BR3: a biologically inspired fish-like robot actuated by SMA-based ...

BR3: a biologically inspired fish-like robot

actuated by SMA-based artificial muscles.

William Hernan Coral Cuellar

Department of Electronics, Informatics and Industrial Engineering

Universidad Politcnica de Madrid, Spain

A thesis submitted for the degree of

Doctor of Philosophy in Robotics

2015

mailto:[email protected]

http://www.disam.upm.es/

http://www.upm.es/

Title:

BR3: a biologically inspired fish-like robot actuated by SMA-based artificial

muscles.

Author:

William Hernan Coral Cuellar, M.Sc

Director:

Prof. Claudio Rossi, Ph.D

Robotics and Cybernetics Group

Tribunal nombrado por el Mgfco. y Excmo. Sr. Rector de la Universidad Politectica de

Madrid, el dıa 25 de Mayo de 2015.

Tribunal

Presidente: D. Manuel Angel Armada Rodrıguez

Vocal: D. Fernando Torres Medina

Vocal: D. Oscar Reinoso Garcia

Vocal: D. Mohamed Abderrahim

Secretario: D. Sergio Dominguez Cabrerizo

Suplente: D. Joao Valente

Suplente: D. Antonio Gimenez Fernandez

Realizado el acto de lectura y defensa de la Tesis el dıa 25 de Mayo de 2015.

Calificacion de la Tesis:

El presidente: Los Vocales:

El Secretario:

ii

Abstract

Fishes are animals where in most cases are considered as highly manoeuvrable and

e↵ortless swimmers. In general fishes are characterized for his manoeuvring skills,

noiseless locomotion, rapid turning, fast starting and long distance cruising. Studies

have identified several types of locomotion that fish use to generate maneuvering

and steady swimming. At low speeds most fishes uses median and/or paired fins

for its locomotion, o↵ering greater maneuverability and better propulsive e�ciency.

At high speeds the locomotion involves the body and/or caudal fin because this can

achieve greater thrust and accelerations.

This can inspire the design and fabrication of a highly deformable soft

artificial skins, morphing caudal fins and non articulated backbone with

a significant maneuverability capacity.

This thesis presents the development of a novel bio-inspired and biomimetic fish-

like robot (BR3 ) inspired by the maneuverability and steady swimming ability of

ray-finned fishes (Actinopterygii, bony fishes). Inspired by the morphology of the

Largemouth Bass fish, the BR3 uses its biological foundation to develop accurate

mathematical models and methods allowing to mimic fish locomotion.

The Largemouth Bass fishes can achieve an amazing level of maneuverability and

propulsive e�ciency by combining undulatory movements and morphing fins. To

mimic the locomotion of the real fishes on an artificial counterpart needs the analysis

of alternative actuation technologies more likely muscle fiber arrays instead of stan-

dard servomotor actuators as well as a bendable material that provides a continuous

structure without joins. The Shape Memory Alloys (SMAs) provide the possibility

of building lightweight, joint-less, noise-less, motor-less and gear-less robots. Thus

a swimming underwater fish-like robot has been developed whose movements are

generated using SMAs. These actuators are suitable for bending the continuous

backbone of the fish, which in turn causes a change in the curvature of the body.

This type of structural arrangement is inspired by fish red muscles, which are mainly

recruited during steady swimming for the bending of a flexible but nearly incom-

pressible structure such as the fishbone. Likewise the caudal fin is based on SMAs

and is customized to provide the necessary work out.

The bendable structure provides thrust and allows the BR3 to swim. On the other

hand the morphing caudal fin provides roll and yaw movements. Motivated by the

versatility of the BR3 to mimic all the swimming modes (anguilliform, caranguiform,

subcaranguiform and thunniform) a bending-speed controller is proposed. The

bending-speed control law incorporates bend angle and frequency information to

produce desired swimming mode and swimming speed. Likewise according to the

biological fact about the influence of caudal fin shape in the maneuverability during

steady swimming an attitude control is proposed.

This novel fish robot is the first of its kind to incorporate only SMAs to

bend a flexible continuous structure without joints and gears to produce

thrust and mimic all the swimming modes as well as the caudal fin to

be morphing. This novel mechatronic design is a promising way to

design more e�cient swimming/morphing underwater vehicles. The

novel control methodology proposed in this thesis provide a totally new

way of controlling robots based on SMAs, making them more energy

e�cient and the incorporation of a morphing caudal fin allows to

perform more e�cient maneuvers.

As a whole, the BR3 project consists of five major stages of development:

• Study and analysis of biological fish swimming data reported in special-

ized literature aimed at defining design and control criteria.

• Formulation of mathematical models for: i) body kinematics, ii) dynam-

ics, iii) hydrodynamics, iv) free vibration analysis and v) SMA muscle-like

actuation. It is aimed at modelling the e↵ects of modulating caudal fin and

body bend into the production of thrust forces for swimming, rotational forces

for maneuvering and energy consumption optimisation.

• Bio-inspired design and fabrication of: i) skeletal structure of backbone

and body, ii) SMA muscle-like mechanisms for the body and caudal fin, iii)

the artificial skin, iv) electronics onboard and v) sensor fusion. It is aimed

at developing the fish-like platform (BR3) that allows for testing the methods

proposed.

• The swimming controller: i) control of SMA-muscles (morphing-caudal fin

modulation and attitude regulation) and ii) steady swimming control (bend

modulation and speed modulation). It is aimed at formulating the proper

control methods that allow for the proper modulation of BR3’s caudal fin and

body.

• Experiments: it is aimed at quantifying the e↵ects of: i) properly caudal fin

modulation into hydrodynamics and rotation production for maneuvering, ii)

body bending into thrust generation and iii) skin flexibility into BR3 bending

ability. It is also aimed at demonstrating and validating the hypothesis of

improving swimming and maneuvering e�ciency thanks to the novel control

methods presented in this thesis.

This thesis introduces the challenges and methods to address these stages. Water-

channel experiments will be oriented to discuss and demonstrate how the caudal fin

and body can considerably a↵ect the dynamics/hydrodynamics of swimming/ma-

neuvering and how to take advantage of bend modulation that the morphing-caudal

fin and body enable to properly change caudal fin and body’ geometry during steady

swimming and maneuvering.

Resumen:

Los peces son animales, donde en la mayorıa de los casos, son considerados como

nadadores muy eficientes y con una alta capacidad de maniobra. En general los

peces se caracterizan por su capacidad de maniobra, locomocion silencioso, giros

y partidas rapidas y viajes de larga distancia. Los estudios han identificado varios

tipos de locomocion que los peces usan para generar maniobras y natacion constante.

A bajas velocidades la mayorıa de los peces utilizan sus aletas pares y / o impares

para su locomocion, que ofrecen una mayor maniobrabilidad y mejor eficiencia de

propulsion. A altas velocidades la locomocion implica el cuerpo y / o aleta caudal

porque esto puede lograr un mayor empuje y aceleracion.

Estas caracterısticas pueden inspirar el diseo y fabricacion de una piel

muy flexible, una aleta caudal morfica y una espina dorsal no

articulada con una gran capacidad de maniobra.

Esta tesis presenta el desarrollo de un novedoso pez robot bio-inspirado y bio-

mimetico llamado BR3, inspirado en la capacidad de maniobra y nado constante

de los peces vertebrados. Inspirado por la morfologıa de los peces Micropterus

salmoides o tambien conocido como lubina negra, el robot BR3 utiliza su funda-

mento biologico para desarrollar modelos y metodos matematicos precisos que per-

miten imitar la locomocion de los peces reales. Los peces Largemouth Bass pueden

lograr un nivel increıble de maniobrabilidad y eficacia de la propulsion mediante la

combinacion de los movimientos ondulatorios y aletas morficas.

Para imitar la locomocion de los peces reales en una contraparte artificial se nece-

sita del analisis de tecnologıas de actuacion alternativos, como arreglos de fibras

musculares en lugar de servo actuadores o motores DC estandar, ası como un mate-

rial flexible que proporciona una estructura continua sin juntas. Las aleaciones con

memoria de forma (SMAs) proveen la posibilidad de construir robots livianos, que

no emiten ruido, sin motores, sin juntas y sin engranajes. Asi es como un pez robot

submarino se ha desarrollado y cuyos movimientos son generados mediante SMAs.

Estos actuadores son los adecuados para doblar la espina dorsal continua del pez

robot, que a su vez provoca un cambio en la curvatura del cuerpo. Este tipo de

arreglo estructural esta inspirado en los musculos rojos del pescado, que son usa-

dos principalmente durante la natacion constante para la flexion de una estructura

flexible pero casi incompresible como lo es la espina dorsal de pescado. Del mismo

modo la aleta caudal se basa en SMAs y se modifica para llevar a cabo el trabajo

necesario.

La estructura flexible proporciona empuje y permite que el BR3 nade. Por otro lado

la aleta caudal morfica proporciona movimientos de balanceo y guiada. Motivado

por la versatilidad del BR3 para imitar todos los modos de natacion (anguilliforme,

carangiforme, subcarangiforme y tunniforme) se propone un controlador de doblado

y velocidad. La ley de control de doblado y velocidad incorpora la informacion del

angulo de curvatura y de la frecuencia para producir el modo de natacion deseado

y a su vez controlar la velocidad de natacion. Ası mismo de acuerdo con el hecho

biologico de la influencia de la forma de la aleta caudal en la maniobrabilidad durante

la natacion constante se propone un control de actitud.

Esta novedoso robot pescado es el primero de su tipo en incorporar solo

SMAs para doblar una estructura flexible continua y sin juntas y

engranajes para producir empuje e imitar todos los modos de natacion,

ası como la aleta caudal que es capaz de cambiar su forma. Este

novedoso diseo mecatronico presenta un futuro muy prometedor para el

diseo de vehıculos submarinos capaces de modificar su forma y nadar

mas eficientemente. La nueva metodologıa de control propuesto en esta

tesis proporcionan una forma totalmente nueva de control de robots

basados en SMAs, haciendolos energeticamente mas eficientes y la

incorporacion de una aleta caudal morfica permite realizar maniobras

mas eficientemente.

En su conjunto, el proyecto BR3 consta de cinco grandes etapas de desarrollo:

• Estudio y analisis biologico del nado de los peces con el proposito de

definir criterios de diseno y control.

• Formulacion de modelos matematicos que describan la: i) cinematica del

cuerpo, ii) dinamica, iii) hidrodinamica iv) analisis de los modos de vibracion

y v) actuacion usando SMA. Estos modelos permiten estimar la influencia de

modular la aleta caudal y el doblado del cuerpo en la produccion de fuerzas de

empuje y fuerzas de rotacion necesarias en las maniobras y optimizacion del

consumo de energıa.

• Diseno y fabricacion de BR3: i) estructura esqueletica de la columna verte-

bral y el cuerpo, ii) mecanismo de actuacion basado en SMAs para el cuerpo y

la aleta caudal, iii) piel artificial, iv) electronica embebida y v) fusion sensorial.

Esta dirigido a desarrollar la plataforma de pez robot BR3 que permite probar

los metodos propuestos.

• Controlador de nado: compuesto por: i) control de las SMA (modulacion

de la forma de la aleta caudal y regulacion de la actitud) y ii) control de nado

continuo (modulacion de la velocidad y doblado). Esta dirigido a la formulacion

de los metodos de control adecuados que permiten la modulacion adecuada de

la aleta caudal y el cuerpo del BR3.

• Experimentos: esta dirigido a la cuantificacion de los efectos de: i) la cor-

recta modulacion de la aleta caudal en la produccion de rotacion y su efecto

hidrodinamico durante la maniobra, ii) doblado del cuerpo para la produccion

de empuje y iii) efecto de la flexibilidad de la piel en la habilidad para doblarse

del BR3. Tambien tiene como objetivo demostrar y validar la hipotesis de

mejora en la eficiencia de la natacion y las maniobras gracias a los nuevos

metodos de control presentados en esta tesis.

A lo largo del desarrollo de cada una de las cinco etapas, se iran presentando los

retos, problematicas y soluciones a abordar. Los experimentos en canales de agua

estaran orientados a discutir y demostrar como la aleta caudal y el cuerpo pueden

afectar considerablemente la dinamica / hidrodinamica de natacion / maniobras y

como tomar ventaja de la modulacion de curvatura que la aleta caudal morfica y el

cuerpo permiten para cambiar correctamente la geometrıa de la aleta caudal y del

cuerpo durante la natacion constante y maniobras.

A Dios, mi Madre, mis hermanos (Adonis, Ivonne, Cesar), ma petite chouchou

(Laura), mis primos, mis hermanos de corazon y mi nueva familia Coral.

Acknowledgements

The author would like to thank to professor Oscar M. Curet for providing the

support and useful knowledge about the hydrodynamics e↵ects on fish robots. To

the Curet Lab team of Florida Atlantic University for providing the water-channel

facility, and their support with the experiments.

Contents

List of Figures ix

List of Tables xvii

1 Introduction 1

1.1 The problem and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Original Contributions of this Work . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Literature Review 19

2.1 General Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Shape memory alloys background . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Improving the performance of SMA actuators . . . . . . . . . . . . . . . . 23

2.2.3 Modeling and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3 Bio-inspired robots with SMA muscle-like actuation . . . . . . . . . . . . . . . . 25

2.3.1 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.1 A micro-robot fish with embedded SMA wire actuated by flexi-

ble biomimetic fin . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1.2 Towards a biologically inspired small-scale water jumping robot 26

2.3.1.3 A micro biomimetic manta ray robot fish actuated by SMA . . . 27

2.3.1.4 Controlling a lamprey-based robot with an electronic nervous

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iii

CONTENTS

2.3.1.5 A biomimetic robotic jellyfish (Robojelly) actuated by shape

memory alloy composite actuators . . . . . . . . . . . . . . . . . 29

2.3.2 Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.2.1 Recent progress in developing a beetle-mimicking flapping-wing

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3.2.2 BATMAV-a biologically inspired micro-air vehicle for flapping

flight: artificial-muscle based actuation . . . . . . . . . . . . . . 30

2.3.3 Ground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3.3.1 Sensor fusion in a SMA-based hexapod bio-mimetic robot . . . . 31

2.3.3.2 Omegabot: Crawling robot inspired by Ascotis Selenaria . . . . 31

2.3.3.3 An earthworm-like micro robot using shape memory alloy actuator 32

2.3.4 Other SMA-based actuation systems . . . . . . . . . . . . . . . . . . . . . 33

2.3.4.1 Research on Development of a Flexible Pectoral Fin Using Shape

Memory Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.4.2 Development of a dexterous tentacle-like manipulator using SMA-

actuated hydrostats . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.3.4.3 Development of a Shape-Memory-Alloy actuated biomimetic hy-

drofoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.4 Review on recent advances: iTuna and BaTboT . . . . . . . . . . . . . . . . . . . 34

2.4.1 iTuna: a bending structure swimming robotic fish . . . . . . . . . . . . . 34

2.4.1.1 SMA control in the iTuna . . . . . . . . . . . . . . . . . . . . . . 36

2.4.1.2 Control architecture . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 BaTboT: a biologically-inspired bat-like aerial robot . . . . . . . . . . . . 37

2.5 Advantages and drawbacks of using SMAs . . . . . . . . . . . . . . . . . . . . . . 38

2.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3 From Ray-Finned Fishes to BR3: Mimicking biology 41

3.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Body and/or Caudal Fin Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Body Undulations and Friction Drag . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4 Wake Structure and Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 Overview of fish fin structure and function . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

iv

CONTENTS

4 BR3 modeling 50


4.2 First approach to kinematic, dynamic and hydrodynamic analysis . . . . . . . . . 50

4.2.1 Overview of the propulsive mechanism . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.3 Kinematics Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.4 Hydrodynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.5 3-D Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Final approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.1 System modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3.2 Kinematic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.3 Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.4 Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3.5 SMA phenomenological model . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Geometry of bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.1 Bend angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.4.2 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.5 Swim patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5.1 Steady swimming (cruise straight) . . . . . . . . . . . . . . . . . . . . . . 69

4.5.2 Cruise-in turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5.3 C-starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.6 Simulation and experimental results . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.6.1 Open-loop simulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6.2 Bio-hydrodynamics simulator . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.6.3 Steady swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.6.4 Morphology parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.6.5 In-cruise turning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.6.6 C-starts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.7 Final remarks and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

v

CONTENTS

5 BR3 design and Fabrication 79

5.1 The general method for BR3 design . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.2 Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Ribs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2.2 Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.2.2.1 Swimming modes . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.3 Skin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3 Caudal fin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4 Mechatronics concept design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.1 Biological foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.4.2 Design Concepts and Modelling . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.3 Bending Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.5 Fabrication and assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.6 BR3 electronics and sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6.1 Arduino controller-board . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.6.2 The Inertial Measurement Unit (IMU) . . . . . . . . . . . . . . . . . . . . 95

5.6.3 Flex Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6.4 Current sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.6.5 Temperature and Humidity sensor . . . . . . . . . . . . . . . . . . . . . . 97

5.6.6 SMA power drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.7 BR3 consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.8 BR3 costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.9 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6 Free vibration analysis based on a continuous and non-uniform flexible back-

bone with distributed masses 101


6.2 Di↵erential quadrature method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.3 Vibration analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.3.2 Compatibility conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6.3.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

6.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.4.1 Numerical comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

vi

CONTENTS

6.4.2 Practical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.5 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 BR3 Control 113

7.1 Control goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2 Electrical resistance control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.2.1 Design of PID by Ziegler-Nichols tuning rule for an SMA wire . . . . . . . 114

7.2.2 SMA control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.2.3 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.3 Bending control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.3.1 Controller setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3.2 Passive Noise Reduction System . . . . . . . . . . . . . . . . . . . . . . . 119

7.3.3 Control Schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8 General experimental results 121

8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.1 Methods and goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.1.2 The water-channel setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.1 Air: spine, without ribs and skin . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.2 Air: spine with ribs and skin . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.2.3 water channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.2.4 Free swimming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

9 Conclusions and Future Work 131

9.1 General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

9.3 Thesis schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

10 Publications 135

10.1 Journals, book chapters and conference proceedings . . . . . . . . . . . . . . . . 135

10.2 Technical And Technological Manufacturing . . . . . . . . . . . . . . . . . . . . . 136

References 137

vii

CONTENTS

11 Annexes 144

11.1 Model converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11.1.1 Importing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11.2 SMA control electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

11.2.1 PWM to DC converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

11.2.2 Electronic design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

11.2.3 Voltage gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

11.2.4 Design for the first filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

11.2.5 cuto↵ frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

11.2.6 Design for the filters 2, 3 and 4 . . . . . . . . . . . . . . . . . . . . . . . . 151

11.2.7 Transfer function for the PWM/DC converter . . . . . . . . . . . . . . . . 151

11.2.8 Matlab Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

11.2.9 SMA Power driver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

11.3 SMA phenomenological model Matlab-code . . . . . . . . . . . . . . . . . . . . . 154

viii

List of Figures

1.1 BR3. This robot has the ability to swim frilly or stationary to do PIV tests. For the

first case, the fish carries inside all the electronics devices and the battery. The weight

of the fish out of the water is 2.5kg, length is 45cm, width 8cm. Inside it has 4 SMAs,

4 Current sensor, 4 flex sensors, 1 temperature sensor, a 6-dof IMU and 4 SMA drivers. 3

1.2 Terminology used in the text to identify the fins and other features of fish. . . . . 6

1.3 (a) The forces acting on a swimming fish. (b) Pitch, yaw, and roll definitions. . . 7

1.4 Diagram showing the relative contribution of the momentum transfer mecha-

nisms for swimming vertebrates, as a function of Re. The shaded area corre-

sponds to the range of adult fish swimming. . . . . . . . . . . . . . . . . . . . . . 8

1.5 Diagram showing the relation between swimming propulsors and swimming func-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Swimming modes associated with (a) BCF propulsion and (b) MPF propulsion.

Shaded areas contribute to thrust generation. . . . . . . . . . . . . . . . . . . . . 12

1.7 Structural steps to be followed during the thesis aimed at the development of

BR3. The pictures depicted herein, correspond to the final BR3 prototype. The

forthcoming chapters will introduce each step with all the details. Source: The

author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Microscopic viewpoint of the Shape Memory E↵ect . . . . . . . . . . . . . . . . . 21

2.2 Fabricated water jumping robot with six legs. LL is the left latch and LR is the

right latch. RLR and RLL are the rear legs, MLL and MLR are the middle legs,

and FLL and FLR are the front legs. Scale bar, 10 mm (1). . . . . . . . . . . . . 27

2.3 Micro biomimetic manta ray robot fish (2). . . . . . . . . . . . . . . . . . . . . . 28

ix

LIST OF FIGURES

2.4 (a) Lamprey Robot with sonar array, (b) Lateral view of tail segment showing

nitinol actuator, Teflon vertebra and tensioning nuts and (c) Lateral view of

pitch mechanism (3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Prototype of the robotic beetle and detail of the unfolding of the artificial wing

(4). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.6 BATMAV. Dual Role of Shape Memory Alloy wires: as actuation muscles, and

super elastic joints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 SMABOT IV, a SMA based hexapod robot with the IMU module, compass

sensor and step touch sensors (5). . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Omegabot, a biomimetic inchworm robot, grasps the branch of a wood, raises its

head, and turns right. Bottom right: Proleg of Omegabot (6). . . . . . . . . . . . 32

2.9 Biomimetic pectoral fin driven by eight couples of SMA plates (7). . . . . . . . . 33

2.10 The SMA-based tentacle (See http://www.octopusproject.eu/). . . . . . . . 34

2.11 Main structure of the iTuna robot fish. a=8.5 cm. Under nominal operatrion,

b ⇠= 96% a = 8.16 cm, h=1.02 cm, b=28� (8). . . . . . . . . . . . . . . . . . . . . 35

2.12 Bending under SMA overloading (8). . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.13 Flow-process for SMA evaluation in the BaTboT prototype (9). . . . . . . . . . . 38

3.1 Thrust generation by the added-mass method in BCF propulsion . . . . . . . . . 42

3.2 Gradation of BCF swimming movements from (a) anguilliform, through (b) sub-

carangiform and (c) carangiform to (d) thunniform mode. . . . . . . . . . . . . . 44

3.3 The Karma street generates a drag force for either (a) blu↵ or (b) streamlined

bodies, placed in a free stream. (c) The wake of a swimming fish has reverse

rotational direction, associated with thrust generation. . . . . . . . . . . . . . . . 46

x

LIST OF FIGURES

3.4 Structure of the fin skeleton in bony fishes. (a) Skeleton showing the positions of the

paired and median fins and their internal skeletal supports. Note that each of the

median fins has segmented bony skeletal elements that extend into the body to support

the fin rays and spines, and that muscles controlling the fin rays arise from these skeletal

elements. (b) Bluegill sunfish hovering in still water with the left pectoral fin extended.

(c) Structure of the pectoral fin and the skeletal supports for the fin; bones have been

stained red. This specimen had 15 pectoral fin rays that articulate with a crescent-

shaped cartilage pad (tan color) at the base of the fin. The smaller bony elements to

the left of the cartilage pad allow considerable reorientation of the fin base and hence

thrust vectoring of pectoral fin forces (10) (11). (d) Anal fin skeleton (bones stained

red and muscle tissue digested away) to show the three leading spines anterior to the

flexible rays, and the collagenous membrane that connects adjacent spines and rays.

(e) Close view of pectoral fin rays (stained red) to show the segmented nature of bony

fish fin rays and the membrane between them. Images in panels A and B modified from

(12). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.5 Pectoral fin structure in bluegill sunfish. (a) Schematic view of the pectoral fin

which typically has 12-15 fin rays. (b) Cross-section through fin rays at the level

of the blue plane shown in panel A obtained with lCT scanning (see (13)) in

which bone is whitish color, and fin collagen and membrane are gray. Cross-

sectional image of rays (top) and close view of two adjacent rays (below). Each

fin ray is bilaminar, with two curved half rays termed hemitrichs. (c) Schematic

of the mechanical design of the bilaminar fin ray in bony fishes. Each fin ray has

expanded bony processes at the base of each hemitrich to which muscles attach

(blue arrows). Di↵erential actuation of fin ray muscles (red arrows) results in

curvature of the fin ray. Fish can thus actively control the curvature of their fin

surface. (d) Frame from high-speed video of a bluegill sunfish during a turning

maneuver, showing the fin surface (outlined in yellow) curving into oncoming flow. 49

4.1 Precession � angle and Nutation � angle, representation . . . . . . . . . . . . . . 51

4.2 X, Y and Z coordinates system representation . . . . . . . . . . . . . . . . . . . . 52

4.3 (a) Link-fixed coordinate system. (b) Three Eulerian angles. . . . . . . . . . . . . 52

4.4 Planar configuration for the robot fish. . . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Planar configuration for the robot fish. . . . . . . . . . . . . . . . . . . . . . . . . 57

xi

LIST OF FIGURES

4.6 Bend angle for the head and tail segments. (a) Reference tail, angle 0o, (b) Tail

left, angle 10o, (c) Tail right, angle 10o, (d) Reference head, angle 0o, (e) Head

left, angle 10o, (f) Head right, angle 10o . . . . . . . . . . . . . . . . . . . . . . . 67

4.7 The geometry of bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8 Approximation of fT

(x, t) (solid lines) with circle arcs (dotted lines). ! =

�⇡, c1 = 4.5479L

, c2 = 0,� = 2⇡4L . The blue circles represent the end point po-

sition of the fish bone segment of L=8.5 cm. c1 corresponds to the maximum

achievable bending, and � has been set for subcarangiform swimming, where

half a wave length is reproduced by the body consisting of two segments. The

trajectory of the end point of the fishbone segment is shown by the arrow. . . . 70

4.9 SimMechanics open-loop simulator for dynamics and SMA actuation. . . . . . . . 73

4.10 3D escenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.11 a) Steady forward swimming (atail

= 0.54, abody

= atail

/2,�tail

= �⇡/4,�body

=

0, tail-beat frequency=1/2 Hz). b) Simulation of the torsional torque ⌧�

required

to bend the polycarbonate structure by using the V-shaped SMA actuators at

tc

= 0.5s. c) Top view of the antagonistic V-shaped wires fixed to the backbone.

For modeling, the bending property of the backbone is considered as a spring. d)

Bending angle � profile during SMA contraction: during t1, the active actuator

contracts upon heating, achieving a bending angle of 36o, subsequently, during

td

= 200ms, both pair of antagonistic actuators remain passive, and the decrease

of the bending angle is provided by the restoring force caused by the polycar-

bonate structure trying to recover its original shape (i.e. spring- damping force).

During t2 the antagonistic actuator turns active providing the opposite motion. . 75

4.12 Cruise-in turning. Labels refer to the desired turning radius (meters), corre-

sponding (from left to right) to bj

= 0.0375, 0.05, 0.075, 0.15, 0.3, (see (4.71)). . . 76

4.13 Stills of the C-start maneuver of the simulation and with the real prototype . . . 77

4.14 Comparison of qualitative assessment, numerical simulations and experimental

results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.1 Backbones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

xii

LIST OF FIGURES

5.2 Simulated BR3 in SimMechanics. (a) Top view: the red line represents the

backbone, while the dotted black lines represent the contracted SMAs. The

angles shown are related to the number and thickness of the Ribs, inter-Rib

spaces and SMA length when contracted. (b) FrontView,(c) CrossSection Rib

(see also Tab. 5.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Operation mode flow chart. ”SMA Wire Left” (SWL), ”SMA Wire Right” (SWR) 83

5.4 Principle of the bendable structure. The SMA wires are parallel to the backbone

segment. As a SMA contracts, it causes the polycarbonate backbone to bend

(angles ↵ and �) the antagonist SMA generates the angles � and �. L1 is the

length segment of the ”Polycarbonate Backbone”. L2 and L3 are the length of

the contracted and relaxed SMA respectively. . . . . . . . . . . . . . . . . . . . . 84

5.5 Components of the resultant force R. F4 = F2 = 321gf , F2x = F2cos(↵), F2y =

F2sin(↵), and the resultant forces areRy

=P

Fy

= �F2y, Rx

=P

Fx

= F2x � F4 84

5.6 Evolution of the forces corresponding to the resultant forces Rx

and Ry

. . . . . 85

5.7 Simulated Swimming patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.8 Building process for the di↵erent skin trials . . . . . . . . . . . . . . . . . . . . . 88

5.9 Sample of Silicone Rubber Skin Tissue . . . . . . . . . . . . . . . . . . . . . . . . 88

5.10 Representation of the intrinsic caudal muscles. Flexor dorsalis (FD, green), flexor

ventralis (FV, blue), hypochordal longitudinalis (HL, purple), infracarinalis (IC,

gray), interradialis (IR, red) and supracarinalis (SC, yellow). The color coding

of the muscles is the same used for the bluegill sunfish (Lepomis) in Flammang

and Lauder (14). (figure adapted from (15)) . . . . . . . . . . . . . . . . . . . . . 89

5.11 Representative examples of caudal fin shape modulation for (a), Steady Swim-

ming (b), Braking (c) Kick (d) Kick and Glide. Tail outlines closely follow the

distal margin of the caudal fin and fin ray position. Arrows indicate the major

direction of movement of the dorsal and ventral lobes of the caudal fin. Bar

(yellow), 2 cm. (figure adapted from (15)) . . . . . . . . . . . . . . . . . . . . . . 90

5.12 (a) The concept of a novel Bio-inspired Morphing Caudal Fin using shape mem-

ory alloys (SMA). (b) Cross-Section basic concept. Note that the SMAs are

embedded (sandwiched) between the Cellulose Acetate Film and the Silicone

Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.13 Representation of the bendable structure with the SMA contracted . . . . . . . . 92

xiii

LIST OF FIGURES

5.14 Beam kinematics concept diagram showing the (a) undeformed and (b) deformed

configuration. Distance d for both SMA wires is less than a 1mm. Distance e

represents the thickness of the silicone rubber layer . . . . . . . . . . . . . . . . . 92

5.15 Final Bending Design. RSMA

m

is the radius of the middle SMA (SMAmiddle

) . . 93

5.16 First Design with asymmetrical SMAs and clear edges. Noted that the SMAmiddle

is not circular, this because the first tests used only the SMA placed at the upper

and lower segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.17 Final design of the Caudal fin. Noted that all SMAs wires are tight and the

presence of the flex sensor used to measure the bend. . . . . . . . . . . . . . . . . 95

5.18 (a) Arduino Micro Front, (b) Arduino Micro Rear, (c) Pin Mapping of the Ar-

duino Micro displays the complete functioning for all the pins . . . . . . . . . . . 96

5.19 (a) Razor IMU Rear size, (b) Razor IMU Front, (c) Razor IMU Rear . . . . . . 96

5.20 Conductive particles (a) close together and (b) further apart, (c) size . . . . . . 97

5.21 Current sensor (a) Front view (b) Rear view . . . . . . . . . . . . . . . . . . . . . 97

5.22 Temperature and Humidity sensor (a) Front view (b) Rear view . . . . . . . . . 98

5.23 Miga analog driver V5 pinout diagram. Source: The author. . . . . . . . . . . . . 99

5.24 Percentage of current consumption per component. . . . . . . . . . . . . . . . . . 100

6.1 Real fish-robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.2 Non-uniform fish-robot backbone with distributed masses. . . . . . . . . . . . . . 104

6.3 Real fish-robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.4 Free Vibration analysis of the backbone. The white spots are the marks for the

tracking, the red line is the trajectory of the tail. . . . . . . . . . . . . . . . . . 111

6.5 Experimental Results. The Natural frequency obtained was 2.1249Hz . . . . . . . 111

6.6 Real fish-robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.1 Histeresis of the SMA. (As

, the austenite start temperature; Af

, the austenite

finish temperature; Ms

, the martensite start temperature; and Mf

, the marten-

site finish temperature.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2 Voltage SMA vs. a given set point. y1 = 0.258, y0 = 0.232, t1 = 2.725 t0 = 2.2

t2 = 4 u1 = 1 u0 = 0, experimentally determined.) . . . . . . . . . . . . . . . . . 114

7.3 Block diagram of the PID controller used . . . . . . . . . . . . . . . . . . . . . . 115

7.4 Block diagram of the PID controller used . . . . . . . . . . . . . . . . . . . . . . 116

xiv

LIST OF FIGURES

7.5 (a) Transient performance for a 11.46⌦ set point. (b) Comparison of the transient

period at 450mA and 500mA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

7.6 Input and Output Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.7 Noise signal (blue) compared with filtered signal (red). (a) Current Sensor Sig-

nals. (b) Flex Sensor Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.8 Passive Low-Pass filters. (a) Current Sensor Low-pass filter. (b) Flex Sensor

Low-pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.9 A bend feedback control schema for a single SMA actuator. . . . . . . . . . . . . 120

8.1 Air-test sets-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.2 Water channel set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3 250 250 mm recirculating water tunnel . . . . . . . . . . . . . . . . . . . . . . . 124

8.4 Swimming modes (a) Thunniform Tail-Up, (b) Thunniform Tail-Down, (c) Carangi-

form Tail-Up, (d) Carangiform Tail-Down, (e) Sub-Carangiform Tail-Up, (f) Sub-

Carangiform Tail-Down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.5 Response for the sub-carangiform swimming mode. Tracked trajectory for the

tail segment. Theoretical trajectory (Red line), Measured trajectory (Blue line) . 126

8.6 Swimming modes (a) Thunniform Tail-Up, (b) Thunniform Tail-Down, (c) Sub-

Carangiform Tail-Up, (d) Sub-Carangiform Tail-Down . . . . . . . . . . . . . . . 126

8.7 Response for the sub-carangiform swimming mode in air with ribs and skin.

Tracked trajectory for the tail segment. Theoretical trajectory (Red line), Mea-

sured trajectory (Blue line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.8 Response for the sub-carangiform swimming mode in water. Tracked trajectory

for the tail segment. Theoretical trajectory (Red line), Measured trajectory (Blue

line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.9 PIV visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

8.10 PIV visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

8.11 Testing bending in water (two segments, overloaded SMAs, open loop) . . . . . . 129

8.12 Linear swimming with f=0.5 Hz, atail

=0.49 at t=1, t=2, t=3, t=4 seconds (two

tail beats). The distance travelled is approximately 7 centimeters. Notice the

reduction of the bending with respect to Figure 5.13 . . . . . . . . . . . . . . . . 129

8.13 Stills of the C-start maneuver of the simulation and with the real prototype . . . 130

11.1 Extrusion select . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

xv

LIST OF FIGURES

11.2 Solidify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

11.3 Solidify window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

11.4 Screen captures. (a) Blender, solid selection (b) Blender, Face extrude (c)

Blender, extrusion parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

11.5 Solidify window . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

11.6 PWM / DC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

11.7 output with f=200Hz and 4 low-pass filters . . . . . . . . . . . . . . . . . . . . . 149

11.8 PWM / DC full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

11.9 Simulink. (a) PWM/DC Simulation Matlab - Simulink (b) Subsystem 1 (c)

Subsystem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.10time response for the PWM / DC . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

11.11Time response to other values of duty cycle . . . . . . . . . . . . . . . . . . . . . 153

11.12(a) Voltage-controlled current source (b) Test circuit . . . . . . . . . . . . . . . . 154

xvi

List of Tables

2.1 Characteristics of NiTinol®SMA wires (16). . . . . . . . . . . . . . . . . . . . . 22

4.1 Parameters for SMA phenomenological model . . . . . . . . . . . . . . . . . . . . . . 66

4.2 Summary of the performances of the simulations on steady swimming . . . . . . 76

5.1 Size Comparison for the rib number 20 and 7 . . . . . . . . . . . . . . . . . . . . . . 83

5.2 Comparison chart between the materials used for the skin. (g=Good gg=

Betterggg= Best) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3 General values of current consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Fabrication costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.1 Comparison between this method and the exact method proposed by (17) for the

first three non-dimensional frequencies (�2) . . . . . . . . . . . . . . . . . . . . . 109

7.1 Summary of the performances of the control . . . . . . . . . . . . . . . . . . . . . 117

7.2 Fit to estimation data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.3 PID controller characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8.1 Comparison of the simulation and experimental results for steady swimming.

(atail

= 0.49, abody

= 0.27, f = ⇡/2 Hz) . . . . . . . . . . . . . . . . . . . . . . . 127

8.2 Comparison of the simulation and experimental results for steady swimming

(performance and morphology parameters) . . . . . . . . . . . . . . . . . . . . . 128

xvii

GLOSSARY

Nomenclature

Modelling. Section 4.

� = Precession angle� = Nutation angleybody

= Transverse displacement of the fish bodyx = longitudinal position with respect to the head of the fishk = body wave number� = body wave lengthc1 and c2 = wave amplitude! = body wave frequency✓f

(t) = pitch angle of the caudal fin relative to main axis✓max

= amplitude of pitch angle' = phase angle between heave and pitchM

i

� ⇠i

⌘i

⇣i

= coordinate systemxF

= x-component of the position of the oscillatory foil pivot�o

= slope angle↵o

= attack angle of the caudal finSt

= Strouhal number↵max

= maximum attack angle�,� and = Eulerian anglesu(t) = forward speedv(t) = sway velocityM = inertia matrixU = acceleration/angular acceleration vectorF = resultant forces/moments vectorF a

j

= active forcesF c

j

= constraint forcesLa

j

= active momentsF

Ij

= added mass forces due to the surrounding fluidF

V j

= hydrodynamic dragF

V j

= hydrodynamic force due tof the caudal fin⌧I

j = added moment induced by the surrounding fluid⌧j�1,j and ⌧

j+1,j = output torques of SMAsC

f

= friction coe�cient⇢ = density of the fluidC

d

= cross flow drag coe�cientSj

= wet surface areaA

j

= area of the cross-sectionC

T

= thrust coe�cient relative to attack angle of the caudal finT = temperature✏ = strain⇠ = martensite fractionIsma

= applied electrical current⇢ = density of wire2r

j

= diameter of wire

xviii

GLOSSARY

cp

= specific heatR

sma

= electrical resistance per unit length of the wireTo

= ambient temperaturehc

= heat convection coe�cient✓s

= thermal expansion factor of the wire⌦ = phase transformation factorA

f

, As

= austenite final and initial temperaturesC

m

= stress coe�cient on martensite temperature� = stress rateT = temperature rate" = strain rateE

A

= Young’s modulus⇠ = phase transformation rateM

f

,Ms

= martensite phase final and initial temperature� = Bending angle

Free vibration analysis. Section 6.

x = Global Spatial Coordinate⇠ = Dimensionless Global Spatial Coordinatex(i) = Local Spatial Coordinate of Element i⇣(i) = Dimensionless Local Spatial Coordinate of Element ivL = Total Length of the Beamt = Timew (x,t) , W (x) = Transverse Displacement (x,t) , (x) = Rotation Angle Due to BendingW (i) = Transverse Displacement of Element i (i) = Rotation Due to Bending of Element iv(i) = Dimensionless Transverse Displacement of Element il(i) = Dimensionless Length of Element iA(x) = Cross Sectional Area of the BeamI(x) = Moment of Inertia About the Neutral AxisA0 = Values of the Cross-Section at the Clamped EdgeI0 = Values of the Moment of Inertia at the Clamped Edgek = Shear Correction FactorE = Youngs Modulus of ElasticityG(x) = Shear Modulus of Beam Material# = Poissons Ratio⇢ = Mass Density! = Angular Natural Frequency of Vibration� = Dimensionless Natural Frequency of Vibrationr = Slender RatioN = Number of Grid PointsM (i) = Bending Moment in Element iV (i) = Transverse Force in Element i

xix

GLOSSARY

↵i

= Dimensionless Value of the ith Concentrated Attached MassJ(x) = Mass Moment of Inertia of the Beam per Unit LengthA(x) = Cross Sectional Area of the BeamQ(x) = Beam Shear Rigidity, kG(x)A(x)m

i

= Translational Inertia of the ith Concentrated Masss = Elastic Section Modulus

xx

GLOSSARY

Acronyms

ABS Acrilonitrilo butadieno estireno

AUV Autonomous Underwater Vehicles

CAD Computer-Aided Design

CM Center of Mass

DC Direct Current

DH Denavit-Hartenberg

DQEM Di↵erential Quadrature Element Method

DQ Di↵erential Quadrature

EoM Equations of Motion

PID Proportional-Integral-Derivative

PWM Pulse Width Modulation

SMA Shape Memory Alloy

SME Shape Memory E↵ect

SWL SMA Wire Left

SWR SMA Wire Right

BCF body-caudal fin

LMM Lycra Microfiber Mesh

FD Flexor dorsalis

FV flexor ventralis

HL hypochordal longitudinalis

IC infracarinalis

IR interradialis

SC supracarinalis

CAF cellulose acetate film

xxi

1

Introduction

Dream On, Dream On; Dream yourself a dream come true; Dream On, Dream On; Dream

until your dream come true; Dream On, Dream On, Dream On...

Aerosmith (Dream On)

Underwater creatures are capable of high performance movements in water. Thus, underwa-

ter robot design based on the mechanism of fish locomotion appears to be a promising approach.

Over the past few years, researchers have been developing underwater robots inspired by the

swimming mechanism of fish (18), (19), (20), (21), (22), (23). Yet, most of them still rely on

servomotor technology and a structure made of a discrete number of linear elements, exceptions

being the Airacuda by FESTO, which adopts pneumatic actuators, and the MIT fish (24), that

has a continuous soft body, and a single motor produces a wave that is propagated backward

in order to generate propulsion.

