Electroactive Polymer Gel Robots: Modelling and Control of Artificial Muscles (Springer Tracts in Advanced Robotics, 59) 3540239553, 9783540239550

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Springer Tracts in Advanced Robotics Volume 59 Editors: Bruno Siciliano · Oussama Khatib · Frans Groen

Mihoko Otake

Electroactive Polymer Gel Robots Modelling and Control of Artificial Muscles

ABC

Professor Bruno Siciliano, Dipartimento di Informatica e Sistemistica, Università di Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy, E-mail: [email protected] Professor Oussama Khatib, Artificial Intelligence Laboratory, Department of Computer Science, Stanford University, Stanford, CA 94305-9010, USA, E-mail: [email protected] Professor Frans Groen, Department of Computer Science, Universiteit van Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands, E-mail: [email protected]

Author Dr. Mihoko Otake University of Tokyo Research into Artifacts, Center for Engineering University of Tokyo 5-1-5 Kashiwa-no-ha, Kashiwa-shi, 277-8568 Japan E-mail: [email protected]

ISBN 978-3-540-23955-0

e-ISBN 978-3-540-44705-4

DOI 10.1007/978-3-540-44705-4 Springer Tracts in Advanced Robotics

ISSN 1610-7438

Library of Congress Control Number: 2009941047 c 2010 

Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed in acid-free paper 543210 springer.com

Editorial Advisory Board

EUR ON

Oliver Brock, TU Berlin, Germany Herman Bruyninckx, KU Leuven, Belgium Raja Chatila, LAAS, France Henrik Christensen, Georgia Tech, USA Peter Corke, CSIRO, Australia Paolo Dario, Scuola S. Anna Pisa, Italy Rüdiger Dillmann, Univ. Karlsruhe, Germany Ken Goldberg, UC Berkeley, USA John Hollerbach, Univ. Utah, USA Makoto Kaneko, Osaka Univ., Japan Lydia Kavraki, Rice Univ., USA Vijay Kumar, Univ. Pennsylvania, USA Sukhan Lee, Sungkyunkwan Univ., Korea Frank Park, Seoul National Univ., Korea Tim Salcudean, Univ. British Columbia, Canada Roland Siegwart, ETH Zurich, Switzerland Guarav Sukhatme, Univ. Southern California, USA Sebastian Thrun, Stanford Univ., USA Yangsheng Xu, Chinese Univ. Hong Kong, PRC Shin’ichi Yuta, Tsukuba Univ., Japan

European

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Research Network

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STAR (Springer Tracts in Advanced Robotics) has been promoted un- ROBOTICS der the auspices of EURON (European Robotics Research Network)

Foreword

By the dawn of the new millennium, robotics has undergone a major transformation in scope and dimensions. This expansion has been brought about by the maturity of the field and the advances in its related technologies. From a largely dominant industrial focus, robotics has been rapidly expanding into the challenges of the human world. The new generation of robots is expected to safely and dependably co-habitat with humans in homes, workplaces, and communities, providing support in services, entertainment, education, healthcare, manufacturing, and assistance. Beyond its impact on physical robots, the body of knowledge robotics has produced is revealing a much wider range of applications reaching across diverse research areas and scientific disciplines, such as: biomechanics, haptics, neurosciences, virtual simulation, animation, surgery, and sensor networks among others. In return, the challenges of the new emerging areas are proving an abundant source of stimulation and insights for the field of robotics. It is indeed at the intersection of disciplines that the most striking advances happen. The goal of the series of Springer Tracts in Advanced Robotics (STAR) is to bring, in a timely fashion, the latest advances and developments in robotics on the basis of their significance and quality. It is our hope that the wider dissemination of research developments will stimulate more exchanges and collaborations among the research community and contribute to further advancement of this rapidly growing field. The monograph written by Mihoko Otake combines ideas from chemistry and physics, material science and engineering for the revolutionary development of the so-called “gel robots”. Electroactive polymers are introduced to build new types of muscular-like actuation for deformable robots. The coverage spans from modelling and design to the development, control and experimental testing. A number of methods are proposed for describing the shapes and motions of such systems. The results are demonstrated for beamshaped gels curling around an object and starfish-shaped gel robots turning over.

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This book is the outcome of the author’s doctoral work and constitutes the first comprehensive monograph ever published in this fascinating new area. As such we warmly welcome it into the STAR series!

Naples, Italy August 2009

Bruno Siciliano STAR Editor

Preface

Electroactive polymers are candidate material to build artificial muscles. Performance of electroactive polymers is improving rapidly from the early 1990s, which attracts wide attention especially in the fields of robotics and polymer science. Artificial muscles, which generate force and change shapes, can potentially form a basis for the development of deformable robots. This will change the assumption of robotic design. For example, traditional robots with rigid bodies basically should not hit upon outer world, while deformable robots can interact with the environment gently and safely. Power assist suits, which are consisting of artificial muscles, will have affinity to human bodies. Different kinds of intelligence should be implemented for robots with active materials, because they are open system capable of exchanging matter and energy with external environments. The main focus of this book is to propose methods for deriving a variety of shapes and motions of such machines, using a particular electroactive polymer gel. Mechanisms consisting of the gel, hereafter called ‘gel robots’, were designed, developed, and controlled experimentally. It includes: (1) a mathematical model of the gel to be applied for design and control of distributed mechanisms, (2) gel robots manufacturing and their driving systems, (3) control of gel robots for dynamic deformations. The results are demonstrated for beam-shaped gels curling around an object and starfish-shaped gel robots turning over. This book is the first comprehensive monograph in the world on deformable robots utilizing electroactive polymers with full of original simulation and experimental results. Most studies on electroactive polymers (EAP) have been focusing on improving the material properties in the field of polymer science or on replacement of the conventional actuators in the field of robotics. In contrast, this book discusses the technical problems for developing novel class of deformable robots utilizing EAP instead of just replacing actuator parts. Varieties of shape and motion patterns are derived without changing material properties. Original model is proposed by the author and examined through numerous

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simulations and experiments. The proposed motion control method based on the model actively using the nonlinearity of the material, which is unique. This book is based on the Ph.D. dissertation of the author ”Modeling, Design and Control of Electroactive Polymer Gel Robots” supervised by Professor Hirochika Inoue with the Department of Mechano-Informatics, the University of Tokyo. This book is open to wide range of readers who are interested in advanced science and technology. The concept of gel robots was proposed by the author, whose results have been presented at related research area: cybernetics, robotics and automation, autonomous systems, polymer science, electroactive polymers, smart materials and structures, and nonlinear chemical dynamics. The research was conducted by the author, from setting up the chemical facility for fabrication, to developing original hardware and software for building and controlling gel robots. Therefore, this book helps researchers in chemistry and physics, material science and engineering, mechanical, electrical, computer science and engineering. This book is intended to be used for references of classes and courses which include but not limited to: physically based modelling, modelling and simulation, advanced robotics in computer science classes; advanced motion control, system identification in mechanical engineering classes; smart materials and structures in aerospace engineering classes; advanced materials, electroactive polymers in material science classes; electrochemistry in physical chemistry classes; nonlinear phenomena, nonlinear chemical dynamics in chemical physics classes. Of course, researchers and practitioners who start or have been to work on electroactive polymers will be benefited from this book. This book will also be utilized for computer aided design software which can design mechanical devices utilizing electroactive polymers. The author is looking forward to seeing and collaborating with the readers of this book in the future.

Kashiwa, Japan August 2009

Mihoko Otake The University of Tokyo, Japan

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Natural and Artificial Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Natural Muscles Made of Motor Proteins . . . . . . . . . . . 1.1.2 Artificial Muscles Other Than Electroactive Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Artificial Muscles Made of Electroactive Polymers . . . 1.1.4 Research Issues on Electroactive Polymers . . . . . . . . . . 1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Deformability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Activeness Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Combination of Activeness and Deformability . . . . . . . 1.3 Gel Robot Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Scope of Machines and Selection of Materials . . . . . . . 1.4 From Modelling, Design to Control . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Necessity of the Modelling Process . . . . . . . . . . . . . . . . . 1.4.2 From Modelling to Design . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 From Modelling to Control . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 From Design to Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Part I: Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Part II: Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Part III: Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Synthesis and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 4 5 8 8 9 9 10 10 10 11 11 12 12 13 13 13 14 14 14

Part I: Modelling 2

Adsorption-Induced Deformation Model of Electroactive Polymer Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19 19

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2.2 Modelling of Nonequilibrium System . . . . . . . . . . . . . . . . . . . . . 2.2.1 Modelling of Gels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Reaction-Diffusion Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Adsorption-Induced Deformation Model of the Gel . . . . . . . . . 2.3.1 Migration of Surfactant Molecules Driven by the Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Adsorption of Surfactant Molecules to the Gel . . . . . . 2.3.3 Gel Deformation Caused by Adsorption of Surfactant Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Extention to Physical and Mathmatical Model . . . . . . . . . . . . 2.4.1 Deformable Lattice Model . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Dynamical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Extension to Mechanical Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Multi-link Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Mass-Spring Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Combining the Active and Passive Deformation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Roadmap of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Parameter Identification by One Point Observation . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Parameter Identification and Calibration . . . . . . . . . . . . . . . . . 3.2.1 Parameter Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Role of Parameters to the Deformation Response . . . . 3.2.3 Spatio-temporal Calibration Method . . . . . . . . . . . . . . . 3.3 Evaluation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Response to Uniform Electric Field . . . . . . . . . . . . . . . . 3.3.2 Response to Alternating Electric Field . . . . . . . . . . . . . 3.4 Hypothesis and Limitation of the Model . . . . . . . . . . . . . . . . . . 3.4.1 Long Time Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Strong Electric Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Examination of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Analysis of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Required Resolution of Time and Space . . . . . . . . . . . .

21 21 22 23 23 24 25 26 26 26 28 28 28 31 31 32 35 35 35 35 36 37 40 40 41 46 46 46 46 46 48

Part II: Design 4

Interaction-Based Design of Deformable Machines . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Measurement of Activeness and Deformability . . . . . . . . . . . . . 4.2.1 Elasticity and Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Generated Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Role of Design and Control Variables . . . . . . . . . . . . . . . . . . . .

63 63 65 65 67 70

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4.3.1 Thickness Dependence on Step and Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Voltage Dependence on Step and Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Time Constant of the System . . . . . . . . . . . . . . . . . . . . . 4.4 Roadmap from Materials to Machines . . . . . . . . . . . . . . . . . . . . 4.4.1 Prototype of Muscle-Hydraulic System . . . . . . . . . . . . . 4.4.2 Material-Field Respective Design . . . . . . . . . . . . . . . . . .

70 70 72 72 76

Spatially-Varying Electric Field Design by Planer Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Previous Driving System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Separate or Composite Electrodes . . . . . . . . . . . . . . . . . 5.2.2 Conversion from Bending to Contracting . . . . . . . . . . . 5.3 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Multiple Electrodes in a Plane . . . . . . . . . . . . . . . . . . . . 5.3.2 Removal of Fixing Point . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Development of Electric Field Generation System . . . . . . . . . . 5.4.1 Hardware System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Software System: Simulation . . . . . . . . . . . . . . . . . . . . . . 5.5 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . 5.5.1 Electrodes in a Plane with Fix-Ended Gel . . . . . . . . . . 5.5.2 Electrodes in a Plane with Free-Ended Gel . . . . . . . . .

77 77 78 78 80 80 80 83 84 85 89 95 95 97

Shape Design through Geometric Variation . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Development of Gel Manufacturing System . . . . . . . . . . . . . . . 6.3 Gels with Cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Deformation Response of Square-Shaped Gel . . . . . . . . 6.3.3 Deformation Response of Cross-Shaped Gel . . . . . . . . . 6.4 Gels with Wave-Shaped Surfaces . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Shapes and Sizes of the Gels for Experiment . . . . . . . . 6.4.4 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.5 Fixed-Ended Gel with Wave-Shaped Surfaces . . . . . . . 6.4.6 Free-Ended Gel with Wave-Shaped Surfaces . . . . . . . . 6.4.7 Effect of Varying Moment of Inertia . . . . . . . . . . . . . . . 6.5 Gels with Various Widths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Deformation Response of Width Varying Gels . . . . . . .

117 117 118 122 123 123 123 124 124 124 126 127 128 129 133 133 133 134

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Part III: Control 7

8

Polarity Reversal Method for Shape Control . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Tip Position Control of Gel Manipulator . . . . . . . . . . . . . . . . . 7.2.1 Inverse Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Dynamic Change of Configuration of the Gel . . . . . . . . 7.2.3 Slight Change of Configuration of the Gel . . . . . . . . . . 7.2.4 Selection of the Path to Reach the Desired Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 Tip Position Control of Gel Manipulators . . . . . . . . . . 7.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Wave-Shape Pattern Formation of Electroactive Polymer Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Application of the Constant Uniform Electric Fields to the Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Simulation of Wave-Shape Pattern Formation . . . . . . . 7.3.3 Mechanism of Wave-Shape Pattern Formation . . . . . . 7.4 Wave-Shape Pattern Control of Electroactive Polymer Gel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Numerical Simulation for Experiment . . . . . . . . . . . . . . 7.4.2 Performance Function for Objective Forms . . . . . . . . . 7.4.3 Pattern Control of Gels with Varying Curvatures . . . . 7.5 Pattern Formation in Variety of Gels . . . . . . . . . . . . . . . . . . . . .

137 137 139 139 142 145

Lumped-Driven Method for Motion Control . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Design of Electric Operator to the Gels . . . . . . . . . . . . . . . . . . . 8.2.1 Electric Fields as Operator . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 2D Operator for Array of Electrodes . . . . . . . . . . . . . . . 8.2.3 3D Operator for Matrix of Electrodes . . . . . . . . . . . . . . 8.3 Lumped Representation of Whole Body Motions . . . . . . . . . . 8.3.1 Conversion of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Turning over Motion Generation of Real Starfishes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Motion Generation of a Starfish-Shaped Gel Robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Objective Motion Generation with Operators . . . . . . . . . . . . . 8.4.1 Operators to Generate Coordinated Motions . . . . . . . . 8.4.2 Selection of a Set of Operators . . . . . . . . . . . . . . . . . . . . 8.4.3 Phase Diagram for Switching of Operators . . . . . . . . . . 8.4.4 Phase Diagram in Other Conditions . . . . . . . . . . . . . . . 8.5 Application of the Lumped-Driven Methods . . . . . . . . . . . . . .

165 165 167 167 168 170 177 177

146 147 147 149 149 151 152 155 155 156 158 161

178 178 183 183 183 184 192 192

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8.5.1 Turning over Motion Control of Starfish-Shaped Gel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 8.5.2 Curling around Motion Control of Gel Manipulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 9

Conclusion and Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Part I: Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Part II: Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Part III: Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 From Control to Design and Modelling . . . . . . . . . . . . . . . . . . . 9.2.1 Requirement for the Model . . . . . . . . . . . . . . . . . . . . . . . 9.2.2 From Design to Modelling . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 From Control to Modelling . . . . . . . . . . . . . . . . . . . . . . . 9.2.4 From Control to Design . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Programming of Gel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Integration of Modelling, Design and Control . . . . . . . 9.3.2 Operators and Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 On the Problems of Activeness and Deformability . . . . . . . . . 9.4.1 Complementary Approach . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 Independent Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Agent Approach to Electroactive Polymer Gel Robots . . . . . . 9.5.1 Agent Model of Electroactive Polymers . . . . . . . . . . . . . 9.5.2 Control System Design Based on the Agent Model . . . 9.5.3 From Gel to Gel Robots . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

201 201 202 202 204 206 206 207 208 208 210 210 210 211 211 211 212 213 213 214 215

A

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

1 Introduction

This book challenges to design and control robots made entirely of electroactive polymer gels. They are deformable like mollusk that can locomote dynamically or manipulate things softly. Such a machine has been a dream in the past but is now experimentally possible. Gel robots are made from electroactive polymer gels. The purpose in creating these machines is to investigate fundamental principles in the design and control of machines comprising of soft and active materials. Ideas are being explored independent of material types, in the hope that the methodology being investigated will be applicable to other materials that may appear in the near future - material developments being an ever-ongoing field of discipline. The advances in electroactive polymers make it a highly attractive candidate for building of artificial muscles machines. The universal difficulty to embody deformable machines made from electroactive polymers has been organized into two parts. One originates in deformability of the machine, and the other derives from activeness of the material being applied. These problems were addressed in this book by utilizing computational models of materials that comprise the machines. Driving systems were explored extensively, designed based on their ability to eliminate difficulty of control. Methods to generate variety of objective shapes and motions are proposed by this research.

1.1 1.1.1

Natural and Artificial Muscles Natural Muscles Made of Motor Proteins

Living animals show elastic motions with flexible bodies[1, 2, 3, 4]. Cheetah runs at full speed after the prey, bending its backbones like an arch. Dolphins flip over while flying out of the water. These vertebrates are elastic while invertebrates like mollusks have more flexible bodies. Starfishes fold their bodies for turning over. Octopus twists objects with their tentacles. Jellyfishes swim in the M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 1–15. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

2

1 Introduction

water by swinging the bell-shaped bodies. The secret for enabling these elegant motions is muscles. Muscles are neither pure force generators nor pure motion generators[5]. They behave like springs with variable original length and tunable elastic parameters. This is an important characteristic in order to achieve versatility and robustness. Typical motility units at molecular level in the living systems: actin and myosin, kinesin and tubulin, dynein and tubulin[6]. They are motor proteins with flexible structures. The first one, a pair of actin and myosin is a building block for muscles[7, 8, 9, 10]. There are other interesting microscopic structures which move inside the living systems. For example, plants, which appear to be still, but there are flows named plasma streaming inside the cells in order to transfer the nutrition and waste products[11]. The flow is driven by special kinds of myosins[12] and actins[13, 14]. Neurons of animals extend their synapses whose components are carried along the axons. Inside the axons, there are microtubules like rails and the kinesins move the vesicles like trams[15, 16]. Single cellular organisms have motility function as well as multi cellular organisms. Interaction between dynein and tubulin causes swimming of paramecium by cilia or that of sperm by flagella[17, 18]. Energies are provided by the ATP, adenosine triphosphate, in all systems. Chemical energies are converted to mechanical energies through interaction of motor proteins and ATP[19]. However, the conversion mechanism of free energy is still a burning issue, a tight coupling or a loose coupling. Tight coupling theories describe that the molecular motors convert a certain amount of chemical energy provided by single ATP molecule into a fixed amount of mechanical energy[20]. Loose coupling theories argue that the energy conversion is not oneto-one, but many-to-many[21, 22] based on the ratchet model[23]. The feasibility of the loose coupling theories is based on the fact that the energy provided by ATP is in the same order as the thermal energy around the molecules[24, 25]. Both of them are far smaller than the energy provided by combustion. They work at room temperatures. We can find muscles in a variety of forms inside our bodies. Some are skeletal muscles and hearts, namely active moving units. Others organize external wall of organs like livers and kidneys, making use of their flexible mechanical properties, which do not move. Among them, skeletal muscles show dynamic motion at macroscopic level because the motility at molecular scale is amplified effectively. The secret is the hierarchical structure of the skeletal muscles: whole skeletal muscles are composed of numerous fascicles of muscle fibers; muscle fibers are composed of myofibrils arranged in parallel; myofibrils are composed of sarcomeres arranged in series; sarcomeres are composed of interdigitating actin and myosin filaments; myosins form myosin filaments while actins form actin filaments with troponin and tropomyosin. Myosin fibers are thick about 15nm in diameter, while actin fibers are thin, whose diameter is about 9nm. The diameter of myofibrils is about 2 to 3 μm, and that of muscle fibers are 50 to 100 μm. The length of sarcomeres at neutral state is about 2 μm. Slipping motion of

1.1 Natural and Artificial Muscles

3

15μm/s between actin and myosin filaments is amplified to contractile motion at 0.75m/s for muscles consisting of 50 thousands sarcomeres. In skeletal muscles, energy conversion process is regulated by the concentration of calcium ions released by neurons. Electric signals trigger this process. Nervous system controls multiple muscles. Dynamic motions like walking, running, jumping, flying and swimming are generated by the combination of coordinated neural inputs and flexibility of the muscles. Another characteristics of muscles are that they are always regenerated and do self-repair. About 60% of muscles cells are replaced in one month. Protein synthesis reaches maximum three hours after training and decreases after that. As a result, they are always new and strong during the lifetime. The typical properties of the natural skeletal muscles are summarized for comparison with artificial muscles. 1. Chemical energies are converted to mechanical energies at room temperature by muscles. 2. Muscles are elastic and distributable, which can be shaped into various forms. 3. Hierarchical and organized structure of muscles amplifies molecular motility effectively, which realizes the high performance and mechanical strength of the muscles. 4. Muscles exhibit contractile motions caused by the interaction of molecules. 5. Output of muscles is controlled by electric signals from the nervous system. 6. Lifetime of muscles is very long because of their metabolism and regeneration. 1.1.2

Artificial Muscles Other Than Electroactive Polymers

Materials or devices which have muscle like elastic structures are called artificial muscles. Their shapes and size of the elastic parts are controllable. Pneumatics and hydraulics actuators, or Nitinol, a shaped memory alloy (SMA), have been popular for alternatives of electric motors because the robotic designs have been constrained by the high weight, limited sizes, complex transmissions and restrictive shapes of electric motors. They have been popular, since they work in dry environment. Active catheters and guide wires for surgery[26, 27, 28, 29], sarcomeres units[30, 31] and cellular actuators[32, 33] have been developed with SMA. Robotics aid system to assist the physically disabled[34], skeletal type of robots[35, 36, 37] were developed using pneumatics and their control method was studied[38]. However, high power sources which supply enough thermal energy into mechanical energy conversion is required for SMA, and large and noisy compressor is necessary for pneumatics and hydraulics actuators. Their driving systems limit the freedom of machine design. A series of linear actuators based on polymer hydro gel was developed[39, 40]. The actuators used arrays of pH sensitive gel fibers together with a fluid irrigation system to locally and rapidly regulate the composition of the solution. Their driving system was similar to that of hydraulic actuators since they were not electroactive, which required regulation of fluids.

4

1 Introduction

Early artificial muscle “mechano-chemical engine” was presented in 1950s[41]. The engine was too large to utilize for robotic actuators at that time. Since then, functional materials have been developed and the whole system has become smaller. Some are packaged into devices such as sensors and actuators. These devices enabled mobile robots in 1980s. Others are composed into structures, which are called intelligent or smart structures and materials. Such structures are experimentally used for aerospace applications in 1990s. 1.1.3

Artificial Muscles Made of Electroactive Polymers

Over a few decades, a considerable number of studies have been made on artificial muscles made of chemomechanical polymer. Among them, electroactive polymers are currently as promising intelligent or smart materials for transducers. Electroactive polymers generate stress and strain based on particular electrochemical or electrical reactions. The performance of electroactive polymers has been improving rapidly in 1990s[42], and newsletters on research and development of electroactive polymers have been published twice a year since 1999[43]. Generating stress, strain and response speed are comparable to animal muscles [44, 45]. Typical materials are conducting polymers (polypyrrole[46, 47], polyaniline[48]), carbon nanotube[49, 50, 51], electrostrictive polymers or dielectric elastomers (both passive[52, 53, 54] and active[55]), piezoelectric polymers[56, 57], ion conducting polymers[58, 59], and gels (both ionic[60, 61, 62] and non-ionic[63, 64, 65]). Polymers that exhibit shape change in response to electrical stimulation can be divided into two distinct groups: ionic and electronic [42]. Ionic EAPs include conjugated polymers, also known as conducting polymers[66, 67, 68, 69], ionic polymer-metal composites (IPMCs) [58, 70, 71, 72, 73, 74], and ionic gels[61, 75]. They rely on ion or solvent transport to effect volume change. Electronic EAPs include piezoelectric polymers, electrostrictive polymers, nonionic gels, and dielectric elastomers. They are driven by electric field or Coulomb forces. Carbon nanotube actuators[49] operate in an electrolyte but rely on capacitive charging rather than ion transport. The difference between ionic and electronic EAP is made clear by comparing the necessary driving system. Ionic EAPs require low voltages for actuation, typically 1 V or less, but they consume relatively large current (102 mA to A order). Their driving system requires current amplifier. Electronic EAPs operate at high voltages up to a few kV, and consume small current whose order is of μA. Their driving system requires voltage amplifier. In the former case, the polymers consume large current when large amount of internal ions or electrons are driven. Electric potential difference causes inhomogeneous distribution of ions or electrons inside the polymers or increase or decrease of charges inside the polymers. Distributions of charges determine the shape of the polymers. Volume changes occur when charged molecules go inside or outside of the polymers. In the latter case, high voltage is applied to the polymers in order to cause structural change of the molecules constituting the materials or to generate electrostatic attractive force between the electrodes clipping the materials. The generating stress and strain become larger when

1.1 Natural and Artificial Muscles

5

the applied voltage is higher. Electroactive polymers, which are driven at both low voltage and consuming small currents, are unavailable in the beginning of the twenty first century. This is because the source of energies which should be converted to mechanical energies is provided by electricity for recent EAPs. In contrast, natural muscles are driven at both low voltages with small current at mV and mA order, because the source of energies is supplied by chemical energy and the electric signal from the nervous system triggers the mechanochemical reaction. Natural muscles are like engines rather than conventional actuators, which convert chemical energy to mechanical energy, triggered by electric stimulus. The engines are driven by combustion at high temperature while the muscles are driven in the process of metabolism at low temperature. We have to keep in mind that recent EAPs require amplifier for operation. Currently available EAPs have neither hierarchical structure, nor metabolic mechanism. Therefore, they lack the mechanical strength and durability, which limits the real applications. If we compare the characteristics of EAPs to natural muscles, elasticity (2), motion based on molecular interaction (4) and controllability by electric signal (5) are common, while direct energy conversion from chemical to mechanical (1), hierarchical structure (3), and long lifetime (6) are different. Direct energy conversion (1) was realized before[41, 76, 77, 78] but not applied recently, because of its low response speed. Hierarchical structure (3) was synthesized with actins and myosins from single molecules to fibrous gel bundles[79]. Ordered structure (3) with topological networks [80] was obtained for increasing mechanical strength. These studies are aimed to overcome the limitation of the current soft materials. Comparison of artificial muscles and natural muscles are discussed in the literatures[81, 45, 68, 82]. 1.1.4

Research Issues on Electroactive Polymers

As mentioned above, problems of electroactive polymers are strength and durability, which limits their current applications. Also, the material which exceeds a real muscle in every characteristic does not yet exist. If the application is fixed and one of three material properties is critical among stress, strain and response speed, specific materials can be selected. For example, if large stress is expected: conducting polymers are appropriate. If large strain is important, dielectric elastomers, gels and ion conducting polymer metal composites are suitable. If high-speed response is required, then electrostrictive polymers or piezoelectric polymers are preferred. There are several ways to overcome the limitations. One is seeking the novel mechanism for actuation. Baughman et al. reported that electromechanical actuators based on sheets of single-walled carbon nanotubes were shown to generate higher stresses than natural muscle and higher strains than high-modulus ferroelectrics[49]. Molecular actuators have been proposed[83, 84, 85]. Zrinyi et al. developed magnetic field sensitive polymer gels[86, 87]. In this approach, we don’t have to stick to polymers. Weissmuller et al. showed that metal also serve as artificial muscles. Electrochemical charge injection into porous nanostructured metals produces dimensional changes large enough to do mechanical work[88].

6

1 Introduction

The second direction is developing novel fabrication techniques. Zhang et al. introduced defects into the crystalline structure of electrostrictive polymer, P(VDF-TrFE) copolymer, which improved generated strain up to 5%[89]. Electrical actuators made from films of dielectric elastomers such as silicones coated on both sides with compliant electrode material[53]. Actuated strains were up to 117% for silicone elastomers, and 215% for acrylic elastomers. Their success was due to the development of compliant electrodes made of carbons, which can expand and shrink without losing conductivity. Plating methods are very important for composite type materials so that chemical plating and grafting techniques are studied for IPMC[71, 90]. Hara et al. showed fiber-like polypyrrole (PPy) actuators deposited electrochemically on metal coils. Even if the PPy exhibits more than 10% strain, the metal coils do not interfere very much the actuation of the PPy because the metal coils themselves are capable of stretching. The idea came from a Japanese paper lantern stretching without much resistance[91]. Different type of conducting fibers have been fabricated with conducting polymers[92, 93, 94, 95]. Nano and micro fabrication methods have been investigated with carbon nanotubes[96, 97, 98, 99], fullerene films[100], conducting polymers[46, 101, 102] and gels[103, 104]. The third is exploring combination of materials. Kim et al. developed an electro-active paper (EAPap) actuator, which is a paper that produces large displacement with small force under an electrical excitation[105]. EAPap is made with a chemically treated paper by constructing thin electrodes on both sides of the paper. Kyokane et al. doped C60 derivatives (fullerenol) into polyurethane elastomers (PUE) so that the actuators could operate under a low voltage[106, 107]. Without doping, the material requires high voltage for operation. The bends of fullerenol doped actuators were larger than non-doped actuators, and the working voltage was small. Varieties of blends have been tested for piezoelectric polymers[108, 109, 110], electrostrictive polymers[111, 112, 113] and conducting polymers[114, 115, 116, 117, 118, 119]. The forth direction, analytical modeling for understanding the behaviors of these materials, has been popular approach. Testing and characterization have been conducted for developing the models. Such attempts have been done especially for ionic polymer metal composites (IPMCs)[58, 70, 72, 120]. NematNasser and his co-workers carried out extensive experimental studies on both Nafion- and Flemion-based IPMCs consisting of a thin perfluorinated ionomer in various cation forms, seeking to understand the fundamental properties of these composites, to explore the mechanism of their actuation, and finally, to optimize their performance for various potential applications[121]. They also performed a systematic experimental evaluation of the mechanical response of both metal-plated and bare Nafion and Flemion in various cation forms and various water saturation levels. They attempted to identify potential micromechanisms responsible for the observed electromechanical behavior of these materials, model them, and compare the model results with experimental data[122]. A computational micromechanics model has been developed to model the initial fast electromechanical response in these ionomeric materials[123]. A number

1.1 Natural and Artificial Muscles

7

of models have been proposed for other class of materials such as conducting polymers[124, 125, 93, 126] and ionic gels[127, 128, 129, 130, 131]. Device development has been active recently because of the high performance of advanced materials. Most studies deal with one or some of these research topics looking for achievable applications. Prototypes are: micro robot[101] that can manipulate cells, tactile display[132], haptic interface[133], catheter[134], surface wiper and rover for space application[42, 135], propulsion mechanisms [136, 137], crowing and jumping mechanisms[138], high dimensional manipulators[139], actuators of high power to weight ratio[140, 141] and linear actuation mechanism with originally bending material[142, 143]. Ingan¨ as et al. have been focusing on the fabrication of nanostructured conducting polymer materials. They fabricated microactuators and microrobots[46, 101]. Several types of microactuators was fabricated by Smela et al., from simple paddles to self-assembling and disassembling cubes[46]. Conducting bilayers made of a layer of polymer and that of gold were used as hinges to connect rigid plates to each other and to a silicon substrate. Jager et al. demonstrated microactuators for biomedical applications based on a PPy/Au bilayer: simple arrays of fingers for positioning or holding small fibers; a microvial that can be closed by a lid operated by the microactuators; and a microrobotic arm for the manipulation of micrometer sized particles[101]. Guo et al. demonstrated underwater micro robot[144]. S-K Lee et al. designed a structure of micro pump of which actuation principle is the volume change of polypyrrole[145]. Tadokoro et al. developed an elliptic friction drive element[146], a distributed actuation device[147], and artificial tactile feel display[132] consisting of ionic polymer metal composite. They also proposed linear approximate dynamic model[148] and microscopic model[149]. Perline et al. materialized dielectric elastomers with compliant electrodes. Prototyped actuators were in a variety of configurations such as stretched films, stacks, rolls, tubes, and unimorphs[52]. A novel electroacoustic transducer that uses the electrostrictive response of a polymer film was developed[150]. Control issues are beginning to be studied[151, 152]. Research issues on electroactive polymers are summarized below. 1. 2. 3. 4. 5.

Discovery of a novel actuation material Proposal of a fabrication technique Doping, blending, and combination of materials Theoretical or analytical modeling, testing and characterization Device and application development utilizing the materials

However, methodology for design and control, which is universal to these materials, has not been well studied. Existing devices have been developed considering the applications and limitations of the currently available materials, and not considering the future materials so much. Purpose of this study is to investigate fundamental principle to design deformable robots like living animals by making use of currently available materials in order to prepare for the future materials. This study is on a novel machine rather than material, which is different from most studies on artificial muscles. Original rules were applied to achieve this goal: never to change material composition or polymerization method, on the

8

1 Introduction

other hand, to focus on the surrounding field and shape of the material, because both of them are universal to other materials.

1.2

Problem Statement

What is the fundamental problem essential to deformable machines consisting of actively deformable material? It is organized into two major problems, which originate from activeness and deformability of the machines. deformability problem Deformable machines have conceptually infinite numbers of control points while numbers of input are finite. activeness problem Machines made of active material have inevitable unreliability because properties of active material scatter because of its activeness. 1.2.1

Deformability Problem

The former one, deformability problem is a problem of redundancy. It has been studied to solve the problem on controlling redundant manipulators [153] in the field of robotics. The degree of freedom of proposing deformable machines is totally different from the conventional machines with rigid structures. Previous machines have finite numbers of joints while deformable machines in this study have virtually infinite numbers of actively deformable elements. Its order is similar to biological systems. If we look at the study of physiology and psychology, it is called “Bernstein’s problem”. It is considered that co-ordination plays an important role to generate organized motion with deformable living organ. Russian physiologist, Nicolas Bernstein, who thoroughly studied the co-ordination defined the problem as follows [154]. The co-ordination of a movement is the process of mastering redundant degrees of freedom of the moving organ, in other words its conversion to a controllable system. Later, American Psychologists, Michael M. Turvey and his colleagues have been developing the studies on this problem further. They named the problem as “degrees of freedom” problem stated as [155]: If the regulation is to be achieved through sequential computation, then the problem is especially acute. Many degrees of freedom is a curse. If the regulation is to be achieved thorough parallel and distributed computation, then the problem is potentially more manageable. They have been trying to understand the mechanism of regulating an artifact of very many independent variables without ascribing to any one subsystem excessive responsibility. They noted the similarity between the self-organizing capabilities of systems in nonequilibrium[156] and coordination. Their main focus has been on temporal co-ordination, namely rhythm generation, but their outlook is stimulating.

