space norm to the induction of a joint-space norm in task-space as a measure of performance;

however, their measure suffers from arbitrariness as well. The reciprocal of the Frobenius

condition number of the Jacobian matrix was proposed as a measure of isotropy in [60]

and [61]. In an isotropic configuration, all of the singular values of the Jacobian matrix are

equal and nonzero.

As a matter of fact, none of the robot capability measures found in the literature meet all

of the requirements of an objective criterion. While frame invariance has been met in most

of the recently proposed measures, unit invariance has not been achieved yet. In [62], it was

argued that the condition number and the generalized inverse of the Jacobian matrix are

not naturally invariant to the change of units and/or reference frame, and hence measures

based on these quantities are unlikely to be otherwise. In [55], the concept of natural (char-

acteristic) length was proposed for rendering dimensionally homogeneous Jacobin matrices,

and in [63] scaling of the Jacobian matrix by the maximum available torque and force was


Table 2.3: Significant robot manipulator performance measures

Measure Frame Invariance Unit Invariance

Yoshikawa’s Manipulability [54] Yes No

Dexterity [56] Yes No

Generalized Manipulability [58] Yes No

Global Conditioning Index I [60] Yes No

Manipulability [57] Yes No

Global Conditioning Index II [65] Yes No

Kinematic Sensitivity [53] Yes No

recommended. However, so far these methods have not been justified or even heuristically

accepted. Major performance measures are outlined in Table 2.3. These measures were

utilized in [15,42,63,64] to evaluate the performance of robots in surgical applications.

Thus, to accurately assess the performance of a surgical manipulator, an objective per-

formance measure that is capable of reflecting the impact of the kinematic constraints

under which the surgical manipulator operates is required. This is supported by the fact

that the performance of an unconstrained manipulator is expected to be superior to that of

a constrained manipulator.

2.4.2 A Modified Measure of Dexterity

So far, kinetostatic dexterity has been defined in terms of either distance from singularity

or proximity to isotropy. It can be understood that it is ideal to find the configuration

which is the closest to isotropy and yet the furthest away from singularity. According

to [55], isotropy and singularity are the two ends of one spectrum, and this spectrum can be

quantified using the Jacobian condition number. However, Singularity is explicitly defined

in terms of the volume of the manipulability ellipsoid (the determinant of the Jacobian

matrix), and therefore the distance from singularity cannot be accurately quantified by the

conditioning index. On the other hand, conditioning of an ellipsoid cannot be indicated

by the determinant of the Jacobian matrix. An ill-conditioned ellipsoid could still have a

large volume. However, the condition number can only reflect the proportion of the largest

and smallest singular values and discards the information that can be acquired from other


singular values of the Jacobian matrix. Therefore, herein a modified measure of dexterity is

proposed that not only can it be used to interpret isotropy and singularity as the opposite

sides of one spectrum, but also can quantify the conditioning of an ellipsoid more accurately.

One such measure can be obtained by incorporating all the singular values of the Jacobian

matrix, normalized by the maximum singular value, i.e.

D(J) =

(

σ1

σn

σ2

σn· · · σn

σn

)1n

, (2.37)

where σ1 < σ2 < ... < σn are the singular values of J . In fact, (2.37) can provide a more

accurate indication of the conditioning of the ellipsoid, and can be expressed in terms of

the Jacobian matrix as:

D(J) =

2n

√

|JJT |‖J‖2

, (2.38)

which can be considered as a normalized version of Yoshikawa’s manipulability (the volume

of the manipulability ellipsoid). One benefit of using this modified measure, when compared

to Yoshikawa’s manipulability, is that it is able to differentiate two manipulability ellipsoids

with identical volumes but different conditioning.

Some of the interesting properties of the proposed measure can be outlined as follows:

• normality: 0 ≤ D(J) ≤ 1

• lower limit (singular configurations): lim σ1σn

→0D(J) = 0

• upper limit (isotropic configuration): lim σ1σn

→1D(J) = 1

• frame-invariance D(BJ) = D(AJ)

By optimizing the proposed measures one can improve dexterity in the sense of proximity

to isotropy as well as avoidance from singularity.

In order to demonstrate how the proposed modified measure can outperform the tra-

ditional dexterity and manipulability measures consider the four manipulability ellipsoids

of order 3 (i.e. n = 3) that are illustrated in Fig 2.13, referred to as E1, E2, E3, and E4


E1 E2 E3 E4

Figure 2.13: Different manipulability ellipsoids to examine the effectiveness of the proposedmeasure of dexterity: four manipulability ellipsoids, denoted by, E1, E2, E3,and E4 whose singular values are given as {2, 5, 6}, {2, 3, 6}, {3, 3, 4}, and{1.5, 1.5, 2}, respectively.

Table 2.4: Comparison of different measures in quantification of different manipulabilityellipsoids

Measure E1 E2 E3 E4

Yoshikawa’s manipulability 60 36 36 368

Condition Number 13

13

34

34

Minimum Singular Value 2 2 3 1.5

Modified Dexterity3√606

3√366

3√364

3√364

whose singular values are given as {2, 5, 6}, {2, 3, 6}, {3, 3, 4}, and {1.5, 1.5, 2}, respectively.

The values of several traditional measures corresponding to these ellipsoids are compared

in Table 2.4. Despite obvious differences between E1 and E2 it can be observed that the

condition numbers of the ellipsoids are equal. Similarly, Yoshikawa’s manipulability is iden-

tical for E2 and E3 despite their obvious differences. Finally, despite the similarity of E3, E4

(only different in scale), their minimum singular values evaluate them differently. The mod-

ified dexterity measure, however, describes E1 more desirable than E2. Based on the same

measure E3 and E4 are equally desirable and also more desirable than the other ellipsoids.

In conclusion, neither of the traditional measures are able to manifest the similarities as

well as the differences between the ellipsoids in accordance with our expectations. In fact,

the comparison provided in Table 2.4 demonstrates that the traditional measures fail to

recognize the differences and similarities of the ellipsoids as is expected. Base on the same

comparison, the proposed dexterity measure seems more compatible with the common sense


in evaluation of manipulability ellipsoids.

2.4.3 Dexterity in Positioning Tasks

In a positioning task, one is concerned about the ability of the wrist in exerting forces

(as opposed to torques) and/or translating (as opposed to rotating) in different directions.

Such capabilities are directly related to the properties of the translational submatrix of the

Jacobian matrix, Jv ∈ R3×n. Hence, positioning dexterity can be quantified as:

Dp = D(Jv), (2.39)

where Jv = Jv(I−J+c Jc). Not only is the dexterity measure (2.39) frame invariant, but it

is also invariant with respect to the choice of the length unit when applied to homogeneous

manipulators5. The Mitsubishi PA10-7C arm studied in the previous section as well as the

DLR MIRO [29] are two examples of homogeneous manipulators with revolute joints.

2.4.4 Dexterity in Manipulation Tasks

In a manipulation task, a combination of rotational and translational motions can be iden-

tified, i.e., J ∈ R6×n; hence, invariance with respect to the choice of the unit length is a

critical issue even for homogeneous manipulators used for manipulation tasks. One trivial

workaround is to consider the translational and rotational components individually, assum-

ing redundancy in each of them. However, this assumption is unrealistic due to the fact that

even in a purely translational (rotational) motion, orientation (position) is constrained and

no redundancy can be identified. Consideration of the coupling between the translational

and rotational measures was asserted in [53] as well.

The surgical motion analysis reported in [66] suggests that surgical manipulation is

mostly dominated by reach and orient motions. A reach motion is a translation while the

orientation is fixed, and an orientation motion is a rotation while the position is fixed. These

5In this context, the term “homogeneous manipulator” refers to a manipulator constructed from jointsof similar types.


motions can be described in terms of a set of tasks in order of priority as follows:

Reach :

Jcq = 0,

Jωq = 0,

Jvq = v,

(2.40)

Orientation :

Jcq = 0,

Jvq = 0,

Jωq = ω,

(2.41)

where Jv ∈ R3×n and Jω ∈ R3×n are the translational and rotational submatrices of the Ja-

cobian matrix, and v and ω are the desired translational and angular velocities, respectively.

By utilizing the concept of a restricted Jacobian, the Jacobian matrices corresponding to

reach and orientation motions are rendered as:

J r = Jv(I − J+ω Jω), (2.42)

Jo = Jω(I − J+v Jv), (2.43)

where Jv ∈ R3×n and Jω ∈ R3×n are the translational and rotational submatrices of the

constrained Jacobian matrix (2.12). In a similar fashion, it can be verified that both of the

above Jacobian matrices are dimensionally homogeneous (for homogeneous manipulators)

and so the reach and orientation dexterity measures may be defined as:

Dr = D(J r), (2.44)

Do = D(Jo). (2.45)

It can be shown that Dr ≤ D(Jv) and Do ≤ D(Jω) due to the implicit constraints in

the reach and orientation Jacobian matrices. Therefore, the proposed measures can give a

more realistic lower bound for dexterity in the rotations and translations involved in surgical


manipulation.

2.4.5 Optimal Placement of Surgical Cavity

Consider the positioning task example described in the previous section. It is desired to find

the optimal location of the box in the plane y = 0 and inside the admissible rectangle so

that the dexterity and accuracy are maximized. This forms a multi-objective optimization

problem, i.e.,

minx,z

A−1

D−1p

, (2.46)

and a Pareto-optimal solution should be sought.

The topography of accuracy for positioning tasks is illustrated in Fig. 2.10. In a

similar fashion, the topography of dexterity can be generated as shown in Fig. 2.14. The

optimum box location for maximal accuracy and dexterity can be determined from the

overlaid topographies illustrated in Fig. 2.15. Note that the intensity of the contours shows

the normalized value of the measures inside the rectangle. As expected, the Pareto-optimal

solution is located on the boundary of the feasible space as illustrated in Fig. 2.15. From

the overlaid topography it can be observed that dexterity and accuracy can be improved

up to 80% when the box is placed at the Pareto-optimal location (a comparison between

the measures pertaining to the Pareto-optimal location and the location with the worst

measures inside the rectangular region), while either of them can be further improved only

up to 20%.

Now consider a manipulation task in which both position and orientation of the wrist

are equally important. Consider a similar scenario as in the previous experiment; however,

this time the target fixture is represented by a position as well as an approach vector. This

will impose five constraints on the intracorporeal kinematics. Therefore, a modified scheme

for intracorporeal inverse kinematics is proposed. The corresponding inverse kinematics

solution has been provided in Appendix A. This treatment can be justified by the fact that

surgical fixtures can be easily represented by a position and an approach vector. The sixth


0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.7

0.70.8

0.9

z(m

)

x (m)0.35 0.4 0.45 0.5 0.55

0.35

0.4

0.45

0.5

0.55

Figure 2.14: Topography of normalized dexterity in the admissible rectangle for positioningtasks

0.1

0.1

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

z(m

)

x (m)

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.60.7

0.70.8

0.9

0.35 0.4 0.45 0.5 0.550.35

0.4

0.45

0.5

0.55

Figure 2.15: The overlaid accuracy and dexterity topographies and the Pareto optimal lo-cation of the box inside the admissible region (marked by a circle)

constraint for solving the intracorporeal inverse kinematics is imposed by considering the

yaw joint at zero position. This can also be justified by the fact that the highest dexterity

is achieved when the yaw joint is at zero position. Using this inverse kinematics solution

the topography of the dexterity measures Dr and Do can be evaluated inside the box for a

given approach vector. Fig. 2.16 illustrates the reach and orient dexterity topographies for

four different wrist orientations inside the box.

It can be observed that the dexterity measures drastically change for different wrist


0.350.4

0.450.5

0.55

0.3

0.4

0.5

0.6

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

x (m)z (m)

Rea

chD

exte

rity

Dr

00−1

0.7070

−0.707

00.707−0.707

−0.7070

−0.707

(a) Reach Dexterity

0.350.4

0.450.5

0.55

0.35

0.4

0.45

0.5

0.550.5

0.6

0.7

0.8

0.9

x (m)z (m)

Orien

tD

exte

rity

Do

00−1

−0.7070

−0.707

00.707−0.707

0.7070

−0.707

(b) Orient Dexterity

Figure 2.16: The reach dexterity Dr and orient dexterity Do calculated inside the box forvarious wrist orientations

orientations. Although the wrist orientation is dictated by the task requirement, it can be

demonstrated that by proper orientation of the surgical cavity within the robot workspace

reach and dexterity measures can be improved.

In a similar fashion as in the positioning task, the accuracy profile for different wrist

orientations are calculated and illustrated in Fig. 2.17. As the figure suggests, the accu-

racy is also highly dependent on the wrist orientation. It is worth mentioning that the

accuracy depends on the length unit used for computing the Jacobin matrix. As was men-

tioned previously, unlike dexterity, which is an intrinsic measure of robot capability, the

proposed accuracy depends on the control scheme utilized. Therefore, the accuracy of the

manipulation task is rendered in terms of the Jacobian matrix utilized for Cartesian control.

The dexterity and accuracy topographies are overlayed to find the Pareto-optimal loca-

tion of the box for each of the above manipulation tasks. Due to the low sensitivity of the

orientation dexterity with respect to the box location, only the reach dexterity and accuracy

are overlaid. The corresponding multi-objective optimization problem can be stated as:

minx,z

A−1

D−1r

. (2.47)

2.5 Summary 39

0.350.4

0.450.5

0.55

0.35

0.4

0.45

0.5

0.550.5

1

1.5

2

2.5

3

x (m)z (m)

Acc

ura

cyA

00−1

−0.7070

−0.707

00.707−0.707

0.7070

−0.707

Figure 2.17: The accuracy A of manipulation tasks calculated inside the box for variouswrist orientations

The results show that the Pareto-optimal solutions can result in up to 80% improvement

in both accuracy and dexterity in manipulation tasks.

This example demonstrates that optimal placement of the cavity can significantly im-

prove various indices such as dexterity and accuracy that can impact RAMIS outcomes.