In fact, actuation technology in robotics is dominated by two kind of actuators: electric

motors/servomotors and pneumatic/hydraulic. In mobile robotics, the former is mostly used,

with exceptions being e.g. large-legged robots. The (rotatory) motion of the motors is trans-

mitted to the e↵ectors through gearboxes, bearings, belts and other mechanical devices where

linear actuation is needed. Although applied with success in uncountable robotic devices, such

systems can be complex, heavy and bulky1. In underwater robots, propellers are mostly used

for locomotion and maneuvering. However, propellers may have problems of cavitation, noise,

e�ciency, can get tangled with vegetation and other objects, and can be dangerous for sea life.

1Robotuna, a robot fish developed at MIT in 1994, has 2843 parts controlled by six motors (source: MIT

News, http://web.mit.edu/newso�ce/2009/robo-fish-0824.html).

1

Alternative actuation technology in active or ’smart’ materials has opened new horizons as

far as simplicity, weight and dimensions. Materials such as piezoelectric composites, electroac-

tive polymers and shape memory alloys (SMAs) are being investigated as promising alternatives

to standard servomotor technology. The potential gain in weight and dimension would allow

for building lighter, simpler and smaller robots.

In order to create the undulatory body motion of fishes, smart materials appear to be

extremely suited. In fact, over the last few years, there has been increasing activity in this

area. Within the field of underwater locomotion, research about the use of smart materials is

mainly focused on mechatronics design and actuation control. As far as mechatronic design

is concerned, much work is devoted to building hydrofoils using, e.g., piezoelectric composite

(25), (26) or embedding SMA wires into an elastic material such as silicone (27), (28), (29),

(30). SMAs are also used as linear actuators in articulated structures (31), (32), (33). Finally,

emerging materials such as electroactive polymers are adopted in (34), (35), (36), (37).

The SMAs were adopted as actuation technology mainly due to their advantage of working

at low currents and voltages. SMAs are also extremely cheap and easily available commercially.

Due to the relative novelty of smart material technology, the literature, state-of-the-art and the

know-how regarding their use is not consolidated yet. The accurate control of such materials

still remains an important challenge to tackle. In terms of control, excellent results have been

achieved in (38), (39), demonstrating that using the proper control, NiTi SMA wires can be

surprisingly fast in contrast with previous approaches found in the literature, and the general

belief that their response speed is limited because of slow heat transfer characteristics (40), (41)

and the long transient associated with the phase transformation process (42), (43).

Motivated by the potential behind fish swimming and maneuvering and the lack of highly

deformable soft artificial skin, morphing caudal fin and non articulated fish like robot, this

thesis presents a novel fish-like robot inspired by the Largemouth Bass fish: Bioinspired and

Biomimetic Fish-Like Robot (BR3) (cf. Figure 1.1). This thesis is about:

The design and fabrication of the first (i) fish-like robot based on a continuous

bendable structure capable of mimicking any undulatory swimming (ii) morphing

caudal fin, and (iii) highly deformable soft artificial skin. A novel strategy for the

swimming control will allow BR3 to e�ciently swim and maneuver by means of

modulating bend angle on the backbone as well as the caudal fin shape, without

the need of any extra mechanism such as gears, joints, or motors.

2

1.1 The problem and motivations

(a)

(b)

(c)

Figure 1.1: BR3. This robot has the ability to swim frilly or stationary to do PIV tests. For

the first case, the fish carries inside all the electronics devices and the battery. The weight of the

fish out of the water is 2.5kg, length is 45cm, width 8cm. Inside it has 4 SMAs, 4 Current sensor,

4 flex sensors, 1 temperature sensor, a 6-dof IMU and 4 SMA drivers.


The problem

Fish-like locomotion for underwater vehicles has emerged as a direction to enhance the e�ciency

of underwater propulsion. There is growing interest in the energy cost of underwater propul-

sion and learning from nature is the key to optimise e�ciency. However, fishes have extreme

complexity in their swim apparatus and much more complexity in their manoeuvre apparatus

and attempting to mimic part of that complexity using artificial counterparts presents several

and complex challenges. More important, biologist have discovered that evolutionary patterns of

intrinsic caudal musculature in ray-finned fishes show that fine control of the dorsal lobe of the

tail evolved first, followed by the ability to control the ventral lobe. This progression of increasing

di↵erentiation of musculature suggests specialisation of caudal muscle roles. Fine control of fin

elements is probably responsible for the range of fin conformations observed during di↵erent ma-

noeuvring behaviours. Swimming locomotion has been classified into two generic categories on

the basis of the movements’ temporal features: 1) Periodic (or steady or sustained) swimming,

characterised by a cyclic repetition of the propulsive movements. Periodic swimming is em-

ployed by fish to cover relatively large distances at a more or less constant speed. 2) Transient

3


(or unsteady) movements that include rapid starts, escape manoeuvres, and turns. Transient

movements last milliseconds and are typically used for catching prey or avoiding predators.

Natural selection has ensured that the mechanical systems evolved in fish, although not

necessarily optimal, are highly e�cient with regard to the habitat and mode of life for each

species. Their often remarkable abilities could inspire innovative designs to improve the ways

that man-made systems operate in and interact with the aquatic environment. An example

application that could substantially benefit are autonomous underwater vehicles (AUV’s). As

research and use of AUV’s are expanding, there is increased demand for improved e�ciency to

allow for longer missions to be undertaken. The highly e�cient swimming mechanisms of some

pelagic fish can potentially provide inspiration for a design of propulsors that will outperform

the thrusters currently in use. For maneuvering or hovering purposes, the existing systems are

insu�cient when it comes to demanding applications, such as dextrous manipulation, and coarse

compared to the abilities of fish. The advantages of noiseless propulsion and a less conspicuous

wake could be of additional significance, particularly for military applications. Robotic devices

are currently being developed to assess the benefits and study the ways of ”porting” mechanisms

utilized by fish and other aquatic animals to artificial systems (for examples, see (44), (45),

(46), (47), (48), (49), (50), (51), (52), (53), (54), (55), (56), (57)). Under this perspective,

engineers working in this area should have a background knowledge of the swimming abilities

and performance of fish that provide benchmarks for evaluating our own designs and drive

further theoretical developments. Biologists have shown a much renewed interest in the area

over the last five years, owing largely to the advent of improved experimental techniques that

have shed new light on a number of the fish swimming mechanisms.

The main problem to tackle in this thesis is how to create thrust and

manoeuvring only by bending a structure. This by developing a novel fish-like

robot prototype with unprecedent morphing caudal fin and flexible

continuous-becakbone inspired by the ray-finned fishes.

Solving this problem can help further the continued development in soft robotics and un-

derwater vehicles that moves e�ciently with low energy consumption and with high manoeu-

vrability ability.

4


The hypothesis

Fish swim bending their body, producing a backward-propagating propulsive wave. This in-

volves the transfer of momentum from the fish to the surrounding water (and vice versa). The

main momentum transfer mechanisms are via drag, lift, and acceleration reaction forces. Based

on this biological fact the following question is formulated:

Could an underwater vehicle inspired by the biomechanics of ray-finned fishes take advantage

of a bendable body to produce trust while using a morphing-caudal fin to manoeuvre?

To this purpose, the following hypothesis is proposed:

Understanding the e↵ect of the bend angle and frequency in terms of steady

swimming (acting the backbone) and manoeuvring (acting the caudal fin) and

therefore including bend angle and frequency information into the swimming

controller will allow for proper modulation of the backbone and caudal fin

kinematics that finally would produce and increase net forces, thereby improving

on swimming e�ciency.

Motivations: learning from ray-finned fishes

Forces Acting on a Swimming Fish

The main properties of water as a locomotion medium that have played an important role in the

evolution of fish are its incompressibility and its high density. Since water is an incompressible

fluid, any movement executed by an aquatic animal will set the water surrounding it in motion

and vice versa. Its density (about 800 times that of air) is su�ciently close to that of the

body of marine animals to nearly counterbalance the force of gravity. This has allowed the

development of a great variety of swimming propulsors, as weight support is not of primary

importance (58).

To aid in the description of the fish swimming mechanisms, Fig. 1.2 illustrates the terminol-

ogy used to identify morphological features of fish, as it is most commonly found in literature

and used throughout this text. Median and paired fins can also be characterized as either

short-based or long- based, depending on the length of their fin base relative to the overall fish

5


dorsal fin

fin base

main axis

caudal fin (tail)

caudal peduncle

anal finpelvic finspectoral fins

paired

spiny dorsal fin

median

Figure 1.2: Terminology used in the text to identify the fins and other features of fish.

length. The fin dimensions normal and parallel to the water flow are called span and chord,

respectively. Terminology used in the text to identify the fins and other features of fish.

Swimming involves the transfer of momentum from the fish to the surrounding water (and

vice versa). The main momentum transfer mechanisms are via drag, lift, and acceleration

reaction forces. Swimming drag consists of the following components:

1. Skin friction between the fish and the boundary layer of water (viscous or friction drag):

Friction drag arises as a result of the viscosity of water in areas of flow with large velocity

gradients. Friction drag depends on the wetted area and swimming speed of the fish, as

well as the nature of the boundary layer flow.

2. Pressures formed in pushing water aside for the fish to pass (form drag). Form drag is

caused by the distortion of flow around solid bodies and depends on their shape. Most of

the fast-cruising fish have well streamlined bodies to significantly reduce form drag.

3. Energy lost in the vortices formed by the caudal and pectoral fins as they generate lift or

thrust (vortex or induced drag): Induced drag depends largely on the shape of these fins.

The latter two components are jointly described as pressure drag. Comprehensive overviews

of swimming drag (including calculations for the relative importance of individual drag com-

ponents) and the adaptations that fish have developed to minimize it can be found in (59) and

(60).

6


Like pressure drag, lift forces originate from water viscosity and are caused by assymetries

in the flow. As fluid moves past an object, the pattern of flow may be such that the pressure

on one lateral side is greater than that on the opposite. Lift is then exerted on the object in a

direction perpendicular to the flow direction.

Acceleration reaction is an inertial force, generated by the resistance of the water surround-

ing a body or an appendage when the velocity of the latter relative to the water is changing.

Di↵erent formulas are used to estimate acceleration reaction depending on whether the water

is accelerating and the object is stationary, or whether the reverse is true (61). Acceleration

reaction is more sensitive to size than is lift or drag velocity and is especially important during

periods of unsteady flow and for time-dependent movements (62), (63).

thrust drag

weight

boyancy plus hydrodynamic lift

(a)

rollpitch

yaw

(b)

Figure 1.3: (a) The forces acting on a swimming fish. (b) Pitch, yaw, and roll definitions.

The forces acting on a swimming fish are weight, buoyancy, and hydrodynamic lift in the

vertical direction, along with thrust and resistance in the horizontal direction [Fig. 1.3(a)].

For negatively buoyant fish, hydrodynamic lift must be generated to supplement buoyancy

and balance the vertical forces, ensuring that they do not sink. Many fish achieve this by con-

tinually swimming with their pectoral fins extended. However, since induced drag is generated

as a side e↵ect of this technique, the balance between horizontal forces will be disturbed, calling

7


10210 103 104 105 106 Reynolds

Lift Pressure dragAc

celeration

V iscous

drag

number

reaction

Figure 1.4: Diagram showing the relative contribution of the momentum transfer mechanisms

for swimming vertebrates, as a function of Re. The shaded area corresponds to the range of adult

fish swimming.

for further adjustments for the fish to maintain a steady swimming speed. For a discussion on

this coupling of the forces acting on a swimming fish, see (59). The hydrodynamic stability and

direction of movement are often considered in terms of pitch, roll, and yaw [Fig. 1.3(b)]. The

swimming speed of fish is often measured in body lengths per second (BL/s).

For a fish propelling itself at a constant speed, the momentum conservation principle requires

that the forces and moments acting on it are balanced. Therefore, the total thrust it exerts

against the water has to equal the total resistance it encounters moving forward. Pressure

drag, lift, and acceleration reaction can all contribute to both thrust and resistance. However,

since lift generation is associated with the intentional movement of propulsors by fish, it only

contributes to resistance for actions such as braking and stabilization rather then for steady

swimming. Additionally, viscous drag always contributes to resistance forces. Finally, body

inertia, although not a momentum transfer mechanism, contributes to the water resistance as

it opposes acceleration from rest and tends to maintain motion once begun. The main factors

determining the relative contributions of the momentum transfer mechanisms to thrust and

resistance are: 1) Reynolds number; 2) reduced frequency; and 3) shape (63).

The Reynolds number (Re) is the ratio of inertial over viscous forces, defined as:

Re =LU

v(1.1)

where L is a characteristic length (of either the fish body or the propulsor), U is the swim-

ming velocity, and u is the kinematic viscosity of water. In the realm of Re typical of adult

fish swimming (i.e., 103 < Re < 5 · 106), inertial forces are dominant and viscous forces are

usually neglected. At those Re, acceleration reaction, pressure drag, and lift mechanisms can

all generate e↵ective forces (Fig. 1.4).

8


The reduced frequency � indicates the importance of unsteady (time-dependent) e↵ects in

the flow and is defined as:

� = 2⇡fL

U(1.2)

where f is the oscillation frequency, L is the characteristic length, and U is the swimming

velocity. The reduced frequency essentially compares the time taken for a particle of water to

traverse the length of an object with the time taken to complete one movement cycle. It is used

as a measure of the relative importance of acceleration reaction to pressure drag and lift forces.

For � < 0.1, the movements considered are reasonably steady and acceleration reaction forces

have little e↵ect. For 0.1 < � < 0.4, all three mechanisms of force generation are important,

while for larger values of � acceleration reaction dominates. In practice, for the great majority

of swimming propulsors, the reduced frequency rarely falls below the 0.1 threshold (63).

Finally, the shape of the swimming fish and the specific propulsor utilized largely a↵ect the

magnitude of the force components. The relationship is well documented for steadystate lift

and drag forces, but relatively little work has been done on the connection between shape and

acceleration reaction.

A common measure of swimming e�ciency is Froude e�ciency ⌘, defined as:

⌘ =hT iUhP i (1.3)

where U is the mean forward velocity of the fish, hT i is the time-averaged thrust produced,

and hP i is the time-averaged power required.

Main Classifications

Fish exhibit a large variety of movements that can be characterized as swimming or nonswim-

ming. The latter include specialized actions such as jumping, burrowing, flying, and gliding, as

well as jet propulsion. Swimming locomotion has been classified into two generic categories on

the basis of the movements’ temporal features (57):

1. Periodic (or steady or sustained) swimming, characterized by a cyclic repetition of the

propulsive movements. Periodic swimming is employed by fish to cover relatively large

distances at a more or less constant speed.

9


2. Transient (or unsteady) movements that include rapid starts, escape maneuvers, and

turns. Transient movements last milliseconds and are typically used for catching prey or

avoiding predators.

Periodic swimming has traditionally been the center of scientific attention among biolo-

gists and mathematicians. This has mainly been because, compared to sustained swimming,

experimental measurements of transient movements are di�cult to set up, repeat, and verify.

Therefore, periodic swimming and transient movements will inevitably be the main focus of this

thesis1. However, given the significant aspects of locomotion associated with transient move-

ments, which provide fish with unique abilities in the aquatic environment and the more recent

interest among scientists in describing them, reference will also be made to transient propulsion

where possible.

The classification of swimming movements presented here adopts the (expanded) nomencla-

ture originally put forth by Breder in (64) and Gray in (65). Breder’s nomenclature has recently

been criticized as oversimplified and ill-defined (see, for example, (66) and (67)) in describing

fish swimming. Nevertheless, since this thesis is mainly concerned with descriptions of the fish

propulsors, on which Breder’s classification is based, it serves as a convenient reference frame,

provided its limitations are held in mind. The interested reader is referred to (66), where a

more holistic classification scheme of swimming is proposed, relating the swimming propulsors,

kinematics, locomotor behavior, and muscle fiber used to the notion of swimming gaits.

Most fish generate thrust by bending their bodies into a backward-moving propulsive wave

that extends to its caudal fin, a type of swimming classified under body and/or caudal fin (BCF)

locomotion. Other fish have developed alternative swimming mechanisms that involve the use of

their median and pectoral fins, termed median and/or paired fin (MPF) locomotion. Although

the term paired refers to both the pectoral and the pelvic fins 1.5, the latter (despite providing

versatility for stabilization and steering purposes) rarely contribute to forward propulsion and

no particular locomotion mode is associated with them in the classifications found in literature.

An estimated 15% of the fish families use non-BCF modes as their routine propulsive means,

while a much greater number that typically rely on BCF modes for propulsion employ MPF

modes for maneuvering and stabilization (67).

A further distinction, and one that is common in literature, made for both BCF and MPF

propulsion is on the basis of the movement characteristics: undulatory motions involve the

1Part of this research has been done with the collaboration of the Curet Lab at Florida Atlantic University.

10


Figure 1.5: Diagram showing the relation between swimming propulsors and swimming functions

passage of a wave along the propulsive structure, while in oscillatory motions the propulsive

structure swivels on its base without exhibiting a wave formation. The two types of motion

should be considered a continuum, since oscillatory movements can eventually be derived from

the gradual increase of the undulation wavelength. Furthermore, both types of motion result

from the coupled oscillations of smaller elements that constitute the propulsor (i.e., muscle

segments and fin rays for BCF and MPF propulsion, respectively).

Generally, fish that routinely use the same propulsion method display similar morphology.

However, form di↵erences do exist and these relate to the specific mode of life of each species.

Webb (68) identified three basic optimum designs for fish morphology, derived from specializa-

tions for accelerating, cruising, and maneuvering. It should be pointed out that they are closely

linked to the locomotion method employed (Fig. 1.5). Also, since they are largely mutually

exclusive, no single fish exhibits an optimal performance in all three functions. But neither are

all fish specialists in a single activity; they are rather locomotor generalists combining design

elements from all three specialists in a varying degree. Further details on the relation between

function and morphology in fish swimming can be found in (67) and (68).

Within the basic grouping into MPF and BCF propulsion, further types of swimming (often

referred to as modes) can be identified for each group, based on Breder’s (64) original classifica-

tion and using his nomenclature (Fig. 1.6). These modes should be thought of as pronounced

points within a continuum, rather than discrete sets. Fish may exhibit more than one swim-

ming mode, either at the same time or at di↵erent speeds. Median and paired fins are routinely

11

1.2 Objectives

Figure 1.6: Swimming modes associated with (a) BCF propulsion and (b) MPF propulsion.

Shaded areas contribute to thrust generation.

used in conjunction to provide thrust with varying contributions from each, achieving smooth

trajectories. Also, many fish typically utilize MPF modes for foraging, as these o↵er greater

maneuverability, the ability to switch to BCF modes at higher speeds, and high acceleration

rates.

1.2 Objectives

This thesis presents a novel fish-like Autonomous Underwater Vehicles (AUV’s) BR3 with

actuated morphing caudal fin that can be e�ciently modulated by a novel swim controller that

uses bend angle and frequency information to that purpose. The goal, to improve on swim and

manoeuvre performance in terms of yaw-roll/thrust production.

In brief, this thesis provides both theoretical and experimental foundations for designing

fish-like robots AUVs aimed at enhancing swim performance via proper body and caudal fin

modulation.

The specific objectives of this research are:

1. To analyze and select which fish-specie would be suitable to be mimicked by an artificial

counterpart.

2. To study the mechanistic basis of fish swim. Based on published biological data that

unveils key aspects of fish morphology, physiology and hydrodynamic performance, to

define a biological-based framework useful for designing a bio-inspired fish-like robot.

12

1.3 Methods

3. To formulate mathematical models for: i) body and caudal fin kinematics, ii) dynamics

(inertial contribution), iii) hydrodynamics (lift and yaw-roll production), and iv) SMA

muscle-like actuation.

4. To validate mathematical models against experimental data.

5. To design and fabricate BR3 using the proposed biological-based framework.

6. To formulate a morphing-caudal(fin)-body controller for the proper regulation of SMA

actuators.

7. To formulate an attitude controller for the proper morphing-caudal fin modulation that

produces forward and turning swim.

8. To analyze and discuss the performance of BR3 in terms of: i) accurate and fast SMA

actuation of morphing-caudal fin, ii) inertial contribution on thrust production, iii) bend-

to-yaw/roll ratio, and iv) power consumption.

9. To demonstrate BR3 would be capable of forward and turning swim via water-channel

testing without the need of external appendices such as rudders, ailerons, propellers, etc.

10. To discuss about the potential of the proposed methodologies towards real swim.

1.3 Methods

This subsection briefly summarizes the methods used for the development of BR3. The follow-

ing procedures will be approached aimed at achieving the main goals of this thesis. Figure1.7

graphically details these procedures which will be all cover within each chapters of this docu-

ment.

1. Biological study of fish swimming:

It presents a detailed study of the most relevant issues that describe fish swimming modes:

i) bio-mechanics, ii) morphology, iii) physiology, iv) muscle actuation, v) kinematics, and

vi) hydrodynamics performance. This study has been based on the most specialized

biological literature review from (69), (70), (66), (67), (68).

2. Bio-inspired design criteria:

It quantifies key design criteria based on the studied biological data analyzed in the

previous procedure. It summarizes these criteria into three fields: i) morphology, ii)

kinematics and iii) hydrodynamics. Most relevant morphological parameters are: caudal

fin area, body length and body mass. Kinematics parameters are: flapping caudal fin

13

1.3 Methods

1. Biological study of fish swim 2. Bio-inspired design criteria

3. Methods: modelling + simulation

SMA actuation

Swim Dynamics

CAD Model

4. BR3 fabrication 5. Control (measurements)

!"#$%!&'$"(

)#*+,&-"

!"#$%&'("))$*

'+)#%+*&',%-.,#&/+"%0

/12

!"#"$"%&"'()*+,'

6. Experimental results

Figure 1.7: Structural steps to be followed during the thesis aimed at the development of BR3.

The pictures depicted herein, correspond to the final BR3 prototype. The forthcoming chapters

will introduce each step with all the details. Source: The author.

frequency and body-stroke trajectories. Hydrodynamics parameters are: lift and drag

forces and body phase angle.

3. Modeling:

It defines morphology and kinematics frameworks of BR3. Fish kinematics are formulated

using modified Denavit-Hartenberg (DH) convention frames (71), whereas body kinemat-

ics is designated by roll and yaw motions with respect to the body-frame. Basic rotation

14

1.3 Methods

matrix are used to express how kinematics variables are propagated from the body to the

tail. This allows for the formulation of an integrated inertial model that mainly consists

on: i) Newton-Euler dynamics equations of motion expressed by spatial algebra notation

(72), and ii) SMA thermo-mechanical actuation based on existing phenomenological mod-

els that describe the shape memory e↵ect (73). Here, SMA performance is quantified in

order to assess the limits of this actuation technology. Also, the influence of body inertia

on robot’s maneuverability is analyzed using the inertial model.

4. Design-Fabrication:

It approaches the design/fabrication problem. It shows a detailed description for the

bio-inspired development of: i) body and caudal fin skeleton, ii) body and caudal fin skin

membrane, iii) actuation mechanisms, and iv) hardware components.

5. Control (measurements):

It tackles the control problem. Two control layers are developed: i)body bend controller,

i) morphing-caudal fin controller and ii) attitude controller. The former regulates the

amount of input heating power to be delivered to the SMA muscles. SMAs actuate to

change the shape of the body and caudal fin (contraction/extension). The latter drives

the former. It regulates the attitude motion of the robot (roll and yaw) by means of

proper caudal fin modulation.

The novelty of the attitude control strategy is due to the incorporation of bend and

frequency information within the control strategy. The idea behind this approach is

aimed at improving the attitude response of the fish-like AUV. The proposed controller

is called ADEX (Adaptive Predictive Expert Control). Such enhancement is based on

the assumption (motivated by the cited biological studies) that fish e�ciently generate

forward thrust by means of bend body modulation, taking advantage of relevant bend-

to-frequency ratio.

6. Experimental results:

It concludes with experiments aimed at:

• assessing the performance of the SMAmuscles driven by the body-bend and morphing-

caudal(fin) control. Performance will be quantified in terms of actuation speed,

output torque and fatigue,

• evaluating the accuracy of the bend controller for tracking swimming references under

the presence of external disturbances caused by hydrodynamics loads,

15

1.4 Original Contributions of this Work

• evaluating the accuracy of the attitude controller for tracking pitch and yaw refer-

ences under the presence of external disturbances caused by hydrodynamics loads,

and

• showing the potential of the proposed methodologies toward achieving the first fish-

like AUV capable of autonomous high e�ciency and maneuverable swimming.

1.4 Original Contributions of this Work

The original contributions of this work cover four areas:

Bio-inspired design and modeling

I BR3 is the first Autonomous Underwater Vehicle composed by only one continuous and

flexible backbone actuated only by Shape Memory Alloys. No other platforms in the

literature have a similar structure without joints or gears in their body-system.

II BR3 is the first real-sized fish-like robot capable of swim in any swimming mode depending

of the need and the circumstances.

III The artificial skin membrane is the first bio-inspired in the literature which allows high

flexibility in the movements of the fish while prevents water seepage.

IV BR3 incorpores the first highly morphing caudal fin capable of passive and active move-

ments.

V The design process of BR3 has been entirely conceived based on a comprehensive analysis

of biological data. The data from in-vivo experiments reported in the specialized literature

allowed for the definition of a bio-inspired design framework which defines every aspect

related to morphology of body and caudal fin, and bio-mimetic behaviour which defines

every aspect related to kinematics and hydrodynamics.

VI BR3 is the first fish-like robot to incorporates a flexible body and a morphing caudal fin

at same time in the same robot.

VII BR3 is the first bio-inspired Autonomous Underwater Vehicle capable of manoeuvring just

by using only a morphing caudal fin. Most of the concepts found in the literature make

use of extra mechanical parts that are based on oar or rigid blades, such as: ailerons,

propellers, joint, gears, etc.

16

1.5 Thesis outline

VIII An bend model aimed at studying the influence of body bending on the production of net

forces for maneuvering and steady swimming.

SMA actuation and power

IX Identified linear models for a NiTi SMA actuator relating output torque with input power

and temperature.

X Quantification of SMA limitations in terms of fatigue and actuation speed. It defines

the trade-o↵ between input power, output torque, and actuation speed. This trade-o↵ is

essential for the designing process of SMA muscle-like actuation mechanisms in similar

applications.

XI Accurate and faster position control of the SMAs (up to 2.5Hz in actuation speed) thanks

to re-adapted anti-overload and anti-slack mechanisms from (74). Normal rates of SMA

actuation speed range between 1 � 2Hz. It also uses SMAs as sensors, saving on weight

and energy.

Attitude control

XII An enhanced ©Adaptive Predictive Expert control law denoted as ADEX. This improves

on attitude tracking and increments the production of thrust.

1.5 Thesis outline

Each chapter of this thesis beings with a General Overview about the problems to be addressed

and a brief description of the methods to be introduced. Thereby the end of each chapter con-

cludes with brief remarks about the topics presented. This document is organized as follows:

Chapter 2 is about a Literature Review. State-of-the-art research is introduced from

specialized literature covering areas such as: i) Shape Memory Alloys as an alternative for

actuation, and ii) Current SMA-Based Robotic platforms.

Chapter 3 is about biological inspiration. Key parameters of biomechanics basis of fish

swimming are studied and unified into a bio-inspired framework for robot design. Relevant

biological data is also highlighted aimed at the proper formulation of robot’s models.

17

1.5 Thesis outline

Chapter 4 is about Modeling. Based on biological data, BR3’s morphology, kinematics,

dynamics, identified hydrodynamics, and SMA actuation are defined and modeled using math-

ematical frameworks. Basic maneuvers are defined by showing the influence of caudal fin and

body modulation on robot’s maneuverability and swimming.

Chapter 5 is about Design. Here, the design process and fabrication of BR3’s components

are introduced. It shows novel approaches for bio-inspired design and for the development of

low-mass high-power circuits. Here is presented as well the bio-inpired design of the artificial

skin membrane and the caudal fin.

Chapter 6 is about Free Vibration Analysis. This is exposed the Free Vibration Analysis

of a Robotic Fish based on a Continuous and Non-uniform Flexible Backbone with Distributed

Masses. I present a Di↵erential Quadrature Element Method for free transverse vibration of a

robotic-fish based on a continuous and non-uniform flexible backbone with distributed masses

(represented by ribs) based on the theory of a Timoshenko cantilever beam, To help improving

the energy e�ciency helped by the resonance frequency of the fish.

Chapter 7 is about Control. It presents novel control techniques to: i) approach faster

SMA morphing-caudal(fin)-body modulation and ii) enhance attitude regulation that allows for

more e�cient swimming control.

Chapter 8 presents the experimental tests carried out. Experiments are conducted to

demonstrate: i) morphing-caudal(fin) and bending backbone control accuracy and speed, iii)

hydrodynamics performance, and iv) overall swimming control.

Chapter 9 concludes the thesis with important remarks on the obtained results. Conclu-

sions are focused on the ares of: i) bio-inspired fish design, ii) SMA as muscle-like actuators,

iii) BR3’s overall control, and iv) General performance of the platform.

18

2

Literature Review

Detras de todos estos anos, detras del miedo y el dolor vivimos anorando algo, algo que nunca

mas volvio. Detras de los que no se fueron, detras de los que ya no estan hay una foto de

familia donde lloramos al final.

Carlos Varela (Foto de familia)

2.1 General Overview

New actuation technology in functional or ”smart” materials has opened new horizons in

robotics actuation systems. Materials such as piezo-electric fiber composites, electro-active

polymers and shape memory alloys (SMA) are being investigated as promising alternatives to

standard servomotor technology (75). This paper focuses on the use of SMAs for building

muscle-like actuators. SMAs are extremely cheap, easily available commercially and have the

advantage of working at low voltages.

The use of SMA provides a very interesting alternative to the mechanisms used by conven-

tional actuators. SMAs allow to drastically reduce the size, weight and complexity of robotic

systems. In fact, their large force-weight ratio, large life cycles, negligible volume, sensing ca-

pability and noise-free operation make possible the use of this technology for building a new

class of actuation devices. Nonetheless, high power consumption and low bandwidth limit this

technology for certain kind of applications. This presents a challenge that must be addressed

from both materials and control perspectives in order to overcome these drawbacks. Here, the

latter is tackled. It has been demonstrated that suitable control strategies and proper mechani-

cal arrangements can dramatically improve on SMA performance, mostly in terms of actuation

speed and limit cycles.

19

2.1 General Overview

Due to their limitations, SMAs have not raised the attention of the robotics technology for

several years. However, recent studies have demonstrated that by (i) finding suitable niches of

application, (ii) dedicated mechatronics design, and (iii) ad-hoc control strategies, SMAs can

e↵ectively be used as an alternative actuation technology in a wide spectrum of applications

and robotic systems. Indeed, as it will be introduced in this chapter, careful control design that

takes into account the particular characteristics of the material coupled with proper mechanic

design, play a significant role for an e�cient use of SMAs. Even so, it is clear that SMAs

(and smart materials in general) cannot, at this stage, be thought as a universal substitute

for classical servomotor technology. However, niches of applications can be found that greatly

benefit from this technology. Bio-inspired artificial systems are one such niche.

Although SMAs are mostly used as actuators, they also have sensing capabilities. Despite

most of the SMA physical parameters are strongly related in a nonlinear hysteresis fashion,

the electrical resistance varies linearly with the strain of the alloy. Because strain is kinemat-

ically related to the motion of the actuator (either linear motion or rotational), the electrical

resistance and the motion produced by the actuator are both linearly related. This linear rela-

tionship between resistance variation and motion is achieved because the martensite fraction is

kinematically coupled to the motion, and the martensite fraction is what drives the resistance

changes. This issue is an advantage for developing closed-loop position controllers that regulate

the SMA actuation. In fact, most of the applications involving position linear control of SMAs,

feedback electrical resistance measurements to estimate the motion generated by the actuator.

This avoids the inclusion of external position sensors for closing the control loop.

SMAs are used in a variety of applications (76),(77),(78),(79),(80),(81). Their special prop-

erties have aroused great expectations in various technologies and industries; it can be used

to generate a movement or storing energy. In addition, its scope covers many sectors ranging

from the use in deployable satellite antennas for di↵erent sensors to machinery, to materials

for the construction of suspension bridges or anti-seismic devices. In general, all applications

somehow depend on the e↵ect of action-reaction of the material and the conditions under which

particular application takes place, which make the SMAs a functional material.

For instance, they are being used in many non-invasive surgery devices (82),(83),(84),(85),(86)

and biomedicine, taking advantage of their large strains and their capability to recover the shape

when the load is removed. This property allows applications in devices such as stents, tubu-

lar prosthetic devices, because it restores the ability of flow of any bodily duct a↵ected by a

narrowing.

20

2.2 Shape memory alloys background

Figure 2.1: Microscopic viewpoint of the Shape Memory E↵ect

In classical robotic systems, linear actuation systems have been proposed using SMAs. The

focus of this chapter is on bio-inspired robotics. SMA-based actuators provide a suitable technol-

ogy as muscle-like actuation mechanisms, which resemble the mechanics of muscles in biological

systems. For this reason in the last years a number of bio-inspired robots have been designed

adopting SMA technology. Next section, review the main prototypes, organizing them accord-

ing to the mean (water, air, ground), and on main morphological characteristics (full body

actuation or appendices only).


2.2.1 Principle of operation

Shape Memory Alloys are metallic materials with the ability to ”remember” a determined shape,

even after a severe deformation produced by a thermal stimulus. In the case of metallic alloys,

the shape memory e↵ect consists on a transition that occurs between two solid phases, one of

low temperature or martensitic and other of high temperature or austenitic. The material is

deformed in the martensitic phase and retrieves, reversibly, its original dimensions by heating

above a critical transition temperature. The terms martensite and austenite originally referred

only to the steel phases, however these terms have been extended referring not only to the

material but also the kind of transformation. Thereby, the martensite steel involves a change

of volume and shape, while the SMA has basically a change of length.

In general, NiTi (Nickel-Titanium) SMAs are the most common alloys used. This is basically

because these materials are intrinsically susceptible of use both as sensors and actuators, which

makes them suitable for integration in smart structures. NiTi SMAs work based on the shape

21


Diameter Size

inches(mm)Resistance

Pull Force

pounds

(grams)

Approximate

Current for 1

Second

Contraction

(mA)

Cooling Time

158�F , 70�C

”LT” Wire

(seconds)

Cooling Time

194�F , 90�C

”HT” Wire

(seconds)

0.001 (0.025) 36.2 (1425) 0.02 (8.9) 45 0.18 0.15

0.0015 (0.038) 22.6 (890) 0.04 (20) 55 0.24 0.2

0.002 (0.050) 12.7 (500) 0.08 (36) 85 0.4 0.3

0.003 (0.076) 5.9 (232) 0.18 (80) 150 0.8 0.7

0.004 (0.10) 3.2 (126) 0.31 (143) 200 1.1 0.9

0.005 (0.13) 1.9 (75) 0.49 (223) 320 1.6 1.4

0.006 (0.15) 1.4 (55) 0.71 (321) 410 2 1.7

0.008 (0.20) 0.74 (29) 1.26 (570) 660 3.2 2.7

0.010 (0.25) 0.47 (18.5) 1.96 (891) 1050 5.4 4.5

0.012 (0.31) 0.31 (12.2) 2.83 (1280) 1500 8.1 6.8

0.015 (0.38) 0.21 (8.3) 4.42 (2250) 2250 10.5 8.8

0.020 (0.51) 0.11 (4.3) 7.85 (3560) 4000 16.8 14

Table 2.1: Characteristics of NiTinol®SMA wires (16).

memory e↵ect, which essentially takes place by the influence of temperature change of the

material; i.e. the temperatures at which the martensitic and austenite phase transformations

begin and end. Figure 2.1 depicts how these changes occur at the microscopic level of the

material. The phase transition occurs when the material is heated or cooled. In general, there

is a certain temperature range for the transition, which is mainly defined by the manufacturer.

SMAs normally exhibit one-way shape memory e↵ect, also called memory e↵ect in a simple

manner. The alloy deforms upon heating but cooling does not change the shape unless it is

stressed again. The percentage of deformation of NiTi alloys (% of strain) is about five percent,

a range considerably higher if one considers that the deformation of common steel allows only

an average of two percent. Currently, SMAs that exhibit two-way shape memory e↵ect are

also manufactured. In this case, the alloy expands by heating above the range of transition

temperature and spontaneously contract when cooled again below this temperature (87). To

produce the double shape memory e↵ect, the material is subjected to heat treatment, also called

training. This training-phase forces the material to remember both heating and cooling states.

From the microscopic viewpoint (Figure 2.1), all the physical properties of the alloy vary

depending on the phase, i.e. from cooling to heating and vice versa. Some of these properties

refer to corrosion resistance, elasticity, damping capacity, strain, stress, electrical resistance,

22


and temperature. Therefore, shape memory alloys behave in a thermo-mechanical way, with

all these variables strongly coupled within a nonlinear hysteresis fashion.

Table 2.1 shows the commercial characteristics of SMAs depending on the diameter of the

wires (NiTiNol®). From the table it can be noticed their high electrical power consumption.

In robotics applications, power consumption is a critical issue due to the level autonomy of the

robotic system is fully dependent on the capacity of the onboard batteries.

2.2.2 Improving the performance of SMA actuators

One of the main limitations in SMA actuation speed is due to high latency that the the cooling

time of the wire implies. Despite increasing the input heating power can reduce the heating

time, large cooling times limit the operation frequency of the actuator. On average, NiTi wires

with a diameter of 127µm typically requires an electrical current input about 320mA to contract

in about 1s (nominal heating time) and relax in approximately 1.4s (nominal cooling time). In

this case both contraction and recovery times would set a nominal actuation frequency about

0.416Hz, quite slow for many applications requirements.

Research to overcome this limitation has been oriented towards developing cooling systems

for SMAs, aimed at decreasing the nominal cooling time involved during the recovery process.

In this direction, temperature control methods have been proposed in (88). Cooling systems

based on Peltier cells (89) or active cooling (90), have been commonly used. However, nowa-

days bio-inspired robotic systems tend to be small and light, therefore other methodologies for

enhancing SMA actuation speed must be addressed. For several years di↵erent strategies have

been proposed to implement rapid control in the SMA wires (91),(92),(93),(94),(95),(96),(97).

A system consisting of rapid heating of the SMA was proposed by (91) aimed at increasing

the overall actuation frequency by means of overloading the operation of SMAs. The term

overloading refers to increasing the amount of input heating power to be delivered to the SMA

wires. In (98) experiments carried out using a two degree-of-freedom Pantograph robot ac-

tuated by an antagonistic pair of SMA wires acting as linear actuators have shown how the

nominal actuation frequency was increased from 0.416Hz to 1Hz.

Overloading should be monitored in order to avoid overheating problems that may cause

physical damage of the shape memory e↵ect. In (96), further research in this direction allowed

for the introduction of a force control architecture with the proper mechanisms for safe overload

the operation of SMA actuators. In the prototypes described in Section 2.4, was used a control

architecture similar to the one described in (96), which makes use of proper mechanisms to

23


overload the operation of SMAs. However, these mechanisms have been adapted to work within

a position control scheme, avoiding the need of including external force sensors. Section 2.4.2

will detail on this issue.

Besides rapid heating techniques to overload SMA operation, further investigations have

been also carried out to verify whether SMAs can respond to high frequencies. In (99) and (100)

experiments have demonstrated that NiTi SMA wires with a diameter of 0.1mm can respond

up to frequencies of 2KHz. This high-frequency response corresponds to small-signal heating

currents inputs with frequencies of that magnitude. These results allow for the development

of small-signal high-bandwidth controllers capable of improving SMA performance, but more

important, eliminating the limit cycles of operation of SMAs. In other approaches, 20� 30Hz

limit cycles have been observed, whereas in (92), (101) at approximately 100� 200Hz. In this

regard, the use of high-bandwidth force sensors might be suitable for developing a SMA force

feedback control system.

2.2.3 Modeling and control

The physical behavior of SMAs is more complex than many common materials: the stress-strain

relationship is nonlinear, hysteresis is presented, large reversible strains are exhibited, and it is

temperature dependent. This thermo-mechanical relationship can be described by formulating

phenomenological models. Tanaka in (102) was one of the pioneers to study a stress-induced

martensite phase transformation, proposing an unified one-dimensional phenomenological model

that makes use of three state variables to describe this process: temperature, strain, and marten-

site fraction. His main contribution was to demonstrate that the rate of stress is a function of

strain, temperature and martensite fraction rates. Later, Elahinia (103), (104) proposed an en-

hanced phenomenological model compared to other works (77), (105), (102) and also addressed

the nonlinear control problem. This model was able to better describe the behavior of SMAs

in cases where the temperature and stress states changed simultaneously. Their model was

verified against experimental data regarding a SMA-actuated robotic arm (106).

Phenomenological models may provide some insights of SMA thermo-mechanical behavior

that facilitate the development of control procedures. To control purposes, parameters’ tuning

is highly dependent of a modeling stage, but definitively phenomenological models are not the

best choice for control design, especially if the goal is related to improving actuation speed.

