1 ver1 hyungsuck cho optomechatronics fusion of Optica(BookZZ org)

Fusion of Optical and Mechatronic Engineering © 2006 by Taylor & Francis Group, LLC Mechanical Engineering Series Frank Kreith & Roop Mahajan - Series Editors Published Titles Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely & Jan F Kreider Elastoplasticity Theor y Vlado A Lubarda Energy Audit of Building Systems: An Engineering Approach Moncef Krarti Engineering Experimentation Euan Somerscales Entropy Generation Minimization Adrian Bejan Finite Element Method Using MATLAB, 2nd Edition Young W Kwon & Hyochoong Bang Fluid Power Circuits and Controls: Fundamentals and Applications John S Cundiff Fundamentals of Environmental Discharge Modeling Lorin R Davis Heat Transfer in Single and Multiphase Systems Greg F Naterer Introductor y Finite Element Method Chandrakant S Desai & Tribikram Kundu Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ilegbusi, Manabu Iguchi & Walter E Wahnsiedler Mechanics of Composite Materials Autar K Kaw Mechanics of Fatigue Vladimir V Bolotin Mechanics of Solids and Shells: Theories and Approximation Gerald Wempner & Demosthenes Talaslidis Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai Multiphase Flow Handbook Clayton T Crowe Nonlinear Analysis of Structures M Sathyamoorthy Optomechatronics: Fusion of Optical and Mechatronic Engineering Hyungsuck Cho Practical Inverse Analysis in Engineering David M Trujillo & Henry R Busby Pressure Vessels: Design and Practice Somnath Chattopadhyay Principles of Solid Mechanics Rowland Richards, Jr Thermodynamics for Engineers Kau-Fui Wong Vibration and Shock Handbook Clarence W de Silva Viscoelastic Solids Roderic S Lakes © 2006 by Taylor & Francis Group, LLC Fusion of Optical and Mechatronic Engineering Hyungsuck Cho Boca Raton London New York A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc © 2006 by Taylor & Francis Group, LLC Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S Government works Printed in the United States of America on acid-free paper 10 International Standard Book Number-10: 0-8493-1969-2 (Hardcover) International Standard Book Number-13: 978-0-8493-1969-3 (Hardcover) Library of Congress Card Number 2005050570 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Library of Congress Cataloging-in-Publication Data Cho, Hyungsuck Optomechatronics / by Hyungsuck Cho p cm Includes bibliographic references and index ISBN 0-8493-1969-2 (alk paper) Mechatronics Optical detectors TJ163.12.C44 2005 670.42'7 dc22 2005050570 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com Taylor & Francis Group is the Academic Division of Informa plc © 2006 by Taylor & Francis Group, LLC and the CRC Press Web site at http://www.crcpress.com Author Hyungsuck Cho gained his B.S degree at Seoul National University, Korea in 1971, an M.S degree at Northwestern University, Illinois in 1973, and a Ph.D at the University of California at Berkeley, California in 1977 Following a term as Postdoctoral Fellow in the Department of Mechanical Engineering, University of California, Berkeley, he has joined the Korea Advanced Institute of Science and Technology (KAIST) in 1978 He was made a Humboldt Fellow in 1984-1985, won Best Paper Award at the International Symposium on Robotics and Manufacturing, USA in 1994, and the Thatcher Brothers Awards, Institute of Mechanical Engineers, UK in 1998 Since 1993, he has been an associate editor or served on the editorial boards of several international journals, including IEEE Transactions on Industrial Electronics, and has been guest editor for three issues, including IEEE Transactions IE Optomechatronics in 2005 Dr Cho wrote the handbook Optomechatronic Systems: Technique and Application, has contributed chapters to 10 other books and has published 435 technical papers, primarily in international journals He was the founding general chair for four international conferences and the general chair or co-chair for 10 others, including the SPIE Optomechatronic Systems Conference held in Boston in 2000 and 2001 His research interests are focused on optomechatronics, environment perception and recognition for mobile robots, optical vision-based perception, control, and recognition, and application of