1.2 Problem Statement

9

Table 1.1. Major problems of deformable machines consisting of actively deformable materials problem components goal alternatives proposers

activeness problem deformability problem actively deformable material deformable machine to obtain reliable output from un- to control conceptually infinite dereliable units grees of freedom reliability problem degrees of freedom problem Von Neuman Nicolas Bernstein

The simplicity can derive from low-dimensional dynamical laws and lowdimensional informative structures. Actively deformable materials work under their dynamical laws, which should help to solve the degrees of freedom problem of deformable machines. Here, the relationship between deformability problem and activeness problem is suggested. 1.2.2

Activeness Problem

The latter problem originates from the activeness of materials. Active materials have some input-output characteristics, which are varied by surrounding environments. History of input also changes the characteristics. Because of these, the response of materials scatters and unreliable. If we broaden our eyes from active material to active units, the same kinds of problems are studied. It is generally called reliability problem in computer science. The initial computers were consisting of unstable units, which made Von Neumann to study reliability problem on computation. He proposed probabilistic logics and a method to synthesize reliable machine from unreliable components [157]. The importance of thermodynamical methods was emphasized. Multiplexing makes the whole system reliable. The subject was on digital or analog automata, it is also applicable to actively deformable materials. Even if the materials are unreliable, they will work in high reliability by connecting them in large numbers and bring out overall deformation for output. In the course of this process, they will form machines with virtually infinite degrees of freedom. 1.2.3

Combination of Activeness and Deformability

Now we can see that activeness problems and deformability problems are related. This relationship has not been aware of before. These problems are summarized in table 1.1. Activeness of units will help to solve the deformability problem of whole system, and deformability (redundancy) of the system will help to solve the activeness problem of units, if we organize machine in an appropriate manner. A method to solve activeness problem and deformability problem simultaneously has not been established nor investigated. These problems were investigated in this study, because they are significant and generic to realize deformable machines independent of kinds of materials or machines.

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1 Introduction

1.3

Gel Robot Approach

The best way to discover concrete methods is to prototype machines from existing material. Real things tell us many things. Constitutive approach was taken for solving the above problems. Deformable robots were experimentally developed made entirely of electroactive polymer gel, hereafter called, “gel robots”. The purpose of gel robots is to discover fundamental principles for designing and controlling deformable machine made of actively deformable materials. Ideas are explored which solve the major problems of activeness and deformability. 1.3.1

Terminology

First, we briefly describe gel and robot, since both of them are technical terms, which are not common to the people outside of their original research fields. gel. Polymer gel is a material, consisting of polymer network filled with medium [158, 81]. Gel tries to balance the internal pressure and external pressure by exchanging matter and energy. Because of this characteristic, we can express that gel has a feedback mechanism. Specifically, electroactive polymer (EAP) gel is a kind of material, which senses electric field and generates force and deforms. robot. The term robot ranges from automatic machine to autonomous machine. Its definition reflects their research philosophy. It is defined in a broad sense in the context of artificial intelligence: robot is an active, artificial agent whose environment is the physical world; an agent is anything that can be viewed as perceiving its environment through sensors and acting upon the environment through effectors [159]. 1.3.2

Scope of Machines and Selection of Materials

As stated in the section 1.1, the materials with one of stress, strain and speed in high performance are realized. Mechanical strength, safety and operating time prevent practical applications. Common approach considering these constraints is to prototype devices. For example, catheter for medical applications does not depend on operating time because they are only used in one operation and disposed afterwards. They came out to market while this study was conducted by spin out company of national research institute. When we started this study in 1998, we selected far-sighted approach. We assumed the time when the ideal material is developed and the above limitations will disappear. Even if these limitations are removed, there remain fundamental difficulties, which were stated in section 1.2. We decided to focus on theoretical aspect and not stick to current applications. There are two directions of machines with actively deformable materials. One is to replace conventional actuators to muscle-like bundles that shrink or stretch. Another direction is to build machine with distributed muscle like structures. The latter is not an extension of conventional machines, and requires methodology to solve major problems. The latter direction is focused on in this study.

1.4 From Modelling, Design to Control

11

With these policies described above, we selected materials to build deformable machines. From among variety of actively deformable materials, we selected electroactive polymer gel, Poly (2-acrylamido-2-methylpropane sulfonic acid) gel (PAMPS gel) [61, 128] which is capable of large deformations although the response speed is not so fast. The main advantage of this material is that transformation can be halted if required. In order to remove the effects of wires and to focus on the major problems, gel robots were controlled by electric fields.

1.4

From Modelling, Design to Control

The goal of this study is to propose methods to solve the activeness and deformability problems. It is to find methods of controlling system with activeness and deformability. Modelling, design and control are comprehensively studied in order to achieve goal (Figure 1.1). In this section, the following questions are answered: Why the modelling is required? Why the problem is decomposed into three processes, and what is the point for modelling, design and control? 1.4.1

Necessity of the Modelling Process

In general, modelling is not necessary and not always preferred. Model based control is so common that sometimes sounds obsolete. Control based on dynamics is preferred in modern control theory[160]. Also, modelling everything is impossible. It is sometimes nonsense to prepare for every situation beforehand, which is called the frame problem[161]. In spite of these, modelling was focused on because of the following reasons. Firstly, modelling is a good way to understand the controlled object. Thorough modelling, it was gradually made clear what is activeness and deformability. I believe that it forms the fundamentals to construct theory. Also, the necessary accuracy and precision of the model was unclear. Detailed model is not always the best model for design and control. The levels of accuracy and precision for the model ought to be cleared. Secondly, modelling and dynamics based control is not a conflict. Good control law would be derived from good model. It was made clear that model describes dynamics, and model is applicable for dynamics based control. This will be made clear in this book.

Modeling

Design

Control

Fig. 1.1. The process to investigate modelling, design, and control methodology

12

1 Introduction

Thirdly, generic modelling framework that describes some classes of materials is universal, independent of the kinds of material. It was developed considering the material that will appear in the future, because such modelling framework or meta-model was unclear. Also, the object is focused on some class of material and not on everything. This is different from environmental modelling for robot that must deal with many situations. The above statements are summarized in the form of three questions, which will be answered in section 9.2. 1. What is the required accuracy and precision of the model? 2. Is model necessary for control? How about dynamics based control? 3. Isn’t it endless to model everything? 1.4.2

From Modelling to Design

There didn’t exist deformable machine made entirely of actively deformable material. This is the reason why the design stage was required. Model enabled me to design innovative mechanisms. There are two advantages for modelling based design. The first advantage is a prospect for design free from existing structures. Designers are sometimes constrained by precedents. The known method is to organize design process in three steps: functional design, mechanical design and structural design. It is encouraged to write down specification before making drawings. Different approach is shown for developing innovative machine based on the model. This was required to conduct this study, since experimental setup different from conventional one was required. There exist prototype devices utilizing electroactive polymers. They are example structures derived from function, but once they are developed, designers tend to design based on existing structure. This pitfall could be avoided by modelling based design. The second advantage is capability to explore design principle for deformable machine. If we look at the design of microscale and nanoscale machines, there exist unique principles different from normal scale machines. It is called the scale effect. This effect is clearly described with the scale parameters and variables utilizing simple equations. Likewise, there should exist principle to design deformable machines different from conventional machines with rigid structures. Since such principles did not exist, the effect of activeness and deformability is discussed in this book. Design method is proposed making use of the advantage of activeness and deformability. 1.4.3

From Modelling to Control

Before this study, the materials of bending type were driven oscillatory. Deformation response to the variety of input was not investigated thoroughly. This is partly because the previous experiments were done for material design and optimization or to present performance of materials or devices. There are two advantages starting from modelling to control.

1.5 Outline

13

The first advantage is that model enables us to reduce the numbers of experiments. It is impossible to test all cases. With simulation based on the model, enumerate approach is enabled. Based on the simulation results, we can narrow down the test cases. In other words, we can design experiments by simulation. The next advantage is that we can do planning and test control system with simulator. This popular method is possible if the simulation results are reliable. For example, motion planning is the traditional approach to generate objective trajectory for robots and other kinds of mechanisms. Simulation results are usually applicable to real world for ordinary robots consisting of reliable elements. It not necessary applicable to deformable machines consisting of unreliable elements, like gel robots. Therefore, the model should have enough accuracy for planning and control. The level of accuracy should be considered since building perfect model is impossible. 1.4.4

From Design to Control

There are two problems to be solved on design and control. First problem is that requirement for design from the viewpoint of control is unclear. If we knew the control method before designing total system, the design might have been different. The designed system was totally new and there exist no control law for them. Therefore, we decided to look for control method utilizing designed system. Fortunately, we discovered control law for designed system, and the requirement for design from the viewpoint of control was made clear. They are summarized in section 9.2. Second problem is to clarify the responsibility of design and control. There are usually several options to achieve the same function. It is inevitable to distinguish what should be prepared beforehand and what should be changed afterward.

1.5

Outline

This book is organized in three parts corresponds to three steps which are stated in the previous section. Part I deals with modelling (in chapter 2 and 3). Part II studies design (in chapter 4 to 6). Part III addresses the issue of control (in chapter 7 and 8). This chapter (chapter 1) is introductory while last chapter (chapter 9) is concluding remark. The typical challenges are as follows on modelling, design and control from the viewpoint of activeness and deformability. 1.5.1

Part I: Modelling

Purpose of chapter 2 is to understand the characteristics of actively deformable materials through modelling. Since electroactive polymer gel is dealt with, its activeness comes from the electrochemical property. It has been believed that electrochemical reaction must be modeled thoroughly to predict the process precisely. This common sense stated in this chapter is challenged. In chapter 3, methods of parameter identification and calibration are proposed. The model proposed in chapter 2 is verified by comparing simulation and

14

1 Introduction

experimental results. The necessary resolution for the model is discussed in order to implement the model. The convention was challenged that deformable objects must be modeled with minute elements. 1.5.2

Part II: Design

Chapter 4 provides principle to design deformable machines consisting of actively deformable materials. Role of design and control parameters are analyzed, referring to the scale analysis of microscale and nanoscale machine. Active materials require driving part to generate field. It has been unconsciously believed that materials to be driven by fields and the fields to drive the materials should be designed together. Opposite policy is announced in this chapter. In chapter 5, driving electrodes to generate electric field are designed. Previous set up had limitation that electrodes must be placed in parallel and the material must be fixed to more than one point. Novel installation of electrodes and material is proposed which remove this limitation, based on the functional model proposed in chapter 2. This is the first step from material to machine. After concept design, hardware and software system was implemented. Chapter 6 deals with shape design of deformable machines. Manufacturing process is developed to form actively deformable materials as desired. Previous approach to design deformable machine was to divide the stiff element into small pieces and add controllable or passive joints[162, 163]. The reversed approach was taken. 1.5.3

Part III: Control

In chapter 7, a method to control shape of deformable machine is studied. Control law is derived from the model. Experimental setup is of Part I in order to keep the condition simple. By carefully changing the normal condition of input, undiscovered shapes are derived. The assumption was challenged that controlling active material requires sequential feedback to counteract the unreliability. Chapter 8 provides method to control motion of deformable machine. Control rule is derived from the result of chapter 5, 6 and 7. In this chapter, experimental setup is of Part II. With this setup, indirect control is only possible. The common idea was challenged that direct controller is necessary to control deformable machine with many degrees of freedom. 1.5.4

Synthesis and Analysis

Aside from the above categories, characteristics of each chapter are classified into two. Synthetic chapters are chapter 2, 5, 6, and 8. Analytic chapters are chapter 3, 4, and 7. Synthetic chapters are organized as follows. In chapter 2, the model was designed from observation. Novel driving and manufacturing systems are developed in chapter 5 and 6. In chapter 8, dynamic motions are derived based on the results of chapter 5, 6, and 7.

1.5 Outline

Modeling

Synthetic

Analytic

Chapter 2

Chapter 3

Chapter 5

Chapter 4

15

Design Chapter 6

Control

Chapter 8

Chapter 7

Fig. 1.2. Structure of this book

Analytic chapters provide fundamentals to synthetic chapters. Chapter 3 evaluates and examines the model proposed in chapter 2. In chapter 4, it is studied that the effects of material parameters to design and control variables for chapter 5 and 6. In chapter 7, methods to reach desired position and to derive objective shapes are proposed, which are integrated in chapter 8. The structure of this book is illustrated in Figure 1.2.

Part I

Modelling

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

2.1

Introduction

This chapter proposes a model for predicting deformation response of deformable machines consisting of actively deformable materials. This is an attempt to understand activeness and deformability through description. The performance measure of the trial is simplicity and outlook of the model. We try to make up strategy to convert uncontrollable material into controllable mechanism. Soft materials that generate force and change shape can potentially form a basis for the development of deformable machines. Electroactive polymer gel is one of such promising materials. Most studies have focused on analyzing the material properties of the materials. It was not yet clear how to design effective shapes or to generate dexterous motions of mechanisms made of materials. It is difficult to apply the conventional models used to control rigid, articulated mechanisms for controlling gels because the material is too elastic and conceptually has infinite DOF. In this chapter, modelling framework is proposed that deals with activeness and deformability respectively, and describe a method for modelling each process. Previous models and proposed models are stated comparatively. Detailed evaluation and examination of the model are done in the next chapter and in the course of this book. 

This chapter was adapted from in part, by permission, M. Otake, M. Inaba and H. Inoue, “Kinematics of Gel Robots made of Electro-Active Polymer PAMPS Gel”, Proceedings of IEEE International Conference on Robotics and Automation, pp.488– 493, 2000; M. Otake, Y. Kagami, M. Inaba, and H. Inoue, “Dynamics of Gel Robots made of Electro-Active Polymer Gel”, Proceedings of IEEE International Conference on Robotics and Automation, pp.1458–1462, 2001; M. Otake, Y. Nakamura, and H. Inoue, “Pattern Formation Theory for Electroactive Polymer Gel Robots”, Proceedings of IEEE International Conference on Robotics and Automation, pp.2782–2787, 2004.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 19–33. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

20

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

Problem statement It has been proposed numerous models to describe actively deformable materials, especially electroactive polymers. Novel model is proposed because of the characteristics of the previous one. 1. The purpose of most model is to design and optimize the material. 2. They are not perspective for the purpose of design and control of machines. 3. Most of them are dependent on kinds of material. Chemists and material scientists proposed most of the models. They tend to describe the mechanisms of deformation in detail, since purpose of their models is different from this study. The papers introducing developed materials focus on differentiation from existing materials, which is apparent from the nature of research. Therefore, most of the models depend on the kinds of material. Some models proposed by physicists are applicable to some class of material, for example, polymer gels. They are aimed to describe critical phenomena like volume phase transition[164]. The required resolution for the model is high in order to express such phenomena. Robotic researchers proposed some models, such as linear approximate model based on system identification theory[165] and continuum mechanics[136] for other electroactive polymers. Their models were specific to particular kinds of material and not considered scalability to others. They also consider specific application rather than general machine design and control. Requirement for modelling It has been believed that reaction process must be modeled thoroughly to predict precisely. This idea is challenged. Requirement for novel model is summarized as follows. 1. The purpose of the models is to design and control system. 2. The model is an approximate one based on universal modelling framework. 3. The model is arranged into parts independent of material and dependent on material. The deformation process is described as follows. The model should be applicable to complex shaped materials and distributed field generated by driving electrodes in order to apply design and control deformable machines made of electroactive polymers. Approximation for effective implementation is focused. We refer to systematic reduction methods, which are proposed in different research fields. The question we must consider here is how to connect chemical theory to mechanical theory, since the deformation of gel material is driven by chemical reactions. It was explored not only chemical and mechanical but also physical, mathematical and computational models. We tried to make clear the interface of models in different fields. Modelling framework is proposed for the purpose

2.2 Modelling of Nonequilibrium System

21

Table 2.1. Specification for the model compared to previous models model universality application simplicity

specification material compatible mechanical design and control simple

previous material dependent material design and optimization complicated

of integrating the idea, which will be stated in the next subsection. The model is carefully divided into material independent part and dependent part. In this way, scalability to other kinds of material that will be developed in the future is considered. We also bring into sight other kinds of mechanisms instead of considering particular kinds of devices. Above discussions on problems of the existing models and specification for the new models are stated in Table 2.1. Modelling framework We propose modelling framework to deal with deformable machines made of actively deformable materials. Hypothesis is that activeness and deformability can be modeled respectively and interface them afterwards. This is because activeness and deformability come from different physical phenomena. It is an orthodox method to describe natural phenomena by carving substance and field, and determining local interaction between the substance and the fields. We adopt this method to keep comprehensive viewpoint. The requirement is that both active and passive deformations are described simultaneously. Active deformations arise from generated stress when electric fields are applied, while passive deformations are due to external forces, such as gravity and friction. Both of them are represented in the same framework, considering them as the result of either electrochemical or mechanical interaction between the materials and the environment. Active deformation is divided into two process, internal state change due to activeness and structural change due to deformability. If we model the phenomena locally, it is applicable to machines with complicated shapes and distributed electric fields for driving. The proposed modelling framework is shown in Figure 2.1. Electrochemical interaction is universal to active materials while mechanical interaction is generic to deformable materials, although the detail constitutive equations are dependent on kinds of materials.

2.2 2.2.1

Modelling of Nonequilibrium System Modelling of Gels

First, we pay attention to models of gels rather than electroactive polymers. This is because most of the models on electroactive polymers are particular to driving mechanism and dependent on kinds of material while models of gels

22

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

activeness

deformability

electrochemical field (a)

mechanical field

material/ machine

(c)

(b)

local interaction

Fig. 2.1. Modelling framework for describing machine made of actively deformable materials: (a) local interaction between electrochemical fields and machines; (b) local interaction between mechanical fields and machines; (c) interface of electrochemical and mechanical interaction

are universal and independent of kinds of monomers. Models on gels focus on cooperative diffusion process driven by osmotic pressure. Flory-Huggins model[166] is the start point of the models on gels. Most of the models are derivatives or extension of the model for describing critical phenomena[164, 60]. Doi et al. studied deformation process of ionic gels in electric fields[127]. They carefully extended the Flory’s model in variety of conditions. Osada et al. studied cooperative binding of surfactant molecules into the ionic gels[128, 129]. They are also based on Flory’s model. In this study, the gel which was developed by Osada et al. was selected[61]. The models of the gel were already proposed. Previous studies focused on molecular binding without electric fields and did not estimate the deformation process in electric fields. The reason is that the binding based deformation was novel and its mechanism was in question. 2.2.2

Reaction-Diffusion Model

Existing models were starting from Flory’s equation and added other equations to meet the numbers of variables. This is the reason why they are complicated. On the other hand, reverse approach has not been investigated, in order to reduce the numbers of variables by focusing only on major variables. This approach is called reductive method[167], which is commonly used in the field of nonequilibrium thermodynamics. For example, Belousov-Zhabotinsky (BZ)

2.3 Adsorption-Induced Deformation Model of the Gel

23

reaction[168] is described in either FKN model[169], Oregonator model[170] and Keener-Tyson model[171]. First one contains many equations and variables while latter two models contain small numbers of equations and variables. This is the reason that the latter models are reductive format of the former one. Regardless of their simplicity, they describe overall process. There should exist reductive format to describe deformation process of the gel. Reaction-diffusion system is known as universal modelling framework to describe natural phenomena with local interaction. The above models that describe BZ reaction are not exception. Likewise, the phenomena are described in reductive format in this book.

2.3

Adsorption-Induced Deformation Model of the Gel

An ionic polymer gel in an electric field deforms through penetration of the surfactant solution [172]. This process is characterized by the following three steps. 1. Migration of surfactant molecules into the gel driven by the electric field 2. Adsorption of surfactant molecules to the polymers 3. Gel deformation caused by adsorption of surfactant molecules The model of the first step is well established in electrochemistry, which we briefly describe later in this book. Osada and co-workers extensively studied the second step in detail [172, 129, 173]. Both electrostatic and hydrophobic interactions were assumed in the adsorption process. These models did not take an account of the electric fields and volume change for simplicity. Kitahara proposed such a mathematical model for the third step[174], that plate deformation is caused by the surface stress due to the adsorption of atoms on the surface. Although Doi et al. theoretically demonstrated the physics of ionic polymer gels in an electric field, their study did not assume the surfactant solution[127]. In this book, we derive a constitutive equation of the second step to contain common variables to those of the first and third steps. This is to demonstrate consistent model from voltage input to gel deformation. Complex adsorption reaction [173] is approximated in a simple manner. The adsorption-induced deformation model gives a theoretical foundation. 2.3.1

Migration of Surfactant Molecules Driven by the Electric Field

The surfactant molecules move toward the cathode by electrophoresis when dc voltage is applied. Under static or quasi-static conditions, we assume current density i is proportional to electric field E, i = εE,

E = −∇Φ,

where electric conductivity ε and electric potential Φ appear.

(2.1)

24

2.3.2

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

Adsorption of Surfactant Molecules to the Gel

Gong et al.[173] theoretically analyzed the cross-linking effect on the polyelectrolyte - surfactant interaction. They derived stability constant K on the basis of the free-energy minimum principle to predict concentration of bound surfactant molecules at equilibrium.   β 4β(1 − β)(u − 1) + 1 + 1 − 2β  , K = K0 u (2.2) (1 − β) 4β(1 − β)(u − 1) + 1 + 2β − 1 where K0 is the binding constant of a surfactant molecule bound to an isolated binding site on a polymer; u is the cooperativity parameter characterizing the interaction between adjacently bound surfactants; β is the degree of binding defined as the molar ratio of bound surfactant to total sulfonate group in the gel. Narita et al. analyzed kinetics of the cationic surfactant uptake into an anionic polymer network in order to discuss transient states[129]. Their theory estimates time profiles of surfactant uptake into the gel when no volumetric change is induced. Although the previous models well describe the binding of surfactant with the polymer network, they do not explain the issue of the electric field and deformation in the first and third steps. We propose and introduce the adsorption equation based on the Langmuir’s theory for the second step. The model of simplicity explains the connection between the first and third steps. The following assumptions have been made to apply the theory to the gel: a) the gel is the porous plate made of polymers like activated carbon; b) the effective surface of the polymer network is the total area of the fine pores; c) the bound molecules do not affect the free molecules once the pore is occupied by a certain numbers of molecules. Accordingly, the polymer network is approximated by a three-dimensional monolayer. Adsorption speed va is proportional to the collision rate of the molecules to empty surfaces. Namely, va = a (1 − α)i (2.3) with constant a . Note that α is the adsorption rate defined as the molar ratio of bound surfactants to the local sulfonate group of the polymer chains. Adsorption rate α is the local variable of the degrees of binding β. The average adsorption rate of the whole gel is the degrees of binding in the Gong’s theory. The normal component of the molecular density to the surface of the gel is obtained by the following equation: i = −si · n, (2.4) where n is the normal vector of the gel surface and s is the ratio of the density of ionized surfactant molecules to the current density. When the adsorption rate is small (α ≈ 0), we can approximate equation (2.3) by substituting equation (2.4) and using a = a s, va = a i = −ai · n. (2.5)

2.3 Adsorption-Induced Deformation Model of the Gel

25

y +

20[mm] φ

5[mm] O

-

x Gel

-20[mm]

Electrode

Fig. 2.2. Deformation process of a surfactant driven ionic polymer gel illustrated with experimental setup.

On the other hand, dissociation speed vd is proportional to the adsorption rate, α, and to exp(−E/RT ). Namely, vd = d α exp(−E/RT ),

(2.6)

where E and d are the adsorption energy and the dissociation constant respectively. When the temperature is constant, exp(−E/RT ) becomes constant. Using equations (2.5), (2.6) and d = d exp(−E/RT ), we have dα = va − vd = −ai · n − dα. dt

(2.7)

Equation (2.7) approximately describes the binding process in a simple manner. 2.3.3

Gel Deformation Caused by Adsorption of Surfactant Molecules

Kitahara derived a formula for the plate deformation due to surface stress caused by atom adsorption on the surface[174] . We replace atoms with plates by surfactant molecules with gels respectively. On one side of the gel, adsorbed molecules give rise to surface stress. The formula for the bending in the case of uniform surface stress was provided by Stoney[175] as follows: 6(1 − ν)σΔ 1 , = R Eh2

(2.8)

26

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

where R is the radius of curvature, ν is the Poisson’s ratio, σ is the surface stress, Δ is the width of the adsorbate, E is the Young modules, and h is the width of the substrate. We approximate that the width of adsorbate and that of substrate are the same. Substituting ν = 0.5 and Δ = h into equation(2.8), we have 3σ 1 = . (2.9) R Eh We approximate that surface stress is proportional to the adsorption rate, and obtain with constant b, σ = bα. (2.10) Equation (2.10) shows a linearized form of the stress as a nonlinear function of the adsorption rate in the neighborhood of α = 0. 2.3.4

Summary

To summarize, in order to determine the deformation of the gel, we have to solve the set of equations (2.1), (2.7), (2.9) and (2.10). We obtain 1) current density from the voltage of electrodes, 2) the adsorption rate from the current density, 3) surface stress and strain from the adsorption rate, respectively. Fig. 2.2 describes the deformation process of the beam of gel in uniform electric fields. Three arrows are current density vectors. Molecular density on the gel is large near the root and small near the tip of the gel. We can estimate that deformation speed near the root of the gel is larger than that near the tip due to the adsorption-induced deformation model. The model is evaluated and examined in the next chapter. Before evaluation, the model is extended from continuous to discrete for numerical simulation in the next section.

2.4 2.4.1

Extension to Physical and Mathematical Model Deformable Lattice Model

There are three ways to simulate reaction-diffusion system. The traditional method is to solve partial differential equation directly. Another way is to divide system into cells, which is called cell dynamic scheme (CDS). Typical models are cellular automata (CA)[176] and coupled map lattice (CML)[177]. In cellular automata model, each value of the cell (lattice) is digital. On the other hand, in coupled map lattice model, each value of the lattice (cell) is continuous. CA model is microscopic while CML model is mesoscopic. The advantage of the CML is compatibility with the physical phenomena by smaller number of cells and numerical stability. Therefore, the model based on CML is developed. Each cell has continuum state and the time step is discrete. Generally, each cell is static and not deformable. Deformable cell (lattice) is supposed in order to represent deformation process of the gel. Each cell deforms based on the internal state, which is determined by the reaction between the cell and the environment.

2.4 Extension to Physical and Mathematical Model

27

θ

electric field i (r

t j)

j+1 j j-1

gel

Fig. 2.3. Deformable lattice model of electroactive polymer gel

Suppose that gel is a one-dimensional cell, which deform in two-dimensional space (Figure 2.3). Cjt is taken to denote the value of site j in a one-dimensional cellular automaton at time step t. Each site value is specified as an integer in the range 0 through n-1. The site has three variables Cjt = [rtj , θtj , atj ],

(2.11)

with position vector r tj , tangent vector θ tj , adsorption state parameter atj . The adsorption state characterizes the electrochemical reaction on the surface of the cells. In general, each cell interacts with neighboring cells, but in this system, cells interact with outside of the cell, namely, the electric fields. This is because the gel is an open system capable of exchanging matter and energy with its environment. It is approximated that each site is mechanically connected to one another, and the electrochemical interaction between the cells is ignored, since the interaction between the cell and the electric field is dominant. Cellular automaton rule is composed of two processes, adsorption and propagation, according to the mechanism. First, initiation process is formulated in the following form, t−1 · i(r t−1 . atj = εθt−1 j j ) + (1 − D)aj

(2.12)

The variable i(rtj ) is a current density vector at position rtj . The parameter ε determines the effect of input to adsorption rate change. The parameter D is a dissociation parameter. This recurrent formula is a discrete expression of equation (2.7). The next step is propagation process. The curvature of the surface of the gel ρtj is approximated so as to be proportional to the adsorption rate atj , ρtj = μatj .

(2.13)

The parameter μ determines the ratio of adsorption rate and curvature. This equation is same as equation (2.10). The angle of the tip of the gel ϕ is a

28

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

summation of the curvature from the root to the tip, which is proportional to the whole amount of adsorbed molecules. ϕ=

n−1  0

ρtj = μ

n−1 

atj .

(2.14)

0

curvature of the gel, which is proportional to the whole amount of adsorbed molecules. Now, we obtained discrete representation of adsorption-induced deformation model. It is named deformable lattice model, since the cell (lattice) is deformable. 2.4.2

Dynamical System

The constitutive equations (2.7) and (2.12) of derived adsorption-induced deformation model and deformable lattice model are regarded as dynamical systems. Its definition[178] is as: We refer to equation (2.15) as a vector field or ordinary differential equation and to equation (2.18) as a map or difference equation. Both will be termed dynamical systems. x˙ = f (x, t; μ)

(2.15)

x → g(x; μ)

(2.16)

with x ∈ U ⊂ R , t ∈ R , and μ ∈ V ⊂ R where U and V are open d sets in Rn and Rn , respectively. The overdot in (2.15) means ” dt ”, and we view the variables μ as parameter. In the study of dynamical systems the independent variable is often referred to as ”time”. n

1

p

The equations (2.7) and (2.12) are rewritten in dynamical systems form with variable x, and control parameter μ, dissociation parameter D and D = 1 − D, x˙ = μ · x − Dx, 

x → μ · x + D x.

(2.17) (2.18)

Internal state θ and a corresponds to variable x, and current density vector i corresponds to control parameter μ. With this perspective, applying electric field to the gel is a kind of mathematical operations. This operation is examined in chapter 7 to achieve shape control of the gel.

2.5 2.5.1

Extension to Mechanical Model Multi-link Mechanism

In the previous models, design parameters such as shape parameter were not considered. For the purpose of designing gel robots, the proposed models were

2.5 Extension to Mechanical Model

29

Fig. 2.4. Multi-link model of electroactive polymer gel

interfaced to mechanical model. Firstly, kinematic model was derived from deformable lattice automata model. The objective was to simulate shape of the gel in a simple manner. Suppose that gels are articulated links made of polymer chains (Figure 2.4). The j th link is formulated using the following parameters: position vector r[j], orientation vector θ[j], thickness h[j], adsorption state parameter a[j]. Based on this formulation, the gel is expressed as: gel = [r, θ, h, a].

(2.19)

When the voltage is applied on the electrodes, the resulting electric field drives the surfactant molecules. The density of the molecules, which is nearly equal to the electric current densities, affects the probability of binding to the gel surface. As the current density increases, the probability of binding rises. Thus, the adsorption process is generally expressed as: a[j] = f (θ[j], i(r[j])) + g(a[j]),

(2.20)

where i(r[j])) is the current density at r[j]. It is calculated from equations (2.21) and (2.22), i(r[j]) = σE(r[j]) (2.21) E(r[j]) = −∇φ(r[j])

(2.22)

where σ is the conductivity, E(r[j]) is an electric field, and φ(r[j]) is an electrostatic potential at r[j]. Specific instance of f and g in equation (2.20) is expressed as: ads[j] = −pele (θ[j] · i(r[j])) + pads ads[j], (2.23) pele and pads are effect parameters of electric field and the previous state of adsorption. pele corresponds to ε and pads corresponds to (1 − D) of deformable lattice automata model. If pele is large, the electric field produces large effects on the adsorption of the molecules. Once the surfactant molecules are adsorbed, it takes a long time to desorb without reversing the electric field. This phenomenon is expressed by setting pads slightly smaller than 1. The next step is the propagation process. The adsorption state affects the joint angle of each link and can be expressed generally as follows:

30

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

dv[j − 1, j] = h(ads[j − m], · · · , ads[j + m − 1]),

(2.24)

v[j] = v[j − 1] + dv[j − 1, j].

(2.25)

Adsorbed molecules aggregate to each other causing the joint angles to change. For simplicity, equation (2.24) is expressed by equation (2.26), which means adsorbed molecules interact only with the molecules on the next link. The parameter, pdv , which corresponds to μ of the previous model, calculates the joint angle from the adsorption state and the thickness of the gel. dv[j − 1, j] =

2pdv (ads[j − 1] + ads[j]) h[j − 1] + h[j]

(2.26)

Using equation (2.25) and the boundary condition, the orientation of each link is calculated. Then the positions are determined by equation (2.27) and the boundary condition. The boundary condition decides which side of the strip is fixed or left free. j−1  r[j] = v[k]. (2.27) k=1

In this way, we can reduce the complexity of the calculation. Even if the electric field is complicated, or the deformation of the gel is very large, we can approximate the deformation by substituting for the current density i(r[j]) which is the main component of the electrochemical field in this case: elechemf ield = [i(r[j])].

(2.28)

The problem then reduces to one of obtaining the current density on the surface of the gel given the applied voltages at the electrodes. This is a kind of boundaryvalue problem. We can solve this problem numerically using the method proposed via charge simulation in chapter 5.

surfactant

gel

Fig. 2.5. Mass-spring model of electroactive polymer gel

2.5 Extension to Mechanical Model

2.5.2

31

Mass-Spring Mechanism

As described, active deformations are generated as a result of the local interaction between the gel and the electrochemical field. The environment is considered as mechanical field in order to model passive deformations in the same way. The mechanical field is a source of a variety of forces. Then we can describe the passive deformations arising from the local interaction between the gel and the mechanical field. The mechanical field is expressed as: mechf ield = [G, B, Fr , R]

(2.29)

with gravity force G, buoyancy force B, friction force Fr and resistance force R of surrounding objects. Recent years, physically based modelling becomes popular and common in computer graphics. Especially, modelling of deformable object has been discussed for more than a decade[179, 180]. This is because deformable model is required for cloth simulation[181] and surgery simulation[182]. Several models are known for simulating deformable objects: a finite element model [183], a boundary element model [184], a mass-spring model [181]. The point of selecting methods is speed, because the model is used for deformation control. And sure, the method should be applicable to large deformations. Mass-spring model was selected because of the above requirements. The massspring model only guarantees the result that looks realistic. It does not guarantee the result corresponding to experimental results. Even though, we found that this approximation is enough for motion design and control of gel robots, which will be revealed in chapter 8. The gel is expressed as an aggregate of masses and springs (Figure 2.5): gel = [M, P, K, l0 ]

(2.30)

M and P represent the weight and the position of each mass. K is the elasticity matrix representing the elasticity of each spring. M and K are calculated from the density and Young’s modulus of the gel. l 0 is the vector representing the original length of each spring. We can calculate the elastic force of the gel Felast = −K(l − l0 ). The length of each spring l is calculated from the positions of mass P. Local interaction between the gel and the mechanical field is expressed as: M P¨ = Felast + (R + Fr ) + (G + B)

(2.31)

Note that the resistance force R and friction force Fr arise only there is a contact between the gels and the surrounding fields. On the other hand, gravity force G and buoyancy force B always work on the gel. 2.5.3

Combining the Active and Passive Deformation Model

When modelling the active deformation, the adsorption rate is directly substituted into the equation (2.23) that defines the deformation. The deformation is

32

2 Adsorption-Induced Deformation Model of Electroactive Polymer Gel

Adsorption-induced deformation model Dynamical system

Multi-link mechanism Deformable lattice automata model Mass-spring mechanism

(mathematical)

(physical) (computational)

(mechanical)

Fig. 2.6. Roadmap of the models of electroactive polymer gel

caused by the stress generation on the surface of the gel. If we consider that the stress is caused by a decrease in the rest length of each spring l0 , then l0 is a function of the adsorption rate. It is approximated that the adsorption rate is proportional to the rest length of the spring l0 = l0init (1 − pl ads).