In this direction system identification is a promising alternative. As noted by (77), (92), (95),

24

2.3 Bio-inspired robots with SMA muscle-like actuation

(96), (107), identified linear models for SMA can be developed. It has been demonstrated that

the AC response of NiTi SMA wires behave as a first order low-pass filter.

Section 2.4 of this paper details two di↵erent approaches for modeling SMAs; one based

on identifying how electrical resistance change as a function of the input current (8), and the

other based on identifying how the output torque produced by an antagonistic pair of SMA

actuators change as as a function of the applied power (9). Furthermore, (9) details how to take

advantage of phenomenological models for simulating overheating problems when a SMA wires

are overloaded. Attempting to perform this analysis on the real SMA actuators might cause

several damage to the structure. Phenomenological models are really useful for determining the

upper limits of applied input heating currents.

The control methods presented in (8) and (9) have been conceived for controlling a pair of

antagonist SMA actuators. The antagonistic configuration is useful for having SMA actuators

where each direction of motion can be controlled. In (8), the antagonistic pair of actuators

must bend the structure of the fish robot, whereas in (9) the antagonistic SMA actuators are

connected to a joint for providing the rotational motion. Other approaches in (95), (108),

(88), (109) have demonstrated the advantages of using an antagonistic arrangement in terms

of controllability. When the active actuator is being heated while the passive (antagonistic) is

cooling, hysteresis e↵ects are reduced due to the external stress that the active actuator applies

on the inactive one above the austenite finish temperature.


The use of SMAs as artificial muscles allows for more realistic bio-inspired actuation pre-

sented in nature (110). SMA wires acting as muscle fibers can respond upon electrical sig-

nals, taking advantage of the large pull force and its excellent strength-weight tradeo↵. Cur-

rently, the use of SMAs in biomimetic robotic systems (111),(112),(113) can be found in

ground, water and air robots, in many sizes, including those micro-robots or microstructures

(114),(115),(116),(117),(1),(118),(119),(119). The following sections, describes the most rep-

resentative bio-inspired robots and structures that integrate SMAs as muscle-like actuation

mechanisms.

25


2.3.1 Water

Biologically inspired robots that operate in water can be found in two categories. Firstly, robots

that use SMAs for actuating appendices (fins), and secondly, robots that use SMAs to actuate

the robot’s body. In the latter body actuation is used for undulatory motion (fish-like robots).

Some animals can move by bending their body in such a way to produce a backward-propagating

propulsive wave. The movement obtained by bending a continuous structure is much more

natural than others where joints are presented. In Section 2.4.2 our bending structure prototype

is presented.

2.3.1.1 A micro-robot fish with embedded SMA wire actuated by flexible biomimetic

fin

In (115) it has been proposed a micro robot fish that uses a flexible biomimetic fin propeller

with embedded SMA wires to mimic the musculature and flexible bending of squid fin. The

propulsion consists of an active component (the biomimetic fin) and a passive component (the

caudal fin). The biomimetic fin-based propulsion mechanism is an actuator that combines the

SMA wire and an elastic substrate.

This micro-robot fish introduces a new concept in the world of biomimetic robotics due to its

ability to swim noiseless. This means the robot avoid the use of any traditional components like

gears, bearings and joints, only using the SMAs as actuators that produce the propulsion. The

robot is able to achieve a swimming speed of 112mm/s when the SMAs actuate at 2.1Hz (con-

tracting upon electrical heating), and a minimum turning radius of 136mm, which makes the

robot the fastest micro robot-fish compared to other prototypes that use IPMCs like actuators

(120),(121),(122). The authors have measured the robot performance based on the Strouhal

number (123), (124), which typically varies between 0.25 to 0.35 for the biological counterparts.

Their robot has a Strouhal number of 0.58 (at maximum swimming speed). This upper value

highlights the optimal movement of the robot, however, high amounts of input power have been

required to actuate the SMAs.

2.3.1.2 Towards a biologically inspired small-scale water jumping robot

In (1), the locomotion description of a water-jumping robot that mimic the ability of the water

striders and the fishing spider to jump on the water surface. This biomimetic robot achieves

a vertical jumping motion by pushing the water surface. The motion is triggered with a latch

driven by the SMA actuator.

26


Figure 2.2: Fabricated water jumping robot with six legs. LL is the left latch and LR is the

right latch. RLR and RLL are the rear legs, MLL and MLR are the middle legs, and FLL and

FLR are the front legs. Scale bar, 10 mm (1).

As a result of the research, quantification of Re = 260 (Reynolds number is the ratio of

inertial over viscous forces), Bo = 0.0054 (Bond number is the ratio of the buoyancy to the

surface tension) and We = 4.7 (Weber number is the ratio of the inertia to the surface tension)

and the Ba

(Baudoin number is the ratio of the body weight to the surface tension) suggest

that the physics of jumping in this robot is similar to those of the fishing spider. The Bond,

Weber number and Baudoin numbers are explained by (123), (124). In terms of actuation,

the SMA allows the robot to be extremely light (mass of 0.51g), which it is essential to ensure

the buoyancy on water. The maximum jumping height is 26mm, 26% of the height reached

when jumping on ground (53.1mm). This prototype is the first concept of jumping robot that

integrates SMAs within a structure with an overall mass of 1g. The robot requires 2W of power

consumption in order to generate a force of 1.35mN .

2.3.1.3 A micro biomimetic manta ray robot fish actuated by SMA

In (2) a manta ray robot fish actuated by SMA wires is designed. Figure 2.3 shows the prototype

of the robot. Two pectoral fins arranged in triangular-shaped made of latex with a thickness

of 0.2mm form the fin surface.

This micro manta ray was the first prototype that uses SMAs to generate thrust. This

robot is capable to swim forward and turn. The sweep back angle of the pectoral fins is 20�.

A maximum swimming speed of 57mm/s was achieved and the maximum amplitude of the

motion was 40mm. All the biomimetic fins are open-loop controlled.

27


Figure 2.3: Micro biomimetic manta ray robot fish (2).

2.3.1.4 Controlling a lamprey-based robot with an electronic nervous system

In (3) a sea Lamprey has been developed. The robot consists of a cylindrical electronics bay

propelled by an undulatory body axis. SMA actuators generate propagating flexion waves in

five undulatory segments of a polyurethane strip. The lamprey robot Figure 2.4(a) consists on

a cylindrical hull that houses the electronics and battery pack. In this application, the authors

use a neuronal network that allows the robot to be controlled in real time. This neuronal

network also drives the control of the SMAs. The results have shown the system can reject

disturbances thanks to the robustness of the nonlinear controller (125). Each SMA wire drains

1.5A of electrical current when activated.

Figure 2.4: (a) Lamprey Robot with sonar array, (b) Lateral view of tail segment showing nitinol

actuator, Teflon vertebra and tensioning nuts and (c) Lateral view of pitch mechanism (3).

28


2.3.1.5 A biomimetic robotic jellyfish (Robojelly) actuated by shape memory alloy

composite actuators

The newest and more advanced aquatic robot that uses SMA actuators is a jellyfish robot

designed by (126). The hydrogen-fuel-powered robot called ”Robojelly” mimics the propulsion,

morphology, kinematics and physical appearance of a medusa (jellyfish); the Aurelia aurita

species. The bio-inspired actuators are made of silicone, SMA wires and spring steel.

The development of Robojelly has introduced a systematic method for the design and fab-

rication of SMA-based actuators called BISMAC (bio-inspired shape memory alloy composite).

This method allows for bending the structure of the robot by means of SMA contraction (127).

Thanks to the BISMAC SMA arrangement, this robot was capable to mimic the physics and

swimming characteristics of jellyfish in terms of A. aurita’s bell geometry, passive relaxation

mechanism, neutral buoyancy, frequency of motion, and deformation-to-flap motion profiles.

The structure can be bended by the SMAs actuators (deformation), and then a flap motion of

the bell-segment structures provide the propulsion. The Robojelly was able to produce enough

thrust to propel itself and achieve a proficiency of 0.19s� 1 which is comparable to the natural

medusa at 0.25s�1. The robot consumes an average of 16.74W over its 14th cycle of actuation.

This robot confirms the fact that most aquatic biomimetic robots use SMA wires combined with

other materials to create SMA-based actuators. This characteristic shows the flexibility of the

SMA to work in combination with other materials.

2.3.2 Air

In aerial bio-inspired robots most of the applications are appendices. Here, here the two main

categories are identified: insects and birds. To the best of the authors’ knowledge only one

robotic flying insect has been developed, apart form the jumping robot described earlier. This

can be explained by the flapping frequency needed, far form the SMA’s capabilities, and also by

their power requirements. For these reasons insect-like flying robots mostly adopt piezo-electric

actuators. In fact, the flying insect prototype described below uses SMA to fold and unfold

the wings, and not for the primary flapping motion. Despite SMA actuation speed does not

allow the actuation of flapping wings, it could allow for other kind of wing actuation, such as

morphing-wings.

29


2.3.2.1 Recent progress in developing a beetle-mimicking flapping-wing system

In (4) a beetle-like insect robot inspired by the Allomyrina Dichotomapresents is presented.

This robot features a morphing-wing airfoil capable of folding and unfolding the hind wing

using SMA wires. A single small size DC motor drives the flapping mechanism. Figure 2.5

shows the prototype and the unfolding of the artificial flapping/morphing wing device.

Similar folding ratio of the robot’s wings has been observed in comparison with the biological

counterpart, accounting for 1.7 of value. On average, wing unfolding was completed within

about 3s and the wing folded in about 4s.

Figure 2.5: Prototype of the robotic beetle and detail of the unfolding of the artificial wing (4).

2.3.2.2 BATMAV-a biologically inspired micro-air vehicle for flapping flight: artificial-

muscle based actuation

The BATMAV is a biologically inspired bat-like Micro-Aerial Vehicle (MAV) with flexible and

foldable wings capable of flapping flight (128). The robot features bat-inspired wings with a

large number of flexible joints that allow mimicking the kinematics of a real bat flyer. Figure

2.6 details the overall structure of the robot, and the main connections of the SMA-like muscles.

BATMAV is the first robot that uses the SMA wires to play a dual role: first, as muscle-like

actuators that provide the flapping and morphing wingbeat motions of the robot, and second,

as super-elastic flexible hinges that join the wing’s bone structure. Most of the experiments

were carried out with a two-degree of freedom wing capable of flapping at 3Hz. Despite the fact

that their robot is able to achieve accurate bio-inspired trajectories, the results presented lack

30


Figure 2.6: BATMAV. Dual Role of Shape Memory Alloy wires: as actuation muscles, and super

elastic joints.

experimental evidence of aerodynamics measurements that might demonstrate the viability of

their proposed design.

2.3.3 Ground

Ground bio-inspired robots have been divided in two categories: the ones that uses actuated

appendices (i.e, legged robots) and those that actuate the whole body, i.e. crawling robots such

as snakes and worms.

2.3.3.1 Sensor fusion in a SMA-based hexapod bio-mimetic robot

In (5) SMABOT is presented, a hexapod biomimetic robot with two SMA actuators that allow

for the motion of the two degree-of-freedom robot. Each SMA actuator produces 300gram �

force of pull force. Figure 2.7 shows the SMABOT IV. SMABOT IV incorporates two-

dimensional inertial navigation system for position control. The average speed when moving

with tripod gait is 30cm/min. Its maximum power consumption is about 25W (the mass is

290g).

2.3.3.2 Omegabot: Crawling robot inspired by Ascotis Selenaria

In (6) a robot inspired by the inchworm Ascotis Selenaria is presented. The robot, called

Omegabot, is named after the omega (⌦) shape of the crawling motion of the inchworm. Figure

2.8 shows the Omegabot platform. Previous work about this robot can be also found in (119).

Experimental results report the first step for establishing an inchworm-like robot that can

crawl on various terrains where conventional robots cannot move. The Omegabot uses a SMA

31


Figure 2.7: SMABOT IV, a SMA based hexapod robot with the IMU module, compass sensor

and step touch sensors (5).

coil actuator that requires a current of 200mA for activation. The frequency of motion is

about 1Hz, limited by the response time of the SMA wires. The inchworm robot is manually

controlled by an IR remote operation, and it achieves a maximum linear velocity of 5mm/s.

The robot travels a distance of 5mm per stroke.

Figure 2.8: Omegabot, a biomimetic inchworm robot, grasps the branch of a wood, raises its

head, and turns right. Bottom right: Proleg of Omegabot (6).

2.3.3.3 An earthworm-like micro robot using shape memory alloy actuator

In (129) a bio-mimetic micro earthworm-like robot with wireless control is proposed. The

actuation mechanism consists on a SMA spring that contract and extend the earthworm muscle

respectively. The proposed mechanism is simple but e↵ective when traveling in narrow and

rough environments, such as human digestive organs, bended long pipeline and so on. Also,

this micro robot incorporates both control and power supply onboard. The theoretical speed

of the micro robot is approximately 3.4mm/cycle, where the total time per cycle is 8s (the

32


contraction time of the SMA is 2s, whereas the recovery time is 6s). The fabricated micro

robot can move with the velocity of 10mm/min during 8 minutes. The stroke per cycle is

2.0mm.

2.3.4 Other SMA-based actuation systems

This section presents two works that do not address the development of a full robot, but rather

studying and developing appendices to be added to future full robotic systems.

2.3.4.1 Research on Development of a Flexible Pectoral Fin Using Shape Memory

Alloys

In (7), experimental research on pectoral fin structure is presented. The design of the pectoral

fin actuator is based on SMAs wires composed by a couple of plates with the opposite functions.

Figure 2.9 shows the biomimetic pectoral fin.

Figure 2.9: Biomimetic pectoral fin driven by eight couples of SMA plates (7).

This pectoral fin was the first and today continues being the only designed with only SMA

wires. In fact, most research on fish-like robots is focused on studying propulsion (how to

generate thrust), while maneuvers is largely unexplored.

2.3.4.2 Development of a dexterous tentacle-like manipulator using SMA-actuated

hydrostats

Novel design principles and technologies for a new generation of high dexterity soft-bodied

robots inspired by the morphology and behavior of the octopus are being developed in the

framework of the OCTOPUS-IP project1.

The imitation of the internal muscular structure of octopuses’ tentacles is being studies

and imitated. Longitudinal cables and transverse SMA imitate the arrangement of muscle

1http://www.octopusproject.eu/

33

2.4 Review on recent advances: iTuna and BaTboT

fibers, controlling contractions as soft actuators within the robot arm (130). Moreover, this

manipulator is surrounded by a sensitive skin, with contact sensors embedded into silicone

rubber, equipped with passive suckers that allow the grasping of objects. SMA actuators are

used to change the section of the tentacle in several locations, inducing its bending.

Figure 2.10: The SMA-based tentacle (See http://www.octopusproject.eu/).

2.3.4.3 Development of a Shape-Memory-Alloy actuated biomimetic hydrofoil

The development and testing of a biomimetic active hydrofoil using Shape Memory Alloy (SMA)

actuators is presented in (131). This work describes the development and testing of a six-

segment demonstration foil and the control schemes used.


This section reports the most recent results on two SMA-actuated bio-inspired robots. The

first, called iTuna, an underwater robot that according to our classification falls into the ”full-

actuated-boy” category. The second is an aerial robot, which implements the concept of mor-

phing wings by means of SMA-based muscles.

2.4.1 iTuna: a bending structure swimming robotic fish

The iTuna (8) is a swimming fish-like robot that apart from the external appearance, imitates

some key features of the internal morphology of fishes.

This mechatronic concept takes inspiration from the arrangement of the red or slow-twitch

muscles (see inset in Figure 2.11). In live fishes, such muscles are used for bending a flexible but

nearly incompressible axis. Such axis is either composed of a (visco) elastic beam (notochord)

34


or a series of vertebrae connected through intervertebral discs. The main structure of the iTuna

robot fish is inspired by the former solution, and is composed by a continuous flexible backbone.

The backbone is composed of polycarbonate of 1mm thickness actuated by SMA muscles acting

as red muscles.

8.3 cm

7 cm

6 cm

Fixed ends: 1.6 mm screws

SMA wire

Backbone

Red muscles

SMA twist point

polycarbonate(1 mm thick)

b

h

Active SMA wire

a

Figure 2.11: Main structure of the iTuna robot fish. a=8.5 cm. Under nominal operatrion,

b ⇠= 96% a = 8.16 cm, h=1.02 cm, b=28� (8).

Six SMA-based actuators whose length is 1/3 of the body length are positioned in pairs,

parallel to the body in such a way to produce an antagonistic movement on three body segments

of 8.5cm length. This antagonistic configuration of SMA wires has some advantages in terms

of increasing the range of controllable actuation, since both directions of motion (contraction

and elongation) can be actively controlled. Figure 2.11 shows the location of the SMA wires

within the skeleton structure of the prototype.

A V-shaped configuration of the wires, where each artificial muscle is composed of a single

V-shaped SMA wire, twisted around the tension screw, allows to double the pull force without

a significant increase of power consumption. NiTi SMA wires with a diameter size of 150µm

have been adopted. These have a pull force of 230grams� force at consumption of 250mA at

room temperature, and a nominal contraction time of 1 second.

Under nominal operation such SMAs can bend the body segments up to 28 degrees (angle �

of Figure 2.11), even if SMA wires only contract approximately 4% of their length. By increasing

the input electrical current and including a suitable control that handles an overloaded SMA

operation, contraction time of 0.5s was achieved, and strain could be increased up to 6%,

corresponding to a bending of approximately 36� (Fig. 2.12).

35


2.4.1.1 SMA control in the iTuna

After identification, a low-level PID controller has been designed to address two main limitations

of SMAs: slack in the fibers, and limited actuation speed. Slack issues appear when SMA wires

develop a two-way memory e↵ect during operation (95). Limitation in actuation speed occurs

due to the large switching time between cooling and heating phases. To address such problems,

a pre-heating mechanism has been developed that works in conjunction to the antagonistic

arrangement. The pre-heating avoids the temperature on both wires drops below the 10% of

the maximum applied electrical current, preventing the inactive alloy from complete cooling.

On the other hand, the antagonistic arrangement provides an external stress to the cooling wire

(provided both by the elastic backbone and by the active antagonistic wire). Working with an

already-warm wire allows for a faster stretch and slack issues are avoided. Note that the PID

controller is based on the experimental observation that the hysteresis on the electrical resistance

curve was smaller than the hysteresis on the temperature curve. Resistance measurements are

used as a feedback signal for closed-loop control (see (8) and (132) form more details).

The control developed allows overloading the SMA with up to 350mA peak current (note

that power signals are sinusoidal, hence overloading only lasts a brief period of time). Overload-

ing has allowed for achieving a 1Hz oscillation time (i.e. 0.5 seconds contraction and cooling

times) and a bending angle of 36 degrees of each body segment.

=36deg

Figure 2.12: Bending under SMA overloading (8).

2.4.1.2 Control architecture

A key feature of SMAs is the possibility to develop closed loop control systems without the

need of external sensor hardware. The feed back signal is provided by the detection of inner

electrical resistance, that allows an indirect measurement of the temperature.

36


The main components are described in the following. A micro controller implements the

PID algorithm. The PID controller receives the input reference position (set point) and the

feedback of SMA’s voltage and current that allows calculating the heating current to drive the

SMA actuator. The digital output of the PID controller is converted to a reference current in

two steps. First, it is converted into an analog signal using a 2-wire serial 8-Bit DAC (Digital

to Analog Converter) with Rail-to-Rail outputs. Then, a Voltage Controlled Current Source

(VCCS) transforms the DC voltage in a constant current that feeds the SMA. This stage has a

power consumption of less than 10mA.

The DAC output ranges from 0 to 5 Volts with a resolution of 0.02V . The measured voltage

(VSMA) and current (ISMA) on the SMA fiber are fed-back to the micro controller in order

to close the control loop. The hardware used (16F690-PIC) had a 12-bits A/D converter with

a resolution of 0.537mV (considering the maximum voltage measured at the SMA V SMA =

0.55V .

On the other hand, taking into consideration the maximum current through the wire

(500mA), SMA resistance variations about 1.074m⌦ can be measured. Therefore, since the

maximum variation in the SMA length is 0.34cm, and the maximum variation of the resis-

tance is 1.6⌦, the theoretical position error of the system based on the SMA length is 0.067%.

i.e.,0.12mm.

2.4.2 BaTboT: a biologically-inspired bat-like aerial robot

BaTboT is a bio-inspired bat robot that uses Shape Memory Alloys (SMAs) as artificial muscles

for powering the morphing motion of the wings. The morphing motion is related to the capacity

of the robot to modulate its wings by contracting and extending the membrane in sync with

the flapping motion. It is precisely this characteristic what makes biological bats more agile to

maneuver than any other flying creature within the same Reynolds number range (103 � 104)

(133), (134). In addition, biological studies in (135), (136) have revealed that real bats are able

to maneuver because of the inertial changes produced by the wings’ modulations. Attempting

to mimic this functionality using an artificial counterpart -BaTboT- mainly presents a twofold

challenge: i) biomechanical design of the wings, and ii) proper control/actuation to module

BaTboT’s wings.

Prior work in (9) presented experimental results regarding both challenges. The investiga-

tion carried out in (9) does not only describes the biomechanical design of BaTboT’s wings,

but also focuses on evaluating the implications of using SMAs as artificial muscles to power the

37

2.5 Advantages and drawbacks of using SMAs

change of wing’s morphology. Figure 2.13 shows the design-flow process to evaluate key issues

of SMA performance and their implications to the application at hand.

Plagiopatagium skin

(0.1mm silicon wing

membrane)

R/C transmission link

49MHz

Antenna

motor+

electronics

SMA

morphing muscles

step 1. SMA wing actuationstep 2. Robot assembly

control encoded.

Re-designingprocess and adjustments

step 3. Wind-tunnel measurements.Implications of SMA performance

Bio-inspired insights

Elbow Joint

3 ~ 60º

Migamotor SMA musclesSMA_1

SMA_2-+

P1 F1

F2

3

P2

Antagonistic configuration

Figure 2.13: Flow-process for SMA evaluation in the BaTboT prototype (9).

The use of SMAs as artificial muscles has been concretely evaluated in terms of two issues:

• Functionality: SMA Power-to-Force.

• SMA Performance: actuation speed and fatigue.


Most relevant advantages and drawbacks of using SMA technology for actuation are highlighted

as follows:

Advantages

• Size and weight : SMAs can be directly used as linear actuators. There is no need for

additional motion components or hardware, which permits easy miniaturizations of the

actuation system. SMA wires have a negligible volume (e.g., 3 ⇥ 10�9m3), allowing for

extremely light wings.

38


• High Force-to-weight ratio: SMA actuators have a large force-weight ratio of ⇠ 8N/1 ⇥

10�5Kg, using a wire with thickness of 150µm, and 0.1m long. SMAs also present large

life cycles (3⇥ 106).

• Noise-free operation: Because SMA actuators do not require friction mechanisms such as

reduction gear, it avoids the production of dust particles, sparks and noise. These merits

make SMA actuators extremely suitable for areas such as microelectronics, biotechnology

and biomimetics applications (high bio-compatibility).

• Sensing properties: Although SMAs are mostly used for actuation, they also have sensing

capabilities. Several properties of the SMAs change as it undergoes martensite phase

transformation. Among these properties is the resistivity that decreases as the tempera-

ture of the wire increases and hence its phase transforms to austenite. A liner relationship

between electrical resistance change and SMA strain can be derived.

Drawbacks; challenges to tackle

• Slow speed : SMA actuators have generally been considered to have slow response due to

restrictions in heating and cooling, and also due to the inherent thermal hysteresis. The

common method in actuation is by electrical heating. Although applying larger electrical

currents can increase the speed, this may also overheat and damage the actuator without

monitoring. Most research so far has investigated SMA position control at generally low

tracking speeds of less than 1Hz. Rise times for step responses usually took more than 1

second.

• Fatigue: Long-term performance of the Shape Memory e↵ect could decrease over time if

the material is expose to large external stress or overheating temperatures resulting from

large input currents.

• Low energy e�ciency : The maximum theoretical e�ciency of SMAs is of the order of

10% based on the Carnot cycle, according to (137). In reality, the e�ciency is often

less than 1%, since the SMA actuator can be considered a heat engine operating at

low temperatures. This means that the conversion of heat into mechanical work is very

ine�cient. Most of the heat energy is lost to the environment. Hence SMA actuator

applications must be limited to areas where energy e�ciency is not an issue (cf. (74)).

39

2.6 Remarks

2.6 Remarks

In terms of actuation, Shape Memory Alloys (SMAs) enable the fabrication of lighter wins

with muscle-like actuation but some challenges should be addressed. Section 2.5 highlighted

the advantages and drawbacks of this material. This thesis will present feasible solutions to

minimize the e↵ects of SMA limitations and it will give an insight into the performance of the

material acting as actuators. The goal is not only to evaluate the use of this actuation technology

for the application at hand but also on providing a formal quantification of performance that

would allow others to drive this technology forward.

By reviewing the state of the art in Section 2.3, one can note the lack of biologically-inspired

robots that explore alternative actuation mechanisms more likely to those found in nature. The

field of bio-inspired AUVs that use smart materials for actuation is still in an early stage. Most

of the works have investigated how to fabricate e�cient robot models, but few have achieved to

develop a complete bio-AUV platform capable of sustained movement. This thesis embarks into

this potential field by presenting the first fish-like AUV that can maneuver and swim by means of

changing caudal fin and backbone morphology and also it takes advantage of the improvements

in swimming e�ciency that caudal fin and backbone modulation enables.

40

3

From Ray-Finned Fishes to BR3:

Mimicking biology

”Que bonito seria poder volar, y a tu lado ponerme yo a cantar, como siempre lo hacamos los

dos, Que mi cuerpo no para de notar, que tu alma conmigo siempre esta, y que nunca de mi

se apartara”

Rosario Flores (Que bonito)

3.1 General overview

This chapter presents insights of in-vivo fish swimming. BR3’s morphology and biomechanics

are based on the fish specie Largemouth Bass physiology. This section describes why the

selection of this specie to be mimicked with BR3. This selection has been based on criteria

regarding: i) fish morphology (i.e., caudal fin, aspect ratio, body and caudal fin mass, etc), ii)

swimming kinematics, and iii) swimming dynamics (propulsive mechanism, flapping caudal fin,

etc).

3.2 Body and/or Caudal Fin Propulsion

In undulatory BCF modes, the propulsive wave traverses the fish body in a direction opposite

to the overall movement and at a speed greater than the overall swimming speed.

The four undulatory BCF locomotion modes identified in Fig. 1.6(a) reflect changes mainly

in the wavelength and the amplitude envelope of the propulsive wave, but also in the way

thrust is generated. Two main methods have been identified: an added-mass method and

41


propulsive element

propulsive element

U

V

FT

FR

FL

F 0L

F 0T

F 0R

propulsivewavespeed

overallswimming

speed

Figure 3.1: Thrust generation by the added-mass method in BCF propulsion

a lift-based (vorticity) method. The latter is primarily used in thunniform swimming, while

anguilliform, subcarangiform, and carangiform modes have long been associated with the added-

mass method. However, recent studies suggest that vorticity mechanisms are also important

for subcarangiform and carangiform swimming (see text below).

A qualitative description of the added-mass method is given by Webb in (68) (see also (138)

for a more mathematical description) and is summarized here. As the propulsive wave passes

backward along the fish, each small body segment (called propulsion element) generates a force

that increases the momentum of the water passing backward. An equal opposing force (the

reaction force FR

) is subsequently exerted by the water on the propulsive element. For most

fish, the magnitude of FR

can be approximated (neglecting viscous e↵ects) as the product of

the water mass accelerated and its acceleration. FR

is normal to the propulsion element and

is analyzed into a lateral FL

and a thrust FT

component (Fig. 3.1). The thrust component

contributes to forward propulsion, while FL

sheds water laterally and can lead to significant

energy losses. Furthermore, the lateral component induces tendencies for the anterior part

of the body to sideslip and yaw (recoil tendencies). FT

is larger for the propulsive elements

near the tail, since the rear elements traverse greater distances and have larger speeds, hence

accelerating the water more. Furthermore, since the amplitude of the propulsive wave increases

toward the caudal fin, the propulsion elements there are oriented more toward the overall

direction of movement, ensuring that the reaction force F 0R

has a larger thrust component F 0T

(Fig. 3.1).

The ratio U/V (where U is the overall fish swimming speed and V is the wave propagation

speed) has long been used as an indication of swimming e�ciency.

Body movements are particularly significant during unsteady swimming actions, like fast

starts and rapid turns, that are characterized by high accelerations. Relatively few kinematic

data have been available for these, due to the di�culties in setting up repeatable experiments

and the complexity and speed of the movements involved.

42


In anguilliform mode, the whole body participates in large-amplitude undulations (Fig.

3.2(a)). Since at least one complete wavelength of the propulsive wave is present along the

body, lateral forces are adequately cancelled out, minimizing any tendencies for the body to

recoil. Many anguilliform swimmers are capable of backward as well as forward swimming

by altering the propagation direction of the propulsive wave. Backward swimming requires

increased lateral displacements and body flexibility (139). Typical examples of this common

locomotion mode are the eel and the lamprey. See (140) for a summary of existing kinematic

data on anguilliform locomotion. Similar movements are observed in the subcarangiform mode

(e.g., trout), but the amplitude of the undulations is limited anteriorly, and increases only in

the posterior half of the body (Fig. 3.2(b)). For carangiform swimming, this is even more

pronounced, as the body undulations are further confined to the last third of the body length

(Fig. 3.2(c)), and thrust is provided by a rather sti↵ caudal fin. Carangiform swimmers are

generally faster than anguilliform or subcarangiform swimmers. However, their turning and

accelerating abilities are compromised, due to the relative rigidity of their bodies. Furthermore,

there is an increased tendency for the body to recoil, because the lateral forces are concentrated

at the posterior. Lighthill (141) identified two main morphological adaptations that increase

anterior resistance in order to minimize the recoil forces: 1) a reduced depth of the fish body

at the point where the caudal fin attaches to the trunk and 2) the concentration of the body

depth and mass toward the anterior part of the fish.

Thunniform mode is the most e�cient locomotion mode evolved in the aquatic environ-

ment, where thrust is generated by the lift-based method, allowing high cruising speeds to be

maintained for long periods. It is considered a culminating point in the evolution of swimming

designs, as it is found among varied groups of vertebrates (teleost fish, sharks, and marine mam-

mals) that have each evolved under di↵erent circumstances. In teleost fish, thunniform mode

is encountered in scombrids, such as the tuna and the mackerel. Significant lateral movements

occur only at the caudal fin (that produces more than 90% of the thrust) and at the area near

the narrow peduncle. The body is well streamlined to significantly reduce pressure drag, while

the caudal fin is sti↵ and high, with a crescent-moon shape often referred to as lunate (Fig.

3.2(d)). Despite the power of the caudal thrusts, the body shape and mass distribution ensure

that the recoil forces are e↵ectively minimized and very little sideslipping is induced. The de-

sign of thunniform swimmers is optimized for high-speed swimming in calm waters and is not

well-suited to other actions such as slow swimming, turning maneuvers, and rapid acceleration

from stationary and turbulent water (streams, tidal rips, etc.).

43

3.3 Body Undulations and Friction Drag

Figure 3.2: Gradation of BCF swimming movements from (a) anguilliform, through (b) sub-

carangiform and (c) carangiform to (d) thunniform mode.

Ostraciiform locomotion is the only purely oscillatory BCF mode. It is characterized by

the pendulum-like oscillation of the (rather sti↵) caudal fin, while the body remains essentially

rigid. Fish utilizing ostraciiform mode are usually encased in inflexible bodies and forage their

(usually complex) habitat using MPF propulsion (142). Caudal oscillations are employed as

auxiliary locomotion means to aid in thrust production at higher speeds, to ensure that the body

remains adequately rigid, or to aid prey stalking (143). Despite some superficial similarities

with thunniform swimmers, the hydrodynamic adaptations and refinements found in the latter

are missing in ostraciiform locomotion, which is characterized by low hydrodynamic e�ciency.

3.3 Body Undulations and Friction Drag

Swimming viscous drag is calculated using the standard Newtonian equation:

Dv

=1

2C

f

SU2⇢ (3.1)

where Cf

is the drag coe�cient (which depends on the Reynolds number and the nature

of the flow), S is the wetted surface area, and ⇢ is the water density. Flexing the body to

achieve propulsion is expected to increase viscous drag by a factor of q compared to that for

an equivalent rigid body, since the motion of the propulsive elements increases their velocity

with respect to the surrounding fluid. This is known as the ”boundary layer thinning” e↵ect,

44

3.4 Wake Structure and Generation

as lateral body movements reduce the boundary layer, resulting in increased velocity gradients

and, hence, shear stress. Exactly how extensive the increase in viscous drag is has long troubled

scientists. Originally, indirect estimations suggested (see, for example (144) and (141)) that q

lies between 4 and 9. Webb in (145) indicates that this must be a significant overestimation,

placing a greater importance on the energy losses arising from recoil forces. A value of q = 18

for a swimming tadpole has been calculated in (146) using three-dimensional (3-D) numerical

simulation, at Re

= 7200. The rather low Re prohibits safe application of this value of q to adult

fish swimming. In the same study, it is shown that the relative amplitude of body undulations

in tadpoles is significantly larger than those observed in fish. When the model was adapted to

swim using the kinematics of a saithe, q was reduced to 1.12, stressing the connection between

large lateral motions and increased friction drag (146).

3.4 Wake Structure and Generation

The wake left behind the tail of undulatory BCF swimmers is a staggered array of trailing

discrete vortices of alternating sign, generated as the caudal fin moves back and forth. A jet flow

with alternating direction between the vortices is also visible [Fig. 3.3(c)]. The structure of the

wake is of a thrust-type, i.e., has a reversed ratational direction compared to the welldocumented

drag-producing Karman vortex street. The latter is typically observed in the wake of blu↵

(nonstreamlined) objects [Fig. 3.3(a)] for a specific range of Reynolds numbers (roughly 40 <

Re < 2 · 105), but also in the wake of stationary [Fig. 3.3(b)] or low-frequency-heaving aerfoils

(see (147)).

The main parameter characterizing the structure of such wakes is the Strouhal number,

defined, for a fish swimming by BCF movements, as:

St =fA

U(3.2)

where f is the tail-beat frequency in hertz, A is the wake width (usually approximated as the

tail-beat peak-to-peak amplitude, and U is the average forward velocity. The Strouhal number

is essentially the ratio of unsteady to inertial forces. Triantafyllou et al. (147) concluded that,

in oscillating foils, thrust development is optimal for a specific range of St (namely 0.25 <

St < 0.40). Existing data on a number of fish species revealed that, for high-speed swimming,

their calculated St values lie within this predicted range. Interestingly, this was valid for

species representing not just thunniform (traditionally associated with oscillating foils) but

45

3.5 Overview of fish fin structure and function

U

U

Uf

A

Figure 3.3: The Karma street generates a drag force for either (a) blu↵ or (b) streamlined

bodies, placed in a free stream. (c) The wake of a swimming fish has reverse rotational direction,

associated with thrust generation.

also subcarangiform and carangiform modes, at a range of 104 < Re < 106. These results

have placed increased significance to vorticity e↵ects and established the Strouhal number as a

prominent factor when analyzing BCF modes. Detailed data on the morphology of the wave

shed behind a mullet (swimming at Re = 22 · 103) can be found in (148).


Fish fins are supported by flexible bony or cartilaginous fin rays that extend from the fin

base into the fin surface and provide support for the thin collagenous membrane that connects

adjacent fin rays (Figs. 3.4, 3.5). Fin rays articulate with the fin skeleton located inside the

body wall which supports musculature that allows the fin rays to be actively moved from side

to side and elevated and depressed (Fig. 3.4, (149), (150), (151), (152)). Many fish also have

dorsal and anal fins which have leading spiny portions of the fin (Fig. 3.4, (153)), and fin

spines typically can only be elevated and depressed; they possess limited sideways mobility.

The posterior region of the dorsal and anal fins is known as the ”soft” dorsal or anal fin and

is supported only by flexible fin rays. Recordings from fin musculature, which is distinct from

the body muscles, unequivocally show that fins are actively moved during swimming, and that

this active movement generates a vortex wake that passes downstream toward the tail, which

thus intercepts the flow that is greatly altered from the free-stream (153), (154), (151), (155).

The modulus of elasticity of bony fin rays is about 1GPa, while the membrane in between fin

46


Figure 3.4: Structure of the fin skeleton in bony fishes. (a) Skeleton showing the positions of the

paired and median fins and their internal skeletal supports. Note that each of the median fins has

segmented bony skeletal elements that extend into the body to support the fin rays and spines,

and that muscles controlling the fin rays arise from these skeletal elements. (b) Bluegill sunfish

hovering in still water with the left pectoral fin extended. (c) Structure of the pectoral fin and the

skeletal supports for the fin; bones have been stained red. This specimen had 15 pectoral fin rays

that articulate with a crescent-shaped cartilage pad (tan color) at the base of the fin. The smaller

bony elements to the left of the cartilage pad allow considerable reorientation of the fin base and

hence thrust vectoring of pectoral fin forces (10) (11). (d) Anal fin skeleton (bones stained red

and muscle tissue digested away) to show the three leading spines anterior to the flexible rays, and

the collagenous membrane that connects adjacent spines and rays. (e) Close view of pectoral fin

rays (stained red) to show the segmented nature of bony fish fin rays and the membrane between

them. Images in panels A and B modified from (12).

rays has a modulus of about 0.3� 1.0MPa (156).

The hallmark of fish fin functional design is the bending of the fin rays which permits

considerable flexibility of the propulsive surface. The fin rays of the large fish group termed ray-

finned fishes (but not sharks), possess a remarkable bilaminar structure and muscular control

47

3.6 Remarks

that allows fish to actively control fin surface conformation and camber during locomotion. As

illustrated in Fig. 3.5, each bony fin ray is composed of two halves (termed hemitrichs) which

are connected along their length by short collagen fibers and may be attached at the end of the

ray. Each fin ray is actuated by four separate muscles, and thus a single fin such as the pectoral

fin of a bluegill sunfish (Lepomis macrochirus), which has about 14 fin rays, potentially has over

50 separate actuators that allow the fin to be reoriented in three dimensions with control over

the position of each ray. Neural control of fin ray motion has yet to be studied in detail, and

the extent to which anatomically homologous muscles on neighboring fin rays can be controlled

independently is unknown. Most importantly, displacement of the two ray halves through the

contraction of fin ray musculature at the base of the fin causes the fin ray to curve. Fish can thus

actively alter the conformation of their propulsive surface by actively bending fin rays, and can

resist hydrodynamic loading, a phenomenon that is observed most clearly during maneuvering

(Fig. 3.5(d)). One result of the complex control and bilaminar fin ray design in fish fins is that,

fins can undergo rather complex three-dimensional changes in shape during locomotion.

3.6 Remarks

This chapter has allowed the understanding of biological parameters that directly a↵ect fish

swim and manoeuvre and provides the foundations and criteria for robot design. Analyses of

biological experiments described in (157) allowed for a complete definition of a set of key issues

to consider during the designing process of BR3. These issues show how morphology, kinematics

and aerodynamics can be related to each other into a bio-inspired designing framework. The

following chapter introduces the mathematical formulation for kinematics, dynamics and body-

caudal(fin)-actuation using SMA-like muscles.

48

3.6 Remarks

Figure 3.5: Pectoral fin structure in bluegill sunfish. (a) Schematic view of the pectoral fin

which typically has 12-15 fin rays. (b) Cross-section through fin rays at the level of the blue

plane shown in panel A obtained with lCT scanning (see (13)) in which bone is whitish color, and

fin collagen and membrane are gray. Cross-sectional image of rays (top) and close view of two

adjacent rays (below). Each fin ray is bilaminar, with two curved half rays termed hemitrichs.

(c) Schematic of the mechanical design of the bilaminar fin ray in bony fishes. Each fin ray has

expanded bony processes at the base of each hemitrich to which muscles attach (blue arrows).

Di↵erential actuation of fin ray muscles (red arrows) results in curvature of the fin ray. Fish can

thus actively control the curvature of their fin surface. (d) Frame from high-speed video of a

bluegill sunfish during a turning maneuver, showing the fin surface (outlined in yellow) curving

into oncoming flow.

49

4

BR3 modeling

”Sometimes I give myself the creeps, Sometimes my mind plays tricks on me”

Green Day (Basket Case)


This chapter presents the modeling of the most important components involved within the de-

sign process of BR3: i) kinematics, ii) dynamics, iii) hydrodynamics and iv) SMA for body and

caudal fin muscle-like actuation.

4.2 First approach to kinematic, dynamic and hydrody-

namic analysis

This first approach incorporates the hydrodynamic analysis but the hydrodynamic interaction

between the di↵erent components around the anterior body, the oscillatory links, and the caudal

fin are ignored. Next section (Sec. 4.3) takes in count all the hydrodynamics e↵ects and

interactions.

4.2.1 Overview of the propulsive mechanism

The carangiform and thunniform swimming modes has the ability to keep high speeds swim-

mings, while the angilliform and subcarangiform swimming exhibits remarkable maneuver-

ability. For this reason the dynamic and kinematic model develop as a whole package the

carangiform and angilliform swimming modes.