By the same token, the location of the access port with respect to the desired task inside

the body cavity can impact the indices. The latter is the main motivation for selective

placement of the patient within the workspace of the robot in RAMIS.

2.5 Summary

In this chapter, general kinematics of compliant motion under trocar constraints were stud-

ied, and both analytical and closed-loop solutions for the inverse kinematics problem of

manipulation under trocar constrains were presented. The proposed solutions are not

manipulator-specific and can be easily extended to different types of mechanisms. It was

also demonstrated that the accuracy of the closed-loop inverse kinematics scheme depends

upon the manipulator’s configuration.

A modified dexterity measure was proposed that is more compatible with common

sense in evaluating various manipulability ellipsoids. Furthermore, Reach and Orientation

2.5 Summary 40

0.1

0.2

0.3

0.4

0.5

0.6

0.70.8

0.1

0.2

0.3

0.4

0.5

0.6

0.60.7

0.80.9

x (m)

z(m

)

0.35 0.4 0.45 0.5 0.550.35

0.4

0.45

0.5

0.55

0.10.20.30

.4

0.4

0.5

0.5

0.6

0.6

0.7

0.80.9

0.9

0.1

0.2

0.3

0.4

0.5

0.60.70.8

0.9

x (m)

z(m

)

0.35 0.4 0.45 0.5 0.550.35

0.4

0.45

0.5

0.55

0.1

0.2

0.3

0.4

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.8

0.9

0.1

0.10.2

0.20.3

0.30.4

0.4

0.5

0.50.60.70.80.9

x (m)

z(m

)

0.35 0.4 0.45 0.5 0.550.35

0.4

0.45

0.5

0.55

0.1

0.20.

3

0.4

0.4

0.5

0.6

0.7

0.8

0.9

0.9

0.10.

2

0.3

0.30.

4

0.4

0.5

0.5

0.60.70.80.9

x (m)

z(m

)

0.35 0.4 0.45 0.5 0.550.35

0.4

0.45

0.5

0.55

0.7070

−0.707

00−1

00.707−0.707

−0.7070

−0.707

Figure 2.18: The Pareto-optimal port locations for different manipulation tasks: The rect-angles highlight the regions where the Pareto-optimal port locations can befound

Jacobian matrices were introduced that are dimensionally homogeneous for manipulators

bearing similar joints, and therefore measures articulated in terms of these matrices are

invariant with respect to the length unit.

Finally, it was demonstrated that the placement of the surgical cavity in the robot’s

workspace affects both dexterity and accuracy of manipulation under trocar constraints.

The optimal cavity placement problem was articulated as a Pareto-optimal solution of a

multi-criteria optimization problem. The preliminary results showed upto 80% improvement

in both accuracy and dexterity. The results further sustain the hypothesis that preoperative

planning can significantly improve the success rate of RAMIS. Extracorporeal collisions and

geometric uncertainty are among the other aspects that should also be taken into account

for optimal planning of actively-constrained RAMIS.

Chapter 3

A Deterministic Approach to

Preparative Planning of RAMICS

Under Uncertainty

3.1 Synopsis

A generic computational framework for robust patient-specific preoperative planning of

RAMICS is presented in this chapter. The work extends the results reported in [22], [2], and

[3]. The proposed framework attempts to enhance robustness to variability of intraoperative

conditions (with respect to preoperative models) by maximizing the tolerance with respect

to uncertainty in preoperative data in the neighborhood of surgical targets.

An overview of the proposed framework is shown in Fig. 3.1. Given a robotic system and

based on preoperative images of the patients, it is desired to find at least a locally optimal

plan such that certain optimality indices, as well as robustness to intraoperative geometric

uncertainties, are improved. Fig. 3.1 illustrates how the preoperative planning problem is

rendered as a robust optimization problem given the images of the patient, the procedure

and the robot. At the highest level, are the inputs to the framework and at the lowest

level is a multi-criteria robust optimization problem. The intermediate blocks illustrate the

information that must be extracted from the inputs to create the final optimization problem.

41

3.1 Synopsis 42

robot

trocarkinematics

forwardkinematics

inversekinematics/joint limits

hollow robot/collisiondetection

patient

surgicalprocedure

task parser

multi-criteria robust optimization

robotperformance

index

robustnessindex

approachanglescriteria

targetfixtures/port loci

Jacobianmatrix

Figure 3.1: A hierarchical diagram of the proposed framework for robust preoperative plan-ning of RAMICS

Trocar kinematics entail the kinematics of any tool under motion constraints imposed by

a trocar. The generic motion constraints equations were derived in Chapter 2. Using

the equations governing the trocar kinematics along with the Denavit-Hartenberg (DH)

parameters of the robot, the forward and inverse kinematics of the robot can be extracted.

The Jacobian matrix of the robot is extracted from the forward kinematics of the robot, to be

used for quantification of the robot performance as a function of the joint values. In order to

address intraoperative geometric uncertainties, a Cartesian robustness index is defined using

the inverse kinematics of the robot, the mechanical joint limits of the robot, and the collision

3.2 Coronary Artery Bypass Surgery 43

detection among the hollow robot arms (created by modeling the robot using geometric

primitives). These measures, along with the measures that define the optimal approach

angles at the surgical sites, must be computed for all surgical fixtures. These surgical fixtures

are determined by parsing the procedure into task frames and correlating these frames to

the patient’s anatomy using the preoperative images of the patient. Moreover, the port loci,

extracted from the images, define the feasible space of the optimization problem. At the

lowest level, the planning problem is rendered as a constrained multi-criteria optimization

problem whose objective function and constraints are determined using the information

compiled from the upper levels in the hierarchy.

The rest of this chapter is organized as follows: A description of the coronary artery

bypass surgery is presented in Section 3.2. This is followed by an overview of the da Vinci

surgical system in Section 3.3. In Section 3.4, the problem formulation is presented, and

the proposed formulation is used for preoperative planning of totally endoscopic coronary

artery bypass using the da Vinci system. Finally, the system performance is evaluated by

several case studies.

3.2 Coronary Artery Bypass Surgery

3.2.1 Background

Coronary arteries supply blood from the Aorta to heart tissues. They are divided into three

branches: the left coronary artery, the circumflex artery, and the right coronary artery (see

Fig. 3.2). The Left Anterior Descending (LAD) artery starts from the bifurcation of the

left coronary artery and continues to the heart apex. The chance that the LAD is occluded

by fat and cholesterol increases with age. This is critical since it can block one of the major

blood supplies for cardiac tissue. Coronary Artery Bypass Grafting (CABG) is a treatment

to provide an alternative blood supply for the clogged LAD. This alternative is usually the

Left Internal Mammary Artery (LIMA). It comes off the subclavian artery close to the first

rib and extends along the sternum to the sixth rib.

Higher resistance to cholesterol buildup and proximity to the LAD makes the LIMA a

3.2 Coronary Artery Bypass Surgery 44

perfect candidate for bypass grafting. This is accomplished by preserving the upper end of

the LIMA at the subclavian artery and suturing the lower end to the LAD on the surface

of the heart. Alternatively, grafting can be done by taking down the LIMA entirely and

suturing it between another major inflow artery and the LAD.

LIMA

Figure 3.2: Left: The location of coronary arteries on the heart (source: [4])). Right: Theleft internal mammary artery (LIMA) located on the chest wall close to thesternum, extended from the first rib to the sixth rib

3.2.2 Endoscopic Coronary artery Bypass Grafting

Conventional single vessel CABG procedure consists of two major steps including: taking

down the LIMA from the chest wall (harvesting), and suturing the harvested LIMA to the

LAD on the anterior surface of the heart (anastomosis) [67]. Both of the above-mentioned

stages could be performed endoscopically referred to as Totally Endoscopic Coronary Artery

(TECAB) surgery. Alternatively, only the LIMA is harvested using an endoscope, and the

anastomosis is performed through a minithoracotomy under direct vision.

3.2.3 Patient Selection and Positioning

Patients’ suitability for coronary artery bypass surgery can be assessed based on their

anatomical geometry. As indicated in [9], patients with body-mass index of smaller than

32 are more suitable for robotics-assisted LIMA harvesting. The author has indicated that

3.3 The da Vinci Surgical System 45

robotics-assisted LIMA harvesting on patients with large hearts is typically more challeng-

ing. The author has also mentioned that harvesting the LIMA in patients with oval axial

shape thoraxes is easier than in those with rectangular axial shape thoraxes. Moreover, as

instructed by [5], patients with target vessel diameters larger than 1.7 mm can be considered

for totally endoscopic coronary bypass. Patients deemed eligible for robotics-assisted CABG

must go through certain preparation protocols preoperatively. Even though there are no

standardized protocols, less or more identical guidelines are followed for patient prepara-

tions. The patient’s left side should be elevated 30 degrees, and the left arm is extended on

an arm board [5]. For LIMA harvesting, on the other hand, [9] has proposed a 20-degree

elevation of the left side and raising the left arm over the patient’s head.

Before harvesting the LIMA, the lungs are collapsed and the chest is insufflated with

carbon dioxide. It has been recommended by [5] and [9] that the ports are placed in the

third, fifth and seventh intercostal spaces along the mid-auxiliary line. Also other qualitative

guidelines have been proposed for port placement with application to endoscopic coronary

artery bypass, e.g., [8] proposed a golden pyramid for optimal port placement.

The reliability of the above-mentioned methods is questionable due to the fact that

they do not accurately contemplate patient-specific dimensions nor robot kinematics. This

is the main motivation for the work presented in the rest of this thesis, i.e., to provide

a computational framework for reliable patient-specific preoperative planning for robotics-

assisted cardiac surgery.

3.3 The da Vinci Surgical System

The da Vinci surgical system (Intuitive Surgical, Inc., Sunnyvale, CA) is currently the

only approved tele-manipulation platform being utilized for RAMICS worldwide. The com-

plete platform consists of two subsystems: the surgeon’s console and the multi-arm robotic

system [28] (see Fig. 3.3). The console is designed to deliver an efficient and reliable

tele-manipulation experience for the surgeon.

The patient-side sub-system has four mechanical arms consisting of three instrument

arms and an endoscopic arm. Only two of the instrument arms are active at a time (referred

3.4 A Deterministic Approach 46

to as left and right). Each instrument arm consists of a setup arm with six passive joints,

denoted by qRpassive ∈ R6, and qL

passive ∈ R6 for the right and left arms, respectively, and a

double parallelogram with three active joints. Three additional Degrees Of Freedom (DOFs)

are provided by roll-pitch-yaw instruments, attached to the arm, rendering six active joints

for each arm denoted by qRactive ∈ R6 and qL

active ∈ R6. The RCM of the spherical mechanism

is positioned at the port of entry (where the trocar is located) to avoid exerting excessive

forces at the incision point. The active section forward kinematics are defined by the

kinematics of the intracorporeal mechanism with a roll-pitch-yaw wrist given in Table 2.1.

The endoscope arm is slightly different and consists of a holder with four passive joints,

qEpassive ∈ R4, and a 4-DOF active section, i.e., qE

active ∈ R4.

3.4 A Deterministic Approach to Robust Preoperative Plan-

ning

In this section, the contributions of various sources of spatio-temporal uncertainty in surgi-

cal manipulation are represented by a unified geometric uncertainty model. This approach

circumvents the requirement of a priori identification of the individual sources of uncer-

tainty and is crucial for avoiding the need to solve a hard problem. The main idea is that

these contributions can be represented in terms of the instrument tip position and orienta-

tion uncertainties. Correspondingly, these two entities can be represented by the instrument

wrist position uncertainty in a neighborhood of surgical targets. In the proposed formu-

lation, the instrument wrists are treated as agents with motion uncertainty. While their

locality is known (in a neighborhood of the target), no presumption of their specific motion

in that vicinity will be made.

3.4.1 Problem Formulation

Let each individual target fixture be represented by a unique pair {pt, nt}, where pt ∈

R3 and nt ∈ R3 represent the target point and the normal vector to the tangent plane

at that point, respectively. It is assumed that at a given target fixture, the endoscope


Figure 3.3: The da Vinci surgical system

remains stationary pointing at the target, while the instrument wrists dynamically move in

a neighborhood of the target. In that neighborhood, one can find a p-norm tolerance ball

with radius r given as:

B , {p ∈ R3| p = pt + rRz, z ∈ R3, ‖z‖p ≤ 1}, (3.1)

within which the right and left instrument wrists can freely move while the chances of

collisions among the extracorporeal bodies and of joint limit violation are guaranteed to be


left instrument right instrument

endoscope

Tolerable wrist position uncertainty at target fixture {pt, nt}

Figure 3.4: A schematic illustration of the surgical workspace and wrist uncertainty withp = 1

eliminated (Fig. 3.4). In (3.1), ‖ · ‖p is the p-norm operator, z is a point inside the p-norm

unit ball in R3 defined by ‖z‖p ≤ 1, and R ∈ SO(3) is a rotation matrix.

In general, a procedure is characterized by N individual target fixtures, denoted by

{pt(k), nt(k)} for k = 1, ..., N , that must be reachable intraoperatively, and it is desired

to find the optimal plan P ∈ Rn such that the tolerable uncertainty volumes, denoted by

B(k) for k = 1, ..., N , over all of the target fixtures are maximally enlarged. The resulting

planning problem is rendered as a mathematical programming problem with the following

general form:

minP

U(P) (3.2)

subject to

gi(P,pRwrist,p

Lwrist) ≤ 0 for i = 1, ...,m1, ∀ pR

wrist ∈ B(P, k), pLwrist ∈ B(P, k) for k = 1, ..., N,

hi(P, k) ≤ 0 for i = m1 + 1, ...,m1 +m2, for k = 1, ..., N,

where pRwrist ∈ R3 and pL

wrist ∈ R3 are the right and left wrist position vectors respectively,

U : Rn → R is the objective function; gi : Rn × R3 × R3 → R and hi : Rn × R → R are

constraint functions; and m1 and m2 are scalars representing the number of constraints.


The mathematical programming problem (3.2) belongs to a class of Generalized Semi-

Infinite Programming (GSIP) problems referred to as design centering (DC) problems [68].