(2.32)

pl is a parameter, which defines the effect of adsorption on shrinkage of the original rest length of the spring. l0init is the initial length of the spring. We can obtain the parameter pl by pdv , considering that shrinkage of the spring in the passive deformation model causes a rotation of each link in the active deformation model.

2.6

Roadmap of the Model

We briefly summarize the development of the model, which is shown in Figure 2.6. The first model was the adsorption-induced deformation model based on the proposed mechanism. We extended it in different formats in order to increase the usability of the model. Starting from adsorption-induced deformation model, five models were derived. At first, discrete model, deformable lattice model, were proposed from adsorption-induced deformation model, utilizing cell dynamic scheme. Both adsorption model and cellular model are ubiquitously used in the field of physics. Then, these models were abstracted to mathematical model. Both of them are regarded as dynamical system. We rewrote the equation into the typical format of dynamical system and characterized the basic property of the model. Thirdly, the discrete model was extended to mechanical models, multi-link mechanism (kinematic model) and deformable objects (dynamic model) to consider design parameters and boundary condition of the gels. If one side of the gel is fixed, the former kinematic model is selected. If both sides of the gel are free

2.6 Roadmap of the Model

33

Table 2.2. Description of activeness and deformability activeness chemical, electrical or electrochemical interaction field electric fields in electrolyte internal state molecular adsorption model adsorption interaction

deformability mechanical interaction mechanical and gravitational field elastic energy, stress deformable object

or contacting with outer world, the latter dynamic model is selected. Kinematic model estimates only ’shape’, while dynamic model can deal with ’force’. The former implementation is utilized in analytic chapters: chapter 3, 4, and 7. The latter implementation is done in synthetic chapters: chapter 5, 6 and 8. For Further Studies It is well known that the volume change mechanisms in conducting polymers are complex because the electrical, mechanical, and chemical properties of the material, all of which influence the behavior, are closely coupled [67, 69]. This applies to other kinds of electroactive polymers, especially ionic ones. For precise prediction of the deformation response of the materials, these inextricably linked characteristics should be considered. Such model would be useful for long term motion planning of the machines made of such materials. Simple decoupled models which are proposed in this book are aimed to be utilized for machine design and control. Combination of simple and complex models would help us further understanding of the materials and making use of them for machines.

3 Parameter Identification by One Point Observation

3.1

Introduction

In this chapter, the model is evaluated and examined that was proposed in the previous chapter. Sticking to the model might be boring to mechanical engineers and robotics researchers, this process is especially important for solving activeness problem. Because of its simplicity, the model is sensitive to the difference of parameters. The system is extended from material to machine in the next part, design. Before that, it is essential to make clear compensation method and level of unreliability. Aside from the activeness problem, there is a fundamental problem for modelling. Most of the polymer scientists believe that the model should not be so much simple, because gel is a complex system. Some researchers noted that the model might be reasonable because it went well with experimental results. The descriptive power of the model is made clear as a result in the course of this book. We believe that the model is acceptable for polymer scientists at certain approximation level. In this chapter, the condition is kept simple. Deformation response of the gel to spatially uniform static electric field is investigated. This strategy was selected to remove the effect of other factors like modelling error of electric field or meshing of the gel.

3.2 3.2.1

Parameter Identification and Calibration Parameter Identification

The parameters to be identified are a, d, and b in equations (2.7) and (2.10). Parameter a represents the adsorption speed of molecules while parameter d represents the dissociation speed of molecules. We name a for adsorption parameter and d for dissociation parameter. Parameter b represents the stress generation when the adsorption rate is given. We normalized and set parameter b so that M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 35–60. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

36

3 Parameter Identification by One Point Observation

Table 3.1. Parameters of electroactive polymer gel a and d, which were calculated from the measured half time period T1/2 and thickness of the gel h

sample 1 sample 2 sample 3 average

T1/2 [s] 265 345 394 324

h d ×10−3 a ×10−2 [mm] [mA/mm2 ] 1.17 1.90 1.82 1.27 2.00 2.21 1.31 2.13 2.49 1.25 2.01 2.19

adsorption rate α directly expresses curvature 1/R. In this way, we can reduce the numbers of parameters from three to two. We identified parameters a and d by experiments. The same experimental setup was used. Contrary to the previous experiments, we applied a square wave electric field. We applied current density 0.1[mA/mm2 ] for 30[s] and observed the angle of the tip of the gel φ for 1800[s]. It is an integral of the curvature from the root to the tip. By substituting equation (2.10) into equation (2.9) and defining the r-axis along the curve of gel with the origin at the fixed root, we have  l  3b l 1 dr = φ= αdr. (3.1) Eh 0 0 R Fig. 3.1 shows the angle of the tip during and after the square wave electric field. After applying the electric field, the gel gradually deformed back to the original shape. The angle of the tip φ decreased. At this process, the equation (2.7) becomes dα = −dα (3.2) dt by substituting i = 0. The dissociation parameter d is obtained by the half-life period T1/2 of the tip during relaxation process. After identifying dissociation parameter d, the adsorption parameter a is calculated from the first period of the graph. The obtained parameters are shown in Table 3.1. The average from three samples were a = 2.19 × 10−2 , d = 2.01 × 10−3 . 3.2.2

Role of Parameters to the Deformation Response

From Table 3.1, the estimated range of adsorption parameters a is from 1.8 × 10−2 to 2.5 × 10−2 . The range of dissociation parameters d is from 1.9 × 10−3 to 2.1 × 10−3 . The simulated deformation response to the same electric field in 30[s] was compared in order to understand the role of parameters. Both parameters were altered from 1.5 to 2.5 in each order. We can see from Figure 3.3 that the final angle of the tip of the gel is sensitive to the scatter of adsorption parameters. Then, the deformation response of the simulated gel are compared whose adsorption parameters are 1.5 × 10−2 and 2.5 × 10−2 . Figure 3.4 illustrates both of them. The position of the tip of each gel is totally different. The difference is 2[mm] along y-axis and 4[mm] along x-axis.

3.2 Parameter Identification and Calibration

(a)

+ r

(b) Adsorption Gel

37

φ

O

Dissociation Electrode

-

Fig. 3.1. Identification of adsorption and dissociation parameters : The angle of the tip of the gel φ is observed.

1.6 1.4 angle [rad]

1.2 1 0.8 0.6 0.4 0.2 0 0

300

600

900 1200 1500 1800 time [s]

Fig. 3.2. Angular change of the tip of the gel for parameter identification

The parameter to be calibrated for each time should be adsorption parameter. It is important for deformation response of the gel. 3.2.3

Spatio-temporal Calibration Method

Then, the parameter identification method was extended to calibration method. For calibration, which coordinate to calculate parameter is very important. The ideal coordinate is that the small difference of adsorption parameter is amplified to large difference of its value. The adsorption parameter a was changed ranging from 1.5 × 10−2 and 2.5 × 10−2 and estimated the displacement and angle of the tip of the gel, which

38

3 Parameter Identification by One Point Observation

2

angle [rad]

1.8 1.6 1.4 d a

1.2

10 3 10 2

1 1.5

2 parameters

2.5

Fig. 3.3. Parameters and angle of the tip of the gels

16 a = 1.5 a = 2.5

y-axis [mm]

14

10 2 102

12 10 8 6 4 2 0 0

2

4

6 8 10 12 14 16 x-axis [mm]

Fig. 3.4. Parameters and shape of the gels in 30[s]

is shown in Figure 3.5. The estimated parameters in 10, 20, 30[s] by different coordinate is summarized in Table 3.2. The simulated range of position or angle of the tip in 10, 20, 30[s] is plotted in Figure 3.6. As time grows, the difference of the position and angle is amplified. The difference is most amplified to the x coordinate in 30[s]. Therefore, it is preferable to observe the point for calibration. Period for reaching the same position or position to reach in the same period should be measured for the calibration of adsorption parameter. If we hypothesize that dissociation parameter d is fixed to 2.0 × 10−3 , we can identify adsorption parameter a by only one observation.

displacement [mm]/angle [rad]

3.2 Parameter Identification and Calibration

15 (a) 10[s] 10

5

x y θ

displacement [mm]/angle [rad]

0

1.5

15

2 parameters

2.5

2 parameters

2.5

2 parameters

2.5

(b) 20[s]

10

5

0 displacement [mm]/angle [rad]

39

1.5

15

(c) 30[s]

10

5

0

1.5

Fig. 3.5. Adsorption parameters (a × 102 ) and displacement or angle of the tip of the gel in (a)10[s], (b)20[s], (c)30[s]

3 Parameter Identification by One Point Observation

displacement [mm]/angle [rad]

40

6

x y θ

4

2

0

20 time [s]

10

30

Fig. 3.6. Difference of the displacement or angle of the gel Table 3.2. Estimated adsorption parameters a by different values at 10, 20, 30[s] x [mm] y [mm] θ [rad]

3.3 3.3.1

10[s] 2.03×10−2 1.96×10−2 1.63×10−2

20[s] 2.25×10−2 2.14×10−2 1.95×10−2

30[s] 2.09×10−2 2.00×10−2 1.91×10−2

Evaluation of the Model Response to Uniform Electric Field

Two cases were tested in order to evaluate the model. Firstly, we describe the case in the same condition for parameter identification. Spatially uniform electric field was applied. The parameter was calibrated with the deformation response in 30[s] by x-coordinate. Then, the results were compared at different time: 10, 20, 30, 40, 50 60 [s] after applying the electric field. The amplitude of the current density was 0.1[mA/mm2 ]. Snapshot of deformation response of the gel to the uniform electric field is shown in Figure 3.7. Simulation in the same condition is plotted in Figure 3.8. Figure 3.9 is a time series of tip position and angle of the simulated and real gels. The simulation well describes the experimental results. The result suggests that the model describes the fundamental response to the electric field. It also suggests the efficiency of calibration method. Once we calibrate the parameter by one point at an arbitrary time, positions and angles at different time also went well. Obtained adsorption parameter a was 2.0 ×10−2 . This was calculated from x-coordinate of the tip of the gel in 30[s] (in Figure 3.9) and mapping of parameters and response (in Figure 3.5).

3.3 Evaluation of the Model

0 [s]

10 [s]

20 [s]

30 [s]

40 [s]

50 [s]

41

Fig. 3.7. Deformation of the beam-shaped gel in spatio-temporally uniform electric field. The current density was 0.1[mA/mm2 ].

3.3.2

Response to Alternating Electric Field

Secondly, alternating electric field was applied. The period before switching of polarity was altered after started applying the electric field for 30, 45, 60[s]. The parameter was calibrated as well as the first case. Then, the results were compared for every 10 [s] after applying the electric fields. The amplitude of the current density was 0.1[mA/mm2 ]. Snapshot of deformation response of the gel to the alternating electric field is shown in Figure 3.10. The polarity of the electric field was switched after 60[s]. Simulation in the same condition is plotted in Figure 3.11. Obtained adsorption parameter a was 2.4 ×10−2 . Deformation speed obtained through experiment was faster than simulation after switching the polarity. This caused the difference of shape in 90[s] and 120[s]. Simulation response caught up in 150[s].

42

3 Parameter Identification by One Point Observation

15

0 [s] y-axis [mm]

y-axis [mm]

15 10 5 0

5

5 10 15 x-axis [mm]

0

20 [s]

10 5 0

5 10 15 x-axis [mm] 30 [s]

15 y-axis [mm]

15 y-axis [mm]

10

0 0

10 5 0

0

15

5 10 15 x-axis [mm]

0

15

40 [s] y-axis [mm]

y-axis [mm]

10 [s]

10 5 0

5 10 15 x-axis [mm] 50 [s]

10 5 0

0

5 10 15 x-axis [mm]

0

5 10 15 x-axis [mm]

Fig. 3.8. Simulated deformation of the beam-shaped gel in spatio-temporally uniform electric field. The current density is 0.1[mA/mm2 ].

displacement[mm]

3.3 Evaluation of the Model

15

displacement[mm]

simulation experiment

(a) 10 5 0 0

10

20

30 40 time[s]

50

60

10

20

30 40 time[s]

50

60

10

20

30 40 time[s]

50

60

15 (b) 10 5 0 0

orientation[rad]

43

2

(c)

1.5 1 0.5 0

0

Fig. 3.9. Displacement and angle of the tip of the gel. (a) x co-ordinate of the tip of the gel, (b) y co-ordinate of the tip of the gel, (c) angle of the tip of the gel

The estimated reason for the difference of speed is that the model doesn’t consider elasticity of the gel. Elastic energy should be accumulated just before switching. After releasing the elastic energy, the condition of the gel might have recovered which met the hypothesis of the model.

44

3 Parameter Identification by One Point Observation

0 [s]

30 [s]

60 [s]

90 [s]

120 [s]

150 [s]

Fig. 3.10. Experimental deformation of the beam-shaped gel in spatio-temporally uniform electric field. The current density is 100[mA]

3.3 Evaluation of the Model

0 [s]

45

30 [s]

5 [mm]

60 [s]

90 [s]

120 [s]

150 [s]

Fig. 3.11. Simulated deformation of the beam-shaped gel in spatially uniform temporally alternating electric field. The current density is 100[mA]

46

3 Parameter Identification by One Point Observation

3.4

Hypothesis and Limitation of the Model

3.4.1

Long Time Response

Adsorption is a process of molecular binding ’on’ the surface of the gel. Absorption is that of molecular binding ’into’ the gel. The model only considers the former case, adsorption. If we take time, absorption process grows. It becomes difficult to ignore the effect of absorption. This is one of the biggest limitations of the model. Simple experiment was conducted for estimating the effect of absorption to the deformation response. Oscillating electric field was applied to the gel (16[mm] long, 1[mm] thick) to keep the amplitude of deformation along oscillating direction to 5[mm]. Amplitude of the current density was 0.3[mA/mm−2 ]. Period of one cycle was measured, which is summarized in Table 3.3. The required time became larger as the time goes. In other word, the response speed became slower. From this simple testing, we can see that absorption of molecules reduces the activeness of the gel. Osada et al, inventor of this material, studied on this absorption process thoroughly[128, 129]. Modification of the model considering their theory is required for describing long time response. 3.4.2

Strong Electric Field

The reaction that occurs inside and outside of the gel is not only binding process of surfactant molecules. If we apply weak electric field, the dominant process is only surfactant binding. In this case, we can ignore other reactions. On the contrary, if we apply strong electric field, other processes like hydrogen ion exchange or transfer are unable to be disregarded. The model considers surfactants in which other molecules are not considered. If the deformation speed is far faster than estimated, other reactions might dominate. Table 3.3. Period for one cycle: The amplitude along oscillating direction was kept 6[mm] for the gel with 16[mm] long. elapsed time [min] period for one cycle [s]

3.5 3.5.1

0 20 40 60 80 100 10 12 16 23 69 710

Examination of Parameters Analysis of Parameters

For the purpose of confirming the obtained parameters, the adsorption parameter a was varied from 10−5 to 10 and studied deformation response. The dissociation parameter d was kept at 10−3 and bending parameter b at 1. Spatially uniform static electric field was applied to the simulated gel for T = 1000[s]. From Figure 3.12 to 3.15 illustrate final shape of the gel. Trajectories of the tip of the gels

3.5 Examination of Parameters

47

yaxis [mm]

3 2 1 0 1

0

2

4

6

8 xaxis [mm]

0

2 xaxis [mm]

10

12

14

16

14

12

yaxis [mm]

10

8

6

4

2

0 4

Fig. 3.12. Final shape of the gel (a = 10−5 (top), a = 10−4 (bottom); T = 1000[s])

are shown from Figure 3.16 to 3.19. Time series of y co-ordinate of the tip of the gel are plotted in Figure 3.20 to 3.23. When the adsorption parameter a exceeds 1, the deformation response becomes chaotic, which is visible from final shape in Figure 3.14 and 3.15. It is required to analyze time series of y co-ordinate in frequency space, to make clear whether the response is chaotic or not. Since the focus of this book is shape and motion control, it will be discussed in further studies. What we learned from the result is: even with this simple model, its behavior might become complex. Final shape of the gel at a = 10−3 is wave shaped. This phenomenon was observed in simulation at first and observed in experiments afterward. We will deal with waving pattern formation in chapter 7, in the context of shape control.

3 Parameter Identification by One Point Observation

16

16

14

14

14

12

12

12

10

10

10

8

8

yaxis [mm]

16

y−axis [mm]

y−axis [mm]

48

8

6

6

6

4

4

4

2

2

2

0

0

0

−1 0 1 x−axis [mm]

1 0 1 xaxis [mm]

−1 0 1 x−axis [mm]

Fig. 3.13. Final shape of the gel (a = 10−3 (left), a = 10−2 (center), a = 10−1 (right); T = 1000[s])

3.5.2

Required Resolution of Time and Space

The results suggest different perspective at the same time from the viewpoint of temporal resolution of the model. If the adsorption parameter is small enough to be ignored, it is equivalent to keep the adsorption parameter small and to make time step small. Adsorption parameter below 10−1 , the difference of each response is speed only. It is apparent by comparing the response of a = 10−4 in T = 1000[s] and that of a = 10−5 in T = 10000[s]. Final shapes (Figure 3.12, Figure 3.24 (top)), trajectories (Figure 3.16, Figure 3.24 (bottom)), time series of y co-ordinate of the tip (Figure 3.20, Figure 3.25) are the same. Therefore, we can take large time step to keep the adsorption parameter below 10−1 . The obtained adsorption parameter was of 10−2 with time step 1[s], namely its order is in stable domain. In this way, we can identify the reasonable approximation of the time step. Even though the resolution of time is rough compared to ordinary simulation, simulation results well describe the experimental results. After discussing the temporal resolution, we refer to spatial resolution. Beamshaped gel is divided into distributed mechanism in the model. The length of each link was 1[mm] for gels with 1[mm] thick. Such large element is impossible

3.5 Examination of Parameters

49

6

5

4

yaxis [mm]

3

2

1

0

1

1

0

1 xaxis [mm]

2

3

Fig. 3.14. Final shape of the gel (a = 1, T = 1000[s])

with normal finite element method. This is the utility of rigid approximation. Such approximation keeps the calculation time small, which enables testing of large numbers of simulations for planning and control.

3 Parameter Identification by One Point Observation

1

0.5

0

yaxis [mm]

50

0.5

1

1.5

2

2.5

3

1.5

1

0.5

0

0.5 1 xaxis [mm]

1.5

2

2.5

Fig. 3.15. Final shape of the gel (a = 10, T = 1000[s])

3

3.5 Examination of Parameters

51

4

yaxis [mm]

3 t=800 t=600 t=400 t=200 t= 0

2 position[15.65, 3.00] orientation 0.337645

1 0 1 2

2

0

2

4

6

8 xaxis [mm]

10

12

14

16

18

15 t=800 t=600

10

yaxis [mm]

t=400

t=200 5

position[1.69, 13.76] orientation 2.083780 t= 0

0

0

2

4

6

8 10 xaxis [mm]

12

14

16

Fig. 3.16. Trajectory and angle of the tip of the gel (a = 10−5 (top), a = 10−4 (bottom); T = 1000[s])

52

3 Parameter Identification by One Point Observation

16

t=800 t=400t=600 t=200

14

12

y−axis [mm]

10

8

6

4

2 position[−0.68, 15.82] orientation 1.670478

t= 0

0

−2

−4

−2

0

2

4

6 8 x−axis [mm]

10

12

14

16

18

t=800 t=600 t=400 t=200

16

14

12

y−axis [mm]

10

8

6

4 t= 0 2 position[0.00, 16.00] orientation 1.570804 0

−2

−5

0

5 x−axis [mm]

10

15

Fig. 3.17. Trajectory and angle of the tip of the gel (a = 10−3 (top), a = 10−2 (bottom); T = 1000[s])

3.5 Examination of Parameters

53

18 17 t=800 t=600 t=400 t=200

16 15

yaxis [mm]

14 13 12 11 10 t= 0 9 8 7 10

5

0

5

xaxis [mm]

8

6

4

2 t=600 position[1.82, 5.19] orientation 208.610388

yaxis [mm]

t=200 t=400 0

t= 0

2

t=800

4

6

8 10

8

6

4

2

0 xaxis [mm]

2

4

6

8

Fig. 3.18. Trajectory and angle of the tip of the gel (a = 10−1 (top), a = 1 (bottom); T = 1000[s])

3 Parameter Identification by One Point Observation

10

t=600

5

yaxis [mm]

54

position[0.49, 0.05] orientation 1762.006586 t=800 t= 0 t=400

0

5 t=200

10

10

8

6

4

2 0 xaxis [mm]

2

4

6

8

Fig. 3.19. Trajectory and angle of the tip of the gel (a = 10; T = 1000[s])

3.5 Examination of Parameters

55

3.5 3 2.5

yaxis [mm]

2 1.5 1 0.5 0 0.5 0

200

400

600

800

1000

time [s] 14 12

yaxis [mm]

10 8 6 4 2 0 0

200

400

600

800

1000

time [s]

Fig. 3.20. Displacement of the tip of the gel (a = 10−5 (top), a = 10−4 (bottom); T = 1000[s])

56

3 Parameter Identification by One Point Observation

16 14 12

yaxis [mm]

10 8 6 4 2 0 0

200

400

600

800

1000

600

800

1000

time [s] 16

14

yaxis [mm]

12

10

8

6

4

0

200

400 time [s]

Fig. 3.21. Displacement of the tip of the gel (a = 10−3 (top), a = 10−2 (bottom); T = 1000[s])

3.5 Examination of Parameters

57

17 16 15

yaxis [mm]

14 13 12 11 10 9 8 0

200

400

600

800

1000

600

800

1000

time [s] 8 6 4

yaxis [mm]

2 0 2 4 6

0

200

400 time [s]

Fig. 3.22. Displacement of the tip of the gel (a = 10−1 (top), a = 1 (bottom); T = 1000[s])

3 Parameter Identification by One Point Observation

10 8 6 4 2 yaxis [mm]

58

0 2 4 6 8 10 0

200

400

600

800

1000

time [s]

Fig. 3.23. Displacement of the tip of the gel (a = 10, T = 1000[s])

3.5 Examination of Parameters

59

14

12

y−axis [mm]

10

8

6

4

2

0 0

2 x−axis [mm]

4

14 t=8000 t=6000

12

y−axis [mm]

10

t=4000

8

6

t=2000

4

2 position[1.70, 13.76] orientation 2.082991 t= 0

0

−2 −2

0

2

4

6

8 10 x−axis [mm]

12

14

16

18

Fig. 3.24. Final shape (top) and trajectory of the tip (bottom) of the gel (a = 10−5 ; T = 10000[s])

3 Parameter Identification by One Point Observation

14 12 10 y−axis [mm]

60

8 6 4 2 0 0

2000

4000

6000

8000

10000

time [s]

Fig. 3.25. Displacement of the tip of the gel (a = 10−5 ; T = 10000[s])

Part II

Design

4 Interaction-Based Design of Deformable Machines

4.1

Introduction

This chapter proposes design principle inevitable to build deformable machines. It is well known that different physical phenomena dominate in microscale and nanoscale world. This makes the design of micro-sized and nanosized machines different from large-scale machines. Likewise, the design of deformable machines should require methods different from rigid machines. Since highly deformable machines like gel robots did not exist before, design principle was not exactly investigated. As the beginning of Part II, this chapter explores method to develop deformable machines from actively deformable materials. Hypothesis is proposed to solve activeness and deformability problems through design. The strategy is to divide whole system into two parts, material and electric field interfaced by local interaction between them. Electric field generator is designed in chapter 5, and shape of the gel in chapter 6, respectively. In this chapter, it is examined that the effect of activeness and deformability on deformation response of the gels both in simulation and experiments. Through varying design and control parameters, Scale effect of activeness and deformability is explored. Before conducting experiments, we explain the procedure of material preparation and experimental setup. Results of this chapter would serve as foundation to design mechanical system containing actively deformable materials like electroactive polymers. Problem Statement Common approach to design deformable machines is to divide rigid structures into small pieces and connect them by active or passive joints. In other case, 

This chapter was adapted from in part, by permission, M. Otake, M. Inaba, and H. Inoue. “Development of Gel Robots made of Electro-Active Polymer PAMPS Gel”, Proceedings of IEEE International Conference on Systems Man and Cybernetics, vol.2, pp.788–793, 1999.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 63–76. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

64

4 Interaction-Based Design of Deformable Machines

passively deformable materials are added into rigid structures. However, with this method, it is impossible to design whole-body deformable robots like sea creatures. A large numbers of motors make the hardware too heavy to move. And also, it is not deformable as a whole. In this book, totally different approach is proposed for the design of deformable machines. Actively deformable materials are distributed to form deformable machines. Design principle which realizes such machines did not exist, because of a lack of precedent. The problems on design are listed below. 1. Design principle particular to deformable machines was not clear, especially the one which was derived from the physical law. 2. A method to define design and control parameters from parameters which represent activeness and deformability was not clear. 3. A roadmap to develop whole-body deformable machines from actively deformable materials was unknown. The first problem comes from the physical property of activeness and deformability. Both of them should work as a constraints or possibility to realize novel structure of machines. The detailed analysis of activeness and deformability from the viewpoint of machine design was not conducted. Therefore, we decided to analyze utilizing electroactive polymer gel. The second problem comes from the fact that general mechanical design method was not investigated before. Design and control parameters were optimized particular to the kinds of materials to develop specific mechanisms. There should exist methods to decide design and control parameters to overcome the problem of activeness and deformability. The third problem comes from the traditional way of machine design. Actively deformable materials have been developed since 1950s, but they have been used as devices. Most of them were packaged into rigid structures and not consisted full-body deformable machines. A roadmap to design deformable machines from materials is required. Approach to deformable machine design For the purpose of solving the above problems on deformable machine design, we explored method to design gel robots from electroactive polymer gel. This approach corresponds to the above problems is summarized below. 1. The role of activeness and deformability is analyzed, and scale effect of design and control parameters through the proposed model and simulation. 2. The activeness and deformability parameters are measured experimentally, changing the design and control parameters. 3. Procedure that designs material-field respectively for building deformable machines. For the first problem, we refer to the scale analysis of microscale and nanoscale machines. For the second problem, simulation and experimental results give

4.2 Measurement of Activeness and Deformability

65

Fig. 4.1. Deformation of beam-shaped gel in the air. The scale bar is 3[mm].

us guideline to determine design and control parameters. The changing role of muscle in living animals is also studied. For the final problem, we show novel approach different from the previous ones. The common way of designing deformable machine has been to build driving elements (electric field generator) and driven elements (material) simultaneously. On the other hand, we design material and electric field respectively, and unify them afterwards. This is because both of them can be separated by the model that describe local interaction between them, and be dealt with in particular. In this way, perspective of deformable machine design becomes clear.

4.2 4.2.1

Measurement of Activeness and Deformability Elasticity and Density

Measurement method Firstly, the deformability of the material is measured and estimated. Deformability is characterized by Young’s modulus. The experimental apparatus are illustrated in Figure 4.2. Strips were placed horizontally in the air and observed from side for the measurement of Young’s modulus(Figure 4.1). Poisson’s ratio was obtained by measuring height and diameter of cylinders when they were put weight on the top. Other measurements were conducted in the solution. The elastic modulus was estimated by the following procedure. Simple experiment was conducted without testing machine. When beam-shaped gel was placed horizontally in the air, deformation of the gel caused by its own weight was observed. It was hypothesized that surface curve of the gel represented elasticity of the gel.

66

4 Interaction-Based Design of Deformable Machines

g v 0

x gel

Fig. 4.2. Measuring the curve of beam-shaped gel to calculate Young’s modulus

Calculation method Here, we describe calculation method from deformation of the gel. Given that the strip of the gel is an elastic beam. Draw a horizontal line (x − axis) from the root of the beam, a vertical line (v − axis) from the tip of the beam, intersecting at the point O (the origin). Positive values for v are measured upward on the v −axis, negative values downward (Figure.4.2). Deflection curve is described as: v=

4x x4 wL4 (1 − + ), 8EI 3L 3L4 bh3 , 12 w = ρd bh, I=

(4.1) (4.2) (4.3)

with static deflection v, distributed pressure of its own weight w, length of beam L, thickness h, width b, Young’s modulus E, moment of inertia of area I, density ρd . Then, maximum displacement of the tip of beam vmax can be obtained by substituting 0 into x. wL4 (x = 0) (4.4) vmax = 8EI The parameters h, L, were selected to satisfy that h is small enough to compare with L, h ≈ vmax in order to consider as an elastic body. With strip satisfying above condition (6.23[mm]×5[mm]×1.31[mm], the first row of Table 4.1), deformation curve is measured every 1[mm]. Measured curve is plotted in Figure.4.3(1). Approximated curve is plotted in two ways. One is plotted by a least squares method with use of measured 8 points (Figure 4.3(2)). Another is calculated on hypothesis that the maximum displacement is satisfies equation (4.4) (Figure 4.3(3)). Mutual relationship is sufficiently estimated between measured curve and approximated curve in this condition. Correlation coefficients were over 99 %. Therefore, maximum displacements were measured and Young’s modulus was estimated by equation (4.4).

4.2 Measurement of Activeness and Deformability

0

1

2

3

4

5

6

67

7

0 -0.1 v-axis [mm]

-0.2 -0.3 -0.4 1: measured curve

-0.5 -0.6

2: calculated from the maximum displacement

-0.7

3: obtained by a least squares method

-0.8 -0.9

x-axis [mm] Fig. 4.3. Measured and approximated curve of the beam surface (1: measured curve, 2:obtained by a least squares method, 3:calculated from the maximum displacement) Table 4.1. Measured maximum displacement vmax and calculated Young’ s modulus E h 1.31 1.34 1.36 1.36 1.15

L 6.23 8.18 6.78 7.80 6.71

vmax 0.80 2.51 1.02 1.76 1.58

3.6 3.2 3.7 3.7 3.2

E × × × × ×

104 104 104 104 104

Deformability of the gel Measurement results and estimated Young’s modulus are shown in Table.4.1. Density was determined beforehand, ρd = 2.2[mg/mm2]. The average of five measurement, calculated Young’s modulus of the gel was 3.5 × 104 [Pa], which is very small. Both Young’s modulus and density depend on a swelling rate, which is affected by a polymerizations condition but their order is accurate. Young’s modulus was a dimension of 104 [Pa]. Poisson’s ratio is 0.5, which means the material doesn’t change volume when the loads are applied. This is typical of soft material like gel whose molecular attraction is small. 4.2.2

Generated Stress

Generating stress could be calculated by measuring the radius of curvature of the beam-shaped material. Following is the reason.

68

4 Interaction-Based Design of Deformable Machines

(a)

0[V]

(b)

3[V]

(c)

4[V]

(d)

5[V]

(e)

7[V]

(f)

10[V]

Fig. 4.4. Step voltage response of the PAMPS gel(25[mm] × 5[mm] × 1.0 [mm]). The scale bar is 5 [mm].

Based on the theory of bending mechanism, surfactants bound on the gel surface generate stress on the surface, because of anisotropic contraction. Stress cause strain on the surface of gel. Before clarifying generating stress, look at Figure 4.4. Convergent state of step voltage response is shown. Voltage under 5[V], a strip bended slowly and stopped at the balanced state, in which the surface curve was uniform. Voltage over 5[V], a strip became perpendicular state and stopped with small oscillation, which was a limit, in which the surface curve was not uniform.

4.2 Measurement of Activeness and Deformability

69

anode(+) solution

r

electrode v gel cathode(-) Fig. 4.5. Measuring the curvature or maximum displacement to calculate stress and strain

3000

stress[Pa]

2500 1.0mm thick

2000

0.6mm thick 1500 1000 500 0 0

2

4 voltage [V]

6

8

Fig. 4.6. Applied voltage versus generated stress and strain on the surface of PAMPS gel

Let us suppose that the radius of curvature is uniform at low voltage, generating stress σ is given by the following equation with radius of curvature ρr (Figure 4.5),Young’s modulus E, thickness of the beam h. σ=

Eh 2ρr

(4.5)

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4 Interaction-Based Design of Deformable Machines

Calculated stress and strain are shown in Figure 4.6. At higher voltage generates larger stress. Even if the thickness is different, the generating force at same voltage is the same. And the order of generating stress is from 102 to 103 [Pa].

4.3 4.3.1

Role of Design and Control Variables Thickness Dependence on Step and Frequency Response

Step and frequency responses are measured. At first, thickness dependency is observed. A deformation response for a step voltage output at 5[V] for 13.5[mm]× 5[mm]× 1.2, 1.8, 2.9[mm] strip are shown in Figure 4.7. Both bending speed and maximum deformation showed dependency on thickness. Each strip deforms until saturation. Due to the low voltage (5[V]), thick one sloped bending before it becomes vertical to the electrode, while thin one bent until it became straight to it. This is because generating force on the surface is equal at the same time, the radius of the curvature of a saturation point is proportional to thickness according to equation (4.5). Deformation speed of thin one is faster than that of thick one. Binding speed becomes faster as the radius of curvature becomes smaller, based on the mechanism of binding. Based on our analysis, the thinner one is supposed to respond to higher frequency, which is observed in Figure 4.8. It shows displacement of free end of 16[mm]×5[mm]×1[mm] membrane for frequency range of 0.1-10[Hz] for square input voltage at 10[V] amplitude. 4.3.2

Voltage Dependence on Step and Frequency Response

Next, voltage dependency is observed. Deformation response for step voltage input at 4, 5, 6, 7[V] for the same strip (13.5[mm]×5[mm]×1.2) are shown in Figure 4.9. Electric current is proportional to input voltage, because the solution between two electrodes works as a resistance, which means the higher the voltage, the faster the speed of binding becomes. Each one bends to the maximum and stopped bending at the same place. Quantities of the maximum binding surfactant are the same according to the surface area of the strip. Difference is just speed to reach its maximum. It is appropriate to suppose that a high input voltage cause faster response, observed in Figure 4.10. Figure 4.10 indicates that difference of amplitude is large between thick part and thin part, when frequency is below 0.5[Hz] and input voltage is high around 10[V]. 4.3.3

Time Constant of the System

Material property and dynamic property are interfaced by the concept of time constant. Referring to the scale analysis of MEMS[185], Time constant T of the system is defined as required time of tip of the gel becomes vertical to the original angle. Then, the time constant is described with the parameters and variables as:

4.3 Role of Design and Control Variables

71

displacement[mm]

10 8 6 1.2mm thick 1.8mm thick 2.9mm thick

4 2 0 0

20

40

60

80

100

time[s] Fig. 4.7. Thickness dependence of bending speed and maximum deformation step response (5[mm]×13.5[mm],voltage:5[V] d.c.)