50

4.2 First approach to kinematic, dynamic and hydrodynamic analysis

S

X

n

�

�

O

ZY

Head

Tail Body

Figure 4.1: Precession � angle and Nutation � angle, representation

Physical model of robotic fish synthesizing both carangiform-anguilliform mode is shown in

Fig. 4.1, which consists of three parts: sti↵ anterior body (head) with a pair of pectoral fins for

up-and-down motion, flexible rear body and an oscillating caudal fin. The flexible rear body

can be designed as a multi-link mechanism which consists of several oscillating hinge joints.

The motion of the multi-link is expected to match an approximate wave Eq. 4.1 to obtain a

forward thrust (158).

ybody

(x, t) = [(c1x+ c2x2)][sin(kx+ !t)] (4.1)

where ybody

represents the transverse displacement of the fish body, x is the longitudinal

position with respect to the head of the fish, k indicates the body wave number (k = 2⇡/�),

� is the body wave length, c1 and c2 are the parameters defining the wave amplitude, and !

is the body wave frequency (! = 2⇡f = 2⇡/T ). Since the flexible rear body fish is composed

of a number of segments, similarly, the designed oscillatory part of a robotic fish consists of

several rotating hinge joints, as shown in Fig. 4.2. It is modelled as a planar serial chain of

links along the axial body displacement, and the positions of the links in the chain are achieved

by numerical fitting. See (159), (160) for details determining and optimizing the link length

ratio l1 : l2 : ... : lN

, where l1 � l2 � lN

. The caudal fin is attached to the last link by the foil

pivot about which the caudal fin rotates in a sinusoidal manner:

✓f

(t) = ✓max

sin[kxF

+ !t+ '] (4.2)

where ✓f

(t) indicates the pitch angle of the caudal fin relative to main axis, ✓max

is the

amplitude of pitch angle, ' is the phase angle between heave and pitch, and xF

is the x-

51


y

xPo

z

P�1

M0

Mr

Body

Head

Pectoral finCaudal fin

P1

P2

Mi

PN+1

⇠i

⌘i

⇣i

Z

X

Y

Figure 4.2: X, Y and Z coordinates system representation

component of the position of the oscillatory foil pivot. Practically, ✓max

can be achieved by:

✓max

= �o

� ↵o

= arctan(@y

@x(x

F

o

, t))� ↵o

(4.3)

where �o

is the slope angle of ybody

(xF

, t) at yF

= 0, ↵o

corresponds to attack angle of the

caudal fin at yF

= 0, yF

is the y-component of the position of the foil pivot, and xF

o

is the

value of xF

at yF

=0.

Mi

⇠i

⌘i

⇣i

(a)

⇠i

⌘i

⇣i

�

�

⇥ Si

Z

mj

⇥

�Y

ni�X

O

(b)

Figure 4.3: (a) Link-fixed coordinate system. (b) Three Eulerian angles.

As shown at (147), (161), oscillating foil is an e↵ective device for propulsion and maneuver-

ing. They concluded that the Strouhal number St

and the maximum attack angle ↵max

have

52


o(P�1)

P0

y

✓1

P1✓2

lj

Pj

Mj

⌘i ⇠

i

Oscillating hingejoints

Mb

PN�1

PN

xF

Anterior Body

(Head)Flexible rear body

First Segment Second Segment Caudal fin

Travelingbody � wave

X

⌘N+1

↵

⇠N+1

✓f

= � � ↵

�

PN

✓f

Figure 4.4: Planar configuration for the robot fish.

direct relevance to the thrust coe�cient and the wake dynamics. To accomplish an e�cient

locomotion, in the following dynamic modelling and analysis,the hydrodynamic parameters are

chosen according to their results, e.g., St ⇡ 0.3, ↵max

⇡ 25o, and ' = 75o.

4.2.2 Dynamic Model

Three di↵erent coordinate systems are defined as:

• space-fixed (inertial) coordinate system O �XY Z

• head-fixed coordinate system Po

�XY Z

• link-fixed coordinate system Mi

� ⇠i

⌘i

⇣i

Set that (~i,~j,~k) and (~ei1,~ei2,~ei3) are base vectors of system O � XY Z and M

i

� ⇠i

⌘i

⇣i

respectively Fig. 4.3(a). The translating velocity vector of the point o(P�1) described by

system O � XY Z is ~U = ~U�1 = (~i,~j,~k)(u(t), v(t), w(t))T and Eulerian angles Fig. 4.3(b)

of the anterior body are �,� and , where, u(t) and v(t) denote the forward speed and sway

velocity respectively. To determine the motion of the whole fish u(t), v(t), w(t), �, �, are chosen

as outputs of the dynamic system. Note that to unify the notations of forces/moments, the

anterior body is considered as the O� th link and the caudal fin as the N +1� th link. In this

sense, the whole robotic fish can be viewed as a realistic multi-link mechanism. All links have

same angles of precession and nutation, shown in Fig. 4.4.

53


4.2.3 Kinematics Analysis

In the head-fixed frame, a pair of end points of the i� th link are Pi�1(xi�1, yi�1) and P

i

(xi

, yi

)

respectively, and the angle between the i� th link and x� axis is ✓. Note |Pi�1Pi

| = li

.

As mentioned previously, the motion of the links is expected to match a traveling body wave

Eq. 4.1 to obtain a forward thrust. Thus, we can derive ✓i

(t) for (i = 1, 2, ..., N) by Eq. 4.4.

The derivatives of ✓i

, i.e., ✓i

(t) and ✓i

(t) are then easily achieved.

8>>>><

>>>>:

xo

= yo

= 0✓1 = arctan( @y

@x

( l12 , t))xi

= xi�1 + l

i

cos✓i

yi

= yi�1 + l

i

sin✓i

✓i+1 = arctan(@y

@

x

(xi

+ l

i+1

2 cos✓i

, t))

(4.4)

For the i� th link, the angle of rotation is ⇥i

= + ✓i

(t), where ✓o

(t) = 0, ✓N+1(t) = 0.

According to Eq. 4.6, the velocity ~U⇤i

and acceleration ~U⇤i

vector of the every center of

mass, can be derived.

~Ui

= ~Ui�1 + ~⌦

i

⇥ ~ri�1,i, (i = 0, 1, ..., N + 1)

~U⇤i

=1

2(~U

i�1 + ~Ui

) = (~i ~j ~k)(u⇤i

u⇤i

w⇤i

) (4.5)

= (~ei1 ~ei2 ~ei3)(u

⇤i

v⇤i

w⇤i

)T

Where ~⌦i

= (~ei1 ~ei2 ~ei3)(!i1 !i2 !i3)T is the angular speed vector of the i � th link and

~r(i�1,i) =��!Pi�1Pi

.

4.2.4 Hydrodynamic Analysis

For this first hydrodynamic analysis, the hydrodynamic interaction between the di↵erent com-

ponents around the anterior body, the oscillatory links, and the caudal fin are ignored.

For (~i ~j ~k) and (~ei1 ~ei2 ~ei3), there is a transform as the following equation:

(~ei1 ~ei2 ~ei3) = (~i ~j ~k)Q

i

(4.6)

where

Qi

=

0

@cos⇥

i

cos� � sin⇥i

cos�sin� �sin⇥i

cos� � cos⇥i

cos�sin� sin�sin�cos⇥

i

cos� + sin⇥i

cos�sin� �sin⇥i

cos� + cos⇥i

cos�sin� �sin�sin�sin⇥

i

sin� cos⇥i

sin� cos�

1

A (4.7)

54


The force and moment exerted on the head and flexible rear body are considered as added

mass force,drag, and added moment, expressed by Eq. 4.8, Eq. 4.9, Eq. 4.10 respectively.

~Fi

= (~i ~j ~k)Qi

��m

i1 0 00 �m

i1 00 0 �m

i1

�QT

i

0

@u⇤i

v⇤i

w⇤i

1

A (4.8)

~Di

= (~i ~j ~k)Qi

✓�1

2⇢ |u⇤

i

| u⇤i

CD1Si1 � 1

2⇢ |v⇤

i

| v⇤i

CD2Si2 � 1

2⇢ |w⇤

i

| w⇤i

CD3Si3

◆(4.9)

~LIi

= (~ei1 ~ei2 ~ei3)(�j

i1!i1 � ji2!i2 � j

i3!i3)T (4.10)

Where ~mij

and jij

(j = 1, 2, 3) denote added mass and added moment of inertia in the

Mi

� ⇠i

⌘i

⇣i

frame.

The fluid forces acting on the caudal fin are described by Eq. 4.11.

~Fcf

=⇣~i ~j ~k

⌘Q

N+1

0

BBB@

12⇢SN+1CT

⇣�u⇤N+1

�2+�v⇤N+1

�2+�w⇤

N+1

�2⌘

12⇢SN+1CL

⇣�u⇤N+1

�2+�v⇤N+1

�2+�w⇤

N+1

�2⌘

12⇢SN+1CD

⇣�u⇤N+1

�2+�v⇤N+1

�2+�w⇤

N+1

�2⌘

1

CCCA(4.11)

Where thrust coe�cient CT

= fT

(↵, St

), lift coe�cient CL

= fL

(↵, St

), and ↵ is attack

angle of caudal fin.

Finally, the fluid forces acting on the pectoral fins are decided simply by lift theorem

(Joukowski), which is described by equation 4.12.

~Fpf

=⇣~i ~j ~k

⌘(sin�sin�F

pf

� sin�cos�Fpf

cos�Fpf

)T (4.12)

Where, Fpf

= 2⇡cLpf

⇢ sin ↵pf

is attack angle of pectoral fins, and ~U⇤pf

is center of mass

velocity of the left (or right) pectoral fin when Eq. 4.12 is used to denote fluid force acting on

the left (or right) pectoral fin.

4.2.5 3-D Dynamic Model

According to the analysis explained above, for every link the Newton-Euler equation can be

obtained by summarising these equations. The 3-D dynamic model of the whole robotic fish

can be obtained using the equations 4.13 to 4.15.

N+1X

i=0

mi

~U⇤i

=NX

i=0

⇣~Fi

+ ~Di

⌘+ ~F

pf l

+ ~Fpf r

+ ~Fcf

(4.13)

55


Where ~Fpf l

and ~Fpf r

denote fluid forces acting on the left and right pectoral fin respec-

tively.

N+1X

i=0

((Ji1 + j

i1) !i1 + !i2!i3 (Ji3 + J

i2)) =1

2(L

pf

Fpf l

� Lpf

Fpf r

) (4.14)

N+1X

i=0

((Ji2 + j

i2) !i2 + !i3!i1 (Ji1 + J

i3)) =1

2lo

Fl

+1

2+

N+1X

i=0

✓1

2li

T(i�1,i)3 �1

2li

T(i+1,i)3

◆

N+1X

i=0

((Ji3 + j

i3) !i3 + !i1!i2 (Ji2 + J

i1)) =N+1X

i=0

✓�1

2li

T(i�1,i)2 +1

2li

T(i+1,i)2

◆

!i1 = �sin⇥

i

sin�+ �cos⇥i

!i2 = �cos⇥

i

sin�+ �sin⇥i

(4.15)

!i3 = ⇥

i

+ �cos�

For the it

h link, ~T(i�1,i) denote the force from the i � 1 � th link, whereas ~T(i+1,i) denote

the force from the i+ 1� th. So, ~T1,0 = ~TN+2,N+1 = 0 and ~T

i�1,i = �~Ti,i�1.

It is necessary that u, v, w, �, �, are taken out of these equations. So we can get a new

dynamic model as:

0

BBBBBB@

m11 m12 m13 m14 m15 m16

m21 m22 m23 m24 m25 m26

m31 m32 m33 m34 m35 m36

0 0 0 0 m45 m46

m51 m52 m53 m54 m55 m56

m61 m62 m63 m64 m65 m66

1

CCCCCCA

0

BBBBBB@

uvw ��

1

CCCCCCA=

0

BBBBBB@

f1f2f3f4f5f6

1

CCCCCCA(4.16)

Equation Eq. 4.16 can be written as:

MU = F (4.17)

Where M is an inertia matrix; U represents an acceleration/angular acceleration vector, and

F is a resultant forces/moments vector. As equation 4.16 is a nonlinear ordinary di↵erential

equation, to obtain (u, v, w, , �, �), we can use a standard numerical integration method to

solve it for each time. The correlative dynamic simulation can ultimately be implemented in

Matlab/Simulink environment.

56

4.3 Final approach

o(P�1)

P0

✓1

P1✓2

lj

Pj

Mj

Oscillating hingejoints

Mb

PN�1

PN

Anterior Body

(Head)Flexible rear body

First Segment Second Segment Caudal fin

Travelingbody � wave

↵

XN+1

✓N+1 = � � ↵

�

PN

✓f

y

yj x

j

xN+1 X

yN+1

X

Y

ZO

Figure 4.5: Planar configuration for the robot fish.

4.3 Final approach

This final approach uses the Schiehlen and Eberhard method (162),(163), (164), (165) (one tool

widely used for analyzing multi-body dynamics.) to derive the dynamic equations including the

e↵ect of the hydrodynamic interaction between the di↵erent components around the anterior

body, the oscillatory links, and the caudal fin.

4.3.1 System modelling

Based on the Schiehlen and Eberhard method the fish is modelled as shown below. The fish

movement in this dynamic analysis is assumed as a planar motion.

As shown in Fig. 4.5, three coordinate systems, are defined as earth-fixed inertial reference

frame O � XY Z, link-fixed reference frame Mj

� xj

yj

zj

along the central principal axis of

the j � th link (j=0,1,...,N+1), and head-fixed one Po

� xyz respectively. Before building the

mathematical model, two limiting assumptions are firstly made for practical reasons. 1) The fish

composed of head, oscillatory part, and caudal fin forms multiple rigid bodies such that elastic

e↵ects can be ignored. 2) The fish body is well-balanced so that the center of gravity and the

center of buoyancy lie in the XZ plane and coincide with each other at most time of steady

swimming and even turning maneuvers. Letting the head be the first link (the 0� th link), the

flexible body has N links, and the caudal fin is the last link (the N+1-th link), thus, the robotic

fish has N +2 links in total, corresponding to 6N +12DOFs. But, the angle ✓j

(j = 1, 2, .., N)

between the link lj

and x � axis as well its derivative ✓j

and ✓j

can be determined by fitting

57

4.3 Final approach

the chosen body wave. Also the pitch angle of the caudal fin relative to the main axis, ✓N+1, is

predetermined. Further, is considered that ✓0 = ✓0 = ✓0. In this way, DOFof the planar motion

of the robotic fish is reduced to three ((6N + 12)� (6N + 9)), i.e., the number of independent

kinematic parameters is three. Therefore, the generalized coordinates are defined as:

q = [q1 q2 q3]T = [x y ] (4.18)

Where x, y, and denote the position of the point o (i.e. P1) in the system OXY Z and

the angle between the head and OX-axis, respectively.

4.3.2 Kinematic Analysis

To aid describing the kinematics behavior, a kinematic analysis is was implemented. A pair of

end points of the j � th link are Pj�1(xj�1, yj�1) and P

i

(xj

, yj

) in the earth-fixed frame, and

|Pj�1Pj

| = li

.The position of the center of mass Mj

(C.M.) of the j � th link is then given by:

rj

=

xj�1 + x

j

2

yj�1 + y

j

20

�T

=

2

6666664

x+NX

i=0

lk

cos( + ✓k

)� 1

2lj

cos( + ✓j

)

y +NX

i=0

lk

sin( + ✓k

)� 1

2lj

sin( + ✓j

)

0

3

7777775(4.19)

The linear velocity and acceleration of the C.M. of the j�th link, expressed in the earth-fixed

frame, are given by Eq. 4.20 and Eq. 4.21, respectively.

vj

= Hj

(q, t)q + vj

(q, t) (4.20)

aj

= vj

= Hj

(q, t)q +Kj

(q, q, t)q + aj

(q, q, t) (4.21)

Where,

Hj

(q, t) =

2

6666664

1 0jX

k=0

lk

sin( + ✓k

) +1

2lj

sin( + ✓j

)

0 1jX

k=0

lk

cos( + ✓k

) +1

2lj

cos( + ✓j

)

0 0 0

3

7777775(4.22)

58

4.3 Final approach

vj

(q, t) =

2

6666664

�jX

k=0

lk

sin( + ✓k

)✓k

+1

2lj

sin( + ✓j

)✓j

jX

k=0

lk

cos( + ✓k

)✓k

� 1

2lj

cos( + ✓j

)✓j

0

3

7777775(4.23)

Kj

(q, q, t) =dH

j

(q, t)

dt(4.24)

aj

(q, q, t) =dv

j

(q, t)

dt(4.25)

Notice that the detailed expressions of kj

(q, q, t) and aj

(q, q, t) are not expended for the

limitation of the page layout.

The angular velocity and acceleration of the C.M. of the j � th link with respect to the

earth-fixed frame are described by ~!j

= ( + ✓j

)~k = !j

~k and ~!j

= ( + ✓j

)~k = !j

~k, where ~k is

the base vector of Mj

zj

, i.e. OZ axis.

4.3.3 Dynamic Analysis

To acquire the full dynamics of the robotic fish, is necessary to consider that each part is rigid

and the forces generated by oscillation of the rear body and the caudal fin comprise external

forces such as drag, added mass force, and so on. The forces acting on each element of a multi-

body system, within the framework of Schiehlen method, include active forces F a

j

and constraint

forces F c

j

expressed in the earth-fixed reference frame. Similarly, the moments on each rigid

element with respect to the link-fixed frame comprises active moments La

j

and constraint ones

Lc

j

= [0 0 Lc

j

]T . In detail, they present in a mathematical manner shown as:

F a

j

= FIj

+ FV j

(4.26)

F c

j

= Tj�1,j + T

j+1,j (4.27)

La

j

= ⌧j�1,j + ⌧

j+1,j + ⌧I

j = [0 0 La

j

]T (4.28)

Where, FIj

is the added mass forces due to the inertia of the surrounding fluid; FV j

is

the hydrodynamic drag when j = 0, 1, ..., N , whereas FV j

is the hydrodynamic force resulting

from the oscillating motion of the caudal fin; Tj�1,j and T

j+1,j indicate the constraint forces

59

4.3 Final approach

acting on the j � th link from the j � 1� th and the j + 1� th respectively; ⌧I

j, is the added

moment induced by the inertia of the surrounding fluid; ⌧j�1,j and ⌧

j+1,j actually are the

output torques of SMAs denote the moments acting on the j� th link. Note that the definitive

form of Lc

j

coming from constraint forces is not provided just for, it could be eliminated during

the subsequent derivation.

Having analyzed forces and moments acting on each link, a basic dynamic equation incor-

porating these components can be obtained:

⇢m

j

aj

= F a

j

+ F c

j

Jj3!j

= La

j

+ Lc

j

(4.29)

where, jj3 indicates the principal moment of inertia matrix.

Furthermore, a more concise form, as a whole, can be stated as:

MH(q, t)q + K(q, q, t) = F a + F c (4.30)

where, each of the components is described in the sequence, except that K(q, q, t) is shown

in Eq. 4.37.

M = diag(Mo

M1 · · · Mj�1 M

j

... MN+1) (4.31)

Mj

= diag(mj

mj

mj

0 0 jj3) (4.32)

H(q, t) = [Ho

A H1 A · · · · · · HN+1 A]T (4.33)

A = diag(0 0 1) (4.34)

F a = [(F a

0)T (La

0)T · · · · · · (F a

N+1)T (La

N+1)T ] (4.35)

F c = [(F c

0)T (Lc

0)T · · · · · · (F c

N+1)T (Lc

N+1)T ] (4.36)

60

4.3 Final approach

K(q, q, t) =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

...

mj

j

X

k=0

⇣

�lk

cos( + ✓k

)( + ✓k

)2 � lk

sin( + ✓k

)✓k

⌘

+12m

j

lj

sin( + ✓k

)✓j

mj

j

X

k=0

⇣

�lk

sin( + ✓k

)( + ✓k

)2 � lk

cos( + ✓k

)✓k

⌘

+12m

j

lj

cos( + ✓k

)✓j

0

0

0

Jj3✓j...

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(4.37)

4.3.4 Dynamic Model

Equation 4.30 contains the terms associated with both the active and constraint forces and

moments, which is not convenient for motion control. Some reduction should be sought. Taking

into consideration that the exerted constraints are regarded as ideal constraints, as suggested

by the Schiehlen method, the following equation can then be inferred as:

HT (q, t)F c = [0 0 0]T (4.38)

Incorporating 4.38 into 4.30, we get a compact dynamic equation of the whole robotic fish:

M(q, t)q = K(q, q, t) = Q (4.39)

where,

M(q, t) = HT (q, t)MH(q, t) (4.40)

K(q, q, t) = HT (q, t)K(q, q, t) (4.41)

Q = HT (q, t)F a (4.42)

Below shown the development of expressions for the force induced by active forces. The

added mass force FIj

is expressed in the frame O �XY Z as:

FIj

= �MIj

(q, t)q + FIj

(q, q, t) (4.43)

61

4.3 Final approach

Where,

MIj

(q, t) = R( + ✓j

)MIj

RT ( + ✓j

)Hj

(4.44)

FIj

(q, q, t) = �R( + ✓j

)MIj

RT ( + ✓j

)(Kj

q + aj

) (4.45)

MIj

= diag(�i1mi

�i2mi

0) (4.46)

Note that R( + ✓j

) is a transformation matrix from the reference frame O � XY Z to

the reference frame Mj

� xj

yj

zj

. Similarly, the added moment relative to the reference frame

Mj

� xj

yj

zj

can be given by:

⌧Ij

= �JIj

q + ⌧Ij

(q, q, t) (4.47)

where,

JIj

= diag(0 0 �j6Jj3) (4.48)

⌧Ij

(q, q, t) =h0 0 � �

j6Jj3✓jiT

(4.49)

With regard to the term FV

j(q, q, t), it has that:

FV j

= R( + ✓j

)

�1

2⇢��u⇤

j

�� u⇤j

Cf

Sf

� 1

2⇢��v⇤

j

�� v⇤j

Cd

Aj

0

�T

(4.50)

for j = 0, 1, ..., N , and

FV (N+1) = R( + ✓

N+1)

1

2⇢S

N+1CT

v2N+1

1

2⇢S

N+1CL

v2N+1 0

�T

(4.51)

for j = N + 1, where (u⇤i

v⇤i

0)T = R�1( + ✓j

)vj

, ⇢, is the density of the fluid, Cf

is

the friction coe�cient, Cd

is the cross flow drag coe�cient, Sj

is the wet surface area of the

j � th propulsive component, Aj

is the area’ cross-section of the j � th propulsive component,

CT

is the thrust coe�cient relative to attack angle of the caudal fin ↵ and Strouhal number

(St

) (i.e. CT

= fT

(↵, St

)), and CL

is the lift coe�cient, which is also a function of ↵ and

St(CL

= fL

(a, St

). Note that where, all these hydrodynamic parameters may be determined

from experimental testing techniques.

62

4.3 Final approach

Through some superposition and reduction operations, we can obtain:

Q = �MI

(q, t)q + Q(q, q, t) (4.52)

where,

MI

(q, t) = HT (q, t)hM

I0 JI0 · · · · · · M

I(N+1) JI(N+1)

i(4.53)

and Q = HT (q, t)

0

BBBB@

2

66664

...FIj

(q, q, t)⌧Ij

(q, q, t)...

3

77775+

2

66664

...F

V j

(q, q, t)⌧ (j�1,j) + ⌧

j+1,j...

3

77775

1

CCCCA(4.54)

In particular, the terms ⌧ (j�1,j) and ⌧ (j+1,j) can be cancelled, according to the form of

H(q, t). As a consequence, Eq. 4.54 can be written as:

Q = Q(q, q, t) (4.55)

Synthesising Eq. 4.39 and Eq. 4.55, finally the following dynamic model is obtained:

(M(q, t) +MI

(q, t))q +K(q, q, t) = Q(q, q, t) (4.56)

where, M(q, t) + MI

(q, t) is the mass matrix incorporating all masses and inertias of the

robotic fish,which also includes the virtual terms associated with the accelerated surrounding

fluid, and K(q, q, t) contains the matrix of Coriolis/Centripetal term.

The Eq. 4.56, in essence, is a nonlinear ordinary di↵erential equation, is possible to apply a

standard integration technique to solve it time step by step. Based on the presented dynamic

model, is possible to predict the propulsive characteristics of the robotic fish by using Matlab-

Simulink.

4.3.5 SMA phenomenological model

SMAs exhibit an unique thermomechanical property due to the phase transformation of the

material, from austenite phase to martensite phase and vice versa. These transformations

mainly occur due to changes in temperature and stress. Extensive research has been devoted

to model these properties. Tanaka in (166) was one of the pioneers to study a stress-induced

martensite phase transformation, proposing an unified one-dimensional phenomenological model

63

4.3 Final approach

that make use of three state variables to describe that process: temperature T , strain ✏, and

martensite fraction ⇠. His main contribution was to demonstrate that the rate of stress is a

function of strain, temperature and martensite fraction rates. Later, Brinson (167) improved

on Tanaka’s model by separating the calculation of the martensite fraction into two parts,

one induced by stress and the other one induced by temperature. This issue allowed for the

description of the shape memory e↵ect at low temperatures.

Elahinia (73),(168) proposed an enhanced phenomenological model compared to the previous

ones, and also addressed the nonlinear control problem. This model was able to better describe

the behavior of SMAs in cases where the temperature and stress states changed simultaneously.

Their model was verified against experimental data regarding a SMA-actuated robotic arm. As

a result, the phenomenological model was able to predict SMA behavior also under complex

thermomechanical loadings. Further experiments were also carried out in (169).

In this thesis, Elahinia’s phenomenological model (168) has been used for assessing the

limits of SMA operation. The model consists of four parts: i) heat transfer, ii) mechanics

model, iii) forward/reverse phase transformation, and iv) kinematics model. The input of

the model is the electrical current Isma

to drive the SMA and the output is the bend angle

that is produced by the strain rate of the SMA when bend the polycarbonate segment. This

model allows for determining proper parameters to safe overload SMA performance without

compromising physical damage to the shape memory e↵ect or overheating issues when subjected

to high amount of input power..

Heat transfer model

The SMA wire heat transfer equation consists of electrical (Joule) heating and natural convec-

tion:

msma

cp

T = I2sma

Rsma

� hc

Ac

(T � To

) (4.57)

SMA NiTi wires have a diameter of 150µm, a mass per unit length of msma

= ⇢⇡r2j

where ⇢

is the density of wire, 2rj

is diameter of wire, Ac

= ⇡2rj

is circumferential area of the unit length

of the wire, cp

is specific heat, Isma

is applied electrical current, Rsma

is electrical resistance per

unit length of the wire, T is temperature of the wire, To

is the ambient temperature, and hc

is

the heat convection coe�cient. Although in Eq. (4.57) is assumed that hc

and Rsma

are both

constant. Using Eq. 4.57 is possible to model how the NiTi wire would heat upon electrical

64

4.3 Final approach

current and by removing the term I2sma

Rsma

(heating power), the equation can be also used to

model how the NiTi wire cools in the absence of heating power.

Mechanical model

SMA mechanical model was firstly introduced by Tanaka in (166). It relates stress rate (�)

with temperature rate (T ) as:

� = ✓

s

�⌦(Af

�A

s

)�1

1�⌦(Af

�A

s

)�1C

m

T (4.58)

Where ✓s

corresponds to the thermal expansion factor of the wire, ⌦ is the phase transfor-

mation factor, Af

, As

are the austenite final and initial temperatures and Cm

is the e↵ect of

stress coe�cient on martensite temperature. Also, the strain rate (") during heating phase can

be calculated as:

" = ��✓s

T�⌦⇠E

A

(4.59)

Where EA

is the austenite the Young’s modulus and ⇠ is the phase transformation rate

which is presented in the following.

Phase transformation model

The reverse transformation equation that describes the phase transformation from martensite

to austenite during heating is:

⇠ = ⇠

m

2 [cos (aA

(T �As

) + bA

�) + 1] (4.60)

where ⇠ is martensite fraction that has a value between 1 (martensite phase) and 0 (austenite

phase). The terms aA

= ⇡(Af

� As

)�1 and bA

= �aA

C�1A

are the curve-fitting parameters

of the phase transformation. Also, the forward transformation equation describing the phase

transformation from austenite to martensite during cooling is:

⇠ = 1�⇠a

2

hcos (a

M

(T �Mf

) + bM

�) + 1+⇠a

2

i(4.61)

Where aM

= ⇡(Mf

� Ms

)�1 and bM

= �aM

C�1M

are the curve-fitting parameters, where

Mf

,Ms

are the martensite phase final and initial temperature respectively.

65

4.4 Geometry of bending

Table 4.1: Parameters for SMA phenomenological model

Variable Model Parameters Value [unit]

Temperature

Heating: m

sma

,R

sma

, I

sma

1.14 ⇥ 10�4 [Kg], 8.5 [⌦]

m

sma

c

p

T = I

2sma

R � h

c

A

c

(T � T

o

) A

c

1.76 ⇥ 10�8h

m

2i

Cooling: h

c

150h

Jm

�2�C

�1s

�1i

(T ) m

sma

c

p

T = �h

c

A

c

(T � T

o

) C

p

0.2h

KcalKg

�1�C

�1i

Stress (�)

Heating: ⌦ �1.12 [GP

a

]

� =✓

s

�⌦⇣

A

f

�A

s

⌘�1

1�⌦⇣

A

f

�A

s

⌘�1C

m

T ✓

s

0.55h

MP

�a

C

�1i

Cooling: C

m

,C

a

10.3h

MP

�a

C

�1i

� =✓

s

�⌦⇣

M

s

�M

f

⌘�1

1�⌦⇣

M

s

�M

f

⌘�1C

a

T A

s

,A

f

,M

s

,M

f

68, 78, 52, 42⇥�

C

⇤

Strain (✏)

Heating:

" = ��✓

s

T�⌦⇠

E

A

E

A

75 [GP

a

]

Cooling: E

M

28 [GP

a

]

" = ��✓

s

T�⌦⇠

E

M

FM (⇠)

Heating: ⇠

m

, ⇠

a

1, 0 [dimensionless]

⇠ = ⇠

m

2

⇥

cos�

a

A

(T � A

s

) + b

A

�

�

+ 1⇤

a

A

0.31h�

C

�1i

Cooling: a

M

0.31h�

C

�1i

⇠ = 1�⇠

a

2

h

cos⇣

a

M

⇣

T � M

f

⌘

+ b

M

�

⌘

+ 1+⇠

a

2

i

b

A

, b

M

�0.03h�

C

�1i

Table 4.1 summarizes the parameters used for the simulation of the thermo-mechanical

equations. Further details on the values assigned to most coe�cients can be also found in (73)

and (168).


The BR3 movement is based on a flexible and continuous backbone as well as for the caudal

fin. Measure its bending angle can be done by using Flex Sensors. However to measure speed

and acceleration commonly Accelerometer Sensors are used. Due to the movement of the body

and caudal fin (as well as for the limitation in the thickness of the caudal fin) is not possible to

use this kind of sensors. For this reason a solution based on the flex sensor is proposed in this

section.

4.4.1 Bend angle

The bend angle � (figure 5.4 shows this bend angle) was determine experimentally by setting

the relation between the measures coming from the flex sensor (in values of electrical resistance)

and the bend angles measured. Due to the linear variation of the resistance in the Flex sensor,

wit only one measure is possible to estimate this relation. Figure 4.6 shows the backbone

performing a bending angle of 10o for both segments.

Based on the Marten Nettelbladt studies (170) about the Geometry of Bending is possible

to use the Pythagorean theorem to formulate a mathematical relationship between (Figure 4.7)

66


(a) (b) (c)

(d) (e) (f)

Figure 4.6: Bend angle for the head and tail segments. (a) Reference tail, angle 0o, (b) Tail

left, angle 10o, (c) Tail right, angle 10o, (d) Reference head, angle 0o, (e) Head left, angle 10o, (f)

Head right, angle 10o

67


Figure 4.7: The geometry of bending

the original length L, distance d between end points and curve height h:

h2 =

✓2L

5

◆2

� d� L

5

2

!2

(4.62)

4.4.2 Acceleration

In physics, acceleration, a, is the amount by which the velocity changes in a given amount of

time. Given the initial and final velocities, vi

and vf

, and the initial and final times over which

the speed changes, ti

and tf

, the equation can be written as:

a =�V

�t=

vf

� vi

tf

� ti

(4.63)

Likewise is possible to relate displacement s, acceleration a, and time t as follows:

s = vi

t+1

2at2 (4.64)

the initial velocity (vi

) is 0 because the bending movement is not allowed to take a running

start due to the e↵ect of the antagonistic SMA. The equation 4.64 can be rearranged to solve

for acceleration as:

a =2s

t2(m/s2) (4.65)

The displacement s is determined by using the information coming from the flex sensor.

The same technique shown in section 4.4.1 was used to find the relation between displacement

and resistance.

68

4.5 Swim patterns

4.5 Swim patterns

Swim patterns can be divided into two categories: periodic and aperiodic. Periodic swimming

refers to cruise (steady) swimming and in-cruise turns, while aperiodic swimming refers to

sudden changes of directions (also referred to in the literature as ”snap”-turns) and fast starts.

4.5.1 Steady swimming (cruise straight)

Pioneer work on swimming patterns is due to (171, 172, 173). For steady forward swimming,

the body motion function can be described by the following equation:

y = fS

(x, t) = (c1x+ c2x2)sin(

2⇡

�x+ !t) (4.66)

where x is the longitudinal position with respect to the head of the fish and y is the lateral

displacement. The c1 and c2 parameters define the wave amplitude, � is the wave length, and

! the wave frequency. The curve fS

is transformed into the curve fT

, representing the position

relative to the head (174):

y = fT

(x, t) = (c1x+ c2x2)sin(

2⇡

�x+ !t)� c1xsin(!t) (4.67)

For modeling purposes, the robot fish is implemented as a discrete number of elements,

and the propagated wave function must be discretized (approximated by segments) in order

to be reproduced. Therefore, the function that describes the wave propagation is defined as a

sinusoidal-based time-dependent joint angle function qj

, where j is the joint index (174):

qj

(t) = aj

· sin(!t+ �j

), j = 1..number of joints (4.68)

In our system, only the last two divisions are used for steady swimming (j 2 {body, tail});

(number of joints = 2). In order to analyze and simulate the system, it is assumed that these

two virtual joints are governed using (4.68), where bending �j

⌘ qj

. Since the actuators bend

the structure into arcs, the curve in (4.66) is approximated by circles (see Fig. 4.8).

4.5.2 Cruise-in turning

During steady swimming, smooth changes of direction, referred to as cruise-in turning, can be

modeled as an asymmetry on the undulation with respect to the longitudinal axis. This can be

modeled adding a bias function that defines a deflection curve:

y = fS

(x, t) + d(x) (4.69)

69

4.5 Swim patterns

Figure 4.8: Approximation of fT

(x, t) (solid lines) with circle arcs (dotted lines). ! = �⇡, c1 =4.5479

L

, c2 = 0,� = 2⇡4L . The blue circles represent the end point position of the fish bone seg-

ment of L=8.5 cm. c1 corresponds to the maximum achievable bending, and � has been set for

subcarangiform swimming, where half a wave length is reproduced by the body consisting of two

segments. The trajectory of the end point of the fishbone segment is shown by the arrow.

On the practical side, this implies that the joint equation in (4.68) becomes:

�j

(t) = qj

(t) = aj

· sin(!t+ �j

) + bj

, (4.70)

where the quantity bj

is related to the curvature radius of the turn. For articulated bodies, it

is easy to see that the bias bj

for each joint and the direction h of the last body w.r.t the first

is: b = h/n, being n the number of joints. On the other hand, for a circular arc of length L,

the relationship between its radius r and the central angle ✓ is r = L/✓. Since h = ✓, we have

that:

bj

=L

n · r . (4.71)

In our case, L = 0.3m, n = 2 (virtual) joints, and the bias parameters bj

in (4.71) can be

easily calculated given the desired turning radius r.

4.5.3 C-starts

This kind of aperiodic pattern is used for fast turns in response to external stimuli (e.g. for

escaping from a menace or for capturing a prey). It comes into two ways: C-turns and S-turns,

70

4.6 Simulation and experimental results

referring to the shape the fish takes during the maneuver. In real fishes, such maneuvers take

the order of milliseconds, and are activated by white (fast-twitch) muscles. It must be pointed

out that our SMA actuator takes about 0.5 seconds (overloaded) to achieve the maximum

curvature of 36o. This is why it makes sense to adopt them as slow-twitch (red) muscles for

steady swimming. Nonetheless, in order to test the limits and possibilities of this concept

regarding SMA technology, it was implemented and simulated one of such ”fast” stars; the

C-shaped.


Simulations and experiments are aimed at analyzing three issues:

1. Bend forces for actuation: It quantifies the required forces to properly bend the body at

each segment. It allows for the characterization of actuators (simulation).

2. Bend-angle to torque-force converter : It explores the ability to convert the swimming

patterns into bend angle for the antagonistic actuation at each segment and then trans-

late this angles to a singular torque-force to properly move the joints (simulation and

experimental).

3. Swimming pattern generation: It explores the ability to create automatically a desir-

able swimming pattern depending on the hydrodynamic characteristics (simulation and

experimental).

4. Body torques for maneuvering : It quantifies the influence of body bend into the production

of yaw torques for maneuvering (simulation and experimental).

5. SMA actuation limits: It explores the limits to safe overload the response of the SMA

actuators by defining the maximum value of input electrical current that achieves the

fastest bend speed of the segments (simulation).

An open-loop Matlab-based simulator has been implemented using the SimMechanics toolbox

of Simulink1. Figure 4.9 details the main modules that compose the simulator. One key

advantage of the simulator consists on the possibility of using the CAD model of the robot

exported directly from Autodesk Inventor2. This allows to include the mechanical assembly

together kinematics and dynamics properties of the robot into the simulation environment.

The simulator will be extended to a closed-loop architecture in Chapter 7.

1http://www.mathworks.es/products/simmechanics/index.html

2http://www.autodesk.es/products/inventor/overview

71

http://www.mathworks.es/products/simmechanics/index.html

http://www.autodesk.es/products/inventor/overview


4.6.1 Open-loop simulator

This simulator is composed by the following modules (cf. Figure 4.9):

• Bio-inspired Trajectories Body Segment 1, 2 : It generates trajectory patterns for each

joint of the body at specified bend angle � and bend frequency f . These patterns are

similar to those shown in Figure 4.8.

• Joint Actuator 1, 2 : This contains the kinematic, dynamic and hydrodynamic modules.

• Mechanics module: It contains the mechanical assembly and properties of robot’s CAD

exported directly from Autodesk Invenotr (figure 4.10).

4.6.2 Bio-hydrodynamics simulator

In order to assess the e↵ectiveness of the proposed structure, extensive numerical simulations

was conducted, using the Bio-hydrodynamics Toolbox (175) for Matlab(TM) (BhT). Such tool-

box provides a simple but thorough simulation tool. It allows to perform numerical simulation

involving 2D motions of rigid bodies in an ideal fluid. BhT is based on the Lagrangian formalism

(least action principle).

The toolbox requires the model to be composed of articulated bodies. Shape-changes be-

tween the bodies generate hydrodynamic forces and torques by which the bodies propel and

steer themselves. Such a physical system based on both solid mechanics and fluid mechanics is

called fluid-structure interaction system. The fluid model of BhT is the one for a perfect fluid,

and water’s pressure does not cause the body to bend.

For the purpose of the simulation, the body of the fish was discretized into nine bodies (four

for each fish body segment), with a mass proportional to the corresponding section of the fish

body. The density of the material was set equal to the density of the water (neutral buoyancy),

whereas model’s weight is 200 grams.

In order to model the torsional torque ⌧�

required to bend the polycarnobate backbone’s

structure and thus achieving the angle �, the V-shaped SMA actuators and the polycarnobate

structure have been modeled as a spring-like mechanism that provides a restoring force F =

�kX after bending. The torsional torque is described as: ⌧B

= Ji

�+c�+Fpull,max

cos��khL1,

being the structure’s moment of inertia Ji

= 2.21⇥ 10�5 Kgm2, and the bending angular rate

and acceleration �, � respectively. The term c is the torsional damping coe�cient obtained

from the average values of computed strain forces of the SMAs (see phenomenological model

in Appendix), k = 0.2987 Nm�1 correspond to the spring constant property of the structure

72


Figure 4.9: SimMechanics open-loop simulator for dynamics and SMA actuation.

73


Figure 4.10: 3D escenario.

when recovering its shape after bending, h = 0.01 m is the maximum displacement of the

structure when bending, and L1 is the length of the polycarbonate segment. The term Fpull,max

correspond to the theoretical maximum pull force of the SMA wire, about 230g � F . Figure

4.11c shows the model, and Figures 4.11b and 4.11d shows a plot of the required bending torque.

4.6.3 Steady swimming

Given the kinematic characteristics of our system, and based on the observation of fish swim-

ming, the quantities described in (4.68) correspond to: abody

= atail

/2,�tail

= �⇡/4,�body

= 0.

atail

= 0.49, for nominal SMA contraction (4%), and atail

= 0.54 for overloaded SMA contrac-

tion (6%).

Figure 4.11a shows an example of the trajectory followed by the fish. Table 4.2 reports the

simulation results for steady swimming for various combinations of the maximum amplitude

and frequency of the undulation (linear speed). The best values for linear speed V are achieved

for the largest amplitudes, which can be obtained overloading the SMA as described earlier. As

expected, such currents induce a further stress on the SMAs, but it is worth noting that, due

to the oscillatory nature of the actuation, high peaks are maintained only for short periods of

time. On the other hand for low amplitudes, the speed achieved has no significant di↵erences

as far as tail-beat frequency.