artificial intelligence/ machine intelligence He has supervised 136 M.S theses and 50 Ph.D theses For the achievements in his research work, he was made POSCO professor from 1995 to 2002 © 2006 by Taylor & Francis Group, LLC Preface In recent years, optical technology has been increasingly incorporated into mechatronic technology, and vice versa The consequence of the technology marriage has led to the evolution of most engineered products, machines, and systems towards high precision, downsizing, multifunctionalities and multicomponents embedded characteristics This integrated engineering field is termed optomechatronic technology The technology is the synergistic combination of optical, mechanical, electronic, and computer engineering, and therefore is multidisciplinary in nature, thus requiring the need to view this from somewhat different aspects and through an integrated approach However, not much systematic effort for nurturing students and engineers has been made in the past by stressing the importance of the multitechnology integration The goal of this book is for it to enable the reader to learn how the multiple technologies can be integrated to create new and added value and function for the engineering systems under consideration To facilitate this objective, the material brings together the fundamentals and underlying concepts of this optomechatronic field into one text The book therefore presents the basic elements of the engineering fields ingredient to optomechatronics, while putting emphasis on the integrated approach It has several distinct features as a text which make it differ somewhat from most textbooks or monographs in that it attempts to provide the background, definition, and characteristics of optomechatronics as a newly defined, important field of engineering, an integrated view of various disciplines, view of systemoriented approach, and a combined view of macro– micro worlds, the combination of which links to the creative design and manufacture of a wide range of engineering products and systems To this end a variety of practical system examples adopting optomechatronic principles are illustrated and analyzed with a view to identifying the nature of optomechatronic technology The subject matter is therefore wide ranging and includes optics, machine vision, fundamental of mechatronics, feedback control, and some application aspects of micro-opto-electromechanical system (MOEMs) With the review of these fundamentals, the book shows how the elements of optical, mechanical, electronic, and microprocessors can be effectively put together to create the fundamental functionalities essential for the realization of optomechatronic technology Emphasizing the interface between the relevant disciplines involving the integration, it derives a number of basic optomechatronic units The book © 2006 by Taylor & Francis Group, LLC then goes on in the final part to deal, from the integrated perspectives, with the details of practical optomechatronic systems composed of and operated by such basic components The introduction presents some of the motivations and history of the optomechatronic technology by reviewing the technological evolution of optoelectronics and mechatronics It then describes the definition and fundamental concept of the technology that are derivable from the nature of practical optomechatronic systems Chapter reviews the fundamentals of optics in some detail It covers geometric optics and wave optics to provide the basis for the fusion of optics and mechatronics Chapter treats the overview of machine vision covering fundamentals of image acquisition, image processing, edge detection, and camera calibration This technology domain is instrumental to generation of optomechatronic technology Chapter presents basic mechatronic elements such as sensor, signal conditioning, actuators and the fundamental concepts of feedback control This chapter along with Chapter outline the essential parts that make optomechatronics possible Chapter provides basic considerations for the integration of optical, mechanical, and electrical signals, and the concept of basic functional modules that can create optomechatronic integration and the interface for such integration