8 displacement[mm]

7 6 5 4 3

0.4mm thick 0.6mm thick 1.0mm thick 2.9mm thick

2 1 0 0.1

1 frequency[Hz]

10

Fig. 4.8. Thickness dependence of bending deformation frequency response (5[mm]×14[mm],voltage of 10[V] square wave)

t E (4.6) LV ε with thickness of the gel t, length of the gel L, amplitude of the electric field V , elasticity of the gel E and reaction parameter ε. It is derived from equation (2.7), equation (2.10) and equation (4.5). Additional description is required for length of the gel L. The orientation of the tip of the gel is a spatial integration of the curvature of the gel. Therefore, if the length of the gel becomes twice, the time to reach the same orientation of the tip becomes one half. T ∝

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4 Interaction-Based Design of Deformable Machines

displacement[mm]

10 8

7V 6V 5V 4V 3.5V

6 4 2 0 0

20

40

60

80

100

time[s] Fig. 4.9. Voltage dependence of bending speed step response (5[mm]×13.5[mm] ×1.2[mm]thick)

displacement[mm]

7 6 5 4 3

10V 8V 6V 4V

2 1 0 0.1

1 frequency[Hz]

10

Fig. 4.10. Voltage dependence of bending deformation frequency response (5[mm]×14[mm]×0.6[mm]thick, square wave)

4.4 4.4.1

Roadmap from Materials to Machines Prototype of Muscle-Hydraulic System

Electroactive polymers are expected to organize muscular hydrostats systems, in which the muscle provides both the mechanical support for position control as well as the force for motion. Examples are arms for locomotion (octopus), fins for swimming (cuttlefish) and limbs for manipulation (trunks of elephants, tentacles of squid, arms of cuttlefish, and the tongues of frogs)[186]. For the purpose of designing such robots, material properties were examined. The point was the scale effect of its softness. Different from solid materials, soft materials amplify

4.4 Roadmap from Materials to Machines

73

Fig. 4.11. Prototype of gel robots: Lizard robot. The scale bar is 5[mm].

small differences of the dimension to a large difference of deforming response, due to small Young’s modulus. This is the meaning of deformability. Prototype of gel robots are developed using the examined material property. For example, a sea butterfly robot that swims in the solution (Figure 4.14), sea anemone robot that swings tentacles reversibly (Figure 4.13). The bottom shape of the sea anemone robot is octagon, 5[mm] each side. Tentacles are 20[mm] long, 3[mm] wide. Bottom of them are 2[mm] thick and becomes thinner to the top. In the center, cylinder-shaped carbon electrode is placed with a diameter of 5[mm], height of 10[mm]. Platinum plate electrode which are used for measurement are put from both side of the cylinder, distance of 30[mm]. By applying low voltage high frequency electricity (4[V] 0.5[Hz]), thin parts (tip) swing and thick parts (root) are static. When applying high

Fig. 4.12. Prototype of gel robots: Shrimp robot. The scale bar is 5[mm].

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4 Interaction-Based Design of Deformable Machines

Fig. 4.13. Prototype of gel robots: Sea anemone robot. The scale bar is 5[mm].

voltage low frequency input (10[V] 0.1[Hz]) thin parts and thick parts bend together. Shape and input signal parameters are selected based on Figure 4.8 and Figure 4.10. The sea butterfly robot is designed to swim under the water, with thin wings and a thick body (Figure 4.14). Thickness of the wings are 0.4[mm] and their shapes are crescent shape, fore ones are 15[mm] wide, rear ones are 10[mm] wide. Body is 20[mm] long, 10[mm] wide and 3[mm] thick. Body fixed the angle of the wings and the pair of platinum electrodes (distance: 65[mm]). When saw wave signal (from 0[V] to 10[V] ascend gradually and descend rapidly from 0[V] to 10[V], frequency of 0.5[Hz]) is applied, the wing bends reversibly and the body receives force, the butterfly moves forward to the anode. Speed of locomotion

Fig. 4.14. Prototype of gel robots: Sea butterfly robot. The scale bar is 10[mm].

4.4 Roadmap from Materials to Machines

25[mm]

40[mm] electrode(anode)

3[mm]

5[mm]

20[mm]

65[mm]

7.5[mm]

5[mm]

surface of the solution

10[mm] electrode

(b)Side view

(a)Top view

Fig. 4.15. Schematic illustration of sea butterfly robot

(Space) uniform

varying

static

(Time)

Previous

Chap 5

Chap 7

Chap 8

alternating

Fig. 4.16. Roadmap of electric field design and control

75

76

4 Interaction-Based Design of Deformable Machines

(Space) uniform

varying

static

(Time)

Previous

Chap 6

Chap 7

Chap 8

alternating

Fig. 4.17. Roadmap of shape design and control

is 1[mm/s]. Without the body, wings could not keep angle to the electrode and became parallel to the electrode and stopped. 4.4.2

Material-Field Respective Design

From the analysis in the previous section, role of material and field variables are relative. It is equivalent for deformation speed that making the gel thinner and applying the electric field larger. Taking account of this relativity, and considering su-field analysis[187], we developed systematically from material to machine, which is shown in Figure 4.16 and 4.17. Electric field evolves from uniform static state to spatially varying, temporally alternating, both spatially and temporally changing one. In the next chapter, spatially varying electric field is applied to the gel. Likewise, shape of the gel evolves from beam-shaped gel to complex shape, both spatially and temporally. In chapter 6, shape design of the gel is studied.

5 Spatially-Varying Electric Field Design by Planer Electrodes

5.1

Introduction

This chapter describes the design and implementation of fields generation system to drive deformable machine consisting of actively deformable materials. The purpose is developing a system to control the shapes and motions of both simulated and real gel robots using electric fields. Such system has not been investigated before. In chapter 4, the method was proposed which designs materials and fields respectively, considering the local interaction equation that defines the relationship between materials and fields. We focused primarily on the electric fields that drive the material rather than the material itself, since the electric fields are common to all material. It was assumed that there must be an optimal method that brings out the performance of material. In this study, performance was measured by the variety of shapes and motions of machines made from material. In order to control gel robots, estimates of the set of possible transformations that result from actuation are needed. The model was proposed which describes the deformation process of the gel in chapter 2, and identified model parameters in chapter 3. The model supposes that the state of the gel is calculated from the local interactions between the gel and the molecules driven by the electric field. Since the chemical reaction is located on the surface of the gel, we need 

This chapter was adapted from in part, by permission, M. Otake, M. Inaba, and H. Inoue. “Development of Electric Environment to control Mollusk-Shaped Gel Robots made of Electro-Active Polymer PAMPS Gel”, Proceedings of SPIE vol.3987 Electroactive Polymer Actuators and Devices (EAPAD) Y. Bar-Cohen (ed.), pp.321– 330, 2000; M. Otake, Y. Kagami, M. Inaba, and H. Inoue, “Dynamics of Gel Robots made of Electro-Active Polymer Gel”, Proceedings of IEEE International Conference on Robotics and Automation, pp.1458–1462, 2001; M. Otake, Y. Kagami, M. Inaba, and H. Inoue, “Motion design of a starfish-shaped gel robot made of electro-active polymer gel”, Robotics and Autonomous Systems, vol. 40, pp. 185–191, 2002.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 77–116. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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5 Spatially-Varying Electric Field Design by Planer Electrodes

to compute the electric field on the surface. Once we obtain an estimate of the electric field on the surface, the state and the shape of the gel can be calculated. The developed system of this chapter is utilized in the next chapter. The system provides fundamental environment to study motion control of deformable machines, which is studied in chapter 8. Problem statement The major problems are stated on designing driving system. 1. We cannot prepare same numbers of input as degrees of freedom of the body. 2. Design principle of driving field is not established. 3. Real performance of the material is not investigated. The first problem arises from deformability. It is impossible to prepare same numbers of input to degrees of freedom of the body, since the deformable machines like gel robots have virtually infinite degrees of freedom. In the case of electroactive polymers, the materials require driving electrodes. This applies to any kinds of materials driven by electric fields. The body becomes full of wires if we prepare driving electrodes as many as possible. A method to reduce the numbers of input effectively is required without decreasing the controllability. The second and third problem is related to each other. The resulting shapes and motions are determined by the kinds of materials and driving system of materials. Machine design principle is not fully investigated because the development of actively deformable materials is ongoing. The maximum performance, in this case, reachable shapes and motions has not fully examined. Approach to electric field design Approach to solve the problems is as follows. 1. A method which generates spatially varying electric fields to drive multiple points is proposed. 2. The real condition is made clear for driving the material based on the model. 3. Novel driving system is developed and it experimentally generate possible deformations and motions.

5.2 5.2.1

Previous Driving System Separate or Composite Electrodes

Electroactive polymer systems consist of polymer and electrodes. Electrodes are necessary to generate electric fields to drive polymers. Polymers generate stress and strain according to the magnitude and direction of the electric fields. From the mechanism, the driving forces are: ionic or electronic transfers or exchanges, or molecular structural changes. We can classify electroactive polymers driven by ions or electrons from the standpoint of electrodes configuration, 1) separate

5.2 Previous Driving System

79

electrolyte

electrode

electrode electrode polymer

electrode

polymer

1a)

2a)

electrolyte

polymer

polymer polymer

electrode 1b)

electrolyte 2b)

Fig. 5.1. Classification of electroactive polymers based on electrodes configuration

or 2) composite, and a) polymers require a pair of electrodes or b) polymers work as electrodes. We can classify them into following four categories. In composite type, electrodes are 1a) electroplated or mechanically coated on the surface of the polymers, or 1b) a pair of polymer sheets works as electrodes, which sandwiches electrolyte. On the other hand, in separate type, electrodes and polymers are immersed in electrolytes. In this case, 2a) a pair of electrodes generates electric fields, which indirectly drive a polymer between them, or 2b) a pair of polymer

80

5 Spatially-Varying Electric Field Design by Planer Electrodes

and an electrode generates electric field in the electrolytes, which alter the electrochemical state of the polymer. All classes are illustrated in Figure 5.1. Specific materials for each categories are 1a) for ionic gel, 2a) for ion conducting polymer metal composite and carbon plated elastomers, 1b) and 2b) for conducting polymers and carbon nanotube sheets. Composite types take advantage of separate type for artificial muscle application, because they can work in the air. In contrast, separate types run ahead of composite types for robotic application, especially 1a) types, because they don’t require direct wiring for each polymers. Composite type requires many wires to represent complex shape, which prevent dynamic motion of materials. Therefore, type 1a) ionic gels were selected. This type ensures the flexibility of total system design because polymers and electrodes can be designed separately. We can universally use the same electrodes system for different set of polymer materials. Also, we can change polymer if the lifetime of the material is expired. The problem then reduces to the configuration of electrodes and application of the voltages to each electrode to generate electric fields. 5.2.2

Conversion from Bending to Contracting

Bending mechanisms were referred in the previous subsection. It is possible to convert some of them into contractile mechanism. For example, polypyrrole, a typical conducting polymer of 1b) or 2b) type, was converted to work as contracting mechanism by mechanical constraint[141]. The gel, which originally exhibited volumetric change, was converted to contractile mechanism by layering multiple gels and electrodes[188]. We can learn from these studies that there should exist undiscovered driving setup to bring out maximum performance or different characteristics of materials.

5.3 5.3.1

Design Process Multiple Electrodes in a Plane

What will happen if we move one of the electrodes next to the facing electrode? Distribution of current density vectors are shown in Figure 5.2. A method for simulating electric field is stated in section 5.4. Top figure of Figure 5.2 shows the current density vectors generated by parallel electrodes, while bottom figure shows the ones generated by electrodes in a plane. Contours represent magnitude of current density vectors. There are 20 levels for both graphs. From the bar on the right, maximum value is 9 for parallel placed electrodes and 8 for electrodes in a plane. Center of parallel electrodes is on the 4th level. These numbers do not correspond to the physical values. They are relative numbers but the ratio of values is reliable. The point in the same level by electrodes in a plane is near electrodes. Numbers are added from first to fourth level and a circle was put on the level. From the graph, we can see that the magnitudes of current density vectors are almost in the same order, although the one generated by electrodes

5.3 Design Process

81

1 9 8

1

0.5

7

2

6 5

3

4

0

4 3 2

-0.5

1 -1 -1

-0.5

0

0.5

1 8

1

7 6

0.5 5

4 3

0

4

2

3

1

-0.5

2 1

-1

-1.5

-1

-0.5

0

0.5

1

1.5

Fig. 5.2. Electric field generated by parallel electrodes (top), by two electrodes in the same plane (bottom). Arrows are current density vectors.

in a plane is smaller. It was assumed that electric field generated by multiple electrodes could drive the materials. For comparison, the conventional configuration of electrodes with fix-ended gel is illustrated in Figure 5.3. Cationic surfactant molecules adsorb on the surface of the anionic gel facing the anode electrode, which cause the gel to bend toward

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5 Spatially-Varying Electric Field Design by Planer Electrodes

Fig. 5.3. Conventional configuration of the parallel electrodes with fix-ended gel

(a)

(b)

Fig. 5.4. New configuration of the electrodes in a plane with fix-ended gel. The electrode above the gel is cathode for a) and anode for b).

5.3 Design Process

83

(a)

(b)

Fig. 5.5. New configuration of the electrodes in a plane with free-ended gel. The electrode above the center of the gel is cathode for a) and anode for b).

the anode electrode. From the mechanism and model, the true requirement for electrodes is to generate enough current density perpendicular to the surface of the gel. Figure 5.4 shows the assumed deformation below a pair of electrodes in a plane. Beam-shaped gel below the cathode would deform against the electrode while the gel below the anode would deform toward the electrode. 5.3.2

Removal of Fixing Point

The second constraint that we tried was removal of the fixing point. It was unconsciously done already at preliminary trial. But, now we can remove the electrodes on the bottom from the assumption. If the gel is placed below the multiple electrodes in a plane, the gravity force will keep the distance between the electrodes and the gel. Therefore, the fixing point should be removed.

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5 Spatially-Varying Electric Field Design by Planer Electrodes

Figure 5.5 shows the assumed deformation of the gel below multiple electrodes. If the electrode above the center of the gel is anode, cationic surfactant molecules adsorb on the reverse side of the anionic gel facing the electrodes, which cause the gel to deform into concave shape. This is because the anode electrode above the center of the gel repels the surfactant molecules. On the other hand, if the electrode above the center of the gel is cathode, cationic surfactant molecules adsorb on the surface of the anionic gel facing the ground; which cause the center of the gel to move toward the cathode electrode. This is because the cathode electrode above the center of the gel attracts the surfactant molecules.

5.4

Development of Electric Field Generation System

Based on concept design, a software and hardware system were developed for studying the motion of gel robots by generating electric fields. The overall setup is shown in Figure 5.6. From applied voltage data, the system convert the electric signals from digital to analog, amplify and generate the electric field using multiple electrodes. It simultaneously calculates the distribution of the electric field and approximates the transformation of the gel. The simulator displays both of them in real time. Fig.5.6(a) shows the generator of the real electric field while Fig.5.6(b) shows the monitor of the electro-active polymer system. With this system, experiments were performed using gel robots in the presence of spatially varying electric fields

(a) Generator Amplifier D/A Board

Electrodes

Voltage Data Electric Field Polymer (b) Simulator

Monitor

Fig. 5.6. Overview of electric field generation system

5.4 Development of the System

85

Personal Computer D/A Board

Amplifier

Fig. 5.7. 4ch electrodes system

Fig. 5.8. A pair of 4ch electrodes: One side of electrodes was used for experiments. The scale bar is 10[mm].

generated by multiple electrodes in planar configuration. Now that outline of the system is explained, hardware and software system are described respectively. 5.4.1

Hardware System

Driving system Firstly, driving system which applies voltages to multiple electrodes was designed and developed. In chapter 4, potentiostat and galvanostat were utilized to drive the gel, because they set current density and potentials in high accuracy. We

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5 Spatially-Varying Electric Field Design by Planer Electrodes

Electrode

Camera

Fig. 5.9. 16ch array of electrodes system (top view)

Amplifier

D/A Board Personal Computer Gel Fig. 5.10. 16ch array of electrode system (side view)

could not apply different voltages to multiple electrodes with them because they have limited numbers of channels (in this case, two channels). The original driving setup for removing the limitation of numbers of channels was developed for design flexibility. Driving system for conventional robots was modified to drive electrochemical setup. Applied voltages were controlled by a PC with D/A board (RIF01, Fujitsu Corporation), or serial-parallel converter (BlackBox, Felixstyle Inc.). Amplifier circuit amplifies the inputs with D.C. power supply (PW18-1.8Q, KENWOOD). Two kinds were prepared. One is analog operational amplifier (LM6321, National Instruments) and another is motor driver IC (TPD4000K,

5.4 Development of the System

87

Fig. 5.11. Photo of 16ch array of electrodes system. The scale bar is 10[mm].

Personal Computer

Gel

D/A Board

Amplifier Fig. 5.12. 4 × 4 matrix 16ch electrodes from the side

Toshiba Semiconductor). The requirements for circuits are possibility to drive capacitive load without oscillation and to output large current. Multiple electrodes 4-channel array of electrodes A setup consisting of multiple electrodes with fixended gel was materialized. For prototype, four electrodes were arranged in a same plane. Schematic illustration and its photo of 4-channel electrodes are shown in Figure 5.7 and 5.8. Each electrode in this case is 4[mm] in width, 20[mm] in length. The space between two electrodes in the array is 2[mm]. There are a total of four electrodes with long side next to each other. 16-channel array of electrodes Then, the numbers of electrodes were increased from 4 to 16, after it was confirmed that the multiple electrodes in a plane

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5 Spatially-Varying Electric Field Design by Planer Electrodes

Gel

Camera

Mirror

Fig. 5.13. Observation from side using mirror

Electrode

Fig. 5.14. 4 × 4 matrix 16ch electrodes from the top

could drive the gel through conducting experiments with 4-channel electrodes. Multiple electrodes arranged in an array that looks like a ladder generate spatially varying electric fields. This setup is a materialization of multiple electrodes with free-ended gel. The size of each electrode is the same (4[mm] × 20[mm]), and the space between the electrodes is the same (2[mm]). The gel is located under the array of electrodes, with the space between the gel and the array being 5[mm] and 10[mm]. Figure 5.9 and 5.10 are top view and side view of the system. Figure 5.11 is a photo of 16-channel array of electrodes. Note that the circles between the electrodes in Figure 5.9 are holes to prevent bubbles to collect and cover the electrodes. Figure 5.16 is an estimated electric field generated by this setup.

5.4 Development of the System

89

Fig. 5.15. Photo of matrix of electrodes. The scale bar is 10[mm].

16-channel matrix of electrodes In the 4-channel electrodes, the four electrodes were arranged in a line. Another dimension was added by arranging the electrodes into a matrix of four rows and four columns (also in the same plane). The size of each electrode is 10[mm] square. The space between each electrode is 5[mm]. The electrodes were placed parallel to the surface of the water and the gel robot was placed under the electrodes at the bottom. The space between the electrodes and the ground is 10 and 15[mm] (Figure 5.15). Figure 5.12 illustrates the setup for 16ch matrix of electrodes system. A mirror was placed below the electrodes for measuring position of the gel (Figure 5.13). Figure 5.14 is a schematic illustration of electrodes. The point is to make holes between electrodes to remove the bubbles. Figure 5.17 is an estimated electric field generated by this setup. 5.4.2

Software System: Simulation

Applying static voltages to multiple electrodes A uniform electric field is not sufficient to simultaneously actuate different parts of a gel robot in opposing directions. We can use spatially varying electric fields generated by multiple electrodes. By applying different electric signals to each of the electrodes, various parts of the gel robot can be moved individually. In electro-chemical analysis, potentiostat is used to maintain a constant electric potential of the electrodes, while galvanostat is used to maintain a constant electric current. By comparison, traditional robots are typically controlled by varying voltage or electric current. We selected the most basic method, voltage control in order to enable scaling up of a gel robot system. In order to actuate the robot, we need to determine the electric field of the liquid solution and to calculate the density of the current on the surface of the gel in response to specific electrode voltages. In this way, we can simulate the behavior of the gel and obtain an estimate of its motion and changes in shape.

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5 Spatially-Varying Electric Field Design by Planer Electrodes

1 0.8 0.6 0.4 0.2

1.5 1 0.5 0

0 -0.2 -0.4 -0.6 -0.8 -1 -5

-4

-3

-2

-1

0

1

2

3

4

5 1 0.8 0.6 0.4

1.5 1 0.5 0

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 -5

-4

-3

-2

-1

0

1

2

3

4

5

Fig. 5.16. Electric field generated by 16 electrodes in the same plane: top shows anode electrode in center, bottom shows cathode electrode in center

The deformation of the gel we used is caused by surface shrinking; which is proportional to the amount of molecules adsorbed on the surface. Roughly speaking, the whole transformation is proportional to the amount of voltage applied over time. When we stop applying voltage to the electrodes, the gel stops changing shape.

5.4 Development of the System

91

1 0.8 0.6 2

0.4

1.5

0.2 0

1

-0.2

0.5

-0.4

0

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0.4

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-0.6 -0.8 0

0.5 1

1.5 2

2.5 3

3.5 4

4.5 5

5.5 6

6.5

-1

Fig. 5.17. Electric field generated by matrix of electrodes in the same plane: top shows anode electrode in center, bottom shows cathode electrode in center

In the experiments, we applied different constant voltages over given time intervals, which causes periods of constant electric current. Because of the time needed to propagate the charge to the gel, these time intervals cannot be too short (i.e. rapidly changing voltages are ineffectual). Applying slowly changing voltages or cyclic voltages of low frequency are possible. We can regard the potential and the electric field as the same as that generated by constant a voltage, if the transition of the voltage is slow enough to ignore the resistance of

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5 Spatially-Varying Electric Field Design by Planer Electrodes

the solution. In this way, we can utilize simple models in order to solve for the generating electric field. Solving for the static and quasi-static electric current There are several methods for solving either numerically or analytically for the electric field and static or quasi-static electric current. We solve Poisson’s equation inside and outside of the electrodes under the constraint that the potential on the surface of the electrodes is constant. This is a type of boundary-value problem. The application of analytical methods is limited to the case when the arrangement of the electrodes is simple. Instead, we use a numerical solution method which is applicable to multiple electrodes. There are four kinds of numerical methods which are generally used: finite difference method, finite element method, charge simulation method, and surface charge method. We apply the third one, charge simulation method, to our problems because it has following three advantages: 1. Dimension of the equation is low. 2. Errors are small because the equation doesn’t contain differentiation. 3. We can obtain the solution of the potential or electric field directly at any point. We don’t have to calculate the value of the entire region in order to find out the value of a partial region. The charge simulation method models exact physical phenomena in a very simple way. Electrodes are modeled as finite numbers of charges which are mapped on the surface of the electrodes. These charges generate the electric field. When voltages are applied to the electrodes, the charges distribute on the surface of the electrodes to satisfy the boundary condition that the potential of electrode is uniform. Unknown charges are obtained by solving simultaneous equations which satisfy the above boundary conditions. The charge simulation method is established within static electric fields which satisfy the Laplace equation: (5.1) ∇2 φ(r) = 0 for an electric potential φ at r. Because we generate static electric fields, we can apply this method to our problem. Calculation of the electric field and current density from potentials of the Electrodes The method which we use requires potentials of the electrodes but we don’t specify the absolute potentials. We just specify relative voltages between the electrodes. Initially, we assume that the absolute potential is given. Later, we calculate the potential from the voltages using other constraints. In this book, we consider the two dimensional electric field generated by square plate electrodes. We regard a plane electrode as an array of infinite line charges. Electrodes are divided in the model in the same manner as the gel is divided into micro chains. Let the coordinate of one of the infinite charges be [X, Y ]

5.4 Development of the System

93

whose charge density per unit length is q. The electric potential φ of position [x, y] generated by the charge is expressed as: φ=−

q ln |(x − X)2 + (y − Y )2 |, 4π

(5.2)

where is a dielectric constant in the medium. Electric field [Ex , Ey ] is calculated as: x−X q Ex = , 2π (x − X)2 + (y − Y )2 (5.3) y−Y q . Ey = 2π (x − X)2 + (y − Y )2 We plot the number of n counter points on the edge of the electrodes which are spaced equally dφφ relative to each other. We assume the potentials at these points are given φ. Then we draw lines perpendicular to the edge line from each counter point, and place an infinite line charge along each perpendicular line. The distance between each charge and counter point is dφq . The potential at the ith counter point is calculated from the equation (5.2) by substituting the coordinate of the ith counter point [x[i], y[i]] and the coordinate of the jth charge [X[j], Y [j]] and its charge density q[j]. We rearrange Equation (5.2) with the parameter P (i, j) which is defined as: P (i, j) = −

1 ln |(x[i] − X[j])2 + (y[i] − Y [j])2 |, 4π

(5.4)

and the potential at the ith counter point φ[i]is expressed as: φ[i] =

n 

P (i, j)q[j].

(5.5)

j=1

Because the parameter P (i, j) doesn’t depend on q[j], we obtain n first-order simultaneous equations with n unknown charges q[j]. ⎤ ⎤ ⎡ ⎡ ⎤⎡ φ[1] q[1] P11 P12 · · · P1n ⎢ P21 P22 · · · P2n ⎥ ⎢ q[2] ⎥ ⎢ φ[2] ⎥ ⎥ ⎥ ⎢ ⎢ ⎥⎢ (5.6) ⎥ ⎥ = ⎢ .. ⎢ .. .. . . .. ⎥ ⎢ .. ⎦ ⎦ ⎣. ⎣ . . . ⎦⎣. . φ[n] q[n] Pn1 Pn2 · · · Pnn Unknown charges q[j] are obtained by solving equation(5.6). Potentials which are generated by the charges q[j] at any point [x, y] in the solution is also expressed using Equation (5.6) by exchanging [x[i], y[i]] for [x, y]. The electric field [Ex , Ey ] is also obtained from Equation (5.3) with [x, y] and the parameters [Fx (j), Fy (j)] defined as: x − X[j] 1 Fx (j) = , 2π (x − X[j])2 + (y − Y [j])2 (5.7) y − Y [j] 1 Fy (j) = , 2π (x − X[j])2 + (y − Y [j])2

94

5 Spatially-Varying Electric Field Design by Planer Electrodes

Ex = Ey =

n  j=1 n 

Fx (j)q[j], (5.8) Fy (j)q[j].

j=1

The current density i at r which is required to calculate the state of the gel is obtained by the equation i(r) = σE(r), (5.9) where σ is the electric conductivity. Another constraint to obtain absolute potentials from applied voltages Current densities in the solution are obtained in the way. This method assumes that the potentials of the electrodes are given. This requires absolute potentials while we have relative potentials among the electrodes. If the voltage of the electrode is V , the potential of the electrodes φ is φ = V + φ0

(5.10)

where φ0 is constant parameter which is unknown and common to the electrodes. The parameter φ0 requires one constraint because it is a scalar. The electric current is static and quasi static in our problem. That means that when we look at the charges of the electrodes at a given time t, the sum of those charges equals to zero. n  q[j] = 0 (5.11) j=1

The charge q[j] is expressed using φ, which is expressed with V and φ0 . Because V is known, we can obtain φ0 . We substitute φ0 into the Equation (5.10) and we can obtain each q[j]. Evaluation of the result We can calculate the electric field and current density at certain points. On the other hand, the potentials of all points along the edge of the electrodes do not necessary equal the given potential φ, because the counter points are discontinuous. The accuracy was evaluated in the following way: We plot the test points between the counter points on the edge of the electrodes. The potentials at the test points are calculated by examining the error. We calculated the potential of a pair of electrodes spaced 30[mm] and 30[mm] wide relative to each other when dφφ = 0.1, dφq = 0.12. The calculated potentials of the electrodes (5[V] applied) are shown in Figure 5.18. The simulated results satisfy the boundary condition well. We have described a method to obtain the current density and electric potential at arbitrary points given applied voltages at the electrodes. This means

5.5 Simulation and Experimental Results

95

5.1

Calculated Voltages [V]

5.08 5.06 5.04 5.02 5 4.98 4.96 4.94 4.92 4.9 0

5

10

15 [mm]

20

25

30

Fig. 5.18. The calculated potentials of the electrodes (5[V] applied).

we can calculate the current density on the surface of the gels even if the gels undergo large changes in shape. We have simulated the electric potentials generated by pairs of electrodes arranged in two different configurations: 1) electrodes 15[mm] wide placed along a line, and 2) electrodes 30[mm] wide placed facing each other, separated by a distance of 30[mm]. In both cases the applied voltages were 5[V] and -5[V]. The results are shown in Figure 5.19. We can hypothesize that the electric field generated by two electrodes in a line will also drive the gel as that generated by parallel electrodes.

5.5 5.5.1

Simulation and Experimental Results Electrodes in a Plane with Fix-Ended Gel

For verification of this hypothesis, we applied voltages to four electrodes in a line and observed the movements of the simulated and real gel. Each electrode in this case is 4[mm] in width, 20[mm] in length. The space between two adjacent electrodes in the array is 2[mm]. The strip of the gel and the electrode array were parallel to each other. The space between the gel and the array is 15[mm]. We applied the following set of voltages to the electrode array from left to right: [V1 , V2 , V3 , V4 ] = [10 0 0

− 10][V ]

The left side of the gel strip was fixed while the right side was free.

(5.12)

96

5 Spatially-Varying Electric Field Design by Planer Electrodes

10

Voltages [V]

5 0 -5 -10 20 0 -20 -20 [mm]

0

20

40

10

Voltages [V]

5 0 -5

-10 20 0 -20

-20

0

20

[mm] Fig. 5.19. Calculated electric potentials generated by electrodes

40

5.5 Simulation and Experimental Results

97

Table 5.1. Experimental conditions of free-ended gel case (1) Polarity of central electrodes (2) Numbers of central electrodes (3) Distance between the electrodes and the gel [mm] (4) Horizontal distance of the center of gels and electrodes [mm]

a+ +

a-

b+ +

b-

c+ +

c-

d+ +

d-

1

1

3

3

1

1

1

1

10

10

10

10

5

5

10

10

0

0

0

0

0

0

2

2

A new kind of motion was created using electrodes in a linear configuration, which is shown in Figure 5.21 and 5.20. Experimental results were obtained after applying an electric field for one minute. Figure 5.20 is of the simulated gel. The generated electric potential and current density are displayed. Figure 5.21 is of the real gel. Both of them are consisting of three figures. The top figure shows the initial state. The middle figure shows the shape of the gel after applying the electric field for 30 seconds. The bottom figure shows the gel state after applying electric field for 60 seconds. Although the speed of the motion was slow, the results show that large transformations are possible to use a linear array of electrodes. The gel bends towards the anode electrodes, and away from the cathode electrodes. This is because an anode repels surfactant molecules, which adhere to the surface of the gel on the same side as the electrode. In contrast, the cathode electrode attracts the surfactant molecules away from the gel surface. The adsorption speed is much faster than desorption speed, which causes the molecules on the surface to accumulate. Hence the curvature becomes larger. 5.5.2

Electrodes in a Plane with Free-Ended Gel

Experimental methods The array of electrodes was used to observe deformation process of free-ended gels. Both simulation and experiments were performed in the same condition and compared the results. Fundamental set was tested, which is expected to deform gels into convex shape and concave shape. Then, conditions were changed compared to the first cases. The varied conditions were: numbers of anionic or cationic electrodes in the center, distance between the electrodes and the gel, relative position of gels and electrodes. The conditions are summarized in table 5.1. The case of (a-), (b-), (c-) and (d-) are illustrated in Figure 5.22. Common condition was the size of the gel, 16[mm] in length, 4[mm] in width and 1[mm] in thickness.

98

5 Spatially-Varying Electric Field Design by Planer Electrodes

20 [mm]

10

0

0

-10[V]

10 0[s] 0

-10

-10

0

10

20

30 [mm]

-10

0

10

20

30 [mm]

-10

0

10

20

30 [mm]

20 [mm] 10 30[s] 0

-10 20 [mm] 10 60[s] 0

-10

Fig. 5.20. Simulated electric field and deformation of the beam-shaped gel with one end fixed: The gel is driven by 4ch electrodes. The surface of the gel facing the anode shrinks, while that facing the cathode expands. This makes the gel to bend toward the anodic electrodes.

5.5 Simulation and Experimental Results

10

0

0

99

-10

0[s]

30[s]

60[s] Fig. 5.21. Experimental result of the beam-shaped gel with one end fixed. The gel was driven by 4ch electrodes. The scale bar is 5[mm].