74


! " # $ % & ' ( )ï%

ï$

ï#

ï"

!

"

#

$

%

00.050.10.150.2

-0.05-0.1

-0.15-0.2 0

0.050.10.150.20.250.30.350.40.450.5

0 1 2 3 4 5 6 7 80 1 2 3 4 5 6 7 8

!"#[N

m]

I [A

]

t [s] t [s]

t1

td!

t2 t

! x

y!!

(c)

L1

(a)

(d)

damping contribution

SMA contribution

(b)

2 4 6 8 10 12 14 16

Figure 4.11: a) Steady forward swimming (atail

= 0.54, abody

= atail

/2,�tail

= �⇡/4,�body

=

0, tail-beat frequency=1/2 Hz). b) Simulation of the torsional torque ⌧�

required to bend the

polycarbonate structure by using the V-shaped SMA actuators at tc

= 0.5s. c) Top view of the

antagonistic V-shaped wires fixed to the backbone. For modeling, the bending property of the

backbone is considered as a spring. d) Bending angle � profile during SMA contraction: during

t1, the active actuator contracts upon heating, achieving a bending angle of 36o, subsequently,

during td

= 200ms, both pair of antagonistic actuators remain passive, and the decrease of the

bending angle is provided by the restoring force caused by the polycarbonate structure trying to

recover its original shape (i.e. spring- damping force). During t2 the antagonistic actuator turns

active providing the opposite motion.

4.6.4 Morphology parameters

Besides the linear speed V , some other parameters are useful for evaluating quantitatively the

soundness of the design. Speed is expressed as body lengths (L) / second, VBL

= V/L, and

results are reported in the last column of Table 4.2.

The Head swing factor Sh

is the ratio between the head oscillation’s amplitude induced by

body motion, and the tail tip oscillations amplitude (Sh

= Ahead

/Atail

) . It ranges between

0.15 and 0.4 in real fishes. High values indicate that a large amount of energy is wasted because

the head oscillates too much and has to push water to the sides. The best result achieved has

been of Sh

= 0.19, using the parameters configuration: atail

= 0.49, abody

= 0.27, f = 1/2 Hz.

Finally, the Strouhal number St

= ftail

·Atail

/V , where ftail

is the tail-beat frequency, Atail

75


Table 4.2: Summary of the performances of the simulations on steady swimming

Maximum Tailbeat V VBL

amplitude (atail

) freq. (Hz) (m/s) (Hz)

1/8 0.025 0.083

Half (0.25) 1/4 0.025 0.083

1/2 0.025 0.083

1/8 0.046 0.15

Nominal (0.49) 1/4 0.048 0.16

1/2 0.052 0.17

1/8 0.093 0.31

Overloaded (0.54) 1/4 0.105 0.35

1/2 0.114 0.38

is the tail-beat peak-peak amplitude and V is the linear speed, refers to the wake vortexes

generated by the fish tail. Its value for real fishes lays in the range 0.25 to 0.35. In the

simulations, our robot fish achieved values in the range [0.12 � 0.73], depending on tail beat

frequency and amplitude. The value corresponding to Fig. 4.11 is St

= 0.41.

Figure 4.12: Cruise-in turning. Labels refer to the desired turning radius (meters), corresponding

(from left to right) to bj

= 0.0375, 0.05, 0.075, 0.15, 0.3, (see (4.71)).

4.6.5 In-cruise turning

Figure 4.12 shows the result for the cruise-in turning maneuver. The turning radius resulting

from the simulations is in well agreement with the theoretical values calculated in (4.71). Using

the same relationship, is possible to compute the minimum turning radius of the robot. Given

that the maximum biases btail

and bbody

are equal to the maximum amplitudes atail

, abody

, i.e.

the oscillation is all at one side (e.g., 0 to atail

instead of �atail

to atail

) the minimum radius will

be rmin

= L

n·bmax

. This corresponds to a minimum theoretical turning radius of 0.83 meters.

76


Figure 4.13: Stills of the C-start maneuver of the simulation and with the real prototype

In the simulations, the fastest turn (minimum turning radius r) has been achieved for the case

of r = 0.5m, which corresponds to a maximum angular speed of 6 deg/s.

4.6.6 C-starts

In this maneuver all the two segments of the fish are bent, the head also takes part to the

maneuver. A third virtual joint located onto the fish ”neck” (right behind the head) also

activates. A sharp turning C-start maneuver implies contracting the muscles on the same side in

a synchronized way (i.e. �tail

= �body

= �head

). The velocity and amplitude of the contraction

induces the turning angle. Figure 8.13 (left) shows various stages of the best performing turn

that can be obtained with a 1 second stroke of 28 degrees, corresponding to nominal SMA

current (300 mA current pulse, for a contraction time of 1 second) for the three segments.

From the Figure, it can be appreciated that the fish turns of about 45� in one second during

the start stroke. After the starting stroke, the fish recovers the straight shape and begins steady

swimming. During this period of time (approximately two tail beats) the fish holds the angular

momentum generated with the initial stroke, and turns further 15�, escaping with a final angle

of approximately 60�.

Figure 8.13 shows a comparison, where only a small delay of about 0.5 seconds can be

noticed.

77

4.7 Final remarks and conclusions

2 bendingsegments

(a) (b) (c)

2 bendingsegments

! ~36[deg]

Figure 4.14: Comparison of qualitative assessment, numerical simulations and experimental

results

4.7 Final remarks and conclusions

The results of the modeling and of the numerical simulations confirm that the concept allows

the robot to perform the main maneuvers according to the theory and models of fish swimming.

I would like to point out that the aim of this work is not to build a ”better” robot fish, but

rather to experiment and elaborate on the concept of gear-less and motor-less robots, in order to

develop the understanding of a technology that may be applied to the next generation of robots.

In terms of the mechatronics design, the simplicity, extremely light weight and practically null

volume of the actuators (at advantage of the available payload), make the fish-like robot suitable

for biological applications. The actuators are absolutely silent and do not produce any vibration,

a feature which can be exploited, e.g. in the observation of sea wild life since the robot would

not disturb in any way (besides its presence). Beyond that, the novelty of using a bendable

structure as the backbone of the fish based on the V-shape configuration of antagonistic SMA-

based actuation muscles has great potential for improving the maneuverability of the fish while

performing the aforementioned swim patterns underwater. In conclusion, I believe that smart

actuators and flexible continuous structures can be a promising field for making alternative

bio-inspired robots, devoid of rotating parts and that are simpler and lighter, and that can

have interesting application domains.

78

5

BR3 design and Fabrication

”We get it almost every night, When that moon is big and bright, It’s a supernatural delight,

Everybody’s dancing in the moonlight”

Toploader (Dancing in the Moonlight)

5.1 The general method for BR3 design

This chapter introduces the design and fabrication process of BR3. This chapter is not

intended to cover performance of the mechanical designs presented, but only the

design process, main functions and the criteria involved. Refer to Chapters 4

(modelling) and 7 (Control) for detailed experiments regarding the mechanical

approaches introduced herein. In this chapter, one of the most important challenges to

be tackled concerned to the design criteria to specify morphological and actuation parameters.

Because the novelty of the robot and the lack of information regarding design issues, this

thesis was meant to design BR3 from the analysis of a specific fish specimen aimed at carefully

mimicking each detail related to morphology, biomechanics, kinematics and even the muscle-

like body and caudal fin actuation system. Chapter 3 already presented this biological analysis

by concluding the chapter with key issues or foundations to the design process. Here, those

foundations are bring back and incorporated into the design framework that brings BR3 to live.

The entire fish was designed based on a 3D model from a real Black-Bass that was scanned

using a 3D laser scanner and reproduced in the Autodesk Inventor (176) CAD software.

Annexes Section 11.1 shows the methodology developed to convert the scanned 3D model of fish

in a 3D model suitable to be used, edited and simulated using Inventor and Matlab. Currently

existing CAD software allows you to import 3D models with di↵erent extensions, but not allow

79

5.2 Body

these to be modified. i.e. what you have import is what you have. Due to my model had to be

extensively modified to allow docked the spine as well as to bring inside the sensors, actuators

and batteries, have a model that can not be changed was something I could not a↵ord. To

perform this task many software were used.

5.2 Body

The robotic black bass is inspired by the European Sea Bass. This species exhibits a Carangi-

form swimming in cruiser and sprint swimming mode.

The main component of the robot is a continuous structure made of polycarbonate of 1mm

thickness, that represents the fish backbone and main spines (Fig. 11.4). This material was

chosen for its flexibility and temperature resistance, since SMAs can heat up to 90ocelsius. Note

that the Backbone has two rectangular holes in correspondence to the SMAs. In this way, the

actual cross section of the polycarbonate structure is reduced approximately 50% reducing its

resistance to bending and optimising the pull force on the SMA. Additionally, it prevents the

external SMA wires to be over stretched when the structure bends.

The backbone of the fish has a 263mm length (note that the first section corresponds to the

head and accounts for approximately 35% of total body length).

The spine is used to support a set of 20 solid sections made of ABS plastic called Ribs1.

Based on my own previous experience i used solid ribs to give more resistance, and a high

number of them in order to achieve a more realistic movement, since the more sections have,

the smoother curves will be obtained and less flow perturbation will be created. Solid Ribs also

add more weight to the robot, which helps to reduce its buoyancy.

5.2.1 Ribs

The entire fish was designed based on a 3D model from a real Black-Bass that was scanned

using a 3D laser scanner and reproduced in the Autodesk Inventor (176) CAD software.

For simulation purposes, the model was exported from Auodesk Inventor to MATLAB

using the SimMechanics toolbox. Due to limitations of the software to simulate complex

flexible structures, the entire body was linked using primitive joints with one rotational degree

of freedom of the same length of the Ribs. Figure 5.2 shows the simulated model.

1Most fishes do not actually have ribs. This term is used here only in metaphorically.

80

5.2 Body

TailSMA’s

HeadSMA’s

TailFlex Sensors

HeadFlex Sensors

TailCurrent Sensors

TailPower Drivers

HeadCurrent Sensors

HeadPower Drivers

TemperatureSensor

Polycarbonate Backbone

6DOF IMU

(a) Backbone and electronic components

HeadFlex Sensors Tail

Flex SensorsTail

SMA’sHeadSMA’s


(b) Backbone with attached flex sensors

Figure 5.1: Backbones

Using the CAD model is possible to estimate the maximum rotational angle for each section

for various Ribs thicknesses, inter-Rib space and number of Ribs, as shown in Figure 5.2. Note

that in order to allow to the movement of the SMAs inside the fish, the inner section of each

Rib must be partially hollow (Fig. 5.2(b)). Examples of the Ribs’ dimensions is show in Table

5.1. The last section (Rib number 20) is used as a dock to attach di↵erent caudal fins.

5.2.2 Actuation

The backbone of the fish is divided in two sections of di↵erent lengths. Each section has two

antagonist SMA wires actuators attached that, when powered, bends the corresponding section

of the backbone. The SMA wires are 234mm and 212mm long and are arranged in a V-shaped

configuration (8) (177), in order to double the pull force. Figure 11.4 shows the structure

81

5.2 Body

~37º~38º

(a)

A

B

C

D

H1

H2

EF

G

H

I

J

K

(b)

L

(c)

Figure 5.2: Simulated BR3 in SimMechanics. (a) Top view: the red line represents the backbone,

while the dotted black lines represent the contracted SMAs. The angles shown are related to

the number and thickness of the Ribs, inter-Rib spaces and SMA length when contracted. (b)

FrontView,(c) CrossSection Rib (see also Tab. 5.1).

of the backbone and the location of the SMA wires. This arrangement achieves a maximum

bending of 40 degrees, regardless the fact that SMA wires only contracts a maximum of 4% of

their length. The diameter of the SMA wires adopted is 0.2mm, which provides a good trade

o↵ between current consumption (410mA nominal), and pull force (321grams). The Cooling

time is 1.7 Seconds when the wire is heated to 90�C. Using higher currents than 410mA the

contraction time can be reduced, at the cost of inducing fatigue (resulting in a reduced strain)

in the actuator after some operation time (see (9)).

The Head and Tail section has the same operation mode. Depending on the direction of the

82

5.2 Body

Table 5.1: Size Comparison for the rib number 20 and 7

Section Number A B C D E F G H H1 H2 I J K L

7 38 43.5 50.5 56 1.5 3.331 4.469 25 12.5 12.5 11 31 11 10.5

20 6 7 18.5 19.5 1.5 3.557 2.559 25 12.5 12.5 11 12 11 8

Measures are in millimetres (mm)

Start

SML On

Angle is Zero

DP reach

Desired Position

(DP)

SML Off

Angle SML Off

SMR Off

Angle SML Off

SMR Off

SMR On

Desired Position

(DP)

SMR On

A

A

B

B

Angle is Zero

DP reach

YES

NO

YES

NOYES

NO

YES

NO

Figure 5.3: Operation mode flow chart. ”SMA Wire Left” (SWL), ”SMA Wire Right” (SWR)

movement (left and right), the SMA’s corresponding to each same side is actuated. the Figure

5.3 show the flow chart for the Tail section movement (the head section has the same operating

principle).

When the SWL is contracted (the backbone is bent to the left side) the SWR passes through

the hole between them.

Note that when a SMA is contracted, due to the gaps in the backbone, the SMA antagonist

SMA passes to its opposite side can’t be actuated because its contraction would obstruct the

relaxing of the backbone. This singularity is shown in Figures 5.4. Angle � is know and ↵ can

be calculated as ↵ = 180o � 90o � �.

The singularity can be analysed considering the force components involved (see Fig. 5.5) and

visualized as shown in the Figure 5.6, where in can be noticed that the antagonistic SMA

should not be actuated until the angle � is zero otherwise the structure will remains in the

same position (bend side) or will contract in the opposite direction to the SMA that is being

83

5.2 Body


SMA Wire Left

SMA Wire Right

h

L1

L2L3

↵ �

� �

Figure 5.4: Principle of the bendable structure. The SMA wires are parallel to the backbone

segment. As a SMA contracts, it causes the polycarbonate backbone to bend (angles ↵ and �)

the antagonist SMA generates the angles � and �. L1 is the length segment of the ”Polycarbonate

Backbone”. L2 and L3 are the length of the contracted and relaxed SMA respectively.

F1 F2

F3 F4 F2x

F2y

↵

�

F1x

F1y

↵

�

RR

Figure 5.5: Components of the resultant force R. F4 = F2 = 321gf , F2x = F2cos(↵), F2y =

F2sin(↵), and the resultant forces areRy

=P

Fy

= �F2y, Rx

=P

Fx

= F2x � F4

actuated.

F4 = F2 = 321gf (5.1)

F2x = F2Cos(↵) (5.2)

F2y = F2Sin(↵) (5.3)

The Resultant forces are:

Ry

=X

Fy

= �F2y (5.4)

Rx

=X

Fx

= F2x � F4 (5.5)

84

5.2 Body

0 50 100 150 200 250 300

−300

−200

−100

0

100

200

300

Resultant Force

Rx (Grams Force)

Ry

(Gra

ms

Forc

e)

Figure 5.6: Evolution of the forces corresponding to the resultant forces Rx

and Ry

R =qR2

x

+R2y

(5.6)

angle formed by the resultant force R with the x-axis

' = tan�1(R

y

Rx

) (5.7)

5.2.2.1 Swimming modes

Fishes swim in a variety of modes. My robot is designed to be capable of reproducing dif-

ferent body-caudal fin (BCF) swimming modes, namely Thunniform, Carangiform and Sub-

Carangiform1, as its predecessor iTuna (8). Such swimming modes have been simulated using

the CAD model described above. Figure 5.7 shows some sample stills of swimming modes.

5.2.3 Skin

All the robot fish except its head is covered with a synthetic skin. The physical characteristics

of this skin makes it one of the most complex part in the fish design. It must satisfy three

main objectives. The first, mobility, refers to allowing the robot to bend his body without

resistance. Clearly, the induced resistance in the fish movement due to the skin e↵ect should

be as small as possible. For a good mobility, the elasticity is the most important characteristic

of the material used. The second objective is impermeability, to protect all the electronic

componentes (sensors, batteries and actuators). Water Impermeability is the most important

characteristic of the material used for the skin, also when the material is stretched. The third

objective refers to durability, i.e. mechanical resistance.

1For more details about swimming modes and their modelling refer to (69) and (174).

85

5.2 Body

(a) Thunniform Up (b) Thunniform Down

(c) Caranguiform Up (d) Caranguiform Down

(e) Subcaranguiform Up (f) Subcaranguiform Down

Figure 5.7: Simulated Swimming patterns

A variety of materials and techniques was tested in order to produce a skin with good

values for the three parameters described above. Three di↵erent materials (Latex, Liquid

Silicone Rubber1 and Rubber paint2) were used for producing a protective (waterproof) layer.

However, these materials by themselves could not reach the impermeability target, because

after some stretching they allow water to leak in. A way to solve this problem is to increase the

amount of layers applied, at the cost of reducing the elasticity and thus a↵ecting skin mobility.

The results of these early tests highlighted the importance of including a material that serves

as support for the protective material. Such a ”carrier” material should provide the function of

preventing the formation of holes in the external layer, preventing localised over-stretching, and

helping a more uniform distribution of the protective material. Furthermore, it helps giving

the skin a specific shape.

Following a bio-inspiration leitmotiv, was analyzed how this problem is solved in Nature.

Biological skin consists of two layers (epidermis, the outer layer and dermis, the interna layer),

plus a hypodermic subcutaneous layer (180). The epidermis of fish consists of an stratified

squamous epithelium non-keratinised epithelium this is the outer skin layer. The number of

cell layers can vary from two in larvae to ten or more in adults. The function of the epidermis

1The Liquid Silicone Rubber used was Dragon Skin 10 Medium, High Performance Silicone Rubber (178)2The Rubber paint used was P lasti�Dip Multi-purpose Rubber Coating Aerosol Spray (179)

86

5.2 Body

(including the mucus secreted by the mucous glands that covers the entire body of the fish)

is to protect against the growth of bacteria, fungi and other microorganisms. The dermis has

several fa structural function, it gives support, density and strength to the epidermis.

Alike Its high concentration of collagen and elastic (collagen and elastin) fibers also provides

elasticity to the entire skin. Besides the dermis is composed by dense connective tissue or

fibrous; has several layers of pigment cells in the marginal portion between the epidermis and

the subcutaneous layer (hypodermis), which gives the color to the fishes, macrophages and mast

cells (181).

In a similar fashion, I have designed a synthetic skin which is composed of two main com-

ponent: a structural component and a protective component. As carrier component (dermis)

I have adopted a Lycra Microfiber Mesh (LMM) due to its flexibility (so it does not a↵ect

mobility) and mechanical resistance (enhanced durability) and because it allows a good adher-

ence of the protective component (latex, liquid silicone and rubber paint). Using such substrate,

I have tested again the three protective components mentioned earlier (representing the epi-

dermis). Figure 5.8 shows the building process. First, a mold (Figure 5.9(a)) is produced by

3D printing. This is then covered with the carrier LMM (Figure 5.9(b)). Finally, I have apply

over this the protective material, which comes in either liquid or spray form (Figure 5.8(c)).

A sample of the ’Silicone Rubber Skin Tissue’ can be seen in Figure 5.9. Table 6.1 shows a

qualitative comparison between the six resulting skin models

The properties of this new material needs to fulfil two characteristics, Flexibility (don’t

reduce the Mobility) and Integration (stick to other material and serves as support). Figure

5.9(a) shows the body fish mold used to create the models for each material. this molde was

printed in a 3d printer and its made of ABS plastic. The first models were made using only one

layer of structural material. however due to its low waterproofing and its fragile consistency is

increased to three layers but because of its thickness the mobility is drastically reduced. Finally

i have chosen chose two layers due to its good relation waterproofing-flexibility.

The tests carried out for the parameters of Mobility and Durability conclude for the case

of Rubber Paint a N/A (not applicable) due to his rigidity the skin never stretched.

87

5.2 Body

(a) Mold

(b) Mold + LMM (c) Mold+LMM+protective element

Figure 5.8: Building process for the di↵erent skin trials

ProtectiveLayerCarrierMaterial

(a) Cross Section (b) Plan View Section

Figure 5.9: Sample of Silicone Rubber Skin Tissue

Table 5.2: Comparison chart between the materials used for the skin.

(g=Goodgg= Betterggg= Best)

Parameters Mobility Impermeability Durability

Material

Latex ggg g gRubber paint N/A ggg N/A

Liquid Silicone ggg g ggLatex & LMM ggg gg gg

Rubber paint & LMM N/A ggg N/A

Liquid Silicone & LMM ggg gg ggg

88

5.3 Caudal fin

5.3 Caudal fin

This section shows a novel bioinspired model for an actuated soft caudal fin. This mimics the

behaviour of a real one including his physical characteristics of softness and its capability to be

actuated adding a new level of maneuvering to the fish swimming.

5.4 Mechatronics concept design

5.4.1 Biological foundations

Frequently the steady swimming in fishes is described emphatically by body undulations of

exemplar species, maneuvering produces a kinematic repertoire that does not conform to the

fin shapes observed in stereotypical steady swimming behaviors. During maneuvers, the motion

of the caudal fin often changes irrespective of the motion of the body Teleost fishes (category

which is part the ray-finned fishes as the Largemouth Bass) are defined as a monophyletic group

by their characteristic caudal skeleton. The skeletal and muscular structure is shown in Figure

5.10.

Figure 5.10: Representation of the intrinsic caudal muscles. Flexor dorsalis (FD, green), flexor

ventralis (FV, blue), hypochordal longitudinalis (HL, purple), infracarinalis (IC, gray), interradi-

alis (IR, red) and supracarinalis (SC, yellow). The color coding of the muscles is the same used

for the bluegill sunfish (Lepomis) in Flammang and Lauder (14). (figure adapted from (15))

The morphing of the caudal fin is studied through examine its kinematics and activity

of the intrinsic caudal musculature during a diversity of unsteady locomotor behaviors: kick-

and-glide swimming, braking and backing maneuvers (14). Analysis of the ability of fish to

actively control tail shape with intrinsic musculature during unsteady locomotion is important

to understand the complete function of the fish tail. Often analyzes consider the caudal fin as a

rigid plate, but, this is capable of substantial shape change and hence modulation and vectoring

of force. Perceive and recognize the caudal fin as capable of maneuvering is very interesting

in the light of recent developments in fish robotics and modelling, for which data on fin ray

89


control could be used to construct more accurate biomimetic models of fin function and fish

robots (182); (183). Largemouth Bass fish are able to modulate the shape of their tail fins into

a variety of configurations di↵erent than that exhibited during steady swimming (Fig.5.11(a)),

depending on the behavior being performed. Braking maneuvers followed acceleration towards

prey and were characterized by a rapid flaring of the dorsal and ventral lobes of the caudal fin in

opposite directions (Fig.5.11(b)). Backing maneuvers often followed braking maneuvers. Kick

maneuvers (Fig.5.11(c)) were characterized by sudden rapid lateral excursion of the caudal fin

and were followed by a forward glide (’kick-and-glide’) (Fig.5.11(d)) (see also (15)).

Figure 5.11: Representative examples of caudal fin shape modulation for (a), Steady Swimming

(b), Braking (c) Kick (d) Kick and Glide. Tail outlines closely follow the distal margin of the

caudal fin and fin ray position. Arrows indicate the major direction of movement of the dorsal

and ventral lobes of the caudal fin. Bar (yellow), 2 cm. (figure adapted from (15))

5.4.2 Design Concepts and Modelling

For this model, I have chose to divide the caudal fin in two segments (upper and lower) of the

same length as shows in figure 5.12(a). The SMA wires attached on either side of each flexure

drive the motion of the caudal fin (see Fig. 5.12(b)).

The forces acting on a swimming fish are weight, buoyancy, and hydrodynamic lift in the

vertical direction, along with thrust and resistance in the horizontal direction (Fig. 1.3(a)).

The hydrodynamic stability and direction of movement are often considered in terms of pitch,

90


UpperSegment

LowerSegmentSMA

CaudalF in

(a)

Silicone Rubber

Silicone Rubber

SMA

SMA

(b)

Figure 5.12: (a) The concept of a novel Bio-inspired Morphing Caudal Fin using shape memory

alloys (SMA). (b) Cross-Section basic concept. Note that the SMAs are embedded (sandwiched)

between the Cellulose Acetate Film and the Silicone Rubber

roll, and yaw (Fig. 1.3(b)). The actuation of the SMAs can produce trust as well induce a

roll and yaw movement at steady swimming. According to figure 5.11 this is able to produce

(with sour model of caudal fin) the ’Steady Swimming’ and ’Braking’ manoeuvring, by acting

the SMAs in the upper and lower segment. To produce the ’Kick and Glide’ manoeuvring is

necessary to include one more SMA wire placed between the dorsal and ventral lobes.

5.4.3 Bending Design

As known for the section 5.4.1 the caudal fin is composed by muscles and fin rays. The design

presented on this work, moves this muscles from his natural position at the base of the caudal

fin to over the fin rays. According to other related works (126, 127, 184) the best way to increase

the e↵ect of the SMA contraction over the bendable structure is to take the SMA as close as

possible to the structure. Thus a small change in wire tension has a strong impact on the final

deformation of the structure since a certain fraction of the 4% contraction is lost in recovering

tension before the structure can be further actuated. Figure 5.13 shows a representation of this

bendable structure, were one end of the SMA wire was attached to the tip while the other end

was anchored to the base, with no gap between them, this to maximize movement.

In this testbed the single layer of cellulose acetate film (CAF) acts as an incompressible

beam and the interaction between the beam and the SMA can be explained by kinematics.

Initially, The lengths of the SMA and the CAF are equals, as shown in figure 5.14(a). Where

the actuation of the SMA starts and assuming that this shrinks only 4% and the CAF is

incompressible, giving the following relation: L’SMA

=0.96LSMA

, where LSMA

is the initial

91


celulose acetate film

sma

connection point

Figure 5.13: Representation of the bendable structure with the SMA contracted

length of the SMA and L’SMA

is the length of the SMA after contraction. Assuming the CAF

beam bends kinematically without any resistance then the following can be obtained:

RSMA

RCAF

=

0

@L

0SMA

↵1

L

CAF

↵1

1

A =0.96L

SMA

LCAF

= 0.96 (5.8)

where RSMA

and RCAF

represent the radius of curvature of the SMA and CAF as described

in figure 5.14(b). The distance d between the SMA and CAF beam is a known parameter and

correlates the two radii of curvatures as:

RCAF

= RSMA

+ d (5.9)

Using equations 5.8 and 5.9, is possible to show that:

RCAF

= 25d (5.10)

SMA wiresdd

e

e CAF layer

L

(a)

d dR

SMA

RCAF

(b)

Figure 5.14: Beam kinematics concept diagram showing the (a) undeformed and (b) deformed

configuration. Distance d for both SMA wires is less than a 1mm. Distance e represents the

thickness of the silicone rubber layer

Equations 5.10 shows the kinematic relation between d and the radius of curvature for an

actuation of 4% related to the maximum shrink of the SMA, which indicates that RCAF

can be

92


manipulated by changing d and further implies that RCAF

can be varied as a function of span (s)

to achieve complex deformation profiles. As noted above this will have two SMAs wires working

in an antagonistic configuration. Knowing that the symmetry between the movements must be

the same to produce the same amount of bending in both directions, the SMAs wires length

must be the same. Likewise both pairs of SMAs wires needs to be placed in the same location,

one pair for the upper segment and the other for the lower segment to avoid asymmetrical

bends. As shown in figure 5.15 the solution adopted was to place the SMA wire responsible for

the left bend inwards the SMA wire responsible for the right bend. Due to this solution, the

length of the inner SMA is less than the outer SMA, to compensate is necessary to add more

length to the inner SMA by attaching it to the base in a farther position than the outer SMA.

Likewise the figure 5.15 shown the SMA responsible for the ’Kick and Glide’ maneuver (see

Sec. 5.4.2). This SMA has a circular shape because is not required to bend the structure as

the others SMAs do, but it is necessary to compress it. Thus, when the SMA contracts all the

Y-Components of the Force Vector came to the center of the circle.

a

b

a0

b0

a

a0

b

b0

SMAinner

SMAouter

RSMA

m

SMAmiddle

Caudal F in

Connection points

Figure 5.15: Final Bending Design. RSMA

m

is the radius of the middle SMA (SMAmiddle

)

The length ratio between the SMAinner

and SMAouter

is estimated by knowing that length

of the outer SMA (Lo

) is equal to the inner SMA length (Li

). Lo

= a + 2b and Li

= a0 + 2b0.

now assuming that a = 1.5a0, It has:

b0 =3L

o

� 2a

6(5.11)

93

5.5 Fabrication and assembly

The equation 5.11 shows the length of the b0 in relation with Lo

and a. these two last values

are chosen according to the design criteria.

5.5 Fabrication and assembly

The results obtained in Sec.5.4 were used to arrive at the fabrication of the caudal fin. The

key design parameter found to dictate the final deformation was the SMA guides (distance d),

the thickness of the silicone rubber layer (e), the connection points and the length of the SMAs

(Lo

). the SMAs inside the sandwich (see Fig. 5.12(b)). Due to the CAF surface is smooth

glossy it was sanded to create better adhesion between it and the silicone rubber. Figure 5.16

shown the first model with asymmetrical SMAs and with the edges of the silicone unbound.

After a while acting the SMAs, the edges of the silicone rubber starts to unstick from the CAF

and a time after all the silicon layers was peeled.

AsymmetricalSM

As

Clear

Edg

es

Figure 5.16: First Design with asymmetrical SMAs and clear edges. Noted that the SMAmiddle

is not circular, this because the first tests used only the SMA placed at the upper and lower

segment.

To overcome this problem, in the second design both layers of the silicone rubber stuck to

itself at the edges of the CAF as shown in the figure 5.17.

At the base of the caudal fin all the SMAs wires were attached using nuts and bolts (con-

nection points), but at the top of the caudal fin the SMAs wires pass through holes in the CAF.

These holes allows to connect to the power supply all the SMAs at the base of the caudal fin.

To measure the bend, a flex sensor was embedded into the silicone rubber layer.

To maximize the bend, all the SMA were pre-tightened between the connection points at the

base of the caudal fin and the holes at the top of this same and after a thin layer of silicon

rubber is applied. At least two silicon layers are needed to fit the SMAs in their places.

94

5.6 BR3 electronics and sensors

Cau

dalFin

Con

nection

points

SM

Ainner

SM

Aouter

FlexSen

sor

Figure 5.17: Final design of the Caudal fin. Noted that all SMAs wires are tight and the presence

of the flex sensor used to measure the bend.


This section details the onboard hardware architecture, its components, functions, and the

power consumption of the robot.

5.6.1 Arduino controller-board

Arduino is one of the most extended, simple and robust commercial solutions of micro-controllers.

The ArduinoMicro version (http://www.arduino.cc/en/Main/arduinoBoardMicro) is one of

the lightest chips powered by an Atmel ATmega32u4 running at 16MHz. It has an operation

voltage of 5V , 20 I/O digital pins (7 are PWM) and 12 analog pins. The ATmega32u4 has 32KB

of flash memory for storing code (of which 4KB is used for the bootloader). Arduino provides

a software (http://arduino.cc/en/Main/Software) for programming the micro-controller.

Figure 5.18 shows the arduino board.

5.6.2 The Inertial Measurement Unit (IMU)

The Inertial Measurement Unit (6DOF IMU ) is a digital combo board of 6 Degrees of Free-

dom with gyros: ITG-3200 (MEMS triple-axis gyro), accelerometers: ADXL345 (triple-axis

accelerometer) (https://www.sparkfun.com/products/10121). IMU readings are vital for

the feedback of attitude variables that enable roll and yaw control of BR3 This chip is ready to

be connected to the Arduino board, the unique requirement is to filter the IMU data for reduc-

ing noise. Kalman filtering technique is used for both reducing noise and predicting attitude

95

http://www.arduino.cc/en/Main/arduinoBoardMicro

http://arduino.cc/en/Main/Software

https://www.sparkfun.com/products/10121


(a) (b) (c)

Figure 5.18: (a) Arduino Micro Front, (b) Arduino Micro Rear, (c) Pin Mapping of the Arduino

Micro displays the complete functioning for all the pins

(a) (b)

Figure 5.19: (a) Razor IMU Rear size, (b) Razor IMU Front, (c) Razor IMU Rear

motions. Figure 5.19 shows the IMU board’ physical characteristics.

5.6.3 Flex Sensor

The flex sensor (https://www.sparkfun.com/products/10264) has one side of the sensor is

printed with a polymer ink that has conductive particles embedded in it. When the sensor is

straight, the particles give the ink a resistance of about 30k Ohms. When the sensor is bent

away from the ink, the conductive particles move further apart, increasing this resistance (to

about 50k Ohms when the sensor is bent to 90, as in the diagram below). When the sensor

straightens out again, the resistance returns to the original value. By measuring the resistance,

you can determine how much the sensor is being bent. Figure 5.20 shows the flex sensor’

working principle and physical characteristics.

Flex sensors are used to measure the bend angle as well as acceleration. Section 4.4 shows

the mathematical analysis carried out.

96



(a) (b) (c)

Figure 5.20: Conductive particles (a) close together and (b) further apart, (c) size

5.6.4 Current sensor

This current sensor (https://www.sparkfun.com/products/8883) gives precise current mea-

surement for both AC and DC signals. These are good sensors for metering and measuring

overall power consumption of systems. The ACS712 current sensor measures up to 5A of DC

or AC current. Also its have an opamp gain stage for more sensitive current measurements. By

adjusting the gain (from 4.27 to 47) is possible to measure very small currents.

The bandwidth on the ACS712 Low Current Sensor Breakout has been set to 34Hz to reduce

noise when using at high gains. However is possible to set the maximum bandwidth of 80KHz.

(a) (b)

Figure 5.21: Current sensor (a) Front view (b) Rear view

5.6.5 Temperature and Humidity sensor

This is a simple breakout board for the SHT15 humidity sensor from Sensirion. The SHT15

digital humidity and temperature sensor (https://www.sparkfun.com/products/8257) is fully

calibrated and o↵ers high precision and excellent long-term stability at low cost. The digital

CMOSens Technology integrates two sensors and readout circuitry on one single chip.

97



5.7 BR3 consumption

(a) (b)

Figure 5.22: Temperature and Humidity sensor (a) Front view (b) Rear view

5.6.6 SMA power drivers

The Miga Analog Driver V5 (MAD-V5) is a MOSFET switch designed to safely power the Mig-

amotor SMA actuators across a wide range of speeds or input voltages. This driver generates

the current signal based on the digital control command sent from the Arduino board. The

schematic of the circuit is described in Figure 5.23. The MAD-V5 allows either push-button

operation, or external GATE (CNTL) signals to actuate the Migamotor SMA actuator until

the END limit is reached (goes LOW). The MAD-V5 then cuts power momentarily, preventing

overheating of the SMA wires. The Gate transistor allows up to 30V input, but it is rec-

ommended to use logic (2.5 to 5-Volt) levels. Pulse-Width-Modulated (PWM) signals can be

applied at the Gate to control the actuation speed for a set voltage, or even an AC driven

current signal mounted on a DC level. For instance, the application of +28VDC power to the

Migamotor SMA actuator would result in very fast actuation (⇠ 70ms). In Figure 5.23, JP1 is

the power supply and/or micro-controller connector, JP2 is connected to the Migamotor SMA

actuator. Maximum peak current: 7A, maximum continuous current: 5A.

5.7 BR3 consumption

Consumption is measured in terms of required electrical currents. Figure 5.24 shows the percent-

age of current consumption of each electronics component of BR3, whereas Table 5.3 consigns

the numerical values. As expected, the SMA actuators require most of the input current, about

91, 12%. However, the total SMA wires can’t be actuated at same time due to its antagonistic

configuration, only two SMA wires will be actuated at same time.

98

5.8 BR3 costs

Figure 5.23: Miga analog driver V5 pinout diagram. Source: The author.

Table 5.3: General values of current consumption

Component Quantity Required current [mA] % of consumption

SMA Wires 4 1600 91, 12%

SMA Drivers 4 40 2, 28%

Flex Sensor 4 10 0, 57%

Current Sensor 4 52 2, 96%

Temperature Sensor 1 21 1, 20%

IMU 1 18 1, 03%

Arduino Board 1 15 0, 85%

Total consumption = 1756mA

5.8 BR3 costs

Table 5.4 details the costs of the components that are involved in the fabrication process of the

robot.

5.9 Remarks

This Chapter has completely described the bio-inspired design-flow applied to the development

and fabrication of BR3. The criteria for design have been classified in terms of morphology,

kinematics, dynamics and aerodynamics parameters extracted from the analysis of biological

data. Most importantly, the data has been related as a function of design parameters such as

99

5.9 Remarks

Figure 5.24: Percentage of current consumption per component.

Table 5.4: Fabrication costs

Component Manufacturer/seller Item cost Quantity used Total

SMA Wires a Dynalloy 30.28e 1 30.28eMiga Analog Driver V5 (MOSFET) Migamotor 10.75e 4 43e

Flex Sensor Sparkfun 7.00e 4 28.00eCurrent Sensor Sparkfun 8.76e 4 35.04e

Humidity and Temperature Sensor Sparkfun 36.92e 1 36.92eIMU 6DOF (ITG3200/ADXL345) Sparkfun 35.16e 1 35.16e

ArduinoMicro Arduino 19.90e 1 19.90e(ABS) plastic material - - - 60e

Dragon Skin Silicone Rubber for wing membrane Smooth-on 23.20e 2 46.40eMicro Fibber Lycra Mesh - 3.00e 1 3.00e

Total costs =337.7e

aSMA Wires are bought in rolls of 5 meters

overall mass, morphing caudal fin, body mass, bending mechanism, etc. These relations have

enabled the definition of a framework that can be applied to future designs of BR3.

100

6

Free vibration analysis based on

a continuous and non-uniform

flexible backbone with

distributed masses

Tonight I’m gonna have myself a real good time, I feel alive and the world turning inside out,

And floating around in ecstasy, So don’t stop me now, don’t stop me...

Queen (Don’t Stop Me Now)


This section presents a Di↵erential Quadrature Element Method for free transverse vibration of

a robotic-fish based on a continuous and non-uniform flexible backbone with distributed masses

(represented by ribs) based in the theory of a Timoshenko cantilever beam. The e↵ects of the

masses (Number, Magnitud and position) on the value of natural frequencies are investigated.

Governing equations, compatibility and boundary conditions are formulated according to the

Di↵erential Quadrature rules. The compatibility conditions at the position of each distributed

mass are assumed as the continuity in the vertical displacement, rotation and bending moment

and discontinuity in the transverse force due to acceleration of the distributed mass. The

convergence, e�ciency and accuracy are compared to other analytical solutions proposed in the

literature. Moreover, the proposed method has been validate against the physical prototype of

a flexible fish backbone.

101


The main advantages of this method, compared to the exact solutions available in

the literature are twofold: first, smaller time-cost and second, it allows analysing

the free vibration in beams whose section is an arbitrary function, which is

normally di�cult or even impossible with analytical other methods.

The literature of robotic fishes and bio-inspired robots has several examples of structures

using flexible materials that bend to produce thrust, for maneuvering, and even for energy har-

vesting. However, few works analyze the normal frequency of vibration of the flexible structures

employed as a way to maximize and optimize the use of energy. This allows a relatively small

force applied repeatedly to make the amplitude of the oscillating system become very large.

The purpose of this section is to study the resonance in structures to create a steady motion

with low energy in order to create robots with improved energy e�ciency. Concretely, It was

considered a fish robot composed of a flexible backbone, made of polycarbonate, and a series

of relatively heavy (i.e. whose weigh is non negligible) ribs (Fig. 6.1).

The dynamic characteristics of systems with flexible components is a very important issue

that allows the study of robots based on a jointless structure. Some researchers have addressed

the problem of vibration analysis of structures with distributed masses located at arbitrary

positions using the Delta Dirac function (185), introducing the mass in the boundary conditions

(186), using the Rayleigh-Ritz method (187). or analyzing the case of flexible structures carrying

distributed mass along the structure (including a free end) (188), (189), (190), (191). In all

cases such cases, the authors use the Bernoulli-Euller beam theory to model simple structures,

which is reliable just for slender beams. In order to increase the accuracy and reliability of

studies, especially for the beams with low length-to-thickness ratio, the study of the natural

frequencies of a Timoshenko beam with a central point mass using coupled displacement field

method has been proposed (see, e.g., (192)).

The Di↵erential Quadrature Element Method (DQEM), provides a powerful numerical

method to analyze the behavior (both static and dynamic) in structures with some discon-

tinuities in loading, material properties or in its geometry. Thus, this method is applied to

solve many problems especially in vibration analysis.

102

6.2 Di↵erential quadrature method

Rib

Caudal Fin

Data Cables

Head


Figure 6.1: The fish-robot prototype is composed of 19 ribs (excluding the head and the tail)

made of 3D-printed ABS plastic that form the distributed masses.