In Chapter 6, basic optomechatronic functional units that can be generated by integration are treated in detail The units are very important to the design of optomechatronic devices and systems, since these produce a variety of functionalities such as actuation, sensing, autofocusing, acousticoptic modulation, scanning and switching visual feedback control Chapter represents a variety of practical systems of optomechatronic nature that obey the fundamental concept of the optomechatronic integration Among them are laser printers, atomic force microscopes (AFM), optical storage disks, confocal microscopes, digital micromirror devices (DMD) and visual tracking systems The main intended audiences of this book are the lower levels of graduate students, academic and industrial researchers In the case of undergraduate students, it is recommended for the upper level since it covers a variety of disciplines, which, though fundamental, involve various different physical phenomena On a professional level, this material will be of interest to engineering graduates and research/field engineers who function in interdisciplinary work environments in the fields of design and manufacturing of products, devices, and systems Hyungsuck Cho © 2006 by Taylor & Francis Group, LLC Acknowledgments I wish to express my sincere appreciation to all who have contributed to the development of this book The assistance and patience of Acquiring Editor Cindy Renee Carelli, have been greatly appreciated during the writing phase Her enthusiasm and encouragement have provided me with a great stimulus in the course of this book writing In addition, I would like to thank Jessica Vakili, project coordinator, Fiona Woodman, project manager, and Tao Woolfe, project editor of Taylor and Francis Group, LLC, for ensuring that all manuscripts were ready for production I am also indebted to my former Ph.D students, Drs Won Sik Park, Min Young Kim and Young Jun Roh for their helpful discussions Special thanks go to Hyun Ki Lee and all my laboratory students, Xiaodong Tao, Deok Hwa Hong, Kang Min Park, Dal Jae Lee and Xingyong Song who have provided valuable help in preparation of the relevant materials and proofreading the typed materials Finally, I am grateful to my wife, Eun Sue Kim, and my children, Janette and Young Je, who have tolerated me with patience and love and helped make this book happen © 2006 by Taylor & Francis Group, LLC Contents Introduction: Understanding of Optomechatronic Technology Fundamentals of Optics 31 Machine Vision: Visual Sensing and Image Processing .105 Mechatronic Elements for Optomechatronic Interface 173 Optomechatronic Integration 255 Basic Optomechatronic Functional Units .299 Optomechatronic Systems in Practice .447 Appendix A1 Some Considerations of Kinematics and Homogeneous Transformation 565 Appendix A2 Structural Beam Deflection 573 Appendix A3 Routh Stability Criterion .577 © 2006 by Taylor & Francis Group, LLC Introduction: Understanding of Optomechatronic Technology CONTENTS Historical Background of Optomechatronic Technology Optomechatronics: Definition and Fundamental Concept Practical Optomechatronic Systems Basic Roles of Optical and Mechatronic Technologies 12 Basic Roles of Optical Technology 13 Basic Roles of Mechatronic Elements 15 Characteristics of Optomechatronic Technology 16 Fundamental Functions of Optomechatronic Systems 20 Fundamental Functions 21 Illumination Control 21 Sensing 24 Actuating 24 Optical Scanning 24 Visual/Optical Information Feedback Control 24 Data Storage 25 Data Transmission/Switching 25 Data Display 25 Optical Property Variation 26 Sensory Feedback-Based Optical System Control 26 Optical Pattern Recognition 26 Remote Operation via Optical Data Transmission 27 Material Processing 27 Summary 27 References 28 Most engineered devices, products, machines, processes, or systems have moving parts and require manipulation and control of their mechanical or dynamic constructions to achieve a desired performance This involves the use of modern technologies such as mechanism, sensor, actuator, control, microprocessor, optics, software, communication, and so on In the early © 2006 by Taylor & Francis Group, LLC Appendix A1 Some Considerations of Kinematics