100

5 Spatially-Varying Electric Field Design by Planer Electrodes

+

(a-)

-

+

-

-

10[mm]

-

(b-)

(c-)

(d-)

5[mm]

2[mm]

Fig. 5.22. Varied experimental conditions of electrodes and the gel whose central electrodes are cathodic: (a-) fundamental set (b-) numbers of central electrodes are three (c-) distance between the electrodes and the gel is 5[mm] (d-) horizontal distance of the center of the gel and the electrodes is 2[mm]

5.5 Simulation and Experimental Results

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5

0

0

15

10 20[s]

20

0

10

5

5 0

15

10 40[s]

20

0

10

5

5 0

10 60[s]

20

0

10 30[s]

20

0

10 50[s]

20

0

10 70[s]

20

15

10

0

10 10[s]

15

10

0

0

15

10

20

0

101

Fig. 5.23. (a-) The gel deforms into convex shape symmetrically. : The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (10 -10 10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

102

5 Spatially-Varying Electric Field Design by Planer Electrodes

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5 0

15

10 20[s]

20

0

10

5

5 0

15

10 40[s]

20

0

10

5

5 0

10 60[s]

0

10 30[s]

20

0

10 50[s]

20

0

10 70[s]

20

15

10

0

20

15

10

0

10 10[s]

15

10

0

0

20

0

Fig. 5.24. (a+) The gel deforms into concave shape symmetrically.: The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (-10 10 -10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

5.5 Simulation and Experimental Results

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5

0

0

15

10 20[s]

20

0

10

5

5 0

15

10 40[s]

20

0

10

5

5 0

10 60[s]

20

0

10 30[s]

20

0

10 50[s]

20

0

10 70[s]

20

15

10

0

10 10[s]

15

10

0

0

15

10

20

0

103

Fig. 5.25. (b-) The gel deforms into convex shape symmetrically. The width of cathode electrodes is larger than the previous one. : The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (-10 -10 -10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

104

5 Spatially-Varying Electric Field Design by Planer Electrodes

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5 0

15

10 20[s]

20

0

10

5

5 0

15

10 40[s]

20

0

10

5

5 0

10 60[s]

0

10 30[s]

20

0

10 50[s]

20

0

10 70[s]

20

15

10

0

20

15

10

0

10 10[s]

15

10

0

0

20

0

Fig. 5.26. (b+) The gel deforms into concave shape symmetrically. The width of cathode electrodes is larger than the previous one. : The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (10 10 10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

5.5 Simulation and Experimental Results

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5

0

0

15

10 2[s]

20

0

10

5

5 0

15

10 4[s]

20

0

10

5

5 0

10 6[s]

20

0

10 3[s]

20

0

10 5[s]

20

0

10 7[s]

20

15

10

0

10 1[s]

15

10

0

0

15

10

20

0

105

Fig. 5.27. (c-) The gel deforms into convex shape symmetrically. The distance between the electrodes and the gel is smaller than the previous one. : The distance between the gel and the electrodes is 5 [mm]. The combination of applied voltages is (10 -10 10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

106

5 Spatially-Varying Electric Field Design by Planer Electrodes

15

15

10

10

5

5

0

0

15

10 0[s]

20

0

10

5

5 0

15

10 2[s]

20

0

10

5

5 0

15

10 4[s]

20

0

10

5

5 0

10 6[s]

0

10 3[s]

20

0

10 5[s]

20

0

10 7[s]

20

15

10

0

20

15

10

0

10 1[s]

15

10

0

0

20

0

Fig. 5.28. (c+) The gel deforms into concave shape symmetrically. The distance between the electrodes and the gel is smaller than the previous one. : The distance between the gel and the electrodes is 5 [mm]. The combination of applied voltages is (-10 10 -10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

5.5 Simulation and Experimental Results

15

15

10

10

5

5

0

0

5

15

10 0[s]

15

20

0

10

5

5

0

0

5

15

10 20[s]

15

20

0

10

5

5 0

5

15

10 15 40[s]

20

0

10

5

5 0

5

10 10[s]

15

20

0

5

10 30[s]

15

20

10 15 60[s]

0

5

10 15 50[s]

20

0

5

10 15 70[s]

20

15

10

0

5

15

10

0

0

15

10

20

0

107

Fig. 5.29. (d-) The gel deforms into convex shape asymmetrically. The distance between the center of the electrodes and the center of the gel is 2[mm]. : The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (10 -10 10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

108

5 Spatially-Varying Electric Field Design by Planer Electrodes

15

15

10

10

5

5

0

0

5

15

10 0[s]

15

20

0

10

5

5 0

5

15

10 15 30[s]

20

0

10

5

5 0

5

15

10 60[s]

15

20

0

10

5

5 0

5

10 90[s]

15

20

0

5

10 15 45[s]

20

0

5

10 75[s]

15

20

0

5

10 15 105[s]

20

15

10

0

10 15[s]

15

10

0

5

15

10

0

0

15

20

0

Fig. 5.30. (d+) The gel deforms into concave shape asymmetrically. The distance between the center of the electrodes and the center of the gel is 2[mm]. : The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (-10 10 -10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

5.5 Simulation and Experimental Results

0 [s]

5 [s]

10 [s]

15 [s]

20 [s]

25 [s]

30 [s]

109

35[s]

Fig. 5.31. (a-) The gel deformed into concave shape symmetrically.: The distance between the gel and the electrodes was 10 [mm]. The combination of applied voltages was (10 -10 10) [V]. The size of the gel was 16[mm] in length, 1[mm] in thickness. The scale bar of 0[s] is 5[mm].

110

5 Spatially-Varying Electric Field Design by Planer Electrodes

0 [s]

10 [s]

20 [s]

30 [s]

40 [s]

50 [s]

60 [s]

70[s]

Fig. 5.32. (a+) The gel deformed into convex shape symmetrically.: The distance between the gel and the electrodes was 10 [mm]. The combination of applied voltages was (-10 10 -10) [V]. The size of the gel was 16[mm] in length, 1[mm] in thickness. The scale bar of 0[s] is 5[mm].

5.5 Simulation and Experimental Results

111

(a)

0[s]

3[s]

(b)

0[s]

15[s] Fig. 5.33. (c-) The gel deformed into convex shape symmetrically (left column of the figures). (c+) The gel deformed into concave shape symmetrically (right column of the figures): The distance between the gel and the electrodes was 5 [mm]. The combination of applied voltages was (10 -10 10) [V]. The size of the gel was 24[mm] in length, 1[mm] in thickness. The scale bar of 0[s] is 5[mm].

Simulation results Simulation results are illustrated from Figure 5.23 to Figure 5.30. Deformation responses to cathode conditions (a-), (b-), (c-), (d-) were convex while those to anode conditions (a+), (b+), (c+), (d+) were concave. By increasing the numbers of cathodic or anodic electrodes, the curvature in the center of the gel became larger. It is clear through comparing the results

112

5 Spatially-Varying Electric Field Design by Planer Electrodes

0 [s]

5 [s]

10 [s]

15 [s]

20 [s]

25 [s]

30 [s]

35[s]

Fig. 5.34. (d-) The gel deformed into convex shape asymmetrically. The distance between the center of the electrodes and the center of the gel was 2[mm]. : The distance between the gel and the electrodes was 10 [mm]. The combination of applied voltages was (10 -10 10) [V]. The size of the gel was 16[mm] in length, 1[mm] in thickness. The scale bar of 0[s] is 5[mm].

5.5 Simulation and Experimental Results

0 [s]

15 [s]

30 [s]

45 [s]

60 [s]

75 [s]

90 [s]

105[s]

113

Fig. 5.35. (d+) The gel deformed into concave shape asymmetrically. The distance between the center of the electrodes and the center of the gel was 2[mm]. : The distance between the gel and the electrodes was 10 [mm]. The combination of applied voltages was (-10 10 -10) [V]. The size of the gel was 16[mm] in length, 1[mm] in thickness. The scale bar of 0[s] is 5[mm].

114

5 Spatially-Varying Electric Field Design by Planer Electrodes

(a) Electrodes

Current Density i(r[j])

Potential Contour

Gel

(b)

Fig. 5.36. Simulated deformation of the free-ended beam-shaped gel with electric field (top) and the photo of convex-shaped gel (bottom), in the case of (a-) in 40[s]. : The gel deforms into convex shape symmetrically. The distance between the gel and the electrodes is 10 [mm]. The combination of applied voltages is (10 -10 10) [V]. The size of the gel is 16[mm] in length, 1[mm] in thickness.

of (a-) and (b-), or (a+) and (b+). Since collision inside the gel has not been implemented, the tips of the gel went through with each other. If the distance between the gel and the electrodes were narrowed down, the bending reaction occurred locally and the deformation speed was faster. It is apparent by comparing the results of (a-) in 10[s] and (c-) in 7[s]. When the center of the gel and the central electrodes was not in a line, the deformation response was asymmetric. The results of asymmetric condition, (d+) and (d-) are illustrated in Figure 5.29 and Figure 5.30. Experimental results Then, experiments were conducted with real gels. Experimental results well corresponds to simulation results. There were some errors especially for ones to deform the gel into convex shape. In fundamental condition of (a-), deformation response to cathode electrodes was almost the same although it reached maximum curvature in 20[s]. After 20[s]

5.5 Simulation and Experimental Results

0.5 20[s]

115

simulation experimental

0

curvature[1/mm]

-0.5 0

5

10

0.5 40[s]

15

simulation experimental

0 -0.5 0

5

10

0.5 60[s]

15

simulation experimental

0 -0.5 0

5

10

15

length[mm] Fig. 5.37. Curvature of the real and simulated gel in the case of (a-) in 20, 40, 60[s] : The curvature near the electrodes of experimental results was larger than simulation.

the gel couldn’t deform into perpendicular to the ground and the electrodes. The deformation speed was twice faster than estimated. On the contrary, fundamental condition of (a+) well corresponds to simulation results. The speed of deformation was almost the same. The deformation of the tip of the gel near the electrode in 60[s] was faster, which made the difference of final shape of the gel. The results of the condition of (c-) and (c+) are shown in Figure 5.33. The error observed for the fundamental set was also observed in these cases. Even with these cases, overall deformation was estimated. Figure 5.34 shows the gel deformed into convex shape asymmetrically (d-). The difference observed in fundamental set was also observed in this case. The deformation speed was twice as faster as estimated. The curvature reached its maximum in 30[s]. The gel deformed into concave shape asymmetrically (d+) is shown in Figure 5.35. It deformed into the shape of ’6’. The deformation response was almost the same as estimated. The difference was the contacting point to the ground. The contacting point of simulated gel is on the left from 15[s] to 45[s], which moves

116

5 Spatially-Varying Electric Field Design by Planer Electrodes

to right in 60[s]. On the contrary, the contacting point of real gel moved to right in 15[s]. The cases of (b+) and (b-), whose numbers of cathodic or anodic electrodes are three, were not tested, because the simulation results are not reliable. The similar case without collision will be thoroughly studied in chapter 8. Supposed modelling errors pH near the electrodes didn’t meet the hypothesis that the molecular adsorption dominates compared to other effects. Ion exchange should have occurred simultaneously, which support the deformation response caused by the surfactant molecular adsorption. It also describes that the deformation speed was twice as faster than estimated. Simulated electric field and deformation of the gel for fundamental case (a-) of 40[s] is shown in Figure 5.36. The curvatures of case (a-) in 20, 40, 60[s] is shown in Figure 5.37. Since the deformation speed was twice as fast as estimated that the simulation time was selected for comparison. Simulation in 20, 40, 60 [s] was compared to experimental results in 10, 20, 30[s], respectively. The curvature near the electrodes of experimental results was larger than simulation. These errors should be avoided by changing the distance between the electrodes and the gel. The distance was increased from 10[mm] to 15[mm] in the case of (b-) in chapter 8.

6 Shape Design through Geometric Variation

6.1

Introduction

Liu and Calvert constructed an electrically driven linear actuator from a series of layers of polyacrylic acid and polyacrylamide hydrogels[188]. In air, a uniform block of gel will bend in response to an applied field or will expel water and contract. The gel stack changes shape as water is driven from one material to the other. They showed that combination of materials convert from the bending to spring like motion. In other context, there have been a number of efforts to make tough synthetic materials using layered structures. Freeform fabrication has been used to make layered structures with a view to introducing similar toughness into brittle materials[189]. Layered technique is useful for increasing the strength of the whole structure, as well as conversion of motions. Using the same material, we can obtain different motions by structural design. This chapter proposes a novel method for designing the shape of small elastic robots made of electroactive polymer gel. It was brought out that directional deformation from originally bending type polymer driven by electric fields. The key idea is partially reducing the structural flexibility through shape design. For the purpose of achieving directional motion, Gels with wave-shaped surfaces and gels with cuttings were designed. Former ones have thick and thin parts with wave-shaped surfaces, distributed in either one or two directions. Latter ones have cutting whose deformation responses are compared to those without cutting. In this way, it is proposed that the method for the design of the desired deformation response by shape design of the material in advance. Several gel robots were prototyped, i.e. butterfly-shaped, lizard-shaped in chapter 4. One of the difficulties with gel robots is their limited controllability 

This chapter was adapted from in part, by permission, M. Otake, Y. Kagami, K. Ishikawa, M. Inaba, and H. Inoue, “Shape Design of Gel Robots made of Electroactive Polymer Gel”, Proceedings of SPIE vol.4234 Smart Materials A. R. Wilson et al. (eds.), pp.194–202, 2001.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 117–134. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

118

6 Shape Design through Geometric Variation

because of their conceptually infinite degrees of freedom. The strategy is to reduce the degrees of freedom from infinite to finite. Mechanics theory suggests that the deformation response of elastic materials is sensitive to their own shapes. It was designed that thin parts to work as actuators, and thick parts to work as structures with the same material. However, we have not yet developed gels whose thick parts and thin parts are distributed. Thick parts and thin parts were designed respectively, assembled both parts by connecting them together afterwards. The purpose of this chapter is to find methods which improve controllability of gel robots by reducing structural flexibility thorough geometric variation. The gels with wave-shaped surfaces were materialized. The effect of thickness, cutting and width through experiments is discussed. Firstly, a method for manufacturing gels into desired form is described.

6.2

Development of Gel Manufacturing System

The gel utilized in this book is made by radical polymerization [61]. Polymerization of the gel is carried out in a hot water of 323[K] for 24 hours. For the purpose of obtaining the desired shaped gel, the monomer solution was put into the mold in desired shape and polymerize. The manufacturing method which we developed is described, focusing on how to make mold in desired shape, which is a key process. The procedure is divided into five steps, first three steps are data generation and conversions, the forth step is milling the mold and the last one is

0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 0

5

10

15

20

25

30

35

40

45

Fig. 6.1. Numeric data of the surface curve of the mold

50

6.2 Development of Gel Manufacturing System

119

Fig. 6.2. VRML 3D data of the surface curve of the mold

Fig. 6.3. 3D Data of the mold in DXF file format

polymerization. Please remind that the cross sections of the surface data are also used for simulations. Step 1. Initially, generate numerical surface data of cross section. Here the data is an array of numbers. (Figure 6.1) Step 2. By extruding the cross-section data in one direction, Three dimensional surface data is obtained. The data is defined in VRML format. (Figure 6.2) Step 3. Add the frames and marks to 3D surface data. Save data in DXF format to be used for milling the mold in the next step(Figure 6.3).

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6 Shape Design through Geometric Variation

Fig. 6.4. Mold for making gels with wave-shaped surfaces

Fig. 6.5. Gels with wave-shaped surfaces

Step 4. Cut the mold with NC machine. (Figure 6.4) Step 5. Put the monomers into the mold and polymerize in a hot water. Obtained gel is shown in Figure 6.5. It is important to peel off the gel carefully from the mold. The gel is fragile and sticks to the mold after polymerization. In this case, the mold was made by cutting, using NC machining tool. Radius of the ditch needs to be larger than the radius of the milling cutter, 0.5[mm]. For the purpose of meeting the above requirement, the edge of circle was cut whose

6.2 Development of Gel Manufacturing System

Ball End Mill

r=0.5

Workpiece

hconcave hconvex

hconcave Fig. 6.6. The radius of machine tool and shape of the mold

Gel

Fig. 6.7. Mold which makes gel whose thickness varies spatially

Fig. 6.8. Gel whose thickness varies spatially

121

122

6 Shape Design through Geometric Variation

Gel

Fig. 6.9. Mold which makes gels with wave-shaped surfaces

Fig. 6.10. Gels with wave-shaped surfaces

radius is larger than 0.5[mm] and connected on the line, which is illustrated in Figure 6.6. We need to make the mold smaller than the desired size, because polymer gels swell in the solution. The gel becomes 1.4 times larger than the mold size. The combination of mold and obtained gels are shown from Figure 6.7 to 6.10. Thickness distributed gels and its mold are shown in Figure 6.7 and 6.8. Gels with wave-shaped surfaces and its mold are shown in Figure 6.9 and 6.10.

6.3

Gels with Cutting

In the previous chapter, starfish-shaped gel robot was prototyped. In this case, the gel was unconsciously cut. Let us reconsider the effect of cutting by comparing deformation response to square-shaped gel and cross-shaped gel. The shape of the gel was changed from beam to square plate. Then problem arose that the deformation response became uncertain. This problem was solved by shape design. First, the experimental setup, problem statement and solution are described. Then we discuss the meaning of the result.

6.3 Gels with Cutting

123

(a)

(b)

Fig. 6.11. Obtained form through shape design: (a) Square-shaped gel, whose height of the center was 7.1[mm]. (b) Cross-shaped gel, whose height of the center was 5.3[mm]. Electric field was applied to matrix of electrodes above the gel for 20[s], respectively. A scale bar is 5[mm].

6.3.1

Experimental Setup

Matrix of electrodes, which was developed in the previous chapter, have total of 16 electrodes with four rows and four columns. The size of each electrode is 10[mm] square. The space between each electrode is 5[mm]. The space between the ground and the matrix is 15[mm]. For description, each matrix of electrodes Ei,j is named by row number i, and column number j. Experiments were conducted with this setup. For first experiment, the square-shaped gel was placed below the electrode E3,2 . The size of the gel is 16[mm] square. The voltage at -10[V] was applied to the electrode above the center of the gel and 10[V] to the other electrodes. 6.3.2

Deformation Response of Square-Shaped Gel

The square-shaped gel deformed into round arch shaped along arbitrary direction, but the direction was unpredictable beforehand (Figure 6.11 (a)). The reason is that the material properties of the gels are unequal because of their heterogeneous polymerization. In the case of square-shaped gel, the small difference of the material properties is amplified into large difference of deformation response, because all units are connected. 6.3.3

Deformation Response of Cross-Shaped Gel

For the purpose of constraining the propagation, the gel was cut into crossshaped. Cross-shaped gel has four tentacles with 6[mm] in length and 4[mm] in

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6 Shape Design through Geometric Variation

uniform width. The deformation should occur along the longitudinal direction mainly. The idea was examined thorough experiment by placing the cross-shaped gel at the same position and applying the same electric field to the gel as the preceding experiment. The gel deformed into dome shaped along longitudinal direction, which is shown in Figure 6.11 (b). They showed stable deformation although the speeds scatter because of the scatter of characteristics. The result suggests that we can bring out different function from the same material by shape design. Once the gel is cut into cross-shaped, the role of each part branches: center of the gel becomes trunk of the body; the tips become tentacles. This helps to generate organized motion.

6.4 6.4.1

Gels with Wave-Shaped Surfaces Design Process

Deformation of the gel is caused by surface stress, which is generated by adsorbed molecules. This means that the proportion of surface area by volume effects the overall deformation of the gel. In a uniform electric field, deformation speed of thin beam-shaped gel is supposed to be faster than that of thick one. This assumption is examined experimentally in chapter 4 and this can be deduced from equation (4.6). Because of the small Young’s modulus, a small difference in thickness causes a large difference in deformation response. It was found that we can design gel robots so that thin parts work as actuators and thick parts work as structures. From this fact, we can predict: Conjecture 1. By distributing thick parts and thin parts in the same structure, we will partially reduce the structural flexibility from infinite to finite and at the same time increase the controllability of the gel. Conjecture 2. If we make the gel with wave-shaped surface, it will deform along the ditch. Conjecture 3. The difference of the direction of the ditch on either side of the gels will cause bi-directional deformation. For the purpose of confirming these hypotheses, it was compared that simulation and experimental results of deformation response among the gel with smooth surface, the one with wave-shaped surface on only one side, or both sides. 6.4.2

Experimental System

Experimental system is consisting of two parts, simulator and generator of the electrochemical field. Input signal is common to simulator and generator, applied voltage data towards pairs or multiple electrodes. Two-dimensional simulator was implemented for examining deformation response of gels and gel robots based on the previously described model. Whole system works as follows: For initial setup, the arrangement of electrodes was selected and the shape and size of the gel for simulation and experiment. The cross section of the surface data was cut into desired length in order to obtain the shape and size data.

6.4 Gels with Wave-Shaped Surfaces

125

(a) Generator Amplifier D/A Board Voltage Data Electric Field Polymer gel

(b) Simulator

electrode

Monitor

Fig. 6.12. Simulator and generator of the electrochemical field

If we set voltage data toward a set of electrodes, the generator converts the electric signals from digital to analog, amplify and generate the electrochemical field using electrodes in the electorate solution. The simulator simultaneously calculates the distribution of the electrochemical field parameter, current density, and approximates the deformation of the gel. Both of them are displayed in real time. The system is shown in Figure 6.12. The top shows generator of the real electric field (a) while the bottom shows simulator of the system (b). The voltages are controlled by a PC I/O board (RIF-01, Fujitsu Corporation) and amplified by amplifier circuits with a D.C. power supply (PW18-1.8Q, KENWOOD). The transformation of the gel is analyzed by a video microscope (VH7000, Keyence). Pairs of parallel electrodes are used for the purpose of generating a uniform electric field for fixed-ended gels. The distance between two electrodes is 30[mm]. Multiple electrodes are used in order to generate a spatially varying electric field to deform gels into concave shapes or convex shapes for free-ended gels. In the latter case, the electrodes were arranged to form a matrix of 3 rows and 3 columns in the same plane. There are a total of nine electrodes. The size of each electrode dele is 10[mm] square. The space between each electrodes dspace is 5[mm] (Figure 6.13). The gel is located under the array of electrodes, with the space between the gel and the array l being 10[mm].

126

6 Shape Design through Geometric Variation

dele

electrode dspace

l

gel Fig. 6.13. Multiple electrodes in a matrix configuration

6.4.3

Shapes and Sizes of the Gels for Experiment

Four different shapes of the gel were prepared, whose surfaces are different on either face side or reverse side. After polymerization, they were cut into pieces, which were to be used in the simulations and experiments. The illustrated shapes of each gel are in Figure 6.14. The top half of each picture is the 3D schematic models of the gel. The bottom half of each picture is side view when it is observed from the direction of each arrow on the top half. (a) Both sides have planar surfaces. The thickness of the gel is uniform h= 0.8[mm]. (b) One side has a planar surface and the other side has a wave-shaped surface. The average thickness is the same as (a), 0.8[mm] and the distance between the ditch and the average plane h(d) is 0.2[mm].The distance between the ridge and the average plane h(r) is also 0.2 [mm]. Radius of cross section of the ditch r(d) is 0.8[mm], and that of ridge r(r) is 2.1[mm]. (c) Both sides have wave-shaped surfaces. The ditch of the face side and the reverse side runs orthogonal to each other. The average thickness h, the distance between the ridge and the average plane h(r), the distance between the ditch and the average plane h(d), a set of radius of cross section of the ditch(r(d)) and ridge(r(r)) are the same as gel (b). (d) Both sides have wave-shaped surfaces. The ditch of the face side and the reverse side runs parallel to each other. The average thickness h, distance between the ridge and the average plane h(r), the distance between the ditch and the average plane h(d) , a set of radius of cross section of the ditch(r(d)) and ridge(r(r)) are the same as gels (b) and (c). Above four specimen are 13[mm] square (size of one side b = 13[mm]) and the numbers of ditch is four and the numbers of ridge is five. From the same mold of gel (b), the beam-shaped gel (b1) was cut vertical to the ditches whose length is 15[mm]. Square-shaped gels with different surfaces were used for experiment and beam-shaped gel was used for experiments and two dimensional simulations.

6.4 Gels with Wave-Shaped Surfaces

(a)

127

z y x z h x

b

(b)

h(d)

h(r)

h b (c)

r(r) r(d)

(d)

r(r) r(d) Fig. 6.14. Shapes and size of the gels with wave-shaped surfaces

6.4.4

Experimental Procedure

With this system, experiments were performed in three steps. First, it was simulated that the deformation of beam-shaped gel (b1) with wave-shaped on the face side and plane surface on the reverse side. A uniform electric field was applied to them by pairs of electrodes. The voltage between a pair of electrodes is 10[V]. The period for applying the electric field is 20[s]. Two cases were examined. In

128

6 Shape Design through Geometric Variation

db1c t=10[s] t=0[s] gel (d) electrode

10[mm] Fig. 6.15. Deformation of the thickness distributed gel

one case, the wave-shaped surface faces to the anode side (b1a). In the other case, the wave-shaped surface faces to the cathode side (b1c). The deformations were compared between these two cases. Evaluation standard is the distance between the fixed-end and the free-end. This is because the distance between the fixed-end and the free-end supposed to become smaller if the beam-shaped gel shows large deformation. Second, based on the basic experiment and simulation, spatially varying electric field was applied generated by a matrix of electrodes to three types of square-shaped gels. Square-shaped gels were placed just below the center cathode electrode so as to be deformed it into a convex shape. It was applied -10[V] to the electrode right above the gel and 0[V] to other 8 electrodes. Then, it was observed that the height of the center of the gel from the bottom h, and at the same time, the curvature of the gel 1/r. We can see what directions the gels are bending along by comparing the curvature at the same height. Every experiments were carried out several times and nearest of the average data are selected because the results are sensitive to the initial shape of the specimen. Finally, -10[V] and 10[V] were applied alternatively right above the (c) and (d) type square-shaped gels, and observed the deformations, which were estimated to show bi-directional motions. 6.4.5

Fixed-Ended Gel with Wave-Shaped Surfaces

At first, the beam-shaped gels are examined and compared the deformation response with different surfaces (Figure 6.15). The distance between the root and the tip of the gel was measured after applying the uniform electric field. It was

6.4 Gels with Wave-Shaped Surfaces

129

7 6 height[mm]

5 4 3 (b) sample1 (b) sample2 (b) sample3 (b) sample4

2 1 0 0

2

4

6 8 time[min]

10

12

Fig. 6.16. Transition of the center height of gel(b)

defined that the distance between the root and tip of the gel of (b1a) as db1a , that of (b1c) as db1c . Simulated results were: (db1a ,db1c )=(13.4, 13.1). Experimental results were: (db1a ,db1c )=(14.2, 13.4). In both cases db1a > db1c . If the curvature is large, the distance is small. We can say that the gel with wave-shaped surface on extending side (b1c) deforms larger than the one with wave-shaped surface on shrinking side (b1a). This result implies that square-shaped gel with ditches on one side deforms along the ditches. Then, what happens to the square-shaped gel with ditches on both sides; the surface side goes along y-axis, the reverse side goes along x-axis? We can hypothesize from the result that it will become concave shape along x-axis and convex shape along y-axis. 6.4.6

Free-Ended Gel with Wave-Shaped Surfaces

For the purpose of verifying the assumption derived from the basic experiment, the electric field was applied to the square-shaped gels with various surfaces. Figure 6.16 illustrates the transition of the (b1) gels’ center height. It shows the speed of overall deformation of the gel. Their speeds are uneven because the material property of the gel is also uneven. But the final height is almost the same, almost the half size of their sides. In finite state, they fold up into doubles. These phenomena happen not only on the gel (b) but (a) and (d). According to these results, the curvatures of the surface at the same height were compared. The large curvature implies the large deformation. The graphs are shown in Figure 6.17. The results of gel (b), (c), and (d) are shown, because the deformation of the gel (a) was unstable and the curvature of the same height cannot be predicted even if we applied the same electric field. Curvatures are illustrated after the heights of the center become more than 2[mm], because the observational error of the curvature is large when the height of the center is small. Figure 6.18 shows final shape of the gel (c).

6 Shape Design through Geometric Variation

curvature[1/mm]

130

0.5

gel(b) x-axis gel(b) y-axis

0.4 0.3 0.2 0.1 0

curvature[1/mm]

2 0.5

4 6 8 height[mm] gel(c) x-axis gel(c) y-axis

0.4 0.3 0.2 0.1 0

curvature[1/mm]

2

4 6 8 height[mm] gel(d) x-axis gel(d) y-axis

0.5 0.4 0.3 0.2 0.1 0 2

4 6 8 height[mm]

Fig. 6.17. Curvature vs. central height of the gels (b),(c), and (d)

Different from gel (a), gel (b) bent along the ditches. The gel (c) bent along y-axis, the same direction of the ditch on extending surface. The gel (d) also bent along y-axis, but the difference of two curvatures of each direction was larger than that of gel(c). In this way, the ideas were checked, of shape design improved controllability (Conjecture 1), and the gel with wave-shaped surface bend along

6.4 Gels with Wave-Shaped Surfaces

131

(a)

z y x

(b)

r z x

(c)

h

z y

Fig. 6.18. Photo of curvature vs. central height of the gel (c): (a)perspective view, (b) front view, (c) side view

the ditch (Conjecture 2). From the latter result, it was presupposed that if we apply the reverse electric field to the gel (c) and (d), especially the gel (d) became concave shape along the x-axis. It was ascertained that the gel (d) bends

132

6 Shape Design through Geometric Variation

(a)

(b)

(c)

(d)

(e)

Fig. 6.19. Deformation of the matrix shaped gel(d). The scale bar is 5[mm].

6.5 Gels with Various Widths

133

as expected. It showed bi-directional motions (Figure 6.19)(Conjecture 3). When the polarity of the electrode above the gel is anionic, the gel becomes convex shape along y-axis. When the polarity of the electrode is reversed, cationic, the gel becomes concave shape along x-axis. We could find the inclination in the gel (c), but the difference was smaller than that of gel (b). 6.4.7

Effect of Varying Moment of Inertia

Preliminary two dimensional simulation and experimental results which compare the deformation of gel (b1a) and (b1c) only support the deformation response of gel (c) bending along the directions of extending surface. The results of bidirectional motions were obtained experimentally. For the purpose of describing the mechanism of directional motions we observed, we need to consider the moment of inertia area I of each cross sections. This is because once the squareshaped gel bends, the I of the cross section which is vertical to the axis of bending becomes larger. It is estimated by simple calculation of I of the gel (a) with uniform thickness. At first, the cross section is rectangle, width b = 15[mm] and thickness h = 0.8[mm]. After applying the electric field, the specimen became convex shape. Final shape is approximated as an arc to be simple, whose central angle is π. Then the radius of its curvature r approximately expressed as b/π because the length of the specimen along the side doesn’t change. From the formula of the moment of inertia area, the moment of inertia area at initial state Iinit and the final state If inal are: Iinit =

1 3 bh , 12

If inal 0.3hr3 =

0.3 3 b h(if r h) π3

(6.1)

Comparing Iinit and If inal , If inal is far larger than Iinit , which means the structural stiffness of this cross section increases as the deformation progresses. By substituting the numbers of this gel(a), the increase of I becomes obvious; Iinit 0.55 and If inal 17.

6.5

Gels with Various Widths

From the experimental results of gels with wave-shaped surfaces, it was compared that deformation response of gels with different width. Narrow gel should be possible to approximate as two-dimensional mechanism. In contrast, it should be difficult to ignore the stiffness change, which occurs by deformation along width direction for wide gel. 6.5.1

Experimental Setup

The array of electrodes system was utilized developed in chapter 5. The beamshaped gels were placed in the same manner as the previous chapter. The voltage at -10[V] was applied to three electrodes above the gel and was kept 10[V] for rest of the electrodes. The distance between the gel and the electrodes was 15[mm]. The width of each gel was 4, 6, 8, and 10[mm] respectively.

134

6 Shape Design through Geometric Variation

(a)

(b)

(c)

(d)

Fig. 6.20. Deformation responses into convex shape whose height in the center are maximum : The width of each gel is (a) 4[mm], (b) 6[mm], (c) 8[mm], (d) 10[mm]. The required times to reach the maximum height are (a) 70[s], (b) 50[s], (c) 35[s], (d) 25[s] respectively. The scale bar is 5[mm].

6.5.2

Deformation Response of Width Varying Gels

Experimental results are shown in Figure 6.20. Narrow gels deformed into hairpin-shaped while wide gels deformed into dome-shaped. After they reached the maximum height, they deformed back to convex shapes with lower height. The difference was speed for reaching the maximum curvature of the center of the gel. The result is reasonable considering the moment of inertia change caused by deformation along width direction.

Part III

Control

7 Polarity Reversal Method for Shape Control

7.1

Introduction

This chapter proposes a foundation to control shape of deformable machines consisting of actively deformable materials. If the problem is stated in inverse kinematics form, the required method is to control joint angles, in this case, curvatures, of arbitrary points to fulfill some condition. It is difficult to control every part of the body directly since the numbers of input is smaller than degrees of freedom of the machines. Part III challenges control problem of gel robots. The problem was arranged into shape control problem and motion control problem and discuss each of them in chapter 7 and chapter 8 respectively. In this chapter, a method to evolve simple shape of the gels into complex shape is studied. The experimental system goes back to the original one before part II temporally, to make the problem simple and to focus on deformation process itself. I apply spatially uniform electric fields to the beam-shaped gel. Only the polarity of the electric fields is altered. Space-time respective control As a beginning of part III, an overall strategy is stated to solve control problem of deformable machine. Fundamental problem is that number of input is smaller 

This chapter was adapted from in part, by permission, M. Otake, Y. Kagami, Y. Kuniyoshi, M. Inaba, and H. Inoue, “Inverse Kinematics of Gel Robots made of ElectroActive Polymer Gel”, Proceedings of IEEE International Conference on Robotics and Automation, pp.3224–3229, 2002; M. Otake, Y. Nakamura, and H. Inoue, “Pattern Formation Theory for Electroactive Polymer Gel Robots”, Proceedings of IEEE International Conference on Robotics and Automation, pp.2782–2787, 2004; M. Otake, Y. Nakamura, M. Inaba, and H. Inoue. Wave-shape Pattern Control of Electroactive Polymer Gel Robots. Proceedings of the 9th International Symposium on Experimental Robotics, ID178, 2004.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 137–164. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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7 Polarity Reversal Method for Shape Control

than degrees of freedom of the body. The strategy of living things is referred that also have deformable body. One strategy is to accumulate the series of input temporally to reach the desired form. Living system takes time to transit the state of the body step by step. This phenomenon is observed in the early stage of life, the developmental process called morphogenesis. Originally uniform structure is converted to complex organ. Degrees of freedom of the body are utilized as maximum. It is applicable to shape control of deformable machines, when the objective is to reach the final form and transitional form is not in question. Another strategy is to couple each part of the body to generate dynamic motion. Animal muscles work cooperatively to achieve objective motion. This is ubiquitous especially for animals with many degrees of freedom like an octopus. In Synergetics, many natural systems are composed of combination of a large number of degrees of freedom and constraints. Constraints are temporally applied to the body in order to generate organized motion. As a result, coupled parts of the body are driven in parallel. Selected degrees of freedom are activated and others are inhibited. It is applicable to motion control of deformable machines, when the objective is the overall transition of the overall state of the body and final form is not in question. Now we can see that shape control and motion control make remarkably contrast, which is summarized in table 7.1. In this study, morphogenetic approach is taken to solve shape control problem and synergetic approach for motion control problem. Table 7.1. Control strategy for deformable machines learning from life control approach stage usage of DOF complement method driving protocol

shape morphogenetic prenatal consumptive input integrated serial

motion synergetic postnatal selective lumped driven parallel

Experimental Setup A simulator was implemented based on the model. The control method is described with simulation results. The simulator setup is shown in Figure 7.1. The length of the gel manipulator is 12[mm] and a pairs of electrodes is placed with 30[mm] spacing which generates spatially uniform electric fields. Draw a horizontal line (x-axis) from the root of the gel, a vertical line (y-axis) from the tip of the gel, intersecting at the point O(the origin). The electrodes are placed at y=15[mm] and -15[mm]. The length of the link is |v⊥ [j]|=1[mm], time step is Δt=1[s]. Although we can apply voltage continuously, discrete voltage was applied to make the discussion simple. The voltage was kept at 5[V] to move the tip to positive direction along y-axis, and which was kept at -5[V] to move it toward negative direction, and was kept at 0[V] to stop it.