6.2 Di↵erential quadrature method

The Di↵erential quadrature method allows expressing function derivatives in x = xi

in terms

of the value of function along the domain as:

d

r

f

dx

r

�

�

�

�

x=x

i

=N

X

j=1

A

ij

(r)f

j

, (6.1)

where A(r) represents the weighting coe�cient associated with the rth order derivative and Nthe number of grid points (193):

A

ij

(1)=

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

N

Q

m=1

m 6=i,j

(xi

�x

m

)

N

Q

m=1

m 6=j

(xj

�x

m

), (i,j=1,2,3, ...N ;i6=j)

N

P

m=1

m 6=i

1

(xi

�x

m

), (i=j=1,2,3, ...N)

A

ij

(r)=

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

r

✓

A

(r�1)ii

A

(1)ij

�A

(r�1)ij

x

i

�x

j

◆

, (i,j=1,2,3, ...N ;i6=j)

�N

P

m=1

m 6=i

A

(r)im

, (i=j=1,2,3, ...N) 1<r (N�1)

(6.2)

Using a set of grid points (blue dots pot Fig. 6.2) following Gauss-Lobatto-Chebyshev points for interval

[0, 1] it has

x

i

=1

2

⇢

1� cos

(i�1)⇡

(N�1)

��

, (i=1,2,3, ...N) . (6.3)

This set of grid point shows the compression of the two end points in the interval [0, 1], providing

in this way high accuracy for estimating the value of the derivative of the function at the boundary

points.

103

6.3 Vibration analysis

Process

x(1)

L

L(1)

L(2)

x, ⇠

1 2

x(2)x(3) x(i) x(i+1)

x(n+1)

i� 1 i n

L(n+1)L(i)


Figure 6.2: Non-uniform fish-robot backbone with distributed masses.


6.3.1 Governing equations

Figure 6.2 shows the fish-robot backbone. Note that the mass of each rib is di↵erent and the backbone

is non-uniform. The entire surface is modelled like a nonuniform cantilever Timoshenko beam. The

Free Vibration governing equations for a Timoshenko beam with distributed masses are written as

(194):@

@x

n

kGA (x)h

@w(x,t)@x

� (x,t)io

�⇢A (x) @2w(x,t)@t

2 =0

@

@x

h

EI (x) @ (x,t)@x

i

+kGA (x)h

@w(x,t)@x

� (x,t)i

�⇢I (x) @2w(x,t)@t

2 =0(6.4)

where w(x, t) is vertical displacement. The term k is introduced to take into account the geometry

dependent distribution of the shear stress and depends on the shape of the section and the Poisson ratio

of the material (195). The displacement w(x, t) and rotation (x, t) can be assumed as the product

of the functions W (x) and (x) which only depend on the spatial coordinate x and a time dependent

harmonic function as

w(x,t)=W (x)ei!t (x,t)= (x)ei!t (6.5)

Substituting the Equations 6.5 into the set of Equations 6.4, is obtained

d

2W (x)dx

2 � d (x)dx

+ 1A

⇤(x)dA

⇤(x)dx

h

dW (x)dx

� (x)i

+ ⇢!

2

kG

W (x)=0,

EI0kA0G

h

d

2 (x)dx

2 + 1I

⇤(x)dI

⇤(x)dx

d (x)dx

i

+A

⇤(x)I

⇤(x)

h

dW (x)dx

� (x)i

+ ⇢I0!2

kA0G (x)=0

(6.6)

The second moment of inertia and cross-sectional area are written in the following dimensionless

form:

I⇤= I(x)I0

A⇤(x)=A(x)A0

(6.7)

where I0 and A0 are values of the moment of inertia and cross-section at the clamped edge of the beam.

104


For the ith sub-beam, the set of Eqs.6.6 are written as

d

2W

(i)(x(i))

d(x(i))2� d (i)(x(i))

dx

(i) + 1A

⇤(x)dA

⇤(x)dx

h

dW

(i)(x(i))

dx

(i) � (i)(x(i))i

+ ⇢!

2

kG

W (i)(xi)=0

EI0kA0G

h

d

2 (i)(x(i))

d(x(i))2+ 1

I

⇤(x)dI

⇤(x)dx

d (i)(x(i))

dx

(i)

i

+A

⇤(x)I

⇤(x)

h

dW

(i)(x(i))

dx

(i) � (i)(x(i))i

+ ⇢I0!2

kA0G (i)(x(i))=0

(6.8)

Introducing the dimensionless parameters:

⇠= x

L

⇣(i)= x

(i)

L

(i) v(i)=W

(i)

L

l(i)=L

(i)

L

(6.9)

the set of Eqs.11.2 can be rewritten as

⇣

1l

(i)

⌘2d

2v

(i)(⇣(i))d(⇣(i))2

�⇣

1l

(i)

⌘

d (i)(⇣(i))d⇣

(i) + 1A

⇤(⇠)dA

⇤(⇠)d⇠

⇣

1l

(i)

⌘

dv

(i)(⇣(i))d⇣

(i) � (i)⇣

⇣(i)⌘

�

+�4s2v(i)⇣

⇣(i)⌘

=0

s2

⇣

1l

(i)

⌘2d

2 (i)(⇣(i))d(⇣(i))2

+ 1I

⇤(⇠)dI

⇤(⇠)d(⇠)

⇣

1l

(i)

⌘

d (i)(⇣(i))d⇣

(i)

�

+A

⇤(⇠)I

⇤(⇠)

⇣

1l

(i)

⌘

dv

(i)(⇣(i))d⇣

(i) � (i)⇣

⇣(i)⌘

�

+�4s2r2 (i)⇣

⇣(i)⌘

=0,

(6.10)

where

�4= ⇢A0L4!

2

EI0, s2= EI0

kA0GL

2=2(1+v)

k

r2, r2= I0A0L

2 . (6.11)

Assuming all grid points are the same for the sub-beams, then

⇣(1)=⇣(2)=⇣(3)=...=⇣(i)=...=⇣(n+1)=⇣. (6.12)

Thus, Eq. 11.4 can be simplified:

⇣

1l

(i)

⌘2d

2v

(i)(⇣)d⇣

2 �⇣

1l

(i)

⌘

d (i)(⇣)d⇣

+ 1A

⇤(⇠)dA

⇤(⇠)d⇠

h⇣

1l

(i)

⌘

dv

(i)(⇣)d(⇣) � (i) (⇣)

i

+�4s2v(i) (⇣)=0

s2

⇣

1l

(i)

⌘2d

2 (i)(⇣)d⇣

2 + 1I

⇤(⇠)dI

⇤(⇠)d⇠

⇣

1l

(i)

⌘

d (i)(⇣)d⇣

�

+A

⇤(⇠)I

⇤(⇠)

h⇣

1l

(i)

⌘

dv

(i)(⇣)d(⇣) � (i) (⇣)

i

+�4s2r2 (i) (⇣)=0

(6.13)

Furthermore, it is introduced a modified form of the weighting coe�cients of element i to simplify

the DQ analogue equations defined as

[A](i)= [A](i)

l

(i) [B](i)= [A](i)

(l(i))2(6.14)

Using Eq. 11.8, is obtained for the governing set of equations of element i the DQ analogue:

[Bve

](i){v}(i)�[Ase

](i){ }(i)+�4s2{v}(i)=0,

[Bse

](i){ }(i)�[Ave

](i){v}(i)+�4s2r2{ }(i)=0(6.15)

where

[Bve

](i)=[B](i)+h

1A

⇤(⇠)dA

⇤(⇠)d⇠

i(i)[A](i) [A

ve

](i)=h

A

⇤(⇠)I

⇤(⇠)

i(i)[A](i)

[Bse

](i)=s2✓

[B](i)+h

1I

⇤(⇠)dI

⇤(⇠)d⇠

i(i)[A](i)

◆

�h

A

⇤(⇠)I

⇤(⇠)

i(i)

[Ase

](i)=[A](i)+h

1A

⇤(⇠)dA

⇤(⇠)d⇠

i(i).

(6.16)

In Eq. 6.16, the terms

105


h

1A

⇤(⇠)dA

⇤(⇠)d⇠

i(i),

h

A

⇤(⇠)I

⇤(⇠)

i(i),

h

1I

⇤(⇠)dI

⇤(⇠)d⇠

i(i)

are geometry-dependent diagonal matrices with values of the geometrical parameters. Now it can

rewrite the motion equations 11.9 for the domain points in order to eliminate the redundant equations

((196),(197) (198)) obtaining:

⇥

Bve

⇤(i){v}(i)�⇥

Ase

⇤(i){ }(i)+�4s2{v}(i)=0⇥

Bse

⇤(i){ }(i)+⇥

Ave

⇤(i){v}(i)+�4s2r2�

(i)

=0(6.17)

where bar signs means truncated non-square matrices. Combining equations Eqs.6.17:

[Bv

] {v}� [As

] { }+�4s2{v}d

=0 [Bs

] { }+ [Av

] {v}+�4s2r2{ }d

=0 (6.18)

where

{v}=

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

v1(1)

v2(1)

.

.

.

v

N�1(1)

v

N

(1)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

v1(2)

v2(2)

.

.

.

v

N�1(2)

v

N

(2)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

· · ·

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

v1(n+1)

v2(n+1)

.

.

.

v

N�1(n+1)

v

N

(n+1)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

T

{ }=

8

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

:

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

1(1)

2(1)

.

.

.

N�1

(1)

N

(1)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

1(2)

2(2)

.

.

.

N�1

(2)

N

(2)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

· · ·

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

1(n+1)

2(n+1)

.

.

.

N�1

(n+1)

N

(n+1)

9

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

;

T

9

>

>

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

>

>

;

T

[Bv

] =diag

✓

h

B

ve

i(1)h

B

ve

i(2)· · ·

h

B

ve

i(n+1)◆

[Av

] =diag

✓

h

A

ve

i(1)h

A

ve

i(2)· · ·

h

A

ve

i(n+1)◆

[Bs

] =diag

✓

h

B

se

i(1)h

B

se

i(2)· · ·

h

B

se

i(n+1)◆

[As

] =diag

✓

h

A

se

i(1)h

A

se

i(2)· · ·

h

A

se

i(n+1)◆

(6.19)

The ”diag” operator provides the diagonal matrices. In order to separate the domain, boundary,

adjacent displacement and rotation components, Equations (6.18) need to be rearranged (199):

[Bv

]b

{v}b

+[Bv

]d

{v}d

+[Bv

]c

{v}c

�[As

]b

{ }b

�[As

]d

{ }d

�[As

]c

{ }c

+�4s2{v}d

=0

[Av

]b

{v}b

+[Av

]d

{v}d

+[Av

]c

{v}c

+[Bs

]b

{ }b

+[Bs

]d

{ }d

+[Bs

]c

{ }c

+�4s2r2{ }d

=0(6.20)

where

{v}b

=

8

<

:

n

v1(1)

o

n

v

N

(n+1)o

9

=

;

{ }b

=

8

<

:

n

1(1)

o

n

N

(n+1)o

9

=

;

{v}c

=

(

n

v

N

(1)o

(

v1(2)

v

N

(2)

)

T

(

v1(3)

v

N

(3)

)

T

· · ·(

v1(n)

v

N

(n)

)

T

n

v1(n+1)

o

)

T

{ }c

=

(

n

N

(1)o

(

1(2)

N

(2)

)

T

(

1(3)

N

(3)

)

T

· · ·(

1(n)

N

(n)

)

T

n

1(n+1)

o

)

T

{v}d

=

8

>

>

>

<

>

>

>

:

8

>

>

<

>

>

:

v2(1)

.

.

.

v

N�1(1)

9

>

>

=

>

>

;

T

8

>

>

<

>

>

:

v2(2)

.

.

.

v

N�1(2)

9

>

>

=

>

>

;

T

· · ·

8

>

>

<

>

>

:

v2(n+1)

.

.

.

v

N�1(n+1)

9

>

>

=

>

>

;

T

9

>

>

>

=

>

>

>

;

T

{ }d

=

8

>

>

>

<

>

>

>

:

8

>

>

<

>

>

:

2(1)

.

.

.

N�1

(1)

9

>

>

=

>

>

;

T

8

>

>

<

>

>

:

2(2)

.

.

.

N�1

(2)

9

>

>

=

>

>

;

T

· · ·

8

>

>

<

>

>

:

2(n+1)

.

.

.

N�1

(n+1)

9

>

>

=

>

>

;

T

9

>

>

>

=

>

>

>

;

T

(6.21)

106


6.3.2 Compatibility conditions

In the following, is analyzed the compatibility conditions that link the inertia and elasticity of the beam

with the distributes masses.

Around of each concentrated mass, e↵ects [x�m

, x+m

], neglecting the moment of inertia of the con-

centrated mass, the compatibility conditions are continuous in the vertical displacement and rotation

due to the acceleration of distributed masses that produce a bending moment and alike discontinuous

in the transverse force.

w

�

x

�m

,t

�

=w

�

x

+m

,t

�

�

x

�m

,t

�

= �

x

+m

,t

�

M

�

x

�m

,t

�

=M

�

x

+m

,t

�

V

�

x

�m

,t

�

�V

�

x

+m

,t

�

=m

i

@

2w(x,t)

@t

2

(6.22)

where mi

, M , V are the translational inertia of the ith concentrated mass, the bending moment

and shear force respectively, which are presented for ith sub-beam as (194)

M

(i)=EI

d (i)

dx

(i) =EI

L

1

l

(i)d (i)

d⇣

V

(i)=kAG

⇣

(i)� dW

(i)

dx

(i)

⌘

=kAG

⇣

(i)� 1

l

(i)d (i)

d⇣

⌘

(6.23)

Compatibility Conditions can be expressed in the DQ form as

v

(i)N

=v

(i+1)1 (i)

N

= (i+1)1

N

P

j=1A

(i+1)1j (i+1)

j

�N

P

j=1A

(i)Nj

(i)j

=0

N

P

j=1A

(i+1)1j v

(i+1)j

�N

P

j=1A

(i)Nj

v

(i)j

+↵

i

s

2�

4

A

⇤(⇠i

)v

(i)N

=0

(6.24)

it can define the dimensionless translational inertias of the ith concentrated mass as

↵

i

=m

i

⇢A0L(6.25)

Eq.6.24 can be rewritten in matrix form as

[Qe

](i)(

{v}(i)

{v}(i+1)

)

+�4[qe

](i)(

{v}(i)

{v}(i+1)

)

=

(

0

0

)

[Qe

](i)(

{ }(i)

{ }(i+1)

)

=

(

0

0

)

(6.26)

where

[Qe

](i)jk

=

8

>

>

>

<

>

>

>

:

��NK

�(N+1)K

�A

(i)Nk

A

(i+1)1k

j=1,1kN

j=1,N+1k2N

j=2,1kN

j=2,N+1k2N

j=1,2

1k2N

[qe

](i)jk

=

8

<

:

↵

i

s

2

A

⇤(⇠i

)0

j=2,k=N

else

(6.27)

Rewriting and composing a new Eq.6.26 for all sub-beams,

[Q] {v}+�4 [q] {v}= {0} [Q] { }= {0} (6.28)

where [Q], [q] are corresponding matrix that contains [Qe

](1) to [Qe

](n+1) and [qe

](1) to [qe

](n+1)

respectively. Eq.6.26 may be rewritten and sectioned to separate the components of domain, boundary,

and adjacent displacement

[Q]b

{v}b

+[Q]d

{v}d

+[Q]c

{v}c

+�4 �

[q]b

{v}b

+[q]d

{v}d

+[q]c

{v}c

�

= {0}[Q

v

]b

{ }b

+[Qv

]d

{ }d

+[Qv

]c

{ }c

= {0}(6.29)

From Eq.6.27, it cab be concluded that [q]b

= [q]d

= 0. Therefore, Eq.6.29 can be summarized as

[Q]b

{v}b

+[Q]d

{v}d

+[Q]c

{v}c

+�4[q]c

{v}c

= {0} (6.30)

107


{ }c

= [Jb

] { }b

+ [Jd

] { }d

(6.31)

where[J

b

] =�[Qv

]�1c

[Qv

]b

[Jd

] =�[Qv

]�1c

[Qv

]d

(6.32)

Substituting Eq.6.31 into Eq.6.20

[Bv

]b

{v}b

+[Bv

]d

{v}d

+[Bv

]c

{v}c

+ [Gsb

] { }b

+ [Gsd

] { }d

+�4s

2{v}d

= {0}[A

v

]b

{v}b

+[Av

]d

{v}d

+[Av

]c

{v}c

+ [Esb

] { }b

+ [Esd

] { }d

+�4s

2r

2{ }d

= {0}(6.33)

where[G

sb

] =[As

]b

+[As

]c

[Jb

] [Gsd

] =[As

]d

+[As

]c

[Jd

] [Esb

] =[Bs

]b

+[Bs

]c

[Jb

]

[Esd

] =[Bs

]d

+[Bs

]c

[Jd

](6.34)

6.3.3 Boundary conditions

The boundary conditions for a cantilever beam (robot fish backbone) depicted in Fig.6.2 can be con-

sidered as

W

(1)�

�

�

x

(1)=0=0

(1)�

�

�

x

(1)=0=0 V

(n+1)�

�

�

x

(n+1)=L

(n+1)=0 M

(n+1)�

�

�

x

(n+1)=L

(n+1)=0 (6.35)

Eq.6.35 can be rewritten using Eq.6.23 as

(1)�

�

�

x

(1)=0=0 1

l

(i)d (n+1)

d⇣

�

�

�

⇣=1=0 v

(1)�

�

�

x

(1)=0=0

⇣

1

l

(i)d (n+1)

d⇣

� (n+1)⌘

�

�

�

⇣=1=0 (6.36)

In the DQ form, Eq. (36) can be indicated as

[m] { }=0 (6.37)

[m] {v}+ [n] { }= {0} (6.38)

where

m

jk

=

8

>

<

>

:

1

A

(n+1)N(k�nN)

0

j=k=1

j=2,nN+1k(n+1)N

else

j=1,2

1k(n+1)N

n

jk

=

(

�1

0

j=2,k=(n+1)N

else

(6.39)

Eqs.6.37 and 6.38 can be rewritten and sectioned in order to separate the components (boundary,

domain, adjacent displacement and rotation)

[m]b

{ }b

+[m]d

{ }d

+[m]c

{ }c

= {0}[m]

b

{v}b

+[m]d

{v}d

+[m]c

{v}c

+[n]d

{ }d

+[n]c

{ }c

= {0}(6.40)

From Eq.6.39, is known that [n]c

= [n]d

= 0; Hence, using Eq.6.31, Eq.6.40 it has

{ }b

= [t] { }d

{v}b

=�[m]�1b

[m]�1d

{v}d

�[m]�1b

[m]�1c

{v}c

�[m]�1b

[n]b

[t] { }d

(6.41)

where[t] =�[r]�1

b

[r]d

[r]b

=[m]b

+[m]c

[Jb

] [r]b

=[m]d

+[m]c

[Jd

] (6.42)

Replacing Eq.6.41 into the set of Eqs.6.20 and 6.30, a new set of equations is obtained

[K]

8

>

<

>

:

{v}d

{v}c

{ }d

9

>

=

>

;

=�4 [M ]

8

>

<

>

:

{v}d

{v}c

{ }d

9

>

=

>

;

(6.43)

108

6.4 Experimental results

where

[K] =

2

6

4

[Bv

]d

�[Bv

]b

[m]�1b

[m]d

[Bv

]c

�[Bv

]b

[m]�1b

[m]c

[Gsd

] + [Gsb

] [t]�[Bv

]b

[m]�1b

[n]b

[t]

[Q]d

�[Q]b

[m]�1b

[m]d

[Q]c

�[Q]b

[m]�1b

[m]c

�[Q]b

[m]�1b

[n]b

[t]

[Av

]d

�[Av

]b

[m]�1b

[m]d

[Av

]c

�[Av

]b

[m]�1b

[m]c

[Esd

] + [Esb

] [t]�[Av

]b

[m]�1b

[n]b

[t]

3

7

5

M=�

2

6

4

s2I(n+1)(N�2)⇤(n+1)(N�2) {0}(n+1)(N�2)⇤2n {0}(n+1)(N�2)⇤(n+1)(N�2)

{0}2n⇤(n+1)(N�2) [q]c

{0}2n⇤(n+1)(N�2)

{0}(n+1)(N�2)⇤(n+1)(N�2) {0}(n+1)(N�2)⇤2n s2r2I(n+1)(N�2)⇤(n+1)(N�2)

3

7

5

(6.44)

Using Eq.6.44, is possible to determine the natural frequencies and corresponding mode shapes.

The corresponding mode shapes can be completed using the Eqs.6.31 and 6.41. It should be noted that

the number of grid points a↵ects the results. The number of grid points must be determined to satisfy

the following relation for convergence of first n frequencies:

�

�

�

�

�

�

l

(N)��l

(N�1)

�

l

(N�1)

�

�

�

�

�

" l=1,2, ...n (6.45)

where " is considered as 0.01 in this study.

6.4 Experimental results

6.4.1 Numerical comparison

In order to assess the e↵ectiveness of the proposed method, was compared its accuracy with the exact

solution obtained with the method proposed by Lee and Len (17).

Were applied both methods to a cantilever Timoshenko beam with (v = 0.25, k = 2/3), an attached

tip mass (↵ = 0.32), beam cross-sectional properties A = A0(1 � 0.4⇠) and I = I0(1 � 0.4⇠)3 for a

slenderness ratio r of 0.1 and 0.04 as shown in Table.6.1. Also it can be observed the good accuracy

for the proposed method.

Table 6.1: Comparison between this method and the exact method proposed by (17) for the first

three non-dimensional frequencies (�2)

Slenderness ratios r 0.04 0.1

Modes of Vibration 1 2 3 1 2 3

Presented Method 2.117 13.42 36.11 1.997 10.69 24.34

Lee and Lin (1995) 2.099 13.55 36.76 2.015 11.07 25.63

6.4.2 Practical application

The ultimate purpose was to analyze the natural frequency in the robot-fish designed, in order to

optimize its energy e�ciency. Therefore, it was applied this method to the physical prototype using the

setup depicted Fig. 6.3. The parameters modelling the robot fish body as a conical Timoshenko beam

where: slender ratio (r = 1.03), elastic section modulus (s = 2.58), diameter variation d = d0(1� 0.5⇠)

109

6.5 Remarks

Figure 6.3: Robot-Fish including a Silicone-rubber-based skin

and uniformly spaced distributed masses (↵ = 1.3). The number of distributed masses (n) are 19. The

first three modes are depicted in Fig.6.6

In order to analyse the behavior of the backbone, the tail was set to its inicial position and then

released. Fig. 6.4 shows the evolution of this movement. A complete tail beat lasted 6 seconds. White

marks in the tail were used to allow a particle tracking software to find the free vibration response (Fig.

6.5a) and the experimental Natural Frequency (Fig. 6.5b), using a camera to capture the movement

at 60 frames per second.

From the analysis of the images recorded, the natural frequency of the structure obtained experi-

mentally was 1.873.

6.5 Remarks

In this section, I have proposed the use of a theoretical model to find the Natural frequency of a

Fish-like Robot with distributed masses along a flexible, continuous and non-uniform backbone. The

Theoretical model proposed can be used for the beams with a large number of sections and capable

of analyse the non-uniform beams with any variation in the cross section and moment of inertia. A

comparison with an exact method for a set of cases where this could be applied assessed the goodness

of the proposed method.

Comparing the data obtained experimentally for a physical prototype is possible to confirm that

this method can e↵ectively be used to analyse the free vibration in beams whose section is an arbitrary

function, and with distributed masses.

Note the small variation of 0.2519 between the theoretical value (1.873) an the one obtained exper-

imentally (2.1249). This is due to the flexible skin, whose e↵ect was not considered.

110

6.5 Remarks

(a) Start position (b) Mid Point

(c) Maximum Height (d) End Position

(e) Reference Fish view

Figure 6.4: Free Vibration analysis of the backbone. The white spots are the marks for the

tracking, the red line is the trajectory of the tail.

0 1 2 3 4 5 6−8

−6

−4

−2

0

2

4Free Vibration

Time (sec)

Dis

pla

cem

en

t (c

m)

(a) Free Vibration (Displacement y axis)

0 2 4 60

50

100

150

Power Spectrum

Frequency (Hz)

Po

we

r

← 2.1249 Hz

(b) Power Spectrum

Figure 6.5: Experimental Results. The Natural frequency obtained was 2.1249Hz

111

6.5 Remarks

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−3

−2

−1

0

1

2

3

4

ζ

No

rma

lize

d m

od

e s

ha

pe

s

Mode 3

Mode 2

Mode 1

Figure 6.6: First three modes of robot-fish with nineteen equally spaced similar concentrated

masses.

112

7

BR3 Control

”There is no escape, From the slave catcher’s songs. For all of the loved ones gone, Forever’s not so

long, And in your soul, They poked a million holes, But you never let them show, Come on its time to

go”

Devotchka (How It Ends)

7.1 Control goal

For practical SMA actuator applications a simple and e↵ective control designs should be employed. In

this thesis a PID controller has been applied using electrical resistance and bending as feedback for the

SMA wire actuators. The control system has been successful in achieving excellent performance and

stability of the SMA bend response.

7.2 Electrical resistance control

The SMA control electronics is conceived to be as simple as possible because of two main reasons. First,

because we have to guarantee a quick answer and a minimum position error for the four actuators,

and second because we do not want dedicate on-board CPU time to low-level control. The control

accuracy of smart SMAs actuators, is limited due to their inherent hysteresis nonlinearities (see Figure

7.1) with a local memory, resulting from the influence of a previous input on subsequent behavior. In

addition, the existence of minor loops in the major loop because of a local memory also makes the

mathematical modeling and design of a controller di�cult for SMA actuators. Therefore, to enhance

the controllability of a smart actuator, the Preisach hysteresis model has emerged as an appropriate

behavioral model.

Nevertheless, the modeling is di�cult and the model equation is very complex. So even though this

model is commonly used, the use of a heat transfer model and sensor hardware has been introduced.

However, SMAs provide the possibility to create controller systems without sensor hardware. The

detection of inner electrical resistance allows to regulate the actuator movement. The method consists

in measuring the electrical resistance of an SMA element, calculating a maximum safe heating current

as a function of measured resistance, and ensuring that the actual heating current does not exceed

this maximum value. In fact, resistance is being used as a form of temperature measurement, and

113


Figure 7.1: Histeresis of the SMA. (As

, the austenite start temperature; Af

, the austenite finish

temperature; Ms

, the martensite start temperature; and Mf

, the martensite finish temperature.)

the maximum safe heating current is designed to prevent overheating. Moreover, the hysteresis on the

resistance curve is smaller than the hysteresis on the temperature curve, as shown in, which makes the

linear approximation more accurate. The maximum contraction of the wire can be measured as

�LA

f

=L

SMA

Mf

LR

�RSMA

Af

LR

(7.1)

where , LSMA

Mf

(cm) is the SMA length in martensite finish temperature (relaxed SMA), LR

(⌦/m)

is the linear resistance and RSMA

Af

(⌦) is the resistance at austenite finish temperature (i.e. at

maximum contraction).

7.2.1 Design of PID by Ziegler-Nichols tuning rule for an SMA wire

In order to tune the control system, the polycarbonate spine was used with the ska wires in a V-shaped

configuration, in order to double the pull force. Figure 7.2 shows the answer to a 260mv step input,

that corresponds to 300mA of SMA arousal. This value was used for tuning the PID controller because

it allows a stable SMA response in open loop.

Figure 7.2: Voltage SMA vs. a given set point. y1 = 0.258, y0 = 0.232, t1 = 2.725 t0 = 2.2

t2 = 4 u1 = 1 u0 = 0, experimentally determined.)

One of the most common controllers that is used in the heat control is the PID (proportional-

114


integral- derivative) controller, and responds to the equation

u(t) = Kp

e(t) +K

p

Ti

Z

t

0

e(t)dt+Kp

Td

de(t)dt

(7.2)

where e(t) is the signal error and u(t) is the control input of the process. Kp

is the proportional

gain, Ti

is the integral time constant and Td

is the derivative time constant. In the s domain, the PID

controller can be written as

U(s) = Kp

1 +1Ti

s+ T

d

s

�

E(s) (7.3)

Following Ziegler-Nichols, we have tuned the values to the three parameters (Kp

, Ti

, Td

) of the PID

controller based on the analysis of the open and close loop of the system to be controlled. At open

loop a lot of systems can be defined according to the following transference function:

G(s) =K0e

�s⌧0

1� s�0(7.4)

where the coe�cients K0, ⌧0 and �0 are obtained from the open loop response to a step input. The

stabilized system starts in y(t) = y0 to u(t) = u0; a step input is applied from u0 to u1. The exit

answer is registered until it stabilizes at the new operation point (see Fig. 7.2). From the experimental

data corresponding to Figure 7.2, we can compute the following parameters: ⌧0 = t1 � t0, �0 = t2 � t1,

K0 = (y1 � y0)/(u1 � u0)

According to Ziegler-Nichols, the relations between these coe�cients and the controller parameters

are: Kp

= (1.2�0)/(K0⌧0), Ti

= 2T0, Td

= 0.5.

Then, according to the Z transform, the discrete PID controller is:

U(z) = E(z)Kp

1 +T

Ti

(1� z�1)+ T

d

1� z�1T

�

(7.5)

also,

U(z)E(z)

= a+b

1� z�1+ c(1� z�1) (7.6)

where: a = Kp

, b =K

p

T

T

i

, c =K

p

T

d

T

The parameters obtained were: ⌧0 = 0.525, �0 = 1.276, K0 =0.03, Kp

=97.22, Ti

=1.05, Td

=0.2625.

The discrete controller parameters are calculated based at the time T = 0.1 < ⌧0/4. Finally, a = 97.22,

b = 1.05, c = 255.202. The PID controller structure used is shown in Figure 7.3.

Figure 7.3: Block diagram of the PID controller used

115


7.2.2 SMA control electronics

The control electronics has two stages. The first one is the conversion of the PWM signal delivered by

the micro-controller to DC tension. The second is a Voltage Controlled Current Source that receives

the DC tension and transforms it in a constant current that feeds the SMA. The second stage has been

designed to have a low power consumption (7mA to 20mA when all the circuits are connected). The

current that passes trough the SMA when it is resting is less than 3mA, which lets it cool quickly.

For more information about the electronic circuit schema and equations refer to the appendix 11.2.

Each SMA wire has an independent electronic circuit feed by the same power supply. The voltage-

controlled current source VCCS is used to generate a constant current in the SMA wire and to heat

it constantly, the current varies according to the control DC tension that is related to the PWM duty

cycle. The PWM duty cycle goes from 5% to 95% with a 1kHz frequency in order to avoid an excessive

current on the SMA. A problem associated to these kinds of designs is the integral windup, which can

provoke long overshoot periods, encouraged by the excessive values that reaches the control sign due

to the accumulation in the integrator. In order to avoid this problem and accelerate the heating, (38)

proposed to feed the SMA with a high pulse of current. However, while in (38) such pulse is maintained

throughout the SMA excitation time, our control systems sends a high pulse of current only for a small

period of time, precisely 500mA for less than 150ms. Then, it feeds the SMA with a normal ramp from

10% of the target current to 100% of the target current. Moreover, to reduce the over and undershoot,

the voltage was limited between a minimum and maximum value of 10mV and 60mV, respectively,

making the integrator act only when these limits are overcome (cf. Fig. 7.4). Such values have been

determined experimentally.

Figure 7.4: Block diagram of the PID controller used

The hardware used (18F458 PIC) has a 10-bits A/D converter. Since in our system the maximum

voltage measured at the SMA is V SMA = 0.55V , with a 10 bits encoding we will have a resolution of

VSMA

210 � 1= 0.537mV (7.7)

Thus, taking into consideration the maximum current through the wire (500mA) we can measure

the SMA resistances variations of 1.074m⌦. Since the maximum variation in the SMA length is 0.34cm

and the maximum variation of the resistance is 1.6⌦, the system theoretical error on the SMA length

is of 0.067% i.e. 0.0023mm.

116


7.2.3 Experimental results

The control system were tested by applying di↵erent target resistances (i.e. desired lengths). Given

the small displacements, it is very di�cult to measure with su�cient precision the final length of the

wire. Therefore, in order to assess the actual precision of the system i had measured the final resistance

of the SMA wire. Figure 7.5(a) shows the transient performance for a 11.46⌦ set point corresponding

to a 3.08mm SMA contraction. With a 450mA current we get a settling time of 0.43s. The overshoot

that can be noticed in the plot is produced by the initial pulse. Note how, thanks to the action of the

integrator, the overshoot is reduced (blue plot) w.r.t the behavior of the SMA without it (red plot). In

the zoomed part the actual precision error of 0.5% can be noticed. Figure 7.5(b) shows the comparison

of the transient period for two di↵erent currents, which is very similar.

(a) (b)

Figure 7.5: (a) Transient performance for a 11.46⌦ set point. (b) Comparison of the transient

period at 450mA and 500mA.

Table 7.1 summarizes the results for various set points (target resistance). As it can be noticed, the

precision obtained is very good, although higher than the one theoretically achievable. On the practical

point of view the precision obtained is satisfactory, since it corresponds to a sub-millimeter position

error. Such error translates into a bending error (height h) which is negligible for the mechanics of

swimming.

Table 7.1: Summary of the performances of the control

Target Resistance (⌦) Settling time (s) Resistance error (%) Length error (mm)

12.5 0.274 3 0.102

12.4 0.302 2 0.068

12.2 0.316 1 0.034

12.0 0.331 5 0.170

11.8 0.353 1 0.034

11.6 0.374 5 0.170

11.46 0.43 0.5 0.017

117

7.3 Bending control

7.3 Bending control

To control the SMA It is very important to have a dynamic model of the SMA plant which involves

the observed output corresponding to the applied input. An SMA input power to output bend model,

in terms of a transfer function, has been obtained experimentally for each SMA. Figure 7.6 illustrates

the input/output signals obtained, where the left column shows the Input signals (Duty Cycle) and the

right column shows the output signals (bend ratio measured in degrees).

0 0.5 1 1.5 2

x 104

0

0.5

Input Left Side Tail

Samples

Du

ty C

ycle

0 0.5 1 1.5 2

x 104

0

50Output Left Side Tail

Samples

De

gre

es

0 5000 10000 150000

0.5

Input Right Side Tail

Samples

Du

ty C

ycle

0 5000 10000 150000

20

40Output Right Side Tail

Samples

De

gre

es

0 0.5 1 1.5 2 2.5

x 104

0

0.5

Input Left Side Head

Samples

Du

ty C

ycle

0 0.5 1 1.5 2 2.5

x 104

0

50Output Left Side Head

Samples

De

gre

es

0 0.5 1 1.5 2 2.5

x 104

0

0.5

Input Right Side Head

Samples

Du

ty C

ycle

0 0.5 1 1.5 2 2.5

x 104

0

50Output Right Side Head

Samples

De

gre

es

Figure 7.6: Input and Output Signals

Using the Ident toolbox of MATLA we can find the Model (Transfer Function) for each pair SMA-

Flex Sensor. Below the transfer functions are shown.

895.2s+169.9s3+7.356s2+4.896s+0.5717

SMAwireTailLeft(STL) (7.8)

358.1s+8532s3+8.939s2+89.41s+69.19

SMAwireTailRight(STR) (7.9)

429.1s+459.7s3+6.055s2+9.526s+3.279

SMAwireHeadLeft(SHL) (7.10)

741.9s+69.01s3+7.118s2+6.128s+0.5014

SMAwireHeadRight(SHR) (7.11)

For each transfer function we can find by using MATLAB the ”Fit to estimation data”. This is the

fit between the simulated response of the model and the measured data. Table 7.2 shows the Fit data

for the chosen transfer function (3 Poles - 1 Zero) and for other configurations (2 Pole - 1 Zero and 3

Pole - 2 Zero).

It can be seen for two cases the Fit to estimation data is around 10% higher than the configuration

chosen but this does not make a big di↵erence in the control of the SMA’s.

118

7.3 Bending control

Table 7.2: Fit to estimation data

Pole & Zero STL STR SHL SHR

2 & 1 62.64% 66.85% 59.96% 53.97%

3 & 1 65.17% 66.85% 63.64% 68.3%

3 & 2 74.03% 66.99% 73.98% 63.45%

7.3.1 Controller setup

For the control strategy, we have used a PID controller. Using the ZieglerNichols methodology, we have

tuned the values of the PID parameters (Kp , Ki , Kd ) based on the analysis of the system under both

opened/closed loops. The system has been represented by the identified model in equations 7.8, 7.9,

7.10 and 7.11. The PID configuration is parallel the equation 7.12 shows this configuration.

Kp

+Ki

1s+K

d

s (7.12)

The PID tuning for all the systems, was made trying to match all the ”Settling times”. This because

is critical to have the same time response over each SMA due to the antagonistic configuration and the

singularity. Table 7.3 shows the values for each gain Kp

Ki

and Kd

, the performance and robustness

find using MATLAB.

Table 7.3: PID controller characteristics

SMA Kp

Ki

Kd

Rise time Settling time Overshoot Peak

STL 1.1442 1.8628 0.15974 0.0155 seconds 0.0271 seconds 0.116% 1

STR 78.7223 618.4335 2.5052 0.00215 seconds 0.0271 seconds 4.3% 1.04

SHL 1.6841 2.1174 0.33485 0.0153 seconds 0.0271 seconds 0% 1

SHR 1.3619 2.1995 0.19387 0.0153 seconds 0.027 seconds 0.0165% 1For all the systems the Closed-loop stability is Stable

7.3.2 Passive Noise Reduction System

Due to an external source (environmental factors such as jerks or vibrations) as well as internal factors,

a considerable amount of noise is added to the signals coming from the sensors. These high frequency

signals (noise) cause the readings to oscillate between considerable high and low values. This noise

must to be filtered by using passive, active or programmable filters. Due to the simplicity we used

passive filters. The figure 7.8 shows the signals (with and without noise) from the Flex and Current

sensors.

Using MATLAB is possible to estimate the Cuto↵ Frequency using the Fast Fourier Transform

(FFT) method. Alike we know what type of filter we need to implement. Figure 7.8 shows the both

filters (low-pass filter) used to reduce the noise on the signals.

The Cuto↵ frequency was set on 3.4361Hz and 23.6177Hz for the current and flex sensor respec-

tively.

7.3.3 Control Schema

A Bend Feedback Control System (BFSC) based on the signals from the Flex sensor is used to control

the shrink on the SMA. Alike as a safety caution to over-current on the SMA and avoid break it, we use

119

7.3 Bending control

0 5000 10000 150000

0.1

0.2

0.3

0.4

0.5

0.6

Current Sensor Noise

Samples

Curr

ent (A

)

(a)

0 0.5 1 1.5 2

x 104

−20

0

20

40

60Flex Sensor Noise

Samples

Angle

(D

egre

es)

(b)

Figure 7.7: Noise signal (blue) compared with filtered signal (red). (a) Current Sensor Signals.

(b) Flex Sensor Signals.

(a) (b)

Figure 7.8: Passive Low-Pass filters. (a) Current Sensor Low-pass filter. (b) Flex Sensor Low-

pass filter.

a current sensor to sense the electrical current on each SMA. Figure 7.9 shows the closed-loop control

block diagram.

Set point(Bend) Microcontroller

PID Controller

To theSMA Wire

Flex Sensor

Current Sensor

SafePWM

10-bit A/D Converter

CurrentBend

Bend

Power Driver

Power Supply

Low-Pass Filter

Low-Pass Filter

Figure 7.9: A bend feedback control schema for a single SMA actuator.

The PID controller receives the input reference Bend (set point) and the feedback of Bend (Flex

sensor); therefore, it calculates the heating current to drive the SMA actuator. The Safe Block works

like a switch, taking the Bend and Current measures to drive the output. Using the technique of

pulse-width modulation (PWM),the digital output of the PID controller is converted to a duty cycle

percentage and send to a Power Drive circuit that safely power the SMA with a current up to 1A. The

signals coming from the sensors are filtered and converted to a digital value using the micro controller’s

(ATmega328) 10-bit D/A converter. The frequency of work is 333.33Hz.

120

8

General experimental results

”This is the end, Beautiful friend, This is the end, My only friend, the end”

The Doors (The End).

8.1 Overview

This chapter presents further experiments aimed at discussing the methods (modelling and control)

proposed in this thesis and assessing their potential for developing bio-inspired fish-like robot. Along

the previous chapters, it is has been presented a detailed workflow that describe all the processes

involved in that goal, from the analysis of biological data that inspire and define the design of BR3, to

the kinematics, dynamics, hydrodynamics, actuation and control methods that enable BR3 to behave

like its biological counterpart. The order of the tests was:

• Air (figure 8.1(a)): spine, without ribs and skin (figure 8.1(b))

• Air: spine ribs and skin (figure 8.1(c))

• Free Swimming

• Water-channel

This allowed me to know the e↵ect of the ribs and skin in the control system.

8.1.1 Methods and goals

Experiments are categorised in three areas:

1. Control performance. It evaluates both steady swimming and morphing caudal fin response. The

goal of the experiments is twofold: (i) to assess the performance of SMA actuators in terms of

accuracy, limitations and impact into the proper modulation of the body and caudal fin. (ii) to

evaluate how the proposed bend controller enables accurate forward and turning swimming.

2. Hydrodynamics. The goal is to demonstrate the impact of body bend controllers into the proper

generation of hydrodynamic forces. It also discusses how to induce accurate hydrodynamic

behaviour for similar robots, not necessarily fish-like, based on kinematics, dynamics and control

parameters.