and Homogeneous Transformation Some Considerations of Kinematics We will consider a rigid body motion in two- and three-dimensional spaces as depicted in Figure A1.1 Let us first consider the planar motion of a rigid body shown in Figure A1.1 The motion is composed of a translation and a rotation motion Suppose that frame xB -yB is a rotating coordinate system attached to the rigid body If rC is denoted by the position vector of a fixed P with respect to OC ; and rO is the position vector of the center OB of the rigid body with respect to OC ; then the combined motion of point P is described by rC ¼ rO þ rB ðA1:1Þ where rB is the position vector of P relative to OB : The velocity of P is then obtained by differentiating both sides of Equation A1.1, giving the equation drC drO ðtÞ dr ðtÞ ¼ þ B dt dt dt ðA1:2Þ The last term of the right-hand side of the above equation is due to the translation of P within the rigid body with respect to OB : When P is fixed in the body , which is translating through space, the above velocity equation can be rewritten as drC drB ¼ dt dt ðA1:3Þ since rB is constant Next, let us consider rotation of the rigid body in a two-dimensional case as shown in Figure A1.2 Assume there is no translation and only pure rotation In this case, we assume that the origins OC and OB coincide When the coordinates frame ðxB -yB Þ rotates about the origin in a counterclockwise direction, we wish to express the xB -yB coordinates in terms 565 © 2006 by Taylor & Francis Group, LLC 566 Optomechatronics yB yC P xB rB ω OB rC rO OC xC FIGURE A1.1 The planar motion of a rigid body of the xC -yC coordinates From the geometry shown in the figure, the unit vectors of the rotating frame rotated by u with reference to xC -yC is expressed by iC ¼ cos u iB sin u jB jC ¼ sin u iB þ cos u jB ðA1:4Þ yC P yB xB rC rB θ OC = OB zC FIGURE A1.2 Rotation of a coordinate system ðxB -yB Þ in a plane © 2006 by Taylor & Francis Group, LLC xC Some Considerations of Kinematics and Homogeneous Transformation 567 where iC and jC are the unit vectors of the xC -yC coordinates, and iB and jB are those of the xB -yB coordinates Utilizing the above relationship, we obtain the following expression between the two coordinates xC -yC and xB -yB such that " # " #" # xC xB cos u 2sin u ¼ ðA1:5Þ yC yB sin u cos u The matrix shown in the above is called a rotation matrix The rotation velocity of the rigid body can be obtained by differentiating Equation A1.5 and setting dxB/dt ¼ dyB/dt ¼ 0, since P has no relative velocity with respect to the xB -yB coordinates Carrying out the differentiation leads to the following relation dxC " #" # dt xB du sin u cos u ðA1:6Þ 7¼2 dyC dt 2cos u sin u yB dt Rewriting the above equation as a vector expression, we have drC du du ¼ xB iB þ yB j B dt dt dt and then finally drC ¼ v rB dt where v ¼ du=dtðkB Þ ¼ du=dtðkC Þ is used, and kB and kC are the unit vectors normal to the plane x-y Combining the translation and rotation of a rigid body motion in the two-dimensional case, we can express the resulting position vector drC drO ðtÞ ¼ þ v rB dt dt ðA1:7Þ where v is the angular velocity of the xB -yB : For the three-dimensional case, shown in Figure A1.3, we can carry out a similar procedure to obtain the rotation matrix and a position vector Referring to Figure A1.3, let {C} be a frame fixed at OC ; {B} be a frame fixed at OB translating with a rigid body without rotating, and {B0 } be a frame fixed at OB0 rotating with the rigid body To obtain the rotational velocity, we assume that the coordinate frames {B} and {B0 } coincide initially When frame {B0 } rotates from frame {B} by u, its unit vectors are expressed by iB ¼ cos u iB0 sin u jB0 þ kB0 kB ¼ u iB0 u jB0 þ kB0 © 2006 by Taylor & Francis Group, LLC jB ¼ sin u iB0 þ cos u jB0 þ kB0 ðA1:8Þ 568 Optomechatronics yB´ zB zB´ yB P zC rB q OB = OB´ rC q rO xB {B} xB´ {B´} yC OC {C} xC FIGURE A1.3 A rigid motion in three-dimensional space Therefore, we obtain the following relationship between two coordinates; 32 3 xB0 xB cos u 2sin u 76 7 6 yB ¼ sin u cos u 76 yB0 ðA1:9Þ 54 5 zB 0 zB0 Carrying out differentiation of the above equation, we can see that the resulting rotational velocity becomes drB du ¼ k ðxB0 iB0 þ yB0 jB0 þ zB0 