In a DC problem, the design parameters are determined such that the tolerance is maxi-

mized; i.e., U must be a decreasing function of the volume of B(P, k). This is equivalent to

designing for the minimum sensitivity to parameter uncertainty.

3.4.2 Solution Method

A GSIP is often solved by replacing the semi-infinite constraints gi by a finite set of con-

straints obtained by grid-based discretization or by bi-level techniques (see [69] and [70]).

The latter solve a GSIP by alternating between the lower-level problem maxpRwrist,p

Lwrist

gi

and the transformed upper-level problem in each iteration. As an alternative, outer ap-

proximation techniques have been proposed for solving DC problems (see [71] and [72]).

In general, solving (3.2) using the above methods can be computationally expensive. For

instance, by discretizing B with an a× a× a grid, each gi in (3.2) would then be replaced

by a3 × a3 ×N constraints. Here, it is demonstrated that by a slight modification of (3.2),

a more tractable problem can be obtained.

Let p-norm tolerance balls B(k) be defined in the space of dual positions p ∈ R6 as:

B(k) ,

p(k) ∈ R6|p(k) =

pt(k)

pt(k)

+ r(k)

R(k) 03×3

03×3 R(k)

z, z ∈ R6, ‖z‖p ≤ 1

,

(3.3)

where z is a point inside the p-norm unit ball in R6. The modified GSIP is given as:

minP

U(P) (3.4)

subject to

gi(P, pwrist) ≤ 0 for i = 1, ...,m1, ∀ pwrist ∈ B(P, k) for k = 1, ..., N,

hi(P, k) ≤ 0 for i = m1 + 1, ...,m1 +m2, for k = 1, ..., N,


where pwrist =

pRwrist

pLwrist

is the dual wrist position vector.

By using the definition of the p-norm, it can be shown that if for a given plan P there

are B(P, k), such that gi(P, pwrist) ≤ 0 for all pwrist ∈ B(P, k) for k = 1, · · · , N , one can

conclude that gi(P, pwrist) ≤ 0 for all pRwrist,p

Lwrist ∈ B(P, k) for k = 1, · · · , N where B(P, k)

has the same r(k) and R(k) as B(P, k). For p =∞ it can be shown that this relationship is

bidirectional, i.e., (3.2) and (3.4) are equivalent. Solving (3.4) with p 6=∞, however, yields

more conservative results, i.e., the actual tolerance will be larger than the tolerance that is

guaranteed by (3.4). In fact, the resulting tolerance can be considered as a lower bound for

the actual tolerance, and this does not risk the optimality of the solution.

In (3.4), ∪Nk=1B(P, k) provides an inner approximation for the feasible subset of the

dual wrist position space, defined as C(P) , {p ∈ R6|gi(P, p) ≤ 0 for i = 1, ...,m1}, in the

neighborhood of surgical targets (see Fig. 3.5). By solving (3.4), one should be able to find a

plan that can maximize the volume of C(P), and therefore ∪Nk=1B(P, k), in the neighborhood

of the targets. It can be shown that with a closed and convex C(P), B(P, k) can be replaced

by their boundaries ∂B(P, k) in (3.4), leading to a more tractable optimization problem.

Moreover, if B(P, k) are convex polytopes with s vertices, it will be sufficient to check the

feasibility of gi(P, pwrist) ≤ 0 at the vertices of those polytopes. In other words, by choosing

p such that B(P, k) are convex polytopes, and starting with a feasible set of polytopes inside

C(P), gi(P, pwrist) can be replaced by gijk(P) = gi(P, pjk) for j = 1, ..., s, k = 1, ..., N in

(3.4), where pjk refers to the position of vertex j of B(P, k). The resulting optimization

problem is rendered as a constrained nonlinear program given as:

minP

U(P) (3.5)

subject to

gijk(P) ≤ 0 for i = 1, ...,m1, j = 1, 2, ..., s, k = 1, ..., N,

hik(P) ≤ 0 for i = m1 + 1, ...,m1 +m2, k = 1, ..., N.

In general, the convexity of (3.5) will be dependent on the convexity of hik, gijk and the


C

B(P0)

B(P1)

B(P∗)

Figure 3.5: A conceptual illustration of design centering

objective function in P.

In (3.3), by choosing p = 1 or p =∞, B(k) will be convex polytopes. By choosing p =∞,

‖z‖p ≤ 1 will be a hypercube with s = 64, while for p = 1 it will be a hyper-octahedron

with s = 12. This suggests that the semi-infinite constraints of (3.4) will be replaced by

12×m1×N and 64×m1×N constraints for p = 1 and p =∞, respectively. It is worth noting

that in (3.2) the respective numbers will be 36×m1×N and 64×m1×N . This proves that

the size of the problem can be significantly reduced by solving the modified GSIP in (3.4)

with p = 1. This results in a significant reduction in the number of function evaluations,

particularly when analytic expressions of the function gradients are not available.

In the upcoming section, preoperative planning of TECAB, assisted by the da Vincir

surgical robotic system, is formulated and solved with the solution method proposed in this

section.

3.4.3 Multi-Criteria Objective Function

The primary objective of preoperative planning of RAMICS is to help the surgeon perform a

robotics-assisted cardiac procedure successfully. This is achieved by improving a number of

performance indices and increasing the chance of success regarding intraoperative uncertain

conditions. In general, reaching an optimal solution may not be feasible and instead a Pareto


optimal solution must be sought. In other words, a trade-off must be made between the

nominal performance and robustness. In the following, an aggregate objective function is

constructed from several performance measures and a measure of robustness.

3.4.3.1 Robustness

The volume of B(P, k), that can be quantified in terms of r(k), is an indication of robustness

of the surgical procedure with plan P. A Robustness Index (RI) can be defined as:

RI = minkr(k). (3.6)

3.4.3.2 Robot Performance

One of the significant aspects of this planning framework is the consideration of the per-

formance of the robotic arms for successful and efficient accomplishment of surgical tasks.

In [56] and [60] robot dexterity was defined in terms of the condition number, κ(J), of

the Jacobian matrix J of the arm. As recommended by [55], a global dexterity measure

can be obtained by averaging the condition number over the entire or a portion of the

arm workspace in Cartesian space. While improving the dexterity of the intracorporeal

mechanism can result in more isotropic capability for exerting forces in Cartesian space,

optimization of other measures such as manipulability [54] and kinematic sensitivity [53] can

improve the performance in other ways. The majority of kinematic performance measures,

including the above-mentioned measures, are defined in terms of the Jacobian matrix, and

as a result they usually suffer from inhomogeneity due to disparate units of the Jacobian

matrix elements. In order to work around this issue, the modified measure D and the ho-

mogeneous Jacobian matrices, that were proposed in Chapter 2, are employed. The global

performance of the instrument arms over a given set of surgical targets pt(k) can be ar-

ticulated in terms of coordinate-free modified global reach and orient conditioning indices,


GRCI and GOCI, given by:

GRCI =1

NΣNk=1

Wk

∫

QD(J reach(k))dq∫

Q dq, (3.7)

GOCI =1

NΣNk=1

Wk

∫

QD(Jorient(k))dq∫

Q dq, (3.8)

where J reach = Jv(I − J+ωJω) and Jorient = Jω(I − J+

v Jv) are the reach and orient Jaco-

bian matrices (Jv and Jω are the translational and rotational submatrices of the Jacobian

matrix), Wk are weights such that ΣNk=1Wi = N , and dq and Q are the infinitesimal volume

element and a portion of the wrist joint space, respectively.

3.4.3.3 Approach Angles

To guarantee that a given target point is reachable by the surgical tool, not only must it

be located within the workspace of the intracorporeal mechanism, but the approach angle

of the tool at the surgical site should also be such that reaching the target is feasible. A

study of the optimal approach angles of surgical tools and the endoscope has been reported

in [10]. In [20] optimal angles of the instruments were taken into account for surgical

planning; however, the recommended angles were measured with respect to the inertial

coordinate frame rather than the target coordinate frame. Generally speaking, to ensure

that the targets are reachable, it is essential that the tools make an acute angle with the

normal vector at the surgical site (Fig. 3.6). Heuristically, an approach angle between 30°

to 60° is recommended. Optimality of the approach angles can be quantified in terms of

the Root Mean Squared Error (RMSE) of the tool approach angle from an optimal angle

φ0:

RMSEφ0 =

(

1

NΣNk=1

(

arccos(nTt m(k))− φ0

)2)

12

, (3.9)

where m(k) ∈ R3 is the unit approach vector of the instrument.


pt

n

m

φ

Figure 3.6: 0 ≤ φ0 ≤ 90° is the optimum approach angle of the tool at the surgical site

3.4.3.4 Hand/Eye Alignment

A better replication of the surgeon’s hands and angle of view by the configuration of the

endoscope and the instruments can assist the surgeon in hand/eye coordination. The align-

ment can be quantified in terms of the angle between the endoscope and the instrument

plane (elevation angle), ψ, and the angle between the instruments (manipulation angle), η

(Fig. 3.7). The associated RMSE criteria are given by:

RMSEψ0 =

(

1

NΣNk=1

(

arccos(

(mL(k)× mR(k))TmE(k))

− π

2+ ψ0

)2)

12

, (3.10)

RMSEη0 =

(

1

NΣNk=1

(

arccos(

mTR(k)mL(k)

)

− η0

)2) 1

2

. (3.11)

where mR, mL and mE are the normal approach vectors of the right tool, left tool and the

endoscope, respectively.

The multi-criteria objective function is defined in terms of the above-mentioned mea-

sures. Table 3.1 outlines the individual criteria incorporated into the objective function to

be minimized and the aggregate objective function is given in terms of the criteria as:

U =10∑

i=1

fi(Ui), (3.12)

where fi : R → R are scaling functions, and Ui are the individual criteria listed in Table


pt

n

30°

mL

mR

mE

ψ

η

Figure 3.7: Hand/eye alignment in terms of the elevation and manipulation angles

Table 3.1: The criteria contributing to the objective function

Criterion Description Index

RI Robustness − mink r(k)

GRCIR Right arm reach dexterity −ΣNk=1

∫

QWkD(J

R

reach(k))dq

N

∫

Qdq

GOCIR Right arm orient dexterity −ΣNk=1

∫

QWkD(J

R

orient(k))dq

N

∫

Qdq

GRCIL Left arm reach dexterity −ΣNk=1

∫

QwkD

−1(JL

reach(k))dq

N

∫

Qdq

GOCIL Left arm orient dexterity −ΣNk=1

∫

QwkD

−1(JL

orient(k))dq

N

∫

Qdq

RMSERφ0

Right arm approach angle(

1N

ΣNk=1

(

arccos(ntT mR(k)) − φ0

)2) 1

2

RMSELφ0

Left arm approach angle(

1N

ΣNk=1

(

arccos(ntT mL(k)) − φ0

)2) 1

2

RMSEEφ0

Endoscope approach angle(

1N

ΣNk=1

(

arccos(ntT mE(k)) − φ0

)2)

RMSEψ0Elevation angle

(

1N

ΣNk=1

(

arccos(

(mL(k) × mR(k))T mE(k))

−π2

+ ψ0

)2) 1

2

RMSEη0Manipulation angle

(

1N

ΣNk=1

(

arccos(

mTR(k)mL(k)

)

− η0

)2) 1

2

3.1. Blending the individual criteria to form an aggregate objective function has been

a topic of research for a long time and is still an open area of research in optimization

theory. The simplest method for constructing an aggregate objective function is the weighted

summation; however, by optimization of a weighted sum of nonconvex functions the Pareto

optimal solutions that reside in the nonconvex portion of the parameter space may not

be reached [73]. Furthermore, this method is only useful whenever the weights can give a

meaningful articulation of preference. Both non-convexity and the difficulty of a meaningful

articulation of preference in terms of weights occur in the problem being addressed in this


chapter. An extensive survey of alternative methods can be found in [74]. However, in

most of the proposed methods an accurate prior articulation of preferences or a knowledge

of individual optima is required. In [75] a more practical and natural way of scaling the

objective function, so called physical programming, has been introduced. This method

only requires a rough articulation of preferences of the individual criteria. The idea is to

simply partition each criterion over its range based on preference and attribute a convex,

unimodal and smooth function to each range and connect the functions to form an overall

smooth, unimodal and convex function. For instance, using piecewise cubic polynomial

interpolation guarantees C 1 smoothness, and preserves unimodality and convexity [76] of

the resulting aggregate objective function. Herein, to reduce the computational burden, a

slightly simplified version of this method is utilized. The function is designed to have only

three different regimes, i.e.,

fi(Ui) =

fi1(Ui) Ui1 ≤ Ui ≤ Ui2 desirable

fi2(Ui) Ui2 ≤ Ui ≤ Ui3 tolerable

fi3(Ui) Ui3 ≤ Ui ≤ Ui4 undesirable

, (3.13)

where fij : R→ R forj = 1, 2, 3 are cubic interpolating polynomials. Obviously the objective

function can be shaped more desirably as the number of partitions increases, yet a trade-off

must be made to keep the complexities low. In order to ensure convexity, the corresponding

function values Yj = fi(Uij) for j = 1, ..., 4, are selected such that ∆Yj > ∆Yj−1.

Finally, using a log-sum smoothing function, as recommended by [75], the aggregate

objective function (3.12) is replaced by:

U = log

(

1

10

10∑

i=1

fi(Ui)

)

. (3.14)

3.4.4 Optimization Problem

The preoperative planning problem implemented in this section includes finding the opti-

mum port locations for the two instrument arms, and the endoscope arm, along with the

optimum orientation of the arms relative to the patient’s body coordinate frame. Normally,


the intercostal spaces (ICS) between the second and eighth ribs on the left side of patient’s

ribcage form the port loci for TECAB. The loci can be represented as a set of parametric

curves (such as splines) extracted from the patient’s CT images:

I = {ci(ui)|ci : R→ R3, 0 ≤ ui ≤ 1 for i = 2, 3, .., 7}, (3.15)

where ui is the curve parameter and ci is the spatial curve representing the ith intercostal

space.