121

8.1 Overview

light

Video Camera

BR3

Control Circuit Board

(a)

(b) (c)

Figure 8.1: Air-test sets-up

3. E�cient swimming. E�ciency is evaluated in terms of net force production. The goal is to

quantify how net forces can be increased by bending the body in a proper way. Here, the

hypothesis introduced at the beginning of this thesis is demonstrated.

8.1.2 The water-channel setup

Quantification of dynamics and hydrodynamics data requires a complex setup. For this, the Florida

Atlantic University’s water-channel facility is used. Figure 8.2 describes the main experimental setups

used in this thesis. Firstly, the design of a morphing caudal fin and swimming by body bending required

the use of a setup specially conceived for assessing SMA actuation. This setup is depicted in Figure

8.2(a) and enables the BR3 to be mounted in. Secondly, Figure 8.2(b) shows the wind-tunnel setup. It

enables the entire robot to be mounted on a 6D force sensor that measures dynamics and hydrodynamics

forces. Water channel experiments have been conducted at the Florida Atlantic University’s Curet lab

facility, which is a 250 250 mm recirculating water tunnel (Figure 8.3). High-resolution/High-speed

CMOS camera (Photron 1024 PCI, resolution 1024x1024 pixels, lens 85mm, f/1.4) allow to capture

122

8.2 Tests

Water Channel

Control Circuit Board

BR3

(a)

PIV particles

Laser beam

Force sensor

Laptop

BR3

Water channelControl circuit board

Power supply

Oscilloscope

(b)

Figure 8.2: Water channel set-up

all markers useful for the kinematics and 2D Particle image velocimetry (PIV) extraction.

8.2 Tests

I had tested the prototype in three di↵erent situations in order to compare the real behaviour with

respect to the theory and the simulations. In all situations the fish was fixed by the head because we

are testing the ability of this structure to moves according to the swimming patterns and to produce

trust by measuring the forward force. To determine the trajectory of the fish body, reference spot was

placed along the fish body and then they were traced using a Particle Tracking Visualisation (PTV)

program. The video was recorded at 60 frames per second thus the time in seconds is Time(Seconds)

= Frames/60

8.2.1 Air: spine, without ribs and skin

This test was made in order to prove the accuracy of the control and to identify the e↵ect of the extra

weight and the tensile strength produce by the Ribs and Skin respectively. Figure 8.4 shows top view

of the backbone running the bend control. The reference position was set to 10o for both sections.

The frequency was set to 1Hz but from the Figure 8.5 we know from that measured frequency was

0.65Hz but according with the cooling time of the SMA wires (1.7 seconds) that sets the Maximum

Theoretical Speed (MTS) we know the SMA frequency (0.588Hz) means that we are 0.06Hz over the

MTS. For the experimental test the bend angle was set to 15o.

8.2.2 Air: spine with ribs and skin

Figure 8.6 shows the real fish performing the thunniform and sub-carangiform swimming mode.

123

8.2 Tests

Figure 8.3: 250 250 mm recirculating water tunnel

When the accuracy of the control was proved by using only the backbone in the air, the ribs and

skin was added and a new test was made. Figure 8.6 shows the experimental setup. Initially this was

set to 1Hz and the angle of movement to 10o (Fig. 8.7) adding both sections bend angles. But the

added masses (ribs) and tensile strength (skin) decreases the frequency up to 0.65Hz according to the

Figure 8.7.

8.2.3 water channel

The BR3 is fixed to the force sensor to measure the thrust and movement. the swimming mode created

was the sub-carangiform type, because this movement implement the contraction of both sections.

However the thrust created by this movement quite similar to the carangiform mode. figure 8.8 shows

the fish swimming movement tracking the caudal fin displacement in x-axis. The angle of the movement

is 15o.

From the figure 8.8 is possible to see the best response of the fish swimming movement compared to

the other two scenarios (Fig. 8.5, Fig. 8.7). This due to the hydrodynamic e↵ect in the control system.

In both cases (Air) the control system includes the hydrodynamics in the model, for these reason the

accurate of the response is lower than the water response. Prove this e↵ect is important because in

this way is possible to know that the hydrodynamic model works well.

Figure 8.9 shows the PIV visualisation for the vortices, moving along the body and finishing at the

caudal fin.

The Figure 8.10 shows the thrust force.

The maximum thrust achieved was 12.5grams-Force with a Sub-carangiform swimming mode.

124

8.2 Tests

(a) (b)

(c) (d)

(e) (f)

Figure 8.4: Swimming modes (a) Thunniform Tail-Up, (b) Thunniform Tail-Down, (c) Carangi-

form Tail-Up, (d) Carangiform Tail-Down, (e) Sub-Carangiform Tail-Up, (f) Sub-Carangiform

Tail-Down

8.2.4 Free swimming

The experimental setting is composed of a pool of 1 ⇥ 0.5 meters, with a depth of about 70 cm and

a grid of 5 cm resolution on the bottom. High-level swim patterns were generated using an o↵-board

laptop computer and were programmed in Matlab(TM). Reference positions (set points for angle �)

were sent to the low-level control electronics via USB interface. This produced the control signals for

the actuators, sent through the tether using a standard I2C bus interface.

Second, I qualitatively tested the e↵ect of water pressure on the bending movement in open loop.

Actuation speed and strain were not significantly a↵ected, demonstrating that the actuators actually

produced enough force to push the water aside (see Figure 8.11).

Then, steady forward swimming tests were performed. The resulting linear speed was measured in

order to calculate the performance parameters described above and reported in Table 8.2. In summary,

125

8.2 Tests

0 50 100 150 200 250 300 350−15

−10

−5

0

5

10

15Tail Movement

Frames

An

gle

Figure 8.5: Response for the sub-carangiform swimming mode. Tracked trajectory for the tail

segment. Theoretical trajectory (Red line), Measured trajectory (Blue line)

(a) (b)

(c) (d)

Figure 8.6: Swimming modes (a) Thunniform Tail-Up, (b) Thunniform Tail-Down, (c) Sub-

Carangiform Tail-Up, (d) Sub-Carangiform Tail-Down

the experimental test have shown a significant di↵erence form the simulation results. Di↵erences were

expected, since the model used for numerical simulation was simplified: as mentioned, it was a 2D

model moving in a perfect fluid (i.e. no vorticity was considered), and the body was discretized into

nine ellipse-shaped bodies. However, a dramatic degradation of the closed loop swimming performances

was observed. An explanation of this is mainly due to the protective skin, that turned out to be not

elastic enough, and by the control tether, whose rigidity hindered the fish movements. The skin caused

the actual bending range to su↵er a reduction of almost 50%, (see Figure 8.12) producing a very slow

linear speed, approximately one fourth to one half of the one obtained in the simulations. Moreover,

126

8.2 Tests

0 50 100 150 200 250 300 350−15

−10

−5

0

5

10

15

20Tail Movement�

Frames

An

gle

Figure 8.7: Response for the sub-carangiform swimming mode in air with ribs and skin. Tracked

trajectory for the tail segment. Theoretical trajectory (Red line), Measured trajectory (Blue line)

the tether caused an unmeasurable lateral and longitudinal drag not considered in the simulations, and

that introduced a noise in the free swimming, with a negative impact on the linear speed.

Table 8.1: Comparison of the simulation and experimental results for steady swimming. (atail

=

0.49, abody

= 0.27, f = ⇡/2 Hz)

Parameter ExplanationValue for iTuna Typical values

Simulation Experiment for real fishes

Head swing factor

Ratio between the tail tip

oscillations and the head 0.19 0.38 0.15-0.4

oscillations

St

= f · atail

/V

0.26 1.32 0.25-0.35Strouhal a

tail

=tail osc. amplitude

number f=tailbeat freq.

V=linear speed

SpeedExpressed in body

0.3 0.1 N/Alengths/second

Finally, the C-start maneuver was performed. In this case, since the control produced a single

strong pulse, the behavior was similar to the open loop test, and performances were much better, in

good accordance with the simulation. Figure 8.13 shows a comparison, where only a small delay of

about 0.5 seconds can be noticed.

127

8.2 Tests

0 500 1000 1500−20

−10

0

10

20Tail Movement

Frames

An

gle

Figure 8.8: Response for the sub-carangiform swimming mode in water. Tracked trajectory for

the tail segment. Theoretical trajectory (Red line), Measured trajectory (Blue line)

Table 8.2: Comparison of the simulation and experimental results for steady swimming (perfor-

mance and morphology parameters)

f=0.5 Hz, atail

=0.49 f=0.5 Hz, atail

=0.56

V (m/s) V (BL/s) St

Sh

V (m/s) V (BL/s) St

Sh

Simulation 0.052 0.17 0.73 0.19 0.114 0.38 0.41 0.19

Real 0.024 0.08 1.67 0.38 0.03 0.1 1.32 0.38

Figure 8.9: PIV visualisation

128

8.2 Tests

0 0.5 1 1.5 2 2.5 3

x 104

−4

−2

0

2

4

6

8

10

12

14

Samples (1kHz)

Fo

rce

(g

ram

s)

Thrust

Figure 8.10: PIV visualisation

Figure 8.11: Testing bending in water (two segments, overloaded SMAs, open loop)

Figure 8.12: Linear swimming with f=0.5 Hz, atail

=0.49 at t=1, t=2, t=3, t=4 seconds (two

tail beats). The distance travelled is approximately 7 centimeters. Notice the reduction of the

bending with respect to Figure 5.13

129

8.2 Tests

Figure 8.13: Stills of the C-start maneuver of the simulation and with the real prototype

130

9

Conclusions and Future Work

”Somewhere over the rainbow, Bluebirds fly, And the dreams that you dreamed of, Dreams really do

come true”

Israel ”IZ” Kamakawiwo’ole (Somewhere over the rainbow)

9.1 General conclusions

BR3 is the first prototype of its kind with the ability to swim (in any swim mode) and manoeuvring

only by using its bendable body and morphing caudal fin. Moreover this prototype has the potencial

to perform deepwater exploration due to its completely absence of movies parts. Alike this robot is

the only of its kind (compare with the AUVs) with the feature to carry large loads within his body

due to the great payload available. All the novel methodologies introduced in this thesis are aimed at

achieving that goal. Motivated by the potential behind fish swimming e�ciency and the lack of highly

manoeuvrable AUVs (not necessarily fish-like), BR3 is definitively a step towards a new generation

of Autonomous Underwater Vehicles with tremendous propulsion e�ciency and manoeuvrability that

bending a flexible body and morphing the caudal fin geometry enables. In pursuing this long-term

vision, a hypothesis was declared:

Understanding the e↵ect of the bend angle and frequency in terms of steady swimming

(acting the backbone) and manoeuvring (acting the caudal fin) and therefore including

bend angle and frequency information into the swimming controller will allow for proper

modulation of the backbone and caudal fin kinematics that finally would produce and

increase net forces, thereby improving on swimming e�ciency.

To demonstrate and validate the aforementioned hypothesis, most of the methods for design, mod-

elling and control were based on an exhaustive and unprecedented analysis of its biological counterpart,

that finally provided the robust foundation to approach each state of BR3’s development.

In conclusion, I believe that smart actuators and flexible continuous structures can be a promising

field for making alternative bio-inspired robots, devoid of rotating parts and that are simpler and

lighter, and that can have interesting application domains.

131

9.1 General conclusions

Design

A novel design framework relating kinematics/hydrodynamics parameters with morphological param-

eters was defined based on validated biological data. It shows how a flexible and continuous body and

caudal fin influence on the proper definition of design criteria. Because BR3 is the first fish-like robot

with a flexible estructure, this framework is key for future developments of similar fish-like robots with

di↵erent morphological parameters.

In terms of the mechatronics design, the simplicity, extremely light weight and practically null

volume of the actuators (an advantage of the available payload) make the fish-like robot suitable for

biological applications. The actuators are absolutely silent and do not produce any vibration, a feature

which can be exploited, e.g., in the observation of sea wild life since the robot would not disturb in any

way (besides its presence). Beyond that, the novelty of using a bendable structure as the backbone

of the fish based on the V-shape configuration of antagonistic SMA-based actuation muscles has great

potential for improving the maneuverability of the fish while performing the aforementioned swim

patterns underwater.

Modeling

Models for kinematics, dynamics, hydrodynamics and free vibrations analysis and actuation were de-

fined and experimentally validated. This models allowed for the quantification of the influence of bend

angle into robot’s manoeuvrability and the key role of proper body and caudal fin modulation aimed

at the production of rolling and yaw torques for turning and forward swim.

Control

A Swim Control Architecture was defined. The proposed control method for attitude control has

demonstrated to be key for achieving proper response in terms of attitude stabilisation and tracking.

More important, the assumptions that,

• it is possible to mimic all kinds of swimming modes using only a continuous and flexible

structure divided into two segments.

• it is possible to generate roll and turn using a morphing caudal fin during steady swimmings

without having to move any other part of the body.

• it is possible to create more energy e�cient robots by using the Modes of vibration. Systems

entry in Resonance when a small periodic driving forces produce large amplitude oscillations.

This is because the system stores vibrational energy.

these assumptions (to mimic, to generate and to create) were proved by using two di↵erent control

theories. Mimicking the way fishes take advantage of inertial and hydrodynamic forces produced by

the body and caudal fin in order to both increase thrust and maneuver is a promising way to design

more e�cient AUVs.

The novel body bend and caudal fin bend modulation strategy and attitude control methodology

presented and validated in this thesis provide a totally new way of controlling swimming robots that

eliminates the need of appendices such as pelvic, anal, and dorsal fins. These developments are a key

step towards achieving the first fish-like robot capable of sustained and energy e�cient swimming. The

132

9.2 Future Work

possibility of controlling the body bend and shape of the caudal fin has great potential to improve the

maneuverability of current Autonomous underwater vehicles.

The accuracy of the actuators has been improved in terms of avoiding the SMA fibers becoming

slack due to the two-way shape memory e↵ect of the antagonistic operation. The pre-heating approach

demonstrated an increase in actuation speed, doubling the actuation frequency. The experimental

results obtained have demonstrated the feasibility of the concept and, although not entirely satisfying,

are very encouraging.

9.2 Future Work

The use of Shape Memory Alloys has been key in achieving light actuated body and morphing caudal

fin but their power consumption and actuation speed are still a radical limitation. This thesis ex-

plored how to speed-up SMA operation while maintaining the limits of power consumption, however,

future work dedicated to improve on SMA performance is required, specially in terms of eliminating

fatigue phenomenon by means of introducing high bandwidth controllers. Methods for embedding force

feedback into a single SMA actuator is a top order for future development of these smart actuators.

9.3 Thesis schedule

The Gantt diagram details the tasks and milestones carried out during the thesis development.

133

9.3 Thesis schedule

2w1.1) Biomechanics6w1.2) Fish physiology4w1.3) Fish hydrodynamics4w1.4) Smart materials

16w1) Nature's Swimmers@UPM

3w2.1) Fish morphology2w2.2) Fish muscles2w2.3) Fish skin3w2.4) Fish hydrodynamics3w2.5) Fish skeleton structure6w2.6) Initial BR3-concept

proposal

19w2) Fish Swimming Kinematics@UPM

3) Milestone 1: Chapter 1-2-3 Doc

2w4.1) Choice of species3w4.2) Morphological

parameters4w4.3) Fish swimming

kinematics2w4.4) Dynamics and

Hydrodynamics5w4.5) Proposed model for

body-bend actuation

16w4) In-vivo fish swim analysis@UPM

5) Milestone 2: Chapter 4 Doc

7w6.1) General morphology and kinematics

12w6.2) Dynamics Modelling15w6.3) Hydrodynamics

Modelling5w6.4) SMA actuation modelling5w6.5) Open-loop testing7w6.6) Model validation against

in-vivo behaviour

51w6) BR3 Modelling@UPM


13w8.1) CAD development5w8.2) Biomechanics insights6w8.3) ABS body ribs

7w 3d8.4) Fish skin membrane6w8.5) Actuators and sensors8w8.6) Hardware onboard

45w 3d8) BR3 design and biomechanics@UPM


19w10.1) Body bend control13w10.2) Morphing caudal fin

control10w10.3) Swimming patterns

control16w10.4) Control architecture23w10.5) Colsed-loop testing

81w10) BR3 Controlin_situ @FAU


37w12.1) Performance analysis (water-channel)

18w12.2) BR3 model identification21w12.3) Investigation of BR3

swimming22w12.4) Free-Vibrations

modelling and analysis25w12.5) swimming control and

manoeuvring

123w12) Experimental resultsIn-situ @FAU

13) Milestone 6: Final Doc

Title E!ortQtr 1 2013 Qtr 2 2013 Qtr 3 2013 Qtr 4 2013 Qtr 1 2014 Qtr 2 2014 Qtr 3 2014 Qtr 4 2014 Qtr 1 2015 Qtr 2 2015

134

10

Publications

The development of this thesis has allowed the following scientific production, including JCR referred

journals, book chapters, conference proceedings and press articles.

10.1 Journals, book chapters and conference proceedings

Referred Journals (ISI-JCR)

1. (Q1) William Coral, Claudio Rossi, Roque Saltaren, Oscar M Curet, Antonio Barrientos, 2015. Free

Vibration Analysis of a Robotic Fish based on a Continuous and Non-uniform Flexible Backbone with

Distributed Masses. The European Physical Journal Special Topics (EPJ ST). To be published (Accepted

2014), Invited Paper. (Topic on Free Vibrations to make more energy e�ciency robots)

2. (Q1) Rossi C, Colorado J., Coral W., Barrientos A., 2011. Bending Continuous Structures with SMAs:

a Novel Robotic Fish Design. Bioinspiration and Biomimetics, vol. 6, No. 4, 15pp. (Topic on SMA

identification and resistance control)

Book chapters

1. Coral W., Rossi C., Colorado J., Barrientos, A., 2012. SMA-based muscle-like actuation in biologically

inspired robots: A state of the art review, in book Smart actuation and sensing systems - Recent advances

and future challenges. In-Tech, ISBN: 978-953-307-990-4. (Literature review on SMA technology

used as actuators in bio-inspired robots

2. Claudio Rossi, William Coral, Antonio Barrientos, 2012. Robotic Fish to Lead the School, in book

Swimming Physiology of Fish: Towards using exercise for farming a fit fish in sustainable aquaculture.

Springer, ISBN: 978-3-642-31048-5. (Using fish-like robots to improve the health and growing

of fishes in aquaculture)

Conferences

1. William Coral, Claudio Rossi, Irene Perrino Martin - Bio-inspired Morphing Caudal Fin Using Shape

Memory Alloy Composites for a Fish-like Robot. Design, Fabrication and Analysis ICINCO-12th Inter-

national Conference on Informatics in Control, Automation and Robotics, Colmar, Alsace - France, 21-23

July, 2015.

135

10.2 Technical And Technological Manufacturing

2. Claudio Rossi, Pablo Gil, William Coral - Evolutionary Training of Robotised Architectural Elements.

18th European Conference on the Applications of Evolutionary Computation, Copenhagen, Denmark,

April 08-10 2015

3. Claudio Rossi, William Coral - Robot Fishes’ Escape From Flatland. 2nd FitFish - Workshop on the

Swimming Physiology of Fish, Barcelona, Spain, October, 2014.

4. Claudio Rossi, Zongjian Yuan, Chao Zhang, Antonio Barrientos, William Coral - Shape Memory Alloy-

Based High Phase Order Motor. ICINCO-12th International Conference on Informatics in Control,

Automation and Robotics, Vienna-Austria, 1-3 September, 2014.

5. Claudio Rossi, William Coral, Julian Colorado, Antonio Barrientos - Towards Motor- less and Gear less

Robots: a bio mimetic Fish Design. International Workshop on bio-inspired robots, Nantes, France, April

6-8, 2011

6. Claudio Rossi, William Coral, Julian Colorado, Antonio Barrientos - A Motor-less and Gear-less Bio-

mimetic Robotic Fish Design. ICRA - IEEE International Conference on Robotics and Automation,

Shanghai, China, May 9-13, 2011

7. Coral Cuellar William, Rossi Claudio, Barrientos Antonio, Colorado Julian - Fish Physiology Put Into

Practice: A Robotic Fish Model. FitFish - Workshop on the Swimming Physiology of Fish, Barcelona,

Spain, July 2-3, 2010.

8. Claudio Rossi, Barrientos Antonio, William Coral Cuellar - SMA Control for Bio-Mimetic Fish Locomo-

tion. ICINCO-International conference on informatics in control, Automation and Robotics, Madeira-

Portugal, June 15-18, 2010

10.2 Technical And Technological Manufacturing

1. PiezoPower High E�ciency Electronic circuit which controls the linear motor PiezoWave micrometer

resolution (0.5µm - 1µm) using low voltages, from 2.7V to 3.3V. 2013

2. PowerSMA Electronic circuit to actuate SMA’s using a PWM signal. The maximum output current is

7000 mA, the maximum power supply voltage is 30V

136

http://www.piezomotor.se

References

[1] Bongsu Shin, Ho-Young Kim, and Kyu-Jin Cho. Towards a bio-

logically inspired small-scale water jumping robot. In

Biomedical Robotics and Biomechatronics, 2008. BioRob 2008. 2nd

IEEE RAS & EMBS International Conference on, pages 127–131.

IEEE, 2008. ix, 25, 26, 27

[2] Zhenlong Wang, Yangwei Wang, Jian Li, and Guanrong Hang. A

micro biomimetic manta ray robot fish actuated by

SMA. In Robotics and Biomimetics (ROBIO), 2009 IEEE Inter-

national Conference on, pages 1809–1813. IEEE, 2009. ix, 27,

28

[3] Anthony Westphal, NF Rulkov, J Ayers, D Brady, and M Hunt.

Controlling a lamprey-based robot with an electronic

nervous system. Smart Structures and Systems, 8(1):39–52,

2011. x, 28

[4] QuocViet Nguyen, Hooncheol Park, Doyoung Byun, and Namseo

Goo. Recent progress in developing a beetle-mimicking

flapping-wing system. In World Automation Congress (WAC),

2010, pages 1–6. IEEE, 2010. x, 30

[5] Shu-Hung Liu, Tse-Shih Huang, and Jia-Yush Yen. Sensor fusion

in a SMA-based hexapod bio-mimetic robot. In Ad-

vanced robotics and Its Social Impacts, 2008. ARSO 2008. IEEE

Workshop on, pages 1–6. IEEE, 2008. x, 31, 32

[6] Je-Sung Koh and Kyu-Jin Cho. Omegabot: Crawling robot

inspired by ascotis selenaria. In Robotics and Automation

(ICRA), 2010 IEEE International Conference on, pages 109–114.

IEEE, 2010. x, 31, 32

[7] Yonghua Zhang, Jianhui He, Jie Yang, and KH Low. Initial re-

search on development of a flexible pectoral fin us-

ing Shape Memory Alloy. In Mechatronics and Automation,

Proceedings of the 2006 IEEE International Conference on, pages

255–260. IEEE, 2006. x, 33

[8] Claudio Rossi, J Colorado, W Coral, and A Barrientos. Bend-

ing continuous structures with SMAs: a novel robotic

fish design. Bioinspiration & biomimetics, 6(4):045005, 2011.

x, 25, 34, 35, 36, 81, 85

[9] J Colorado, A Barrientos, C Rossi, J W Bahlman, and K S Breuer.

Biomechanics of smart wings in a bat robot: morphing

wings using SMA actuators. Bioinspiration & Biomimetics,

7(3):036006–036006, August 2012. x, 25, 37, 38, 82

[10] ELIOT G Drucker and George V Lauder. Locomotor forces

on a swimming fish: three-dimensional vortex wake

dynamics quantified using digital particle image ve-

locimetry. Journal of Experimental Biology, 202(18):2393–

2412, 1999. xi, 47

[11] Eliot G Drucker and George V Lauder. Function of pec-

toral fins in rainbow trout: behavioral repertoire

and hydrodynamic forces. Journal of experimental biology,

206(5):813–826, 2003. xi, 47

[12] George V Lauder, Erik J Anderson, James Tangorra, and Peter GA

Madden. Fish biorobotics: kinematics and hydrody-

namics of self-propulsion. Journal of Experimental Biology,

210(16):2767–2780, 2007. xi, 47

[13] Silas Alben, Peter G Madden, and George V Lauder. The me-

chanics of active fin-shape control in ray-finned fishes.

Journal of The Royal Society Interface, 4(13):243–256, 2007. xi,

49

[14] B E Flammang and G V Lauder. Caudal fin shape modulation

and control during acceleration, braking and backing

maneuvers in bluegill sunfish, Lepomis macrochirus.

The Journal of Experimental Biology, 212(2):277–286, Decem-

ber 2008. xiii, 89

[15] B E Flammang and G V Lauder. Caudal fin shape modulation

and control during acceleration, braking and backing

maneuvers in bluegill sunfish, Lepomis macrochirus.

The Journal of Experimental Biology, 212(Pt 2):277–286, Jan-

uary 2009. xiii, 89, 90

[16] Dynalloy. Dynalloy, Inc. FLEXINOL. http://www.dynalloy.

com/TechDataWire.php. xvii, 22

[17] SY Lee and SM Lin. Vibrations of elastically restrained

non-uniform Timoshenko beams. Journal of sound and vi-

bration, 184(3):403–415, 1995. xvii, 109

[18] H. Hu. Biologically Inspired Design of Autonomous

Robotic Fish at Essex. In IEEE SMC UK-RI Chapter Con-

ference, on Advances in Cybernetic Systems, pages 3–8, 2006. 1

[19] Jamie M. Anderson and Narender K. Chhabra. Maneuvering and

Stability Performance of a Robotic Tuna. Integrative and

Comparative Biolog, 42(1):118–126, 2002. 1

[20] K.A. Morgansen, B.I. Triplett, and D.J. Klein. Geometric Meth-

ods for Modeling and Control of Free-Swimming Fin-

Actuated Underwater Vehicles. IEEE Transactions on

Robotics, 23(6):1184–1199, 2007. 1

[21] Patricio A. Vela, Kristi A. Morgansenand, and Joel W. Burdick.

Underwater locomotion from oscillatory shape defor-

mations. In 2002 IEEE Conference on Decision and Control,

2002. 1

[22] K. H. Low and C. W. Chong. Parametric study of the swim-

ming performance of a fish robot propelled by a flex-

ible caudal fin. Journal of Bioinsp. Biomim, 5, 2010. 1

[23] G. V. Lauder O. M. Curet, N. A. Patankar and M. A. MacIver. Me-

chanical properties of a bio-inspired robotic knifefish

with an undulatory propulsor. Journal of Bioinsp. Biomim,

6, 2011. 1

[24] P. Valdivia y Alvarado and K. Youcef-Toumi. Design of Ma-

chines with Compliant Bodies for Biomimetic Loco-

motion in Liquid Environments. ASME Journal of Dynamic

Systems, Measurement, and Control, 128:3–13, 2006. 1

[25] Aiguo Ming, Seokyong Park, Yoshinori Nagata, and Makoto Shimojo.

Development of Underwater Robots Using Piezoelec-

tric Fiber Composite. In IEEE International Conference on

Robotics and Automation, 2009. 2

[26] Quang Sang Nguyen, Hoon Cheol Park, and Doyoung Byun. Thrust

Analysis of a Fish Robot Actuated by Piezoce-

ramic Composite Actuators. Journal of Bionic Engineering,

(8):158164, 2011. 2

[27] Z Wang, G. Hang, Y Wang, J. Li, and W. Du. Embedded

SMA wire actuated biomimetic fin: a module for

biomimetic underwater propulsion. Smart Materials and

Structures, 17(2):25–39, 2008. 2

[28] Yong hua Zhang, Yan Song, and Jie Yang anbd K. H. Low. Numer-

ical and Experimental Research on Modular Oscillat-

ing Fin. Journal of Bionic Engineering 5, (5):13?23, 2008. 2

[29] Zhenlong Wang, Yangwei Wang, Jian Li, and Guanrong Hang. A

micro biomimetic manta ray robot fish actuated by

SMA. In Proceedings of the 2009 IEEE International Conference

on Robotics and Biomimetics, pages 1809–1813, 2009. 2

137

REFERENCES

[30] Zhiye Zhang, Michael Philen, and Wayne Neu. A biologically in-

spired artificial fish using flexible matrix composite

actuators: analysis and experiment. Smart Mater. Struct.,

(19):111, 2010. 2

[31] O. K. Rediniotis, L. N. Wilson, D. C. Lagoudas, and M. M.

Khan. Development of a Shape-Memory-Alloy Actu-

ated Biomimetic Hydrofoil. Journal of Intelligent Material

Systems and Structures, 13(1):35–49, 2002. 2

[32] Yonghua Zhang, Shangrong Li, Ji Ma, and Jie Yang. Development

of an Underwater Oscillatory Propulsion System Us-

ing Shape Memory Alloy. In Proceedings of the IEEE Inter-

national Conference on Mechatronics and Automation, pages 1878–

1883, 2005. 2

[33] Afzal Suleman and Curran Crawford. Design and testing of

a biomimetic tuna using shape memory alloy induced

propulsion. Computers and Structures, (86):491499, 2008. 2

[34] S. Shatara Zheng Chen and Xiaobo Tan. Modeling of

Biomimetic Robotic Fish Propelled by An Ionic Poly-

merMetal Composite Caudal Fin. IEEE/ASME Transac-

tions on Mechatronics, 15:448–459, 2010. 2

[35] J. Jung B. Kim, D. Kim and J. Park. A biomimetic undulatory

tadpole robot using ionic polymermetal composite ac-

tuators. Smart Mater. Struct, 14:15791585, 2005. 2

[36] D. Zio J. Tangorra, P. Anquetil T. Fofonoff A. Chen and

I. Hunter. The application of conducting polymers to

a biorobotic fin propulsor. Journal of Bioinsp. Biomim, 2,

2007. 2

[37] H Wang, SS Tjahyono, B Macdonald, PA Kilmartin, J Travas-Sejdic,

and R Kiefer. Robotic fish based on a polymer actua-

tor. In Proceedings of the Australasian Conference on Robotics and

Automation, pages 1809–1813, 2007. 2

[38] Yee Harn Teh and Roy Featherstone. An Architecture for

Fast and Accurate Control of Shape Memory Al-

loy Actuators. International Journal of Robotics Research,

27(5):595–611, 2008. 2, 116

[39] H. Meier, A. Czechowicz, and C. Haberland. Control loops with

detection of inner electrical resistance and fatigue-

behaviour by activation of NiTi -Shape Memory Al-

loys. In European Symposium on Martensitic Transformations,

2009. 2

[40] R. Andrew Russell and Robert B. Gorbet. Improving the re-

sponse of SMA actuators. In IEEE International Conference

on Robotics and Automation, page 22992304, 1995. 2

[41] Kotekar P. Mohanchandra Daniel D. Shin and Gregory P. Carman.

High frequency actuation of thin film NiTi. Sensors and

Actuators A: Physical, 2004. 2

[42] Q. Sun Y. Zohar C. Ma, R. Wang and M. Wong. Frequency re-

sponse of tini shape memory alloy thin film micro-

actuators. In 13th International Conference on Micro Electro

Mechanical Systems, page 370374, 2002. 2

[43] Yitshak Zohar Rong Xin Wang and Man Wong. Residual stress-

loaded titanium-nickel shape-memory alloy thin-film

micro-actuators. Journal of Micromechanics and Microengineer-

ing, 12:323327, 2002. 2

[44] Yonas Tadesse, Alex Villanueva, Carter Haines, David Novitski, Ray

Baughman, and Shashank Priya. Hydrogen-fuel-powered bell

segments of biomimetic jellyfish. Smart Materials and

Structures, 21(4):045013–18, March 2012. 4

[45] Alex Villanueva, Colin Smith, and Shashank Priya. A biomimetic

robotic jellyfish (Robojelly) actuated by shape mem-

ory alloy composite actuators. Bioinspiration & Biomimet-

ics, 6(3):036004–17, August 2011. 4

[46] Andrew J Richards and Peter Oshkai. Experimental Study

of a Flexible Oscillating-Foil Propulsion System With

Variable Sti↵ness and Inertia. In ASME 2014 Pres-

sure Vessels and Piping Conference, pages V004T04A016–

V004T04A016. ASME, July 2014. 4

[47] Izaak D Neveln, Yang Bai, James B Snyder, James R Solberg, Oscar M

Curet, Kevin M Lynch, and Malcolm A MacIver. Biomimetic and

bio-inspired robotics in electric fish research. The Jour-

nal of Experimental Biology, 216(13):2501–2514, July 2013. 4

[48] Colin F Smith and Shashank Priya. ¡title¿Bio-inspired un-

manned undersea vehicle¡/title¿. In Zoubeida Ounaies

and Jiangyu Li, editors, SPIE Smart Structures and Materials +

Nondestructive Evaluation and Health Monitoring, pages 76442A–

76442A–9. SPIE, March 2010. 4

[49] Jian Deng, Xue-Ming Shao, and Zhao-Sheng Yu. Hydrodynamic

studies on two traveling wavy foils in tandem arrange-

ment. Physics of Fluids (1994-present), 19(11):113104, Novem-

ber 2007. 4

[50] N Kato and M Furushima. Pectoral fin model for maneuver

of underwater vehicles. In Autonomous Underwater Vehicle

Technology, 1996. AUV ’96., Proceedings of the 1996 Symposium on,

pages 49–56. IEEE, 1996. 4

[51] J M Anderson and P A Kerrebrock. The vorticity control unmanned

undersea vehicle (VCUUV)-An autonomous vehicle employing fish

swimming propulsion and maneuvering. . . . on Unmanned . . . ,

1997. 4

[52] Naomi Kato and Tadahiko Inaba. Guidance and Control of

Fish Robot with Apparatus of Pectoral Fin Motion.

ICRA, 1:446–451, 1998. 4

[53] P R Bandyopadhyay and M J Donnelly. THE SWIMMING HY-

DRODYNAMICS OF A PAH* OF FLAPPING FOILS

ATTACHED TO A RIGD) BODY. . . . Speed Body Motion

in Water, 1998. 4

[54] James Taylor Czarnowski. Exploring the possibility of plac-

ing traditional marine vessels under oscillating foil

propulsion. 2005. 4

[55] K H Low and A Willy. Biomimetic Motion Planning of

an Undulating Robotic Fish Fin. Journal of Vibration and

Control, 12(12):1337–1359, December 2006. 4

[56] K H Low and Junzhi Yu. Development of modular and

reconfigurable biomimetic robotic fish with undulat-

ing fin. In 2007 IEEE International Conference on Robotics and

Biomimetics (ROBIO), pages 274–279. IEEE, 2007. 4

[57] D Weihs and P W Webb. Optimization of locomotion. Fish biome-

chanics, 1983. 4, 9

[58] C C Lindsey. Form, Function, and Locomotory Habits in

Fish. In Locomotion, pages 1–100. Elsevier, 1979. 5

[59] Magnuson, John J. Locomotion by Scombrid Fishes: Hy-

dromechanics, Morphology, and Behavior. In Locomo-

tion, pages 239–313. Elsevier, 1979. 6, 8

[60] George V Lauder and Eric D Tytell. Hydrodynamics of Un-

dulatory Propulsion, 2005. 6

[61] George I Matsumoto. Life in moving fluids. Aquatic Botany,

51(3-4):343–344, 1995. 7

[62] THOMAS L DANIEL. Unsteady Aspects of Aquatic Loco-

motion. Integrative and Comparative Biology, 24(1):121–134,

1984. 7

[63] PAUL W WEBB. Simple Physical Principles and Verte-

brate Aquatic Locomotion. Integrative and Comparative Bi-

ology, 28(2):709–725, 1988. 7, 8, 9

[64] Charles Marcus Breder and New York Zoological Society. The

Locomotion of Fishes, 1926. 10, 11

[65] J Gray. The locomotion of fishes. Essays in marine biology,

1953. 10

138

REFERENCES

[66] Paul W Webb. The biology of fish swimming. In L Maddock,

Q Bone, and J M V Rayner, editors, Mechanics and Physiology of

Animal Swimming, pages 45–62. Cambridge University Press,

Cambridge, 2009. 10, 13

[67] John J Videler. Fish swimming. London ; New York : Chapman

& Hall, 1993. 10, 11, 13

[68] Paul W Webb. Form and Function in Fish Swimming.

Scientific American, 251(1):72–82, July 1984. 11, 13, 42

[69] M. Sfakiotakis, D. M. Lane, and J. B. C. Davies. Review of fish

swimming modes for aquatic locomotion. IEEE journal

of oceanic engineering, 24(2):235–252, 1999. 13, 85

[70] D. J. Ellerby, J. D. Altringham, T. Williams, and B. A. Block. Slow

Muscle Function of Pacific Bonito (Sarda Chiliensis)

During Steady Swimming. The Journal of Experimental Bi-

ology, 203:2001–2013, 2000. 13

[71] Denavit J. and Hartenberg R.S. A Kinematic Notation for

Lower Pair Mechanisms Based on Matrices. Trans.

ASME J. Applied Mechanics, vol. 22, pp. 215221, 1955. 14

[72] R. Featherstone. Rigid Body Dynamics Algorithms. Springer, New

York, 2008. 15

[73] M.H. Elahinia. E↵ect of System Dynamics on Shape Memory Alloy

Behavior and Control. PhD thesis, Virginia Polytechnic Insti-

tute and State University, 2004. 15, 64, 66

[74] T. Yee. Fast Accurate Force and Position Control of Shape Memory

Alloy Actuators. PhD thesis, ANU College of Engineering and

Computer Science, 2008. 17, 39

[75] Jose L Pons. Emerging actuator technologies: a micromechatronic

approach. John Wiley & Sons, 2005. 19

[76] Minoru Hashimoto, Masanori Takeda, Hirofumi Sagawa, Ichiro Chiba,

and Kimiko Sato. Application of shape memory alloy

to robotic actuators. Journal of robotic systems, 2(1):3–25,

1985. 20

[77] Katsutoshi Kuribayashi. A new actuator of a joint mech-

anism using TiNi alloy wire. the International journal of

Robotics Research, 4(4):47–58, 1986. 20, 24

[78] D Raynaerts and H van Brussel. Development of a SMA

high performance robotic actuator. In Advanced Robotics,

1991. ’Robots in Unstructured Environments’, 91 ICAR., Fifth Inter-

national Conference on, pages 61–66. IEEE, 1991. 20

[79] Koji Ikuta. Micro/miniature shape memory alloy actu-

ator. In Robotics and Automation, 1990. Proceedings., 1990 IEEE

International Conference on, pages 2156–2161. IEEE, 1990. 20

[80] Ian W Hunter, Serge Lafontaine, John M Hollerbach, and Peter J

Hunter. Fast reversible NiTi fibers for use in micro-

robotics. In Micro Electro Mechanical Systems, 1991, MEMS’91,

Proceedings. An Investigation of Micro Structures, Sensors, Actua-

tors, Machines and Robots. IEEE, pages 166–170. IEEE, 1991.

20

[81] Tom Waram. Actuator design using shape memory alloys. Hamilton,

Ont.: TC Waram, 1993. 20

[82] Xia Liu, Hong-Yan Luo, Shang-Ping Liu, and De-Feng Wang. Pilot

study of SMA-based expansion device for transanal

endoscopic microsurgery. In Machine Learning and Cybernet-

ics (ICMLC), 2011 International Conference on, 3, pages 1420–

1424. IEEE, 2011. 20

[83] Minoru Hashimoto, Tsuyoshi Tabata, and Takahiro Yuki. Devel-

opment of electrically heated SMA active forceps for

laparoscopic surgery. In Robotics and Automation, 1999. Pro-

ceedings. 1999 IEEE International Conference on, 3, pages 2372–

2377. IEEE, 1999. 20

[84] Zhenyun Shi, Da Liu, Cheng Ma, and Depeng Zhao. Accurate con-

trolled shape memory alloy actuator for minimally in-

vasive surgery. In Mechatronics and Automation (ICMA), 2011

International Conference on, pages 817–822. IEEE, 2011. 20

[85] Mingyen Ho, Alan B McMillan, J Marc Simard, Rao Gullapalli,

and Jaydev P Desai. Toward a meso-scale SMA-actuated

MRI-compatible neurosurgical robot. Robotics, IEEE

Transactions on, 28(1):213–222, 2012. 20

[86] WD Li, W Guo, MT Li, and YH Zhu. Design and Test of a Cap-

sule Type Endoscope Robot with Novel Locomation

Principle. In Control, Automation, Robotics and Vision, 2006.