kB0 Þ ¼ v rB dt dt B ðA1:10Þ Combining this with the translation of the rigid body we have the following position vector of point P, drC dr dr ¼ O þ B dt dt dt ðA1:11Þ where drO =dt is the translation of the rigid body in three-dimensional space, drB /dt, in Equation A1.10 The rotational velocity yields the same equation as given in the two-dimensional case, since we had made the same assumption to derive the rotational speed © 2006 by Taylor & Francis Group, LLC Some Considerations of Kinematics and Homogeneous Transformation 569 Homogeneous Transformation As discussed previously, the rigid body motion in general consists of translation and rotation motions To represent these in a single matrix form, we normally use homogeneous transformation, which is particularly useful in the manipulation of image matrices or coordinate transformation for computer vision Before we elucidate the concept of the transformation, let us consider a pure rotational motion first In this case, the rotation matrix describing the orientation of a frame {xB -yB -zB : ½B } can be determined in a systematic way as shown in Figure A1.4 To start with, the frame is assumed to be initially coincident with a reference frame, {xC -yC -zC : ½C }: If CB Rðg; b; aÞ denotes a resultant rotation matrix due to rotations RðxC ; gÞ about xC axis, RðyC ; bÞ about yC axis, and RðzC ; aÞ about zC axis, they are called, roll, pitch, yaw, respectively The definitions of these rotations are depicted in the figure If the order of the rotations is roll, pitch, and then yaw, the resulting transformation is given by C B Rðg; b; aÞ ¼ Rðz; aÞ Rðy; bÞ Rðx; gÞ 32 cos b cos a 2sin a 76 76 ¼6 sin a cos a 54 0 6 cos g zB sin g 2sin b sin b 0 cos b 7 2sin g ðA1:12Þ cos g zC zB zC zC = zB yB yB β g g OC OC β xC xC (a) yC x –axis : roll (b) α xC xB y-axis : pitch FIGURE A1.4 Rotation about fixed axes in three-dimensional space © 2006 by Taylor & Francis Group, LLC yC = yB (c) α OC xB z –axis : yaw yC 570 Optomechatronics Carrying out multiplication of the above equation, we have C B Rðg; b; aÞ cos a cos b cos a sin b sin g sin a cos g cos a sin b cos g þ sin a sin g 7 ¼ sin a cos b sin a sin b sin g þ cos a cos g sin a sin b cos g cos a sin g 2sin b cos b sin g cos b cos g ðA1:13Þ Let us take some examples of this rotation transformation Consider the rotation of a point defined in the coordinate frame {C} by an angle 908 about zC and then 908 about yC Then Rðy; 908Þ Rðz; 908Þ ¼ 0 32 21 0 76 07 54 7 6 07 ¼ 41 05 0 21 0 Suppose that the point P is given by ½4i þ 2j 3k : The above two rotations of this vector will result in 23 0 32 76 6 ¼ 0 76 54 23 When we reverse the order of the rotation, we can notice that Rðz; 908Þ Rðy; 908Þ – Rðy; 908Þ Rðz; 908Þ Now, let us consider general case of a coordinate transformation involving rotation and translation The transformation equation can be rewritten in a £ matrix form as 32 6 r CR C7 B 6 6 7¼6 6 6 · · · · · ·· · · 76 76 r 76 B 76 76 76 76 76 7 · · ·· · · 54 · · · 1 C B Tr ðA1:14Þ where CB R is the £ rotation matrix of frame {B} with respect to {C}, and CB T is the £ translation vector of {B} with respect to {C} This £ matrix is called homogeneous transformation matrix H The above equation can then © 2006 by Taylor & Francis Group, LLC Some Considerations of Kinematics and Homogeneous Transformation 571 be expressed with H as 3 rC rB 7 7 7 ¼ H6 7 4···5 4···5 1 ðA1:15Þ The transformation H, corresponding to a translation by a vector 2i þ 10j þ 4k; is expressed by 0 6 10 7 H¼6 60 0 If a vector 4i 3j þ 6k is 667 61 0 6 7 6 7¼6 10 0 4···5 4··· ··· ··· ··· 0 translated by the above H, then 32 76 7 76 7 23 10 76 76 7 76 · · · 54 · · · 1 The transformation H, corresponding to a rotation about x-axis by an angle g, is given by 0 6 cos g 2sin g 7 Hðx; gÞ ¼ 6 sin g cos g 0 ðA1:16Þ In a similar way, Hðy; bÞ and Hðz; aÞ can be expressed using Equation A1.14 Let us suppose that a vector 4i þ 12j 3k is rotated by an angle 908 about the x-axis In this case, the transform is obtained from Equation A1.16 as follows 32 0 76 6 0 21 76 12 7 76 7 76 7¼6 76 6 12 0 76 23 54 0 1 We now combine a series of rotations R(z,908), R(y,908) with a translation 2i þ 5j 4k: The combined homogeneous transformation matrix © 2006 by