For a given ICS triad on the patient’s rib cage, c =

ci(ui)

cj(uj)

ck(uk)

∣

∣

∣

∣

∣

ci, cj , ck ∈ I

, the

planning problem can be formally rendered as an instance of (3.2), i.e.,

P∗(c) = arg minP

U (3.16)

subject to

dactive > 0 ∀ pE = pt(k), pRwrist ∈ B(P, k), pLwrist ∈ B(P, k) for k = 1, 2, ..., N,

qactive ∈ Qvalidactive ∀ pE = pt(k), pRwrist ∈ B(P, k), pLwrist ∈ B(P, k) for k = 1, 2, ..., N,

ppassive ∈ c,

dpassive > 0,

qpassive ∈ Qvalidpassive

where P∗(c) ={

q∗passive, p∗

patient, {r∗(k),R∗(k)}1:N

}

is the optimal plan for ICS triad c,

ppatient ∈ R3 is the position vector of the patient with respect to the robot base frame,

pE ∈ R3 is the position of the endoscope, ppassive ∈ R9 is the augmented position vector of

the holders (that must coincide with the ports on the patient’s intercostal spaces), dpassive is

the minimum distance between the passive sections and surrounding static objects, dactive

is the minimum distance between the active sections and all other surrounding objects,

qpassive ∈ R16 is the augmented vector of passive joints belonging to all three arm holders,


qactive ∈ R9 is the augmented vector of the active joints excluding the wrist joints, and

Qvalidactive and Qvalid

passive are the valid active and passive joint spaces determined by the joint

limits. In order to solve (3.16), it has to be transformed into an instance of (3.5) using the

approach proposed in the previous section.

Note that it has been assumed that the orientation of the patient is fixed with respect

to the robot base frame (base of the setup arm), and only the position can be adjusted.

Also it is assumed that the robot is always placed on the right side of the patient.

3.4.5 Collision Avoidance

For the minimum distance calculation, the bodies are modeled using a set of geometric

primitives [77, 78]. Capsules [78] and spheres are utilized for geometric modeling of the

extracorporeal sections of the da Vincir robotic arms. A capsule is simply an extension of

a line segment and has a convex geometry. Due to its smooth surface, the minimum distance

between a capsule and any other smooth convex primitive can be obtained analytically, in

contrast to a cylinder that requires an ad hoc collision detection procedure. Fig. 3.8

illustrates the da Vincir active sections that are modeled by geometric primitives, and the

resulting hollow robot of the da Vinci is shown in Fig. 3.9. As a function of the wrist

positions, the minimum distance between the arms is expressed in terms of the distances

between their primitives using the analytic expressions given in [78] for capsule/capsule and

capsule/sphere cases.

Fig. 3.10 illustrates a wire frame representation of the active section of the da Vinci

hollow arm. The arm is composed of a double parallelogram that creates a remote center

of motion. As it has been depicted in the figure, the two parallelograms are not accurately

aligned and are misaligned by a small angle α. This misalignment is to ensure that the

remote center of motion occurs on the instrument shaft - rather than on the leg of the

parallelogram - which has a small offset l from the leg of the larger parallelogram. This can

be simply satisfied by choosing α = arcsin( lL

). The spatial kinematics of the mechanism


sphere

capsules

sphere

capsules

Figure 3.8: The da Vincir instrument arm (left), and the endoscope arm (middle) modeledby geometric primitives: capsules and spheres

Figure 3.9: The da Vinci hollow robot composed of geometric primitives

can be described in terms of the active joints as:

p1 =

Lc(θ2 + α)c(θ1)

Lc(θ2 + α)s(θ1)

−Ls(θ2 + α)

, (3.17)


x0

z0

p1

l

p2

l1

p3

l2

p4

L

α

θ2 θ1

Figure 3.10: Wireframe representation of the da Vinci active arm

p2 = p1 +

l2cθ2cθ1

l2cθ2sθ1

−l2sθ2

, (3.18)

p3 = p1 +

0

0

−l2

, (3.19)

p4 =

0

0

−l2

. (3.20)

These points are in fact the endpoints of the capsules, and are used to calculate the minimum

distances among the arms.


3.4.6 Coding of Patient-Specific Models

The information extracted from each patient’s CT images are encapsulated into a data

structure. The structure is composed of the following members:

• A pair of spatial curves representing the LIMA extended from the subclavian artery

to the sixth rib and the normal vector to the chest wall along the LIMA

• A pair of spatial curves representing the LAD from the bifurcation to the apex, and

the normal vector to the heart surface along the LAD

• A set of spatial curves representing the intercostal spaces (from the second to the 7th)

and the normal vectors to the rib cage along the intercostal spaces

• A capsule representing the patient’s body

• The orientation of the patient with respect to the CT reference coordinate frame

The structure is passed on to the planner as the input, and the planner returns an output

structure containing the optimal plan.

3.4.7 Implementation

Algorithm 3.1

c← PickTriad(I)P0(c)← ∅{P i0(c)}i=1, 2,..., M ← rand(M)i← 1while i ≤M and P0(c) = ∅ doP0(c)← FindFeas(P i0(c))i← i+ 1

end whileif P0(c) 6= ∅ thenP∗(c)← Optimize(P0(c))

elseP∗(c)← ‘infeasible’

end ifreturn P∗(c)

An overview of the implementation of the planner is given in Algorithm 3.1. Given an

ICS triad c ⊂ I, selected by PickTriad, M initial plans, denoted by P i0(c) for i = 1, · · · ,M ,


are randomly generated. In the proposed solution method in Section 3.4.1, it has been

assumed that the initial plan is feasible; however, the generated plans are not necessarily

feasible. Hence, FindFeas attempts to find a feasible plan by solving a minimax problem

whose initial plan is picked from the generated plans. If a feasible plan is found, FindFeas

returns it, otherwise, ∅ is returned. Once a feasible plan is found, it is passed on as the

initial plan to Optimize, which is an implementation of (3.16) as in (3.5). Otherwise, triad

c is labelled as infeasible.

Due to the nonconvex nature of the resulting nonlinrear programming problem, the

resulting plan is usually only a locally-optimal plan. One solution to address this issue

is convex relaxation, i.e., approximating the original problem with a convexified problem.

However, this approach is not trivial and may result in either fairly conservative or unreliable

solutions. Another more trivial approach is to pick the best plan from the set of plans that

are obtained by running the optimization problem initialized with different feasible plans.

Theoretically, as the number of runs increases, the chance of getting a globally-optimal

solution increases as well.

The resulting optimization problem was implemented in MATLAB using fmincon em-

ploying active-set Sequential Quadratic Programming (SQP) and the Broyden-Fletcher-

Goldfarb-Shanno (BFGS) Hessian matrix update rule. Surgical features, that are significant

for TECAB, were extracted from the CT images, and were stored in a data file to be used

by the planner. N = 15 surgical fixtures along the LIMA and LAD were sampled, and for

the sake of computational simplicity it was assumed that all B(P, k) were identical, i.e.,

R(k) = R and r(k) = r. Not only is this assumption reasonable when all of the targets

are equally significant, but it results in a minimal parameterization of the plan. This is

particularly valid considering that (3.6) only considers the ball with the minimum volume.

Nevertheless, the proposed formulation is capable of incorporating non-uniform task uncer-

tainty at various target locations by using different weightings or using splines to address

variable task requirements at different target locations.

Fig. 3.11 shows the target fixtures and ICSs that were extracted from a patient’s CT

images. This information was then passed on to the algorithm for planning. For each ICS


triad, the planning took roughly 10 minutes on a PC with a quad core Intel i7 2.60 GHz

CPU. According to the results, the best score was obtained by placing the left, endoscope

and right ports in the sixth, fourth and third ICSs, respectively (see Fig. 3.12 (left)). The

tolerance volumes, B(k), in the neighborhood of the surgical targets are also shown in Fig.

3.12 (left) as a union of octahedra with r = 19 mm. The optimal port locations as well as the

tolerance volumes that were found by the algorithm based on the ICS triad recommended

by the surgeon are also shown in Fig. 3.12 (right). As can be seen, the tolerance volumes

(octahedra) pertaining to the surgeon-recommended triad are much smaller than those of

the best plan found (among all possible ICS triads) by the algorithm.

Fig. 3.13 shows the optimal placement of the robot (corresponding to the port locations

shown in Fig. 3.12 (left)) with respect to the patient’s thorax. Also, the snapshots of

the arms while reaching the targets along the LAD and the LIMA, shown in Fig. 3.14,

demonstrate the feasibility of the plan under nominal conditions.in

terc

ost

al

space

s(I

CS

)

LA

D

LIM

A

Figure 3.11: Surgical fixtures (composed of target positions and normal vectors) extractedfrom a patient’s 3D model reconstructed from the patient’s CT images. Thefixtures are determined based on the procedure and are identified by a surgeonor a radiologist. The surface normal vector at each target location is deter-mined by measuring three adjacent points surrounding the target and fittinga plane to those points.


tolerance polytopes

endoscope portleft port right port

tolerance polytopes

endoscope portleft port right port

Figure 3.12: Left: Port locations recommended by the planner and the computed wristuncertainty volume as a union of octahedra. Right: Port locations recom-mended by the planner based on the triad recommended by the surgeon, andthe computed wrist uncertainty volume as a union of octahedra

Figure 3.13: A 3D view (left) and the top view (right) of the positioning of the rib cage andthe arms pertaining to the best plan recommended by the algorithm

3.4.8 System Performance

In order to determine in what way the planning strategy proposed in this chapter can

improve current practice, the quality of the ports selected by an experienced surgeon and

those recommended by the algorithm were compared. Such a comparison was carried out

for four different patients requiring coronary artery bypass. The surgeon was asked to mark

the port locations with fiducial markers on the patients’ rib cages before they were scanned.


1 2 3

4 5 6

Figure 3.14: The da Vincir arms in action: a collision-free trajectory of the active sectionsas they reach the individual target sites inside the thorax

In order to analyze the suitability of the ports selected by the surgeon, first the algorithm

was run to find the best possible robot configuration with the surgeon’s recommended ports

(SRP). Interestingly, in two cases the algorithm was unable to find a feasible plan due to

collisions and/or joint limits. In other words, the ports recommended by the experienced

surgeon based on common practice can be infeasible in terms of target reachability and

collision avoidance. In the second round of experiments, the ports were no longer assumed

known a priori, but they were assumed to be within the same ICS triad that the ports

selected by the surgeon belonged to, i.e., the algorithm was run to find the optimal robot

configuration and port locations based on the surgeon’s recommended ICS triad (SRT).

Finally, for each patient the algorithm was run for all candidate ICS triads and the best

plan was chosen. A comparison between SRP and SRT plans with the plans obtained based

on the algorithm’s recommended ports (ARP) is presented in Figs. 3.15, 3.16, and 3.17.

According to the results, ARP plans generally have larger aggregate scores (normalized

absolute value of (3.14)), and have up to 300% larger robustness indices without risking the

total dexterity (an infeasible plan is given an aggregate score as well as a robustness index of


Case I

Case II

Case III

Case IV

SRP

SRT

ARP

0

0.2

0.4

0.6

0.8

Aggre

gate

Sco

re

Figure 3.15: Comparison of the overall scores of the plans pertaining to ARP, SRP andSRT

zero). The charts also imply that ARP plans can result in higher success rate. In fact, the

actual tolerance that has been achieved by ARP plans for different patients can be evaluated

by Monte Carlo simulations. For each case, a set of wrist positions are randomly generated

with a uniform distribution within the cube circumscribing the octahedron in the vicinity

of each target, and the joint values and minimum distance between the arms are computed

assuming that ARP plans have been chosen for each case. This volume is equivalent to the

volume of B defined with p =∞ and is six times larger than the volume of the octahedron.

The percentage of the constraints in (3.5) that are satisfied are computed at each target

location. Fig. 3.18 shows the computed rate of the constraint satisfaction for different

patients. According to the results, the cube circumscribing the octahedron can be taken

as the tolerance with less than 5% chance of collisions and/or joint limit violation while

this tolerance volume is six times larger than the volume of the octahedron. This result

is particularly significant due to the fact that it has been obtained without compromising

the efficiency of the algorithm by employing computationally expensive sampling-based or

grid-based methods. Furthermore, with a similar simulation in which the wrist positions

3.5 Execution 67

Case I

Case II

Case III

Case IV

SRP

SRT

ARP0

10

20

30

Robust

nes

sIn

dex

(mm

)

Figure 3.16: Comparison of the achieved robustness in the plans pertaining to ARP, SRPand SRT

were confined to be within the volume of the octahedra, constraint satisfaction of 99% was

achieved. This verifies that the convexity assumption of the lower level problem is realistic.

3.5 Execution

In order to transfer the plans obtained using the proposed algorithm into the operating

room the following steps must be taken:

• The patient’s pose must replicate his/her pose during the preoperative scanning as

closely as possible.

• The robot arms passive joint values must be configured as recommended by the plan-

ner. This might require calibrating the joints of the utilized CAD model with those

of the actual robot.

• The patient must be positioned with respect to the robot base as recommended by

the planner. This requires measuring the positions of the robot base as well as the

3.6 Summary 68

Case I

Case II

Case III

Case IV

SRP

SRT

ARP0

0.2

0.4

0.6

Aver

age

Dex

etrity

Figure 3.17: Comparison of the achieved dexterity in the plans pertaining to ARP, SRPand SRT

patient. Note that the algorithm calculates the optimal relative position of the base

with respect to the CT images reference coordinate frame. Therefore, it might be

more reasonable to replace this frame with a rigidly-mounted frame (e.g., a fiducial

frame) on the patient’s chest that can be scanned and used for accurate placement of

the patient with respect to the robot in the operating room.

After taking these steps the remote center of motions on the arms must coincide with the

desired port locations and the tools can be easily inserted.