7.2 Tip Position Control of Gel Manipulator

V1[V]

139

15[mm]

y-axis Gel

x-axis

O 12[mm] Pairs of Electrodes 0[V]

-15[mm]

Fig. 7.1. Structure of the gel and electric field setup

7.2 7.2.1

Tip Position Control of Gel Manipulator Inverse Kinematics

The purpose of this section is to propose an inverse kinematics method which could ultimately be used to control the shape of the gel robot such as tentacle control of octopus-shaped gel robot which was prototyped (Figure 7.2). It was focused on tip (end-effector) position control of a gel manipulator as a first step (Figure 7.3). This will form a foundation for shape control of gel robots. One of the primary difficulties is that the number of degrees of freedom is larger than the number of inputs, since deformable materials have conceptually infinite degrees of freedom. One approach applies spatially varying electric fields generated by multiple electrodes. Another approach, which is proposed in this

Fig. 7.2. Prototype of octopus-shaped gel robot

140

7 Polarity Reversal Method for Shape Control

Fig. 7.3. Gel manipulator

t=15,45[s]

6

y-axis [mm]

4 2 t=0[s] 0

-2 -4 t=30[s]

-6 0

2

4 6 8 x-axis [mm]

10

12

Fig. 7.4. Tip path and final shape of gel manipulator: amplitude of 6 [mm]

section, is to apply alternating electric fields. In order to simplify the problem, it was applied that alternating but spatially uniform electric fields generated by parallel electrodes. We demonstrate the application of this method to move the tip of a gel manipulator to a desired position. In such a constrained system, we cannot measure distance between the current state and goal state by Euclidian measure. Therefore, randomized approach[190] is often taken, especially for motion planning of deformable objects[191]. The conceptual joints of the gel manipulator are constrained and calculated with equation (2.23) and equation (2.26). The above equations are typical

7.2 Tip Position Control of Gel Manipulator

141

voltage

6 4 2 0 -2 -4 -6

5

position -5 0

5

10

15

20 25 time [s]

30

35

40

45

applied voltage [V]

tip position y [mm]

Fig. 7.5. Trajectory of curvature of gel manipulator: amplitude of 6 [mm]

Fig. 7.6. Applied voltage and tip position of y

equations of nonholonomic system, which are called the input-linear or affine systems described as follows[192] x(t) ˙ = f (x(t)) +

m 

gi (x(t))ui (t),

(7.1)

i=1

together with some output equation only depending on the state. Nonholonomic systems are typically controllable in a configuration space of higher dimension than that of the input space[193]. Typical nonholonomic mechanical systems are; rolling contact between objects[194], manipulation by pushing[195],

142

7 Polarity Reversal Method for Shape Control

space robots[196]. Free-joint manipulators are controllable by small numbers of motors[197]. We can assume that the tip of the gel manipulators is reachable not only on the narrow arc. Here, a method to control positions of the tip of gel manipulators through applying oscillatory input considering dynamic property of the system. In general, the gel manipulator swings repeatedly if the polarity of an electric field is altered repeatedly. If the amplitude of the swinging motion is symmetric and stable, the tip of the gel moves on the same path. The tip path of the gel manipulator swings between y=6[mm] and -6[mm], and shape at t=0,15,30,45[s], are shown in Figure 7.4. Trajectory of curvature at the root θ1 , center θ2 and tip θ3 is plotted in Figure 7.5. The applied voltage and the y-coordinate of the tip is shown in Figure 7.6. The polarity of the voltage altered when the tip reached y=6[mm] and -6[mm]. In this chapter, the method is shown to move the tip of the gel from the path to the desired position. Controlling either x or y-coordinate is relatively easy. We need to apply the electric field until the x or y-coordinate of the tip reach the desired position. The coupled deformation of the gel makes it difficult to control both x and y-coordinate at the same time. We designed a procedure based on mechanism in order to change the x and y-coordinate of the tip independently. The strategy is to utilize the nonlinearity of equation (2.23). The final position of the tip depends on the path of each link, because the joint angle is the function of position r[j] and orientation |v⊥ [j]|. We try to change the path of each link by applying timely alternating electric fields. The deformation response of the real gel varies to the same input signal because the properties of the gels are not uniform. The tip position is monitored and the polarity of input electric field is switched based on its position, for reducing the effect of this scatter. 7.2.2

Dynamic Change of Configuration of the Gel

First of all, we describe the way of changing the path of the gel manipulator dynamically. If we keep on applying the electric field, the gel manipulator bends towards the electrodes and goes along the electric field. Once the orientation of the link is perpendicular toward the electrode, the dissociation speed exceeds the adsorption speed of molecules. This make the joint angles of the tip to becomes larger compared to those of the root, which make the x-coordinate of the tip becomes larger while y-coordinate of the tip remains constant (Figure 7.7, t=60 to 100[s]). After keeping the y-coordinate of the tip toward one side, we apply -5[V] until the tip goes back to the original position (Figure 7.7, t=130[s]). The gel becomes convex-shaped which make the x-coordinate of the tip becomes smaller, x=11.3[mm]. Path of the tip and initial (t=0[s]), transitional (t=100[s]) and final (t=130[s]) shape of the gel are illustrated in Figure 7.7. Trajectory of curvature at the root θ1 , center θ2 and tip θ3 is plotted in Figure 7.8. The applied voltage and the y-coordinate of the tip is shown in Figure 7.9. The initial applied voltage was 5[V] until the tip reached y=11[mm], and its polarity was reversed to -5[V] until the tip reached y=0[mm].

7.2 Tip Position Control of Gel Manipulator

12

t= 100[s]

10

t=110[s]

t= 60[s]

8 y-axis [mm]

143

t= 30[s]

6 4 2 t= 0[s] t=130[s]

0 0

2

4 6 8 x-axis [mm]

10

12

Fig. 7.7. Macro position control by large deformation

Fig. 7.8. Curvature trajectory while in large deformation

In this way, we can move the path of the tip by dynamically changing the configuration of the gel manipulator based on deformation mechanism. This trajectory change occurs when the tip of the gel exceeds π/2. This phenomenon

7 Polarity Reversal Method for Shape Control

position

10 5

5

voltage

0 -5 0

-5 20

40

60 80 time [s]

100

120

applied voltage [V]

tip position y [mm]

144

Fig. 7.9. Applied voltage and tip position of y

10 x3

xd x1

yd

x2

y-axis [mm]

5

0

-5

-10 0

5 10 x-axis [mm]

Fig. 7.10. Path of the gel manipulator

is regarded as bifurcation from dynamical systems perspective[178]. Figure 7.10 show the path of the tip which moves between 10[mm] to -10[mm] (dotted line), and 11[mm] to -11[mm] (continuous line). The tip moves on the dotted line for the first time and transit to the continuous line. But still, the x and ycoordinate of the gel is coupled although the combination is different from the original ones. We would like to slightly change the x-coordinate of the tip while maintaining the y-coordinate of the tip. We will describe the method in the next subsection.

7.2 Tip Position Control of Gel Manipulator

145

12 10

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Fig. 7.11. Micro position adjustment by oscillation

7.2.3

Slight Change of Configuration of the Gel

Now that we can change the path dynamically, we would like to adjust the position. By applying oscillating electric field with small amplitude, we can slightly change the configuration. The basic idea is that we need to curve or straighten the overall shape of the gel to adjust tip position. The x-coordinate of the gel at the same y-coordinate becomes relatively smaller if the gel manipulator curves. The method to straighten the curved gel is: • Move the tip of the gel by the electric field to the center of the original position so that the orientation of the gel becomes vertical to the electric field. • Apply oscillatory electric field at that position. • The gel straighten because the molecules adsorb and desorb uniformly along the longitude of the gel. The method to curve the straight gel is: • Move the tip of the gel by the electric field to the anode electrode so that the orientation of the gel becomes parallel with the electric field. • Apply oscillatory electric field at that position. • The mechanism of slight change is almost the same as dynamic change, the desorption of molecules near the tip.

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Fig. 7.12. Curvature trajectory of micro position adjustment by oscillation

We will show examples respectively. For the first case, we move the tip of the gel to y=11[mm] and move back to y=0[mm]. Then apply -5[V] and 5[V] repeatedly for 20 times so that the gel oscillates between y=-1[mm] and 0[mm]. The initial shape and final shape, and the path of the gel are shown in Figure 7.11. The tip position is (11.6, 0) whose x-coordinate became larger. For another case, we start with straight one. Move the tip to y=8[mm] and oscillates between y=8[mm] and 9[mm] for 20 times and move back to 0[mm]. The results are shown in Figure 7.13. The tip position is (11.9, 0). The gel deforms into convexshaped, whose joint angles are smaller compared to the one which is illustrated in Figure 7.7. 7.2.4

Selection of the Path to Reach the Desired Position

There are three ways to reach the tip to the desired position (xd , yd ). The cross section of the path and the line y = yd is ri = (xi , yd )(i = 1, 2, 3) (Figure 7.10). We need to select one of the positions and adjust the gel to the desired position. We name the controller function ui (i = 1, 2, 3) which corresponds to ri (i = 1, 2, 3). Then we adjust the tip by oscillatory motion. In all cases, the controller estimate the x-coordinate at y = yd in each period of oscillation and if the condition is satisfied, oscillation will be stopped and the tip will move to the goal.

7.2 Tip Position Control of Gel Manipulator

147

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Fig. 7.13. Micro position adjustment by oscillation II

We used total time to reach the goal for cost. We named the cost of controller function ui (i = 1, 2, 3) for Ji (i = 1, 2, 3) and calculate Jmin = min[J1 , J2 , J3 ]

(7.2)

Controller function ui was selected that corresponds to Jmin . 7.2.5

Tip Position Control of Gel Manipulators

Experiment 7.1. Tip position control of gel manipulators Experimental methods Experiments were performed to examine the method. It is shown that the simulation and experimental results of the final shape and path to reach the goal position (9.5, 6.0). The criterion function was calculated that Jmin = J1 =416. Experimental results The path of the tip and final shape of the simulation are illustrated in Figure 7.16, and final shape real gel are shown in Figure 7.17. Repeated time of oscillation is 10 times. 7.2.6

Summary

In this section, it was shown that a method to solve inverse kinematics of gel robots made of electro-active polymer gel. As a first step, a method was proposed to control the tip position of a manipulator entirely made of electro-active

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Fig. 7.14. Curvature trajectory of micro position adjustment by oscillation II

polymer gel. It was made clear that the various shapes of the gels can be derived by applying time alternating electric fields even if those are spatially uniform. The problem is we cannot control the gel manipulator directly, because the electro-active polymer gels which we use in this book are driven by separated electrodes and their motions are coupled. The constitutive equation was examined and the method was proposed of changing the configuration of joint angles by altering the orientation of each link to the spatially uniform electric field. The interesting point is that the tip of the gel manipulator can reach finite area in the electric field, not only on the narrow arc. First, the dynamic change of configuration of the gel manipulator was demonstrated when almost all links goes with the electric field. Then, the slight change of configuration with oscillating electric field was presented. Finally, by combining the above methods which generate macro-micro motion, the tip position of the gel manipulator was controlled. Trajectory of the curvature corresponded to series of input. Each input generated particular trajectory. This corresponding structure is utilized for motion control in chapter 8. The method generates subset of possible workspace and moves the tip of the gel in the plane. From the viewpoint of problem solving algorithm, it is evaluated that this inverse kinematic model as practical approach; efficiency is in high

7.3 Wave-Shape Pattern Formation of Electroactive Polymer Gel

149

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Fig. 7.15. Path of the tip and final shape of the gel

priority and optimality and completeness are neglected. This result will be one of the fundamental examples to control deformable robot made of electro-active polymer gel. As a result of tip position control, a variety of shapes was derived. Before moving on to shape control, spontaneous pattern formation was studied. It was found them in simulation first, and discovered experimentally afterward.

7.3 7.3.1

Wave-Shape Pattern Formation of Electroactive Polymer Gel Application of the Constant Uniform Electric Fields to the Gel

In this section, we report and discuss wave-shape pattern formation of surfactantdriven ionic polymer gel in constant electric fields. Experiment 7.2. Wave-shape pattern formation of the gel Experimental methods The experimental setup included a pair of parallel platinum plate electrodes of 25 [mm] wide and 40 [mm] long for each, which were horizontally placed with

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Fig. 7.16. Curvature trajectory of the gel

Fig. 7.17. The gel manipulator which reached the final goal

40 [mm] vertical spacing between them. A beam-shaped gel of 4 [mm] wide, 21 [mm] long, and 1[mm] thick was also horizontally placed in-between with one end fixed for 5 [mm] and the other end free. The vertical section of the experimental setup is seen in Fig. 2.2. The two electrodes and the gel were immersed in the solution. A uniform electric field was applied by the electrodes. The current density was kept constant by a galvanostat at 0.15 [mA/mm2 ] for 600 [s]. The deformation of the gel was monitored and recorded by a video microscope for analysis. The fixture size was set small so as not to disturb the electric field. As shown in Fig.2.2, the x-axis was chosen as the horizontal line going through the fixed end of the gel, while the y-axis was the vertical one also going through

7.3 Wave-Shape Pattern Formation of Electroactive Polymer Gel

151

the fixed end. The electrodes were placed at y=±20 [mm]. Let φ be an angle between the tangential line of the gel at the free end and the x-axis. Experimental results With the constant electric field, the gel showed waving motion and a wave shape was eventually generated after a while. First the gel bent toward the anode side. When φ went over π/2, a portion of gel near the free end started to bend in the other direction. The deformation of root portion remained same. Again, when φ went under π/2, a smaller portion of the gel near the free end started to bend the first direction. Likewise the free end of the gel showed an oscillation. Fig. 7.18 shows the angle of the tip of the gel φ during motion. In the uniform electric field, the beam-shaped gel bent toward the anode side. The tip of the gel showed counterclockwise motion in the x-y plane from the initial point (16, 0) [mm]. Angle φ increased from 0 and reached the local maximum of φ1 =2.11[rad] at t1 = 50[s]. It didn’t stop when the tip became vertical to the electrode and parallel to the electric field. Then the direction of the movement of the tip reversed. The angle decreased to the local minimum of φ2 = 1.02[rad] at t2 = 210[s]. The direction of motion of the tip again reversed to increase φ. The angle of the tip became the local maximum of φ3 = 1.80[rad] at t3 = 420[s], after that it gradually decreased in the course of experiment. The transitional shapes of the gel are shown in Fig. 7.19. Fig. 7.19(a) represents the shape of the gel at t1 when the direction of the tip reversed for the first time. The second extremum of the gel near the tip appeared after t1 . The second extremum grew until the direction of the tip movement reversed in t2 whose shape are shown in Fig. 7.19(b). After that, the third extremum appeared. When the tip of the gel reversed at t3 , the gel had three extrema (Fig. 7.19(c)). The maximum curvature decreased as the numbers of extrema increased. After the third reversing point of the tip, it became difficult to count the numbers of extrema. This is because the curvature became small and exceeded the maximum accuracy of measurement. The whole shape of the gel became mostly vertical to the electrode. In summary, waveshape pattern formation of the gel observed. The waving rhythm generation was accompanied it. 7.3.2

Simulation of Wave-Shape Pattern Formation

Numerical simulation was conducted based on the proposed model and the obtained parameters. The beam-shaped gel was discretized every δl=1 [mm] along the longitudinal direction for numerical integration. We applied the same current density to the gel for 400[s]. The time step for numerical integration was δt=1 [s]. The waving motion of the tip was observed and plotted in Fig. 7.20. The speed was faster than the experimental results. The times of extremum were t1 =43 [s], t2 = 133[s], t3 = 271[s]. The angles of the tip were φ1 =2.13[rad], φ2 =1.31[rad], φ3 =1.68[rad], respectively. The shapes of the gel, which were obtained numerically, are shown in Fig. 7.21. Therefore, the numerical simulations qualitatively confirm the wave-shape pattern formation observed in the experiments.

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orientation[rad]

2 1.6 1.2 0.8 0.4 0

0

200

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600

time[s] Fig. 7.18. The angle of the tip of the beam-shaped gel φ in a spatio-temporally uniform electric field

(a)

(b)

(c)

Fig. 7.19. Deformation of the beam-shaped gel in a spatio-temporally uniform electric field: The scale bar is 5[mm].

7.3.3

Mechanism of Wave-Shape Pattern Formation

We analyze the mechanism of wave-shape pattern formation based on the proposed model and verify it by comparing the numerical computation and the experimental data. The model is represented by a rather simple nonlinear firstorder partial differential equation of equation (2.7). It describes that the effect of the electric field to the gel is determined solely by the geometry of the equipotential surface and the gel surface. We rewrite the equation as follows in order to represent the geometry of the experimental setup.

7.3 Wave-Shape Pattern Formation of Electroactive Polymer Gel

153

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1.6 1.2 0.8 0.4 0

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Fig. 7.20. Simulated angle of the tip of the beam-shaped gel φ in spatio-temporally uniform electric field.

(a)

(b)

(c)

Fig. 7.21. Simulated deformation of the beam-shaped gel in spatio-temporally uniform electric field: The scale bar is 5[mm].

The constant electric field is generated by the pair of parallel electrodes of infinite length. The equipotential surfaces of the electric field run parallel to the electrodes. The current density vector, a gradient of the electric potential, is vertical to the equipotential surface. Since the electrodes were placed parallel to the x-axis, the current density vector is written by: i = (ix , iy ) = ic (0, −1),

(7.3)

with the constant ic = 0.15[mA/mm2 ]. If we take the tangent angle on the gel surface for θ, the normal vector on the surface becomes as follows:

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7 Polarity Reversal Method for Shape Control

n = (nx , ny ) = (− sin θ, cos θ).

(7.4)

We can obtain the specific expression of equation (2.7) by substituting equations (7.3) and (7.4) into it, dα = aic cos θ − dα. (7.5) dt Note that the curvature 1/R is a spatial differentiation along the beam-shaped gel. Equation (2.9) is rewritten by using angle θ and length along the surface r and equation (2.10), 1 3b ∂θ = = α. (7.6) ∂r R Eh By substituting equation (7.6) into equation (7.5) to eliminate α, we have d ∂θ 3abic ∂θ = cos θ − d . dt ∂r Eh ∂r

(7.7)

The second term of the right hand side of equation (7.7) is damping, which originates from the dissociation term in equation (2.7). If the first term of the right hand side of equation (7.7) is constant, the answer of the equation is a simple exponential function, which converses to zero. Inclusion of cosine function to the first-order partial differential equation of equation (7.7) causes rich pattern development under a certain condition. The necessary condition is the beamshaped gel placed parallel to the electrodes with one end fixed. If the beamshaped gel placed vertical to the electrodes initially, pattern doesn’t appear in the electric field. If one end of the gel is not fixed, equation (7.7) is not applicable. Nonlinearity of cosine function, initial and boundary conditions play important roles in the pattern formation. The development of the wave-shape pattern is described as follows. The electric field activates and drives the surfactant molecules, which are adsorbed on the gel and deform it. As the adsorption progresses, the deformation occurs in such a way that the surface normal of the gel approaches parallel to the equipotential surface of the electric field. Fig. 7.22 illustrates the geometry of the gel and the electric field. Horizontal lines are the equipotential surfaces of the electric field. Arrows on the gel surface are normal vectors of the gel. From equation (2.7), the effect of the electric field to the gel disappears when the surface normal of the gel and the equipotential surface of the electric field become parallel (Fig. 7.22(a)). The angle of the tip of the gel φ reaches maximum when the deformation speed near the root and one near the tip balance (Fig. 7.22(b)). The gel deformation works to deactivate the adsorption reaction and causes oscillatory motion. Differences of maximum curvature and deformation speed are seen between the experimental and simulation results. Among several possible reasons for them, we would like to point out the modeling error of higher nonlinearity. We need to consider the effect of electrostatic and hydrophobic interactions between the surfactant molecules and polymer networks[129, 172, 173] for higher precision of simulation. Equation (2.7) was derived assuming that the adsorption rate is small. When the adsorption rate becomes large, the not modeled nonlinearity

7.4 Wave-Shape Pattern Control of Electroactive Polymer Gel

(a)

155

(b)

Fig. 7.22. Mechanism of wave-shape pattern formation

in the equation grows not negligible. We approximated the curvature 1/R being proportional to the adsorption rate α in equation (2.10). The deformation speed slowed down as the adsorption rate became large in the experiments. The gel deformation terminated in a wave-shaped before it reached the theoretical equilibrium of equation (7.7), that is namely a straight line perpendicular to the both electrodes. This is due to the effect of friction neglected for the simplicity of the model. Flexibility of the polymer networks decreases as the density of the polymer networks increases. Stiffness of the polymer network works as friction in the development of wave-shape pattern. The effects of mechanical deformation on the chemical reaction have been discussed for self-oscillating gels[198, 199]. The polymer network in the selfoscillating gel takes part in the BZ reaction, when immersed in an aqueous solution containing the reactants. It is because the catalyst is covalently bonded to the polymer chain. The difference between the self-oscillating gel and our study on wave-shape pattern emergence is that the latter in comparison with the former, has more dependence on the interaction between the chemical reaction and the mechanical response. Adsorption reaction alone would not bring out any pattern without the interaction with mechanical response, which is therefore fundamental to wave-shape pattern formation.

7.4 7.4.1

Wave-Shape Pattern Control of Electroactive Polymer Gel Numerical Simulation for Experiment

Wave-shaped gels with varying curvature were obtained by switching the polarity of a spatially uniform electric field. The period for reversing the polarity was explored through numerical simulation. The polarity of one of the electrodes was either anodic (0) or cathodic (1). A control sequence is described with a time interval and its sequence. A time interval of 10 second was initially selected and its sequence of eight intervals were enumerated from (00000000) to (11111111). Other intervals were also considered every 10 seconds from 20 to 120 seconds.

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7 Polarity Reversal Method for Shape Control

θ1 θ2

3

θ1

y

3

x

θ0

Fig. 7.23. Definition of the objective shapes with tangential angles in the case of three-half-waved shape

We determined (00001111) with 120 seconds time interval as the best input sequence, which generated three-half-waved shape with large curvature. 7.4.2

Performance Function for Objective Forms

The performance function was defined with the tangential angle of the gel for determining input sequence. That of the three-half-waved shape f3 is described by f3 = (θ 23 − θ1 ) + (θ 23 − 13 ) + (θ0 − 13 ) = θ0 − 2θ 31 + 2θ 23 − θ1 ,

(7.8)

where the suffix of the orientation θ is the normalized length from the root to the arbitrary point. The orientation of the root is θ0 , that of the tip is θ1 , that of the center is θ 12 , that of one third from the root is θ 13 , that of two thirds from the root is θ 23 . Performance function of the shape with x half-waves is noted as fx . The objective shape with tangential angles is illustrated in Fig. 7.23. Simulated forms of the gel at every time interval are shown in Fig. 7.28. Different shapes were generated through alternating the performance functions. The performance functions of one-half-waved shape (f1 ) and two-halfwaved shape (f2 ) are defined as same as that of the three-half-waved shape. f 1 = θ0 − θ1

(7.9)

Table 7.2. Combination of time step and symbol array to generate objective shape of the letter letter C S E J L

period[s] 90 90 120 120 105

length[mm] 16 16 16 16 16

symbols 11000011 11111000 00001111 01110001 10000011

7.4 Wave-Shape Pattern Control of Electroactive Polymer Gel

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Fig. 7.24. Simulation of deformation process to form the letter ’C’

f2 = (θ 12 − θ0 ) + (θ 12 − θ1 ) = 2θ 21 − θ0 − θ1

(7.10)

The input sequences which maximize the performance functions were obtained. The performance function of the shape that a half near the root is straight (the number of waves is 0) and another half near the tip is curved (the number of waves is 1) is defined as: f01 = (θ 12 − θ1 ) − (θ 12 − θ0 )2 . The final shape of the gel should resemble to the letter ’J’.

(7.11)

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7 Polarity Reversal Method for Shape Control

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Fig. 7.25. Experimental result of deformation process to form the letter ’C’

Likewise, the performance function of the shape that a half near the root is curved(the number of waves is 1) and another half near the tip is straight(the number of waves is 0) is defined as: f10 = (θ0 − θ 12 ) − (θ 12 − θ1 )2 .

(7.12)

The final shape of the gel should resemble to the letter ’L’. 7.4.3

Pattern Control of Gels with Varying Curvatures

The symbols are selected, which maximize the performance function. The combination of periods, length of the gel and the symbols for each letter is summarized in Table 7.2.

7.4 Wave-Shape Pattern Control of Electroactive Polymer Gel

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Fig. 7.26. Simulation of deformation process to form the letter ’S’

Timing for switching was converted from time to space in order to apply the simulation results to experiments. The polarity of the electric field was switched when the tip of the gel reached the same orientation of reversing period. Experiment 7.3. Wave-shape pattern control of the gel with different numbers of half-waves

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7 Polarity Reversal Method for Shape Control

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Fig. 7.27. Experimental result of deformation process to form the letter ’S’

Experimental Methods Experimental setup was the same as that of wave-shape pattern formation. The current density was kept constant by a galvanostat at 0.10 [mA/mm2 ]. The polarity of the electric field was reversed from anodic (0) to cathodic (1) when the tangential angle at the tip of the gel reached the same values as that of the simulation. Experimental Results Simulation and experimental results are shown from Figure 7.24 to Figure 7.31. Experimental results well confirmed the simulation.

7.5 Pattern Formation in Variety of Gels

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Fig. 7.28. Simulation of deformation process to form the letter ’E’

7.5

Pattern Formation in Variety of Gels

Spontaneous pattern formations are fundamental characteristics of systems under thermodynamically open conditions. Polymer gel of a cross-linked polymer network immersed in liquid shows rich spatiotemporal pattern formations since it is a typical open system exchanging its matter and energy with the surroundings[158]. Gel reversibly swells and shrinks when the external condition, such as temperature or solvent composition, varies[164, 200].

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7 Polarity Reversal Method for Shape Control

(1)

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Fig. 7.29. Experimental result of deformation process to form the letter ’E’

When ionized acrylamide polymer gel undergoes extensive swelling, an extremely fine pattern appears on the surface, and evolves with time[201]. On the other hand, when cylindrical gels of acrylamide shrink, they exhibit bubble pattern and bamboo-like pattern[202]. Under certain conditions, alternate swollen and shrunken portions appear (bubble pattern). In other cases, crosssectional planes made of the collapsed gel membrane, whose thickness comparable to the wavelength of light, appear in the cylinder (bamboo-like pattern). Gel has also played an important role in the study of the Belousov-Zhabotinsky (BZ) reaction[168, 203], which induces spatiotemporal patterns like the Turing pattern[204, 205] and spiral wave[206]. In the reaction, gel works as supporting

7.5 Pattern Formation in Variety of Gels

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Fig. 7.30. Simulation of deformation process to form the letter ’J’

media being inactive to the BZ reagents. In contrast, Yoshida el al. developed a gel exhibiting periodical volume changes without any external stimuli[207, 198] as a result of mechanical response due to the BZ reaction. The polymer gel we utilized has been known for bending mechanism, but has not been investigated from the viewpoint of pattern formation. The patterns generated in this chapter were discovered for the first time. There should be

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7 Polarity Reversal Method for Shape Control

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Fig. 7.31. Simulation of deformation process to form the letter ’L’

other interesting patterns which have not been discovered yet. Other electroactive polymer materials will also bring out interesting patterns because of the nonlinearity of the materials.

8 Lumped-Driven Method for Motion Control

8.1

Introduction

This chapter proposes motion control method for deformable machines consisting of actively deformable materials. If the problem is stated in inverse dynamics form, the required method is to move typical position of the robot on the objective trajectory by deforming its whole body. It was neither the selection of typical points nor objective trajectory was clear for the beginning. In the previous chapter, morphogenetic method was proposed to control shape of gel robots. Their bodies grow from simple to complex forms. Degrees of freedom of the body are utilized at maximum. In this chapter, totally different approach is proposed to utilize deformability of the body. Motion control problem is dealt with referring to the motions of deformable living animals. The turning over motion control problem was selected as a typical example. Learning from a real starfish and other researches, the mechanism is made clear and motion is generated in high reliability. A variety of motions of gel robots is derived by applying either spatially or time varying electric fields in the previous chapters. Both spatially and time alternating electric fields were not yet applied to them. For the purpose of generating turning over motion, both spatially varying time alternating electric fields are applied to them. Typical operators of electric fields are applied and switched to gel robots. This work is the first step towards motion control of gel robots and deformable machines in the future. 

This chapter was adapted from in part, by permission, M. Otake, Y. Kagami, M. Inaba, and H. Inoue, “Motion design of a starfish-shaped gel robot made of electroactive polymer gel”, Robotics and Autonomous Systems, vol. 40, pp. 185–191, 2002; M. Otake, Y. Kagami, Y. Kuniyoshi, M. Inaba, and H. Inoue, “Inverse Dynamics of Gel Robots made of Electroactive Polymer Gel”, Proceedings of IEEE International Conference on Robotics and Automation, pp.2299-2304, 2003.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 165–199. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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8 Lumped-Driven Method for Motion Control

Problem statement Common approach to generate motion is to define series of pose and connect them with dynamically stable trajectory. This approach cannot be applied to gel robots. We cannot predefine series of pose because we cannot control their poses directly. Motion control problem of deformable machine made of actively deformable materials are classified into three, which are summarized below. 1. We cannot control each point directly. 2. A procedure to convert objective behavior to motion control rule is unclear. 3. Whether there exists a solution is unknown. The first problem is a side effect of coupling. Coupling multiple points of the body and move them simultaneously reduce the difficulty of degrees of freedom control, which was originally stated in chapter 5. This method has side effects. Because they are coupled, we cannot control each point individually. The second problem is a general problem. It applies to other kinds of mechanism. There are few motions whose generation mechanisms are clear. Well-known exception is a bipedal locomotion. Zero moment point (ZMP) of the body is regarded as typical control point for motion generation. For other kinds of motion, such point is unknown. The third problem is also common especially for mechanism with constraints. If control points to generate desired motion are found, it is not guaranteed that they can be realized. Reachable area of constrained mechanism is bounded. That of beam-shaped gel uniformly coupled by electric fields was thoroughly studied in chapter 7. Approach to solve motion control problem For the purpose of solving these problems of deformable machines, motion control problem of gel robots was solved. The approach corresponds to the three problems are described. 1. Arrange operators to generate trajectory as invariants. 2. Convert motion generation problem to search problem of state transition. 3. Generate candidate motions considering the dynamical property of the system. Approach to the first problem is based on the fact that we cannot define series of poses, but can define operators to act on the coupled points beforehand. Therefore, operators are regarded as invariants and poses as variable. Poses are generated by a set of initial pose and series of operators. Here, the role of operators and states are reversed. For gel robots system, operators are electric fields defined by typical sets of applied voltages to the electrodes. Instantaneous deformation response of the gel is unique when the following three elements are settled: form of gel, surrounding electric field, and relative position and orientation of the gel and the electric fields. Series of input of electric fields causes

8.2 Design of Electric Operator to the Gels

167

series of deformation, namely motion of gels. Typical operators are selected that should generate typical motions. For the second problem, the problem is converted into somewhat ’chemical’ format. In most chemical systems, quantitative change induces qualitative change at some level. This perspective should be also applicable to motion generation of deformable machines. The hypothesis comes from the similarity of both chemical system and deformable machines that they contain great numbers of components and function as a whole. Needless to say, electroactive polymer gel is both chemical and mechanical material. We describe the overall procedure for conversion of the problem. Focus on qualitative difference of the initial state and final state of the machine before and after certain motion at first. Then, figure out dominant parameter or point to define overall state of the machine. Numbers of candidate positions or parameters are not so many. If the machine has target object to work on, the points to be attended are contacting points. Other positions to be attended are cross section of symmetry axes, fixed-points, or center of motion. Finally, describe the states with selected parameters or points. Look for critical point to divide initial state and final state. Each state is then defined with inequality. In this way, we can convert motion generation into searching critical point. Solution space is defined large at most, which helps to solve the problem. There are numerous movements for solution which meet the purpose of behavior, for example, approaching to some objects. We take computational approach in order to solve the third problem. Candidate motions are generated and solutions are searched to fulfill the condition. This is a common approach to solve the inverse problems if we do not know the direct inverse method to solve them. Essence of the approach is to generate search space with wide variety, and define the solution space as large as possible. For the purpose of avoiding duplication of candidate motions and to keep simplicity of the problem, a set of typical operators for state transition are selected from initial to final state. First operators make the point to approach the critical point. Successive operators make the point to get across the critical point to induce state transition. In the case of gel robot system, we can select operators without difficulty, if we try to move only one point of the whole body. Electric fields act only on the points of gels, which are vertical to the current density vectors with large magnitude. Therefore, we can select operators that act on the target point intensively. After we select operators, the requirement is only looking for switching timing of these. The idea of switching operators is the extension of chapter 7, which contains only two operators, upward or downward electric field.

8.2 8.2.1

Design of Electric Operator to the Gels Electric Fields as Operator

It is a side effect of coupling that we cannot control each point of the body directly. We can reduce the difficulty by considering the local interaction mechanism. Interaction between the gel and the electric field does not occur uniformly.

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4[mm]

2[mm]

Electrodes 20[mm] 15[mm] Gel Fig. 8.1. The array of electrodes

A part of the gel reacts rapidly while most of the gel reacts slowly. This happens by the following reason. A part of the gel deforms rapidly if the absolute value of the current density vector on the point is large and the current density vector is perpendicular to the surface of the point. On the other hand, other parts of the gel deforms slowly if the absolute value of the current density vector is small or the current density is parallel to the surface of the point. Therefore, the effective way to deform a target point rapidly is to generate large current density vector perpendicular the surface of the point. Distribution of the current density vectors is determined by the combination of applied voltages to the electrodes. In this section, the typical set of applied voltages is arranged to the electrodes as electric operator. 8.2.2

2D Operator for Array of Electrodes

Firstly, the array of electrodes which generate two-dimensional electric fields is considered (Figure 8.1). Applied voltages to the electrodes are restricted to E [V] or -E [V] in order to make the problem simple. The absolute values of the voltages are the same. The simplest case to generate electric fields is to divide the array of electrodes in two regions and apply one region to E [V] and another region to -E [V]. The former electrodes become anode and the latter become cathode. Anode electrodes propel the cationic molecules and cathode electrodes repel them. Strong electric fields are generated between the boundary of anode and cathode electrodes. The strength and directions of the molecules are represented as current density vector of the field. At the boundary of anode and cathode, the current density vectors are directed from left to right, when left side of the electrodes is anode and right side of the electrodes is cathode. If we put gels on the boundary between anode and cathode, the left side shrinks and the right side expands, which leads the gel to bend toward left side. On the contrary, the current density vectors are directed from right to left, when the left side is cathode and the right side is anode. If we put gels on the same place, the gel bends toward right side. These generated electric fields are arranged as right or left operator, which represents the direction of current density vectors. The second simplest way to generate electric fields is to separate the array of electrodes in three regions. This case is described in chapter 5, but we describe

8.2 Design of Electric Operator to the Gels

(a) Left

(b) Right

(c) Up

(d) Down

169

Fig. 8.2. 2D operator for array of electrodes, a side view: Arrow represents the direction of current density vectors. Squares above the arrows are electrodes. Red electrodes are anodes that propel the molecules. White electrodes are cathodes that repel the molecules. Each figures show the (a)Left (b)Right (c)Up (d)Down operators respectively.

in the same way as the simplest case. There are two combinations of applied voltages, one is to apply -E [V] to the center of three regions and to apply E [V] to the other regions. Another is to apply E [V] to the central region. The strong electric fields are generated below the electrodes of central region. For former combination, the electrodes in the center become cathode and repel cationic molecules. The molecules are driven in upward directions. For latter combination, the electrodes in the center become anode and propel cationic molecules. The molecules are driven in downward directions. These generated electric fields are named as up and down operators. Now, fundamental set of operators are obtained: left and right, up and down (Figure 8.2 8.3). How they work depends on the relative position and orientation of gels to the generated electric fields. For example, the gel transforms toward right and up direction when we apply ’up’ operator on right side above the gel. This is because the cationic molecules are attracted toward anode electrodes,

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(a) Left

(b) Right

(c) Up

(d) Down

Fig. 8.3. 2D operator for array of electrodes, a top view: Squares are pairs of electrodes. Red electrodes are anodes and white electrodes are cathode. Each figures show the (a)Left (b)Right (c)Up (d)Down operators respectively.