Taylor & Francis Group, LLC 572 Optomechatronics is expressed by 0 32 0 0 76 7 76 0 0 76 7 76 7¼6 Tð2; 5; 24ÞRðy; 908ÞRðz; 908Þ ¼ 6 76 7 0 24 76 0 24 54 5 0 0 0 With the vector above becomes 2 0 6 61 6 7¼6 6 21 4 5i þ 3j 2k, the transformation matrix expression in the 32 25 76 7 76 7 76 76 22 24 54 0 Stretching spreads the points uniformly by a factor sx along the x axis, along the y axis by a factor sy, and along the z axis by a factor sz : The matrix for this transformation is sx 0 6 s 07 y ST ¼ 6 0 sz 0 Scaling is stretching points out along the coordinate directions by the same factor s The scaling matrix is expressed by s 0 60 s 07 7 S¼6 60 s 07 0 When a vector point given by þ bj þ ck is scaled by s, the matrix is obtained by 32 3 a s 0 sa 76 7 6 sb s 0 76 b 76 7 76 7¼6 76 7 6 sc 0 s 76 c 54 5 0 1 © 2006 by Taylor & Francis Group, LLC Appendix A2 Structural Beam Deflection When a structural beam is subjected to lateral loads acting transversely to the longitudinal axis, the loads cause the beam to bend When there is no applied load to the beam, it will not be deflected, its longitudinal, neutral axis being in line with the x-axis, as shown in Figure A2.1 Now, suppose that an initially undeformed beam segment A – B is bent downward due to the applied bending moment M, as shown in the figure Let us derive vðxÞ, the transverse displacement along the x direction due to this moment If r is defined as the radius of curvature, then from the geometry of the bent beam segment A – B, we have rdu ¼ ds or du ¼ r ds ðA2:1Þ where du is the small angle between the normals, AO and BO, and ds is the distance along the curve between the normals The reciprocal of r in the above equation is called curvature which is defined by 1=r: For small deflection, i.e., small u, ds is approximately given by ds dx ðA2:2Þ The transverse displacement vðxÞ is described by tan u u ¼ dv dx ðA2:3Þ Combining Equation A2.2 and Equation A2.3, we obtain du d2 v ¼ dx dx2 ðA2:4Þ From Hook’s law, the curvature 1=r is related to the bending moment M by du M ¼ ¼2 EI r dx ðA2:5Þ if the beam deflection remains within the elastic limit In the above equation E is Young’s modulus of elasticity, I is the moment of inertia of the beam, and 573 © 2006 by Taylor & Francis Group, LLC 574 Optomechatronics x dx v M A x q v + dv ds q + dq B M r dq O FIGURE A2.1 Geometry of a deflected beam segment subjected to a bending moment EI is called flexural rigidity Minus sign in the above is due to the sign convention; “ þ ” sign assigned to bending moment acting on the beam upward, while “ ” sign to the downward one Substituting Equation A2.5 into Equation A2.4, we finally have d2 v M ¼2 EI dx2 ðA2:6Þ This is the fundamental equation governing the beam deflection Once the bending moment M and beam geometry are given, the transverse displacement of the beam along the x direction can be obtained from this equation Now, let us consider a cantilever beam, whose supports are clamped at one end and the other end is free as shown in Figure A2.2, is subjected to an external load The beam is subjected to a uniform load of strength p In this case, the bending moment becomes M¼2 pð‘ xÞ2 Then, substituting this equation into Equation A2.6, we obtain EI d2 v pð‘ xÞ2 ¼ 2 dx2 © 2006 by Taylor & Francis Group, LLC ðA2:7Þ Structural Beam Deflection 575 P d q FIGURE A2.2 A cantilever subjected to a uniform load of intensity p Integrating this equation once we get dv pð‘ xÞ3 ¼2 þ C1 dx 6EI ðA2:8Þ The boundary condition dv=dx ¼ at x ¼ determines the constant C1 , and substitution of C1 into Equation A2.8 results in dv px ¼ ðx 3‘x þ 3‘2 Þ dx 6EI Integration of the above equation yields v¼ px2 ðx 4‘x þ 6‘2 Þ þ C2 24EI Again, using the boundary condition, vð0Þ ¼ 0; we obtain C2 ¼ 0: Finally, the equation describing the beam deflection is obtained by v¼ px2 ðx 4‘x þ 6‘2 Þ 24EI ðA2:9Þ Therefore, the deflection d at the free end can be easily found from the above equation and is obtained by plugging x ¼ ‘ in the above equation vðxÞlx¼‘ ¼ d ¼ p ‘4 ; 8EI uðxÞlx¼‘ ¼ u ¼ p ‘3 6EI ðA2:10Þ In a similar way, the deflection equation of a beam supported under various boundary conditions can be easily obtained Table A2.1 summarizes the deflection ðdÞ and angle of rotation