3.6 Summary

A computational framework for patient-specific preoperative planning of Robotics-Assisted

Minimally Invasive Cardiac Surgery (RAMICS) has been developed. The proposed frame-

work is capable of addressing the geometric uncertainty that is present in RAMICS, and

is generic in the sense that no presumptions about the procedure or the manipulator are

made. The proposed framework can handle the contributions of several sources of geometric

3.6 Summary 69

Case I Case II Case III Case IV90

92

94

96

98

100

Con

stra

int S

atis

fact

ion

Rat

e

Figure 3.18: Actual reliability in terms of the constraint satisfaction rate based on the plansrecommended by the algorithm for four different patients: each bar representsthe constraint satisfaction rate at one target fixture k = 1, · · ·N .

uncertainty (i.e., the mismatches between the preoperative data and intraoperative condi-

tions) without characterizing them directly (i.e., using a non-model-based approach). This

has been accomplished by representing the uncertainty in the intraoperative conditions (in

the vicinity of the targets) with the uncertainty in the position of the instrument wrists

with respect to the surgical targets. The strength of the proposed formulation is that it

provides some degree of robustness with respect to geometric uncertainties without solving

a hard problem, however this comes at the expense of more conservative results.

It was demonstrated that the underlying problem can be exposed as a multi-criteria

GSIP. By assuming that the lower-level problem is convex, the resulting GSIP was efficiently

transformed into a tractable constrained nonlinear programming problem. In order to ensure

that all Pareto optimal solutions of the problem can be found, the multi-criteria objective

function was created using physical programming. The results of planning for a number of

case studies showed up to 300% larger tolerance volume without risking dexterity and task

feasibility, as compared with the tolerance achieved based on the surgeon-recommended

ports. In all cases, the achieved robustness indices have been larger than 15 mm, which

is typically acceptable considering that, for instance, the actual heart displacements in the

lateral and anterior/posterior direction are typically about 17 mm and 9 mm, respectively

3.6 Summary 70

[79]. Moreover, the aggregate objective function can always be fine-tuned to control the

trade-off between different criteria. As a result, the proposed framework can produce plans

that are robust with respect to intraoperative uncertainties due to physiological motions.

Moreover, through Monte Carlo (MC) simulations, the robustness of the plans with

respect to random wrist displacements inside the octahedra was evaluated. Since no pre-

sumptions about the wrist motion in the vicinity of the targets could be made, random local

wrist positions were generated with uniform distributions. The Monte Carlo simulations

proved the validity of the convexity assumption of the lower-level problem in (3.16). They

also demonstrated that the 95% constraint satisfaction volume is up to six times larger than

the deterministically-guaranteed tolerance volume.

Chapter 4

A Stochastic Approach To

Preoperatve Planning of RAMICS

Under Uncertainty

4.1 Synopsis

In the previous chapter, a deterministic method for addressing geometric uncertainty in

preoperative planning was proposed. As was mentioned, identification and incorporation

of spatio-temporal uncertainties in preoperative planning may result in a computationally-

intractable problem. The approach proposed in the previous section could partially alleviate

the computational complexities of planning under uncertainty by unifying the contributions

of these spatio-temporal uncertainties into the uncertainty in the wrist position, at the ex-

pense of yielding more conservative results. The core contribution of the proposed method

is maximizing the robustness of planning with respect to unknown spatio-temporal uncer-

tainties. In this chapter, a new formulation for addressing spatio-temporal uncertainties is

proposed. With the proposed formulation, it is possible to maximize maneuverability of the

robotic arms (and the instruments) by assuming spatio-temporal uncertainty at the task

level.

As an improvement to the algorithm proposed in the previous chapter, the proposed

71

4.2 Problem Formulation 72

formulation presents a formalism for separate treatment of position and orientation uncer-

tainties. This is particularly advantageous considering that different tasks and procedures

have different requirements in terms of translational and rotational motions. In fact, solv-

ing this problem with the deterministic formalism proposed in the previous section is not

recommended for two reasons:

• The resulting optimization problem will have twice more constraints than the problem

in the previous section which results in a huge increase in the computational time.

• The convexity assumption that was employed for deriving the solution method in the

previous section is unlikely to be realistic for orientation uncertainty.

In this chapter, the planning problem is formulated as a chance-constrained program-

ming problem, and, as a solution method, an efficient sampling-based technique is proposed.

Unlike linearization techniques, the proposed technique does not require the calculation of

the constraint Jacobian, nor does it resort to propagating a large number of samples as in

particle methods. In this formulation, robustness with respect to the lack of information at

the task level is increased by maximizing the entropy of the task in Cartesian space, while

the cross entropy of the task in joint space and Cartesian space is maximized as a measure

of kinematic performance.

This chapter is organized as follows: in Section 4.2, a stochastic preoperative planning

scheme is formally presented as a chance-constrained entropy maximization problem, and

an efficient solution method is described. An implementation of the proposed formalism is

presented in Section 4.3. Finally, the suitability of the approach is investigated through a

number of case studies in Section 4.4.

4.2 Problem Formulation

Every surgical procedure can be expressed in terms of several surgical tasks that can be

represented by a number of task frames inside the surgical cavity. Usually, these task frames

can be affixed to certain anatomical features, and therefore surgical tasks can be uniquely

described with respect to the patient’s anatomy. While the correlation between the task


and the position of a specific surgical feature is easily understood, the link between the

instrument orientation and the target may not be as obvious. However, a task can only be

uniquely expressed in terms of the position as well as the orientation of a hand.

In practice, surgical gestures (subtasks) are subject to spatio-temporal uncertainties.

For surgeons with different levels of skill and experience, the uncertainty can become more

significant. In this section, a planning formalism that can address translational and rota-

tional uncertainties at the task level is presented. The main objective of developing such a

formalism is to apply it for finding preoperative plans with minimal susceptibility to a lack

of information at the task level. In other words, the proposed formalism is used to seek

the plan that maximizes the tolerable uncertainty at the task level for which the chance of

success remains sufficiently high.

RAMIS usually requires three arms, including two arms for carrying the right and left

instruments, as well as one arm for carrying the endoscope. Let xR ∈ R6 and xL ∈ R6

be the 6-D poses of the right and left instrument tips given by gR ∈ SE(3) and gL ∈

SE(3), respectively. Throughout this chapter, it is assumed that the desired task frames

for the right and left instruments are given as µRx ∈ R6 and µL

x ∈ R6, and the pose of

the instruments in the vicinity of the task frames is represented by Gaussian distributions

xR ∼ N (µRx ,Q

Rx ) and xL ∼ N (µL

x ,QLx ). In practice, for a given task within the vicinity of

a task frame, the endoscope remains still; therefore, henceforth the pose of the endoscope

is treated as a deterministic variable.

In order to increase the chance of success for the completion of a given surgical task,

the robustness of the plan with respect to uncertainty at the task level must be maximized.

In this chapter, the proposed planning scheme attempts to find an optimal plan that has

minimal susceptibility with respect to the lack of information about the task. With a

task modeled by a Gaussian distribution, this is equivalent to maximizing a norm of the

covariance of the task. Equivalently, the optimal plan may be found by minimizing a norm

of the inverse of the covariance matrix, known as the information matrix. It is well-known

that the information entropy is a measure of the lack of information, and according to the

maximum entropy principle, among several distributions, the distribution with the highest


entropy pertains to the variable with the least information [80].

4.2.1 Objective Function

The objective of the proposed planning formalism is articulated in terms of two criteria: task

entropy and kinematic performance. In the following sections, these criteria are formally

defined.

4.2.1.1 Task Entropy

Let a procedure be stochastically represented with N discrete task frames as:

X =

xR(k)

xL(k)

for k = 1, 2, ..., N

, (4.1)

where each task is represented by the Gaussian distributions xR(k) ∼ N (µRx (k),QR

x (k))

and xL(k) ∼ N (µLx (k),QL

x (k)). Note that, while the mean value of the tasks, and as a

result the procedure, is assumed known a priori and independent of the plan, the tolerable

task uncertainty is determined by the plan. The information entropy of the procedure, as

a measure of the lack of information regarding that procedure, can be expressed as:

Ent1(X ) = − 1

NΣNk=1

(

log(|QRx (k)|) + log(|QL

x (k)|))

, (4.2)

where | · | is the matrix determinant operator, and log(|Qx|) is recognized as the Shannon

entropy of the task. As will be discussed below, the goal is to find the plan that maximizes

the procedure entropy.

4.2.1.2 Performance

In the previous chapter, a Jacobian-based measure of kinematic performance for preoper-

ative planning was used. In this chapter, kinematic performance is considered by incorpo-

rating the cross entropy of the task in Cartesian and joint spaces. As will be seen, this

measure has physical interpretations, and renders the multi-criteria objective function of


the planning scheme as an entropy maximization problem. Given the task frames with

Gaussian distributions xR(k) ∼ N (µRx (k),QR

x (k)) and xL(k) ∼ N (µLx (k),QL

x (k)), assume

that the propagated distribution of the joint vectors can be approximated by Gaussian dis-

tributions qR(k) ∼ N (µRq (k),QR

q (k)) and qL(k) ∼ N (µLq (k),QL

q (k)), where qR and qL are

the right and left instrument arm joint vectors (for the sake of brevity we simply use q for

qactive). Note that these distributions depend upon the configuration of the arms as well as

the distribution of the task frames. In general, it is desired that for a given displacement

in the pose of the end effector, the joint displacement is minimized. This can be expressed

in terms of the cross entropy of the task in joint space and Cartesian space as:

Ent2(X ) =1

NΣNk=1

(

log

(

|QRq (k)|

|QRx (k)|

)

+ log

(

|QLq (k)|

|QLx (k)|

))

. (4.3)

The aggregate objective function can be expressed as:

Ent(X ) = Ent1(X ) + Ent2(X ), (4.4)

and the resulting objective function can be rendered as:

Ent(X ) =1

NΣNk=1 log

(

|QRq (k)||QL

q (k)||QR

x (k)|2|QLx (k)|2

)

. (4.5)

4.2.2 Constraints

The constraints can be classified into three main categories: constraints imposed to avoid

collisions, constraints imposed to avoid joint-limit violations, and constraints imposed by

target reachability. In this section, each one of these categories is formally described in

terms of the planning parameters P and the arm poses xR, xL, and xE.

Given a plan P, the necessary and sufficient conditions for collision avoidance can be

described in terms of the minimum distances among the arms, denoted by

d = fd

(

xR(k),xL(k),xE(k),P)

,


where fd : R6 × R6 × R3 × Rn → Ra, as:

fd

(


> 0 for k = 1, ..., N. (4.6)

The above constraint would only be meaningful if all of the parameters were deterministic;

however, xR and xL are stochastic variables and therefore the above constraint can only

be satisfied probabilistically, i.e., the constraint must be formally replaced with a chance

constraint such as:

P

(

fd

(


> 0

)

> 1− ε for k = 1, ..., N, (4.7)

where 0 < ε� 1, and P (·) is the probability operator.

In a similar fashion, given a plan P, it must be ensured that while performing task x

the joint limits are not violated. This must be articulated using the inverse kinematics of

the arm, q = f IK(x,P), where f IK : R6 ×Rn → R6. The corresponding chance constraints

are given as:

P(

fRIK(xR(k),P) ∈ QR

valid|P)

> 1− ε for k = 1, ..., N, (4.8)

P(

fLIK(xL(k),P) ∈ QL

valid|P)

> 1− ε for k = 1, ..., N, (4.9)

where Qvalid refers to the valid joint space determined by the joint limits.

Another set of constraints is imposed by task feasibility requirements. One trivial re-

quirement for feasibility is reachability. Since the task frames are usually located on (or

close to) surfaces, the arm configuration by which the task frame can be reached, and as

a result the maneuverability of the instrument pertaining to that task, may be confined.

By the same token, the patient’s external anatomical geometry (close to the port location)

can confine the maneuverability of the arm. Herein, we limit the scope to the topological

constraints at the target and the port. The configuration of the arm can be described as

γ = fγ(x,P), where fγ : R6 × Rn → R2. As opposed to the joint limits that confine

reachability, these constraints are imposed by the topology of the environment (in this case


γ1

z3

γ2

(pt, nt)

x5

x6

position uncertainty

maneuverable range

(ptroc

, ntroc)

Figure 4.1: Maneuverability of the tool confined by reachability constraints at the trocarand target(s)

the patient’s anatomy). The reachability requirements can be described in terms of the

reachable subset of the configuration space Creach as:

P(

fRγ (xR(k),P) ∈ Creach|P

)

> 1− ε for k = 1, ..., N, (4.10)

P(

fLγ (xL(k),P) ∈ Creach|P

)

> 1− ε for k = 1, ..., N, (4.11)

Fig. 4.1 shows how the reachability constraints at the trocar and target can confine ma-

neuverability. Later in this chapter, the reachability constraints will be defined in terms of

γ =

γ1

γ2

, where γ1 and γ2 are the approach angles at the trocar and target, respectively.

Generally speaking, finding the exact solution of the above stochastic programming

problem is not trivial. In the following sections, the above chance-constrained program-

ming problem is implemented by transforming it into an ordinary constrained nonlinear


programming problem. It is also demonstrated that the proposed transformation results in

sufficiently accurate solutions.

4.2.3 Problem Transformation

A natural way of solving the above constrained optimization problem is random sampling

(particle) methods. In these methods, for every given set of parameters, a large number

of samples of the stochastic variables are randomly generated and propagated through the

constraints (e.g., see [81] for more details). This process is executed iteratively until the

percentage of the samples for which the constraints are satisfied is larger than 1 − ε. De-

spite their accuracy, a large number of samples are required for accurate estimation of

the distribution as dictated by the central limit theorem. Therefore, these techniques are

computationally expensive. Alternatively, the chance constraints can be approximated and

replaced by a set of deterministic constraints. The latter method requires that the distri-

butions (or at least the first two moments) of the constraints are known. Nevertheless, the

constraints are generally nonlinear functions of the random variables and, in general, their

distributions are unknown and usually non-Gaussian. A trivial workaround is a local linear

approximation of the constraint functions and estimations of the mean and variance of the

constraints using the constraint function Jacobian [82]. However, this method usually fails

to reveal the true statistical properties of the constraint when the function is highly non-

linear [83]. Furthermore, an analytic expression of the Jacobian of the constraint function

may not be known, and the numerical calculation of the Jacobian of the constraints may

be computationally demanding.