10[mm]

5[mm]

Electrodes

15[mm] Gel Robot Fig. 8.4. Matrix of electrodes

which are placed right side of the gel. In this case, ’up’ operator works as upright operator. 8.2.3

3D Operator for Matrix of Electrodes

Secondly, the matrix of electrodes to generate three-dimensional electric fields is considered (Figure 8.4). Applied voltages to the electrodes are restricted to E [V] or -E [V] as well as the case of array of electrodes.

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171

Let us consider the case to divide into two regions. There are 8 horizontal operators considering the direction of the boundary. If we state column direction as north and south, and row direction as east and west, then the operators are classified to north, south, east, west, northeast, northwest, southeast, southwest operators which are illustrated in Figure 8.5. The north operator is described for example. The boundary of the north operator is along the row direction, namely east and west. Applied voltages toward northern part of the electrodes are -E [V] while applied voltages to southern part of the electrodes are E [V]. The current density vectors directed to north, because southern part propels the molecules while northern part repels the molecules. We can think of other operators in the same manner. The directions of boundary are row, column and diagonal (in two ways), and for each case, there are two kinds of polarities. There is another case that divides electrodes into two regions. One region is square-shaped and another region is the rest of the electrodes. This derives from the case to divide into three regions in two dimensions. When the central electrode works as cathode, it is ’up’ operator. When the central electrode works as anode, it is ’down’ operator. The length of the side determines the strength of the generated electric fields. Figure 8.6 shows a set of isotropic up and down operators. Then, a set of electrodes divided into three regions is considered. Anisotropy appears because the width of the central region determines the strength of the electric fields. We restrict the width of operator to one electrode for simplicity. Then, the width of the central region along the boundary becomes larger than the width perpendicular to the boundary. This leads the generated electric fields along the boundary stronger compared to that of perpendicular to the boundary. Figure 8.7 shows the 8 set of directional up and down operators. The direction of the regions is the same as that of horizontal operators: row, column, and diagonal (in two ways), and for each case, there are two kinds of polarities. Experiment 8.1. Raising and righting motion generation of the starfish-shaped gel robots Experimental methods The 16ch matrix of electrodes was utilized for generating electric fields. The voltage applied to the electrode ij is referred to as Vij , which is illustrated in Figure 8.8(a). The electrodes were placed parallel to the surface of the water and the gel robot was placed under the electrodes at the bottom (Figure 8.8(b)). The robot was placed right below the electrode 33. The space between the electrodes and the gel robot was 15[mm]. The experiments were done in two steps. 1. Raising motion Firstly, voltages were applied to deform into convex shape, concave shape, and to make the ray up and down. The combinations of applied voltages to the electrodes are summarized. 1. V23 = 10[V ], Vij = −10[V ] ((i, j) = (2, 3)) 2. V43 = 10[V ], Vij = −10[V ] ((i, j) = (4, 3))

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(a) North

(b) South

(c) East

(d) West

(e) Northeast

(f) Northwest

(g) Southeast

(h) Southwest

Fig. 8.5. 3D horizontal operator for matrix of electrodes, a top view: Squares are multiple of electrodes. Anodes are colored in red and cathodes are colored in white. The arrows represent current densities which across from anodes to cathodes. Each figures show the (a) North (b) South (c) East (d) West (e) Northeast (f) Northwest (g) Southeast (h) Southwest operators respectively.

8.2 Design of Electric Operator to the Gels

(a) Up

173

(b) Down

Fig. 8.6. 3D isotropic vertical operator for matrix of electrodes, a top view: Squares are multiple of electrodes. Anodes are colored in red and cathodes are colored in white. The directions of current densities from the central regions are vertically up if they are cathodes and down if they are anodes. Each figures show the (a) Up (b) Down operators respectively.

3. V33 = −10[V ], Vij = 10[V ] ((i, j) = (3, 3)) 4. V33 = 10[V ], Vij = −10[V ] ((i, j) = (3, 3)) First two cases were assumed to raise the ray of the robot, and last two cases were assumed to deform into convex shape and concave shape. The length of each tentacles was 15[mm] and the width is 10[mm] at the root and narrower at the tip. The gel robot was 40[mm] wide and 1[mm] thick. 2. Righting motion Secondly, righting motion generation was challenged. Voltages are applied to each of the matrix electrodes 10[V] and -10[V] alternatively. The simultaneous applied voltages were: ⎤ ⎡ ⎤ ⎡ 10 −10 10 −10 V11 V12 V13 V14 ⎢ V21 V22 V23 V24 ⎥ ⎢ −10 10 −10 10 ⎥ ⎥ ⎢ ⎥ ⎢ (8.1) ⎣ V31 V32 V33 V34 ⎦ = ⎣ 10 −10 10 −10 ⎦ [V ] . −10 10 −10 10 V41 V42 V43 V44 The length of each tentacles was 10[mm] and the width was 5[mm] at the root and narrower at the tip. The gel robot was 25[mm] wide and 1[mm] thick. Experimental results 1. Raising motion The results of the first experiment is shown in Figure 8.9. In the first two cases, the robot raised their rays as predicted. Side effect of coupling was also observed. Center of the robot became convex shape while raising left or right ray. In the last two cases, the robots deformed into convex shape and concave shape. The results were the extension of that of the previous subsection.

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Upward

Downward

East-West (row)

North-South (column)

Northwest-Southeast (diagonal)

Northeast-Southwest (diagonal)

Fig. 8.7. 3D anisotropic vertical operator for matrix of electrodes, a top view: Squares are multiple of electrodes. Anodes are colored in red and cathodes are colored in white. The directions of current densities from the central regions are vertically up if they are cathodes (left side of the figure) and down if they are anodes (right side of the figure). The directions of the boundary of central regions are along East-West (row), NorthSouth (column), Northwest-Southeast (diagonal), and Northeast-Southwest (diagonal) from top to bottom.

8.2 Design of Electric Operator to the Gels

V11

V12

V13

V14

V21

V22

V23

V24

V31

V32

V33

V34

V41

V42

V43

V44

175

(a)

Electrodes (b) Gel robot

10[mm] Fig. 8.8. Matrix of electrodes and a starfish gel robot

2. Righting motion The obtained series of the starfish gel robot motion is shown in Fig.8.10. Initially, the center of the gel robot was below the cathode electrode of V23 (Fig.8.10, 1). The center of the starfish gel robot assumed a convex shape since the polarity of the electrode V23 was negative (2). After the robot folded and became vertical to the electrodes (3), it lost its balance and turned over (4). The curvature of the robot then reversed (5). The center of the robot then lay just below the electrode V32 whose polarity was also negative, which caused the curvature of the robot to reverse (6). This series of the motions of the gel robot implies that we can bring about dynamic motion such as turning over using spatially varying electric fields. The point is that the set of applied voltages were not changed while the gel robots were moving. Changes in the relative position of the gel robots to the electric field were caused by the deformation of the gel robot itself. The electric field varied spatially since the polarity of the applied voltages to neighboring electrodes was opposite. The amplitudes and directions of the force, which were generated on the gel robot surface varied from place to

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8 Lumped-Driven Method for Motion Control

(a)

(b)

(c)

(d)

(e)

Fig. 8.9. Deformation of the starfish-shaped gel robot. a, Initial state, b, raising left ray, c, raising right ray, d, move center upward and deform into convex-shaped, e, move center down and deform into concave-shaped

8.3 Lumped Representation of Whole Body Motions

1

4

2

5

3

6

177

Fig. 8.10. A starfish gel robot that turns over

place. Therefore, the deformations of the gel robot progress autonomously as a result of the interaction between the gel material and the electric field.

8.3 8.3.1

Lumped Representation of Whole Body Motions Conversion of Problems

For example motion, turn over motion of a starfish was selected to study motion control of deformable robots, because it is a dynamic motion, which is driven by synchronized neuron ring[208]. One of the problems of deformable robots is that motion control is difficult because whose bodies have virtually infinite degrees of freedom. Motion generation of gel robots is not a simple problem if we formulate it in an ordinary manner. Learning from real starfishes and other works, there should exist an alternative methodology. In this section, biologically inspired method to drive deformable robots is proposed. The assumption is that only one or a few points are controlled by whole parts of the robot, which work cooperatively.

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8 Lumped-Driven Method for Motion Control

The point is to convert this ill-posed problem to well-posed problem. This problem is decomposed into inverse dynamics problem considering the trajectory of the center of the robot. 8.3.2

Turning over Motion Generation of Real Starfishes

The turning over movements of starfishes has been studied since 1800s. Thorough observation and discussion on mechanism were done in early 1900s[209, 210]. They can coordinate their behavior in spite of lacking central nervous system. The role of radial and ring nerve were experimentally studied through severing them [211, 212], and studied through simulation[208]. This is the reason their movements have been attracting attention of physiologists and system engineers. Starfish robot was experimentally developed to study autonomous motion control driven by distributed neural system[213]. The major points of the problems were summarized as[214]: 1) As no special static organ is known to be present in that group of animals, how can the sea-star realize its unnatural, inverted state? 2) What controls those unified, coordinated movements as seen when the animal strives to right itself? We shall refer to the study that classified the resulting motions for the purpose of applying to experiments. The turning over movements can be classified into two types, folding over and somersault. The following statements are brief summary of the results by Oshima[214]. Folding over motion is illustrated in Figure 8.11(a). The body at first falls on one side so that the two arms 1 and 2 come to lie parallel to the bottom. In the last stage of the process, the arm 1 glides underneath the body and finally comes to appear between the arms 2 and 5. Somersault motion is illustrated in Figure 8.11(b). All the arms bend down, toward the bottom. Two of these, 1 and 2, glide toward their tips. The body is set upright gradually. At last the balance is lost, and the body turns a somersault. Each arm then stretches out and comes to lie flat and the normal posture is thus resumed. Besides these there is a modification of the former, starting with the ”tulipstate”. Also, there are combination of folding over and somersault. One is starting by folding over to finish by turning a somersault. Another is starting by the somersault and to finish by folding over. 8.3.3

Motion Generation of a Starfish-Shaped Gel Robot

Decomposition of the problem If we look back to the experiment of turning over motion in chapter 5, it corresponds to the somersault type. The somersault motion was focused for turning over motion in this study. The key to achieve turning over motion is to coordinate each part of the body. If we get back to the objective of the motion, we do not have to control every part of the body in each step. The assumption

8.3 Lumped Representation of Whole Body Motions

2

(a)

(b)

2

179

3

1

1 3 2

3

2

1

1 3 2

2

3

1

3

1

2 3

2

(1) 1 3 Fig. 8.11. Two types of turning over movements of a starfish: (a) folding over type, redrawn from Fig. 1 of [214], (b) somersault type, from Fig. 4 of [214]

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8 Lumped-Driven Method for Motion Control

is that only one or a few points are controlled by overall deformation of the other parts, which work cooperatively. We focus on the center position based on the preliminary experiment. If one observes the trajectory of the center of the starfish-shaped gel robot while it turns over, it lifts up and moves above the tip. The orientation of the center rotates and reversed in final state. This observation leads us to regard turn over motion as a kind of locomotion along with rotation. From this point of view, we can define the direction of locomotion. Both the shape of the gel robot and the method to generate electric field will determine this direction. In this manner, we can regard turn over motion control problem as inverse dynamics problem. The problem is then decomposed into three parts. The first problem is how to determine the direction of locomotion. The second one is to generate desired trajectory of the center of the robot along locomotion direction. The third one is to make the robot to follow the desired trajectory. The following subsections solve these problems. Direction of locomotion through symmetry analysis We analyze the shape of the starfish at first in order to identify the potential direction of locomotion. It is known that symmetry is a powerful way to measure both living things and artifacts [215, 216], especially machine that locomotes [217]. The symmetrical aspect of the starfish-shape is examined, in other words, pentagram. Pentagram is carried into itself by 10 kinds of operations; five rotations around its center, the angles of which are multiples of 2π/5, and five reflections in the lines joining center with the five vertices (Figure 8.12). I consider one from five axis of reflection, which will be extended to the rest of four axes, because turn over motion is a kind of reflection by its whole body. It was found that turn over motion is classified into two modes by the number of tips, which slide on the ground. If we consider the symmetry of shape and locomotion, three or four are potential numbers. Three is the minimum number to support its body. In the case of three, one of the sliding tips is on the axis of reflection which we name tip 1 in Figure 8.12(b). Other two are facing to it across the center O, which we name tip 2 and 3 in Figure 8.12(b). In this former case, the direction of locomotion is along the axis direction. One is to move the center above tip 1, another is to move the one above tip 2 and 3 (Figure 8.12(c)). In the case of four, two pairs of tips are slipping facing with each other across the axis; only the tip on the axis is free. In this latter case, the direction of locomotion is vertical to the axis direction. In summary, potential directions of turn over motion are 10, by two modes for each five axes. Trajectory generation of the center of gel robots Now that the nature of turn over motion is abstracted and the potential directions of motion is figured out by symmetric analysis, let’s move on to the desired trajectory generation.

8.3 Lumped Representation of Whole Body Motions

181

Fig. 8.12. Symmetry and vertical axis

(a)

(b)

Fig. 8.13. A top view. Two patterns of grounding of starfish-shaped gel robots with: a) three rays are grounding; b) four rays are grounding

Our strategy to generate objective motion in a simple manner is to divide the motion into two processes, preparation process and transition process. At preparation process, we apply the electric field to move the center upward and to make the distance smaller between the center and the tip along the ground(Figure 8.15(a)). At transition process, the electric field is switched so as to make the center rotates about the tip (Figure 8.15(b)).

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8 Lumped-Driven Method for Motion Control

(a)

(b)

Fig. 8.14. A side view of bending. Two patterns of grounding of starfish-shaped gel robots with: a) three rays are grounding; b) four rays are grounding

(-)

(-)

(-)

(-)

(-)

(-)

(+)

(-)

(-)

(+)

(-)

(-)

(a)

(b)

Fig. 8.15. Process of turn over motion

(b) final initial (a) Fig. 8.16. Trajectory of the center, initial and final shape of the gel robot (×: preparation process, o: transition process)

8.4 Objective Motion Generation with Operators

8.4 8.4.1

183

Objective Motion Generation with Operators Operators to Generate Coordinated Motions

In the previous section, a method to define objective motions is proposed. In this section, objective motion is generated by applying operators, which were arranged in section 8.2. The fundamental idea is common to chapter 7, which is to generate candidate solutions effectively considering dynamical property of the system and search the best solution among them. Following two ideas are unified, which were proposed in the previous chapter: There exist an eigen trajectory which corresponds to input electric field; The internal state of the gel transits at certain timing of switching. In this section, the problem is solved in two steps. 1. The first step is to select a set of operators considering their eigen trajectory. 2. The second step is to search the switching period of operators to generate objective motion. The selection of operators and figuring out the period of switching are possible at planning stage. Once we select a set of operators in off-line, we can focus on the timing of switching in on-line. This is the essential point of this approach. The simulation determines the switching periods. There exist critical points that the effect of switched operator to the gels changes qualitatively rather than quantitatively. The trajectory of the internal state space changes before and after the critical points. The switching periods were altered and conditions to generate objective motions were explored. Turning over motion generation problems in two and three-dimensional space were selected as a typical example. Former one is that of free-ended beamshaped gel utilizing the array of electrodes. Latter one is that of starfish-shaped gel robots utilizing the matrix of electrodes. The latter problem is of threedimensional space, which can be solved by extending the solution to former problem of two-dimensional space. It is shown that the procedures to select a set of operators and to search the switching periods. Switching timing was studied through simulation. 8.4.2

Selection of a Set of Operators

Turning over motion is divided into two processes, approaching process and rotating process. Operators for each process were selected. Firstly, a set of operators for approaching process was selected. The point is to move the center of the gel to approach the edge of it in the direction of advance. The effective operator to approach them is ’up’ operator. Among the up operators, the one whose central region is wider applies better. This is clear from the experimental results in chapter 5. Therefore, ’up’ operator was selected whose central region contains three electrodes as the first operator for approaching process. Secondly, operator for rotating process was selected. If the direction of turning over is right, there are two candidate operators to rotate them. One is to apply

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8 Lumped-Driven Method for Motion Control

10 [V]

-10 [V]

Fig. 8.17. Applied voltage input sets

’right’ operator and the other is to apply ’up’ operator above the right side of the gel. The ’up’ operator generates strong electric fields near the electrodes on the boundary between anode region and cathode region, while ’right’ operator generates relatively uniform electric fields on it. The ’up’ operator was selected because we would like to move only the center of the gel near the electrodes toward right, and to make the edge of the gel on the ground stick to the point. The ’up’ operator was placed next to the original position. Namely, the operator was moved one step toward right direction to work as operator for rotating process. If we make the distance between the gel and the ’up’ operator, the electric field below the anode electrodes deforms the center of the gel downward instead of rotating about the edge. This is the reason for moving the first operator one step to work as second operator. Figure 8.17 illustrates selected operators. 8.4.3

Phase Diagram for Switching of Operators

After a set of operators is selected, the timing for switching these operators was made clear. Condition was: the size of the gel, 16[mm] in length, 4[mm] in width and 1[mm] in thickness; polarity of central electrodes was cationic; distance between the electrodes and the gel was 15[mm]; horizontal distance of the center of gels and electrodes was 0[mm]. Phase diagrams were drawn thorough simulation. The condition was that the timing for switching for 0 to 100 [s], and period to apply electric field after switching for 0 to 100[s]. The graph is shown in Figure 8.18. Top figure illustrates the height of the left tip; middle figure represents orientation of the tip. Turn over state was defined as follows: the state which meets both conditions that left tip is set free from the ground, and right tip facing on the ground, when it turns over toward right direction. The bottom graph shows whether the turn over motion achieved or not.

8.4 Objective Motion Generation with Operators

185

(a) [mm]

6 4 2 0 0

[s]

50

50

100 [s]

50

100 [s]

50

100 [s]

100 0

(b) [rad]

2 0

[s]

-2 0

50

100 0

(c) 1 0.5

[s]

0 0

50

100 0

Fig. 8.18. Phase diagram of a beam-shaped gel: (a) height of the left edge of the gel, (b) angle of the normal vector on the right edge of the gel, (c) final state of the turnover (1: success, 0: failure). Applied electric field was generated by up operator which contains three cationic electrodes. Applied period of each step was from 0 to 100 [s]. X-axis on the left is an applied period of the first operator. Y-axis on the right is an applied period of the second operator.

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8 Lumped-Driven Method for Motion Control

0 [s]

40 [s]

10 [s]

50 [s]

20 [s]

60 [s]

30 [s]

70 [s]

Fig. 8.19. Preparation process of turning over motion generation of beam-shaped gel: The time for switching was 70[s]. The gel deformed into hairpin shape. The horizontal distance between the tip and center of the gel became smaller. The scale bar is 5[mm].

8.4 Objective Motion Generation with Operators

70 [s]

74 [s]

71[s]

75 [s]

72 [s]

80 [s]

73 [s]

90 [s]

187

Fig. 8.20. Transition process of turning over motion generation of beam-shaped gel: The rotating motion occurred just after switching the operator in 70[s]. It took 5[s] to turn over. The scale bar is 5[mm].

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8 Lumped-Driven Method for Motion Control

0 [s]

40 [s]

10 [s]

50 [s]

20 [s]

60 [s]

30 [s]

Fig. 8.21. Failure example of turning over motion generation of beam-shaped gel: The time for switching was 40[s]. The gel deformed into asymmetric convex shape. The scale bar is 5[mm].

If the switching is early (40 [s]), and applying period after switching is short (20[s]), the rotating motion does not occur and only the position of the highest of the gel moves to right. If the operator is switched after the center and the edge of the gel approached enough (100 [s]), the gel should rotate toward right direction. For the purpose of applying the simulation results to experiments, the calibration method was extended, which was originally proposed in chapter 3 and applied in the experiment of chapter 7. The period to reach the switching point

8.4 Objective Motion Generation with Operators

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(a) [mm]

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Fig. 8.22. Phase diagram of a beam-shaped gel: (a) height of the left edge of the gel, (b) angle of the normal vector on the right edge of the gel, (c) final state of the turn over (1: success, 0: failure). Applied electric field was generated by up operator which contains one cationic electrode. Applied period of each step was from 0 to 100 [s]. X-axis on the left is an applied period of the first operator. Y-axis on the right is an applied period of the second operator.

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Fig. 8.23. Phase diagram of a beam-shaped gel: (a) height of the left edge of the gel, (b) angle of the normal vector on the right edge of the gel, (c) final state of the turnover (1: success, 0: failure). Applied electric field was generated by up operator which contains one cationic electrode. Applied period of each step was from 100 to 200 [s]. X-axis on the left is an applied period of the first operator. Y-axis on the right is an applied period of the second operator.

8.4 Objective Motion Generation with Operators

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Fig. 8.24. Phase diagram of a beam-shaped gel in small gravity fields: (a) height of the left edge of the gel, (b) angle of the normal vector on the right edge of the gel, (c) final state of the turnover (1: success, 0: failure). Applied electric field was generated by up operator which contains three cationic electrodes. Applied period of each step was from 0 to 100 [s]. The gravity was 0.01 times of ordinary one. X-axis on the left is an applied period of the first operator. Y-axis on the right is an applied period of the second operator.

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scatters because the parameters of the gel scatter. The model suggests that the deformation response of the same shape of gel is the same. The strategy is to switch the operator based on the position of the center of the gel. In this way, we can apply the simulation results to experimental results. Experiment 8.2. Turning over motion generation of beam-shaped gel Turning over motion generation of free-ended gel was generated successfully based on the proposed method. Figure 8.19 shows preparation process of turn over motion. It took 70[s] to reach timing for switching. It took 5[s] after switching the operator for transition process (Figure 8.20). If the timing for switching was earlier, 40[s] after applying the first operator, objective motion was not generated (Figure 8.21). We can generate objective motion in high reliability through considering the trajectory and transition of internal state corresponds to a set of operators. 8.4.4

Phase Diagram in Other Conditions

Turn over motion generation might look as if it is too common or simple, when we look at Figure 8.18 and experimental results. The truth is, it is a really scare condition. For the purpose of testing the condition, the combination of operators, applying periods, and magnitude of gravity was altered. Phase diagram of other conditions are plotted in Figure 8.22 to 8.24. Figure 8.22 shows the turn over state of final gel whose applied operators contained one cationic electrode. Position of the initial and switched operators was the same as the previous one. The turn state was 0 in every condition. The graph suggests that it is impossible to achieve turn over motion with this condition. Turn over motion is possible in some combination when the timing for switching was longer, from 100 to 200 [s], and period to apply electric field after switching was from 100 to 200[s]. The diagram is shown in Figure 8.23. Figure 8.24 illustrates the turn over state of final gel in small gravity field. The magnitude of gravity was 0.01 times of ordinary one. The period to apply first and second operator, and numbers of operators were the same as the condition of Figure 8.18. It is known that making use of gravity force generates locomotion behavior[218, 219, 220, 221]. With small gravity, it is impossible to achieve turn over motion in the same condition. From simulation results, we can see that combination of operators, applying period, and gravity should be regulated to generate organized motion.

8.5 8.5.1

Application of the Lumped-Driven Methods Turning over Motion Control of Starfish-Shaped Gel Robots

Here, the results in 2D space for beam-shaped gels are extended to apply for starfish-shaped gel robots in 3D space. The major difference between problems

8.5 Application of the Lumped-Driven Methods

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Fig. 8.25. Applied voltage input sets for matrix electrodes

in 2D and 3D space is the anisotropy of the operators. Considering the symmetry of operators, It was studied that the turning over motion toward northeast and north direction in horizontal plane. At first, the candidate operators were selected to generate each motion respectively. Then, the switching period common to both cases was examined. Selection of a set of operators The first case to deal with is the turning over toward northeast direction. We need to approach the tip of the ray and the center of the starfish-shaped gel robot in order to move it toward the direction. The candidate operator for the first step is the directional ’up’ operators whose boundary is along northeast or southeast directions. As was stated in section 8.2, anisotropic ’up’ operators generate strong electric fields along their boundary. The coupling along diagonal direction is not so strong as row or column directions. Preliminary results suggest that we can bend the gel to rotate toward northeast direction thorough applying ’up’ operator whose cathode electrodes are placed along southeast direction, namely diagonal direction. This operator was selected as the first operator. The same operator was selected whose position is moved toward northeast direction for one step, for the second step. The reason is the same with the problem in 2D space. It applies the center of the gel to move toward northeast direction. Then, the turning over toward north direction was considered. For the first step, we need to bend the gel along east-west direction, namely row direction. Based on the mechanism as was stated in section 8.2, the generated electric fields along the boundary are stronger compared to that of perpendicular to the boundary. Therefore, the ’up’ operator whose boundary is along the column direction was selected for the first step. For generation of rotating motion toward north direction, the operator for the second step should be also the ’up’ operator. It was selected that the same operator placing next to the center of the gel along northern direction.

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(a)

(b)

Fig. 8.26. Experimental results. Two patterns of bending with: a) three rays are grounding; b) four rays are grounding were observed. The scale bar is 5[mm].

Switching of operators After the selection of operators, the next task is to figure out the switching period. The condition for switching the two processes was investigated. It was found that the gel robot turns over in high reliability by switching the operators when the orientation of the tip becomes vertical. If the switching timing is earlier, the robot deforms back to the planer shape. The height of the center of the gel robot was 6 [mm]. The period to reach the position differs for different operators, the average preparation process was 40[s] and the transition process was 10[s] for turning over across diagonal direction. Experiment 8.3. Turning over motion generation of starfish-shaped gel robots Experimental methods Three template patterns of applied voltage sequences were prepared; one along diagonal direction, and two along column direction of matrix electrodes (Figure 8.25). The experimental protocol is as follows. Initially, starfish gel robot is placed below the electrode whose applied voltage is -E. The adjoining electrodes along the diagonal direction or column direction are also applied voltage -E, and other electrodes are applied voltage E. Secondly, the first operator was switched to the second operator, when orientation of the tip of the ray became vertical to the ground.

8.5 Application of the Lumped-Driven Methods

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Fig. 8.27. Turning over motion generation, experimental results: a) three rays are grounding; b) four rays are grounding. The scale bar is 5[mm].

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8.5 Application of the Lumped-Driven Methods

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Fig. 8.29. Twisting to M3-size spacer: The length of the spacer is 5[mm]. The scale bar is 5[mm].

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Experimental results We carried out experiments for five times for evaluation of our approach. Five times out of five the starfish-shaped gel robots successfully turned over. Two patterns were observed (Figure 8.26). The case with four rays to turn over was observed one time out of five. Figure 8.27 shows their snapshots of motion. A method aimed at solving the motion control problem of deformable robots made of electro-active polymer gel. As a first step, inversion (turn over) motions of beam-shaped gels and starfish-shaped gel robots were realized. One of the difficulties in controlling deformable robots is that they have conceptually infinite degrees of freedom to control. This problem was solved through selecting an essential point to generate the desired motion, in this case, the center of the body. Then, the difficulty of the problem reduces to trajectory control of this point, which can be formulated as an inverse dynamics problem. The turn over motion generation was generated experimentally in chapter 5. Now that the mechanism and condition were made clear, we can generate the motion reproductively. 8.5.2

Curling around Motion Control of Gel Manipulators

The process to solve other motion generation problems is described, to make clear a procedure to apply proposed methods to other problems, and to prove the effectiveness of the methods. The curling around an object was selected as typical example. This is because deformable machines should be good at manipulating an object without damaging by its nature. The problem is only how we realize it. The two-dimensional system and beam-shaped gel with fixing point placed below the array of electrodes were selected. Since the operators are already arranged, we start from the second step. For the purpose of realizing twisting motion, the most important point should be the point that is about to contact. Once the point is contacted, the requirement of that point is to keep the contact. Therefore, we need to curl the point just contacting and to keep contacted point as before. The critical point is always the contacting point, in this case the same as dominant point of motion. The operator to bend from tip toward root of the gel depends on the position of fixing point. If the left side of the gel is fixed and an object is put on the other side, the operator to apply is left operator. Left operator contains cathode electrodes on the left and anode electrodes on the right of the boundary. The problem is the place of an operator. If the object is just below the edge of electrodes, it is not difficult to place an operator. The boundary between anode and cathode should be set above an object. If the object is just below the middle of electrode, the polarity of the electrode should be cathode to keep the contact below the electrode.

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Experiment 8.4. Curling around an object of the gel manipulator Based on the rules generated from the methods, curling around motion of beamshaped gels was successfully generated. The twisted objects were spacers for M2 and M3 size bolt with 5 [mm] long. The sequence of motions is shown in Figure 8.28 and Figure 8.29. In this case, spatially varying electric fields work as peripheral motor programs, which are observed in the real octopus[222].

9 Conclusion and Future Works

This book proposed novel methods for building deformable machines with conceptually infinite degrees of freedom. The bodies of these machines consist of actively deformable materials, which are generally referred to as artificial muscles. Recent advances in electroactive polymers provided promising classes of materials for these artificial muscles. Despite their lack of durability, strength and safety, preventing them from being of complete practical use, we were still motivated for realizing deformable machines. Rather than solving the material problems, we tackled the following mechanical problems: organized the universal difficulty to embody deformable machines made of active materials into two parts. One originates in deformability of the machines, and the other derives from activeness of the materials applied. It was referred to as the deformability problem and the activeness problem. The major finding of this research is that these problems are complimentary, and one solves the other. Gel robots were prototyped utilizing electroactive polymers in order to solve these problems. Modelling, design and control methodologies were explored, which form the foundations in addressing the key issues. In this chapter, the findings are summarized, and the ideas are integrated from the viewpoint of activeness and deformability. We also discuss the application of the results and future works.

9.1

Summary

In chapter 1 of section 1.5, it was stated that the common sense of modelling, design and control of material and machines is challenged. In this chapter, the results of challenges for each chapter are stated. 

This chapter was adapted from in part, by permission, M. Otake, “Agent Model of Electroactive Polymers: How can we bring out intelligence from smart materials?”, Proceedings of the Third Conference on Artificial Muscles, 2006.