ðuÞ of a beam under various supporting conditions © 2006 by Taylor & Francis Group, LLC 576 Optomechatronics TABLE A2.1 Deflections and Slopes of Beams in Various Support Conditions Beam-Deflection Conditions A Deflection Equation Load Uniform B d v¼ px2 ðx 4‘x þ 6‘2 Þ 24EI Deflection and Slopes q Triangular v¼ po x2 ð2x3 þ 5‘x2 120‘EI 10‘2 þ 10‘3 Þ p‘4 8EI d ¼ vð‘Þ ¼ u¼ dv p‘3 ð‘Þ ¼ dx 6EI d¼ p‘3 30EI u¼ p‘3 24EI d¼ p‘3 3EI u¼ p‘2 2EI d¼ p‘4 384EI P Concentrated v¼ px2 ð3‘ xÞ 6EI P Uniform a v¼ px2 ðx ‘Þ2 24EI b ua ¼ ub ¼ P Uniform a v¼ px ðx3 2‘x2 þ ‘3 Þ 24EI d¼ b ua ¼ P Concentrated a 2 b © 2006 by Taylor & Francis Group, LLC v¼ px ð24x2 þ 3‘2 Þ 48EI 0#x# ‘ d¼ 5p‘4 48EI p‘3 ¼ ub 24EI p‘3 48EI ua ¼ ub ¼ p‘2 16EI Appendix A3 Routh Stability Criterion Stability is the most important characteristic that needs to be analyzed for control systems This is because if a control system becomes unstable, analyzing its controlled performance becomes meaningless There are several ways of checking system stability but we will confine ourselves to the Routh stability criterion Let us consider a closed-loop control system shown in Figure A3.1, whose transfer function is given by GðsÞ ¼ XðsÞ b sm þ b1 sm21 þ · · · þ bm21 s þ bm ¼ n Xd ðsÞ a0 s þ a1 sn21 þ · · · þ an21 s þ an ðA3:1Þ where ai’s and bi’s are constant and n $ m: The Routh stability criterion enables us to find the stability condition under which the system is stable, namely, when the system becomes unstable and how many closed-loop poles lie in the right-half s-plane The advantage of using the criterion is that it does not require factoring of the characteristic polynomial of the denominator of GðsÞ in order to find the closed-loop poles If we write the characteristic polynomial equation DðsÞ ¼ a0 sn þ a1 sn21 þ · · · þ an21 s þ an ¼ ðA3:2Þ the first stability condition is that the above equation must have all a0i s for a0 – 0: This also means that all ’s must have the same sign for a0 – 0: If this condition is met, then we need to proceed ordering the coefficients of the characteristic equation into an array called the Routh array composed of the following rows and columns sn n21 s sn22 sn23 s2 s1 s0 a0 a1 b1 c1 d1 e1 f1 a2 a3 b2 c2 a4 a5 b3 c3 a6 a7 b4 c4 ··· ··· ··· ··· d2 577 © 2006 by Taylor & Francis Group, LLC 578 Optomechatronics X d (s) + − G o (s) X (s) (a) A closed loop system X d (s) G (s) X (s) (b) The equivalent loop transfer function FIGURE A3.1 Transfer function of a closed loop system The coefficients in the above arrays are given by a a a0 a3 b1 ¼ a1 a1 a4 a0 a5 b2 ¼ a1 a1 a6 a0 a7 b3 ¼ a1 This process continues until we get a zero for the last coefficient in the third row Similarly, the coefficients of the remaining rows the 4th, 5th, … and nth are determined in the following way: Here, only the 4th row is shown for illustration c1 ¼ b1 a3 a1 b2 b1 c2 ¼ b1 a5 a1 b3 b1 c3 ¼ b1 a7 a1 b4 b1 This process continues until the nth row is completed In the process of developing the Routh array the missing terms are replaced by zeros The Routh stability criterion is stated as below: The necessary and sufficient condition for a control system to be stable is that each term of the first column of Routh array be positive if a0 0: © 2006 by Taylor & Francis Group, LLC Routh Stability Criterion 579 If this condition is not met, some of the roots of the characteristic equation given in Equation A3.2 lie in the right-half of the s-plane, implying the system is unstable The number of such roots is equal to the number of changes in sign of the coefficients of the first column of the array Consider the forth order-system with the transfer function CðsÞ ¼ RðsÞ sðs2 þ s þ 1Þðs þ 2Þ þ The characteristic equation is given by s4 þ 3s3 þ 3s2 þ 2s þ ¼ Then, the Routh array is constructed below s4 3 7 s s2 s1 s0 1 Since the terms in the first column is all positive, the system is stable Let us take another example for stability test If the transfer function is given by CðsÞ ¼ RðsÞ sðs2 þ s þ 1Þðs þ 2Þ þ Since this system has the characteristic equation s4 þ 3s3 þ 3s2 þ 2s þ ¼ the Routh array is given by, s4 3 24 s s2 s1 s0 Examination of the first column of the Routh array show that there are two changes in sign Therefore, the system under consideration is unstable having two poles in the right half of the s-plane © 2006 by Taylor & Francis Group, LLC [...]