More recently, a theoretical framework has been developed for analysis of the propaga-

tion of probability distributions on motion groups (see [84–86]). In [84], error propagation

in robotic manipulators has been studied. The authors showed that when two members

of SE(3) are multiplied, the resulting error distribution is calculated by convolving their

respective error distributions. In [25], error propagation in needle steering has been studied

using a second-order approximation of the propagated covariance proposed in [85]. The

authors used the formulation to find the optimal insertion location for flexible needle steer-


ing. However, these methods are also computationally demanding and are not suitable for

a complicated surgical manipulation planning problem.

As an alternative approach, herein the unscented transformation, proposed by [83] is

employed to estimate the statistics of the constraints by fitting a Gaussian distribution to

the transformed samples. However, unlike Monte Carlo methods, these samples are not

randomly selected. Instead, by a systematic selection mechanism, it is possible to infer

the statistical properties of the nonlinear function by propagating only a small number of

samples.

Consider a nonlinear mapping of the stochastic variable x with a Gaussian distribution

given as y = f(x), where f : Rm → Rn. Assume that a set of Sigma points are selected

as Xi = {µx,µx ± [√αQx]i} for i = 1, ...,m where α is a scalar, and [·]i represents the ith

column of the argument. The mean and covariance of y can be estimated as:

µy = Σ2mi=0WiY i, (4.12)

Qy = Σ2mi=0Wi(Y i − µy)(Y i − µy)′, (4.13)

where Y i = f(Xi) and Wi are scalar weights that are selected such that the statistics of

the Sigma points and x are identical. The interpretation of this transformation is that

instead of propagating the mean and covariance of x, as in linearization-based methods, or

propagating a large number of samples of x, as in Monte Carlo methods, the Sigma points

are propagated, and the mean and covariance of y are estimated by fitting a Gaussian

distribution to the propagated points. A conceptual comparison between the unscented

transform, linearization and probabilistic sampling methods for distribution approximation

is presented in Fig. 4.2. In the traditional probabilistic sampling methods, provided that

the sample size is sufficiently large, the statistics of the propagated samples can accurately

resemble the statistics of the nonlinear mapping. However, the accuracy of linearization-

based approximation methods is highly dependent upon the degree of nonlinearity as well

as the distribution dispersion. The Unscented transform, on the other hand, is able to yield

relatively accurate approximations, by propagating only a small number of samples that


are selected judiciously.

y=

f(x

)

y=

f(x

)

y=

f(x

)

µy = E(f(Xi))

Qy = Cov(f(Xi))

probabilistic sampling

µy = f(µx)

Qy = ∂f

∂x

TQx

∂f

∂x

linearization unscented transform

µy = Σ2m

i=0WiY i

Qy = Σ2m

i=0Wi(Y i − µy)(Y i − µy)′

Figure 4.2: An illustrative comparison of the unscented transform with linearization andprobabilistic sampling: red dots represent the samples taken from the originaldistribution.

Once the means and the covariance matrices of the constraints are estimated, the chance

constraints can be reduced to deterministic constraints. For ε = 1−Φ(ζ), where ζ ∼ N (0, 1),

and Φ(·) is the standard normal cumulative distribution function, the chance constraints

(4.7)–(4.11) can be substituted by the following deterministic constraints (see Appendix C

4.3 Planning of Endoscopic Coronary Artery Bypass Surgery 81

for more details):

Φ−1(1− ε)(

diag(Qd(k)))

12< µd(k), (4.14)

∣

∣

∣

∣

∣

µRq (k)− qR

max + qRmin

2

∣

∣

∣

∣

∣

+ Φ−1(1− ε)(

diag(QRq (k))

)

12<

qRmax − qR

min

2, (4.15)

∣

∣

∣

∣

∣

µLq (k)− qL

max + qLmin

2

∣

∣

∣

∣

∣

+ Φ−1(1− ε)(

diag(QLq (k))

)

12<

qLmax − qL

min

2, (4.16)

µRγ (k) + Φ−1(1− ε)

(

diag(QRγ (k))

)

12< π/2, (4.17)

µLγ (k) + Φ−1(1− ε)

(

diag(QLγ (k))

)

12< π/2, (4.18)

for k = 1, ..., N , where d ∼ N (µd,Qd(k)), q ∼ N (µq,Qq(k)) and γ ∼ N (µγ ,Qγ(k)) are

the approximated distributions using (4.12) and (4.13).

Finally, the resulting problem is rendered as an ordinary nonlinear programming problem

with the objective function given in (4.5) and subject to the constraints (4.14)–(4.18) along

with additional deterministic constraints pertaining to the endoscope and passive section.

In the following section, the efficacy of the proposed formulation for preoperative plan-

ning of RAMIS is demonstrated through an illustrative example.

4.3 Planning of Endoscopic Coronary Artery Bypass Surgery

In this section, we apply the proposed chance-constrained programming formulation for

planning of robotic-assisted LIMA harvesting. From the complexity point of view, LIMA

harvesting is a relatively simple task and can be easily represented by a minimal number of

task frames. The desired task frame for harvesting can be empirically determined by in vivo

observation of the tool gestures, or more accurately, can be determined by statistical analysis

of the observed tool gestures. The task consists of pulling the tissue with the left instrument

while cutting the tissue along the artery using electrocautery on the other instrument [5]. In

order to avoid ambiguity in representation, each target fixture is represented with a position

vector as well as a normal vector to the surface where the target is located, i.e., x ∈ R5.

The planning problem is rendered as the solution of a constrained entropy optimization


problem, i.e.,

P∗ = arg minP

1

NΣNk=1 log

(

|QRq (k)||QL

q (k)||QR

x (k)|2|QLx (k)|2

)

(4.19)

subject to

Φ−1(1− ε)(

diag(Qd(k)))1

2< µd(k),

∣

∣

∣

∣

∣

µRq (k)− qR

max + qRmin

2

∣

∣

∣

∣

∣

+ Φ−1(1− ε)(

diag(QRq (k))

)

12<

qRmax − qR

min

2,

∣

∣

∣

∣

∣

µLq (k)− qL

max + qLmin

2

∣

∣

∣

∣

∣

+ Φ−1(1− ε)(

diag(QLq (k))

)

12<

qLmax − qL

min

2,

∥

∥

∥

∥

∥

µRγ (k) + Φ−1(1− ε)

(

diag(QRγ (k))

)

12

∥

∥

∥

∥

∥

∞< π/2,

∥

∥

∥

∥

∥

µLγ (k) + Φ−1(1− ε)

(

diag(QLγ (k))

)

12

∥

∥

∥

∥

∥

∞< π/2,

∣

∣

∣

∣

∣

qE(k)− qEmax + qE

min

2

∣

∣

∣

∣

∣

<qE

max − qEmin

2,

∥

∥

∥γE(k)∥

∥

∥

∞< π/2,

dpassive > 0,

qpassive ∈ Qvalidpassive

for k = 1, 2, ..., N,

where{

q∗passive,p

∗patient, {QR

x (k)}∗1:N , {QLx (k)}∗1:N |

}

is the optimal plan, qpassive referes to

the augmented vector of passive joints, Qvalidpassive is the valid joint space for the passive

section, dpassive is the minimum distance between the passive section and other stationary

objects, and qE and γE refer to the endoscope joint vector and approach angles, respectively.

For simplicity, it is assumed that the Cartesian covariance matrices at different targets

are diagonal and identical, i.e., Qx(k) =

σ2pI3×3 0

0 σ2oI3×3

, where σ2p and σ2

o are the

position and orientation variances, respectively. This resembles symmetric position and

orientation uncertainties as illustrated in Fig. A.2. As a result, the covariance matrices

pertaining to the right and left instruments can be conveniently parameterized by only four


parameters σ2p,R, σ2

o,R, σ2p,L, and σ2

o,L. Equations pertaining to uncertainty propagation

through intracorporeal kinematics can be found in Appendix A.2. It must be noted that the

covariance matrices contain nonhomogeneous elements. Hence, it is reasonable to express

the covariance matrices in terms of a length scale (similar to the characteristic length [55]).

The proposed formulation is utilized for preoperative planning of LIMA harvesting for

a case study with N = 6 and ε = 0.02, and the results are given in Figs. 4.3 and 4.4.

The resulting constrained nonlinear programming problem was solved using active-set SQP

using the BFGS Hessian update implemented by fmincon in MATLAB. Fig. 4.3 shows the

da Vinci arms as they reach the LIMA for harvesting, while the robot is placed as suggested

by the planner when the right, left and the endoscope ports are placed in the 3rd, 6th and

5th intercostal spaces on the patient’s rib cage.

1 2

3 4

Figure 4.3: The da Vincir arms in action: a collision-free trajectory of the active sectionsas they reach the individual target sites inside the thorax (for k = 1, 2, 3, 4)

The actual reliability achieved by the proposed plan P∗ is evaluated by a Monte Carlo

simulation. With the resulting task covariance matrices Q∗x,R and Q∗

x,L, a set of random

target fixtures, denoted by xR,i and xL,i, is generated, and the satisfaction of the chance

constraints (4.7)–(4.11) is investigated. The positions of the admissible task fixtures are

4.4 System Performance 84

Figure 4.4: The position of the admissible task fixtures generated in a Monte Carlo simu-lation for a LIMA harvesting task

illustrated in Fig. 4.4. The analysis shows that, with the proposed plan, more than 99%

of the generated task fixtures satisfied the constraints, which exceeds the original design

specification (98%).

4.4 System Performance

In order to evaluate the performance of the proposed planning scheme, an analysis similar

to the one presented in the previous chapter is used here. Given the preoperative CT images

of the same four patients, ARP, SRT and SRP plans were sought for N = 6 and ε = 0.02.

The quality of the resulting ARP, SRT and SRP plans are compared in Figs. 4.5, 4.6, and

4.7. As the figures show, ARP plans provide the best robustness and performance. On the

other hand, SRP plans provide the lowest robustness and performance. This analysis proves

the suitability of the proposed formalism for addressing the uncertainty at the task level in

preoperative planning of RAMICS. In all of the cases, Monte Carlo simulations showed an

actual reliability of 99%, which is slightly above the expected reliability score (98%).

4.5 Summary 85

Case I

Case II

Case III

Case IV

SRP

SRT

ARP0

0.5

1

Agg

rega

teSco

re

Figure 4.5: Comparison of the overall scores pertaining to the SRP, SRT and ARP plans

Case I

Case II

Case III

Case IV

SRP

SRT

ARP

0

0.5

1

Entr

opy

Figure 4.6: Comparison of the (normalized) Cartesian task entropy pertaining to the SRP,SRT and ARP plans

4.5 Summary

In this chapter, a novel formulation for stochastic preoperative planning of robotics-assisted

minimally invasive surgery for addressing uncertainty at the task level was proposed. The

original problem was formulated as a chance-constrained stochastic programming problem.

4.5 Summary 86

Case I

Case II

Case III

Case IV

SRP

SRT

ARP

0

0.5

1

Cro

ssE

ntr

opy

Figure 4.7: Comparison of the (normalized) joint space/Cartesian space cross entropy per-taining to the SRP, SRT and ARP plans

It was shown that the original problem could be modified as a chance-constrained entropy

maximization problem. As an efficient solution method for the proposed formulation, the

unscented transform was used to transform the resulting chance-constrained entropy maxi-

mization problem into a reasonably tractable constrained nonlinear programming problem.

Finally, it was demonstrated that the proposed solution method could efficiently handle an

intrinsically computationally expensive optimization problem without resorting to lineariza-

tion or Monte Carlo methods. The initial results prove that the proposed formalism for

preoperative planning outperforms an experienced surgeon’s recommended plan in terms of

robustness and performance.

Chapter 5

Discussions

5.1 Synopsis

In this section, the contributions of this thesis are stated, and their significance is discussed.

This is followed by concluding remarks, suggestions, and implications for future work.

5.2 Contributions

As in any safety-critical process, robustness is a key success factor in both planning and

execution of robotics-assisted surgical procedures. While a safe execution relies upon the

surgeon’s skill and experience, a safe planning remains dependent upon the level of uncer-

tainty in preoperative data envisaged during the preoperative planning stage. In this thesis,

we have successfully developed a tractable framework for robust preoperative planning of

robotics-assisted minimally invasive cardiac surgery that can be considered as a stepping

stone to the application of machine intelligence and automation in the operating room.

As one of its minor contributions, a modified measure of dexterity has been proposed

in this thesis. Also, two formalisms for robust preoperative planning of RAMICS have

been presented. This includes formulating the problem as robust optimization problems

and proposing efficient solution methods. In the formalism presented in Chapter 3, a deter-

ministic approach has been adopted that is approximate in uncertainty quantification, but

exact in planning. The formalism presented in Chapter 4, however, is exact in uncertainty

87

5.3 Conclusions and Future Work 88

quantification, but approximate in planning.

Despite their novelty, the above techniques can be further extended and improved both in

formulation and in solution methods. This will also enhance their admissibility to operating

rooms.

5.3 Conclusions and Future Work

In general, accurate planning of RAMIS is a complicated problem. In particular, systematic

integration of patient-specific data, robot kinematics and procedure-specific kinematics into

the planning algorithm is nontrivial, and yields cumbersome optimization problems. While

in many similar situations heuristic and so-called global optimization techniques such as

genetic algorithms are employed for solving such problems, this thesis has avoided resorting

to such computationally expensive techniques, and instead has focused on establishing an

analytical framework that can be reproduced and expanded upon in the future. Also, various

aspects of RAMIS were studied and mathematical formalisms were presented wherever

possible.