ICARCV’06. 9th International Conference on, pages 1–6. IEEE,

2006. 20

[87] James M McNANEY, Valentina Imbeni, Youngjean Jung, Panayiotis

Papadopoulos, and RO Ritchie. An experimental study of the

superelastic e↵ect in a shape-memory Nitinol alloy

under biaxial loading. Mechanics of Materials, 35(10):969–

986, 2003. 22

[88] Katsutoshi Kuribayashi. Improvement of the response of

an SMA actuator using a temperature sensor. The In-

ternational Journal of Robotics Research, 10(1):13–20, 1991. 23,

25

[89] Michele Granito. S.M.A. Shape Memory Alloy Cooling System by

Peltier Cells: A Cooling System for Shape Memory Alloy Based on

the Use of Peltier Cells. LAP Lambert Academic Publishing,

2011. 23

[90] Yonas Tadesse, Nicholas Thayer, and Shashank Priya. Tailor-

ing the response time of shape memory alloy wires

through active cooling and pre-stress. Journal of Intelli-

gent Material Systems and Structures, 21(1):19–40, 2010. 23

[91] Roy Featherstone and Yee Harn Teh. Improving the speed

of shape memory alloy actuators by faster electrical

heating. In Experimental Robotics IX, pages 67–76. Springer,

2006. 23

[92] Danny Grant. Accurate and rapid control of shape memory alloy

actuators. 1999. 23, 24

[93] Yee Harn Teh. A Control System For Achieving Rapid Controlled

Motions From Shape Memory Alloy (SMA) Actuator Wires. PhD

thesis, PhD thesis, Dept. Engineering, The Australian Na-

tional University, 2003. 23

[94] Yee Harn Teh and Roy Featherstone. A new control system

for fast motion control of SMA actuator wires. In The

1st International Symposium on Shape Memory and Related Tech-

nologies, 2004. 23

[95] Yee Harn Teh and Roy Featherstone. An architecture for fast

and accurate control of shape memory alloy actuators.

The International Journal of Robotics Research, 27(5):595–611,

2008. 23, 24, 25, 36

[96] Yee Harn Teh. Fast, accurate force and position control of shape

memory alloy actuators. Australian National University, 2008.

23, 25

[97] Yee Harn Teh and Roy Featherstone. Accurate force control

and motion disturbance rejection for shape memory

alloy actuators. In Robotics and Automation, 2007 IEEE Inter-


[98] Yee Harn Teh and Roy Featherstone. Experiments on the

performance of a 2-DOF pantograph robot actuated

by shape memory alloy wires. In Proc. Australasian Conf.

Robotics & Automation, Canberra, Australia, Dec, pages 6–8,

2004. 23

[99] Daniel D Shin, Kotekar P Mohanchandra, and Gregory P Carman.

High frequency actuation of thin film NiTi. Sensors and

Actuators A: Physical, 111(2):166–171, 2004. 24

[100] Yee Harn Teh and Roy Featherstone. Experiments on the au-

dio frequency response of shape memory alloy actua-

tors. In Proc. Australasian Conf. Robotics & Automation, Sydney,

Australia, Dec, pages 5–7, 2005. 24

139

REFERENCES

[101] van der MWM Wijst. Shape control of structures and materials

with shape memory alloys. PhD thesis, Technische Universiteit

Eindhoven, 1998. 24

[102] Kikuaki Tanaka. A thermomechanical sketch of shape

memory e↵ect: one-dimensional tensile behavior. Res

Mechanica, 18:251–263, 1986. 24

[103] H Elahinia Mohammad. E↵ect of System Dynamics on Shape Memory

Alloy Behavior and Control. PhD thesis, PhD thesis, Virginia

Polytechnic Institute and State University, 2004. 24

[104] Mohammad H Elahinia and Mehdi Ahmadian. An enhanced SMA

phenomenological model: I. The shortcomings of the

existing models. Smart materials and structures, 14(6):1297,

2005. 24

[105] Koji Ikuta, Masahiro Tsukamoto, and Shigeo Hirose. Mathe-

matical model and experimental verification of shape

memory alloy for designing micro actuator. In Micro

Electro Mechanical Systems, 1991, MEMS’91, Proceedings. An In-

vestigation of Micro Structures, Sensors, Actuators, Machines and

Robots. IEEE, pages 103–108. IEEE, 1991. 24

[106] Ehsan Tarkesh Esfahani and Mohammad H Elahinia. Stable walk-

ing pattern for an SMA-actuated biped. Mechatronics,

IEEE/ASME Transactions on, 12(5):534–541, 2007. 24

[107] Danny Grant and Vincent Hayward. Variable structure con-

trol of shape memory alloy actuators. Control Systems,

IEEE, 17(3):80–88, 1997. 25

[108] Koji Ikuta, Masahiro Tsukamoto, and Shigeo Hirose. Shape mem-

ory alloy servo actuator system with electric resis-

tance feedback and application for active endoscope.

In Robotics and Automation, 1988. Proceedings., 1988 IEEE Inter-


[109] Istvan Mihalcz. Fundamental characteristics and design

method for nickel-titanium shape memory alloy. Me-

chanical Engineering, 45(1):75–86, 2001. 25

[110] Ian W Hunter and Serge Lafontaine. A comparison of muscle

with artificial actuators. In Solid-State Sensor and Actua-

tor Workshop, 1992. 5th Technical Digest., IEEE, pages 178–185.

IEEE, 1992. 25

[111] DG Caldwell and PM Taylor. Artificial muscles as robotic

actuators. In Robot Control 1988 (SYROCO’88): Selected Papers

from the 2nd IFAC Symposium, Karlsruhe, FRG, 5-7 October 1988,

page 401. Elsevier, 2014. 25

[112] Hiroyuki Fujita. Studies of micro actuators in Japan. In

Robotics and Automation, 1989. Proceedings., 1989 IEEE Interna-

tional Conference on, pages 1559–1564. IEEE, 1989. 25

[113] Yavuz Eren, Constantinos Mavroidis, and Jason Nikitczuk. B-

Spline based adaptive control of shape memory al-

loy actuated robotic systems. In ASME 2002 International

Mechanical Engineering Congress and Exposition, pages 471–478.

American Society of Mechanical Engineers, 2002. 25

[114] Dai Honma, Yoshiyuki Miwa, and Nobuhiro Iguchi. Micro robots

and micro mechanisms using shape memory alloy. In

The Third Toyota Conference, Integrated Micro Motion Systems,

Micro-machining, Control and Applications, 1984. 25

[115] Zhenlong Wang, Guanrong Hang, Jian Li, Yangwei Wang, and Kai

Xiao. A micro-robot fish with embedded SMA wire ac-

tuated flexible biomimetic fin. Sensors and Actuators A:

Physical, 144(2):354–360, 2008. 25, 26

[116] T Nick Pornsin-Sirirak, YC Tai, H Nassef, and CM Ho. Titanium-

alloy MEMS wing technology for a micro aerial vehicle

application. Sensors and Actuators A: Physical, 89(1):95–103,

2001. 25

[117] Byungkyu Kim, Sukho Park, and Jong-Oh Park. Microrobots for

a capsule endoscope. In Advanced Intelligent Mechatronics,

2009. AIM 2009. IEEE/ASME International Conference on, pages

729–734. IEEE, 2009. 25

[118] Yunchun Yang, Xiufen Ye, and Shuxiang Guo. A new type

of jellyfish-like microrobot. In Integration Technology,

2007. ICIT’07. IEEE International Conference on, pages 673–678.

IEEE, 2007. 25

[119] Je-Sung Koh and Kyu-Jin Cho. Omegabot: Biomimetic inch-

worm robot using SMA coil actuator and smart com-

posite microstructures (SCM). In Robotics and Biomimetics

(ROBIO), 2009 IEEE International Conference on, pages 1154–

1159. IEEE, 2009. 25, 31

[120] Shuxiang Guo, Toshio Fukuda, and Kinji Asaka. A new

type of fish-like underwater microrobot. Mechatronics,

IEEE/ASME Transactions on, 8(1):136–141, 2003. 26

[121] Andres Punning, Mart Anton, Maarja Kruusmaa, and Alvo Aabloo.

A biologically inspired ray-like underwater robot with

electroactive polymer pectoral fins. In International IEEE

Conference on Mechatronics and Robotics, 2004, pages 241–245,

2004. 26

[122] Byungkyu Kim, Deok-Ho Kim, Jaehoon Jung, and Jong-Oh Park.

A biomimetic undulatory tadpole robot using ionic

polymer–metal composite actuators. Smart materials and

structures, 14(6):1579, 2005. 26

[123] George Keith Batchelor. An introduction to fluid dynamics. Cam-

bridge university press, 2000. 26, 27

[124] Merle Potter, David Wiggert, and Bassem Ramadan. Mechanics of

Fluids SI Version. Cengage Learning, 2011. 26, 27

[125] Mikhail I Rabinovich, Pablo Varona, Allen I Selverston, and

Henry DI Abarbanel. Dynamical principles in neuro-

science. Reviews of modern physics, 78(4):1213, 2006. 28

[126] Yonas Tadesse, Alex Villanueva, Carter Haines, David Novitski, Ray

Baughman, and Shashank Priya. Hydrogen-fuel-powered bell

segments of biomimetic jellyfish. Smart Materials and

Structures, 21(4):045013, 2012. 29, 91

[127] Alex Villanueva, Colin Smith, and Shashank Priya. A biomimetic

robotic jellyfish (Robojelly) actuated by shape mem-

ory alloy composite actuators. Bioinspiration & biomimetics,

6(3):036004, 2011. 29, 91

[128] Gheorghe Bunget and Stefan Seelecke. BATMAV: a 2-DOF

bio-inspired flapping flight platform. In Mehrdad N

Ghasemi-Nejhad, editor, Proceedings of the SPIE, pages 76433B–

76433B–11. North Carolina State Univ., United States,

SPIE, April 2010. 30

[129] Byungkyu Kim, Moon Gu Lee, Young Pyo Lee, YongIn Kim, and Ge-

unHo Lee. An earthworm-like micro robot using shape

memory alloy actuator. Sensors and Actuators A: Physical,

125(2):429–437, 2006. 32

[130] Cecilia Laschi, Matteo Cianchetti, Barbara Mazzolai, Laura

Margheri, Maurizio Follador, and Paolo Dario. Soft robot arm

inspired by the octopus. Advanced Robotics, 26(7):709–727,

2012. 34

[131] OK Rediniotis, LN Wilson, Dimitris C Lagoudas, and MM Khan.

Development of a shape-memory-alloy actuated

biomimetic hydrofoil. Journal of Intelligent Material Systems

and Structures, 13(1):35–49, 2002. 34

[132] Claudio Rossi, Antonio Barrientos Cruz, and William Coral Cuel-

lar. SMA control for bio-mimetic fish locomotion.

2010. 36

[133] Sharon M Swartz, K Bishop, and Ismael MF Aguirre. {Dynamic

complexity of wing form in bats: implications for

flight performance}. 2006. 37

[134] Anders Hedenstrom, L Christoffer Johansson, and Geoffrey R Sped-

ding. Bird or bat: comparing airframe design and flight

performance. Bioinspiration & Biomimetics, 4(1):015001,

2009. 37

140

REFERENCES

[135] Jose Iriarte-Dıaz, Daniel K Riskin, David J Willis, Kenneth S Breuer,

and Sharon M Swartz. Whole-body kinematics of a fruit

bat reveal the influence of wing inertia on body ac-

celerations. The Journal of experimental biology, 214(9):1546–

1553, 2011. 37

[136] Daniel K Riskin, Jose Iriarte-Dıaz, Kevin M Middleton, Kenneth S

Breuer, and Sharon M Swartz. The e↵ect of body size on

the wing movements of pteropodid bats, with insights

into thrust and lift production. The Journal of experimental

biology, 213(23):4110–4122, 2010. 37

[137] Jan Van Humbeeck. Non-medical applications of shape

memory alloys. Materials Science and Engineering A, 273-

275:134138, 1999. 39

[138] Paul W Webb. 3 Hydrodynamics: Nonscombroid Fish.

Fish physiology, 7:189–237, 1979. 42

[139] John H Long Jr, William Shepherd, and RG Root. Manueu-

verability and reversible propulsion: How eel-like

fish swim forward and backward using travelling

body waves. In Proc. Special Session on Bio-Engineering Re-

search Related to Autonomous Underwater Vehicles, 10th Int. Symp.

Unmanned Untethered Submersible Technology, pages 118–134.

DTIC Document, 1997. 43

[140] Gary B Gillis. Undulatory locomotion in elongate

aquatic vertebrates: anguilliform swimming since Sir

James Gray. American Zoologist, 36(6):656–665, 1996. 43

[141] MJ Lighthill. Hydromechanics of aquatic animal propul-

sion. Annual review of fluid mechanics, 1(1):413–446, 1969. 43,

45

[142] RW Blake. On ostraciiform locomotion. Journal of the Ma-

rine Biological Association of the United Kingdom, 57(04):1047–

1055, 1977. 44

[143] CC Lindsey. Form, function and locomotory habits in

fish. Fish physiology, 7, 1978. 44

[144] R McNeill Alexander. Functional design of fishes. 1974. 45

[145] Paul W Webb. Is the high cost of body/caudal fin un-

dulatory swimming due to increased friction drag or

inertial recoil? Journal of Experimental Biology, 162(1):157–

166, 1992. 45

[146] H Liu, RICHARD Wassersug, and KEIJI Kawachi. The three-

dimensional hydrodynamics of tadpole locomotion.

The Journal of experimental biology, 200(22):2807–2819, 1997.

45

[147] GS Triantafyllou, MS Triantafyllou, and MA Grosenbaugh. Opti-

mal thrust development in oscillating foils with appli-

cation to fish propulsion. Journal of Fluids and Structures,

7(2):205–224, 1993. 45, 52

[148] UK Muller, BLE Van Den Heuvel, EJ Stamhuis, and JJ Videler.

Fish foot prints: morphology and energetics of the

wake behind a continuously swimming mullet (Ch-

elon labrosus Risso). The Journal of experimental biology,

200(22):2893–2906, 1997. 46

[149] PJ Geerlink and JJ Videler. Joints and muscles of the dor-

sal fin of Tilapia nilotica L.(Fam. Cichlidae). Netherlands

Journal of Zoology, 24(3):279–290, 1973. 46

[150] PJ Geerlink. The Anatomy of the Pectoral Fin in

Sarotherodon Niloticus Trewavas*(Cichlidae). Nether-

lands Journal of Zoology, 29(1):9–32, 1978. 46

[151] Bruce C Jayne, Adrian F Lozada, and George V Lauder. Func-

tion of the dorsal fin in bluegill sunfish: motor pat-

terns during four distinct locomotor behaviors. Journal

of Morphology, 228(3):307–326, 1996. 46

[152] Richard Winterbottom. A descriptive synonymy of the

striated muscles of the Teleostei. Proceedings of the

Academy of Natural Sciences of Philadelphia, pages 225–317,

1973. 46

[153] Eliot G Drucker and George V Lauder. Locomotor function of

the dorsal fin in teleost fishes: experimental analysis

of wake forces in sunfish. Journal of Experimental Biology,

204(17):2943–2958, 2001. 46

[154] Eliot G Drucker and George V Lauder. Locomotor function

of the dorsal fin in rainbow trout: kinematic patterns

and hydrodynamic forces. Journal of Experimental Biology,

208(23):4479–4494, 2005. 46

[155] EM Standen and GV Lauder. Hydrodynamic function of dor-

sal and anal fins in brook trout (Salvelinus fontinalis).

Journal of Experimental Biology, 210(2):325–339, 2007. 46

[156] George V Lauder and Eric D Tytell. Hydrodynamics of un-

dulatory propulsion. Fish physiology, 23:425–468, 2005. 47

[157] George V Lauder and Peter GA Madden. Fish locomotion:

kinematics and hydrodynamics of flexible foil-like fins.

Experiments in Fluids, 43(5):641–653, 2007. 48

[158] JA Sparenberg. Survey of the mathematical theory of fish

locomotion. Journal of Engineering Mathematics, 44(4):395–

448, 2002. 51

[159] Junzhi Yu, Min Tan, Shuo Wang, and Erkui Chen. Development

of a biomimetic robotic fish and its control algorithm.

Systems, Man, and Cybernetics, Part B: Cybernetics, IEEE Trans-

actions on, 34(4):1798–1810, 2004. 51

[160] Junzhi Yu and Long Wang. Parameter optimization of sim-

plified propulsive model for biomimetic robot fish.

In Robotics and Automation, 2005. ICRA 2005. Proceedings of the

2005 IEEE International Conference on, pages 3306–3311. IEEE,

2005. 51

[161] JM Anderson, K Streitlien, DS Barrett, and MS Triantafyllou.

Oscillating foils of high propulsive e�ciency. Journal

of Fluid Mechanics, 360:41–72, 1998. 52

[162] Peter Eberhard and Werner Schiehlen. Computational dy-

namics of multibody systems: history, formalisms,

and applications. Journal of computational and nonlinear dy-

namics, 1(1):3–12, 2006. 57

[163] Werner Schiehlen and Peter Eberhard. Applied Dynamics.

Springer, 2014. 57

[164] Werner Schiehlen. Multibody system dynamics: roots

and perspectives. Multibody system dynamics, 1(2):149–188,

1997. 57

[165] Werner Schiehlen. Computational dynamics: theory and

applications of multibody systems. European Journal of

Mechanics-A/Solids, 25(4):566–594, 2006. 57

[166] Tanaka. A thermomechanical sketch of shape memory

e↵ect: one-dimensional tensile behavior. Res. Mech., Int.

J. Struct. Mach. Mater. Sci. 18 251-63, 1986. 63, 65

[167] Brinson L.C. One-dimensional constitutive behavior of

shape memory alloys: thermomechanical derivation

with non-constant material functions and redefined

martensite internal variable. J. Intell. Mater. Syst. Struct. 4

22942., 1993. 64

[168] M.H. Elahinia and M. Ahmadian. An enhanced SMA phe-

nomenological model: I. The shortcomings of the ex-

isting models. Smart Mater. Struct. 14, 1297-1308, pp. 1-13,

2005. 64, 66

[169] E. Esfahani and M. Elahinia. Stable Walking Pattern for an

SMA-Actuated Biped. Mechatronics, IEEE/ASME Transac-

tions on, vol. 12, no. 5, pp. 534-541, 2007. 64

[170] Marten Nettelbladt. The Geometry of Bending. ISBN

978916373329-1, 2013. 66

[171] J. Gray. Studies in Animal Locomotion. Journal of Exper-

imental Biology, 10:88–104, 1933. 69

141

REFERENCES

[172] M. J. Lighthill. Note on the swimming of slender fish.

Journal of Fluid Mechanics, 9:305–317, 1960. 69

[173] T.Y.T. Wu. Swimming of a waving plate. Journal of Fluid

Mechanics, 10:321–344, 1961. 69

[174] Jindong Liu and Huosheng Hu. Methodology of Modelling

Fish-like Swim Patterns for Robotic Fish. In Proceed-

ings of the 2007 IEEE International Conference on Mechatronics and

Automation, 2007. 69, 85

[175] A. Munnier and B. Pincon. Biohydrodynamics Toolbox,

2008. http://bht.gforge.inria.fr/. 72

[176] Autodesk. Inventor. http://www.autodesk.com/. Accessed

October 8, 2014. 79, 80, 144

[177] Rossi C. Coral W. Colorado J. Barrientos A. A Motor-less and

Gear-less Bio-mimetic Robotic Fish Design. In IEEE

International Conference on Robotics and Automation-ICRA, pages

3646–3651, 2011. 81

[178] Inc. Smooth-On. Dragon Skin. http://www.smooth-on.com.

Accessed October 8, 2014. 86

[179] Plasti Dip International. Plasti Dip.

http://www.plastidip.com. Accessed October 8, 2014.

86

[180] Joyce W Hawkes. The structure of fish skin. Cell and Tissue

Research, 149(2):159–172, June 1974. 86

[181] Gene Helfman, Bruce B Collette, Douglas E Facey, and Brian W

Bowen. The Diversity of Fishes. Biology, Evolution, and Ecol-

ogy. John Wiley & Sons, April 2009. 87

[182] J L Tangorra, S N Davidson, I W Hunter, PGA Madden, G V Lauder,

Dong Haibo Dong Haibo, M Bozkurttas, and R Mittal. The De-

velopment of a Biologically Inspired Propulsor for Un-

manned Underwater Vehicles. Oceanic Engineering, IEEE

Journal of, 32(3):533–550, July 2007. 90

[183] Qiang Zhu and Kourosh Shoele. Propulsion performance of

a skeleton-strengthened fin. The Journal of Experimental

Biology, 211(Pt 13):2087–2100, July 2008. 90

[184] Colin Smith, Alex Villanueva, Keyur Joshi, Yonas Tadesse, and

Shashank Priya. Working principle of bio-inspired shape

memory alloy composite actuators. Smart Materials and

Structures, 20(1):012001–8, December 2010. 91

[185] Yu Chen. On the vibration of beams or rods carrying a

concentrated mass. Journal of Applied Mechanics, 30(2):310–

311, 1963. 102

[186] PAA Laura, MJ Maurizi, and JL Pombo. A note on the dy-

namic analysis of an elastically restrained-free beam

with a mass at the free end. Journal of Sound and Vibration,

41(4):397–405, 1975. 102

[187] PAA Laura, P Verniere de Irassar, and GM Ficcadenti. A note

on transverse vibrations of continuous beams subject

to an axial force and carrying concentrated masses.

Journal of Sound and Vibration, 86(2):279–284, 1983. 102

[188] M Gurgoze. A note on the vibrations of restrained beams

and rods with point masses. Journal of Sound and Vibration,

96(4):461–468, 1984. 102

[189] M Gurgoze. On the vibrations of restrained beams and

rods with heavy masses. Journal of Sound and Vibration,

100(4):588–589, 1985. 102

[190] WH Liu, J-R Wu, and C-C Huang. Free vibration of beams

with elastically restrained edges and intermediate

concentrated masses. Journal of Sound and Vibration,

122(2):193–207, 1988. 102

[191] MA De Rosa, C Franciosi, and MJ Maurizi. On the dynamic

behaviour of slender beams with elastic ends carrying

a concentrated mass. Computers & structures, 58(6):1145–

1159, 1996. 102

[192] G Venkateswara Rao, K Meera Saheb, and G Ranga Janardhan.

Fundamental frequency for large amplitude vibra-

tions of uniform Timoshenko beams with central point

concentrated mass using coupled displacement field

method. Journal of sound and vibration, 298(1):221–232, 2006.

102

[193] Charles W Bert and Moinuddin Malik. Di↵erential quadrature

method in computational mechanics: a review. Applied

Mechanics Reviews, 49(1):1–28, 1996. 103

[194] William Weaver Jr, Stephen P Timoshenko, and Donovan Harold

Young. Vibration problems in engineering. John Wiley & Sons,

1990. 104, 107

[195] T Kaneko. On Timoshenko’s correction for shear in

vibrating beams. Journal of Physics D: Applied Physics,

8(16):1927, 1975. 104

[196] H Du, MK Lim, and RM Lin. Application of generalized

di↵erential quadrature method to structural prob-

lems. International Journal for Numerical Methods in Engineering,

37(11):1881–1896, 1994. 106

[197] H Du, MK Lim, and RM Lin. Application of generalized dif-

ferential quadrature to vibration analysis. Journal of

Sound and Vibration, 181(2):279–293, 1995. 106

[198] RM Lin, MK Lim, and H Du. Deflection of plates with nonlin-

ear boundary supports using generalized di↵erential

quadrature. Computers & structures, 53(4):993–999, 1994.

106

[199] G Karami and P Malekzadeh. A new di↵erential quadrature

methodology for beam analysis and the associated dif-

ferential quadrature element method. Computer Methods

in Applied Mechanics and Engineering, 191(32):3509–3526, 2002.

106

142

Declaration

I herewith declare that I have produced this thesis document without the prohibited assistance of

third parties and without making use of aids other than those specified; notions taken over directly

or indirectly from other sources have been identified as such.

The thesis work was conducted from 2012 to 2015 under the supervision of Prof. Claudio Rossi,

PhD.

Madrid, Spain 2015

11

Annexes

11.1 Model converter

The entire fish was designed based on a 3D model from a real Black-Bass that was scanned using a 3D laser

scanner and reproduced in the Autodesk Inventor (176) CAD software. This section shows the methodology

developed to convert the scanned 3D model of fish in a 3D model suitable to be used, edited and simulated

using Inventor and Matlab. Currently existing CAD software allows you to import 3D models with di↵erent

extensions, but not allow these to be modified. i.e. what you have import is what you have. Due to my model

had to be extensively modified to allow docked the spine as well as to bring inside the sensors, actuators and

batteries, have a model that can not be changed was something I could not a↵ord. To perform this task many

software were used.

11.1.1 Importing process

This tutorial lets you convert and edit models in 3ds to Inventor, opening first with Sketchup pro.

1. Import the file with extension .3ds to sketchup pro

2. Edit the file as you wish.

3. Download the plugin to convert files from Sketchup pro to files with extension .stl

(a) In the web page ’convert-sketchup-skp-files-dxf-or-stl’ you can download the plugin, the installation

instruction are inside the file downloaded.

(b) The How to Use instruction for this plugin can be found at ’como-exportar-de-google-sketchup-a-stl’

4. Export the model from Sketchup pro to a .stl file

5. Improt the model using the software called Blender.

6. solidifying the part (mesh) as follows

(a) first switch to the metric units as shown in Figure 11.1

(b) Now make the solidification process as shown in Figure 11.2

(c) Solidify window is shown at Figure 11.3

i. Thickness: is the thickness of the walls in the solid, I am using 5 cm because in this way all

the mesh achieves can be solidify

144

http://www.guitar-list.com/download-software/convert-sketchup-skp-files-dxf-or-stl


Figure 11.1: Extrusion select

Figure 11.2: Solidify

145


Figure 11.3: Solidify window

ii. O↵set: This value can only be -1, 1, 0. when this o↵set is -1 the extrusion is made to the

inside of the the mesh, when the o↵set is 1 the extrusion is made to the outside of the mesh,

and when the o↵set is 0 the extrusion is made from the mesh outward and inward evenly. I

use 0.

iii. Even Thickness: This made the extrusion been uniform same as High Quality Normals y Fill

Rim. I choose the option 3.

iv. For better and more information about solidify you can watch this video, and in this web page

you can find the Wiki of Blender.

7. Export the Blender file using the .stl extension.

8. verify the solidification opening the .stl exported with the software called 3D-Tool (you can download the

program in this web page, it is free). this can be verified because the entire model is green everywhere

and there are no gaps or other things.

9. Import the .stl file to the software Autodesk 3ds Max

10. Extruding the complete model, this is done as follows

(a) Select the solid as shown in the following figure 11.4(a)

(b) From the tab in the figure below 11.4(b) shown in blue, select ’face extrude’

(c) Now select how much would you like to extrude as shown in te figure 11.4(c)

11. Export the model to .stl file type

12. Open the new model with the 3D-Tool program to check the if the thickness of the layers is the desired.

13. When the walls have the desired thickness export the model from Autodesk 3ds Max software to .SAT

format with this kind of file extension is possible to import it to inventor

14. Import the file .SAT to Inventor. take into account that you need to modify the Import options in

Inventor in the window OPEN at the Inventor software.

15. The file generated in point 11 can be opened directly with the program of the 3D printer (CatalystEX)

and clicking the ’Process STL’ button to check that printing is the desired.

146

http://www.youtube.com/watch?v=SSNVJzjjzCs

http://wiki.blender.org/index.php/Doc:2.6/Manual/Modifiers/Generate/Solidify

http://www.3d-tool.de

http://www.3d-tool.de

11.2 SMA control electronics

(a) (b) (c)

Figure 11.4: Screen captures. (a) Blender, solid selection (b) Blender, Face extrude (c) Blender,

extrusion parameters


The figure 11.5 shows complete electronic circuit used to measure and drive the SMA wires. this is composed by

three parts, Micro-controller (used to send the PWM signal to control the SMA wires), PWM-to-DC converter

(used to confer the PWM signal from the micro controller to a DC voltage) and Driver (used to supply the curet

to the SMA wires).

11.2.1 PWM to DC converter

This circuit converts the PWM signal coming from the Micro-controller to a DC voltage. This circuit is based

on a multi-stage low pass filter. Its need to be multi-stage because the output DC voltage needs to be very flat

without ripple voltage. this is due to the control over the SMA wires is made based on the electrical resistances

changes, and as is well known this resistance has little changes and tiny variations in the voltage can cause great

variations in the reading measures of the electrical resistance. The circuit shown in the figure 11.6 is the PWM

to DC converter.

finding the transfer function for the whole system we have:

PWM=vi

DC (out)=vo

Converting the capacitor to Laplace

C1=1

C1s

C2=1

C2sUsing the voltage divider

va

=vi

1C1s

R1+1

C1s

va

=vi

R1C1s+1(11.1)

147


Figure 11.5: Solidify window

By using the Kirchho↵’s law for current and

knowing that the current through R2 is equals to the

current passing through R3, we have:

I= v

a

�0R2

I= v

b

�v

a

R3v

a

R 2= v

b

R 3� v

a

R 3! v

a

R 2+ v

a

R 3= v

b

R 3

va

⇣1R 2

+ 1R 3

⌘= v

b

R 3⇣1R 2

+ 1R 3

⌘=R3+R2

R2R3

va

= v

b

R3

⇣

R3+R2R2R3

⌘

va

= v

b

R2R3+R2

vb

=va

(R3+R2)

R2(11.2)

Using the voltage divider

vc

=vb

1C2s

R4+1

C2s

vc

=vb

R4C2s+1(11.3)


vd

=vc

1C3s

R5+1

C3s

vd

=vc

R5C3s+1(11.4)


vo=vd

1C4s

R6+1

C4s

vo=vd

R6C4s+1(11.5)

Replacing 11.4 in 11.5

vo=

v

c

R5C3s+1

R6C4s+1

vo=vc

(R5C3s+1) (R6C4s+1)(11.6)


vo=

v

b

R4C2s+1

(R5C3s+1) (R6C4s+1)

148


Figure 11.6: PWM / DC

vo=vb

(R5C3s+1) (R6C4s+1) (R4C2s+1)(11.7)


vo=

v

a

(R3+R2)R2

(R5C3s+1) (R6C4s+1) (R4C2s+1)

vo=va

(R3+R2)

R2 (R5C3s+1) (R6C4s+1) (R4C2s+1)(11.8)


vo=

v

i

R1C1s+1 (R3+R2)

R2 (R5C3s+1) (R6C4s+1) (R4C2s+1)

vo=vi

(R3+R2)

(R1C1s+1)R2 (R4C2s+1) (R5C3s+1) (R6C4s+1)

vo

vi

=(R3+R2)

(R1C1s+1)R2 (R4C2s+1) (R5C3s+1) (R6C4s+1)

G (s)=(R3+R2)

(R1C1s+1)R2 (R4C2s+1) (R5C3s+1) (R6C4s+1)(11.9)

Figure 11.7 shows the e↵ect of adding two low-pass filters. note that ripple is inexistent

Figure 11.7: output with f=200Hz and 4 low-pass filters

149


11.2.2 Electronic design

Is well known that:

• the operational amplifier is used only to amplify the input voltage

• there are four low pass filters

• the input frequency can vary from 200Hz to 20kHz

• the input voltage is 5V and is multiplied by a voltage gain (Av

) of 2 and the output is a positive DC

voltage signal with a maximum of 9.9V

• the response time of the circuit must be less than or equal to 0.1s

11.2.3 Voltage gain

we know that voltage gain of the operational amplifier non-inverting configuration will be:

Av

= 1 +R2

R1

R2 = R1 = 1.2k⌦

now

Av

= 1 +1.2k⌦

1.2k⌦! A

v

= 2

11.2.4 Design for the first filter

we know that,

vo

=vip

(1 + (2⇡RC)2)

We assume a value of R of 10k⌦, it is recommended that the input current in an op amp be very small.

finding C:

C =

rv

2i

v

2o

� 1

2⇡R(11.10)

We know that the input voltage is 5V and assume that the output voltage is near 5V but not equal, then

vo

= 4.99 V., then:

C =

q52

4.992� 1

2⇡(10k⌦)! C = 1µF

11.2.5 cuto↵ frequency

We know that the cuto↵ frequency (fc

) corresponding to these values of R and C is:

fc

=1

2⇡RC! f

c

=1

2⇡(10k⌦)(1µF )! f

c

= 15.92Hz

With this fc

which is much less than 200Hz we ensure that the low-pass filter works correctly and attenuates

frequencies higher than 15.92Hz and can make the PWM / DC conversion.

150


11.2.6 Design for the filters 2, 3 and 4

We know the value of Xc should be 10 times smaller than the load resistance to the cuto↵ frequency, this means

that for the design of the second filter, we will take as load resistance the resistance of the third filter and for the

third filter, we will take as load resistance, the resistance of the fourth filter. Thus, we assume for the second

filter a 1k⌦ resistance, for the third 10k⌦ and for the fourth 100k⌦. In this way we have the values of capacitors:

C =

q52

4.992� 1

2⇡(1k⌦)! C = 10µF

C =

q52

4.992� 1

2⇡(10k⌦)! C = 1µF

C =

q52

4.992� 1

2⇡(100k⌦)! C = 100nF

11.2.7 Transfer function for the PWM/DC converter

Figure 11.8 shows the circuit diagram for the PWM / DC converter, with the values of the resistors and capacitors

found.

Figure 11.8: PWM / DC full

With the equation 11.9 and the values of resistors and capacitors calculated, the transfer function found in

this system is:

G(s) =200000000

s4 + 400s3 + 60000s2 + 4000000s+ 100000000(11.11)

11.2.8 Matlab Simulations

Figure 11.9 shows the bloque modules in Matlab Simulink.

using the simulink bloques in the figure 11.9 we can made test to determine its response.

151


(a) (b)

(c)

Figure 11.9: Simulink. (a) PWM/DC Simulation Matlab - Simulink (b) Subsystem 1 (c) Sub-

system 2

The speed with which responds the PWM / DC converter (the time it takes to change the output when

the input changes) is very important to control the position of the SMA wire, because the maximum time that

SMA takes to contract and relax is 1 second, then if the response of this circuit is very slow, we can not control

the position precisely, figure 11.10 shows the response of this circuit to an input signal.

Figure 11.10: time response for the PWM / DC

The characteristics of the input signal correspond to a PWM signal with the following characteristics:

frequency of 10KHz, amplitude 5V and duty cycle 70%

From the Figure 11.10 we can infer that has a maximum voltage of 7V and a response time of 0.1s (seconds),

this means that,

• The output signal (time response) meets with the relationship between the duty cycle and output

voltage, which specifies that the circuit for an input signal having a duty cycle of 75% will have

152


an of 7.5V and according to equation 11.12 the output voltage to a duty cycle of 70% should be

7V

• The time response of the circuit is fast enough to control quickly the SMA

vo

=DutyCycle

100⇤ 10 ! v

o

=70%

100⇤ 10 ! v

o

= 7V (11.12)

The Figure 11.11 shows the output voltages (1V, 2V, 3V, 4V, 5V, 6V, 7V, 9.9V and 8V) for di↵erent values

of duty cycle (10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 99%).

Figure 11.11: Time response to other values of duty cycle

11.2.9 SMA Power driver

The main goal of the power unit is to give enough power to the SMA. SMA movement is based on the shape

memory e↵ect. This process depends only on the temperature. This is accomplished by applying a voltage

across the SMA causing a flow of current through the cable. The result is a joule heating e↵ect generating the

contraction of the SMA wire. To control the amount of current through the wire, the circuit of figure 11.12(a)

was designed. This is a current source (current through the SMA) controlled by voltage (output voltage from

the PWM/DC converter). For each SMA wire a power circuit is required.

By using the figure 11.12(a) is possible to calculate the electrical resistance of the SMA (RSMA

) by using

the measures of the current and voltage in the SMA (ISMA

, VSMA

)

The power circuit allows voltage measurements at two points of the circuit. These points are connected

to the inputs of the analog / digital converter (A / D) 12-bit PIC micro-controller. The data are processed

to obtain information about the status of the SMA. Normally, the resistance of the SMA and power can be

measured using the laws to solve electrical circuits, as described below. Using the circuit of Figure 11.12(b), we

find the corresponding equations.

V1 and V3: voltage measured by the A / D 12-bit PIC. Assuming that the current through the SMA ⇠=current through the resistance R5, as the emitter current is very small, I

E

⇠= 0, then: Current through the SMA

ISMA

:

ISMA

⇠= IR5 =

V1

R5(11.13)

153

11.3 SMA phenomenological model Matlab-code

(a) (b)

Figure 11.12: (a) Voltage-controlled current source (b) Test circuit

Current through the resistance R6, IR6 = current through the R7, I

R7 resistance. Therefore, the current

through the resistance is R6:

IR6 = I

R7 =V3

R7(11.14)

then:

V2 � V3 = IR6R6 (11.15)

Therefore, the voltage at the drain of the MOSFET is:

V2 = IR6R6 + V3 (11.16)

Then, the resistance of the SMA (RSMA

) will be:

RSMA

=(V2 � V1)

ISMA

(11.17)

And the power consumed by the SMA (PSMA

) is:

PSMA

= I2SMA

RSMA

(11.18)


1 function [T,strain,stress,theta] = SMA phenomenologicalModel(I,step,Time)

154


2 To = 20; %ambient temperature [C]3 m = 0.00014; %SMA mass [Kg]4 R = 8.5; %SMA initial resistance [Ohms]5 Lo = 0.085; %link length6 ro = 0.0025; %Link joint radius7 % Fixed Parameters8 Cp = 0.2; %Specific heat of wire9 Ac = 0.0004712; %SMA wire?s circumferential area per unit length (150um)

10 hc = 150; %Heat convection coefficient11 t = 0:step:Time; %Time vector12 %Initial conditions13 T(1) = To(1); %Initial Temperature [C]14 stress(1) = 75; %Initial stress [MPa]15 strain(1) = 0.04; %[MPa]16 Text = To(1);17 %**************************************************************************18 %Evolution during Heating19 %Temperature [C]20 p = length(t); cont2 = 1; cont = 1;21 tempo =1;22 for i=1:p�123 T(i+1) = step*((I*I*R)�hc*Ac*(T(i)�Text))+T(i); %Heating Temperature24 cont = cont+1;25 if tempo (length(To)�1)26 if cont > p/(length(To)�1);27 cont2 =cont2+1;28 Text = To(cont2);29 cont = 1;30 end31 tempo = tempo+1;32 end33 end34 %Stress computing as a function of temperature35 As = T(1); Af = T(i+1); aA = pi/(Af�As);36 bA = �aA/10.3; %10.3 is the effect stress constant on Austenite temperatures [MPa.1/C]37 p = length(T);38 for j=1:p�139 stress(j+1) = step*(((0.55+1120*(1/(Af�As)))*((T(j+1)�T(j))/step))/(1+1120*(1/(Af�As))))+stress(j);

%Computing stress [MPa]40 end41 %Martensite fraction computing and its derivative:42 p = length(stress);43 for k=1:p�144 M(k) = 0.5*(cos(aA*(T(k)�As)+bA*stress(k))+1); %Martensite fraction during heating45 dM(k) = �0.5*(sin(aA*(T(k)�As)+bA*stress(k)))*(aA*((T(k+1)�T(k))/step)+bA*((stress(k+1)�stress(k))/step));46 T h(k) = T(k);47 end48 Austenite = M;49 %Strain computing as a function of stress, temperature, and Marsenite fraction50 p = length(M);51 for u=1:p�152 strain(u+1) = (step/75000)*(((stress(u+1)�stress(u))/step)�0.55*((T(u+1)�T(u))/step)+1120*((M(u+1)�M(u))/step))+strain(u);

%Computing strain [MPa]53 end54 %Kinematics model (SMA attached to a link)55 p = length(strain); Theta(1) = 0; �Y = Lo;56 for w=1:p�157 Theta(w+1) = (�step*((Lo*((strain(w+1)�strain(w))/step))/(2*ro)))+Theta(w);58 end59 Theta = Theta*(180/pi);60 %**************************************************************************

155


61 %Evolution during Cooling62 %Temperature63 i = i+1; i flag = i+1;64 cont = 1; %flag counter used for knowing how many steps are required in cooling phase65 Ms = T(i);66 while (T(i) > (To+0.5))67 T(i+1) = step*(�hc*Ac*(T(i)�To))+T(i); %Cooling temperature68 t(i+1) = t(i)+step; %Filling time vector with the cooling phase69 i = i+1; cont = cont+1;70 end71 %Stress computing as a function of temperature72 Mf = T(i); aM = pi/(Ms�Mf);73 bM = �aM/10.3; %10.3 is the effect stress constant on Austenite temperatures [MPa.1/C]74 j = j+1;75 j flag = j+1;76 for j2=i flag:cont�177 stress(j+1) = step*(((0.55+1120*(1/(Ms�Mf)))*((T(j2+1)�T(j2))/step))/(1+1120*(1/(Ms�Mf))))+stress(j);

%Computing stress [MPa]78 j = j+1;79 end80 %Martensite fraction computing and its derivative:81 k = k+1; j2 = i flag; k flag = k; temp = 1;82 for k2=j flag:cont�183 M(k) = 0.5*(cos(aM*(T(j2)�Mf)+bM*stress(k2)))+0.5; %Martensite fraction during heating84 Martensite(temp) = M(k);85 T c(temp) = T(j2);86 j2 = j2+1; k = k+1; temp = temp+1;87 end88 %Strain computing as a function of stress, temperature, and Marsenite fraction89 u = u+1; j2 = i flag; k2 = j flag;90 for u2=k flag:(cont�3)91 strain(u+1) = (step/28000)*(((stress(k2+1)�stress(k2))/step)�0.55*((T(j2+1)�T(j2))/step)+1120*((M(u2+1)�M(u2))/step))+strain(u);

%Computing strain [MPa]92 j2 = j2+1; k2 = k2+1; u = u+1;93 end94 end

156

BR3: a biologically inspired fish-like robot actuated by SMA-based ...

Documents

Transcript of BR3: a biologically inspired fish-like robot actuated by SMA-based ...