M. Otake: Electroactive Polymer Gel Robots, STAR 59, pp. 201–216. c Springer-Verlag Berlin Heidelberg 2010 springerlink.com 

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9 Conclusion and Future Works

Part I: Modelling

Part I contains chapter 2 and 3, which dealt with modelling problems. It was shown the example implementations of the general modelling framework. In chapter 2, it was proposed that modelling framework for actively deformable materials, and machines whose bodies are organized by distributing the materials. Activeness of the material is written in adsorption equation, while deformability of the material is represented with continuum mechanics. By modelling both properties respectively, we can approximate the system effectively, even though strictly speaking, both processes are coupled. Based on the adsorption-induced deformation model, we derived specific constitutive equation of typical electroactive polymer gel, poly 2-acrylamido-2-methylpropane sulfonic acid (PAMPS) gel, and its co-polymer gel. It contains two equations with two main parameters. Here, the common sense, “ electrochemical process must be modeled precisely and thoroughly ” was questioned. Then, the model was approximated from continuous to discrete and it was named deformable lattice automata model by referring to cell dynamic scheme. For representing machines rather than materials, the model was extended considering boundary conditions and showed example implementations. One is a multi-link mechanism, and the other is a mass-spring mechanism. Both of them are typical approximation of continuum. In chapter 3, the model was evaluated and examined, which was proposed in chapter 2. Firstly, parameter identification method was proposed based on mechanism. We can identify adsorption parameter and dissociation parameter by observing the deformation response of the beam-shaped gel in uniform electric field. The tip position and orientation of beam-shaped gel is a function of internal state of the whole gel. Therefore, we can identify parameters through observation of the tip. Secondly, the method was extended to calibrate the parameters. Adsorption parameter mainly affects the deformation speed of the material, which also scatters. Two methods were considered in order to calibrate reaction parameter. One is to estimate it by the deformation response of the gel for a given period of time. Another is to do it by the time required to deform into the particular shape of the gel. Thirdly, the resolution was changed to digitize spatial and temporal variables. The convention “ deformable objects must be modeled with minute elements ” was broken down. It was made clear that beam-shaped gel whose length is 16 mm could be approximated into multi-link mechanism whose links are 1 mm in length. The model was proposed and extended in chapter 2, and was evaluated and examined in chapter 3. It was proposed modelling framework to divide system into material and field, and deal with electrochemical and mechanical processes respectively. This perspective was taken into Part II. 9.1.2

Part II: Design

Part II continues from chapter 4 to chapter 6, which dealt with design problems. Design principle was proposed for mechanism consisting of active and deformable

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materials. Based on the principle, electric field generator was developed for driving gel robots and manufacturing environment for materializing gel robots. In chapter 4, the method was shown for defining design and control variables based on the model. Design variables mainly belong to materials, while control variables mainly belong to fields. Therefore, the position of these variables in the constitutive equation determines the role of parameters. For PAMPS gel system, control variable is proportional to the deformation response, whereas design variable is inversely proportional to the deformation response. Therefore, it is equivalent to make the gel thinner and to apply electric field stronger. The method was proposed for designing materials and fields respectively, considering the local interaction equation that defines relationship between materials and fields. The characteristics of design and control variables are different, since material variables are invariable while field variables are variable after the total system is set up. Here, the method was proposed different from the previous one; “ materials to be driven by fields and the fields to drive the materials should be composite ”. Firstly, field generation system was designed, which is described in chapter 5. Secondly, material manufacturing system was designed, which is presented in chapter 6. In chapter 5, the electric field generation system was designed and developed based on the model, and a variety of unknown forms and motions of the gels was derived. The known configuration of the electrodes to drive the material was parallel. Also, one or both sides of the tip should have been fixed to keep the distance between the material and the electrodes. It was figured out that the above constraints are not necessity. It is effective to design system based on the model that clearly represent function in order to remove the psychological bias on design. The model suggests that real requirement to drive the material is that enough current density vectors perpendicular to the surface of the material, in the case of PAMPS gel. It was discovered that multiple electrodes in a plane could also drive the material. Gravity field is so common that it is usually overlooked. It was found that it could be utilized to keep the distance between the electrodes and material. Multiple electrodes was horizontally configured, and placed the gel underneath the electrodes. With this setup, the following shapes were derived: convex shape, concave shape, hairpin shape, and heart shape. These results suggest that the possible performance, in this case, shapes and motions, of these materials are not fully investigated. Here, the convention was broken that electrodes to drive the material should be placed parallel by making use of the model that was proposed. With this setup, the required numbers of electrodes decreased drastically, because we don’t have to prepare pairs of electrodes for each part of materials that constitute the deformable machine. In chapter 6, the manufacturing system was designed and developed to form gels into variety of shapes, and showed diverse deformation of the gels. The common way was challenged that “ to design deformable machine is to divide the stiff element into small pieces and add controllable or passive joints ”, like serpentine robots and modular robots. The reversed approach was proposed for designing whole body deformable machine utilizing actively deformable materials, and

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remove extra part. One way is to cut unwanted connection to deform along one direction. Another way is to change thickness of the material so as to make thick parts work as structures and thin parts work as actuators. It is already known that stiffness is determined by the combination of material property of the element and the shape of the element. Changing the design parameters like thickness and width can alter structural stiffness. Firstly, thickness-distributed and cut gels were developed for reducing the relative degrees of freedom, which showed organized motions. Secondly, deformation response of beam-shaped gel with different width was compared, and the importance of structural stiffness change after deformation was discovered. The curvature of the narrow gel was larger than the wide gel in the same electric field. This is because structural stiffness along longitude direction of the former is smaller than the latter. Therefore, the original shape of the material defines initial state of the deformation process. Experimental results suggest that methods of shape design contribute to generate organized motion of the machine, especially for initial direction of the whole process. The electric field for driving gel robots was developed without changing the shape of the gel in chapter 5. Then the shape of the gel was changed to the same electric field and compared the results in chapter 6. With this respective method, we can explore design space systematically and effectively. This method is applicable to design of other machines consisting of active materials that interact with outside environment. 9.1.3

Part III: Control

Part III consists of chapter 7 and chapter 8, which studied control problems. The difficulty arises from that the numbers of input is smaller than the numbers of output. The problems were arranged into shape control and motion control. The goal of shape control is to reach the final shape and the speed or deformation process is not in question. On the other hand, the goal of motion control is to generate desired motion in a required time and the final shape is not considered. Example methods for achieving these goals were shown. In chapter 7, methods for reaching the objective shape of the gel based on the model were proposed. Series of input to the gel was applied to reach the goal in order to compensate the lack of input numbers compared to degrees of freedom. Intuitively, complex shape with varying curvature is achievable by spatially varying electric field, which was generated in chapter 5. Contrary to this expectation, it was found that switching the polarity of spatially uniform electric field at the right moment could derive variety of shapes. Before finding out this fact, it was challenged that tip position control of the gel cantilever in spatially uniform electric field. In common sense, the tip of the gel is achievable along a narrow arc. On the contrary, the simulation and experimental results suggested that reachable area was larger. It was carefully changed the cycle of oscillatory input, and discovered the transition point to make trajectory change of the tip. After the trajectory change, the state of the gel is no longer the same as the initial state. Large deformation induces dynamic configuration change and

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small oscillation makes slight configuration change. In both cases, the trajectories converge in the course of deformation. It was poured out protocol for reaching the desired position by combining large deformation and small oscillation. Based on this protocol, another protocol was proposed for deriving objective shape. Possible shapes were enumerated in finite time steps. It was symbolized that one polarity of electric field for 0, and another for 1. Final shape of the gel is symbolized with the input symbol array. The period of time steps was changed and the final shape was compared in simulation. Based on the simulation, we experimentally derived shapes of the letter ’C’ with one pole, ’S’ with two poles, ’E’ with three poles, ’J’ with tip curved, ’L’ with root curved. To obtain the objective shape, the requirement is to switch the polarity of the electric field at predefined position. Sequential control is not required if the time to reach the goal is not critical. The assumption, “ controlling active material requires sequential feedback to counteract the unreliability ” was denied. In chapter 8, it was proposed that the method for generating objective motion considering dynamics of the system. In chapter 7, the setup was kept simple to focus on the essence of the problem, controlling multiple points by small numbers of input. The gel was beam-shaped, which we referred to it as manipulator, and the electric field was spatially uniform. In this chapter, we again utilized the electric field generation system and gel-manufacturing system developed in chapter 5 and chapter 6. It was integrated that the idea of unique trajectory defined by input and switching point for state transition in chapter 7. It was selected that righting motion generation for a starfish-shaped gel robot as a typical case study. It was shown a systematic method for bringing out objective motion considering dynamics of material, field, and interaction. Firstly, we arranged operators for generating the electric field by the combination of voltages applied to the electrodes. These operators are unique to the electric field generation system, namely array of electrodes and matrices of electrodes. The operators are common to the kinds of objective motions. Secondly, the essence of objective motions and the characteristics of the shape of the robot were studied, in this case, righting motion and asteroid shape. The description of motion was reduced to the symmetric transformation achieved by coupling of the whole body. The procedure to reduce the objective motion to phase-transition problem is universal and is independent of the kinds of motions or control systems. Finally, we showed procedure to select a set of operators to fulfill the condition to achieve the goal. The switching timing of the operators was calculated from the simulator based on the model. The shape of machine, surrounding field, and interaction law determines this step. Righting motion of starfish-shaped gel robots was successfully generated. This method was also applied to different kinds of motion, twisting an object of tentacle-type gel. In this way, the common idea of “ direct controller is necessary to control deformable machine with many degrees of freedom ” was challenged. We challenged to control shape of gel robots in chapter 7, and motion of gel robots in chapter 8. As for shape control, series of input are timely integrated. In this way, the numbers of input virtually increase. As for motion control, each

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part of the body is coupled temporally. Here, the degree of freedom of the body is virtually reduced. This classification is applicable to other deformable machines.

9.2

From Control to Design and Modelling

The purpose of this study is to propose methods, which controls deformable machines consisting of actively deformable materials. Here, we would like to make clear the requirement for modelling and design from the viewpoint of control. We would also like to state the role of design and control. We built up methods of modelling, design and control in this order, since none of them was clear. Now that we went around them, let us explore them in the reverse order in this section. 9.2.1

Requirement for the Model

As a starting point, it was made clear that the requirement for the model. The following questions are answered that were stated in section 1.4. 1. What is the required accuracy and precision of the model? 2. Is model necessary for control? How about dynamics based control? 3. Isn’t it endless to model everything? Answer to the first question The first question was the necessary accuracy and precision of the model. This determines the possible level of approximation. We proposed overall modelling framework for dealing with electrochemical process and mechanical process in parallel, and interface them afterward. With this framework, it was made clear that the required resolution for these processes. The resolution was examined by comparing simulation and experimental results. The simulation results well confirmed the experimental results throughout chapter 3 to chapter 8. • Electrochemical process can be represented with adsorption-induced deformation model, in reductive forms. For digitizing, it is preferable to represent internal state variable as continuum while spatial and temporal variables to be discrete. The size of the unit was 1 mm and the time step was 1 s.

Modeling

Design

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Fig. 9.1. Requirement for modelling and design from control

9.2 From Control to Design and Modelling

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• Mechanical processes are described in Newtonian equation. It is common that deformable object must be consisting of minute element. On the contrary to this convention, it was shown that the required resolution of the model is really rough. For example, the beam-shaped gel of 16 mm long can be represented as multi-link mechanism with 16 links. Answer to the second question The second question was on the necessity of the model itself. Generally, modelbased control opposes to dynamics-based control. This idea was questioned because of the following reasons. • The model which was derived represents dynamics. The model consisting of two parts, one describes electrochemical interaction and another describes mechanical interaction. Both of them are dynamics equation. In other words, the model itself serves as dynamical system, which was pointed out in chapter 2. In this case, it is difficult to separate model and dynamics. • The model is not necessary utilized directly. The model was not referred sequentially for control. In this study, the method was proposed for bringing out global dynamics from local dynamics equation. The input was symbolized in chapter 7, and operators were arranged in chapter 8. With symbols and operators, phase diagram or global map of the system was drawn. In this way, the model can be converted into dynamic rules. Models theoretically support the dynamics-based control. Answer to the third question The third question was on the productiveness of the modelling. There exist numerous numbers of models for many kinds of material. Under this circumstance, the model was proposed because we believed that general description was necessary. • We don’t have to model from scratch for each material, if we have general template for modelling. The modelling framework was proposed that actively deformable materials are approximately modeled as combination of activeness model and deformation model. This framework is applicable to the material, which will be developed in the future. • A method for deriving control rule or protocol from the model is universal, regardless of the kinds of materials, if the model is generic. Universal method of control based on the models remains as part of future works. 9.2.2

From Design to Modelling

We think that the importance of model for design should be emphasized, compared to the importance of model for control. The ideal design process is a conversion process starting from function to mechanism and structure. Starting

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from existing mechanism or structure, the best solution usually remains undiscovered. It is an effective method for designing mechanism and structure based on functional model. In this study, the design of mechanism is to place multiple electrodes in the same plane. The design of structure is to define size of electrodes and space between the electrodes. Therefore, the requirement for the model from design is clarity for designers. The conditions for clarity are organized as: 1. Material, field, and local interaction between them could be respectively described. This is because we take the strategy to develop material and field respectively. 2. Relationship between the design variables and control variables could be clear. This is because the role of design and control variables is comparable. Design variables are pre-defined while control variables are changeable after the total system setup. Designers can select which is preferable. 3. Interaction between material and field could be compressed into small numbers of equations. This is because the global dynamics are visible or easy to conceive from local dynamics described with the model. 9.2.3

From Control to Modelling

Overall requirement for the model from control is its usability rather than precision. The usability for control is summarized as: 1. The constitutive equation could be expressed explicitly rather than implicitly. It is necessary to examine the deformation response to the diverse numbers of input systematically and quickly. Most of the existing models contain implicit equation because the internal state of the active materials is determined by boundary condition. Even though, the approximation to explicit from implicit representation is preferable. 2. Operators could be derived from the model. It is required that rough estimation of the global structure of the system consisting of active elements that interact with the environment. In this study, it was shown that method for defining fundamental set of operators from the model. With these operators, we can semi directly control the machine by field. The requirement for the model is to predict structural change. 3. Phase diagram or map could be derived from the model and the operators. The requirement is to figure out the transition points, which induce qualitative change rather than quantitative change, when we apply operators sequentially. We can regulate the simulation results to apply experiments, if we can predict global structure of the system. Topological accuracy is required for dynamic motion generation. 9.2.4

From Control to Design

Requirements for Design from Control As was stated in chapter 1, design method changes after we understand the control method. Here, the requirement for design from control is summarized. It

9.2 From Control to Design and Modelling

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is reconsidered that field design in chapter 5 and shape design in chapter 6 from the viewpoint of motion control in chapter 8. 1. It is preferable that the driving system which generates field is spatially uniform. This is because we arrange operators to control the motion of machines through coupling. Driving system could be designed to be able to arrange operators. In this study, rectangular electrodes were arranged to form array and square-shaped electrodes to form matrix. It is preferable that the size of each electrode is uniform. 2. It is preferable that shapes of deformable machines have potentially stable mode. For example, thickness-distributed shape in one direction is potentially deformable along ditch. Another example is symmetric shape, which helps to control whole body motion, because center of coupling is usually on intersection of symmetry axis. Shape that constrains deformation reduces the degrees of freedom. This helps precise control because the numbers of input is bounded. 3. Size of machines and that of operators ought to be in the same order. Required spatial resolution of driving system is smaller than the size of machine to arrange operators. In the case of gel robots, the size of electrodes is preferable to be smaller than the size of gels to apply operators for coupling. Precise positioning of operators is required in order to gain controllability. Therefore, when we increase the spatial resolution, the size of operators could be kept in the same order of machines and position of operators could be regulated. Responsibility of Design and Control Another problem was on the responsibility of design and control. It was made clear that the guideline what could be designed beforehand and be controlled afterwards. 1. Arrangement of operators could be at the design stage while selection and switching of operators could be considered at the control stage. The concept of operator was derived when motion control was tried in chapter 8. Once we obtained this idea, the operator could be considered at design stage in chapter 5, since the shape and size of electrodes to generate electric field cannot be changed. On the other hand, combination of voltages to apply electrodes can be altered. If we prepare operators beforehand, we only need to select and switch them at control stage. In this way, the difficulty of control is reduced by design. 2. In summary, design deals with spatial properties whereas control deals with temporal properties. In the previous statement, arrangement of operators is spatial and switching of them is temporal. Design and control of shape is much more complex. Shape design was done in chapter 6, and shape control was done in chapter 7. Original shape (spatial property) is determined at design stage while transition (temporal property) of shape (spatial property) is dealt with at the control stage. Shape control is a continuous process of shape design.

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3. For the purpose of designing and controlling variables, the point is whether they could be altered afterwards. It is clear from the model that there exist several solutions to bring out the same function. For example, it is equivalent to make the gel thinner and to apply the electric fields stronger, if we take output for deformation speed. The amplitude of electric field can be changed while thickness of the gel cannot be changed after the total system is setup.

9.3 9.3.1

Programming of Gel Robots Integration of Modelling, Design and Control

Here, the three processes are combined, modelling, design and control to the activeness and deformability problem from the viewpoint of programming. From the results of chapter 7 and chapter 8, we can regard gels as interpreter and executer of inputs whose output is a deformation. Activeness and deformability problem in the case of gel robots is expressed as: how to program the shapes and motions of gel robots through applying series of electric field? With this perspective, let us reconsider the modelling, design and control process. Then, modelling is a process of identifying internal state transition of the material. Electric field design is that of preparing programming environment to apply operators. Shape design is the initialization of variables. In this case, gel robots are variables whose values are always rewritten by arrays of operators and symbols. Control corresponds to main function. 9.3.2

Operators and Diagrams

The concepts of operators and symbols were proposed when control problems were solved. Once we obtain them, then they could be considered at the design

Modeling

Design

operators symbols 0 or 1

Control

diagram symbol 010101111111 array

Fig. 9.2. Model, operator, and diagram for gel robot programming

9.4 On the Problems of Activeness and Deformability

211

stage. We can extract operators and symbols from model. Phase diagram and spatio-temporal map are derived by combination of operators. Model describes local dynamics of materials while phase diagram represent global dynamics of machines. The point is for connecting different level of dynamics in order to design deformable machine from actively deformable materials. The essential method for connecting them is to define operator and to generate diagram. We can regard operators and diagrams are derivative of symbols and symbol array. The concept of operators in chapter 8 is an extension of symbolization in chapter 7. Likewise, phase diagram generated by combination of operators in chapter 8 is an extension of time constant for trajectory transition in chapter 7. The above statements are illustrated in Figure 9.2. This is a method for realizing controllable deformable machine consisting of actively deformable materials.

9.4 9.4.1

On the Problems of Activeness and Deformability Complementary Approach

First of all, overall solutions are summarized into activeness and deformability problems. Activeness and deformability are complementary to each other; when we solve problems derived from each property. In the previous section, it was described on gel robots in detail. The proposed methods are generalized in the following statements. Activeness to solve deformability problem: It is possible to reduce the difficulty of deformability problem by constructing deformable machines with active material. 1. Small numbers of input can generate driving field for controlling large numbers of active elements in union. The point is to allocate one input to multiple points. 2. Hysteretic properties of active materials also help to solve the problems. They accumulate input continuously. Therefore, we can virtually increase the number of input if we take time. Deformability to solve activeness problem: It is possible to reduce the difficulty of activeness problem by distributing active materials to form deformable machines. 1. If we set an average of active elements as output from whole the system, the scatter of the stimulus response of elements is cancelled. 2. The property of materials can be estimated by the deformation response speed, because they are observable through coupling. Thereby, we can calibrate the internal parameters and regulate inputs. 9.4.2

Independent Approach

Needless to say, we can deal with activeness and deformability problems respectively. Solutions particular to each problem are proposed.

212

9 Conclusion and Future Works

Activeness to solve activeness problem: It is helpful to consider time constant of system which characterizes activeness for solving activeness problem. 1. Response function of active material is unreliable, which is different from traditional mechanism. It was figured out that most of them are reduced to the problem of delay or exceed of response. We can absorb the scatter of response by regulating switching of operators or by applying compensation parameters. The point is to include internal state changes of the materials to control laws of machines. Internal state changes are representation of activeness. 2. In chemical systems, experimental results are always reasonable. If the results are different from the hypothesis, there should be only two reasons. One is that the hypothesis is wrong, and another is the experimental condition doesn’t fit the supposed condition. Since the model is an approximation and the simulation results are references, the hypothesis is more or less wrong. But if we can keep the condition to be within the supposed condition of the model, we can regard the materials as controllable. Then, remaining problems are regulation of response speed and timing for switching. Deformability to solve deformability problem: Maximizing solution space reduces the difficulty of deformability problem. 1. Deformable machines have conceptually infinite degrees of freedom. When the mechanism is coupled and nonlinear, we take the enumerative approach. Then, the problem arises that the size of search space becomes virtually infinite. The size of solution space becomes, at the same time, larger than conventional mechanism of small numbers of degrees of freedom. Number of options becomes larger to realize objective shapes and motions. Considering this characteristic reduces the difficulty of the problem. 2. For shape control, we tried to maximize the differential of angles of arbitrary points, which characterize the objective shape. In this way, objective shapes were derived without the knowledge of reachable shape. For motion control, a set of operators were selected to generate the overall motion of the machine and regulated timing for switching. Phase diagram was drawn by simulator based on the model. The conditions for generating righting motion were systematically derived.

9.5

Agent Approach to Electroactive Polymer Gel Robots

In the literature, ”Are Smart Materials Intelligent?”[223], smart materials and structures are defined that which incorporate one or more of the following features: sensors or actuators which are either embedded within a structural material or else bonded to the surface of that material; control capabilities which permit the behavior of the material to respond to an external stimulus according to a prescribed functional relationship or control algorithm. At a more sophisticated level, such smart materials become intelligent when they have the ability

9.5 Agent Approach to Electroactive Polymer Gel Robots

213

to respond intelligently and autonomously to dynamically-changing environmental conditions. The author concluded that turning to the question posed by the title, ”Are Smart Materials Intelligent?”, they should answer ”not yet”, which was in 1994. How about recently? Various materials have been developed and their performances have been improving every year. However, fundamental theory for bringing out the intelligence from the smart materials has not fully been investigated. In this section, the author proposes agent model which represents the original smartness of the material, and control system structure which is also represented with higher-level agent model. 9.5.1

Agent Model of Electroactive Polymers

In the field of artificial intelligence, agent is defined that anything that can be viewed as perceiving its environment through sensors and acting upon that environment through actuators [159], which was stated in Section 1.3. This definition also applies to smart materials. Among various types of agents, we can model smart materials as model-based reflex agents, which maintain some sort of internal state that depends on the percept history and thereby reflects at least some of the unobserved aspects of the current state. Internal state functions are determined based on the electrochemomechanical property of each material. Figure 9.3 (a) gives the structure of the reflex agent with internal state, showing how the current percept is combined with the old internal state to generate the updated description of the current state. Rectangles denote the current internal state of the agent’s decision process and ovals to represent the background information used in the process. 9.5.2

Control System Design Based on the Agent Model

Knowing about the current state of the environment is not always enough to decide what to do. The agent needs some sort of goal information that describes situations that are desirable. The agent program can combine this with information about the results of possible actions in order to choose actions that achieve the goal. However, goals alone are not really enough to generate high-quality behavior in most environments. Goals just provide a crude binary distinction between desirable and undesirable states, whereas a more general performance measure should allow a comparison of different world state according to exactly how desirable they would make the agent if they could be achieved. If one world state is preferred to another, then it has higher utility for the agent. A utility function maps a state onto a real number, which describes the associated degree of desirability. Control system for electroactive polymers should contain modules which represent goal or utilities, since smart materials are modeled as simple reflex agents with internal state without goal or utility by its nature. Therefore, we model electroactive polymers, and include the model into control systems (Figure 9.3 (b) top half). Then, the utility function is added in order to determine the appropriate action or electrical stimuli to the material from the control system

214

9 Conclusion and Future Works

Real Sensors (Electric Field) What the world is like now

Input (Deformation) State

How the world evolves

What my action do What action I should do now Real Effectors (Deformation)

Real Gel

(a) Step 1: Model of the real gel

State

How the world evolves

What my action do What action I should do now

Condition-action rules

Simulated Effectors (Deformation)

Simulated Gel

What it will be like if I do action A

What it will be like if I do action A

Output (Electric Field)

What the world is like now

Condition-action rules

Input (Deformation)

What action I should do now

Simulated Sensors (Electric Field)

How happy I will be in such a state

Utility

Condition-action rules Condition-action rules generated by Simulated Experiment

(c) Step 3: Controller of the real gel

What action I should do now Output (Electric Field)

Simulated Experiment for generating Condition-action rules

(b) Step 2: Simulation for designing controller

Fig. 9.3. Control system based on the agent model of electroactive polymers

(Figure 9.3 (b) bottom half). Simulation would be conducted in various situations so that condition action rules are determined (Figure 9.3 (c)). The last module of (c), the controller with condition action rules is necessary for real-time operation. 9.5.3

From Gel to Gel Robots

Variety of shapes and motions were derived through developing novel driving and manufacturing system with predictive simulator based on utility based agent model. The shapes and motions of gels and gel robots were sometimes beyond imagination. It should be emphasized that the originally known motion was only swinging motion of beam-shaped gel between pairs of electrodes. • Combination of fix-ended beam-shaped gel and parallel placed electrodes (Ordinary combination): C shape, S shape, E shape, J shape, L shape (chapter 7) • Combination of free-ended beam-shaped gel and arrays of multiple electrodes: Convex shape, asymmetry ’ˆ’ shape, Concave shape, ’6’ shape, Hair-pin shape, righting motion (chapter 5 and 8) • Combination of free-ended beam-shaped gel and arrays of multiple electrodes: twisting of objects (chapter 8)

9.5 Agent Approach to Electroactive Polymer Gel Robots

215

• Combination of gel with wave-shaped surfaces and matrix of multiple electrodes: Uni-directional curving, Bi-directional curving (chapter 6) • Combination of starfish-shaped gel robot and matrix of multiple electrodes: Righting motion (chapter 8) Internal state of electroactive polymers keeps the history of electric input, which is the origin of potential smartness. With this in mind, we can bring out intelligence from smart materials. 9.5.4

Future Works

Polymer Robots and Muscle Suits Starfish-shaped robots which turn over and manipulators which twist around an object were developed by artificial muscles, as we dreamed of in Section 1.1. We have not yet achieved jellyfish robots swim in the water by swinging the bell-shaped bodies. This remains for future work. Future direction of gel robots includes: polymer robots assisting human with physical interaction; muscle suits wrapping around the elders or athletes to support their movements. Artificial muscles will be used for such systems when strength and durability of artificial muscles improve in the future, although existing personal robots and muscle suits are driven by motors or rubber actuators. This study contributes to design and control artificial muscle suits with small numbers of electric wires which enable dexterous and dynamic motions. Nonlinear Chemical Dynamics It is known that thermodynamic system far from equilibrium exhibits pattern formation phenomena. Self oscillations are synthesized in other polymer systems[207, 224]. Some are applied to actuators[104] and robot control[225]. We found that electroactive polymers are not exceptions. Pattern formation phenomena were observed, in the course of exploring shape control method, which has not been expected from the beginning. Through applying uniform electric field, waving pattern formation was observed (Figure 7.19). The results of this study contribute to the study on nonlinear chemical dynamics. Artificial muscle machine will exhibit variety of shapes and motions, by considering nonlinear chemical dynamics of polymeric systems[226] and making use of them. Platform for Artificial Life Studies on artificial life became visible since late 1980’s. Gel robots have potential ability to become virtual life forms[227, 228] in the real world[229, 230] because of their life-like body, and will work as a platform to study above issues, like role of experimental animals in the field of life science. Although understanding life through synthesis is not simple, gel robot approach is applicable to study emergence of shapes and motions of life.

216

9 Conclusion and Future Works

Artificial Neural Systems Natural muscles are controlled by neurons and network of neurons. We can imagine artificial neurons and network of artificial neurons as well. Artificial muscles with motor proteins are studied and attract attention[79]. One direction is to develop deformable machine with real motor proteins, actins and myosins, and neurons. Another direction is to develop neural network software to control distributed artificial muscles. The author has been developing open brain simulator which can emulate the activities of human nervous system for estimating internal state of human through external observation[231]. Such software is also applicable to control artificial muscle systems, which is implemented on the personal robots and humanoid robots in the future.

Acknowledgement

This book was originally written as dissertation under the supervision of Professor Hirochika Inoue, leading researcher of robotics. His encouragement, precise comments and bravery enabled me to complete this adventurous research. When I started up the gel robot project in 1998, he gave me a chance to construct chemical equipments and electrochemical instruments to synthesize and drive gels in his robotics laboratory. He gained permission to learn the synthetic method of the gel from its inventor, Professor Yoshihito Osada of Hokkaido University, a pioneering researcher of gels worldwide. He encouraged me by inviting to Hokkaido University when famous snow festival was held in Sapporo city, which was so beautiful. Dr. Yoshiharu Kagami carefully taught me to synthesize the gel in Hokkaido University. He answered to additional questions by e-mail after I went back to Tokyo. Professor Yukinori Kakazu played an important role as a cross linking agent of Prof. Inoue and Prof. Osada because of his friendship. He is a founder of complex system engineering division in Hokkaido University and his philosophy was of great help to harness the gels, typical complex system. When I visited Hokkaido University, I enjoyed discussions and drinking with members of the Laboratory of soft and wet of Prof. Osada, Autonomous system engineering laboratory of Prof. Kakazu. Professor J. P. Gong, co-director of Osada laboratory taught me about the physical properties of gels. Professor Hiroshi Yokoi, co-director of Kakazu laboratory told his perspective on biological systems. These are the fundamental background to conduct this interdisciplinary research. To summarize the works of this five-year project, five professors gave me exact comments to improve this book as reviewers: Professor Tomomasa Sato, Professor Yoshihiko Nakamura, Professor Isao Shimoyama, Professor Masayuki Inaba, and Professor Hirochika Inoue with department of mechano-informatics, University of Tokyo. Dr. James Kuffner and Dr. Gordon Cheng helped improving the papers written in English. Artificial muscles have been attracting attention in the field of robotics; however, to conduct research on that subject has not been simple, since advanced materials have been commercially unavailable. Senior researchers on this area

218

Acknowledgement

gave me valuable advices and encouragements. Professor Satoshi Tadokoro told me about his development of the ion conducting polymer film devices and their driving system. Professor Koji Ikuta told me about his research experience on shape memory alloy actuators during his graduate studies in 1980s. Since the end of the first year of this research in spring 1999, the extensive symposium on electroactive polymers have been held every year. Dr. Joseph Bar-Cohen of NASA JPL has been organizing the symposium to form the field visible. He provided me wonderful environment to present the results and to ask questions directly to top researchers in this field at once, Dr. Roy Kornbluh, Professor Mohsen Shahimpoor, Professor Tribio Otero, Professor Danilo os Zryni, Professor Toshihiro Hirai. They told me about De Rossi, Professor Mikl¨ their research and development of novel materials and devices. Their comments were of great help to promote studies on gel robots. Dr. Shigenori Kabashima, and Professor Hiroshi Asanuma introduced me historical review and state of the art on smart materials and structures especially for aerospace applications. Professor Makoto Kaneko introduced me dynamic sensing of deformable materials and organs based on robotics. Dr. Kinji Asaka organizes conference on artificial muscle twice a year in Japan, where the author could discuss top researchers, Professor Ray Baughman and Professor Elisabeth Smela, Professor Hidenori Okuzaki. Professor John Pojman and Qui Tran-Cong-Miyata gave me a chance to attend conference on nonlinear chemical dynamics, where the author could discuss important researchers in the field attended, Professor Anatol M. Zhabotinsky, Professor Stefan C. M¨ uller, Professor Ryo Yoshida. The research to complete this book was supported by several projects and funds. First of all, I would like to gratefully acknowledge the support from the Japan Society of the Promotion of Science as a research fellow for the research project ’Gel Robots - shape and motion control of continuum machines made of functional polymer material-’ (PI: M. Otake, University of Tokyo, 2001-2002fy). In the first three years, the fundamental fund for the research activities was provided by the Research for the Future Project of the Japan Society of the Promotion of Science through ’Micro-mechatronics and soft mechanics’ (PI: H. Okamura, Science University of Tokyo, 1996-2000fy). This was the nationwide project, which provided research exchange environment. For the last three years, the Japan Society of the Promotion of Science supported the research through the specific project on ’Gel Robots’ (PI: H. Inoue, University of Tokyo, 2000fy), ’Gel Robots - shape design and motion control of electroactive polymer gel-’ (PI: H. Inoue, University of Tokyo, 2001-2002fy). It took time to revise the thesis into monograph. I would like to thank all the patience and encouragement of Dr. Thomas Ditzinger with Springer-Verlag, and Editor in Chief, Professor Bruno Siciliano, and all editors. Finally, I thank the author’s family for their understanding and support, So’ichiro, Ken’ichiro and Mariko.

A Appendix

Materials and Methods Preparation of materials Two kinds of gels were prepared. One is PAMPS gel and the other is PAMPS copolymer gel. Another is the extension of the former material, which is strengthened by co-monomers. Preparation of PAMPS gel A poly(2-acrylamido-2-methylpropane sulfonic acid) (PAMPS) gel was prepared by radical polymerizations from 1.0M solution of 2-Acrylamido-2-methylpropane sulfonic acid (AMPS) (Tokyo Kasei Co., Ltd.), in the presence of 0.05M N,N’methylenebisacrylamide (MBAA) (Tokyo Kasei Co., Ltd.) and 10−3 M K2 S2 O8 (Wako Pure Chemicals Industries, Ltd.) at 323[K] for 24 hours [61]. AMPS is a monomer, which make the polymer anionic. MBAA is a cross linking agent. Molecular structures are illustrated in Figure A.1 (a) and (b). Bubbling of Nitrogen was done to remove the oxygen in the solution before polymerization, which is shown in Figure A.2. Polymerization was done in molding dyes between a pair of glass plates, which are shown in Figure A.3. The thickness of the gels was determined by the thickness of the silicon rubber sheet between the plates. The thickness of the sheets was 0.3, 0.5, 1, 3[mm] respectively. After the polymerizations, the gel was immersed in a large amount of pure water to remove un-reacted reagents until it reached an equilibrium state. Figure A.4 shows the swelled state of the gel. The degree of swelling of the gel was 79. This number is a typical index to characterize gels. It determines the density of polymer network of the gel. Thickness of PAMPS gel plates were 0.6, 1, 3, 8[mm] obtained by the thickness of 0.3, 0.5, 1, 3[mm] molding dyes made of glass. Thickness of the gels scatters even from the same mold, because it depends on the conditions of polymerizations. They are sliced strips into 5 [mm] wide, 10,

220

Appendix

H

H C=C

H

N+

C=O N-H H3C-C-CH3

Cl

C C

CH2

C

SO3H+

C C

(a)

C H

C

H

C

C=C

C

C=O N-H CH3

H

C C C

N-H

CH3

H

C=O (c)

C=C H

H

(b) Fig. A.1. Monomer, cross linking agent, surfactant for PAMPS gel: (a) AMPS for monomer, (b) MBAA for cross linking agent, (c) DPyCl for surfactant

14, 18 [mm] long. Those lengths are decided for the purpose of measurement. Poisson’s ratio and density are calculated with a cylinder having a diameter of 7.5[mm] and a height of 8[mm]. Before the experiment, it was immersed in a dilute solution of the surfactant, n-dodecylpyridinium chloride (DPyCl) (Tokyo Kasei Co., Ltd.) containing 3×10−2 M sodium sulphate (Wako Pure Chemicals Industries, Ltd.). The structure of this surfactant is illustrated in Figure A.1(c).

Appendix

Fig. A.2. Bubbling

Fig. A.3. Mold for making gels

221

222

Appendix

Fig. A.4. Swelled gel

Fig. A.5. Pairs of electrodes and the beam-shaped gel

Preparation of PAMPS co-polymer gel Preparation of PAMPS co-polymer gel is almost the same as that of PAMPS gel. The difference is just the numbers of monomers. Stearyl acrylate (SA), and acrylic acid (AA) were added to increase the Mechanical strength of the gel. The gel was prepared by radical copolymerization at 323[K] for 48 hours. The total monomer concentration in N, N-dimethylformamide was kept at 3.0M in the presence of 0.01M N,N’- methylenebisacrylamide (MBAA) as a crosslinking agent and 0.01M α, α - azobis (isobutyronitrile) (AIBN) as an initiator. Monomers were 2 - acrylamido -2- methylpropanesulfonic acid (AMPS), n-stearyl acrylate (SA), and acrylic acid (AA) with the composition (AMPS: SA: AA) = (20: 5: 75). After the polymerizations, the gel was immersed in a large amount of pure water to remove un-reacted reagents until it reached an equilibrium

Appendix

223

Electrodes Function Generator Gel Amplifier

Fig. A.6. Basic electrode system (top view)

Electrode Na2SO4,DPyCl aq

Gel

Fig. A.7. Basic electrodes system (side view)

state. The degree of swelling of the gel was 20. This number is determined as a weight ratio of the water-swollen gel to its dry state, which characterizes the gel. The gel was immersed in a dilute solution of 0.01M dodecylpyridinium chloride containing 0.03M sodium sulphate in order to apply the electric field. Driving and measurement setup Fundamental setup A pair of parallel plate platinum electrodes (each platinum plate electrode was 25[mm] wide, 40[mm] long, 0.2[mm] thick) was placed in the cell with 30[mm] spacing with each other. A strip of the gel was placed horizontally, one side of the gel fixed while the other side left free to give a swinging motion and observed from top. Length of fixed area was 5[mm], rest of them are exposed in the solution or in the air. Numbers written in caption would be a length of exposed area. Both of them were immersed in the surfactant solution. Photo of this electrode setup is shown in Figure A.5. Views of this setup from the top and the side are illustrated in Figure A.7 and Figure A.6 respectively.

224

Appendix

Fig. A.8. Generating bubble

Camera

Gel

Fig. A.9. Driving and measurement setup using video microscope

Direct or sine wave voltages were applied through the electrodes from an electric source using a function generator (FG120, Yokogawa Electric Corporation), d.c. power supply (PW18-1.8Q, KENWOOD) and amplifier circuits. Galvanostat

Appendix

225

Gel

Electrode Fig. A.10. Bubble and removing tool

(HABF501, Hokuto Denko) was utilized to apply constant current density. The deforming shape of the PAMPS gel, by the gravity in the air or in the solution were analyzed utilizing video microscope (VH7000, Keyence). All experiments were carried out in the room temperature of 298[K]. Additional setup Aside from the fundamental setup, another setup for measurement was added. If we apply voltages to the pair of electrodes, they generate bubbles caused by electrochemical reaction between the electrodes and the solution. The top view of the electrodes setup with a lot of bubbles is shown in Figure A.8. A bubble cover was placed between the electrodes in order to prevent bubbles hide the gel between them, which is shown in Figure A.10. Overall setup considering the bubble effect is shown in Figure A.9.

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