In Chapter, 2, a framework for the study of actively constrained surgical manipula-

tion was presented. General forward and inverse kinematic solutions for nonredundant

manipulators with generic surgical instruments operating under a trocar constraint were

formulated. While the proposed solution is mostly advantageous for offline applications

such as analysis and planning, it is generic and simpler than solving a high-order set of

simultaneous nonlinear equations. For real-time applications, task priority was exploited

to formulate a CLIK scheme based on trocar differential kinematics. It was demonstrated

that the accuracy of the CLIK scheme relies on the manipulator configuration, and it can

be improved by proper placement of the cavity within the manipulator workspace. Finally,

dexterity measures for actively constrained surgical manipulation were proposed. The mea-

sures are invariant with respect to the choice of the reference coordinate frame as well as

the length unit for homogeneous manipulators.

This chapter showed how the trocar constraints could impact various measures includ-

ing dexterity and accuracy (particularly in active RCM realization). Moreover, through


a number of illustrative examples it was demonstrated how preoperative planning could

improve various indices. This chapter also established a valid basis for the comparison and

optimal design of surgical manipulators. The analysis presented in Section 2.3, for instance,

demonstrated that the Mitsubishi arm with the proposed CLIK exhibits poor accuracy. In

fact, it can be shown that accuracy is dependent on the robot kinematics, therefore it is

worth considering the optimal design of the surgical manipulator and the control scheme at

the same time, i.e., using the paradigm of design for control. Measures such as dexterity,

that are innate indicators of the robot’s capability, can also be considered in the optimal

design of the manipulator. This can be done by articulation of various measures in terms

of various design parameters of the manipulator and seeking the optimal values by solving

an optimization problem.

Incorporation of automatic collision avoidance in multi-arm scenarios, and detailed anal-

ysis of flexible tools under trocar constraints should be considered as future work. The

results obtained in this chapter can also be utilized for preoperative as well as intraopera-

tive planning and control of RAMIS performed by actively-constrained manipulators. It is

expected that planning will improve clinical outcomes as well as the success rate of robotic

procedures. However, reliable planning requires identification of the motions involved in the

task being performed at a surgical site in a given surgical procedure. In positioning tasks,

such as in seed placement, the location of the tip position can be related to the position of

the target; however, in manipulation tasks, such as in suturing and knot tying, identification

of the wrist orientation is not as straightforward.

In Chapter 3, a computational framework for addressing geometric uncertainty in pre-

operative data for patient-specific preoperative planning of RAMICS was developed. The

proposed framework is capable of addressing the geometric uncertainty that is present in

RAMICS, and is generic in the sense that no presumptions about the procedure or the

manipulator are made. The proposed framework can handle the contributions of several

sources of uncertainty without characterizing them directly. By assuming that the lower

level problem is convex, the resulting GSIP was efficiently transformed into a tractable

constrained nonlinear programming problem. In order to ensure that all Pareto optimal


solutions of the problem can be found, the multi-criteria objective function was created

using physical programming. The results of planning for a number of case studies showed

up to 300% larger tolerance volume without risking dexterity and task feasibility. Moreover,

through Monte Carlo simulations, not only could the validity of the convexity assumption

of the lower-level problem in (3.16) be proved, but it could also be demonstrated that the

95% constraint satisfaction volume is six times larger than the deterministically-guaranteed

tolerance volume.

The outcomes of the proposed formulation are highly dependent upon how the individual

criteria are incorporated into a single objective function. As potential future work, the

reliability of the plans can be significantly improved through a systematic tuning of the

multi-criteria objective function. For this purpose, a large database of different cases has

to be created. The selected port locations, as well as metrics that quantify the outcomes

(e.g., surgical time), must be included in the database. Once a sufficiently-large database

is available, statistical inference techniques may be used to finetune the parameters such

that the likelihood of success is further increased. Ultimately, by developing patient-specific

uncertainty models, less conservative plans can be expected.

As another recommendation for future work, the proposed GSIP can be modified as

a mixed-integer programming problem (in which the integers refer to the indices of the

ICS) to consider all potential ICS triads and avoid running the optimization for each triad

separately. Although this may result in a slightly more complicated problem, all potential

ICS triads can be considered in a single run of the new mixed-integer GSIP. This formulation

would also result in a complete optimization problem.

In Chapter 4, the planning of robotics-assisted interventions under task uncertainty was

addressed. In order to accommodate more surgeons with different levels of skill and experi-

ence into the planning, it is essential that the plans accommodate a larger task uncertainty.

The ultimate goal of this approach is to increase the chance of success by ensuring that

the chances of collisions and joint limit violations remain sufficiently small. Therefore, the

planning problem was formulated as a chance-constrained programming problem in terms

of the uncertainty of the instrument tip pose in the vicinity of the desired task frame, mini-


mizing the information regarding the task. In other words, it was assumed that the surgical

procedure could be decomposed into several random task frames (and twists) with Gaussian

distributions. To avoid using sampling-based techniques for solving the resulting stochas-

tic optimization problem, the unscented transformation was utilized. This transformation

yields more accurate estimation of the statistics of the constraints while the complexities

pertaining to the linearization of the nonlinear constraints are avoided. The efficiency of the

proposed formulation was demonstrated by several case studies addressing optimal planning

of robotics-assisted LIMA harvesting in minimally invasive coronary artery bypass surgery

using the da Vinci robotic system.

The proposed stochastic formulation assumes that a given procedure can be modeled

by a set of task frames accurately enough. This requires a systematic method of modeling

and identification of surgical procedures in terms of task frames. However, surgical task

modeling is still an active area of research and such a framework is not available yet.

As a recommendation, the proposed chance-constrained entropy maximization can also

be extended to a mixed-integer stochastic programming problem to consider all of the

potential ICS triads in the planning problem in a single run.

One of the issues of the proposed planners is that the solutions are highly sensitive

to the initialization. As a matter of fact, this problem can be associated with the non-

convex nature of the resulting optimization problems. It is expected that by using convex

relaxation techniques, the resulting nonlinear programming problems will be transformed

into (quasi-)convex optimization problems with reduced sensitivity to the initialization.

The formalisms presented in this thesis can also be modified to accommodate other

procedures, robots and tools so long as the presumptions taken here are also applicable to

those procedures, robots and tools. For instance, using higher dexterity tools might require

imposition of more constraints to ensure that the tools will not collide with themselves

and with internal organs. This is particularly important for Single Port Access (SPA)

procedures that usually employ snake-like robots. In fact, consideration of intracorporeal

collisions might be the main challenge to address in applying the proposed formalisms to

SPA procedures. Another good example of the application of the proposed formalism is


planning for radiation therapy where minimizing the dosage of the radiation as well as

maximizing the chance of success must be concurrently considered.

Apart from its explicit applications in treatment/surgical planning, the proposed frame-

work can also be employed to evaluate the eligibility of the patient, robot and the surgeon

for a given procedure. The proposed framework can be used to improve the manipulator

design as well the surgeon training.

In the implementations of the proposed formalisms, in this thesis, dynamic effects of the

robot (e.g. actuator dynamics and links inertia) have not been addressed directly. In gen-

eral, quasi-static assumption is realistic for tele-operated surgical manipulation particularly

when the manipulation has to be performed in a restricted volume due to either a small

workspace or remaining within he endoscope field of view. However, due to the nonlinear

mapping between the Cartesian and joint spaces, a small displacement in Cartesian space

does not necessarily result in a small displacement in joint space or vice versa. Yet, it

can be shown that a small displacement in Cartesian space leads to a small displacement

in joint space so long as the robot is sufficiently far from a singular configuration. This

has already been addressed by incorporating a dexterity measure in the objective function.

Nevertheless, the proposed formalism can easily be extended to accommodate various dy-

namic effects including tissue deformation, tumor growth dynamics (for radiation therapy)

as well as the dynamic effects of the robot.

References

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Appendix A

Inverse Kinematics

A.1 Intracorporeal (The da Vinci Active Section) Inverse

Kinematics

For a desired fixture represented by gic =

R

u

v

w

0 1

, where R ∈ SO(3), and u, v

and w are scalars, the intracorporeal joint values (see Fig. 2.1) can be calculated by1:

θ6 =atan2(v′,−u′), (A.1)

θ5 =atan2(w′,−a6 − u′c6 + v′s6), (A.2)

d4 =(−a6 − u′c6 + v′s6)c5 + w′s5, (A.3)

θ2 =atan2

(

R33,

√

R213 + R

223

)

, (A.4)

θ3 =atan2

(

−R32

c2,R31

c2

)

, (A.5)

θ1 =atan2

(

−R23

c2,−R13

c2

)

, (A.6)

1ci = cos(θi) and si = sin(θi).

100

A.2 Uncertainty Propagation in Intracorporeal Kinematics 101

θ1

.

z0, z1, x2, x3

.

x0, x1

.

z2θ2

.

z5

θ5

θ3

.

z3, z4, x5

.

x4

.

x6

θ6

.

z6

.

.

d4

.

Figure A.1: Frame attachment in intracorporeal kinematics (the distance between frames 5and 6 is denoted by a6)

where

u′

v′

w′

, −RT

u

v

w

and R , R

c6s5 −s6 c5c6

−s5s6 −c6 −c5s6

c5 0 −s5

.

A.2 Uncertainty Propagation in Intracorporeal Kinematics

Assume that a target fixture is represented by a position vector pt, and a normal vector

nt, and the pose uncertainty in the neighborhood of the target fixture is defined as pt + z,

where ‖z‖2 ≤ δ, z ∈ R3 is the position uncertainty, and R(α, β) =

cαcβ −sα cαsβ

sαcβ cα sαsβ

−sβ 0 cβ

is the approach uncertainty with respect to the normal vector. The intracorporeal joint

A.2 Uncertainty Propagation in Intracorporeal Kinematics 102

ptroc

z3

(pt, nt)

x5

x6

position uncertainty

approach uncertainty

Figure A.2: Target pose uncertainty propagation in intracorporeal kinematics.

values are calculated in terms of the pose uncertainty as:

d4z4 = pt + z + a6R(α, β)nt ,

u′′,

v′′,

w′′

, (A.7)

θ2 = atan2(w′′,±√

u′′2 + v′′2), (A.8)

θ1 = atan2(−c2v′′,−c2u

′′), (A.9)

θ5 = atan2(±√

B211 + B2

31,−B21), (A.10)

θ3 = atan2(s5B31, s5B11), (A.11)

where B ,

c1s2 s1s2 c2 0

c1c2 c2s1 −s2 0

−s1 c1 0 0

0 0 0 1

−R(α, β)nt pt + z

0 1

.

A.3 Holder/Instrument Inverse Kinematics 103

A.3 Holder/Instrument Inverse Kinematics

Based on the convention introduced in [87], the Denavit-Hartenberg (D-H) parameters of

the Mitsubishi PA10-7C arm mounted with a roll-pitch-yaw instrument are given in Table

A.1.

Table A.1: D-H Parameters of the Mitsubishi PA10-7C mounted with a roll-pitch-yaw in-strument

i θ′i di ai αi

1 θ′1 0.317 0 0

2 θ′2 0 0 −π

2

3 θ′3 0.45 0 π

2

4 θ′4 0 0 −π

2

5 θ′5 0.48 0 π

2

6 θ′6 0 0 −π

2

7 θ′7 0.28 0 π

2

8 θ′8 0 0 −π

2

9 θ′9 0 0.008 −π

2

The Mitsubishi PA10-7C arm is a redundant manipulator consisting of 7 revolute joints.

To avoid redundancy, the shoulder joint, θ′1, was locked and the inverse kinematics solution

of θ′i for i = 2, ..., 6 was obtained in terms of the elbow position pelb =

x

y

z

and tool

approach rtool ,

r1

r2

r3

as:

θ′2 =atan2

(

b,±√

1− b2)

− atan2(z − 0.317, x), (A.12)

b =10

9√

(z − 0.317)2 + x2

(

x2 + y2 + (z − 0.317)2 − 0.0279)

, (A.13)

θ′4 =atan2

(

±√

1− f2, f

)

, (A.14)

A.3 Holder/Instrument Inverse Kinematics 104

f =25

12

(

(z − 0.317)c′2 + xs′

2 −9

20

)

, (A.15)

θ′3 =atan2

(

ys′4, (xc

′2 − (z − 0.317)s′

2)s′4

)

, (A.16)

θ′6 =atan2

(

±√

1− r22,−r2

)

, (A.17)

θ′5 =atan2

(

r3

s′6

,r1

s′6

)

. (A.18)

Given gins = [tij ] for an instrument with a roll-pitch-yaw wrist, the joint values θ′i for

i = 7, ..., 9 are calculated as:

θ′8 =atan2

(

t23,−t24 + 7

25

0.008

)

, (A.19)

θ′7 =atan2

(

t34

s′8

,t14

s′8

)

, (A.20)

θ′9 =atan2

(

t22

c′8

,− t21

c′8

)

. (A.21)

Appendix B

Mathematical Preliminaries

B.1 Pseudo-Inverse Properties

For J ∈ Rm×n, the following expressions hold:

JJ+J = J , (B.1)

J+JJ+ = J+, (B.2)

(J+J)T = J+J , (B.3)

(JJ+)T = JJ+, (B.4)

(I − J+J)(I − J+J) = (I − J+J), (B.5)

(I − J+J)T = (I − J+J), (B.6)

J(I − J+J) = 0, (B.7)

(I − J+J)J+ = 0. (B.8)

105

B.2 Range Space and Null Space Properties 106

B.2 Range Space and Null Space Properties

The following properties hold for the range space R(J) and the null space N (J) of J ∈

Rm×n:

R(J) = N (J+)T = R(JJ+) = N (I − JJ+), (B.9)

R(J)T = N (J+) = N (JJ+) = R(I − JJ+), (B.10)

N (J) = R(J+)T = N (J+J) = R(I − J+J), (B.11)

N (J)T = R(J+) = R(J+J) = N (I − J+J). (B.12)

B.3 Convexity

A function f : Rn → R with a convex domain dom(f) is convex if [88]

∀x1,x2 ∈ dom(f), ζ ∈ [0, 1] f(ζx1 + (1− ζ)x2) ≤ ζf(x1) + (1− ζ)f(x2). (B.13)

Now consider the following constrained optimization problem

minxf(x) (B.14)

subject to

gi(x) ≤ 0 for i = 1, ...,m,

where f : Rn → R are the objective function, and gi : Rn → R are the constraints. The

above optimization problem is a convex optimization problem if f and gi are convex.