... (middle) The middle part of the figure implies an employment of combined macro- and micro/nano-systems for the integration Integration of this type can be found from a variety © 2006 by Taylor & Francis Group, LLC Introduction: Understanding of Optomechatronic Technology 19 * atom nanotube virus & bacteria optical lithography human hair optical fiber 10 10 −5 10 10 11 10 10 10 −9 STM AFM x-ray nanomanipulation... consisted of two kinds of components: mechanism, and electronics and electric hardware Because of 10 k 10 k mechanism mechanical automation − 411 + analog control μ-processor embedded M/C optomechatronically embedded system internet based (teleoperation) mechatronic technology optomechatronics technology FIGURE 1. 2 Evolution of machines © 2006 by Taylor & Francis Group, LLC Introduction: Understanding of Optomechatronic... circuits following the invention of the transistor in 19 48 Then the microprocessor was invented in 19 71 with the aid of semiconductor fabrication technology and made a tremendous impact on a broad spectrum of technological fields In particular, the development created a synergistic fusion of a variety hardware and software technologies by combining them with computer technology The fusion made it possible for... plate ring spacing 10 −8 10 −7 10 −6 biological cell FET DNA CNT NEMS visible light −4 −3 MEMS infrared * scale in meters FIGURE 1. 10 Scales and dimensions of some entities that are natural and man-made of physical systems such as an optical disc (macro) having a micro fineadjustment mechanism, a laser printer that may be operated by micro-lens slits, an AFM, etc Figure 1. 10 depicts scales of some natural... control described here [ 21] Basic Roles of Optical and Mechatronic Technologies Upon examination of the functionalities of a number of optomechatronic systems, we can see that there are a number of functions that can be carried © 2006 by Taylor & Francis Group, LLC Introduction: Understanding of Optomechatronic Technology 13 out by optical technology The major functions and roles of optical technology... transmission / switching micro elements Pattern recognition MEMs Sensor fusion FIGURE 1. 11 Enabling technologies for optomechatronics © 2006 by Taylor & Francis Group, LLC Signal processing Sensors & measurement Introduction: Understanding of Optomechatronic Technology 21 There are a number of distinct functions which originate from the basic roles of optical elements and mechatronic elements discussed When these... micro-machining technology Figure 1. 12m shows a laser surface hardening process [46], which is a typical example of an optical-based monitoring and control system The laser of high power (4 kW) focused through a series of optical units hits the surface of a workspace and changes the material state of the workpiece Maintaining a uniform thickness of the hardened surface (less than 1 mm) is not an easy task.. .Optomechatronics value (performance) 2 optical element electrical/ electronics electrical/ electronics optical element software software electrical/ electronics electrical/ electronics optical element software electrical/ electronics mechanical mechanical mechanical mechanical mechanical mechanical element element element element element element 18 00 19 70 2000 year FIGURE 1. 1 Key component... systems Visual servoing of a robot operated in a remote site is a typical example of such a system, as shown in Figure 1. 12‘ In the operation room, the robot controls the position of a vibration sensor to monitor the vibration signal with the aid of vision information and transmits this signal to a transceiver Another typical example is the operation of the visual servoing of mobile robots over the... of the optically integrated mechatronic technology may be easily found when we revisit the historical background of the technological developments of mechatronics and optoelectronics Figure 1. 3 shows the development of mechatronic technology in the upper line above the arrow and that of optical engineering in the lower line [8] The real electronic revolution came in the 19 60s with the integration of