B.4 Pareto Optimality 107

B.4 Pareto Optimality

Consider the following constrained vector optimization problem:

minx

f1(x)

...

fp(x)

(B.15)

subject to

gi(x) ≤ 0 for i = 1, ...,m,

where fi : Rn → R for i = 1, · · · , p are the objective functions, and gi : Rn → R for

i = 1, · · · ,m are the constraints. A feasible point x∗ is a Pareto optimal point, if for any

point x 6= x∗, there is some i such that fi(x) > fi(x∗) [82].

B.5 Logarithmic Smoothing

Consider y = f(x), where f : R→ R is a scalar differentiable function with extrema at x1,

x2 and x3, as illustrated in Fig. B.1. Depending on the initial point, a search for the global

minimum of f(x) using gradient-based optimization algorithms may output x1 or x3 as the

best solution. Fig. B.1 demonstrates how by using a logarithmic mapping, one can smooth

the local minima and as a result decrease the risk that the algorithm gets stuck at the local

minimum.

B.6 Active Set SQP Implementation of GSIP 108

x

y

y = f(x)

x1 x2 x3z

y

z=

αlog(β

y)+

γ

z

x

z=

g(x)

x1

x2

x3

local extrema

smoothed

Figure B.1: Log-smoothing technique: While the convexity of z = g(x) in the neighborhoodof x1 is much smaller than that of y = f(x) within the same neighborhood, theconvexity of the function in the vicinity of x3 is not affected as much. Thislowers the chance that gradient-based minimization techniques getting stuck atx1.

B.6 Active Set SQP Implementation of GSIP

Consider the following transformed GSIP:

minP

U(P) (B.16)

subject to

gijk(P) ≤ 0 for i = 1, ...,m1, j = 1, 2, ..., s, k = 1, 2, ..., N,

hik(P) ≤ 0 for i = m1 + 1, ...,m1 +m2, k = 1, 2, ..., N.


The Karush-Kuhn-Tacker (KKT) necessary optimality conditions can be stated as fol-

lows [89]:

∇U(P∗) + Σm1i=1Σs

j=1ΣNk=1λ

gijk∇gijk(P∗) + Σm1+m2

i=m1+1ΣNk=1λ

hik∇hik(P∗) = 0, (B.17)

gijk(P∗) ≤ 0 for i = 1, 2, · · · ,m1, j = 1, 2, · · · , s, k = 1, 2, ..., N, (B.18)

hik(P∗) ≤ 0 for i = m1 + 1, 2, · · · ,m2, k = 1, 2, ..., N, (B.19)

λgijkgijk(P∗) = 0 for i = 1, 2, · · · ,m1, j = 1, 2, ..., s, k = 1, 2, ..., N, (B.20)

λhikhik(P∗) = 0 for i = m1 + 1, 2, ...,m2, k = 1, 2, ..., N, (B.21)

λgijk ≤ 0 for i = 1, 2, · · · ,m1, j = 1, 2, ..., s, k = 1, 2, ..., N, (B.22)

λhik ≤ 0 for i = m1 + 1, 2, ...,m2, k = 1, 2, ..., N, (B.23)

where λgijk and λh

ik are Lagrange multipliers. It is desired that in a given iteration t, P(t)

is adjusted such that P(t+ 1) satisfies the above conditions. The update term can be given

as:

P(t+ 1) = P(t) + η(t)δ(t), (B.24)

where δ(t) ∈ Rn is the search direction and η(t) is the step length. In order to find the

optimum δ in each iteration, the following quadratic programming problem is solved:

minδ(t)

1

2δ(t)TH(t)δ(t) +∇U(P(t))Tδ(t) (B.25)

subject to

∇gijk(P(t))Tδ(t) + gijk(P(t)) ≤ 0 for i = 1, 2, · · · ,m1, j = 1, 2, ..., s, k = 1, 2, ..., N,

∇hik(P(t))Tδ(t) + hik(P(t)) ≤ 0 for i = 1, 2, · · · ,m1, k = 1, 2, ..., N,

where H = ∇2L(P, λhik, λ

gijk) is the Hessian matrix of the Lagrangian defined as:

L(P, λhik, λ

gijk) = U(P) + Σm1

i+1Σsj=1ΣN

k=1λgijkgijk(P) + Σm1+m2

i=m1+1ΣNk=1λ

hikhik(P). (B.26)


In practice, the Hessian matrix is not calculated directly and instead is approximated.

The Broyden-Fletcher-Goldfarb-Shanno (BFGS) update method is the most popular method

in which a positive definite approximation of the Hessian matrix is provided in each itera-

tion [90].

In order to find the optimum search direction in each iteration, in an active set method,

the search direction is chosen such that it is in the null space of the set of active constraints.

Suppose that the set of active constraints of (B.25) is given by Aa(t)Tδ(t) = 0, where

Aa ∈ Rl×n with l ≤ (m1 × s+m2)×N , then the search direction can be given as:

δ(t) = (I −Aa(t)+Aa(t))w (B.27)

where w ∈ Rn is an arbitrary vector. By incorporating the above expression in (B.25) the

transformed unconstrained quadratic programming problem is given as:

minw

1

2wT (I −Aa(t)+Aa(t))TH(t)(I −Aa(t)+Aa(t))w +∇U(t)T (I −Aa(t)+Aa(t))w.

(B.28)

Solving the above quadratic programming problem for w yields:

w∗ = −(

H(t)(

I −Aa(t)+Aa(t)))+∇U(t). (B.29)

From the KKT conditions for P(t+ 1), it is necessary that ∇L(P(t+ 1), λhik(t+ 1), λg

ijk(t+

1)) = 0. By approximation (see [89]) the condition can be rewritten as:

H(t)δ(t) +∇U(t) + ATa λ(t+ 1) = 0, (B.30)

where λ is the vector of Lagrange multipliers pertaining to the active set. Solving (B.30)

for λ results in:

λ(t+ 1) = −A+a (H(t)δ(t) +∇U(t)) . (B.31)

B.7 Chance Constrained Programming 111

The value of the Lagrange multipliers will be used to update the Hessian matrix using the

BFGS formula and update the quadratic programming problem of (B.25).

B.7 Chance Constrained Programming

Suppose that x is a random variable, and y = g(x), g : Rn → R. Now consider the following

chance constraint

P (y ≤ 0) ≥ p, 0 ≤ p ≤ 1, (B.32)

that can be rewritten in terms of the probability distribution function of y as:

∫ 0

−∞fy(y)dx ≥ p. (B.33)

If x ∼ N (µx,Qx), and the distribution of y can also be approximated by a Gaussian

distribution as y ∼ N (µy, Qy), the chance constraint can be rewritten in terms of the

standard normal cumulative distribution function as:

1− Φ

(

µy√

Qy

)

≥ p. (B.34)

Finally, the constraint can be rewritten in terms of the moments of y as:

µy + Φ−1(p)√

Qy ≤ 0. (B.35)

Ideally, the constraint has to be rewritten in terms of the moments of x. One trivial

solution can be obtained using linearization as follows:

(

∂g

∂x

)

µx + Φ−1(p)

√

(

∂g

∂x

)

Qx

(

∂g

∂x

)T

≤ 0. (B.36)

In deriving the above expression, two assumptions are made. First, we are assuming that

distribution of y can be accurately captured by a Gaussian distribution; second, that the

Gaussian distribution of y can be fairly accurately estimated by linearization.

Appendix C

The Heart Anatomy

The heart is located posterior to the sternum and between the lungs, which are located in

lateral spaces in the thorax called pleural cavities. The cavity between the pleural cavities

is called the mediastinum. The mediastinum is composed of superior, middle and inferior

cavities. The heart is located in the middle mediastinum, and is not aligned with the plane

of thorax but it is located in a plane extended from the right shoulder to the left nipple and

is inscribed by a double-layered cavity called the Pericardium, which has a fluid between its

layers. The heart apex projects close to the fifth rib, making it a suitable place for sensing

the heart beat [4].

The heart is composed of four chambers, two atria and two ventricles (see Fig. C.1). The

atria are located superior and to the right of their ventricles. The ventricle is more powerful

and is responsible for pumping the blood out of the heart, and the atrium is for collection

of blood before pumping it into ventricle. Blood with low oxygen comes into the right

atrium via the Superior Vena Cava and is pumped out to the lungs through the ventricle

and the pulmonary trunk, which branches off to the right and the left pulmonary arteries.

Pulmonary veins bring the oxygenated blood to the left atrium and the left ventricle pumps

it out through the Aorta.

The blood flow through the atria and ventricles is unidirectional as facilitated by a

set of four valves that are controlled cooperatively (see Fig. C.2). The tricuspid valve,

located at the right atrioventricular orifice, is composed of an annulus and three leaflets

112

113

that prevent the blood flowing back (regurgitation) into the right atrium when the right

ventricle contracts during systole. The bicuspid (Mitral) valve, composed of an annulus

and two leaflets is located at the left atrioventrical orifice and prevents regurgitation of

the blood into the left atrium when the left ventricle contracts (see Fig. C.2). Aortic and

pulmonary semilinar valves are another pair of valves that prevent backflow of blood from

the corresponding vessels into the ventricles during diastole.

Figure C.1: The blood flow through the heart atria and ventricles: 1. right atrium 2. rightventricle 3. pulmonary trunk 5. pulmonary veins 6. left atrium 7. left ventricle8. aortic artery 10. Venae Cavae (source: [4])

114

Figure C.2: The heart valves (source: [4])

116

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Dear Sir/ Madam

I would appreciate if you grant me the permission to use the following materials in my

PhD thesis. Further information is provided in below.

Name:

Hamidreza Azimian, University of Western Ontario

The publication info:

Handbook of Cardiac Anatomy, Physiology, and Devices, chapter Anatomy of the

Human Heart. Humana Press Inc., 2005, ISBN 9781592598359

The materials to be used:

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VITA

VITA 119

Name: Hamidreza Azimian

Post-secondary The University of Western Ontario

Education and London, ON, Canada

Degrees: 2007–2012 Ph.D.

Mechanical and Materials Engineering (Automation)

K. N. Toosi University of Technology

Tehran, Iran

2004–2006 M.Sc.

Electrical Engineering (Control Systems)

Iran University of science and Technology

Tehran, Iran

1999-2003 B.Sc.

Electronics Engineering

Honours and Distinguished student

Awards: Dept. of Electrical Engineering


Tehran, Iran, 2006

Related Work Senior Project Manager

Experience: Centre for Image Guided Intervention and Therapeutic Innovation

Hospital for Sick Children

Toronto, ON, Canada

since 2011

Research Assistant

Canadian Surgical Technologies and Advanced Robotics

London, ON, Canada

2008-2011

Research Assistant

Advanced Process Automation and Control Lab

Department of Electrical Engineering


Tehran, Iran

2006

VITA 120

Teaching Assistant

Robotics and Manufacturing Automation

Mechatronic System Design

Applied Sensing and Measurement

Control Systems

Electronics

Other: Reviewer of:

Journal of Control Engineering Practice

Journal of Intelligent and Robotic Systems

IEEE International Conference on Robotics and Automation

Publications: H. Azimian, R. V. Patel, M. D. Naish, B. Kiaii, ‘A Semi-InfiniteProgramming Approach to Patient-Specific PreoperativePlanning of Robotic Cardiac Surgery Under Geometric Un-certainty’, accepted for publication in IEEE Transactions on Infor-mation Technology in Biomedicine

H. Azimian, A. Fatehi, B. N. Araabi, ‘Takagi-Sugeno Controlof The Elevation Channel of A Twin-Rotor System Basedon Closed-Loop Empirical Data’, In Proceedings of AmericanControl Conference, Montreal, Canada, 2012

H. Azimian, R. V. Patel, M. D. Naish, ‘A Chance-ConstrainedApproach to Preoperative Planning of Robotics-AssistedInterventions’, In Proceedings of 33rd Annual International Con-ference of the IEEE Engineering in Medicine and Biology Society,Boston, MA, 2011

H. Azimian, R. V. Patel, M. D. Naish, B. Kiaii, ‘A Frameworkfor Preoperative Planning of Robotics-Assisted MinimallyInvasive Cardiac Surgery Under Geometric Uncertainty’, InProceedings of IEEE International Conference on Robotics and Au-tomation, Shanghai, China, 2011

H. Azimian, R.V. Patel and M.D. Naish, ‘On Constrained Manip-ulation in Robotics-Assisted Minimally Invasive Surgery’,In Proceedings of IEEE RAS/EMBS International Conference onBiomedical Robotics and Biomechatronics, Tokyo, Japan, 2010

VITA 121

H. Azimian, J. C. Breetzke, A. L. Trejos, R.V. Patel, M.D. Naish,T. Peters, J. Moore, C. Wedlake and B. Kiaii, ‘PreoperativePlanning of Robotics-Assisted Minimally Invasive CoronaryArtery Bypass Grafting’, In Proceedings of IEEE InternationalConference on Robotics and Automation, Anchorage, AK, 2010

H. Azimian, A. Fatehi, B. Araabi, ‘Closed-Loop Identificationof a Global Fuzzy Model: a Case Study’, In Proceedings ofMediterranean Control Conference, MED’ 07, Athens, Greece, 2007

R. Adlgostar, H. Azimian, H. Taghirad, ‘H∞ Robust RegulatorDesign for a Rotational/Translational Actuator’, In Proceed-ings of IEEE Conference on Control Applications’06, Munich, Ger-many, 2006

H. Azimian, R. Adlgostar, M. Teshnehlab, ‘Velocity Control ofan Electrohydraulic Servomotor Using Neural Networks’,In Proceedings of Conference on Physics and Control, PhysCon’05,Russia, 2005

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Preoperative Planning of Robotics-Assisted Minimally